Renormalization theory in statistical physics and stochastic ......Abstract In this thesis we study...
155
Renormalization theory in statistical physics and stochastic analysis Hao Shen A Dissertation Presented to the Faculty of Princeton University in Candidacy for the Degree of Doctor of Philosophy Recommended for Acceptance by the Program in Applied and Computational Mathematics Adviser: Weinan E June 2013
Transcript of Renormalization theory in statistical physics and stochastic ......Abstract In this thesis we study...
Renormalization theory in statistical physics
and stochastic analysis
Hao Shen
A Dissertation
Presented to the Faculty
of Princeton University
in Candidacy for the Degree
of Doctor of Philosophy
Recommended for Acceptance
by the Program in
Applied and Computational Mathematics
Adviser: Weinan E
June 2013
c© Copyright by Hao Shen, 2013.
All Rights Reserved
Abstract
In this thesis we study the theory of renormalization from different perspectives. For the first per-
spective, we study the long distance behavior of a model from statistical physics, more precisely the
classical dipole gas. We develop a rigorous renormalization group method based on conditional expec-
tations and harmonic extensions, and show that the dipole interactions result in renormalized Gaussian
behavior at large scales. This large scale Gaussian behavior allows us to control functional integrals
associated with the model; for instance we can study the scaling limit of generating functional. Our
new renormalization group method is implemented purely in real space, as contrast to earlier meth-
ods based on decomposition of Gaussian covariances which usually resort to Fourier space. It has
some advantages than earlier methods such as simpler norms. Estimates for decay of Poisson kernels
and (derivatives of) Green’s functions play the essential role. We can generalize the method to deal
with slightly spatially-inhomogeneous situations, such as systems with a boundary. The main result is
that the scaling limit of the generating function with smooth test function is equal to the generating
function for the the renormalized Gaussian free field.
For the second part, we are concerned with short scale behavior of stochastic partial differential
equations (SPDEs). These SPDEs are of parabolic type and with additive white noises, which are very
singular random inputs as spatial dimension becomes higher. The main problem is to interpret the
nonlinearity at presence of these noices. Renormalization is required to remove the small scale singu-
larities in these cases. We perform a systematic study of renormalized powers of Gaussian processes
associated with the linearized equations. As an example, we study the Ginzburg-Landau equations,
improve the regularity results in earlier works in two dimension, and show local well-posedness for
Ginzburg-Landau equation with quadratic nonlinearity in three dimension. This part is a minor mod-
ified version of a joint work by E, Jentzen and me.
Then we proceed to discuss the shear flow problem modelled by an SPDE. We use exact RG
arguments to recover previous results in all different scaling regimes by Avellaneda and Majda. This
example shows that the RG method, if implemented exactly instead of performing drastic truncations,
can be a powerful tool to obtain the correct large scale behaviors of such systems.
iii
Acknowledgements
I would like to thank my thesis advisor Weinan E. I received wonderful mathematical training from
him during the past five years. He has been my best source of guidance, not only in directly mentoring
and helping me with my specific research projects, but also in directing me to a broader view of
mathematics, good problems, other relevant researchers and so on.
I am greatly indebted to other professors, especially Professor Michael Aizenman, David Brydges,
and Arnulf Jentzen. I thank David Brydges for his kind hospitality of my visits to University of
British Columbia, as well as a lot of encouragement and helpful conversations by him. I also appreciate
the discussions with Professor Michael Aizenman in the seminars and other private communications.
Arnulf Jentzen is one of my collaborators for the project related with Part II of this thesis, who greatly
inspired me by his solid professional background and mathematical sophistication.
Last but not least, I would like to thank my family for their spiritual support. This thesis is
dedicated to them.
iv
To my mother Ping Lv and my father Shuliang Shen.
v
Contents
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv
1 Renormalization group by harmonic extensions and classical dipole gas 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Outline of the method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2.1 Basic settings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2.2 Conventions about notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2.3 Definition of model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2.4 The problem of scaling limit and tuning . . . . . . . . . . . . . . . . . . . . . . . 5
1.2.5 Outline of main ideas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3 The renormalization group steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.3.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.3.2 Renormalization group steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.3.3 Properties about conditional expectation . . . . . . . . . . . . . . . . . . . . . . 18
1.4 Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
1.4.1 Definitions of norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
1.4.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.5 Smoothness of RG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
1.6 Linearized RG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
1.6.1 Large sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
1.6.2 Taylor remainder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
1.6.3 L3 and determination of coupling constants . . . . . . . . . . . . . . . . . . . . . 37
1.7 Proof of scaling limit of the generating function . . . . . . . . . . . . . . . . . . . . . . . 43
vi
1.8 Generalization to dipole system with boundary . . . . . . . . . . . . . . . . . . . . . . . 44
1.8.1 Definition of model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
1.8.2 The a priori tuning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
1.8.3 RG maps and modification of norms . . . . . . . . . . . . . . . . . . . . . . . . . 49
1.8.4 Linearized RG map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
1.9 Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
1.9.1 Sine-Gordon transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
1.9.2 Decay of Green’s functions and Poisson kernels . . . . . . . . . . . . . . . . . . . 53
1.9.3 The initial expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
1.9.4 Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
2 Renormalized powers of Ornstein-Uhlenbeck processes and
well-posedness of stochastic Ginzburg-Landau equations 63
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
2.1.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
2.2 Renormalized powers of Ornstein-Uhlenbeck processes . . . . . . . . . . . . . . . . . . . 67
2.2.1 Setting and assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
2.2.2 Hypercontractivity estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
2.2.3 Estimates for discrete convolutions . . . . . . . . . . . . . . . . . . . . . . . . . . 73
2.2.4 Wick powers of Ornstein-Uhlenbeck processes . . . . . . . . . . . . . . . . . . . . 77
2.2.5 Averaged Wick powers of Ornstein-Uhlenbeck processes . . . . . . . . . . . . . . 86
2.2.6 Convolutional Wick powers of Ornstein-Uhlenbeck processes . . . . . . . . . . . . 91
2.2.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
2.3 Stochastic partial differential equations (SPDEs) . . . . . . . . . . . . . . . . . . . . . . 99
2.3.1 Local existence and uniqueness of mild solutions of deterministic nonautonomous
partial differential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
2.3.2 SPDEs with space-time white noise and polynomial nonlinearities in two space
dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
2.3.3 SPDEs with space-time white noise and quadratic nonlinearities in three space
dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
vii
3 Exact renormalization group study of the shear flow 124
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
3.2 Steady case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
3.2.1 The Polchinski equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
3.2.2 Fixed points for the steady case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
3.3 Unsteady case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
3.3.1 The Polchinski equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
3.3.2 Fixed points for the unsteady case . . . . . . . . . . . . . . . . . . . . . . . . . . 135
3.4 Effective SPDE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
viii
Chapter 1
Renormalization group by harmonic
extensions and classical dipole gas
1.1 Introduction
In this part we develop a renormalization group (RG) method to estimate functional integrals, based
on ideas of conditional expectations and harmonic extensions. We demonstrate this method with the
model of classical dipole gas, which has always been considered as a simple model to start with for
this type of problems. For classical dipole model, earlier important works are [FP78, FS81c]. The
renormalization group approach to this model originated from the works by Gawedzki and Kupiainen
[GK80, GK83], based on Kadanoff spin blockings. A different method by Brydges, Yau, Slade and
so on uses the idea of decomposition of the covariance of the Gaussian field, which was initiated
from [BY90], and was simplified and pedagogically presented in the lecture notes [Bry09], see also
[Dim09]. The latter method has achieved several important applications in other problems such as
Kosterlitz-Thouless transition, φ4 and self-avoiding walks [BMS03, BDH98, BS10, BDH95, Fal12].
Our method is different from the above two methods, and may be as well regarded as a variation of
the method by Brydges et al. Their decomposition of covariance scheme, which was also used by other
people such as [Gal85], could be usually achieved by Fourier analysis. In [BGM04], a decomposition of
Gaussian covariance with every piece of covariance having finite ranges was constructed using elliptic
partial differential equation techniques, which also depends to some extent on Fourier analysis, and this
decomposition is the foundation of the simplified version of their RG method (see also [Bau12, BT06]
1
for alternative constructions of such decompositions). We don’t perform such a decomposition of
covariance. Instead we directly take harmonic extensions as our basic scheme and use the Poisson
kernel to smooth the Gaussian field. We don’t need Fourier analysis; instead, real space decay rates
of Poisson kernels and (derivatives of) Green’s functions are essential. Some complexities in [BGM04]
such as proof of elliptic regularity theorem on lattice are avoided. Many elements of this method
such as the polymer expansions and so on are very close to the method by Brydges et al, especially
to [Bry09], while we also have some new features, such as simpler norms and regulators. We keep
notations as close as possible to [Bry09] for convenience of the readers who are familiar with [Bry09].
One may find that this method also resembles Gawedzki and Kupiainen’s approach [GK83, GK80]
because the Poisson kernel here plays a similar role as their spin blocking operator. However, there’re
many differences; for example our fluctuation fields have finite range covariances. As a matter of fact,
the idea of conditional expectation was initially proposed in Frohlich and Spencer’s work on Kosterlitz-
Thouless transition [FS81b, FS81a] which didn’t take dynamical system viewpoint very explicitly. Very
roughly speaking, our method is to rewrite an expectation (a functional integral over the field φ) w.r.t.
a Gaussian measure into expressions involving a family of conditional expectations at a sequence of
scales parametrized by integer j:
E
[∑
X
eσ∑
x∈Xc (∂φ(x))2
K(X,φ)
]
≈ eEjE
[∑
Y
eσj∑
x∈Y c E[∂φ(x)|Bcx]2E[K ′j(Y, φ)|Y c
]
]
(1.1.1)
where E [F (φ)|Xc] for a function of the field F (φ) means integrating all the variables φ(x) : x ∈ X
with φ(x) : x ∈ Xc fixed, σj is the most important dynamical parameter that reflects the information
of renormalization of the dielectric constant in the dipole model, Bx is a block containing x, and the
expansion over local pieces X or Y will be clear in the content. This idea is close to [FS81b, FS81a]
who take inside an expectation conditional integrations, each over all variables φ(x) : x ∈ Ω where
Ω is a bounded region around a charge density ρ with diameter ∼ 2j (but their implementations are
different).
Such conditional expectations can be carried out by minimizing the quadratic form in the Gaussian
measure with conditioning variables fixed. Since the Gaussian is associated to a Laplacian these
minimizers are harmonic extensions of φ fromXc intoX . These harmonic extensions result in smoother
dependence of the integrand of the expectation on the field. Some simple elliptic PDE methods along
with random walk estimates will be used. We remark that this variational viewpoint also shows up
2
in Balaban’s RG method (see for instance [Bał83] or Section 2.2 - 2.3 of [Dim11]). Hopefully our
approach would to some extent help in understanding those works.
With this method we can study the dipole model with an insulating boundary. The RG approaches
to dipole model in [GK83, GK80] or [BY90] have always been working with periodic boundary condition
(in [Dim09] a formal step was applied for dipoles with a boundary, but it’s not clear for us how to
make it rigorous). But a dipole system with a realistic boundary is certainly more interesting, see
section 6.3 “open problems” (4) in [Bry09]. Because of the boundary, we will need some estimates for
upper bounds on decay of Green’s functions and Poisson kernels for a graph Laplacian with slightly
non-constant edge weights.
In Section 1.2 - Section 1.7 we illustrate our method with periodic boundary and in Section 1.8 we
generalize it to a system with realistic boundary.
1.2 Outline of the method
1.2.1 Basic settings
In this paper we work with lattice Zd. Assume that d ≥ 2. Denote the sets of lattice directions
as E+ = e1, ..., ed and E− = −e1, ...,−ed, where ek := (0, . . . , 0, 1, 0, . . . , 0) with only the k-th
element being 1. Let E = E+ ∪ E−. For e ∈ E , ∂ef(x) = f(x+ e)− f(x) is the lattice derivative. For
x, y ∈ Zd, we say that (x, y) is a nearest neighbor pair and write x ∼ y if there exists an e ∈ E such
that x = y+ e. Denote E(Zd) to be the set of all nearest neighbor pairs of Zd. For X ⊂ Zd, we define
E(X) := (x, y) ∈ E(Zd) : x, y ∈ X.
Define d(x, y) := minn ∈ Z : ∃(a0, . . . , an) ∈ (Zd)n+1, (ak, ak+1) ∈ E(Zd) for all k = 0, . . . , n −
1, x = a0, y = an. Also for x = (x1, . . . , xd) ∈ Zd, we define |x|∞ = max1≤i≤d |xi|. Also define ∂X to
be the “outer boundary”: ∂X = x ∈ Zd : d(x,X) = 1.
Let L be a positive odd integer, and N ∈ N. Thoughout Section 1.2 to Section 1.7, let Λ =
[−LN/2, LN/2]d ∩ Zd and we will consider functions on Λ with periodic boundary condition; in other
words we view Λ as a torus.
3
1.2.2 Conventions about notations
When it doesn’t cause confusions, we sometimes write for short
∑
X
(∂φ)2 =∑
x∈X(∂φ(x))2 :=
1
2
∑
x∈X
∑
e∈E(∂eφ(x))
2 (1.2.1)
and similarly for other such type of summations.
We will use a short-hand notation for conditional expectation
E[−∣∣X]:= E
[−∣∣φ(x)
∣∣x ∈ X
](1.2.2)
where the precise meaning of E will be clear in the content.
For any set X ⊂ Λ and function f on Λ, PXf is the unique function that satisfies (−∆ +
m2)PXf(x) = 0 for all x ∈ X and PXf(x) = f(x) for all x /∈ X . For existence and uniqueness
of PXf that satisfies the above conditions, see [Kum10]. Also, by standard theories in [Kum10], there
exists a function PX(x, y) (x ∈ X , y ∈ ∂X) so that PXf(x) =∑
y∈∂X PX(x, y)f(y) for all x ∈ X . PX
or PX(x, y) (x ∈ X , y ∈ ∂X) is called the Poisson kernel for X . We call PXf the harmonic extension
of f from Xc into X with f∣∣Xc
unchanged.
Also, the Poisson kernels and Green’s functions will depend on m where m is a mass regularization
in −∆+m2 . To simplify notations we will only keep in mind that they depend on m without explicitly
writing it out.
1.2.3 Definition of model
For any X ⊆ Λ, define Xc := Λ\X . Define an operator on space of functions on Λ with periodic
boundary condition Cm := (−∆+m2)−1 with m > 0.
The classical dipole gas can be defined by the grand canonical ensemble over configurations. Each
configuration consists of a number n ∈ N and n couples (xk, pk)nk=1 where xk ∈ Λ, pk ∈ E . The
potential between two dipoles at xj , xk with moments pj , pk is
∂pj∂pkCm(xj , xk) (1.2.3)
4
The energy for this configuration is
Hm((xk, pk)) =1
2
n∑
j,k=1
∂pj∂pkCm(xj , xk) (1.2.4)
and the grand canonical ensemble can be written as
ZN = limm→0
∞∑
n=0
zn
n!
∑
(xk,pk)nk=1xk∈Λ,pk∈E
e−βHm((xk,pk)) (1.2.5)
with β the inverse temperature.
Let φ(x) : x ∈ Λ be the Gaussian free field on the Λ with covariance Cm(x, y), and E be the
expectation over φ. Then by Sine-Gordon transform (see Appendix 1.9.1)
ZN = limm→0
E
[
exp
(
2z∑
(x,y)∈E(Λ)
cos(√
β(φ(x) − φ(y))))]
(1.2.6)
Define for X ⊆ Λ
W (X,φ) =∑
x∈X
∑
e∈Ecos(√
β∂eφ(x))
(1.2.7)
then ZN = limm→0 E [exp (zW (Λ, φ))].
1.2.4 The problem of scaling limit and tuning
Let Λ := [− 12 ,
12 ]d ⊂ Rd. Given a mean zero function f ∈ C∞(Λ),
´
Λf = 0 with periodic boundary
condition, we study the generating function
ZN(f) := limm→0
E[e∑
x∈Λ f(x)φ(x)ezW (Λ,φ)]
E[ezW (Λ,φ)
] (1.2.8)
where f(x) := L−(d+2)N/2f(L−Nx). The main question is the scaling limit of ZN(f) as N → ∞.
As the start of our strategy to study this problem, the first step is an a priori tuning of the Gaussian
measure, which we describe now.
Define for X ⊆ Λ
V (X,φ) :=1
4
∑
x∈X,e∈E(∂eφ(x))
2(1.2.9)
The tuning is to split part of the quadratic form of the Gaussian measure into the integrand, so that
5
the resulting Gaussian field has covariance [ǫ(−∆+m2)]−1, associated with expectation Eǫ:
ZN(f) = limm→0
Eǫ[e∑
x∈Λ f(x)φ(x)e(ǫ−1)V (Λ,φ)+zW (Λ,φ)]
Eǫ[e(ǫ−1)V (Λ,φ)+zW (Λ,φ)
] (1.2.10)
Note that normalization factors caused by re-definition of Gaussian:
Eǫ [exp ((ǫ − 1)V (Λ, φ))] (1.2.11)
appear in both numerator and denominator and are cancelled.
We would like to make the RG map independent of ǫ. So we rescale φ→ φ/√ǫ and let σ = ǫ−1−1,
so that
ZN (f) = limm→0
E[
e∑
x∈Λ f(x)φ(x)/√ǫe−σV (Λ,φ)+zW (Λ,
√1+σφ)
]
E[e−σV (Λ,φ)+zW (Λ,
√1+σφ)
] (1.2.12)
1.2.5 Outline of main ideas
Now we outline the main ideas. As the first step, let −∆m = −∆ + m2 and make a translation
φ→ φ+ ξ where ξ = (−√ǫ∆m)−1f in the numerator in (1.2.12) which becomes
e12
∑
x∈Λ f(x)(−ǫ∆m)−1f(x)E[
e−σV (Λ,φ+ξ)+zW (Λ,(φ+ξ)/√ǫ)]
(1.2.13)
Let −∆m = −∆ + m2, where ∆ is the Laplacian in continuum, Cm := (−∆m)−1 and ξ :=
(−√ǫ∆m)−1f . We can verify that L−2NCL−Nm(LNx) = Cm(x) and L
d−22 Nξ(LNx) = ξ(x). Let
q < dd−1 and
R = supm>0
max(∥∥∥Cm
∥∥∥Lq,∥∥∥∂Cm
∥∥∥Lq) <∞ (1.2.14)
We will assume that∥∥∥f∥∥∥Lp
≤ h/R (p > d), for a constant h to be specified later, so that for α = 0, 1
‖∂αξ‖L∞ ≤ hL−(d−22 +α)N (1.2.15)
by Young’s inequality. Then
ZN (f) = limm→0
e12
∑
x∈Λ f(x)(−ǫ∆m)−1f(x)Z ′N(ξ)
/Z ′N (0) (1.2.16)
6
where
Z ′N (ξ) = E
[
e−σV (Λ,φ+ξ)+zW ((φ+ξ)/√ǫ)]
(1.2.17)
As the next step, we will perform a Mayer expansion (Appendix 1.9.3) so that
Z ′N (ξ) = E
∑
X⊆Λ
I(Λ\X,φ+ ξ)K(X,φ+ ξ)
(1.2.18)
where I(X) =∏
x∈X I(x) and
I(x, φ+ ξ) = e−14σ
∑
e∈E (∂eφ(x)+∂eξ(x))2
(1.2.19)
K(X,φ) =∏
x∈Xe−
14σ
∑
e∈E (∂eφ(x)+∂eξ(x))2(
ezW(x,(φ+ξ)/√ǫ) − 1
)
(1.2.20)
We will prove that3∑
n=0
1
n!
∥∥∥K(n)(X,φ)
∥∥∥ ≤ ‖K‖A−|X|e
κ2
∑
X (∂φ)2 (1.2.21)
where∥∥K(n)(X,φ)
∥∥ is the amplitude of the n-th derivative of K in φ, whose meaning as well as the
constants ‖K‖ , A, κ will be specified later.
For Z ′N (0) we perform (1.2.18)-(1.2.21) with ξ = 0.
Our renormalization group method is based on the idea of rewriting the expectation into an ex-
pectation of an expression involving many conditional expectations. We will carry out a multiscale
analysis; an RG map will be iterated from one scale to the next one, during which we will re-arrange
the conditional expectations. A basic algebraic structure and analytical bound will be propagated to
every scale.
The basic structure that we want to propagate to every scale of the RG iterations is, for j ≥ 0
Z ′N(ξ) = eEjE
[∑
X∈Pj(Λ)
Ij(Λ\X, φ, ξ)Kj(X,φ, ξ)
]
(1.2.22)
Here, eEj is a φ, ξ independent constant factor. This constant will be shown to be the same for Z ′N (ξ)
and Z ′N (0) and thus cancels. Pj is the set of “j-polymers” that are unions of “j-blocks” which are
elements in a set Bj , and for X ∈ Pj , X ∈ Pj is a suitable enlargement of X ; these definitions will
be clear in Section 1.3. Kj(X,φ, ξ) only depends on the values of φ, ξ in a small neighborhood of X .
In between Λ\X and X lives nothing (or, one can think of 1(X\X) lying there) and will be called
7
“corridors” which will be important in our conditional expectation method.
Furthurmore, Ij will have local form in the sense that it factorizes over j-blocks Ij(X,φ, ξ) =
∏
B∈Bj Ij(B, φ, ξ) and
Ij(B, φ, ξ) = e−14σj
∑
x∈B,e∈E (∂ePB+φ(x)+∂eξ(x))2
(1.2.23)
where B+ is a slightly larger box containing B. Ij is essentially determined by the dynamical parameter
σj . Kj will only factorize over “connected components of polymer”.
The basic bounds that hold on every scale about Kj whose form will not be explicit is
3∑
n=0
1
n!
∥∥∥K
(n)j (X,φ, ξ)
∥∥∥ ≤ ‖K‖j A−|X|jG(X,X+) (1.2.24)
where X ⊂ X+ are slightly larger sets containing X , and for X ⊂ Y , G(X,Y ) is a normalized
conditional expectation called regulator
G(X,Y ) = E[
eκ2
∑
X (∂φ)2∣∣φY c
] /N(X,Y ) (1.2.25)
and the normalization factor is
N(X,Y ) = E[
eκ2
∑
X(∂φ)2∣∣φY c = 0
]
(1.2.26)
This form of G is simpler than the regulator defined in [Bry09], and will be shown to have some nice
properties.
The initial structure and bound (1.2.19)-(1.2.21) are not exactly in these forms, therefore the first
RG step is slightly different.
Now we outline the steps to go from scale j to scale j +1 while the structure (1.2.22) is preserved.
1) Extraction and reblocking.
Reblocking is a procedure which rewrites (1.2.22) into an expansion over “j + 1 scale polymers”; and
we extract the components that grow too fast under this reblocking. Before all that we should make a
corridor around Kj , by writing Ij = 1+(Ij −1), and glue some “j blocks” onto X ; these “j blocks” are
the ones where Ij − 1 live on while all the rest “j blocks” have an O(Lj+1) distance away from them
8
as well as X . After these corridors are formed, we will have an expansion
E
[∑
X∈PjIΛ\Yj (φ, ξ)K
j(X,φ, ξ)
]
(1.2.27)
where Y , which depends on X , is a suitably larger set containing X . Now we can write Ij = Ij + δIj
where
Ij(B, φ, ξ) = eEj+1− 14σj+1
∑
x∈B,e∈E (∂eP(B)+φ(x)+∂eξ(x))2
(1.2.28)
is an object which “postulates” the the dynamical parameter σj+1 for Ij+1, and Ej+1 is a constant, B
is a “j + 1 block”, and δIj will be chosen (which amounts to choosing σj+1) to cancel the dangerous
parts extracted from Kj . Then, we will reblock the Ij − 1, δIj and Kj components which all live on
“j scale polymers” into an entire piece that lives on a “j + 1 polymer” and obtain an expansion which
up to some other subtleties almost looks in the form
E
[∑
U∈Pj+1
IΛ\Uj (φ, ξ)K
j(U, φ, ξ)
]
(1.2.29)
where U are “j + 1 polymers”. The precise form is given in Section 1.3.
2) Conditional expectation.
This step is the main difference between this new method and [Bry09]. As above, we will first make
a corridor around Kj by writing Ij = (Ij − 1) + 1 and glue some “j + 1 blocks” onto U ; these “j + 1
blocks” are the ones where Ij − 1 live on while all the rest “j + 1 blocks” are neither touching them
nor touching U . Then we will have a form
E
[∑
U∈Pj+1
IΛ\Uj+1 (φ, ξ)K#
j (U, φ, ξ)
]
(1.2.30)
where U\U is the corridor we just made, of width Lj+1. We then take conditional expectation
E
[∑
U∈Pj+1
IΛ\Uj+1 (φ, ξ)E
[
K#j (U, φ, ξ)
∣∣(U+)c
]]
(1.2.31)
where U ⊂ U+ ⊂ U . For notation conventions, see subsection 1.2.2. This conditional expectation
followed by factoring out φ, ξ independent constant gives Kj+1 and we’re back to the form (1.2.22)
9
with all j replaced by j + 1. In case U = Λ, we just integrate (unconditionally): E[K#j (Λ, φ)
], but
to streamline expressions we still write (1.2.31) keeping in mind the special treatment for the U = Λ
term.
Remark 1. The reason that we have to create corridors before conditional expectation is, obviously,
to make Ij+1 intact, while the conditioning can be a bit away from U , that is (U+)c. As we will see,
an important ingredient that makes the method work is the O(Lj+1) distance between U and (U+)c.
We have to create corridors as well before extraction and reblocking because: K#j (U) is a complicated
product of Kj , Ij − 1, δIj , Ij − 1; each Kj(X) has an Lj corridor created in the previous (j − 1’th
RG step) and depends on φ in an Lj/3 neighborhood of X , but δIj , Ij − 1 both depend on φ in an
O(Lj+1) neighborhood that would intrude into the O(Lj) corridor of Kj(X) which would be bad for
the estimates. Gluing some Ij − 1 onto Kj is unharmful because Ij − 1 only depends on φ in an Lj/3
neighborhood, which can’t penetrate the Lj corridor of Kj(X).
We point out two important facts about the conditional expectation step. The first one is that we
can write the Gaussian field φ into a sum of two decoupled parts. Let PU be the Poisson kernel for U
and recall our convention that PUφ(x) = φ(x) for x /∈ U as in subsection 1.2.2.
Proposition 2. Let U ⊂ V be finite graphs. Define ζ via φ(x) = PUφ(x) + ζ(x). Then the quadratic
form
−∑
x∈Vφ(x)∆φ(x) = −
∑
x∈Uζ(x)∆D
U,mζ(x)−∑
x∈VPUφ(x)∆mPUφ(x) (1.2.32)
where −∆DU,m = −∆D
U +m2 and ∆DU is the Dirichlet Laplacian for U , m ≥ 0.
Notice that x ∈ U don’t contribute to the last summation since ∆mPUφ(x) = 0 in U . By this
proposition, taking expectation of a function K(φ) conditioned on φ(x)∣∣x ∈ U c is simply integrating
out a Gaussian field ζ:
E[K(φ, ξ)
∣∣U c]= Eζ [K(PUφ+ ζ, ξ)] (1.2.33)
where the covariance of ζ is the CDU - the Dirichlet Green’s function for U . In particular, we observe
that Ij defined in (1.2.23) has an alternative representation
Ij(B, φ, ξ) = e− 1
4σj∑
x∈B,e∈E E
[
∂eφ(x)+∂eξ(x)∣∣(B+)c
]2
(1.2.34)
It’s conceptually helpful to keep in mind that we’re just re-arranging the following structure (comparing
10
with (1.2.18)-(1.2.19))
E
[∑
X∈Pje− 1
4σj∑
x/∈X,e∈E E
[
∂eφ(x)+∂eξ(x)∣∣(B+)c
]2
E
[
· · ·∣∣(X+)c
]]
(1.2.35)
namely an outmost (unconditional) expectation of a simple combination of many conditional expecta-
tions.
Remark 3. In the paper, PUφ will always be well-defined: by Prop 1.11 of [Kum10], if the probability
that the random walk starting from any point in U exits U in finite time is 1, then the harmonic
extension exists and is unique. Domains U ( Λ will always satisfy this condition because the random
walk hits any point in Λ in finite time with probability one.
The next fact is as follows:
Proposition 4. Let d ≥ 2, x ∈ X ⊂ U ⊂ Λ. If d(x, ∂X) ≥ cLj, then
|(∂xPX)CDU (∂xPX)⋆(x, x)| ≤ O(1)L−dj (1.2.36)
where O(1) depends on c, and CDU is the Dirichlet Green’s function for U .
See Lemma 9. This result gives the expected scaling for the covariance of ∂PXζ where PX is a
Poisson kernel obtained from the previous RG step. We take a heuristic test to see the necessity of this
proposition: setting ξ = 0, for X ⊂ U , if we perform an expectation conditioned on φ(x)∣∣x ∈ Xc,
followed by another expectation conditioned on φ(x)∣∣x ∈ U c, by (1.2.33)
EζUEζX [K(PX(PUφ+ ζU ) + ζX)] = EζUEζX [K(PUφ+ PXζU + ζX)] (1.2.37)
then we need this proposition to deal with PXζU when integrating over ζU .
Proofs of the above two results are in the following sections.
Linearization and stable manifold theorem
We have just outlined a single RG map (σj , σj+1, Ej+1,Kj) → Kj+1. We will show smoothness of this
map in Section 1.5. Note that two issues haven’t been discussed: 1) choice of σj+1, Ej+1, which should
be a function of (σj ,Kj), so that the RG map becomes (σj ,Kj) → (σj+1,Kj+1) (notice that we won’t
regard Ej+1 as dynamical parameter and we’ll factorize it out); 2) choice of σ in the a priori tuning
11
step. We will outline how to treat these two issues now.
Clearly (σ,K) = (0, 0) is a fixed point of the RG map. In Section 1.6 we show that the linearization
of the map (σj , σj+1, Ej+1,Kj) → Kj+1 around (0, 0, 0, 0) has a form L = L1 + L2 + L3 where L1
captures the “large polymers” contributions to Kj+1, and L2 involves the remainder of second order
Taylor expansion of conditionally expected Kj on “small polymers”, both of which will be shown
contractive with arbitrarily small norm by suitable choices of constants L and A introduced above.
Furthurmore, L3 will roughly have a form
L3(D) ≈ LdEj+1 + σj+1
∑
x∈D(∂PD+φ(x))2 − σj
(∑
x∈D(∂PD+φ(x))2 + δEj
)+ Tay (1.2.38)
where Tay is the second order Taylor expansion of conditionally expected Kj on small polymers, which
consists of constant and quadratic terms, and D is a j + 1 block. Now it’s easy to see that there is
a way to choose Ej+1 and σj+1 so that L3 is almost 0, up to a localization procedure for “Tay”. For
proofs see Section 1.6.
Once we have shown a way to choose the constants σj+1, Ej+1 to ensure contractivity of the above
linear map, a stable manifold theorem can be applied to prove that there exists a suitable tuning of σ
so that
|σj | . 2−j ‖Kj‖j . 2−j (1.2.39)
Main result: the scaling limit
Theorem 5. For any p > d there exists constants M > 0 and z0 > 0 so that: for all ‖f‖Lp ≤M and
all |z| ≤ z0 there exists a constant ǫ depending on z and
limN→∞
ZN(f) = exp
(1
2
ˆ
Λ
f(x)(−ǫ∆)−1f(x)ddx
)
(1.2.40)
where ∆ is the Laplacian in continuum.
The main ingredient of the proof is that at scale N − 1 (we don’t want to continue all the way to
the last step since it would be a bit awkward to define IN−1 and IN ), by eq. (1.2.22) (1.2.39)
Z ′N (ξ) ≈ lim
m→0eEN−1
∑
X∈PN−1
(1 + 2−N)Λ\X2−N (1.2.41)
Bounding the number of terms by 2Ld
we see that it is almost eEN−1 as N becomes large. The
12
constant eEN−1 will be the same for Z ′N (ξ) and Z ′
N (0). So only the exponential factor in equation
(1.2.16) survives in the N → ∞ limit and it goes to the right hand side of (1.2.40). Details are
in Section 1.7. We remark that the assumption on f , which makes f smooth at the scale N is for
simplicity of the demonstration of the method.
1.3 The renormalization group steps
1.3.1 Definitions
Polymers
1. We call blocks of size Lj j-blocks which are translations by vectors in(LjZ
)dof x ∈ Zd : |x| <
12 (L
j − 1). In particular a 0-block is a single site in Zd. A j-polymer X is a union of j-blocks.
In particular the empty set is also a j-polymer. The number of lattice sites in X ⊂ Zd is denoted
by |X |. The number of j-blocks in a j-polymer X is denoted by |X |j .
2. X ⊂ Zd is said to be connected if for any two points x, y ∈ X there exists a path (xi : i = 0, . . . , n)
with |xi+1 − xi|∞ = 1 connecting x and y. Note that (0, 0), (1, 1) is connected. Connected sets
are not empty. A nonempty polymer X can be decomposed into connected components. We let
C(X) be the set of connected components of X . Two sets X,Y are said to be strictly disjoint if
there is no path from x to y when x ∈ X and y ∈ Y ; otherwise we say that they touch.
3. A j-polymer X is called a small set or small polymer if it is connected and |X |j ≤ 2d. Otherwise
it’s called large.
4. For a j-polymer X we have the following notations. Bj(X) is the set of all j-blocks in X . Pj(X)
is the set of all j-polymers in X . Pj,c(X) is the set of all connected j-polymers in X . Sj(X) is
the set of all small j-polymers in X . We sometimes just write Pj ,Pj,c and so on when X = Λ.
Define Sj to be the set of pairs (B,X) so that X ∈ Sj and B ∈ Bj(X). We also introduce a
notation Y ∈X Pj which means Y ∈ Pj and that if X = ∅ then Y = ∅.
5. Let X ∈ Pj . Define its closure X ∈ Pj+1 to be the smallest (j+1)-polymer that contains X .
Define for j ≥ 1
X := ∪B ∈ Bj : B touches X (1.3.1)
X+ = ∪x ∈ Λ : |x,X |∞ ≤ 1
3Lj (1.3.2)
13
X = ∪x ∈ Λ : |x,X |∞ ≤ 1
12Lj (1.3.3)
X = ∪x ∈ Λ : |x,X |∞ ≤ 1
6Lj (1.3.4)
Note that we have X ⊂ X ⊂ X ⊂ X+ ⊂ X and only X, X belong to Pj .
6. For X ∈ P0, define X = X = X+ = X = X , and the Poisson kernel at scale 0 is understood as
PX+ := id.
Functions of the fields
1. Define N to be the set of functions of φ. Define N (X) ⊆ N to be the set of functions of
φ(x)∣∣x ∈ X. NPj is the set of maps K : Pj → N such that K(X) ∈ N (X). We define NBj ,
NPj,c similarly.
2. For I ∈ NBj we write I(X) = IX :=∏
B∈Bj(X) I(B) for X ∈ Pj. For K ∈ NPj we say that K
factorizes over connected components and write K ∈ NPj,c if
K(X) =∏
Y ∈C(X)
K(Y ) (1.3.5)
3. Define for X ∈ PjH K(X) =
∑
Y ∈Pj(X)
H(X\Y )K(Y )
H K(X) =∑
Y ∈Pj(X)
H(X\Y )K(Y )
1.3.2 Renormalization group steps
In this method, we show that at each scale
Z ′N(ξ) = eEjE [Ij Kj(Λ, φ, ξ)] = eEjE
∑
X∈PjIΛ\Xj (φ, ξ)Kj(X,φ, ξ)
(1.3.6)
with Ij ∈ NBj , and Kj ∈ NPj,c , and
Ij(B, φ, ξ) = e−14σj
∑
x∈B,e∈E (∂ePB+φ(x)+∂eξ(x))2
(1.3.7)
Kj(X) depends on φ(x)∣∣x ∈ ∂X+ for X ∈ Pj.
14
Now we focus on a single RG map from scale j to j+1. For simpler notations we omit the subscript
j and objects or sets at scale j + 1 will be labelled by a prime, e.g. K ′, P ′. The guidance principle
will be that for all kinds of I’s below, I − 1 and their difference δI and K will be small, so their
products will be higher order small quantities. These remarks will make more sense after we discuss
the linearization of the smooth RG map.
Extraction and Reblocking
We first define a notation χjA where A is a set of polymers: χjA = 1 if any two polymers in A are
strictly disjoint as j-polymers and χjA = 0 otherwise. Also, if A is a set of polymers, let’s write XA to
be the union of all elements of A.
Define I ∈ NBj as
I(B) = eE′− 1
4σ′ ∑
x∈B,e∈E(∂eP(B)+φ(x)+∂eξ(x))2
(1.3.8)
where E′ and σ′ will be chosen later. Denote
〈X〉 := ∪B ∈ Bj : (B)+ ∩ X 6= ∅ (1.3.9)
Then let
1(B) = (1 − eE′
) + eE′
if B ⊆ X\X
I(B) = (I(B)− eE′
) + eE′
if B ⊆ 〈X〉 \X
I(B) = δI(B) + I(B) if B ⊆ 〈X〉c
K(X) =∑
B∈B(X)1
|X|jK(B,X) if X ∈ S
(1.3.10)
where δIj is defined implicitly, and K(B,X) := K(X). Insert these summations into the product
factors in (1.3.6), and expand, we obtain
Z ′N (ξ) = eEE
[∑
X
IΛ\X1X\X∏
Y ∈C(X)\SK(Y )
∏
Y ∈C(X)∩SK(Y )
]
=eEE
[∑
X ,Y
χX∪Y∑
P,Q,Z
(1− eE′
)P (I − eE′
)Q(eE′
)(〈X〉\X)\(P∪Q)δIZ I〈X〉c\Z
·∏
Y ∈XK(Y )
∏
(B,Y )∈Y
1
|Y |jK(B, Y )
]
(1.3.11)
In the above equation, X is a family of connected large polymers, Y is a family of elements in S i.e.
15
Y =
(Bi, Yi) ∈ Sj
1≤i≤nfor some n ≥ 0, and Y := ∪ni=1Yi, and X := XX∪Y , and P ∈ P(X\X),
Q ∈ P(〈X〉 \X), Z ∈ P(〈X〉c).
Now we make1 a next scale polymer V ∈ P ′ using P ∪Q ∪ Z ∪ (∪iBi) ∪XX ,
Z ′N (ξ) =eEE
[∑
V ∈P′
∑
(P,Q,Z,X ,Y)→V
(1− eE′
)P (I − eE′
)QδIZ∏
Y ∈XK(Y )
∏
(B,Y )∈Y
1
|Y |jK(B, Y )
]
· IV c∩〈X〉c(eE′
)Vc∩(〈X〉\X)IV ∩(〈X〉c\Z)(eE
′
)V ∩(〈X〉\X)\(P∪Q)
](1.3.12)
where, with X , Y,Y, X described above,
∑
(P,Q,Z,X ,Y)→V
:=∑
X ,Y
χX∪Y∑
P∈P(X\X)
∑
Q∈P(〈X〉\X)
∑
Z∈P(〈X〉c)1P∪Q∪Z∪(∪ni=1Bi)∪XX=V
(1.3.13)
Now write I = (I − eE′
) + eE′
, and expand,
IVc∩〈X〉c =
∑
W∈P′(V c)
(I − eE′
)W∩〈X〉c(eE′
)(Vc\W )∩〈X〉c (1.3.14)
For each V and W , define UW,V to be the smallest union of connected components of V ∪W that
contains V :
UW,V := ∩U∣∣U ∈ UC(V ∪W ), U ⊇ V ∈ P ′ (1.3.15)
where UC(V ∪W ) is the set of unions of (j + 1 scale) connected components of V ∪W . Observe that
〈X〉 ⊆ U . Indeed, we can even show that 〈X〉 ⊆ V , since if (B)+ ∩ X 6= ∅ then d(B,X) ≤ Lj
2 so for L
sufficiently large B touches V . So
IVc∩〈X〉c =
∑
W∈P′(V c)
(I − eE)W\U (I − eE)W∩U∩〈X〉c(eE′
)(Vc\W )\U (eE
′
)(Vc\W )∩U∩〈X〉c (1.3.16)
Let R := W\U = W\U , noticing that W ∩ U = U\V and (V c\W )\U = (U)c\R and (V c\W ) ∩ U =
1Formulas are a bit complicated here because of the corridors. Making a next scale polymer V using the closure of
X would ruin the important property |X|j+1 ≤ |X|j . Also, notice that I don’t fill up everywhere of Λ\V .
16
U\U , the above summation over W amounts to a summation over U and R:
IVc∩〈X〉c =
∑
U∈V P′,U⊇V
∑
R∈P′(Λ\U)
(I − eE′
)R(I − eE′
)(U\V )∩〈X〉c(eE′
)(U)c\R(eE′
)(U\U)∩〈X〉c
=∑
U∈V P′,U⊇VIΛ\U (I − eE
′
)(U\V )∩〈X〉c(eE′
)(U\U)∩〈X〉c(1.3.17)
Also, since 〈X〉 ⊆ U
(eE′
)Vc∩(〈X〉\X) = (eE
′
)Vc∩〈X〉(e−E
′
)Vc∩X
= (eE′
)(U\U)∩〈X〉(eE′
)Vc∩〈X〉∩U (e−E
′
)Vc∩X
(1.3.18)
Combine (1.3.12)(1.3.17)(1.3.18),
Z ′N(ξ) =e
EE
[∑
U∈P′
IΛ\U (eE′
)UK#(U)
]
(1.3.19)
where for U 6= ∅
K#(U) :=∑
V⊆U,V 6=∅
∑
(P,Q,Z,X ,Y)→V
(1 − eE′
)P (I − eE′
)QδIZ∏
Y ∈XK(Y )
∏
(B,Y )∈Y
1
|Y |jK(B, Y )
· (I − eE′
)(U\V )∩〈X〉c(eE′
)(〈X〉\X)∩U\(P∪Q)(e−E′
)U∪X IV ∩(〈X〉c\Z)
] (1.3.20)
Factorizing the constant eE′
by letting
E ′ = E + E′|Λ|j (1.3.21)
I ′(D) = e−LdE′ ∏
B∈B(D)
I(B) = e−14σj+1
∑
x∈D,e∈E(∂ePD+φ(x)+∂eξ(x))2
(1.3.22)
for D ∈ B′, we obtain
Z ′N (ξ) =eE
′
E
[∑
U∈P′
(I ′)Λ\UK#(U)
]
(1.3.23)
Conditional expectation
Lemma 6. K# factorizes over j + 1 scale connected components, namely
K#(U) =∏
V ∈Cj+1(U)
K#(V ) (1.3.24)
17
where Cj+1(U) is the set of connected components of U as a j + 1 polymer.
Proof. Let V1, . . . , V|C(U)| be all the connected components of U . For any E which may stand for
U,Z, P,Q, elements of X ∪ Y, one of the Bi, or X = XX∪Y , let E(p) = E\ ∪q 6=p Vq. It’s easy to
check that for i 6= j, E(i) and E(j) are strictly disjoint on scale j. Then the lemma is proved by the
factorization property of I,K on scale j.
We are now ready to take the expectation of K#(V ) conditioned on φ outside V + for each V ∈
C(U)\Λ, because Λ\V and V + don’t touch. In the case V = Λ, we just take expectation of K#(V )
without conditioning, but write E[K#(Λ)
∣∣(Λ+)c
]:= E
[K#(Λ)
]to shorten the notations.
Z ′N(ξ) = eEj+1E
[∑
U∈Pj+1
IΛ\Uj+1
∏
V ∈C(U)
E[
K#j (V )
∣∣(V +)c
]
︸ ︷︷ ︸
=:Kj+1(U)
]
(1.3.25)
Now we come back to the basic structure (1.2.22) or (1.3.6) with j replaced by j + 1. Obviously,
Kj+1(U) ∈ Pj+1,c. In Section 1.4 we give precise definitions for norms and spaces of the Kj above,
and in section 1.5 we prove smoothness of the above map (σj , Ej+1, σj+1,Kj) 7→ Kj+1.
1.3.3 Properties about conditional expectation
The variation principle
One of our main ideas is to write the Gaussian field φ into a sum of two decoupled parts. This is
important for the conditional expectation.
Fact. Given any positive definite quadratic form Q(v) for vector v, if v = (x, y), we can write Q(v) =
Q1(x) + L(x, y) +Q2(y) where Q1,2 are positive definite quadratic forms and L is the crossing term.
Let x(y) be its minimizer with y fixed. We can cancel L by shifting x by x
Q(v) = Q1(x− x) +Q ((x, y)) (1.3.26)
Proposition 7. Let U ⊂ V be finite graphs. Let φU and φUc be the restriction of φ to U and U c.
Let PU be the Poisson kernel for U and recall our convention that PUφ(x) = φ(x) for x /∈ U as in
18
subsection 1.2.2. Write φ(x) = PUφ(x) + ζ(x). Then the quadratic form
−∑
x∈Vφ(x)∆φ(x) = −
∑
x∈Uζ(x)∆D
U ζ(x) −∑
x∈VPUφ(x)∆PUφ(x) (1.3.27)
where ∆DU is the Dirichlet Laplacian for U .
Proof. We can apply the Fact (1.3.26) for φ = (φU , φUc), where U c = V \U , and
Q(φ) = −∑
x∈Vφ(x)∆φ(x) = −
∑
x∈UφU (x)∆
DU φU (x) + L(φU , φUc)−
∑
x∈UcφUc(x)∆
DUcφUc(x) (1.3.28)
where L is the crossing term, and ∆DUc is the Dirichlet Laplacian for U c. Since the minimizer with φUc
fixed is PUφ,
−∑
x∈Vφ(x)∆φ(x) = −
∑
x∈U(φU − PUφ) (x)∆
DU (φU − PUφ) (x) −Q ((PUφ, φUc))
= −∑
x∈Uζ(x)∆D
U ζ(x) −∑
x∈VPUφ(x)∆PUφ(x)
(1.3.29)
By this proposition, taking expectation of a function K(φ) conditioned on φ(x)∣∣x ∈ U c is simply
integrating out ζ:
E[K(φ)
∣∣U c]= Eζ [K(PUφ+ ζ)] (1.3.30)
where the covariance of ζ is the Dirichlet Green function for U .
The important scaling
We first prove some general results about harmonic functions, such as averaging properties and that
the derivative of a harmonic function is bounded by itself with a factor of dimension [1/length].
For R > 0 we call KR is a cube of size R centered at a if
KR :=y ∈ Zd
∣∣ |y − a|∞ ≤ R
(1.3.31)
for some a ∈ Zd.
Lemma 8. Let KR and KR2
be cubes of sizes R, R2 respectively centered at the same point. Assume
19
that u be harmonic in a cube KR. Let X = KR\KR/2, x ∈ KR/2 and d(x, ∂KR/2) > R/6. Then
u(x) ≤ O(R−d)∑
y∈Xu(y) (1.3.32)
u(x)2 ≤ O(R−d)∑
y∈Xu(y)2 (1.3.33)
and for e ∈ E,
|∂eu(x)| ≤ O(R−1) supy∈X
|u(y)| (1.3.34)
Proof. For any integer R2 ≤ b < R, let Kb be cubes co-centered with KR. Let w be the random walk
starting from x and τb := inft > 0
∣∣wt ∈ ∂Kb
. By Lemma 45, there exists a constant c so that
Px(wτb = y) ≤ cb−(d−1) (1.3.35)
for all y ∈ ∂Kb. Then since u is harmonic,
u(x) = Ex[u(wτS(b)
)]≤ cb−(d−1)
∑
y∈∂Kbu(y) (1.3.36)
Multiply both sides by bd−1 and sum over R2 ≤ b ≤ R, we have
Rdu(x) ≤ c′∑
y∈Xu(y) (1.3.37)
which proves (1.3.32). By Cauchy-Schwartz,
u(x) ≤ O(R−d)( ∑
y∈Xu(y)2
)1/2|X |1/2 (1.3.38)
which proves (1.3.33). Let Xo be the interior of X , namely Xo ∪ ∂Xo = X . In (1.3.37) replace u by
∂eu, which is harmonic in Xo, and apply summation by parts along each line parallel to e,
Rd |∂eu(x)| ≤ c′
∣∣∣∣∣∣
∑
y∈X,y+e/∈Xou(y + e) +
∑
y∈X,y−e/∈Xou(y − e)
∣∣∣∣∣∣
≤ O(Rd−1) supy∈X
|u(y)| (1.3.39)
which proves (1.3.34).
The next Lemma plays an important role in controlling the scalings. The Poisson kernels CX and
20
Dirichlet Green’s functions CU below are associated to ∆m and thus depend on m, see Subsection
1.2.2.
Lemma 9. Let d ≥ 2, x ∈ X ⊂ U ⊂ Λ. If d(x, ∂X) ≥ cLj, then
∑
y1,y2∈∂X(∂x,ePX)(x, y1)CU (y1, y2)(∂x,ePX)(x, y2) ≤ O(1)L−dj (1.3.40)
for all e ∈ E ,m > 0 where the constant O(1) depends on c. Here ∂x,e means taking discrete derivative
w.r.t. the argument x to direction e.
Proof. Notice that CU ≤ CΛ as quadratic forms, so it’s enough to prove the statement with CU replaced
by CΛ. Since y2 ∈ ∂X and CΛ(x− y2) is −∆m-harmonic in x ∈ X
∑
y1∈∂XPX(x, y1)CΛ(y1, y2) = CΛ(x, y2) (1.3.41)
Taking derivative w.r.t. x on the above equation we obtain that the left hand side of eq. (1.3.40)
equals∑
y2∈∂X∂x,eCΛ(x, y2)(∂x,ePX)(x, y2) (1.3.42)
Now let R = cLj/3 and define a cube KR centered on x. Apply lemma 8,
|∂x,ePX(x, y2)| ≤ O(L−j) |PX(x⋆1, y2)| (1.3.43)
where x⋆1 ∈ KR. By Corollary 44
|∂x,eCΛ(x, y2)| ≤ O(L−(d−1)j) (1.3.44)
so (1.3.42) is bounded by O(L−dj) since∑
y2∈∂X PX(x⋆1, y2) ≤ 1 for all m > 0.
21
1.4 Norms
1.4.1 Definitions of norms
Define hj = hL−(d−2)j/2 for some constant h > 0. We first define the norm for the fields. For j > 0
and X ⊂ Y define
‖(f, λξ)‖Φj(X,Y ) := h−1j sup
x∈X,e
∣∣Lj∂e(PY f(x) + λξ(x))
∣∣ (1.4.1)
‖f‖Φj(X,Y ) will be understood as ‖(f, 0)‖Φj(X,Y ). As a special case, if X ∈ Pj then we write
‖(f, λξ)‖Φj(X) := ‖(f, λξ)‖Φj(X,X+) (1.4.2)
We then define differentials for functions of the fields, and their norm. For test functions (f, λ)×n
:=
(f1, λ1ξ, · · · , fn, λnξ), the n-th differential of K(X,φ, ξ) is
K(n)(X,φ, ξ; (f, λ)×n
) :=∂n
∂t1 . . . ∂tnK(X,φ+
n∑
i=1
tifi, ξ +
n∑
i=1
tiλiξ)
∣∣∣∣ti=0
(1.4.3)
It is normed with a space of test functions Φ by
‖K(n)(X,φ, ξ)‖Tnφ (Φ) := sup‖(fi,λiξ)‖Φ≤1
∣∣K(n)(X,φ, ξ; (f, λ)
×n)∣∣ (1.4.4)
In most of our discussions Φ above will be chosen to be Φj(X). We then measure the amplitude of
K(X,φ, ξ) at a fixed function φ by incorporating all its derivatives at φ that we want to control:
‖K(X,φ, ξ)‖Tφ(Φ) :=
3∑
n=0
1
n!‖K(n)(X,φ, ξ)‖Tnφ (Φ) (1.4.5)
Define “regulators”:
G(X,Y ) := E[
eκ2
∑
x∈X,e∈E (∂eφ(x))2∣∣Y c] /N(X,Y ) (1.4.6)
for X ⊂ Y where the normalization
N(X,Y ) := E[
eκ2
∑
x∈X,e∈E (∂eφ(x))2∣∣φY c = 0
]
(1.4.7)
22
Define
‖K(X)‖j := supφ
‖K(X,φ, ξ)‖Tφ(Φj(X))G(X,X+)−1 (1.4.8)
Finally, for A > 0,
‖K‖j := supX∈Pj
‖K(X)‖jA|X|j (1.4.9)
The case j = 0 is treated separately: (1.4.1)-(1.4.5) are still defined for j = 0 with PY deleted and
X replaced by X . (1.4.8) is defined with G replaced by
G0(X) := eκ2
∑
x∈X,e∈E (∂eφ(x))2
(1.4.10)
and (1.4.9) is defined with A replaced by another constant A0 > A. See Appendix 1.9.3. no need
1.4.2 Properties
Lemma 10. Let F be function of φ, X ⊂ Y ⊂ U . We have the following property for the Tφ(Φ)
norms:
‖F (n)(φ)‖Tnφ (Φj(Y,U)) ≤ ‖F (n)(φ)‖Tnφ (Φj(X,U)) (1.4.11)
which also holds without n.
Proof. The proof is immediate because ‖f‖Φj(Y,U) ≥ ‖f‖Φj(X,U).
For furthur properties we first exploit a kind of functions K(X,φ, ξ) with an “special structure”: it
depends on φ, ξ via PX+φ+ ξ; in other words there exists a function K(X,ψ) so that
K(X,φ, ξ) = K(X,PX+φ+ ξ) (1.4.12)
In view of this special structure we define new function spaces Φj(X,Y ) for all X ⊂ Y
Φj(X,Y ) := functions harmonic on Y ⊕ Rξ (1.4.13)
23
(this is really a direct sum since ξ is either zero or non-harmonic) equiped with norm
‖g ⊕ λξ‖Φj(X,Y ) := h−1j sup
x∈X,e
∣∣Lj∂e(g(x) + λξ(x))
∣∣ (1.4.14)
The following result roughly says that conditional expectation of a product followed by taking norm
is bounded by the other way around, with norms taken on each factor.
Lemma 11. Let Xk ⊂ Yk ⊂ U , k = 1, 2, . . . , n, suppose that Kk(φ, ξ) = Kk(PYkφ + ξ). Then
E[∏
kKk(φ, ξ)∣∣U c]
depends on φ, ξ via PUφ+ ξ.
Furthermore, let ψ = PUφ + ξ and F (ψ) = Eζ
[∏
k Kk(ψ + PYkζ)]
where ζ is the Gaussian field
with Dirichlet Green’s function on U as covariance, then
∥∥∥F (ψ)
∥∥∥Tψ(Φj(X1∪X2,U))
≤ E
[∏
k
‖Kk(φ, ξ)‖Tφ(Φj(Xk,Yk))∣∣U c
]
(1.4.15)
Proof. The first statement holds because
E
[∏
k
Kk(φ, ξ)∣∣U c
]
= Eζ
[∏
k
Kk(PYk(PUφ+ ζ) + ξ)
]
(1.4.16)
and PYkPU = PU . For the second statement, without of generality let n = 2. By definition of Φj
norm, we have
∥∥∥F (n)(ψ)
∥∥∥Tnψ(Φj(X1∪X2,U))
≤ sup‖gi⊕λiξ‖Φj(X1∪X2,U)≤1
∣∣∣∣∣∂nti
∣∣∣∣ti=0
Eζ
[∏
k
Kk(PUφ+ PYkζ +
n∑
i=1
tigi + ξ +
n∑
i=1
tiλiξ)
]∣∣∣∣∣
≤Eζ
[
sup‖gi⊕λiξ‖Φj(X1∪X2,U)≤1
∣∣∣∣∣∂nti
∣∣∣∣ti=0
∏
k
Kk(PUφ+ ζ +
n∑
i=1
tigi, ξ +
n∑
i=1
tiλiξ)
]∣∣∣∣∣
(1.4.17)
By product rule of derivatives, the fact that harmonic functions on U are harmonic on Yk, and Lemma
24
10,
3∑
n=0
1
n!
∥∥∥F (n)(ψ)
∥∥∥Tnψ (Φj(X1∪X2,U))
≤3∑
n=0
1
n!E
[ n∑
r=0
(nr) sup‖gi⊕λiξ‖Φj(X1,Y1)≤1,i=1,...,r
∣∣∣∣∣∂rti
∣∣∣∣ti=0
K1(φ+
n∑
i=1
tigi, ξ +
n∑
i=1
tiλiξ)
∣∣∣∣∣
sup‖gi⊕λiξ‖Φj(X2,Y2)≤1,i=r+1,...,n
∣∣∣∣∣∂n−rti
∣∣∣∣ti=0
K2(φ+
n∑
i=1
tigi, ξ +
n∑
i=1
tiλiξ)
∣∣∣∣∣
∣∣U c]
≤E
[∏
k
‖Kk(φ, ξ)‖Tφ(Φj(Xk,Yk))∣∣U c
]
(1.4.18)
where in the last step we used
Kk(φ+
n∑
i=1
tigi, ξ +
n∑
i=1
tiλiξ) = Kk(PYkφ+
n∑
i=1
tigi + ξ +
n∑
i=1
tiλiξ) (1.4.19)
so that we’re effectively deforming (φ, ξ) using test functions in Φj(Xk, Yk) to come back to Tφ(Φj(Xk, Yk))
norm. Summing over n leads to the inequality without n.
Before the next lemma we introduce a short notation
(∂mf)2 := (∂f)2 +m2f2 (1.4.20)
Lemma 12. We have the following properties for the regulator:
1. G(X,Y, φ = 0) = 1
2. If X1 ⊂ Y1, X2 ⊂ Y2, and Y1, Y2 are strictly disjoint, then
G(X1, Y1)G(X2, Y2) = G(X1 ∪X2, Y1 ∪ Y2) (1.4.21)
3. We have an alternative representation of G(X,Y )
G(X,Y ) = exp
(
κ
2
∑
X
(∂ψ1)2 − 1
2
∑
Y
(∂mψ1)2 +
1
2
∑
Y
(∂mψ2)2
)
(1.4.22)
where ψ1 is the minimizer of∑
Y (∂mφ)2 − κ
∑
X(∂φ)2 with φY c fixed, and ψ2 is the minimizer
of∑
Y (∂mφ)2 with φY c fixed.
25
4. Fixing Y , G(X,Y ) is monotonically increasing in X for all X ⊂ Y .
5. With ψ1,2 defined in (3),
exp
(
κ
2
∑
X
(∂ψ2)2
)
≤ G(X,Y ) ≤ exp
(
κ
2
∑
X
(∂ψ1)2
)
(1.4.23)
Proof. (1)(2) hold by definition and the fact that G(X,Y ) is a function of φ on ∂Y . For (3),
G(X,Y ) =
´
eκ2
∑
X(∂φ)2− 12
∑
Λ(∂mφ)2
dY φ
/´
e−12
∑
Λ(∂mφ)2
dY φ
´
eκ2
∑
X(∂φ)2− 12
∑
Y (∂Dmφ)2dY φ
/´
e−12
∑
Y (∂Dmφ)2dY φ
(1.4.24)
where dY φ is the Lebesgue measure on φ(x) : x ∈ Y ∼= RY , ∂D takes Dirichlet boundary condition
on ∂Y . Using Fact (1.3.26) for both quadratic forms −κ2
∑
X(∂φ)2 + 12
∑
Λ(∂mφ)2 and 1
2
∑
Λ(∂mφ)2,
we obtain (3), where the quantity´
eκ2
∑
X(∂φ)2− 12
∑
Y (∂Dmφ)2
dY φ appears in both numerator and de-
nominator and thus cancels, and so does the quantity´
e−12
∑
Y (∂Dmφ)2
dY φ.
(4) holds because of (3) and that
infφ
∑
Y
(∂mφ)2 − κ
∑
X
(∂φ)2∣∣Y c
(1.4.25)
is monotonically decreasing in X . The two inequalities in (5) hold by replacing ψ1 by ψ2 or replacing
ψ2 by ψ1, and using definitions of ψ1, ψ2.
Before proving a furthur property we recall a formula. If U is a finite set and ψ = ψ(x) : x ∈ U is a
family of centered Gaussian random variables with covariance identity, and T : l2(U) → l2(U) satisfies
‖T ‖ < 1 then
E
[
exp
(1
2(ψ, Tψ)l2(U)
)]
= det (1− T )−1/2
= exp
(
1
2
∞∑
n=1
1
nTr(T n)
)
(1.4.26)
The next lemma shows that the conditional expectations almost automatically do the work when
one wants to see how the regulators undergo integrations, except that we need to manually control a
ratio of normalizations.
26
Lemma 13. For X ⊂ Y ⊂ U , and d(X,Y c) = c0Lj
E[G(X,Y )
∣∣U c]≤ cL
−dj|X|G(X,U) E[G(X,Y )
∣∣(Λ+)c
]:= E [G(X,Y )] ≤ cL
−dj|X| (1.4.27)
for some constant c only depending on c0. In particular if X = X0 for some X0 ∈ Pj, then the factor
cL−dj|X| can be written as c|X0|j with a suitable change of constant c → c2d. Furthurmore, G0 also
satisfies the same bound.
Proof. By definition
E[G(X,Y )
∣∣U c]= E
[
eκ2
∑
x∈X,e∈E (∂eφ(x))2∣∣U c] /N(X,Y ) = G(X,U)
N(X,U)
N(X,Y )(1.4.28)
Define φ = C1/2Y ψ so that ψ has covariance identity, where CY is the Dirichlet Green’s function for
Y . Then define TY = 12
∑
e∈E(∂eC1/2Y )⋆1X(∂eC
1/2Y ) as an operator on l2 = l2(Λ). We define similar
operators CU , TU in the same way for U . Let ∂De , −∆Y take Dirichlet boundary condition on ∂Y .
Because CY is the inverse of −∆Y
(f, TY f)l2 =1
2
∑
x∈X,e∈E(∂eC
1/2Y f(x))2 ≤ 1
2
∑
x∈Y,e∈E(∂De C
1/2Y f(x))2
≤∑
x∈YC
1/2Y f(x)(−∆Y )C
1/2Y f(x) ≤ (f, f)l2
(1.4.29)
we have ‖TY ‖ ≤ 1. Similarly ‖TU‖ ≤ 1. By (1.4.26)
N(X,U)
N(X,Y )=
E[eκ2 (ψ,TUψ)
]
E[eκ2 (ψ,TY ψ)
] =
(det(1− κTU )
det(1− κTY )
)−1/2
(1.4.30)
Taking logarithm, we need to compute
Tr (log(1− κTU )− log(1 − κTY )) ≤ O(1)Tr (κTU − κTY ) (1.4.31)
where we have used ‖TY ‖ ≤ 1, ‖TU‖ ≤ 1, κ is small, and log(1− x) is Lipschitz on x ∈ [− 12 ,
12 ]. Since
27
CU − CY = PY CU ,
Tr (TU − TY ) =1
2
∑
e∈E,x∈X∂e(CU − CY )∂
⋆e (x, x)
=1
2
∑
e∈E,x∈X
∑
y∈∂Y∂x,ePY (x, y)∂x,eCU (y, x)
(1.4.32)
By Lemma 8 and proceed similarly as eq. (1.3.42) in proof of Lemma 9, making use of the O(Lj)
distance between x and y, the above expression is bounded by O(L−jd) |X | which concludes the proof.
The bound on E[G(X,Y )
∣∣(Λ+)c
]is similar. The only modification is to replace CU by CΛ which
satisfies periodic instead of Dirichlet boundary condition. For G0, we can directly bound
E[
eκ2
∑
x∈X,e∈E (∂eφ(x))2∣∣U c]
(1.4.33)
by c|X|.
1.5 Smoothness of RG
First of all, we need some geometric results from [Bry09].
Lemma 14. (Brydges [Bry09]) There exists an η = η(d) > 1 such that for all L ≥ 2d + 1 and for all
large connected sets X ∈ Pj, |X |j ≥ η|X|j+1. In addition, for all X ∈ Pj, |X |j ≥ |X |j+1, and
|X |j ≥1
2(1 + η)|X |j+1 −
1
2(1 + η)2d+1|C(X)| (1.5.1)
Proof. The lemma is the same with [Bry09] (Lemma 6.15 and 6.16), so we omit the proof.
Proposition 15. Let B′(NPj+1,c
j+1 ) be a ball centered on the origin in NPj+1,c
j+1 . There exists A(d, L,B′)
and A⋆(d,A) such that for A > A(d, L,B′) and A⋆ > A⋆(d,A), the map (σj , Ej+1, σj+1,Kj) 7→ Kj+1
defined above is smooth from (−A⋆−1, A⋆−1)3 × BA⋆−1(NPj+1,c
j+1 ) to B′(NPj+1,c
j+1 ) where BA⋆−1(NPj,cj )
is a ball centered on the origin in NPj,cj with radius A⋆−1.
Proof. We omit subscript j for objects at scale j and write a prime for objects at scale j + 1, as in
Section 1.3. Let
A⋆−1 ≪ κ (1.5.2)
28
For U ∈ P ′c, U 6= ∅, by definition of K#,
K ′(U) =∑
V⊆U,V 6=∅
∑
(P,Q,Z,X ,Y)→V
(1 − eE′
)P (eE′
)(〈X〉\X)∩U\(P∪Q)(e−E′
)U∪X︸ ︷︷ ︸
≤(A⋆/2)−|P |j 2|(〈X〉\X)∩U\(P∪Q)|j 2|U∪X|j
EU+
(1.5.3)
where, with∏K :=
∏
Y ∈X K(Y )∏
(B,Y )∈Y1
|Y |jK(B, Y ) for short,
EU+
:= E
[
(I − eE′
)(U\V )∩〈X〉c IV ∩(〈X〉c\Z)δIZ(I − eE′
)Q∏
K∣∣(U+)c
]
=E
[
E
[
(I − eE′
)(U\V )∩〈X〉c IV ∩(〈X〉c\Z)δIZ(I − eE′
)Q∣∣(W+)c
]
︸ ︷︷ ︸
=:EW+
∏
K∣∣(U+)c
]
(1.5.4)
where W = U\X (recall that X := XX∪Y) and the last step used the corridors around K(Y ) in order
to make sense of the (W+)c conditional expectation. In the above W+ is a + operation at scale j and
U+ is a + operation at scale j + 1.
We first control EW+
. With φ = PW+φ+ ζ and (a+ b)2 ≤ 2a2+2b2, and using assumption (1.5.2),
Lemma 49, we list the estimates for each factors.
‖(I − eE′
)(B)‖Tφ(Φj(B)) ≤ 5(κA⋆)−1eκ2
∑
B(∂PW+φ)2+κ
2
∑
B(∂PB+ζ)2
(1.5.5)
for all B ∈ Q, where B+ ⊆W+ since Q ⊆ 〈X〉 \X; and,
‖(I − eE′
)(B)‖Tφ(Φj(B)) ≤ 5(κA⋆)−1eκ2
∑
B(∂PW+φ)2+κ
2
∑
B(∂P(B)+ζ)2
(1.5.6)
for all B ∈ Bj((U\V ) ∩ 〈X〉c), where (B)+ ⊆W+ since 〈X〉 is designed to ensure that; and
‖I(B)‖Tφ(Φj(B)) ≤ 2eκ2
∑
B(∂PW+φ)2+κ
2
∑
B(∂P(B)+ζ)2
(1.5.7)
for all B ∈ Bj(V ∩ (〈X〉c \Z)), where (B)+ ⊆W+ since B ⊆ 〈X〉c; and
‖δI(B)‖Tφ(Φj(B)) ≤ ‖I(B)− 1‖Tφ(Φj(B)) + ‖I(B)− 1‖Tφ(Φj(B))
≤ 8(κA⋆)−1eκ2
∑
B(∂PW+φ)2
eκ2
∑
B(∂PB+ ζ)2+κ
2
∑
B(∂P(B)+ζ)2
(1.5.8)
by ea + eb ≤ 2ea+b (a, b > 0) for all B ∈ Bj(Z), where (B)+ ⊆ W+ since Z ⊆ 〈X〉c. Combining all
29
above estimates, together with Lemma 11, we have
‖EW+‖Tφ(Φj(W )) ≤ (κA⋆/8)−|Q∪Z∪((U\V )\〈X〉)|jeκ2
∑
W (∂PW+φ)2M (1.5.9)
where
M ≤Eζ
[
eκ2
∑
B∈Bj(W )
∑
B(∂PB+ζ)2
eκ2
∑
B∈Bj (W )
∑
B(∂P(B)+ ζ)2]
(1.5.10)
In the next Lemma we show that M ≤ c|U|j .
Now we proceed to control EU+
. Instead of (a+ b)2 ≤ 2a2 +2b2 we use properties of the regulator
established in Section 1.4. Since for all X ∈ Pj,c
‖Kj(X)‖Tφ(Φj(X)) ≤ A⋆−1G(X,X+)A−|X|j (1.5.11)
By Lemma 12 (2)(4)(5) and Lemma 13
‖EU+‖Tφ(Φj(W )) ≤c|U|j · (κA⋆/8)−|Z∪Q∪((U\V )\〈X〉)|j−|X |−|Y|E
[
eκ2
∑
W (∂PW+φ)2
∏
Y ∈XG(Yk, Y
+k )
∏
Y ∈YG(Yi, Y
+i )∣∣(U+)c
]
A−|XX∪Y |j
≤c|U|j · (κA⋆/8)−|Z∪Q∪((U\V )\〈X〉)|j−|X |−|Y|G(U , U+)c′|W |j (A/c′)−|XX∪Y |j
(1.5.12)
We can bound the number of terms in the summation in (1.5.3) by k|U|j with k = 27, because
every j-block in U either belongs to V or V c, and the same statement is true if V is replaced by
P,Q,Z,XX , YY , and if it’s in Y ∈ Y it’s either the B of (B, Y ) ∈ Y or not. By Lemma 14, for
a = 12 (1 + η), with X = Xk, Y = (Bi, Yi)
a|U |j+1 ≤ a|Z|j+1 + a| ∪i Bi|j+1 + a| ∪k Xk|j+1 + a|Q|j+1 + a|(U\V ) ∩ 〈X〉c |j+1
≤(|Z|j + a2d+1|C(Z)|
)+ a|Y |+
(∑
k
|Xk|j + a2d+1|X |)
+(|Q|j + a2d+1|C(Q)|
)+ aLd|(U\V ) ∩ 〈X〉c |j
≤ (1 + a2d+1)(|Z|j + |Q|j) + a|Y |+ (|XX |j + a2d+1|X |) + aLd|(U\V ) ∩ 〈X〉c |j
(1.5.13)
Then we can easily check that with A,A⋆ sufficiently large as assumed in the proposition
‖K ′‖j+1 = supU∈P′
‖K ′(U)‖j+1Aa|U|j+1A(1−a)|U|j+1 < r (1.5.14)
30
where r is the radius of B′(NPj+1
j+1 ), because A|XX |j is cancelled by it’s inverse in (1.5.12), and
limA→∞
A(1−a)|U|j+1A−|XY |jk|U|jc|U|jc′|W |j+|XX∪Y |j2|(〈X〉\X)∩U\(P∪Q)|j2|U∪X|j = 0 (1.5.15)
limA⋆→∞
(κA⋆/8)−|Q∪Z∪((U\V )\〈X〉)|j−|X |−|Y|
·A(1+a2d+1)|Q∪Z|j+a|Y|+a2d+1|X |+aLd|(U\V )∩〈X〉c|j = 0
(1.5.16)
The derivatives of the map (σj , Ej+1, σj+1,Kj) 7→ Kj+1 are bounded similarly.
Lemma 16. There exists a constant c independent of L,A,A⋆ such that
M ≤ c|U|j (1.5.17)
Proof. Defining ζ = C1/2W+ψ where CW+ is the Dirichlet Green function for W+ and ψ ∈ L2(W+), M
is bounded by
Eψ exp
4κ∑
x∈Wψ(x)Tψ(x)
(1.5.18)
where ψ has identity covariance and
T =1
4
∑
B∈Bj(W ),e∈E
(
C1/2U+ P
⋆B+∂⋆e1B∂ePB+C
1/2U+ + C
1/2U+ P
⋆(B)+∂
⋆e1B∂eP(B)+C
1/2U+
)
=: T1 + T2 (1.5.19)
is a linear map from L2(W+) to itself. T1, T2 are defined to be the two terms respectively. We have
by Lemma 9,
Tr(T ) =1
4
∑
B∈Bj(W ),e∈E
(∑
x∈B∂ePB+CU+ (∂ePB+)
⋆(x, x) +
∑
x∈B∂eP(B)+CU+
(∂eP(B)+
)⋆(x, x)
)
≤ O(1)(L−dj + L−d(j+1))|W | ≤ O(1)|W |j(1.5.20)
For the next step we bound ‖T ‖. In fact,
(f, T1f)l2 =1
4
∑
B∈Bj(W )
∑
x∈B,e
(
∂ePB+C12
U+f(x))2
≤1
4
∑
B∈Bj(W )
∑
x∈B+,e
(
∂eC12
U+f(x))2
≤ cd∑
x∈W,e
(
∂eC12
U+f(x))2
(1.5.21)
where we used the fact that the harmonic extension minimizes the Dirichlet form to get rid of the
31
Poisson kernels. The constant cd comes from overlapping of B+’s. Then we can proceed as (1.4.29)
to bound the above expression by cd(f, f)l2 . T2 is bounded in the same way. Now by |Tr(T n)| ≤
|Tr(T )| ‖T ‖n−1, and formula (1.4.26) the proof of the lemma is completed.
1.6 Linearized RG
Having established smoothness, in this section we study the linearization of the RG map.
In view of Lemma 11, we can show, by induction along all the RG steps, that Kj(X) depends on
φ, ξ via PX+φ + ξ (at scale 0, I0,K0 depend on φ, ξ via φ + ξ). We write TayE [Kj(X)|(U+)c] to be
the second order Taylor expansion of E [Kj(X)|(U+)c] in PU+φ+ ξ.
Proposition 17. The linearization of the map (σj , Ej+1, σj+1,Kj) → Kj+1 around (0, 0, 0, 0) is
L1 + L2 + L3 where
L1Kj(U) =∑
X∈Pj,c\Sj ,X=U
E[Kj(X)
∣∣(U+)c
](1.6.1)
L2Kj(U) =∑
B∈Bj ,B=U
∑
X∈Sj,X⊇B
1
|X |j(1− Tay)E
[Kj(X)
∣∣(U+)c
](1.6.2)
L3 (σj , Ej+1, σj+1,Kj) (U) =∑
B∈Bj,B=U
(
Ej+1(B) +σj+1
4
∑
x∈B,e
(∂eP(B)+φ(x) + ξ(x)
)2
− σj4
∑
x∈BE[
(∂PB+φ(x) + ξ(x))2 ∣∣(U+)c
]
+∑
X∈Sj ,X⊇B
1
|X |jTayE
[Kj(X)|(U+)c
]) (1.6.3)
Proof. In Proposition 15 we proved that the map (σj , Ej+1, σj+1,Kj) → Kj+1 is smooth around
(0, 0, 0, 0) so that we can linearize the map. In (1.3.20) since V 6= ∅, Ij−eEj+1 factor doesn’t contribute
to the linear order. Also if X = ∅ then X = 〈X〉 = ∅, so 1 − eEj+1 and Ij − eEj+1 don’t contribute to
the linear order either. The terms that contribute to the linear order correspond to (Z, |X |, |Y |) equal
to (∅, 0, 1) or (∅, 1, 0) or (B, 0, 0) where B ∈ Bj. Grouping these terms into large sets part and small
sets part with Taylor leading terms and remainder we obtain the above linear operators.
1.6.1 Large sets
Lemma 18. Let L be sufficiently large and A be sufficiently large depending on L. Then L1 in
Proposition 17 is a contraction. Moreover, limL→∞ limA→∞ ‖L1‖ = 0.
32
Proof. The proof essentially follows the arguments of [Bry09] Lemma 6.18. By Lemma 13
‖L1Kj(U)‖j+1 ≤∑
X∈Pj,c\Sj ,X=U
‖Kj‖jc|X|jA−|X|j (1.6.4)
therefore by Lemma 14,
‖L1Kj‖j+1 = supU∈Pj+1
‖L1Kj(U)‖j+1A|U|j+1
≤[
supU∈Pj+1
A|U|j+1
∑
X∈Pj,c\Sj ,X=U
c|X|jA−|X|j]
‖Kj‖j
≤[
supU∈Pj+1
A|U|j+12Ld|U|j+1(A/c)−η|U|j+1
]
‖Kj‖j
(1.6.5)
where η > 1 is introduced in Lemma 14. The bracketed expression goes to zero as A→ ∞.
1.6.2 Taylor remainder
We prepare to show contractivity of L2. We first show that the Taylor remainder after the second
derivative is bounded by the third derivative. It’s a general result about the Tφ(Φ) norm with no need
to specify the test function space Φ.
Lemma 19. For F a function of φ let Tayn be its n-th order Taylor expansion about φ = 0, and Φ be
a space of test functions, then
‖(1− Tay)F (φ)‖Tφ(Φ) ≤ (1 + ‖φ‖Φ)3 supt∈[0,1]
∥∥∥F (3)(tφ)
∥∥∥T 3tφ
(Φ)(1.6.6)
Proof. The proof essentially follows as Lemma 6.8 in [Bry09]. By Taylor remainder theorem,
‖(1 − Tay2)F (φ)‖Tφ(Φ) =
3∑
n=0
1
n!sup
(f1,...,fn):‖fi‖Φ≤1
∣∣∣(F − Tay2F )
(n)(φ; f×n)
∣∣∣
=
3∑
n=0
1
n!sup
(f1,...,fn):‖fi‖Φ≤1
∣∣∣
(
F (n) − Tay2−n(F(n)))
(φ; f×n)∣∣∣
=
3∑
n=0
1
n!sup
(f1,...,fn):‖fi‖Φ≤1
∣∣∣∣1n<3
ˆ 1
0
(1 − t)2−n
(2− n)!∂3−nt F (n)(tφ; f×n) + 1n=3F
(3)(φ; f×n)
∣∣∣∣
=
3∑
n=0
1
n!sup
(f1,...,fn):‖fi‖Φ≤1
∣∣∣∣1n<3
ˆ 1
0
(1 − t)2−n
(2− n)!F (3)(tφ;φ×(3−n), f×n) + 1n=3F
(3)(φ; f×n)
∣∣∣∣
(1.6.7)
33
where φ×(3−n) means 3− n test functions φ. Calculating the time integrals,
‖(1− Tay2)F (φ)‖Tφ(Φ) ≤3∑
n=0
1
n!sup
(f1,...,fn):‖fi‖Φ≤1
∣∣∣∣
1
(3− n)!supt∈[0,1]
F (3)(tφ;φ×(3−n), f×n)
∣∣∣∣
≤ (1 + ‖φ‖Φ)3 supt∈(0,1)
∥∥∥F (3)(tφ)
∥∥∥T 3tφ(Φ)
(1.6.8)
where in the last step binomial theorem is applied.
The proof of next Lemma heavily depends on Lemma 8. Recall the “cube” KR and “cube-couple”
KR\KR/2 discussed there.
Lemma 20. If (B,X) ∈ Sj, B = U , and h is large enough depending on κ and L, then
(
2 + ‖φ‖Φj+1(X,U+)
)3
G(X, U+) ≤ qG(U , U+) (1.6.9)
for a constant q, where the dots operations on X are at scale j, and + operation on U is at scale j+1.
Proof. We summarize in advance all the sets introduced below: X ⊂ Y , and TY is a translation of Y
that doesn’t touch Y , and Y ∪ TY ⊂ Z with d(∂Z, Y ∪ TY ) = O(Lj), and W intersects with ∂Z but
doesn’t intersect with Y or TY . All these sets have size of O(Lj) and are in U . They can be chosen
because of O(Lj+1) distance between U and (U)c and smallness of X .
For the first step, let ψ2 = PU+φ. For each e ∈ E , ∂eψ2 is harmonic in U+ ∩ (U+ − e). Let Y ⊃ X
be a cube of size c1Lj. By (1.3.33)
supe∈E,x∈X
|∂eψ2(x)|2 ≤ O(L−dj)∑
e∈E(Y )
(∂eψ2(y))2
(1.6.10)
where ∂ef = f(x)− f(y) for e = (x, y); namely,
‖φ‖2Φj+1(X,U+)
≤ O(Ld)h−2∑
e∈E(Y )
(∂eψ2(y))2 (1.6.11)
So there exists q so that
(
2 + ‖φ‖Φj+1(X,U+)
)3
≤ q exp
κ′∑
e∈E(Y )
(∂eψ2(y))2
(1.6.12)
where κ′ = O(Ld)h−2.
34
As the next step, by Lemma 12 (3)(4) the left hand side of (1.6.9) times q−1 is bounded by
exp
κ′
2
∑
e∈E(Y )
(∂ψ2)2 +
κ
2
∑
X
(∂ψ′1)
2 − 1
2
∑
U+
(∂mψ′1)
2 +1
2
∑
U+
(∂mψ2)2
≤ exp
κ′
2
∑
e∈E(Y )
(∂eψ2)2 +
κ
2
∑
e∈E(Y )
(∂eψ′1)
2 − 1
2
∑
U+
(∂mψ′1)
2 +1
2
∑
U+
(∂mψ2)2
≤ exp
κ′
2
∑
e∈E(Y )
(∂eψ2)2 +
κ
2
∑
e∈E(Y )
(∂eψ1)2 − 1
2
∑
U+
(∂mψ1)2 +
1
2
∑
U+
(∂mψ2)2
(1.6.13)
where
ψ′1 maximizes κ
∑
X
(∂φ)2 −∑
U+
(∂mφ)2 fixing φ
∣∣(U+)c
ψ1 maximizes κ∑
e∈E(Y )
(∂eφ)2 −
∑
U+
(∂mφ)2 fixing φ
∣∣(U+)c
.
(1.6.14)
Note that ψ1 solves an inhomogeneous elliptic equation so that maximal principle holds. With these
definitions we observe the quantity
κ∑
e∈E(Y )
(∂eψ1)2 −
∑
U+
(∂ψ1)2 +
∑
U+
(∂ψ2)2 (1.6.15)
Replacing ψ2 by ψ1 makes it larger and replacing ψ1 by ψ2 makes it smaller. Therefore
∑
e∈E(Y )
(∂eψ2)2 ≤
∑
e∈E(Y )
(∂eψ1)2 (1.6.16)
As the next step we take a translate TY of Y so that: Y and TY don’t touch and are both in a
connected set Z ⊂ U and diam(Z) = c2Lj and d(∂Z, Y ∪ TY ) = c3L
j. For any points x ∈ Y, y ∈ TY
apply Newton-Leibniz formula along a curve in Z connecting x, y, and average x ∈ Y, y ∈ TY , we
obtain∑
e∈E(Y )
(∂eψ1)2 ≤ 2
∑
e∈E(TY )
(∂eψ1)2 + 2(diamZ)2 max
x∈Z,e1e2∈E(∂2e1e2ψ1)
2|Y | (1.6.17)
Let e1, e2 and z ∈ ∂Z maximizes ∂2e1e2ψ1 on Z ∪ ∂Z, by maximum principle. Note that ψ1 is ∆m-
harmonic outside Y (though not exactly ∆m-harmonic as we cross into Y ). (1.3.34) followed by (1.3.33)
35
can be applied with R = O(Lj) and we obtain
(diamZ)2 maxx∈Z,e1e2∈E
(∂2e1e2ψ1)2|Y | ≤ c4
∑
W
(∂ψ1)2 (1.6.18)
whereW ⊂ U is a cube of size c4Lj that doesn’t intersect with Y ∪TY . Then by (1.6.16)(1.6.17)(1.6.18),
equation (1.6.13) is bounded by
exp
cκ′
∑
e∈E(W∪TY )
(∂eψ1)2 +
κ
2
∑
e∈E(Y )
(∂eψ1)2 − 1
2
∑
U+
(∂mψ1)2 +
1
2
∑
U+
(∂mψ2)2
≤ exp
κ
2
∑
U
(∂ψ1)2 − 1
2
∑
U+
(∂mψ1)2 +
1
2
∑
U+
(∂mψ2)2
≤ exp
κ
2
∑
U
(∂ψ3)2 − 1
2
∑
U+
(∂mψ3)2 +
1
2
∑
U+
(∂mψ2)2
= G(U , U+)
(1.6.19)
where
ψ3 maximizes κ∑
U
(∂φ)2 −∑
U+
(∂mφ)2 fixing φ
∣∣(U+)c
(1.6.20)
and h is large enough so that 2cκ′ < κ.
Before the next Lemma we define
FX(U, φ, ξ) := E[Kj(X,φ, ξ)
∣∣(U+)c]
(1.6.21)
and we know that it depends on φ, ξ via ψ := PU+φ + ξ, i.e. there exists a function FX such that
FX(U, φ, ξ) = FX(U,ψ).
Lemma 21. Let L be sufficiently large. Then L2 in Proposition 17 is a contraction.
Proof. By Lemma 10 and Lemma 19 with test function space Φ = Φj(X, U+) we have
‖(1 − Tay)FX(U, φ, ξ)‖Tφ(Φj+1(U)) ≤ ‖(1− Tay)FX(U, φ, ξ)‖Tφ(Φj+1(X,U+))
=∥∥∥(1− Tay)FX(U,ψ)
∥∥∥Tψ(Φj+1(X,U+))
≤(
1 + ‖ψ‖Φj+1(X,U+)
)3 ∥∥∥F
(3)X (U,ψ)
∥∥∥T 3ψ(Φj+1(X,U+))
(1.6.22)
where Tay always means second order Taylor expansion in ψ = PU+φ + ξ so that the equality above
36
holds. Now by linearity of F(3)X in test functions and Lemma 11, we have
∥∥∥F
(3)X (U,ψ)
∥∥∥T 3ψ(Φj+1(X,U+))
≤ L− 32d∥∥∥F
(3)X (U,ψ)
∥∥∥T 3ψ(Φj(X,U
+))
≤L− 32dE
[
sup‖fi⊕λξ‖Φj(X)≤1
∣∣∣∣∂3t1t2t3
∣∣ti=0
Kj(X,φ+
3∑
i=1
tifi, ξ +
3∑
i=1
tiλiξ
∣∣∣∣
∣∣(U+)c
]
≤L− 32d3!E
[‖Kj(X,φ, ξ)‖Tφ,ξ(Φj(X))
∣∣(U+)c
]≤ O(L− 3
2d)‖Kj(X)‖jc|X|jG(X, U+)
(1.6.23)
where in the last step Lemma 13 is applied. Next we estimate
‖ψ‖Φj+1(X,U+) ≤ h−1j sup
x∈X,e
∣∣Lj∂ePU+φ(x)
∣∣ + h−1
j supx∈X,e
∣∣Lj∂eξ(x)
∣∣ ≤ ‖φ‖Φj+1(X,U+) + 1 (1.6.24)
by (1.2.15). By (1.6.22) (1.6.23) and Lemma 20 and (4) of Lemma 12
‖(1− Tay)FX(U)‖j+1 ≤ O(L−3d/2)c|X|j‖Kj(X)‖j ≤ O(L−3d/2)(A
c)−|X|j‖K‖j (1.6.25)
thus by Lemma 14,
‖L2Kj‖j+1 = O(L−3d/2)
[
supU∈Pj+1
A|U|j+1
∑
B∈Bj ,B=U
∑
X∈Sj,X⊇B
1
|X |j(A
c)−|X|j
]
‖K‖j
≤ O(L−3d/2)
[
supU∈Pj+1
A|U|j+1O(Ld)A−|U|j+1c2d
]
‖K‖j = O(L−d/2)‖K‖j(1.6.26)
1.6.3 L3 and determination of coupling constants
We now localize the last term in L3, which is the second order Taylor expansion of FX(U,ψ) in ψ (recall
that FX(U,ψ) and ψ are introduced before Lemma 21). To do this we fix a point z ∈ B, and replace
ψ(x) by x · ∂ψ(z) (which according to our convention means 12
∑
e∈E xe∂eψ(z)), and then average over
z ∈ B. We will show that the error of this replacement is irrelevant. Then
1
2F
(2)X (U, 0;ψ, ψ) = LocKj(B,X,U) + (1− Loc)Kj(B,X,U) (1.6.27)
37
where
LocKj(B,X,U) :=1
8|B|∑
z∈B,µ,ν∈E∂2t1t2
∣∣∣∣ti=0
Eζ [Kj(X, t1xµ + t2xν + ζ)] ∂µψ(z)∂νψ(z) (1.6.28)
and
(1− Loc)Kj(B,X,U) :=1
2|B|∑
z∈B
(
∂2t1t2
∣∣∣∣ti=0
Eζ [Kj(X, t1ψ + t2ψ + ζ)]
− ∂2t1t2
∣∣∣∣ti=0
Eζ [K(X, t1x · ∂ψ(z) + t2x · ∂ψ(z) + ζ)]
)
=1
2|B|∑
z∈B
(
F(2)X (U, 0;ψ − x · ∂ψ(z), ψ) + F
(2)X (U, 0;ψ − x · ∂ψ(z), x · ∂ψ(z))
)
(1.6.29)
We show that ψ − x · ∂ψ(z) gives additional contractive factors as going to the next scale:
Lemma 22. If ψ = PU+φ+ ξ ∈ Φj(X, U+),
‖ψ − x · ∂ψ(z)‖Φj(X,U+) ≤ O(L− d2−1)
(
‖φ‖Φj+1(U) + 1)
(1.6.30)
Proof. Since PU+x = x,
‖ψ − x · ∂ψ(z)‖Φj(X,U+) = h−1j sup
x∈X,eLj |∂ePU+φ(x) + ∂eξ(x) − ∂ePU+φ(z)− ∂eξ(z)|
=h−1j sup
x∈X,eLj (|∂ePU+φ(x) − ∂ePU+φ(z)|+ |∂eξ(x) − ∂eξ(z)|)
(1.6.31)
For the first term we apply Newton-Leibniz formula along a curve connecting x, z, and then apply
(1.3.34) with R = O(Lj+1) using the distance O(Lj+1) between X and ∂U ,
h−1j sup
x∈X,eLj |∂ePU+φ(x) − ∂ePU+φ(z)|
≤h−1j sup
x∈ULjdiam(X)O(L−j−1) |∂PU+φ(x)| ≤ O(L− d+2
2 ) ‖φ‖Φj+1(U)
(1.6.32)
where diam(X) = O(Ld) since X is small. The second term in (1.6.31) can be bounded by
h−1j sup
x∈X,eLj |∂eξ(x)− ∂eξ(z)| ≤ O(L− d
2 (N−j)) ≤ O(L− d+22 ) (1.6.33)
38
as long as j + 1 < N , and by d ≥ 2 and (1.2.15). Therefore
‖ψ − x · ∂ψ(z)‖Φj(X,U+) ≤ O(L− d2−1)
(
‖φ‖Φj+1(U) + 1)
(1.6.34)
Lemma 23. If L be sufficiently large and define
L′3Kj(U) =
∑
B=U
∑
X∈Sj,X⊇B(1− Loc)Kj(B,X,U) (1.6.35)
then L′3 is contractive with arbitrarily small norm; namely, ‖L′
3‖ → 0 as L→ ∞.
Proof. Recall that ψ = PU+φ+ ξ and let
Hz,X(U, φ, ξ) = F(2)X (U, 0;ψ − x · ∂ψ(z), ψ) (1.6.36)
then with f := PU+f + λξ
H(1)z,X(U, φ, ξ; (f, λξ)) = F
(2)X (U, 0;ψ − x · ∂ψ(z), f) + F
(2)X (U, 0; f − x · ∂f(z), ψ)
H(2)z,X(U, φ, ξ; (f1, λ1ξ), (f2, λ2ξ)) = F
(2)X (U, 0; f1 − x · ∂f1(z), f2) + F
(2)X (U, 0; f2 − x · ∂f2(z), f1)
(1.6.37)
and H(3)z,X = 0. In the calculations here, though z is fixed, PU+φ(z) should also participate in the
differentiations: PU+φ(z) → PU+(φ+ tf)(z).
Similarly with the previous lemma,
‖PU+f − x · ∂PU+f(z)‖Φj(X,U+) ≤ O(L− d+22 ) ‖f‖Φj+1(U) (1.6.38)
Since ‖−‖Φj(X,U+) ≤ ‖−‖Φj(U,U+) ≤ L−d/2 ‖−‖Φj+1(U) we also have estimates
‖ψ‖Φj(X,U+) ≤ O(L−d/2)(
‖φ‖Φj+1(U) + 1)
; ‖PU+f‖Φj(X,U+) ≤ O(L−d/2) ‖f‖Φj+1(U) (1.6.39)
39
Combine (1.6.37) (1.6.38) (1.6.39) we have for n = 0, 1, 2
∣∣∣H
(n)z,X(U, φ, ξ; (f, λξ)×n)
∣∣∣ ≤O(L−d−1)
∥∥∥F
(2)X (U, 0)
∥∥∥T 20 (Φj(X,U+))
·(
‖φ‖Φj+1(U) + 1)2−n n∏
i=1
‖(fi, λiξ)‖Φj+1(U)
(1.6.40)
So by the same arguments as (1.6.12) and Lemma 12(5),
‖Hz,X(U, φ, ξ)‖Tφ(Φj+1(U)) ≤ O(L−d−1)∥∥∥F
(2)X (U, 0)
∥∥∥T 2φ(Φj(X,U
+))
(
1 + ‖φ‖Φj+1(U)
)2
≤ O(L−d−1)∥∥∥F
(2)X (U, 0)
∥∥∥T 2φ(Φj(X,U
+))G(U , U+)
(1.6.41)
By Lemma 11, Lemma 13, Lemma 12(1) and X ∈ Sj
∥∥∥F
(2)X (U, 0)
∥∥∥T 2φ(Φj(X,U
+))≤ E
[
‖Kj(X,φ, ξ = 0)‖Tφ(Φj(X,U+))
∣∣φ(U+)c = 0
]
≤E[
‖Kj(X)‖j G(X,X+)∣∣φ(U+)c = 0
]
≤ ‖Kj(X)‖j c|X|j ≤ O(1)A−1 ‖Kj‖j(1.6.42)
Combining the above inequalities, we obtain
‖Hz,X(U)‖j+1 ≤ O(L−d−1)A−1 ‖K‖j
The other term in (1.6.29) is similar.
Therefore
‖L′3K(U)‖j+1 ≤
∑
B=U
∑
X∈Sj ,X⊇B
1
|B|∑
z∈BO(L−d−1)A−1 ‖K‖j ≤ O(L−1)A−1 ‖K‖j (1.6.43)
Since L′3Kj(U) = 0 unless U is a block, ‖L′
3Kj‖j+1 ≤ O(L−1) ‖K‖j .
Remark 24. From the previous two lemmas, we see the necessity of having anO(Lj+1) distance between
∂X and ∂X.
Now we turn to LocKj. We observe that the coefficient of ∂µψ(z)∂νψ(z) is 14 times
αµν(B) :=1
2|B|∑
X∈Sj,X⊇B∂2t1t2
∣∣∣∣ti=0
Eζ [Kj(X, t1xµ + t2xν + ζ)] (1.6.44)
Noticing ‖xµ‖Φj ≤ h−1Ldj/2 we have |αµν(B)| ≤ O(1)h−2 ‖Kj‖j A−1.
40
Now for a fixed D ∈ Bj+1, and for all B = D, αµν(B) depends on the position of B in D because ζ
is not translation invariant. This problem wasn’t present in the method [Bry09]. We cure this problem
by the following lemma.
Lemma 25. Let D ∈ Bj+1, and let Bct ∈ Bj be the j-block at the center of D. Then with definition
(1.6.44),
|αµν(B) − αµν(Bct)| ≤ O(L−d)h−2 ‖Kj‖j A−1 (1.6.45)
for all B ∈ Bj such that B = D.
Proof. Let T be a translation so that TB = Bct, and ζD+ , ζTD+ be Gaussian fields on D+, TD+ with
Dirichlet Green’s functions CD+ , CTD+ as covariances respectively.
|αµν(B)− αµν(Bct)|
≤ 1
8|Bct|∑
X∈Sj ,X⊇Bct
∣∣∣∣∂2t1t2
∣∣∣∣ti=0
(EζTD+ [Kj(X, t1xµ + t2xν + ζTD+)]− EζD+ [Kj(X, t1xµ + t2xν + ζD+)]
)∣∣∣∣
≤O(1)∑
X∈Sj ,X⊇Bcth−2 ‖Kj‖j A−1
∣∣∣N(X,D+)−N(X, TD+)
∣∣∣
N(X,X+)
(1.6.46)
As in the proof of Lemma 13, define TD+ = 12
∑
e∈E(∂eC1/2D+ )
⋆1X(∂eC1/2D+ ) and similarly TTD+ , we can
prove that their norms are both bounded by 1, and
N(X,D+)
N(X,X+)= e−
12Tr(log(1−κTD+ )−log(1−κTX+ )) (1.6.47)
Following the proof of Lemma 13, the trace in the above exponential is O(1), then since ex and
log(1− κx) are Lipschitz if x = O(1),
∣∣∣N(X,D+)−N(X, TD+)
∣∣∣
N(X,X+)≤ O(1)
∣∣Tr(log(1− κTD+)− log(1− κTTD+)
)∣∣
≤O(1)∣∣Tr(κTD+ − κTTD+
)∣∣ = O(1)
∣∣∑
x∈X,e∈E
∂e (CTD+ − CD+) ∂⋆e (x, x)∣∣
(1.6.48)
Let E = TD+∪D+, clearly for any X ∈ Sj , X ⊇ Bct, we have X+ ⊂ TD+∩D+ and d(∂E, ∂X+) =
41
O(Lj+1). Fix x ∈ X, without loss of generality assume that ∂ (CTD+ − CD+) ∂⋆(x, x) ≥ 0, then
∂ (CTD+ − CD+) ∂⋆(x, x)
≤∂ (CE − CD+) ∂⋆(x, x) = ∂PD+CE∂⋆(x, x)
(1.6.49)
The key observation is that d(x, ∂D+) = O(Lj+1). Then we proceed as the arguments following
(1.3.42) in proof of Lemma 8 of the arguments following (1.4.32) in proof of Lemma 13, the above
expression is bounded by O(L−d(j+1)). Since |X | = O(Ldj), we complete the proof.
Let D ∈ Bj+1. Define αµν := αµν(Bct) where Bct ∈ Bj is at the center of D. Clearly it’s
well defined (independent of D). By reflection and rotation symmetries, there exists an α so that
αµν = 12α(δµν + δµ,−ν).
Lemma 26. With ψ := PU+φ+ ξ
L′′3 :=
1
4
∑
B=D
(∑
x∈B,e∈Eα (∂eψ(x))
2 −∑
x∈B,e∈Eαµν (∂eψ(x))
2
)
(1.6.50)
is contractive.
Proof. This is essentially Lemma 10 of [Dim09], so the proof is omitted.
Proposition 27. We can choose Ej+1 and σj+1 so that if L be sufficiently large then L3 in Proposition
17 is contractive.
Proof. As the first step with D = B ∈ Pj+1(Λ), φ = PD+φ+ ζ we compute
E
[∑
x∈B,e∈E(∂ePB+φ+ ∂eξ(x))
2∣∣(D+)c
]
=∑
x∈B,e∈E(∂ePD+φ(x) + ∂eξ(x))
2 + δEj (1.6.51)
where δEj =∑
x∈B,e∈E Eζ[(∂ePB+ζ)2
]= O(1) by Lemma 9.
Let ψ = PD+φ+ ξ. By Lemma 232625, it remains to show the contractivity of
L3 =∑
B=U
[
Ej+1(B) +σj+1
4
∑
x∈B,e∈E(∂eψ(x))
2 − σj4
(∑
x∈B,e∈E(∂eψ(x))
2 + δEj
)
+ Eζ [Kj(X, ζ)] +α
4
∑
x∈B,e∈E(∂eψ(x))
2
] (1.6.52)
42
Choose
σj+1 = σj − α
Ej+1 = σjδEj − Eζ [Kj(X, ζ)]
(1.6.53)
then we actually have L3 = 0.
By the above choice of Ej+1 we can easily see that it’s the same number for Z ′N(ξ) and Z ′
N(0).
Therefore eEj is the same for Z ′N(ξ) and Z ′
N (0), for all j.
1.7 Proof of scaling limit of the generating function
Proposition 28. Let L be sufficiently large; A sufficiently large depending on L; κ sufficiently small
depending on L,A; h sufficiently large depending on L,A, κ; and r sufficiently small depending on
L,A, κ, h. Then for |z| < r there exists a constant σ depending on z so that the dynamic system
σj+1 = σj + α(Kj)
Kj+1 = LKj + f(σj ,Kj)
(1.7.1)
satisfies
|σj | ≤ r2−j ‖Kj‖j ≤ r2−j (1.7.2)
Proof. By contractivity of L we apply Theorem 2.16 in [Bry09] (i.e. the stable manifold theorem) to
obtain a smooth function σ = h(K0) so that (1.7.2) hold. Since K0 depends on z and σ, we solve σ
from equation σ − h(K0(z, σ)) = 0, using Lemma 51. Noting that this equation holds with (σ, z) = 0,
and that K0(z = 0, σ) = 0, the derivative of left hand side w.r.t. σ is 1. So by implicit function theorem
there exists a σ depending on z so that σ = h(K0(z, σ)). Therefore the proposition is proved.
With the generating function ZN(f) defined in (1.2.8), we have
Theorem 29. For any p > d there exists constants M > 0 and z0 > 0 so that for all ‖f‖Lp ≤ M ,
and all |z| ≤ z0 there exists a constant ǫ depending on z so that
limN→∞
ZN(f) = exp
(1
2
ˆ
Λ
f(x)(−ǫ∆)−1f(x)ddx
)
where ∆ is the Laplacian in continuum.
43
Proof. By (1.2.16),
ZN (f) = limm→0
e12
∑
x∈Λ f(x)(−ǫ∆m)−1f(x)Z ′N(ξ)
/Z ′N (0) (1.7.3)
In fact, since´
Λf = 0
e12
∑
x∈Λ f(x)(−ǫ∆m)−1f(x) → e12
´
Λf(x)(−ǫ∆)−1f(x)ddx (1.7.4)
as m→ 0 followed by N → ∞.
At scale N − 1 (we don’t want to continue all the way to the last step since it would be a bit
awkward to define IN−1 and IN ), by Prop 28 and Lemma 13
∣∣Z ′N (ξ)− eEN−1
∣∣ = eEN−1 |E [IN−1KN−1]− 1|
≤eEN−1
[∣∣∣∣
∑
∅6=X∈PN−1
(1 + 2−N+1)|Λ\X|N−12−N+1EG(X,X+)
∣∣∣∣+
∣∣∣∣IΛN−1 − 1
∣∣∣∣
]
≤eEN−1
[∑
∅6=X∈PN−1
(1 + 2−N+1)|Λ\X|N−12−N+1c|X|N−1 + 2−N+1
]
≤eEN−1
[
2Ld
(1 + 2−N+1)Ld
2−N+1cLd
+ 2−N+1
]
(1.7.5)
Since the constant eEN−1 is identical for Z ′N (ξ) and Z ′
N (0), and Z ′N(0) satisfies the same bound above,
so Z ′N(ξ)
/Z ′N (0) → 1. Therefore the theorem is proved.
1.8 Generalization to dipole system with boundary
1.8.1 Definition of model
Let L be a positive odd integer, and N ∈ N. For simplicity let d > 2. Define a cylinder Θ = Zd/ ∼.
Here ∼ is an equivalent relation and (a1, · · · , ad) ∼ (a′1, · · · , a′d) if there exists a (b2, · · · , bd) ∈ Zd−1
such that for 2 ≤ k ≤ d we have ak = a′k + bkLN and a1 = a′1. For (a1, · · · , ad) ∈ Zd we denote by
(a1, · · · , ad) ∈ Θ the image under the quotient map.
Assume that the dipole system is confined in the domain
Λ = (a1, · · · , ad) ∈ Θ : a1 ∈ (−LN
2,LN
2)
Its boundary consists of two (d − 1) dimensional tori, orthogonal to e1. Physically this describes a
44
region between two plates, which seems for us to be the simplest domain with a boundary (for instance
there’re not edges, corners etc which would be too tedious to be discussed). In the analysis that follows
we also need a domain
Ξ = (a1, · · · , ad) ∈ Zd : a1 ∈ (−LN
2,LN
2) (1.8.1)
and clearly Λ = Ξ/ ∼.
A configuration of the classical dipole gas system consists of n dipoles, with positions xk, k =
1, · · · , n. The dipoles have moments pk ∈ E , k = 1, · · · , n. We have restrictions that for each k,
xk, xk + pk ∈ Λ. This means that both ends of a dipole are in Λ.
In our model, Λ only serves as a restriction of positions of dipoles, while the Coulomb potential
which determines the energy of the configuration will be the one associated with Θ. This means
that ∂Λ is a pair of insulating boundaries physically. When we proceed to define potentials between
dipoles, we have to regularize the Coulomb potential because of recurrence of the random walk in Θ.
Let CΘ(x, y;m) be the Green’s function of −∆m for Θ.
The potential between two dipoles at xj , xk with moments pj , pk is
∂pj∂pkCΘ(xj , xk;m) (1.8.2)
The energy for this configuration is
H((xk, pk);m) =1
2
n∑
j,k=1
∂pj∂pkCΘ(xj , xk;m)
and the grand canonical ensemble can be written as
ZN = limm→0
∞∑
n=0
zn
n!
∑
(xk,pk)nk=1xk,xk+pk∈Λ,pk∈E
e−βH((xk,pk);m)
with β the inverse temperature.
Let φ(x) : x ∈ Θ be the Gaussian free field on the Θ with covariance CΘ(x, y;m). Then by
Sine-Gordon transform (see Appendix 1.9.1)
ZN = limM→∞
E
exp
2z∑
(x,y)∈E(Λ)
cos(√
β(φ(x) − φ(y)))
(1.8.3)
45
Define for X ⊂ Zd
W (X,φ) =∑
x∈X
∑
e∈Ex+e∈Λ
cos(√
β∂eφ(x))
(1.8.4)
then ZN = E [exp (zW (Λ, φ))].
Remark 30. In physics literatures such as [Die], a field theoretic model with a boundary is usually
defined via the field only inside the domain (in our case it would be φ(x) : x ∈ Λ). If we follow this
way, the quadratic form that defines the Gaussian measure will be written as
1
2(φ,∆φ)Θ =
1
2(φ,DNφ)∂(Λc) −
1
2
∑
(x,y)∈E(Λ)
(φ(x) − φ(y))2 (1.8.5)
where DN is the Dirichlet to Neumann map. Observe that we obtain a boundary term in this way,
which looks like |ξ| |φ(ξ)|2 in Fourier space. Our choice in this paper is to define the Gaussian field
outside Λ as well (but interactions are only inside Λ), which is mainly a matter of convenience.
1.8.2 The a priori tuning
Let Θ, Λ, Ξ be the continuum counterparts of Θ,Λ,Ξ. Namely, let Θ = Rd/ ∼ where (a1, a2, . . . , ad) ∼
(a′1, a′2, . . . , a
′d) if and only if there exists a (b2, · · · , bd) ∈ Zd−1 so that a1 = a′1 and ai = a′i + bi for
all i = 2, . . . , d; let Ξ ⊂ Rd be the set of points with first coordinate in [− 12 ,
12 ]; let Λ = Ξ/ ∼; define
ΘM similarly. Given a mean zero function f ∈ C∞(Θ) with compact support, we study the generating
function
ZN (f) := limm→0
E[e∑
x∈Θ f(x)φ(x)ezW (Λ,φ)]
E[ezW (Λ,φ)
] (1.8.6)
where L(d+2)N/2f(LNx) = f(x). The main question is the scaling limit of ZN(f) as N → ∞.
Before we describe the a priori tuning of the Gaussian measure, we gather some basic aspects of
weighted graphs.
preliminaries for weighted graphs
A weighted graph is a triple (X,E, µ) where X is a finite or countably infinite set of vertices, and E
is a subset of (x, y) : x, y ∈ X, x 6= y whose elements are called edges. We write x ∼ y if (x, y) ∈ E.
A nonnegative weight µxy is endowed for each pair x, y and µxy > 0 iff x ∼ y. Let µx =∑
y µxy. For
A ⊆ X , define µ(A) =∑
x∈A µx. The weighted graph (X,E, µ) is associated with a Dirichlet quadratic
46
form E , a Laplacian ∆, and a random walk w:
E(f, g) = 1
2
∑
x∼y(f(x) − f(y))(g(x)− g(y))µxy
for all f, g ∈ H2 where H2 = f : X → R|E(f, f) <∞, and
∆f(x) =∑
y
P (x, y)f(y)− f(x) =1
µx
∑
y
(f(y)− f(x))µxy
for all f : X → R. If ∆f = 0 we say that f is harmonic. The maximum principle holds. The random
walk w = wn on X has transition probabilities P (x, y) =µxyµx
. Its heat kernel is pn(x, y) = Px(wn =
y)/µy. We can also associate a continuous Markov chain wtt≥0 which stays at x for an (independent)
exponential time with parameter 1 and then jumps to y with probability P (x, y). Its heat kernel is
pt(x, y) =
∞∑
n=0
e−ttn
n!pn(x, y)
We refer to [Kum10] for these definitions.
Now we define a typical kind of weights that we’ll consider in this section for Zd and Θ. Given
parameters (α1, α2) with |αi − 1| < 1/2, define weighted graphs Zd(α1,α2)= (Zd, E(Zd), µ(α1,α2)) with
weights
µ(α1,α2)xy =
α1 x ∼ y, and, x or y ∈ Λ
α1+α2
2 (x, y) ∈ E(∂Λ)
α2 x ∼ y, and, x or y ∈ (Λ ∪ ∂Λ)c
(1.8.7)
We’re particularly interested in cases such as (α1, α2) = (ǫ, 1) or (α1, α2) = (1, 1/ǫ) etc. with parameter
ǫ s.t. |ǫ − 1| is small. Define Θ(α1,α2) := Zd(α1,α2)/ ∼ with induced weights. Let ∆(α1,α2), D(α1,α2) be
the Laplacian operator and Dirichlet quadratic form associated with (Zd, E(Zd), µ(α1,α2)).
Define for X ⊆ Λ
V (X,φ) :=1
4
∑
x∈X,e∈E(∂eφ(x))
2(1.8.8)
U∂(X,φ) :=1
8
∑
(x,y)∈E(Θ),y∈∂Λx∈∂X∩∂Λ
(φ(x) − φ(y))2+
1
4
∑
(x,y)∈E(Θ)x∈∂Λ,y∈X
(φ(x) − φ(y))2
(1.8.9)
With these definitions, our tuning is to split part of the quadratic form of the Gaussian measure
47
into the integrand, so that the resulting Gaussian is associated to a Laplacian with weights µ(ǫ,1):
ZN (f) = limm→0
E(ǫ,1)[
e∑
x∈ΘMf(x)φ(x)
e(ǫ−1)V (Λ,φ)+(ǫ−1)U∂(Λ,φ)+zW (Λ,φ)]
E(ǫ,1)[e(ǫ−1)V (Λ,φ)+(ǫ−1)U∂(Λ,φ)+zW (Λ,φ)
] (1.8.10)
Note that normalization factors caused by re-definition of Gaussian:
E(ǫ,1) [exp ((ǫ − 1)V (Λ, φ) + (ǫ − 1)U∂(Λ, φ))] (1.8.11)
appear in both numerator and denominator and are cancelled.
Remark 31. 1) The motivation is that zW (Λ, φ) behaves as O(z)∑
(x,y)∈E(Λ)(φ(x)−φ(y))2, so a part
of the Gaussian measure is split out to “cancel” this interaction. However we didn’t split out a term
from Gaussian such like ǫ−12
∑
(x,y)∈E(Λ)(φ(x) − φ(y))2, but instead we defined Gaussian free field
via a Laplacian with weight in the form of (1.8.7), mainly because our Laplacian here is good for a
separation of variable analysis for the Green’s function. 2) In definition of V , the derivative is allowed
to be taken towards all directions, even if x + e /∈ Λ, because we don’t want to distinguish the forms
of V (X) for X in the deep bulk of Λ and for X close to ∂Λ. We will let V close to ∂Λ flow as if they
were in the bulk. 3) Finally, these considerations about E(ǫ,1) and V give us a boundary term U∂ as
above, which is “irrelevant” in RG terminology.
We would like to make the RG map for the bulk of Λ independent of ǫ. So we rescale φ → φ/√ǫ
and let σ = ǫ−1 − 1 , so that
ZN (f) = limm→0
E(1,1/ǫ)[
e∑
x∈Θ f(x)φ(x)/√ǫe−σV (Λ,φ)−σU∂ (Λ,φ)+zW (Λ,
√1+σφ)
]
E(1,1/ǫ)[e−σV (Λ,φ)−σU∂ (Λ,φ)+zW (Λ,
√1+σφ)
] (1.8.12)
We make a translation φ→ φ+ ξ where ξ = (−∆(1,1/ǫ)m )−1f in the numerator, which becomes
e12
∑
x∈Λ f(x)(−∆(1,1/ǫ)m )−1f(x)E
[
e−σV (Λ,φ+ξ)−σU∂ (Λ,φ+ξ)+zW (Λ,(φ+ξ)/√ǫ)]
(1.8.13)
Then
ZN (f) = limm→0
e12
∑
x∈Λ f(x)(−∆(1,1/ǫ)m )−1f(x)Z ′
N(ξ)/Z ′N (0) (1.8.14)
where
Z ′N(ξ) = E
[
e−σV (Λ,φ+ξ)−σU∂(Λ,φ+ξ)+zW ((φ+ξ)/√ǫ)]
(1.8.15)
48
Simplification of notations
In this Section, for simpler notations, we write ∆ := ∆(1,1/ǫ), E = E(1,1/ǫ) keeping in mind that it
depends on the weight (1, 1/ǫ). Also, within this Section, “harmonicity” is always respect to −∆(1,1/ǫ)m =
−∆(1,1/ǫ)+m2, and Poisson kernels and Green’s functions are all associated with ∆(1,1/ǫ)m and we won’t
specify explicitly ǫ,m for them.
1.8.3 RG maps and modification of norms
With all the definitions for polymers in 1.3.1, we have the following polymer expansion
Z ′N (ξ) = E
∑
X⊆Λ
I(Λ\X,φ+ ξ)K(X,φ+ ξ)
(1.8.16)
where I is defined the same as (1.2.19) and
K(X,φ) =∏
x∈Xe−
14σ
∑
e∈E (∂eφ(x)+∂eξ(x))2
·(
e−σU∂(x,φ+ξ)+zW(x,(φ+ξ)/√ǫ) − 1
)(1.8.17)
We define the RG map (σ,E′, σ′,K) → K ′ the same as subsection 1.3.2, except that now E′ depends
on B ∈ Bj.
We show that with boundary, Lemma 8, 9 still hold. For this purpose, for R > 0 consider cube of
size R
KR :=y ∈ Zd
∣∣ |y − a|∞ ≤ R
(1.8.18)
for some a ∈ Zd. We say that KR is Λ-adapted if either KR ⊂ Λ or ∂Λ separate KR into two half
cubes with the same size.
Lemma 32. Let KR and KR2
be Λ-adapted cubes of sizes R, R2 respectively centered at the same point.
Assume that u be harmonic in a cube KR. Let X = KR\KR/2, x ∈ KR/2 and d(x, ∂KR/2) > R/6.
Then (1.3.32)(1.3.33)(1.3.34) still hold.
Proof. For any integer R2 ≤ b < R, Kb is Λ-adapted. The statement holds by Lemma 47 and the same
arguments in proof of 8.
A very useful argument is as follows.
49
Fact 33. One of the following two cases must happen. The first case is that there exists a cube KRwith R = 1
3c0Lj that contains x and d(x, ∂KR) > R/3. The second case is that there doesn’t exist such
a cube, which means that x is too close to ∂Λ, and therefore we can find a Λ-adapted cube KR which
is crossed from the middle by ∂Λ and x ∈ ∂KR and d(x, ∂KR) > R/3.
Lemma 34. Under the setting of Subsection 1.8.1, Lemma 9 still holds.
Proof. It’s enough to prove the statement with CU replaced by CΘ. Following the argument of Lemma
9, we only need to show the bound of ∇xPX(x, y2) and ∇xCΘ(x, y2) where y2 ∈ ∂X . Using Fact 33,
we can find a Λ-adapted cube KR with R = O(Lj) so that
|∂x,ePX(x, y2)| ≤ O(L−j) |PX(x⋆1, y2)| (1.8.19)
where x⋆1 ∈ KR. The bound for ∇xCΘ(x, y2) is immediate by Lemma 48.
For the boundary problem, we modify the regulators:
G(X,Y ) := E[
eκ2
∑
x∈X,e∈E µe(∂eφ(x))2∣∣Y c] /N(X,Y ) (1.8.20)
for X ⊂ Y where the normalization
N(X,Y ) := E[
eκ2
∑
x∈X,e∈E µe(∂eφ(x))2∣∣φY c = 0
]
(1.8.21)
G0(X) := eκ2
∑
x∈X,e∈E µe(∂eφ(x))2
(1.8.22)
where µe := µ(x,x+e) is the weight for weighted graph (Θ, α(1,1/ǫ)).
Proposition 35. With the modified regulators defined above, Lemma 10 - Lemma 12 still hold. In
particular, Lemma 12 (3)(5) are restated as
G(X,Y ) = exp
κ
4
∑
X,e
µe(∂eψ1)2 − 1
4
∑
Y,e
µe(∂e,mψ1)2 +
1
4
∑
Y,e
µe(∂e,mψ2)2
(1.8.23)
where ψ1 is the minimizer of∑
Y,e µe(∂e,mφ)2−κ∑X,e(∂eφ)
2 with φY c fixed, and ψ2 is the minimizer
50
of∑
Y,e µe(∂mφ)2 with φY c fixed; and
exp
κ
4
∑
X,e
µe(∂eψ2)2
≤ G(X,Y ) ≤ exp
κ
4
∑
X,e
µe(∂eψ1)2
(1.8.24)
Also, the Lemma 13 holds without the special treatment of E[G(X,Y )
∣∣(Λ+)c
](i.e. regard Λ+ ⊂ Θ).
Proof. Some modifications for proof of Lemma 13 is needed. We have to define TY , TU to be operators
on l2 = l2(Θ) where
TY =1
2
∑
e∈Eµe(∂eC
1/2Y )⋆1X(∂eC
1/2Y ) (1.8.25)
TU =1
2
∑
e∈Eµe(∂eC
1/2U )⋆1X(∂eC
1/2U ) (1.8.26)
where µe := µ(x,x+e) is the weight for weighted graph (Θ, α(1,1/ǫ)). Now since CY and −∆Y are both
associated with the weighted graph (Θ, α(1,1/ǫ)), they’re still inverse of each other, and Lemma 13
follows by the same arguments.
1.8.4 Linearized RG map
Proposition 36. Let B′(NPj+1,c
j+1 ) be a ball centered on the origin in NPj+1,c
j+1 . There exists A(d, L,B′)
and A⋆(d,A) such that for A > A(d, L,B′) and A⋆ > A⋆(d,A), the map (σj , σj+1, Ej+1,Kj) 7→ Kj+1
defined above is smooth from (−A⋆−1, A⋆−1)2 × (−A⋆−1, A⋆−1)Bj × BA⋆−1(NPj+1,c
j+1 ) to B′(NPj+1,c
j+1 )
where BA⋆−1(NPj,cj ) is a ball centered on the origin in NPj,c
j with radius A⋆−1.
Proof. The proof is essentially the same with Proposition 15 and thus omitted.
Proposition 37. The linearization of the map (σj , Ej+1, σj+1,Kj) → Kj+1 around (0, 0, 0, 0) is
L1 + L2 + L3 where L1,L2,L3 are given by the same forms as in Prop 17.
The discussions about L1,L2 are the same as the torus case. For the boundary problem, there’s
an important difference for the analysis of L3: the αµν defined before Lemma 26 does depend on Bct
(in other words, on D). This is due to two reasons: firstly,
αµν(B) :=1
2|B|∑
X∈Sj(Λ),X⊇B∂2t1t2
∣∣∣∣ti=0
Eζ [Kj(X, t1xµ + t2xν + ζ)] (1.8.27)
so if B touches ∂Λ, the set X ∈ Sj(Λ), X ⊇ B becomes smaller; secondly, the expectation is different
51
since ∂Λ crosses (B)+.
Define α := α(Bct) where Bct is in the bulk of Λ. If B ∈ Bj touches ∂Λ, define
α′µν(B) :=
1
2|B|∑
X∈Sj(Θ),X⊇B∂2t1t2
∣∣∣∣ti=0
Eζ [Kj(X, t1xµ + t2xν + ζ)] (1.8.28)
Following the same proof of Lemma 25 we can prove that α − α′µν(B) for B ∈ Bj touching ∂Λ is
negaligible. So we only need to prove that αµν(B)− α′µν(B) for B ∈ Bj touching ∂Λ is negaligible:
Lemma 38. Define
∆K(U) =∑
B=U
V (αµν − α′µν , B) (1.8.29)
where V (α,B) :=∑
B α(∂ψ)2, with ψ := PU+φ + ξ, then for L sufficiently large, ∆ is a contraction
with arbitrarily small norm.
Proof. Observe that (αµν − α′µν)(B) = 0 if B is away from ∂Λ. Combining this observation with
|αµν |+ |α′µν | ≤ O(1)h−2‖K‖jA−1 we obtain the result.
Proposition 39. Under the same assumptions of Proposition 28, the estimates (1.7.2) still hold.
Proof. The arguments in the proof of Prop 28 allow us to show the estimates (1.7.2) in the bulk of Λ.
It remains to use |σj | ≤ r2−j , which holds even near the boundary, to show that near the boundary
‖Kj‖j ≤ r2−j is also true. Suppose this is true for j, then for some constant M and sufficiently small
r
‖Kj+1‖j+1 ≤ 1
4‖Kj‖j +M
(
σ2j + ‖Kj‖2j
)
≤ r2−j−1 (1.8.30)
which is the desirable bound at scale j + 1.
With the generating function ZN(f) defined in (1.8.6), we have
Theorem 40. Under the same assumptions of Theorem 29 there exists a constant ǫ depending on z
so that
limN→∞
ZN (f) = exp
(1
2
ˆ
Λ
f(x)(−∆ǫ)−1f(x)ddx
)
where ∆ǫ is an operator in continuum: ∆ǫ = ∇(χ(x)∇) with χ(x) = 1 if x ∈ Λ and χ(x) = 1/ǫ if
x /∈ Λ.
Proof. Since
ZN(f) = limm→0
e12
∑
x∈Θ f(x)(−∆ǫm)−1f(x)Z ′N (ξ)
/Z ′N(0) (1.8.31)
52
In fact, since´
Λf = 0
e12
∑
x∈Λ f(x)(−∆ǫm)−1f(x) → e12
´
Λf(x)(−∆ǫ)−1f(x)ddx (1.8.32)
as m→ 0 followed by N → ∞.
At scale N , by Prop 39 and Lemma 13
∣∣Z ′N(ξ)− eEN
∣∣ = eEN |E [IN KN ]− 1|
≤eEN[∣∣∣∣2−NEG(Λ,Λ+)
∣∣∣∣+
∣∣∣∣IΛN − 1
∣∣∣∣
]
≤c · eEN2−N
(1.8.33)
Since the constant eEN is identical for Z ′N (ξ) and Z ′
N (0), and Z ′N(0) satisfies the same bound above,
so Z ′N(ξ)
/Z ′N (0) → 1. Therefore the theorem is proved.
1.9 Appendices
1.9.1 Sine-Gordon transformation
Define f(x) =∑n
k=1 ∂pkδxk(x). Then
e−βHm(xk,pk) = E[
ei∑
x∈Λ
√βφ(x)f(x)
]
= E[
ei√β∑nk=1 ∂pkφ(xk)
]
(1.9.1)
where i =√−1 . Thus
ZN = limm→0
E
[ ∞∑
n=0
zn
n!
(∑
(x,p)∈Λ×Eei
√β∂pφ(x)
)n]
= limm→0
E
exp
2z∑
(x,y)∈E(Λ)
cos(√
β(φ(x) − φ(y)))
(1.9.2)
Define W (X,φ) as (1.2.7) then ZN = E [exp (zW (Λ, φ))].
1.9.2 Decay of Green’s functions and Poisson kernels
The decay rates of Green’s functions and their derivatives are essential in our method.
53
Homogeneous case
First of all consider the Green’s function of −∆m = −∆+m2 on Zd. If d ≥ 3, let Gm = (−∆m)−1.
If d = 2 let Gm(x) = (−∆m)−1(x) − (−∆m)−1(0) for m > 0 and from [Law91] we know that
limm→0Gm(x) exists. Write G = Gm=0.
Lemma 41. Let G(x) = ad|x|2−d if d ≥ 3 or G(x) = ad log |x| if d = 2 where ad only depends on d.
Let k = 2γ+log 8π if d = 2 where γ is Euler’s constant and k = 0 if d ≥ 3. As |x| → ∞
G(x) = G(x) + k +O(|x|−d) (1.9.3)
Furthermore, for all e ∈ E
∂eG(x) = ∂eG(x) +O(|x|−(d+1)) (1.9.4)
where ∂eG(x) is also discrete derivative.
Proof. See [LL10] Theorem 4.3.1, 4.4.4, Corollary 4.3.3, 4.4.5. The only difference here is a sharper
estimate of the error term for ∇G, which is remarked after those corollaries and thus the proof is
omitted.
Lemma 42. Let d ≥ 2. For all e ∈ E, x ∈ Λ where Λ is the torus defined in subsection 1.2.3 and
m ≥ 0,∣∣∣∣
∑
y∈Zd\0∂eGm(x+ LNy)
∣∣∣∣≤ cdL
−(d−1)N (1.9.5)
where cd only depends on d.
Remark 43. Note that the left hand side is not absolutely summable uniformly in m ≥ 0.
Proof. It’s enough to show the proof for m = 0. Denote Dµ to be the smooth derivative. Without loss
of generality assume e = e1. The term O(|x|−(d+1)) in (1.9.4) is summable:
∣∣∣∣
∑
y∈Zd\0O(|x+ LNy|−(d+1))
∣∣∣∣= O(L−(d+1)N ) (1.9.6)
54
Up to this term, ∂e1G(x+ LNy) is equal to
G(x+ e1 + yLN)− G(x + yLN)
=
(
G(yLN) + (x+ e1) ·DG(yLN ) +1
2(x+ e1)
2 ·D2G(yLN ) +O(L2N sup∣∣D3G
∣∣)
)
−(
G(yLN ) + x ·DG(yLN ) +1
2x2 ·D2G(yLN) +O(L2N sup
∣∣D3G
∣∣)
)
=D1G(yLN) + (x ·DD1G(yLN) +
1
2D2
1G(yLN)) +O(L2N sup
∣∣D3G
∣∣)
(1.9.7)
where the last term comes from Taylor remainder theorem. It’s a straightforward calculation to see
that the summation over y 6= 0 of the first three terms is zero due to cancellations. The summation
over y 6= 0 of the last term gives O(L−(d−1)N).
Corollary 44. Let d ≥ 2 and Cm be the Green’s function of −∆+m2 on the torus Λ. For all e ∈ E,
x ∈ Λ and m ≥ 0,
|∂eCm(x)| ≤ cd|x|−(d−1) (1.9.8)
where cd only depends on d.
Proof. The statement is immediately shown by
∂eCm(x) =∑
y∈Zd
∂eGm(x+ LNy) (1.9.9)
and Lemma 41, 42.
Lemma 45. Define a cube KR = x ∈ Zd : xk ∈ [1, R− 1]. If d(x, ∂KR) > R/3, y ∈ ∂KR then
PKR(x, y) ≤O(1)
Rd−1(1.9.10)
Proof. It’s enough to prove it for large R. Without loss of generality we assume that y ∈ ∂1KR = x ∈
∂KR : x1 = R. The Poisson kernel PKR(x, y) associated to the standard Laplacian is, by Prop 8.1.3
of [LL10], equal to
1
nd−1
∑
z∈∂1KR
sinh(αzx1π/(2n))
sinh(αzπ)
d∏
i=2
sin(zixiπ
2n)
d∏
i=2
sin(ziyiπ
2n) (1.9.11)
55
where αz is the unique nonnegative number satisfying
cosh(αzπ
2n) +
d∑
i=2
cos(ziπ
2n) = d (1.9.12)
If d(x, ∂KR) > R/3, we have αzx1π/n1 ≤ 56αzπ, so
PKR(x, y) ≤1
nd−1
∑
z∈∂1KR
sinh(αzx1π/(2n))
sinh(αzπ)≤ C
Rd−1(1.9.13)
for some C depending on d, because sinh grows exponentially and αz grows at least linearly in z.
Inhomogeneous case
After the tuning the Gaussian is determined by a Laplacian with non-constant coefficient. We are
interested in the decay rate of its Green function. In a quite general setting, we have the following
Theorem 46. ([Del99])For a finite or countably infinite weighted graph (Γ, µ), suppose that it has
(1) (Doubling volume property) V (x, 2r) ≤ C1V (x, r) for all x ∈ Γ, r ≥ 0;
(2) (Poincare inequality) for any function f on Γ,
∑
x∈B(x0,r)
µx|f(x)− fB|2 ≤ C2r2
∑
x,y∈B(x0,2r)
µxy(f(x) − f(y))2 (1.9.14)
for all x0 ∈ Γ, r ≥ 0, where
fB =1
V (x0, r)
∑
x∈B(x0,r)
µxf(x) (1.9.15)
(3) (Local ellipticity) for all x ∈ Γ, x ∼ x, and if x ∼ y then µxy ≥ αµx where α > 0.
Then the Green function C(x, y) = 1µy
∑∞n=0 pn(x, y) is finite if and only if
∞∑
n=0
n
V (x, n)< +∞ (1.9.16)
and it satisfies
C−1∞∑
n=d(x,y)
n
V (x, n)≤ C(x, y) ≤ C
∞∑
n=d(x,y)
n
V (x, n)(1.9.17)
In our case of (Zd, µ), d > 2, where µxy = 1 if (x, y) ∈ E(ΛN ) and µxy = σ if (x, y) ∈ E(Zd)\E(ΛN ),
where |σ − 1| < 1/2, we have the property that if x ∼ y then µxy ≥ αµx with α = 14d , but we don’t
56
have x ∼ x for all x ∈ Zd. To cure this problem, consider the weighted graph (Zd, µ(2)), where µ(2) is
the iterated weight:
µ(2)xy =
∑
z∈Zd
µxzµzyµz
(1.9.18)
Then
µ(2)xx =
∑
z
(µxz)2
µz≥∑
z
αµxz = αµx > 0 (1.9.19)
Also, µ(2)x =
∑
y µ(2)xy =
∑
y
∑
zµxzµzyµz
=∑
zµxzµzµz
= µx, and since µx ∈ (d, 3d)
1
6<
1
2
∑
z∈Zd
µzy3d
<1
2
∑
z∈Zd
µzyµz
< µ(2)xy =
∑
z∈Zd
µxzµzyµz
<3
2
∑
z∈Zd
µzyµz
<3
2
∑
z∈Zd
µzyd
<9
2(1.9.20)
thus the property (3) holds. Write the heat kernel and Green function for (Zd, µ(2)) as p(2)(x, y) and
C(2)(x, y). We have
p(2)(x, y) = p2(x, y) =∑
z
p(x, z)p(z, y) (1.9.21)
and
C(2)(x, y) =1
µy
∞∑
n=0
p(2)n (x, y) (1.9.22)
It’s obvious that the doubling volume property holds. By the standard Poincare inequality on Zd
2d∑
x∈B(x0,r)
|f(x)− fB|2 ≤ C′2r
2∑
x,y∈B(x0,2r)
(f(x)− f(y))2 (1.9.23)
and µ(2)x ∈ (d, 3d), µ
(2)xy ∈ (1/6, 9/2), we know that Poincare inequality holds in our case, with C2 = 9C′
2.
For d > 2∞∑
n=0
n
V (2)(x, n)< +∞ (1.9.24)
so C(2)(x, y) <∞ and
C(2)(x, y) ≤ C
∞∑
n=d(x,y)
n
V (2)(x, n)≤ C1
∞∑
n=d(x,y)
n
nd≤ C2n
2−d (1.9.25)
and similarly C(2)(x, y) ≥ C−12 n2−d with possibly change of C2.
If x and y have the same parity, C(x, y) =∑∞
n=0 p2n(x, y) = C(2)(x, y), so it enjoys the same bound
57
as C(2). If x and y have different parity, then by harmonicity
C(x, y) =1
µy
∑
e
C(x, y + e)µy(y+e) (1.9.26)
also enjoys the same bound, with possibly change of constant.
The derivatives of the Green function are more subtle. In general it could have very poor behavior
as discussed in e.g. [DD05][CS11]. But in our boundary problem we can bound ∇f by O(R−1)f where
R has dimension of length when f is harmonic. This is done by showing that the exit distribution
for a random walk from deep interior of a cube of size O(R) is bounded by O(R−(d−1)), even if our
boundary crosses the cube in some nice ways.
In the simplest situation where only one insulating plate crosses the middle of a cube, we have
simple arguments.
Lemma 47. Suppose that R′ = 2R+ 1 and KR′ is a Λ-adapted cube of size R. If d(x, ∂KR′ ) > R′/3,
y ∈ ∂KR′ then PKR′ (x, y) ≤ O(1)R′d−1 .
Proof. We cut KR into 2 equal sub-cubes, each with size R × (2R + 1)d−1, and let S ⊂ KR ∪ ∂KRbeing the d − 1 dimensional cube that cuts through KR. Let K′ = KR\S be the union of the two
sub-cubes. Consider the Markov chain starting from x ∈ KR with d(x, ∂KR) > R/3. Let pxy be its
transition kernel. Write PK′(x, y) with x ∈ K′ and y ∈ ∂K′ to be the Poisson kernel of K′, and since K′
is disconnected, PK′(x, y) = 0 if y is not on the boundary of the connected component that contains
x. Extend PK′ to a function HK′(x, y) with x ∈ K′ ∪ S and y ∈ ∂K′ by HK′(x, y) = δxy if x ∈ S and
HK′(x, y) = PK′(x, y) if x ∈ K′. We have
PKR(x, y) =HK′(x, y) +∑
z1∈S,µ1∈EHK′(x, z1)pz1,z1+µ1HK′(z1 + µ1, y)
+∑
z1,z2∈S,µ1,µ2∈EHK′(x, z1)pz1,z1+µ1HK′(z1 + µ1, z2)pz2,z2+µ2HK′(z2 + µ2, y) + · · ·
=HK′(x, y) +HK′
( ∞∑
n=0
(pHK′)n
)
pHK′(x, y)
(1.9.27)
Now if for x ∈ S, pxy are not all equal to 12d , we have HK′(z + e1, y) = HK′(z − e1, y) where e1 is
the positive direction of the first coordinate of Zd and z ∈ S, because the two sub-cubes are identical
and the Markov chain within them are both the standard random walk. If furthurmore we have
58
px,x+e1 + px,x−e1 = 1d for all x ∈ S, and px,x+µ = 1
2d for all µ ∈ E±\e1,−e1, we have then
pHK′(z1, z2) =∑
µ∈e1,−e1pz1,z1+µHK′(z1 + µ, z2) +
∑
µ∈E±\e1,−e1pz1,z1+µHK′(z1 + µ, z2) (1.9.28)
is unchanged under the deformation from µxy ≡ 1 for all (x, y) ∈ E(KR) to µxy = 1+α2 if (x, y) ∈ E(S)
and µxy = 1 or µxy = α if (x, y) ∈ E(Ki ∪ S)\E(S) with i = 1, 2 respectively, where Ki(i = 1, 2) are
the two components of K′ and α is small. The only changed quantity is the p in the last pHK′ factor.
But the p after this deformation is bounded by its value before deformation times 2d. Therefore
PKR(x, y) ≤ CRd−1 still holds provided we have shown it holds for standard random walk, which is
Lemma 45.
Lemma 48. Let d > 2. For all e ∈ E, x, y ∈ Θ, where Θ is defined in subsection 1.8.1 and m ≥ 0, if
d(x, y) = c0Lj
∣∣∣∣∂eCΛ,m(x, y)
∣∣∣∣≤ cL−(d−1)j (1.9.29)
where c only depends on d and c0.
Proof. We apply the Fact 33 to find a Λ-adapted cube of size 13c0L
j and then apply Lemma 32 together
with Theorem 46, which completes the proof.
1.9.3 The initial expansion
Consider equation (1.2.17): following Mayer expansion,
Z ′N (ξ) =E
[
ezW (Λ)−σV (Λ)]
=E
[∏
x∈Λ
(
e−σV (x) +(
ezW (x) − 1)
e−σV (x)) ]
=E
[∑
X∈P0
IΛ\X0 K0(X)
]
= E
[
(I0 K0) (Λ)
]
(1.9.30)
where we have defined I0 ∈ NB0 and K0 ∈ NP0,c as
I0(x) = e−σV (x,φ+ξ) = exp
(
− σ
4
∑
e∈E(∂eφ(x) + ∂eξ(x))
2
)
K0(X) =∏
x∈X
(
ezW (x,(φ+ξ)/√ǫ) − 1)
e−σV (x,φ+ξ)
59
This proves the statement (1.2.18).
1.9.4 Estimates
Lemma 49. There exists a constant c > 0 so that if σ/κ < c and h2σ < c, B ∈ Pj, j < N − 1,
∥∥∥e−
σ2
∑
x∈B,e(∂ePB+φ(x)+∂eξ(x))2∥∥∥Tφ(Φj(B))
≤ 2eκ4
∑
B(∂PB+φ)2
(1.9.31)
∥∥∥e
− σ2
∑
x∈B,e(∂eP(B)+φ(x)+∂eξ(x))2∥∥∥Tφ(Φj(
˙B,B +))≤ 2e
κ4
∑
B(∂P(B)+φ)2
(1.9.32)
And∥∥∥e−
σ2
∑
x∈B,e(∂ePB+φ(x)+∂eξ(x))2 − 1
∥∥∥Tφ(Φj(B))
≤ 4c−1h2|σ|e κ4∑
B(∂PB+φ)2
(1.9.33)
∥∥∥e
− σ2
∑
x∈B,e(∂eP(B)+φ(x)+∂eξ(x))2 − 1
∥∥∥Tφ(Φj(
˙B,B +))≤ 4c−1h2e
κ4
∑
B(∂P(B)+φ)2
(1.9.34)
Proof. Let V = − 12
∑
x∈B,e(∂ePB+φ(x) + ∂eξ(x))2, ‖(f, λξ)×n‖Φj(B) ≤ 1, by |∂ξ|2 ≤ h2L−dN it’s
straightforward to check that if σ/κ is sufficiently small, for n = 0, 1, 2,
∣∣∣(σV )(n)(φ, ξ; (f, λξ)×n)
∣∣∣ ≤ κ
2n+4
∑
x∈B,e(∂ePB+φ(x))2 + 2σh2 (1.9.35)
and for n ≥ 3, V (n) = 0. Therefore for n = 0, . . . , 3
1
n!
∣∣∣
(eσV
)(n)(φ, ξ; (f, λξ)×n)
∣∣∣ ≤ e|σV |e|σV (1)|+|σV (2)|
≤e κ4∑
x∈B,e(∂ePB+φ(x))2+8σh2 ≤ 2e
κ4
∑
x∈B,e(∂ePB+φ(x))2
if h2σ is sufficiently small, where we bounded the polynomials in (σV )(n) by e|σV (1)|+|σV (2)|. So (1.9.31)
is proved. (1.9.32) is proved in the same way.
To prove (1.9.33),
‖eσV − 1‖Tφ(Φj(B)) = ‖ 1
2πi
ˆ
|z|=ch−2
σezV
z(z − σ)dz‖Tφ(Φj(B)) ≤ 4c−1h2|σ|e κ4
∑
B(∂PB+φ)2
(1.9.36)
and (1.9.34) is proved in the same way.
Another example is the esimate of the initial interaction. At step j = 0 a block B is a single lattice
60
point x. Define
W (x, φ, u) = 1
2
∑
e∈E,x+e∈Λ
cos(u∂eφ(x))
Recall definition (1.2.7) then W (x, φ) = W (x, φ,√
β(1 + σ)).
Lemma 50. If κ ≥ h−1
1) W (x, φ, u) satisfies
3∑
n=0
1
n!sup
|∂f(x)|≤h∂t1...tn
∣∣ti=0
∣∣∣∣∣∂mu W (x, φ+
n∑
i=1
tifi)
∣∣∣∣∣≤ d(2h)mehue
κ2
∑
e∈E(∂eφ(x))2
(1.9.37)
for m ∈ 0, 1, 2, . . . .
2) For |z| sufficiently small∥∥∥ezW (B)
∥∥∥s,0
≤ 2 (1.9.38)
Proof. 1) The case m = 0 holds even without eκ2
∑
e∈E(∂eφ(x))2
by straightforward computations and
thus is omitted. For m > 0,
∂mu W = ±1
2
∑
e∈E,x+e∈Λ
sincos(u∂eφ(x)) (∂eφ(x))
m (1.9.39)
and we bound
3∑
n=0
1
n!sup
|∂f(x)|≤h∂t1...tn
∣∣ti=0
(
∂e(φ(x) +
n∑
i=1
tifi(x))
)m
≤ (2h)meκ2
∑
e∈E(∂eφ(x))2
(1.9.40)
The bound for ∂mu W follows by product rule of differentiations and the case m = 0.
2) Let ‖ − ‖00 be the ‖ − ‖0 norm with G = 1. For |z| sufficiently small
∥∥∥ezW (B)
∥∥∥00
≤∞∑
n=0
|z|nn!
‖W (B)‖n00 ≤ exp(4d|z|eh
)≤ 2 (1.9.41)
Lemma 51. Given r > 0, if |z| and |σ| are sufficiently small, then ‖K0‖ < r. Furthuremore, K0 is
smooth in z and σ.
Proof. We have∥∥∥ezW (B) − 1
∥∥∥00
≤ exp(4d|z|eh
)− 1 ≤ c|z| (1.9.42)
61
for some constant c. By Lemma 49,
∥∥∥(ezW (B) − 1)e−V0(B)
∥∥∥0≤ 2c|z| (1.9.43)
therefore
‖K0‖0 = supX∈P0,c
‖K0(X)‖0A|X|0 ≤ supX∈P0,c
(2c|z|A)|X|0 < r (1.9.44)
The derivative of∏
B∈B0(X)(ezW (B) − 1) w.r.t σ is equal to
∑
B0⊆XzW ′(B)
1
2√1 + σ
∏
B⊆X\B0
(ezW (B) − 1) (1.9.45)
therefore its ‖ − ‖0 norm is bounded by c′A|z| for some constant c′. The derivative of e−V0(B) can be
bounded similarly.
62
Chapter 2
Renormalized powers of
Ornstein-Uhlenbeck processes and
well-posedness of stochastic
Ginzburg-Landau equations
2.1 Introduction
This chapter is a minor modified version of the paper [EJS13] by E, Jentzen and me.
The first part of this chapter (see Section 2.2 below) investigates well-definedness and regularity
of suitable renormalized powers of Ornstein-Uhlenbeck processes. More formally, let (Ω,F ,P) be a
probability space, let d ∈ N := 1, 2, . . ., n ∈ 2, 3, 4, . . . and let (Wt)t∈R be a two-sided cylindrical
I-Wiener process on the R-Hilbert space L2([0, 2π]d,R) of equivalence classes of Lebesgue square
integrable functions from [0, 2π]d to R. Moreover, let CP([0, 2π]d,R) be the space of periodic continuous
functions from [0, 2π]d to R, let A : D(A) ⊂ CP ([0, 2π]d,R) → CP([0, 2π]d,R) be the Laplacian with
periodic boundary conditions on CP([0, 2π]d,R) minus the identity operator (see (2.1.5) below for
details) and consider the stationary solution Vt =´ t
−∞ eA(t−s) dWs, t ∈ R, of the SPDE
dVt = AVt dt+ dWt (2.1.1)
63
for t ∈ R. Note that the process Vt, t ∈ R, does in the case d ≥ 2 P-almost surely not take values in
a function space anymore but in D((−A)(2−d)/4−ε) (see, for instance, Da Prato & Zabczyk [DZ92]).
Nonetheless, powers of V are well defined in a suitable sense in the case d = 2. Indeed, n-th renor-
malized power of V , that is, the stochastic process : (Vt)n :, t ∈ R, is well defined and its regular-
ity is analyzed in the case d = 2 in Lemma 3.2 in Da Prato & Debussche [DPD03] (see, e.g., also
[Sim79, GJ87, DT07] for further details on the definition of the n-th renormalized power). Proposi-
tion 65 extends the regularity statement of this result and also establish well definedness of : (Vt)2 :,
t ∈ R, in the case d = 3. Moreover, if d = 3, n ≥ 3 or if d ≥ 4, then : (Vt)n :, t ∈ R, can not be
defined anymore (see Section 7.1 in Da Prato & Tubaro [DT07] in the case d = n = 3 and Lemma 67
below in the general case). Although : (Vt)3 :, t ∈ R, does not make sense in the case d = 3, we
establish in Proposition 70 and Lemma 72 below that the processes´ t
t0: (Vs)
n : ds, t ∈ [t0,∞), t0 ∈ R,
(which we refer as averaged Wick powers) are well defined if and only if n+1n−1 > d
2 (i.e., if and only
if d ∈ 1, 2 or (d = 3 and n ∈ 2, 3, 4) or (d ∈ 4, 5 and n = 2)). The integral thus mollifies the
renormalized power in a suitable sense and allows us to define´ t
t0: (Vs)
3 : ds, t ∈ [t0,∞), t0 ∈ R, even
in the case d = 3. Another possibility to extend the definition of : (Vt)n :, t ∈ R, is to consider the
process´ t
−∞ eA(t−s) : (Vs)n : ds, t ∈ R, which we refer as convolutional Wick power. Proposition 75
and Lemma 76 prove that´ t
−∞ eA(t−s) : (Vs)n : ds, t ∈ R, is (as in the case of averaged Wick powers)
well defined if and only if n+1n−1 >
d2 . Proposition 75 also proves that convolutional Wick powers enjoy
more regularity properties than averaged Wick powers constructed in Proposition 70. Our analysis of
convolutional Wick powers is inspired by a Walsh-expansion for the KPZ equation in the fundamen-
tal recent article Hairer [Hai]. For details on the results on Wick power, averaged Wick powers and
convolutional Wick powers the reader is referred to the summary in Subsection 2.2.7 below.
The above outlined results on the well-definedness and regularity of renormalized powers of V are
used in the second part of this chapter (see Section 2.3 below) to analyzes strong solutions of stochastic
Ginzburg-Landau equations with polynomial nonlinearities. More formally, let η, κ0, κ1, . . . , κn ∈ R,
let x0 ∈ D((−A)η) and consider a solution process (Xt)t∈[0,∞) of the SPDE
dXt =[
AXt+ :(∑n
i=0κi (Xt)
i)
:]
dt+ dWt (2.1.2)
for t ∈ [0,∞) with the initial condition X0 = x0 and where the expression :(∑n
i=0 κi(Xt)i): is a
suitable renormalization of the term∑n
i=0 κi(Xt)i for t ∈ [0,∞) (see Subsections 2.3.2 and 2.3.3 below
for further details). The parameter η ∈ R thus measures the regularity of the initial value. SPDEs
64
of the form (2.1.2) have a strong connection to models from quantum field theory; see [PW81]. Local
and global existence, uniqueness and regularity of solutions of SPDEs of the form (2.1.2) (and suitable
mollified versions of (2.1.2) respectively) have been intensively studied in the last two decades; see,
e.g., the monograph [DZ92] and the references mentioned therein for the one-dimensional case d = 1
and see [JLM85, BCM88, AR91, DZ92, DZ96, GG96, LR98, MR99, DPD03] for the more subtle two-
dimensional case d = 2. In this thesis we are mainly interested in strong solutions of (2.1.2) and we
therefore review results for strong solutions of (2.1.2) in a bit more detail in the following.
In the case d = 1, global existence, uniqueness and regularity of strong solutions follows, e.g.,
from Section 7.2 in Da Prato & Zabczyk [DZ92] if n is odd and if κn < 0. In the case d = 1 the
expression∑ni=0 κi(Xt)
i appearing in (2.1.2) is well defined and it is not necessary to replace it by its
renormalization :(∑n
i=0 κi(Xt)i): for t ∈ [0,∞). Moreover, note that the solution process (Xt)t∈[0,∞)
of the SPDE (2.1.2) satisfies P[Xt ∈ D((−A)1/4−ε) ∪ ∞
]= 1 for all t, ε ∈ (0,∞) in the case d = 1.
The solution process thus takes P-almost surely values in D((−A)1/4−ε)∪∞ in the case d = 1 where
ε ∈ (0,∞) is arbitrarily small. Here and below the solution process takes the value ∞ after its possible
blow up (e.g., if κn > 0).
In the case d = 2 the renormalization is necessary and can not be avoided (see Walsh [Wal86] and,
e.g., Section 1 in Hairer et al. [HRW12]). In the case d = 2 local existence, uniqueness and regularity
of solutions of (2.1.2) have been established in Proposition 4.4 in Da Prato & Debussche [DPD03] if
the condition
η > infp∈(n,∞)
(
max
−2
p (2n+ 1),
−1
(n− 1)
(
1− n
p
))
= − supp∈(n,∞)
(
min
2
p (2n+ 1),
1
(n− 1)
(
1− n
p
))
(2.1.3)
is fulfilled beside other assumptions (see also Theorem 4.2 in [DPD03] for the corresponding global
existence result). The first main result of this chapter, Theorem 82 in Subsection 2.3.2, extends Da
Prato & Debussche’s result by establishing local existence of strong solutions in the case d = 2 for a
larger class of initial values, that is, if the condition
η > − 2
n(2.1.4)
is fulfilled instead of (2.1.3). Clearly, assumption (2.1.4) is less restrictive than assumption (2.1.3). In
addition, under assumption (2.1.4), Theorem 82 establishes more regularity of the solution process of
the SPDE (2.1.2). The reader is referred to (2.3.71) in Subsection 2.3.2 for a detailed comparison of
65
the regularity statement in Proposition 4.4 in Da Prato & Debussche [DPD03] and of the regularity
statement in Theorem 82 below. Under assumption (2.1.4), Theorem 82 also shows that the solution
process (Xt)t∈[0,∞) of the SPDE (2.1.2) satisfies P[Xt ∈ D((−A)−ε) ∪ ∞] = 1 for all t, ε ∈ (0,∞)
and all r ∈ (−∞, 0) in the case d = 2. The solution process thus takes P-almost surely values in
D((−A)−ε) ∪ ∞ in the case d = 2 where ε ∈ (0,∞) is arbitrarily small.
The next main result of this chapter is devoted to the case d = 3 and n = 2. More precisely,
Theorem 83 in Subsection 2.3.3, proves local existence, uniqueness and regularity of strong solutions of
(2.1.2) in the case d = 3 and n = 2 if the condition η > −1 is fulfilled. Under these assumptions, Theo-
rem 83 proves that the solution process of the SPDE (2.1.2) satisfies P[Xt ∈ D((−A)−1/4−ε) ∪ ∞
]=
1 for all t, ε ∈ (0,∞). The solution process thus takes P-almost surely values in D((−A)−1/4−ε)∪∞
in the case d = 3 and n = 2 and η > −1 where ε ∈ (0,∞) is arbitrarily small. To the best of our
knowledge, Theorem 83 is the first result in the literature that establish local existence of solutions
of the SPDE (2.1.2) in the three dimensional case d = 3. The proof of Theorem 83 is based on a
detailed analysis of mild solutions of determinisitic nonautonomous partial differential equations in
Subsection 2.3.1 and on the analysis of : (Vt)2 :, t ∈ R, in three dimensions d = 3 (see Section 2.2).
Acknowledgements
Jan van Neerven, Alesandra Lunardi and Philipp Doersiek are gratefully acknowledged for a number of
quite useful comments and references concerning analytic semigroups and their infinitesmal generators.
2.1.1 Notation
Throughout this chapter the following conventions are used. If Ω is a set and F ⊂ P(Ω) is a subsets
of the power set of Ω, then we denote by σΩ(F) the sigma-algebra on Ω which is generated by F . If
(E, E) is a topological space, then we denote by B(E) := σE(E) the Borel sigma-algebra of (E, E).
Furthermore, if d ∈ N := 1, 2, . . ., then we denote by CP([0, 2π]d,R) the R-Banach space of periodic
continuous functions from [0, 2π]d to R and by Ad : D(Ad) ⊂ CP([0, 2π]d,R) → CP ([0, 2π]d,R) the
generator of a strongly continuous analytic semigroup which satisfies
D(Ad) ⊃
v ∈ CP ([0, 2π]d,R) :
(
∃w ∈ C2(Rd,R) :
[
∀x ∈ Rd : ∀ j ∈ 1, . . . , d : w(x) = w(x+ 2πe
(d)j
)]
∧[
w|[0,2π]d = v])
(2.1.5)
66
and Adv = v− v for all v ∈ D(Ad). The fact that such an operator exists and is unique can, e.g., be
proved by considering the Laplacian on the whole Rd. In addition, if d ∈ N and r ∈ R, then we denote
by
(CrP([0, 2π]d,R), ‖·‖CrP ([0,2π]d,R)
):=(
D((−Ad)
r/2 ),∥∥ (−Ad)
r/2(·)∥∥C([0,2π]d,R)
)
(2.1.6)
the R-Banach space of the domain of the r2 -fractional power of Ad. Finally, we observe that there exist
real numbers c(d)α,β,γ ∈ [0,∞), α, β, γ ∈ R, d ∈ N, such that for every d ∈ N, every α, β, γ ∈ R with
α + β > 0 and γ < min(α, β), every v ∈ CαP([0, 2π]d,R) and every w ∈ CβP([0, 2π]d,R) it holds that
v · w ∈ CγP([0, 2π]d,R) and that
‖v · w‖CγP([0,2π]d,R) ≤ c(d)α,β,γ ‖v‖CαP ([0,2π]d,R) ‖w‖CβP([0,2π]d,R) . (2.1.7)
More details on interpolation spaces and analytic semigroups can, e.g, be found in the excellent books
Lunardi [Lun09], Van Neerven [Nee92] and Sell & You [SY02]. Finally, throughout this chapter, if
(V, ‖·‖V ) is an R-Banach space, then we equip the set V ∪ ∞ with the topology
A ⊂(
V ∪ ∞)
:
(
∀ a ∈ A\∞ :[
∃ ε ∈ (0,∞) : y ∈ V : ‖y − v‖V < ε ⊂ A])
and
(
∞ ∈ A⇒[
∃R ∈ (0,∞) : y ∈ V : ‖y‖V > R ⊂ A])
(2.1.8)
and we observe that the pairing consisting of V ∪∞ and (2.1.8) is a complete metrizable topological
space.
2.2 Renormalized powers of Ornstein-Uhlenbeck processes
2.2.1 Setting and assumptions
Throughout Section 2.2 we will frequently assume that the following setting is fulfilled. Let d ∈ N, let
δ : Zd × Zd → R be a function defined through
δv,w :=
1 : v = w
0 : v 6= w
(2.2.1)
67
for all v, w ∈ Rd and let gv : [0, 2π]d → C, v ∈ Zd, be a family of functions defined through
gv(x) := ei〈v,x〉Rd = ei(v1x1+...+vdxd) (2.2.2)
for all v = (v1, . . . , vd) ∈ Zd and all x = (x1, . . . , xd) ∈ [0, 2π]d. Next let(H := L2((0, 2π)d;C), 〈·, ·〉H ‖·‖H
)
be the C-Hilbert space of equivalence classes of Lebesgue square integrable functions from (0, 2π)d to C
with 〈v, w〉H =´
(0,2π)d v(x) ·w(x) dx for all v, w ∈ H . Observe that (2π)− d
2 gv, v ∈ Zd, is an orthonor-
mal basis of H and that y =∑
v∈Zd1
(2π)d〈gv, y〉H gv for all y ∈ H . Moreover, let N0 := 0, 1, 2, . . .,
let Pm := (i, j) ∈ 1, 2, . . . ,m2 : i < j, m ∈ N, be sets and let Θ: ∪∞m=1 (N0)
Pm → ∪∞m=1 (N0)
mbe
a function defined through
Θ(α) :=
∑
(i,j)∈Pmi=1 or j=1
α(i,j) , . . . ,∑
(i,j)∈Pmi=m or j=m
α(i,j)
∈ (N0)m
(2.2.3)
for all α ∈ (N0)Pm and all m ∈ N. Furthermore, we denote by
Φ :=ϕ : Zd → [0,∞) :
(∀ v ∈ Zd : ϕv = ϕ−v
)(2.2.4)
the set of all functions from Zd to [0,∞) that are symmetric with respect to the origin and equipp it
with the Fréchet metric
dΦ(ϕ, ψ) :=∑
k∈Zd
min(1, ϕk − ψk)
2(|k1|+...+|kd|) (2.2.5)
for all ϕ, ψ ∈ Φ. Next define Φ0 := ϕ ∈ Φ: ϕk = 0 for almost all k ∈ Zd ⊂ Φ and Φ0,≤1 :=
ϕ ∈ Φ0 : (∀ k ∈ Zd : ϕk ∈ [0, 1]) ⊂ Φ. In addition, let (Ω,F ,P) be a probability space and let
βv : R × Ω → C, v ∈ Zd, be a family of jointly Gaussian complex valued stochastic processes with
continuous sample paths and with
βvt = β−vt and E
[
βvt1βwt2
]
=
δv,wmin(|t1| , |t2|) : t1 · t2 ≥ 0
0 : t1 · t2 < 0
(2.2.6)
for all t, t1, t2 ∈ R and all v, w ∈ Zd. Observe that βv, v ∈ Zd, are two-sided complex valued standard
Brownian motions. Moreover, let V ϕ : R × Ω → CP([0, 2π]d,R), ϕ ∈ Φ0, be a family of stochastic
68
processes with continuous sample paths satisfying
V ϕt =∑
v∈Zd
√2ϕv
[ˆ t
−∞e−λv(t−s) dβvs
]
gv (2.2.7)
P-almost surely for all t ∈ R and all ϕ = (ϕv)v∈Zd ∈ Φ0. Observe that
1
(2π)2d
E
[⟨
gv1 , Vϕ(1)
t1
⟩
H
⟨
gv2 , Vϕ(2)
t2
⟩
H
]
=δv1,v2 ϕ
(1)v1 ϕ
(2)v2 e
−λv1 |t2−t1|
λv1(2.2.8)
for all t1, t2 ∈ R, v1, v2 ∈ Zd and all ϕ(1) = (ϕ(1)v )v∈Zd , ϕ
(2) = (ϕ(2)v )v∈Zd ∈ Φ0 and that
E
[
V ϕ(1)
t1 (x1)Vϕ(2)
t2 (x2)
]
=∑
v∈Zd
1
(2π)2d
E
[⟨
gv, Vϕ(1)
t1
⟩
H
⟨
gv, Vϕ(2)
t2
⟩
H
]
gv(x2 − x1)
=∑
v∈Zd
ϕ(1)v ϕ
(2)v e−λv|t2−t1| gv(x1 − x2)
λv
(2.2.9)
for all ϕ(1), ϕ(2) ∈ Φ0, t1, t2 ∈ R and all x1, x2 ∈ [0, 2π]d. Moreover, if n ∈ N, then we denote by
Wn ⊂ L2(Ω;R) the closure in L2(Ω;R) of the set
⋃
k∈N
⋃
p : Rk→R is apolyn. of degree n
⋃
v1,...,vk∈Z
d
⋃
t1,...,tk∈R
p(βv1t1 , . . . , β
vktk
)
. (2.2.10)
Note for every n ∈ N that the R-Hilbert space Wn is the direct sum of the first n Wiener chaoses;
see, e.g., Section 4 in Da Prato & Tubaro [DT07] and Section A.1 in Hairer [Hai]. Furthermore, let
Hn : R → R, n ∈ 0, 1, 2, . . ., be the unique functions satisfying
e−t2
2 +tx =∞∑
n=0
tn
n!·Hn(x) (2.2.11)
for all t, x ∈ R. The functions Hn, n ∈ 0, 1, 2, . . ., are typically referred as (probabilists’) Hermite
polynomials in the literature. Note that H0(x) = 1, H1(x) = x, H2(x) = x2 − 1, H3(x) = x3 − 3x,
H4(x) = x4 − 6x2 + 3, . . . for all x ∈ R. In addition, if Z : Ω → R is a centered real valued Gaussian
random variable and if n ∈ N0, then we denote by :Zn: : Ω → R the n-th Wick power of Z, that is,
69
the random variable given by
:Zn: =
(E[Z2])n
2 Hn
(
Z√E[Z2]
)
: E[Z2] > 0
Zn : E[Z2] = 0
(2.2.12)
(see, e.g., page 9 in Simon [Sim79]). Moreover, we denote by : (V ϕ)n : : R × Ω → CP([0, 2π]d,R),
ϕ ∈ Φ0, n ∈ N0, the stochastic processes with continuous sample paths given by
(: (V ϕt )
n:)(x) = :(V ϕt (x))
n: (2.2.13)
for all t ∈ R, x ∈ [0, 2π]d, ϕ ∈ Φ0 and all n ∈ N0. Note that : (V ϕt )0: = 1, : (V ϕt )
1: = V ϕt ,
: (V ϕt )2: = (V ϕt )2 − E
[(V ϕt )2
]= (V ϕt )2 −∑v∈Zd
(ϕv)2
λv, : (V ϕt )
3: = (V ϕt )3 − 3V ϕt E
[(V ϕt )2
]= (V ϕt )3 −
3V ϕt(∑
v∈Zd(ϕv)
2
λv
), . . . for all t ∈ R and all ϕ ∈ Φ0. In addition, we denote by
(V ϕt0,(·)
)n : [t0,∞)×
Ω → CP ([0, 2π]d,R), ϕ ∈ Φ0, n ∈ N0, t0 ∈ R, the stochastic processes with continuous sample paths
defined by
(V ϕt0,t)n :=
ˆ t
t0
: (V ϕs )n : ds (2.2.14)
for all ϕ ∈ Φ0, n ∈ N0 and all t0, t ∈ R with t0 ≤ t and we denote by •(V ϕ)n• : R×Ω → CP ([0, 2π]d,R),
ϕ ∈ Φ0, n ∈ N0, the stochastic processes with continuous sample paths defined by
• (V ϕt )n• :=
ˆ t
−∞eAd(t−s)
[: (V ϕs )n :
]ds (2.2.15)
for all ϕ ∈ Φ0, n ∈ N0 and all t ∈ R. The readers who are familiar with quantum field theory should
distinguish the concept of the "time-ordered product" in quantum field theory (see, for instance, Peskin
& Schroeder [PS95]) from the averaged and the convolutional Wick power defined above. Finally, note
that (V ϕt (x))n, :(V ϕt (x))
n: ,
(V ϕt0,t(x)
)n , • (V ϕt (x))n • ∈ Wn for all n ∈ N, x ∈ [0, 2π]d and all t0, t ∈ R
with t0 ≤ t.
2.2.2 Hypercontractivity estimates
The following lemma allows us to calculate regularities of suitable stochastic processes by computing
their correlations in Fourier space. It is quite similar to Proposition A.2 in Hairer [Hai].
Lemma 52. Assume the setting of Subsection 2.2.1, let n ∈ N and let a, b ∈ R with a < b. Then there
70
exist real numbers χn,d,p,a,bα,α,β,β
∈ [0,∞), p, α, α, β, β ∈ R, such that
‖X‖Lp(Ω;Cα([a,b],C2βP ([0,2π]d,R))) (2.2.16)
≤ χn,d,p,a,bα,α,β,β
supt1,t2∈[a,b],t1 6=t2
∑
v1,v2∈Z
d
[∣
∣
∣E
[
〈gv1 ,Xt1〉H〈gv2 ,Xt1〉H]∣
∣
∣
(λv1λv2)−β
+
∣
∣
∣E
[
〈gv1 ,Xt1−Xt2〉H〈gv2 ,Xt1−Xt2〉H]∣
∣
∣
(λv1λv2)−β |t1−t2|2α
]
12
for all p ∈ (0,∞), α ∈ (α, 1), α ∈ (0, 1), β ∈ (β,∞), β ∈ R and all stochastic processes X : [a, b]×Ω →
∩r∈RCrP([0, 2π]d,R) with continuous sample paths which satisfy for every t ∈ [a, b] and every x ∈ [0, 2π]d
that Xt(x) ∈ Wn.
Proof of Lemma 52. Hypercontractivity (see, e.g., Lemma A.1 in Hairer [Hai]) ensures that there exist
real numbers κk,p ∈ [0,∞), k ∈ N, p ∈ [2,∞), such that
E[|Y |p
]≤ κk,p
(
E[
|Y |2])p
2
(2.2.17)
for all p ∈ [2,∞), Y ∈ Wk and all k ∈ N. Note that
∥∥(−A)βX
∥∥Cα([a,b],Lp(Ω;Lp((0,2π)d;R)))
= supt∈[a,b]
∥∥(−A)βXt
∥∥Lp(Ω;Lp((0,2π)d;R))
+ supt1,t2∈[a,b]t1 6=t2
∥∥(−A)β(Xt1 −Xt2)
∥∥Lp(Ω;Lp((0,2π)d;R))
|t1 − t2|α
= supt∈[a,b]
ˆ
(0,2π)dE[∣∣((−A)βXt)(x)
∣∣p]
dx
1p
+ supt1,t2∈[a,b]t1 6=t2
´
(0,2π)d E[∣∣((−A)β(Xt1 −Xt2))(x)
∣∣p]
dx 1p
|t1 − t2|α
= supt∈[a,b]
ˆ
(0,2π)dE
∣∣∣∣
∑
v∈Zd
(λv)β 〈gv, Xt〉H gv(x)
∣∣∣∣
p
dx
1p
+ supt1,t2∈[a,b]t1 6=t2
´
(0,2π)d E[∣∣∑
v∈Zd(λv)
β 〈gv, Xt1 −Xt2〉H gv(x)∣∣p]
dx 1p
|t1 − t2|α
(2.2.18)
for all p ∈ (0,∞), α ∈ (0, 1), β ∈ R and all stochastic processes X : [a, b] × Ω → ∩r∈RCrP([0, 2π]d,R).
71
Estimate (2.2.17) hence implies that
∥∥(−A)βX
∥∥Cα([a,b],Lp(Ω;Lp((0,2π)d;R)))
≤ κn,p
(2π)d
[
supt∈[a,b]
ˆ
(0,2π)d
(
E
[∣∣∣
∑
v∈Zd(λv)
β 〈gv, Xt〉H gv(x)∣∣∣
2])p
2
dx
1p
+ supt1,t2∈[a,b]t1 6=t2
´
(0,2π)d
(
E
[
∣
∣
∣
∑
v∈Zd(λv)
β〈gv ,Xt1−Xt2 〉Hgv(x)∣
∣
∣
2])
p2dx
1p
|t1−t2|α
]
=κn,p
(2π)d
[
supt∈[a,b]
ˆ
(0,2π)d
∑
v1,v2∈Zd
E
[
〈gv1 ,Xt〉H〈gv2 ,Xt〉H]
g(v2−v1)(x)
(λv1λv2)−β
p2
dx
1p
+ supt1,t2∈[a,b]t1 6=t2
ˆ
(0,2π)d
∑
v1,v2∈Zd
E
[
〈gv1 ,Xt1−Xt2〉H〈gv2 ,Xt1−Xt2〉H]
g(v2−v1)(x)
(λv1λv2)−β |t1−t2|2α
p2
dx
1p ]
(2.2.19)
for all p ∈ (0,∞), α ∈ (0, 1), β ∈ R and all stochastic processes X : [a, b] × Ω → ∩r∈RCrP([0, 2π]d,R)
which satisfy for every t ∈ [a, b] and every x ∈ [0, 2π]d that Xt(x) ∈ Wn. This implies
∥∥(−A)βX
∥∥Cα([a,b],Lp(Ω;Lp((0,2π)d;R)))
≤ κn,p
(2π)d
[
supt∈[a,b]
ˆ
(0,2π)d
(∑
v1,v2∈Zd
∣
∣
∣E
[
〈gv1 ,Xt〉H〈gv2 ,Xt〉H]∣
∣
∣
(λv1λv2)−β
)p2
dx
1p
+ supt1,t2∈[a,b]t1 6=t2
ˆ
(0,2π)d
(∑
v1,v2∈Zd
∣
∣
∣E
[
〈gv1 ,Xt1−Xt2〉H〈gv2 ,Xt1−Xt2〉H]∣
∣
∣
(λv1λv2)−β |t1−t2|2α
)p2
dx
1p ]
=κn,p
(2π)d
[
supt∈[a,b]
∑
v1,v2∈Zd
∣
∣
∣E
[
〈gv1 ,Xt〉H〈gv2 ,Xt〉H]∣
∣
∣
(λv1λv2)−β
12
+ supt1,t2∈[a,b]t1 6=t2
∑
v1,v2∈Zd
∣
∣
∣E
[
〈gv1 ,Xt1−Xt2 〉H〈gv2 ,Xt1−Xt2 〉H]∣
∣
∣
(λv1λv2)−β |t1−t2|2α
12]
(2.2.20)
and hence
∥∥(−A)βX
∥∥Cα([a,b],Lp(Ω;Lp((0,2π)d;R)))
≤ κn,p
supt1,t2∈[a,b]t1 6=t2
∑
v1,v2∈Z
d
∣
∣
∣E
[
〈gv1 ,Xt1〉H〈gv2 ,Xt1〉H]∣
∣
∣
(λv1λv2)−β
+
∣
∣
∣E
[
〈gv1 ,Xt1−Xt2〉H〈gv2 ,Xt1−Xt2〉H]∣
∣
∣
(λv1λv2)−β |t1−t2|2α
12
(2.2.21)
72
for all p ∈ (0,∞), α ∈ (0, 1), β ∈ R and all stochastic processes X : [a, b] × Ω → ∩r∈RCrP([0, 2π]d,R)
which satisfy for every t ∈ [a, b] and every x ∈ [0, 2π]d that Xt(x) ∈ Wn. Moreover, the Sobolev
embedding theorem ensures that there exist real numbers ρp,α,αβ,β
∈ [0,∞), p, α, α, β, β ∈ R, and ρp,α,α ∈
[0,∞), p, α, α ∈ R, such that
‖X‖Lp(Ω;Cα([a,b],C2βP ([0,2π]d,R))) =
∥∥(−A)βX
∥∥Lp(Ω;Cα([a,b],CP([0,2π]d,R)))
≤ ρp,α,αβ,β
∥∥(−A)βX
∥∥Lp(Ω;W α,p([a,b],Lp((0,2π)d;R)))
= ρp,α,αβ,β
∥∥(−A)βX
∥∥W α,p([a,b],Lp(Ω;Lp((0,2π)d;R)))
≤ ρp,α,αβ,β
ρp,α,α∥∥(−A)βX
∥∥Cα([a,b],Lp(Ω;Lp((0,2π)d;R)))
(2.2.22)
for all stochastic processes X : [a, b] × Ω → ∩r∈RCrP([0, 2π]d,R) with continuous sample paths and all
p ∈ (0,∞), α, α, α ∈ (0, 1), β, β ∈ R with α > α, α − α > 1p and β − β > d
p . Combining (2.2.21) and
(2.2.22) implies (2.2.16) and this completes the proof of Lemma 52.
2.2.3 Estimates for discrete convolutions
We first state three well known lemmas that we will use below.
Lemma 53 (Finiteness of infinite sums). Let d ∈ N, α ∈ R and let λx ∈ [1,∞), x ∈ Rd, be real
numbers with λx = 1 + (x1)2+ . . . + (xd)
2for all x = (x1, . . . , xd) ∈ Rd. Then
∑
k∈Zd
1(λk)
α < ∞ if
and only if α > d2 .
Lemma 54 (Growth rate of finite sums). Let d ∈ N, α ∈ [0, d2 ), β ∈ R, c ∈ (0,∞) and let λx ∈ [1,∞),
x ∈ Rd, be real numbers with λx = 1 + (x1)2+ . . . + (xd)
2for all x = (x1, . . . , xd) ∈ Rd. Then
supv∈Zd
[∑
k∈Zd,‖k‖
Rd≤c‖v‖
Rd
(λv)β
(λk)α
]
<∞ if and only if β ≤ α− d2 .
Lemma 55 (Growth rate of infinite sums). Let d ∈ N, α ∈ (d2 ,∞), β ∈ R, c ∈ (0,∞) and let
λx ∈ [1,∞), x ∈ Rd, be real numbers with λx = 1 + (x1)2 + . . . + (xd)
2 for all x = (x1, . . . , xd) ∈ Rd.
Then supv∈Zd
[∑
k∈Zd,‖k‖
Rd>c‖v‖
Rd
(λv)β
(λk)α
]
<∞ if and only if β ≤ α− d2 .
Lemmas 53–55 can all be proved by estimating the sums through suitable Lebesgue integrals and
then by using polar coordinates. The proofs of Lemmas 53–55 are straightforward and well known and
therefore omitted.
Lemma 56 (Two-sided bounds for discrete convolutions). Let d ∈ N and let λx ∈ [1,∞), x ∈ Rd, be
73
real numbers with λx = 1 + (x1)2+ . . .+ (xd)
2for all x = (x1, . . . , xd) ∈ Rd. Then
4−β
(λv)β
∑
k∈Zd,
‖k‖Rd
≤12‖v‖Rd
1
(λk)α
≤∑
k∈Zd,
‖k‖Rd
≤12‖v‖Rd
1
(λk)α(λv−k)
β≤ 4β
(λv)β
∑
k∈Zd,
‖k‖Rd
≤12‖v‖Rd
1
(λk)α
, (2.2.23)
4−α
(λv)α
∑
k∈Zd,
‖k‖Rd
≤13‖v‖Rd
1
(λk)β
≤∑
k∈Zd, 12‖v‖Rd
<
‖k‖Rd
≤2‖v‖Rd
1
(λk)α(λv−k)
β≤ 4α
(λv)α
∑
k∈Zd,
‖k‖Rd
≤3‖v‖
Rd
1
(λk)β
, (2.2.24)
4−β
∑
k∈Zd,
‖k‖Rd>
2‖v‖Rd
1
(λk)(α+β)
≤∑
k∈Zd,
‖k‖Rd>
2‖v‖Rd
1
(λk)α (λv−k)
β≤ 4β
∑
k∈Zd,
‖k‖Rd>
2‖v‖Rd
1
(λk)(α+β)
(2.2.25)
for all v ∈ Zd and all α, β ∈ [0,∞).
Proof of Lemma 56. First of all, observe that
∑
k∈Zd,
‖k‖Rd
≤12 ‖v‖Rd
3−β
(λk)α (λv)
β≤
∑
k∈Zd,
‖k‖Rd
≤12‖v‖Rd
1
(λk)α ( 9
4λv)β
≤∑
k∈Zd,
‖k‖Rd
≤12‖v‖Rd
1
(λk)α(λv−k)
β≤
∑
k∈Zd,
‖k‖Rd
≤12‖v‖Rd
1
(λk)α (λv
4
)β≤
∑
k∈Zd,
‖k‖Rd
≤12‖v‖Rd
4β
(λk)α(λv)
β
(2.2.26)
74
for all v ∈ Rd. This proves (2.2.23). Furthermore, note that
∑
k∈Zd,
‖k‖Rd<
12‖v‖Rd
4−α
(λk)β (λv)
α≤
∑
k∈Zd, 12‖v‖Rd
<
‖v−k‖Rd
≤ 32‖v‖Rd
4−α
(λk)β (λv)
α=
∑
k∈Zd, 12‖v‖Rd
<
‖k‖Rd
≤2‖v‖Rd
1
(4λv)α (λv−k)
β
≤∑
k∈Zd, 12‖v‖Rd
<
‖k‖Rd
≤2‖v‖Rd
1
(λk)α (λv−k)
β≤
∑
k∈Zd, 12‖v‖Rd
<
‖k‖Rd
≤2‖v‖Rd
1(λv4
)α(λv−k)
β
=∑
k∈Zd, 12‖v‖Rd
<
‖v−k‖Rd
≤2‖v‖Rd
4α
(λk)β(λv)
α≤
∑
k∈Zd,
‖k‖Rd
≤3‖v‖
Rd
4α
(λk)β(λv)
α
(2.2.27)
for all v ∈ Zd. This establishes (2.2.24). Finally, observe that
∑
k∈Zd,
‖k‖Rd>
2‖v‖Rd
3−β
(λk)(α+β)
≤∑
k∈Zd,
‖k‖Rd>
2‖v‖Rd
1
(λk)α[
1 +[‖k‖
Rd+ ‖v‖
Rd
]2]β
≤∑
k∈Zd,
‖k‖Rd>
2‖v‖Rd
1
(λk)α(λv−k)
β≤
∑
k∈Zd,
‖k‖Rd>
2‖v‖Rd
1
(λk)α(
1+[‖k‖Rd
−‖v‖Rd ]
2)β
≤∑
k∈Zd,
‖k‖Rd>
2‖v‖Rd
1
(λk)α (λk
4
)β=
∑
k∈Zd,
‖k‖Rd>
2‖v‖Rd
4β
(λk)(α+β)
(2.2.28)
for all v ∈ Rd. This shows (2.2.25). The proof of Lemma 56 is thus completed.
The next elementary lemma, Lemma 57, is a direct consequence of Lemma 53 and of (2.2.25) in
Lemma 56. The proof of Lemma 57 is clear and therefore omitted.
Lemma 57 (Finiteness of discrete convolutions). Let d ∈ N, α, β ∈ [0,∞), v ∈ Zd and let λx ∈ [1,∞),
x ∈ Rd, be real numbers with λx = 1 + (x1)2 + . . . + (xd)
2 for all x = (x1, . . . , xd) ∈ Rd. Then
∑
k∈Zd1
(λk)α(λv−k)
β <∞ if and only if α+ β > d2 .
The next lemma, Lemma 58, follows from Lemmas 54, 55 and 56.
Lemma 58 (Regularity of discrete convolutions). Let d ∈ N, α, β, γ ∈ [0,∞) be real numbers with
α+ β > d2 6= max(α, β) and let λx ∈ [1,∞), x ∈ Rd, be real numbers with λx = 1+ (x1)
2 + . . .+ (xd)2
75
for all x = (x1, . . . , xd) ∈ Rd. Then
supv∈Zd
∑
k∈Zd
(λv)γ
(λk)α (λv−k)
β
<∞ (2.2.29)
if and only if γ ≤ min(α, β, α + β − d2 ).
Proof of Lemma 58. Note that∑
k∈Zd(λv)
γ
(λk)α(λv−k)
β =∑
k∈Zd(λv)
γ
(λk)β(λv−k)
α for all v ∈ Zd. W.l.o.g. we
assume that α ≤ β. This ensures that β 6= d2 . Moreover, Lemma 54 and (2.2.23) in Lemma 56 prove
that
supv∈Zd
∑
k∈Zd,‖k‖
Rd≤ 1
2‖v‖Rd
(λv)γ
(λk)α (λv−k)
β
<∞
⇔([(
γ ≤ α+ β − d2
)
∧(
α < d2
)]
∨[(
γ < β)
∧(
α = d2
)]
∨[(
γ ≤ β)
∧(
α > d2
)])
⇐(
γ ≤ min(α, β, α + β − d2 ))
.
(2.2.30)
In addition, Lemma 54 and (2.2.24) in Lemma 56 show that
supv∈Zd
∑
k∈Zd, 12‖v‖Rd
<‖k‖Rd
≤2‖v‖Rd
(λv)γ
(λk)α (λv−k)
β
<∞
⇔([(
γ ≤ α+ β − d2
)
∧(
β < d2
)]
∨[(
γ ≤ α)
∧(
β > d2
)])
⇔([(
γ ≤ min(α, α+ β − d2 ))
∧(
β < d2
)]
∨[(
γ ≤ min(α, α+ β − d2 ))
∧(
β > d2
)])
⇔(
γ ≤ min(α, α + β − d2 ))
⇔(
γ ≤ min(α, β, α + β − d2 ))
.
(2.2.31)
Finally, Lemma 55 and (2.2.25) in Lemma 56 prove that
supv∈Zd
∑
k∈Zd,‖k‖
Rd>2‖v‖
Rd
(λv)γ
(λk)α(λv−k)
β
<∞
⇔(
γ ≤ α+ β − d2
)
. (2.2.32)
Combining (2.2.30)–(2.2.32) completes the proof of Lemma 58.
Corollary 59 (Regularity of discrete convolutions). Let d ∈ N, α, β ∈ [0,∞) and let λx ∈ [1,∞),
x ∈ Rd, be real numbers with λx = 1 + (x1)2+ . . . + (xd)
2for all x = (x1, . . . , xd) ∈ Rd. Then
supv∈Zd
[∑
k∈Zd(λv)
γ
(λk)α(λv−k)
β
]
<∞ for all γ ∈[0,min(α, β, α + β − d
2 )).
76
2.2.4 Wick powers of Ornstein-Uhlenbeck processes
The next elementary lemma is, e.g., similar to Lemma 2.4 in Da Prato & Tubaro [DT07] and Corollary
8.3.2 in Glimm & Jaffe [GJ87].
Lemma 60 (Expectations of products of Wick powers of Gaussian random variables). Assume the
setting of Subsection 2.2.1, let m ∈ N and let Z = (Z1, . . . , Zm) : Ω → Rm be a centered jointly
normally distributed random variable. Then
E[
(: (Z1)n1 :) · (: (Z2)
n2 :) · . . . · (: (Zm)nm :)]
=∑
α∈(N0)Pm
Θ(α)=n
n!
α!
∏
(i,j)∈Pm
(E[ZiZj
])α(i,j)
(2.2.33)
for all n = (n1, . . . , nm) ∈ (N0)m.
Proof of Lemma 60. W.l.o.g. we assume that E[(Zi)
2]> 0 for all i ∈ 1, 2, . . . ,m. Next throughout
this proof let Zi : Ω → R, i ∈ 1, 2, . . . ,m, be random variables defined through
Zi :=Zi
(E[(Zi)2
])1/2(2.2.34)
for all i ∈ 1, 2, . . . ,m. The definition of the Hermite polynomials Hn, n ∈ 0, 1, 2, . . ., then proves
that
∑
n=(n1,n2,...,nm)∈N0
(s1)n1 · (s2)n2 · . . . · (sm)nm · E
[∏mi=1Hni
(Zi)]
n1!n2! . . . nm!
= E
[m∏
i=1
( ∞∑
ni=0
(si)ni
ni!Hni
(Zk)
)]
= E
[m∏
i=1
exp
(
− (si)2
2+ siZi
)]
= exp
(
−∑mi=1 (si)
2
2
)
E
[
exp
(m∑
i=1
siZi
)]
= exp
−∑m
i=1 (si)2
2+
1
2E
(m∑
i=1
siZi
)2
= exp
(−∑m
i=1(si)2+
∑mi,j=1 sisjE[ZiZj]2
)
=∏
i,j∈1,2,...,mi<j
exp(
sisjE[
ZiZj
])
(2.2.35)
77
and the identity es1s2c =∑∞
n=0(s1s2)
ncn
n! for all s1, s2, c ∈ R therefore shows that
∑
n=(n1,n2,...,nm)∈N0
(s1)n1 · (s2)n2 · . . . · (sm)nm · E
[∏mi=1Hni
(Zi)]
n!
=∏
(i,j)∈Pm
∞∑
α(i,j)=0
(sisj)α(i,j)
E[
ZiZj
]α(i,j)
α(i,j)!
=∑
α∈(N0)Pm
1
α!
∏
(i,j)∈Pm
(sisj)α(i,j)
E[
ZiZj
]α(i,j)
.
(2.2.36)
This implies
1
n!E
[m∏
i=1
Hni
(Zi)
]
=∑
α∈(N0)Pm
Θ(α)=n
1
α!
∏
(i,j)∈Pm
E[
ZiZj
]α(i,j)
(2.2.37)
and hence
E
[m∏
i=1
(: (Zi)
n :)
]
=E[(Z1)
2]n1
2 · . . . ·E[(Zm)2
]nm2
·∑
α∈(N0)Pm
Θ(α)=n
n!
α!
∏
(i,j)∈Pm
E[ZiZj ]√
E[(Zi)2]E[(Zj)2]
α(i,j)
(2.2.38)
for all n = (n1, . . . , nm) ∈ (N0)m. The definition of the function Θ: ∪∞
m=1 (N0)Pm → ∪∞
m=1(N0)m
therefore completes the proof of Lemma 60.
Remark 61 (Wick’s theorem). Assume the setting of Subsection 2.2.1, let m ∈ N and let Z =
(Z1, . . . , Zm) : Ω → Rm be a centered jointly normally distributed random variable. Then Lemma 60
implies that
E[Z1 · Z2 · . . . · Zm
]=
∑
α∈(N0)Pm
Θ(α)=(1,1,...,1)
n!
∏
(i,j)∈Pm
(E[ZiZj
])α(i,j)
. (2.2.39)
Equation (2.2.39) is often referred as Wick’s theorem in the literature (see, e.g., Proposition 5.2 in
Hairer [Hai]).
The next lemma is a direct consequence of Lemma 60.
Corollary 62 (Products of Wick powers of V ϕ, ϕ ∈ Φ0, in real space). Assume the setting of Subsec-
78
tion 2.2.1, let m ∈ N and let n = (n1, n2, . . . , nm) ∈ (N0)m\0. Then
E
[(
:(V ϕ
(1)
t1
)n1:)
(x1) ·(
:(V ϕ
(2)
t2
)n2:)
(x2) · . . . ·(
:(V ϕ
(m)
tm
)nm:)
(xm)
]
=∑
α∈(N0)Pm
Θ(α)=n
n!
α!
∏
(i,j)∈Pm
∑
k∈Zd
ϕ(i)k ϕ
(j)k gk(xi − xj) e
−λk|ti−tj |
λk
α(i,j)
=∑
α∈(N0)Pm
Θ(α)=n
n!α!
∑
k∈(Zd)
(A,l)∈Pm×N :l≤αA
∏
(i,j,r)∈
(A,l)∈Pm×N :l≤αA
ϕ(i)k(i,j,r)
ϕ(j)k(i,j,r)
e−λk(i,j,r)
|ti−tj |gk(i,j,r) (xi−xj)
λk(i,j,r)
(2.2.40)
for all t1, t2, . . . , tm ∈ R, x1, x2, . . . , xm ∈ [0, 2π]d and all ϕ(1) = (ϕ(1)k )k∈Zd , ϕ
(2)(ϕ(2)k )k∈Zd , . . . ,
ϕ(m) = (ϕ(m)k )k∈Zd ∈ Φ0.
Proof of Corollary 62. Combining Lemma 60 and equation (2.2.9) implies that
E
[(
:(V ϕ
(1)
t1
)n1:)
(x1) ·(
:(V ϕ
(2)
t2
)n2:)
(x2) · . . . ·(
:(V ϕ
(m)
tm
)nm:)
(xm)
]
= E[(
:(
V ϕ(1)
t1 (x1))n1
:)
·(
:(
V ϕ(2)
t2 (x2))n2
:)
· . . . ·(
:(
V ϕ(nm)
tnm(xnm)
)nm:)]
=∑
α∈(N0)Pm
Θ(α)=n
n!
α!
∏
(i,j)∈Pm
(
E[
V ϕ(i)
tni(xni )V
ϕ(j)
tnj(xnj )
])α(i,j)
=∑
α∈(N0)Pm
Θ(α)=n
n!
α!
∏
(i,j)∈Pm
∑
k∈Zd
ϕ(i)k ϕ
(j)k e−λk|ti−tj | gk(xi − xj)
λk
α(i,j)
(2.2.41)
and therefore
E
[(
:(V ϕ
(1)
t1
)n1:)
(x1) ·(
:(V ϕ
(2)
t2
)n2:)
(x2) · . . . ·(
:(V ϕ
(m)
tm
)nm:)
(xm)
]
=∑
α∈(N0)Pm
Θ(α)=n
n!
α!
∏
(i,j)∈Pm
∑
k1,k2,...,
kα(i,j)∈Z
d
α(i,j)∏
l=1
ϕ(i)klϕ(j)kle−λkl |ti−tj | gkl(xi − xj)
λkl
=∑
α∈(N0)Pm
Θ(α)=n
n!α!
∑
k∈(Zd)
(A,l)∈Pm×N :l≤αA
∏
(i,j,r)∈
(A,l)∈Pm×N :l≤αA
ϕ(i)k(i,j,r)
ϕ(j)k(i,j,r)
e−λk(i,j,r)
|ti−tj |gk(i,j,r) (xi−xj)
λk(i,j,r)
(2.2.42)
for all t1, t2, . . . , tm ∈ R, x1, x2, . . . , xm ∈ [0, 2π]d and all ϕ(1), ϕ(2), . . . , ϕ(m) ∈ Φ0. The proof of
79
Corollary 62 is thus completed.
In the special case m = 2, Corollary 62 reduces to the following result.
Corollary 63 (Correlation of Wick powers of V ϕ, ϕ ∈ Φ0, in real space). Assume the setting of
Subsection 2.2.1. Then
E
[(
:(V ϕ
(1)
t1
)n1:)
(x1) ·(
:(V ϕ
(2)
t2
)n2:)
(x2)
]
= n1! δn1,n2
∑
k∈Zd
ϕ(1)k ϕ
(2)k gk(x1 − x2) e
−λk|t1−t2|
λk
n1
=
n1! δn1,n2
[∑
k1,...,kn1∈Zd
∏n1
r=1
ϕ(1)kr
ϕ(2)kr
e−λkr |t1−t2| gkr (x1−x2)
λkr
]
: n1 · n2 6= 0
δn1,n2 : n1 · n2 = 0
(2.2.43)
for all t1, t2 ∈ R, x1, x2 ∈ [0, 2π]d, ϕ(1), ϕ(2) ∈ Φ0 and all n1, n2 ∈ N0.
Corollary 63 investigates correlations of Wick powers of V ϕ, ϕ ∈ Φ0, in real space. The next
lemma studies correlations of Wick powers of V ϕ, ϕ ∈ Φ0, in Fourier space. Its proof makes use of
Corollary 63.
Lemma 64 (Correlation of Wick powers of V ϕ, ϕ ∈ Φ0, in Fourier space). Assume the setting of
Subsection 2.2.1. Then
1
(2π)2d
E
[⟨
gk1 , :(V ϕ
(1)
t1
)n1:⟩
H
⟨
gk2 , :(V ϕ
(2)
t2
)n2:⟩
H
]
=
n1! δn1,n2 δk1,k2
[
∑
l1,...,ln1∈Zd
l1+...+ln1=k1
∏n1
i=1
ϕ(1)li
ϕ(2)li
e−λli
|t1−t2|
λli
]
: n1 · n2 6= 0
δn1,n2 δk1,k2 δk1,0 : n1 · n2 = 0
(2.2.44)
for all t1, t2 ∈ R, k1, k2 ∈ Zd, ϕ(1), ϕ(2) ∈ Φ0 and all n1, n2 ∈ N0.
Proof of Lemma 64. First of all, observe that
E
[⟨
gk1 , :(V ϕ
(1)
t1
)n1:⟩
H
⟨
gk2 , :(V ϕ
(2)
t2
)n2:⟩
H
]
=
ˆ
(0,2π)d
ˆ
(0,2π)dE
[(
:(V ϕ
(1)
t1
)n1:)
(x1) ·(
:(V ϕ
(2)
t2
)n2:)
(x2)
]
g−k1(x1) gk2(x2) dx1 dx2
(2.2.45)
for all t1, t2 ∈ R, k1, k2 ∈ Zd, ϕ(1), ϕ(2) ∈ Φ0 and all n1, n2 ∈ N0. Equation (2.2.45) and Corollary 63
80
prove (2.2.44) in the case (n1, n2) ∈ (N0)2\(k, k) ∈ N2 : k ∈ N. Furthermore, equation (2.2.45),
Corollary 63 and the integral transformation theorem imply that
E
[⟨
gk1 , :(V ϕ
(1)
t1
)n:⟩
H
⟨
gk2 , :(V ϕ
(2)
t2
)n:⟩
H
]
= n!
ˆ
(0,2π)d
ˆ
(0,2π)d
∑
v1,...,vn∈Z
d
∏nr=1[ϕ
(1)vr
ϕ(2)vr
e−λvr |t1−t2| gvr (x1−x2)]λv1 ·...·λvn
g−k1(x1) gk2(x2) dx1 dx2
= n!
ˆ
(0,2π)d
ˆ
(0,2π)d−x2
∑
v1,...,
vn∈Zd
∏nr=1[ϕ
(1)vr
ϕ(2)vr
e−λvr |t1−t2| gvr (y)]λv1 ·...·λvn
g−k1(y + x2) gk2(x2) dy dx2
= n!
ˆ
(0,2π)d
∑
v1,...,
vn∈Zd
ˆ
(0,2π)d−x2
∏nr=1[ϕ
(1)vr
ϕ(2)vr
e−λvr |t1−t2| gvr (y)]λv1 ·...·λvn
g−k1(y) dy
g(k2−k1)(x2) dx2
= δk1,k2n! (2π)d
∑
v1,...,vn∈Zd
ˆ
(0,2π)d
∏nr=1[ϕ
(1)vr
ϕ(2)vr
e−λvr |t1−t2| gvr (y)]λv1 ·...·λvn
g−k1(y) dy
(2.2.46)
for all t1, t2 ∈ R, k1, k2 ∈ Zd, ϕ(1), ϕ(2) ∈ Φ0 and all n ∈ N. This shows that
E
[⟨
gk1 , :(V ϕ
(1)
t1
)n:⟩
H
⟨
gk2 , :(V ϕ
(2)
t2
)n:⟩
H
]
= δk1,k2n! (2π)d
∑
v1,...,vn∈Zd
ˆ
(0,2π)d
∏nr=1[ϕ
(1)vr
ϕ(2)vr
e−λvr |t1−t2| gvr (y)]λv1 ·...·λvn
g−k1(y) dy
= δk1,k2n! (2π)d∑
v1,...,vn∈Zd
ˆ
(0,2π)d
[∏nr=1 ϕ
(1)vr
ϕ(2)vr ] g(−k1+
∑nr=1
vr)(y) e−[
∑nr=1 λvr ]|t1−t2|
λv1 ·...·λvndy
= δk1,k2n! (2π)2d
∑
v1,...,vn∈Zd
v1+...+vn=k1
n∏
i=1
[
ϕ(1)vi ϕ
(2)vi e
−λvi |t1−t2|
λvi
]
(2.2.47)
for all t1, t2 ∈ R, k1, k2 ∈ Zd, ϕ(1), ϕ(2) ∈ Φ0 and all n ∈ N. The proof of Lemma 64 is thus
completed.
The next result, Proposition 65, proves convergence of Wick powers in the case (n, d) ∈ (2, 3, . . .×
2) ∪ (2, 3). The proof of Proposition 65 makes use of Lemma 64.
Proposition 65 (Convergence of Wick powers). Assume the setting of Subsection 2.2.1 and let (n, d) ∈
(2, 3, . . . × 2) ∪ (2, 3). Then there exists an up to indistinguishability unique stochastic process
81
:(V )n : : R × Ω → ∩β∈(−∞,2−d)CβP([0, 2π]d,R) with continuous sample paths which satisfies for every
T, p ∈ (0,∞), α ∈ (0, 4−d4 ), β ∈ R with 2α+ β < 2− d that
‖: (V ϕ)n : − : (V )n :‖Lp(Ω;Cα([−T,T ],CβP([0,2π]d,R))) → 0 as Φ0,≤1 ∋ ϕ→ 1. (2.2.48)
Proof of Proposition 65. We apply Lemma 64 four times to obtain that
E
[⟨
gk1 , :(V ϕt)n: − :
(V ψt)n:⟩
H
⟨
gk2 , :(V ϕt)n: − :
(V ψt)n:⟩
H
]
= n! (2π)2dδk1,k2
∑
l1,...,ln∈Zd
l1+...+ln=k1
n∏
i=1
[ϕli ]2
λli− 2
n∏
i=1
ϕliψliλli
+
n∏
i=1
[ψli ]2
λli
= n! (2π)2d δk1,k2
∑
l1,...,ln∈Zd
l1+...+ln=k1
(∏ni=1 ϕli −
∏ni=1 ψli)
2
(∏ni=1 λli)
(2.2.49)
for all t ∈ R, ϕ, ψ ∈ Φ0 and all k1, k2 ∈ Zd. Next observe that
∣∣∣∣∣
n∏
i=1
ϕli −n∏
i=1
ψli
∣∣∣∣∣=
∣∣∣∣∣∣
n∑
i=1
i∏
j=1
ϕli
n∏
j=i+1
ψli
−
i−1∏
j=1
ϕli
n∏
j=i
ψli
∣∣∣∣∣∣
=
n∑
i=1
i−1∏
j=1
ϕli
︸ ︷︷ ︸
≤1
n∏
j=i+1
ψli
︸ ︷︷ ︸
≤1
|ϕli − ψli | ≤n∑
i=1
|ϕli − ψli |︸ ︷︷ ︸
≤n
(2.2.50)
for all t ∈ R, l1, . . . , ln ∈ Zd and all ϕ, ψ ∈ Φ0,≤1. Combining (2.2.49) and (2.2.50) implies that
supt∈R
∑
k1,k2∈Zd
∣
∣
∣E
[
〈gk1 ,:(V ϕt )n:− :(V ψt )n:〉H〈gk2 ,:(V ϕt )n:− :(V ψt )
n:〉H
]∣
∣
∣
(λk1λk2)−β
≤ n! (2π)2d
∑
k∈Zd
∑
l1,...,ln∈Zd
l1+...+ln=k
(λk)2β
(∑n
i=1 |ϕli − ψli |)2
(∏ni=1 λli)
(2.2.51)
for all β ∈ R and all ϕ, ψ ∈ Φ0,≤1. Morever, combining the identity
∑
l1,...,ln∈Zd
l1+...+ln=k
1
(∏ni=1 λli)
=∑
l1∈Zd
1
λl1
∑
l2∈Zd
1
λl2
. . .
∑
ln−1∈Zd
1
λln−1 · λ(k−l1−...−ln−1)
(2.2.52)
82
for all k ∈ Zd with Corollary 59 and with the assumption (n, d) ∈ (2, 3, . . . × 2) ∪ (2, 3) proves
that supk∈Zd
[
∑
l1,...,ln∈Zd
l1+...+ln=k
(λk)β
(∏
ni=1 λli)
]
<∞ for all β ∈ (−∞, 2− d2 ). This implies that
∑
k∈Zd
∑
l1,...,ln∈Zd
l1+...+ln=k
(λk)2β
(∏ni=1 λli)
<∞ (2.2.53)
for all β ∈ (−∞, 2−d2 ). Dominated convergence and (2.2.51) therefore show for every β ∈ (−∞, 2−d2 )
that
supt∈R
∑
k1,k2∈Zd
∣
∣
∣E
[
〈gk1 ,:(V ϕt )n:− :(V ψt )n:〉H〈gk2 ,:(V ϕt )n:− :(V ψt )
n:〉H
]∣
∣
∣
(λk1λk2 )−β → 0 as (Φ0,≤1)
2 ∋ (ϕ, ψ) → (1, 1).
(2.2.54)
Next observe that Lemma 64 shows that
E
⟨
g−k1 ,[
: (V ϕt1 )n : − : (V ψt1 )
n :]
−[
: (V ϕt2 )n : − : (V ψt2 )
n :]⟩
H
·⟨
gk2 ,[
: (V ϕt1 )n : − : (V ψt1 )
n :]
−[
: (V ϕt2 )n : − : (V ψt2 )
n :]⟩
H
= n! (2π)2dδk1,k2
∑
l1,...,ln∈Zd
l1+...+ln=k1
(
2∏ni=1(ϕli)
2−4∏ni=1 ϕliψli+2
∏ni=1(ψli)
2)(
1−e−∑ni=1 λli
|t1−t2|)
(∏
ni=1 λli)
= 2n! (2π)2dδk1,k2
∑
l1,...,ln∈Zd
l1+...+ln=k1
(∏ni=1 ϕli −
∏ni=1 ψli)
2 (1− e−
∑ni=1 λli |t1−t2|
)
(∏ni=1 λli)
≤ n! (2π)4d δk1,k2
∑
l1,...,ln∈Zd
l1+...+ln=k1
(∑n
i=1 |ϕli − ψli |)2(∑ni=1 λli)
2α
(∏ni=1 λli)
|t1 − t2|2α
≤ n! (2π)4dδk1,k2
∑
l1,...,ln∈Zd
l1+...+ln=k1
(∑n
i=1 |ϕli − ψli |)2(∑n
i=1 (λli)2α)
(∏ni=1 λli)
|t1 − t2|2α
(2.2.55)
for all t1, t2 ∈ R, k1, k2 ∈ Zd, α ∈ (0, 12 ] and all ϕ, ψ ∈ Φ0,≤1 where we used 1 − e−x ≤ x2α for all
α ∈ [0, 12 ] and all x ∈ [0,∞) and∏ni=1 ϕli −
∏ni=1 ψli ≤ ∑n
i=1 |ϕli − ψli | for all l1, . . . , ln ∈ Zd and
all ϕ, ψ ∈ Φ0,≤1 (cf. (2.2.50)) in the last but one line of (2.2.55) and where we used (∑ni=1 λli)
2α ≤
83
∑ni=1 (λli)
2αfor all α ∈ [0, 12 ] in the last line of (2.2.55). Moreover, Corollary 59 proves that
∑
k∈Zd
∑
l1,...,ln∈Zd
l1+...+ln=k1
(λln)2α
(λk)2β
(∏ni=1 λli)
<∞ (2.2.56)
for all α ∈ (0, 4−d4 ), β ∈ R with α+ β < 2−d2 . Dominated convergence and (2.2.55) therefore show for
every α ∈ (0, 4−d4 ), β ∈ R with α+ β < 2−d2 that
supt1,t2∈R
t1 6=t2
∑
k1,k2∈Zd
∣
∣
∣
∣
∣
∣
∣
E
⟨
g−k1,
[
: (Vϕt1
)n
: − : (Vψt1
)n
:
]
−
[
: (Vϕt2
)n
: − : (Vψt2
)n
:
]⟩
H
·
⟨
gk2,
[
: (Vϕt1
)n
: − : (Vψt1
)n
:
]
−
[
: (Vϕt2
)n
: − : (Vψt2
)n
:
]⟩
H
∣
∣
∣
∣
∣
∣
∣
(λk1λk2 )−β |t1−t2|2α
→ 0 as (Φ0,≤1)
2 ∋ (ϕ, ψ) → (1, 1).
(2.2.57)
Combining (2.2.54) and (2.2.57) with Lemma 52 completes the proof of Proposition 65.
The next proposition is well known in the literature (see, for instance, Da Prato & Zabczyk [DZ92]
for related results and references) and its proof is therefore omitted.
Proposition 66 (Ornstein-Uhlenbeck processes). Assume the setting of Subsection 2.2.1 and let d ∈ N.
Then there exists an up to indistinguishability unique stochastic process V : R × Ω → ∩β∈(−∞, 2−d2 )
CβP([0, 2π]d,R) with continuous sample paths which satisfies for every T, p ∈ (0,∞), α ∈ (0, 12 ), β ∈ R
with 2α+ β < 2−d2 that
‖V ϕ − V ‖Lp(Ω;Cα([−T,T ],CβP([0,2π]d,R))) → 0 as Φ0,≤1 ∋ ϕ→ 1. (2.2.58)
Proposition 65 shows convergence of Wick powers in the case (n, d) ∈ (2, 3, . . . × 2)∪ (2, 3).
In the case (n, d) ∈ (3, 4, . . .×3)∪(2, 3, . . .×4, 5, . . .), Wick powers do not converge anymore.
This is the subject of the next lemma. In the case d = n = 3, a statement similar to the next lemma
has been formulated in Section 7 in Da Prato & Tubaro [DT07].
Lemma 67 (Divergence of Wick powers). Assume the setting of Subsection 2.2.1, let d ∈ 3, 4, . . .,
n ∈ 2, 3, . . . be natural numbers with d + n ≥ 6 and let C0, C1, . . . , Cn−1 : Φ0 → R be arbitrary
functions. Then it holds for every v ∈ Zd and every t ∈ R that
E
∣∣∣∣∣
⟨
gv, (Vϕt )
n −n−1∑
k=0
Ck(ϕ) · (V ϕt )k
⟩
H
∣∣∣∣∣
2
→ ∞ as Φ0 ∋ ϕ→ 1. (2.2.59)
84
Proof of Lemma 67. Throughout this proof let C0, C1, . . . Cn : Φ0 → R be the unique functions satis-
fying C0(0) = −C0(0), C1(0) = −C1(0), . . . , Cn−1(0) = −Cn−1(0), Cn(0) = 1 and
xn −n−1∑
k=0
Ck(ϕ) · xk =
n∑
k=0
Ck(ϕ) ·
∑
v∈Zd
(ϕv)2
λv
k2
·Hk
x
√∑
v∈Zd(ϕv)2
λv
(2.2.60)
for all x ∈ R, ϕ = (ϕv)v∈Zd ∈ Φ0\0 and all t ∈ R. This ensures that Cn(ϕ) = 1 and
(V ϕt )n −
n−1∑
k=0
Ck(ϕ) · (V ϕt )k=
n∑
k=0
Ck(ϕ) ·(
: (V ϕt )k:)
(2.2.61)
for all ϕ ∈ Φ0 and all t ∈ R. Lemma 64 hence implies that
E
∣∣∣∣∣
⟨
gv, (Vϕt )
n −n−1∑
k=0
Ck(ϕ) · (V ϕt )k
⟩
H
∣∣∣∣∣
2
= E
∣∣∣∣∣
n∑
k=0
⟨
gv, Ck(ϕ)(: (V ϕt )k :
)⟩
H
∣∣∣∣∣
2
=
n∑
k,l=0
Ck(ϕ) · Cl(ϕ) · E[
〈gv, : (V ϕt )k :〉H⟨gv, : (V
ϕt )l :
⟩
H
]
=
n∑
k=0
∣∣∣Ck(ϕ)
∣∣∣
2
E[∣∣⟨gv, : (V
ϕt )k :
⟩
H
∣∣2]
≥∣∣∣Cn(ϕ)
∣∣∣
2
E[
|〈gv, : (V ϕt )n :〉H |2]
= n! (2π)2d
∑
l1,...,ln∈Zd
l1+...+ln=v
n∏
i=1
(ϕli)2
λli
≥
∑
l1,...,ln∈Zd
l1+...+ln=v
(ϕl1)2 · . . . · (ϕln)2
λl1 · . . . · λln
(2.2.62)
for all v ∈ Zd and all ϕ ∈ Φ0. Next note that the estimate
∑
l1,l2∈Z3
1
λl1λl2λ(v−l1−l2)≥
∑
l1,l2∈Z3
1
3(1+‖l1‖2R3)(1+‖l2‖2
R3)(1+‖l1‖2
R3+‖l2‖2
R3+‖v‖2
R3)= ∞ (2.2.63)
for all v ∈ Zd together with the assumptions d ≥ 3, n ≥ 2 and d+ n ≥ 6 and Lemma 57 implies that
∑
l1,...,ln∈Zd
l1+...+ln=v
1
λl1 · . . . · λln=
∑
l1,...,ln−1∈Zd
1
λl1 · . . . · λln−1 · λ(v−l1−...−ln−1)= ∞ (2.2.64)
for all v ∈ Zd. Combining this with (2.2.62) completes the proof of Lemma 67.
85
2.2.5 Averaged Wick powers of Ornstein-Uhlenbeck processes
In the previous subsection it has been proved in the case d = 3 that for every t ∈ R the family : (V ϕt )3 :,
ϕ ∈ Φ0,≤1, does not converge as Φ0,≤1 ∋ ϕ→ 1 ∈ Φ0,≤1 (see Lemma 67). In this subsection we prove
in the case d = 3 that for every (t0, t) ∈ (s0, s) ∈ R2 : s0 ≤ s the family (V ϕt0,t)3 =´ t
t0: (V ϕs )3 : ds,
ϕ ∈ Φ0,≤1, does converge as Φ0,≤1 ∋ ϕ→ 1 ∈ Φ0,≤1 (see Proposition 70).
Lemma 68 (Correlation of averaged Wick powers of V ϕ, ϕ ∈ Φ0, in Fourier space). Assume the
setting of Subsection 2.2.1. Then
1
(2π)2dE
[⟨
gk1 , (V ϕ
(1)
t0,t1
)n1⟩
H
⟨
gk2 , (V ϕ
(2)
t0,t2
)n2⟩
H
]
=
n1! δn1,n2 δk1,k2∑
l1,...,ln1∈Zd
l1+...+ln1=k1
∏n1i=1 ϕ
(1)li
ϕ(2)li
∏n1i=1 λli
´ t1t0
´ t2t0e−(
∑ni=1 λli)|s1−s2| ds2 ds1 : n1n2 6= 0
δn1,n2 δk1,k2 : n1n2 = 0
(2.2.65)
for all k1, k2 ∈ Zd, ϕ(1), ϕ(2) ∈ Φ0, n1, n2 ∈ N0 and all t0, t1, t2 ∈ R with t0 ≤ min(t1, t2).
Lemma 68 is an immediate consequence of Lemma 64 and the proof of Lemma 68 is therefore
omitted.
Lemma 69 (Time integrals for averaged Wick powers). Assume the setting of Subsection 2.2.1. Then
ˆ t1
t0
ˆ t2
t0
e−c|s1−s2| ds2 ds1 = 2(min(t1,t2)−t0)c +
([
∑2j=1 e
−c(tj−t0)]
−1−e−c(t2−t1))
c2(2.2.66)
and
(t1−t0)(
1−e−c(t1−t0)
2
)
c ≤ˆ t1
t0
ˆ t1
t0
e−c|s1−s2| ds2 ds1 =2
c2
(
e−c(t1−t0) − [1− (t1 − t0) c])
≤ 2 (t1 − t0)θ
c(2−θ)
(2.2.67)
for all c ∈ (0,∞), θ ∈ [1, 2] and all t0, t1, t2 ∈ R with t0 ≤ min(t1, t2).
Proof of Lemma 69. Note that
ˆ t1
t0
ˆ t2
t0
1(u1,u2)∈R2 : u2≤u1(s1, s2) · e−c|s1−s2| ds2 ds1
=
ˆ t1
t0
ˆ s1
t0
e−c(s1−s2) ds2 ds1 =
ˆ t1
t0
(1− e−c(s1−t0)
)
cds1 =
(t1 − t0)
c+
(e−c(t1−t0) − 1
)
c2
(2.2.68)
86
and hence
ˆ t1
t0
ˆ t2
t0
1(u1,u2)∈R2 : u1≤u2≤t1(s1, s2) · e−c|s1−s2| ds1 ds2
=
ˆ t1
t0
ˆ s2
t0
e−c(s2−s1) ds1 ds2 =(t1 − t0)
c+
(e−c(t1−t0) − 1
)
c2
(2.2.69)
for all c ∈ (0,∞) and all t0, t1, t2 ∈ R with t0 ≤ t1 ≤ t2. Furthermore, observe that
ˆ t1
t0
ˆ t2
t0
1(u1,u2)∈R2 : u1≤t1≤u2(s1, s2) · e−c|s1−s2| ds2 ds1
=
ˆ t2
t1
ˆ t1
t0
e−c(s2−s1) ds1 ds2 =
ˆ t2
t1
(e−c(s2−t1) − e−c(s2−t0)
)
cds2
=−(e−c(t2−t1) − 1
)+(e−c(t2−t0) − e−c(t1−t0)
)
c2=
1 + e−c(t2−t0) − e−c(t2−t1) − e−c(t1−t0)
c2
(2.2.70)
for all c ∈ (0,∞) and all t0, t1, t2 ∈ R with t0 ≤ t1 ≤ t2. Combining (2.2.68)–(2.2.70) results in
ˆ t1
t0
ˆ t2
t0
e−c|s1−s2| ds2 ds1
=2 (t1 − t0)
c+
2(e−c(t1−t0) − 1
)
c2+
(1 + e−c(t2−t0) − e−c(t2−t1) − e−c(t1−t0)
)
c2
=2 (t1 − t0)
c+
(e−c(t1−t0) + e−c(t2−t0) − 1− e−c(t2−t1)
)
c2
(2.2.71)
for all c ∈ (0,∞) and all t0, t1, t2 ∈ R with t0 ≤ t1 ≤ t2. In addition, observe that
|y|(1− e
y2
)
2=
ˆ
y2
y
1− ey2 ds ≤
ˆ
y2
y
1− es ds ≤ˆ 0
y
1− es ds
= ey − (1 + y) =
ˆ 0
y
1− es ds ≤ˆ 0
y
[1− es]θds ≤
ˆ 0
y
[ˆ 0
s
eu du
]θ
ds
≤ˆ 0
y
|s|θ ds =ˆ −y
0
sθ ds =|y|(1+θ)(1 + θ)
≤ |y|(1+θ)
(2.2.72)
for all y ∈ (−∞, 0] and all θ ∈ [0, 1]. Combining this with (2.2.71) completes the proof of Lemma 69.
The next result, Proposition 70, establishes convergence of averaged Wick powers under the as-
sumption that n, d ∈ 2, 3, . . . with n+1n−1 > d
2 . The proof of Proposition 70 exploits Lemma 52,
Lemma 68 and Lemma 69.
Proposition 70 (Convergence of averaged Wick powers). Assume the setting of Subsection 2.2.1, let
t0 ∈ R and let n, d ∈ 2, 3, . . . with n+1n−1 >
d2 . Then there exists an up to indistinguishability unique
87
stochastic process
(Vt0,(·)
)n : [t0,∞)× Ω → ∩β∈(−∞,1+ 1n− d
2+(n−1)min(1+ 1n− d
2 ,0))CβP([0, 2π]d,R) (2.2.73)
with continuous sample paths which satisfies for every T ∈ (t0,∞), p ∈ (0,∞), α ∈ (0, 1) and every
β ∈ (−∞, 1 + 1n − d
2 + (n− 1)min(1 + 1n − d
2 , 0)) that
‖ (V ϕt0,(·))n − (Vt0,(·))n ‖Lp(Ω;Cα([t0,T ],CβP([0,2π]d,R))) → 0 as Φ0,≤1 ∋ ϕ→ 1. (2.2.74)
Proof of Proposition 70. Lemma 68 and Lemma 69 imply
1
(2π)2d
E
[⟨
gk1 , (V ϕt,t
)n − (V ψt,t
)n⟩
H
⟨
gk2 , (V ϕt,t
)n − (V ψt,t
)n⟩
H
]
= n! δk1,k2∑
l1,...,ln∈Zd
l1+...+ln=k1
∏ni=1
[ϕli ]2
λli− 2
∏ni=1
ϕliψliλli
+∏ni=1
[ψli ]2
λli
·´ t
t
´ t
te−[
∑ni=1 λli ]|s2−s1| ds2 ds1
= n! δk1,k2∑
l1,...,ln∈Zd
l1+...+ln=k1
(∏ni=1 ϕli −
∏ni=1 ψli)
2
(∏ni=1 λli)
ˆ t
t
ˆ t
t
e−[∑ni=1 λli ]|s2−s1| ds2 ds1
≤ n! 2 δk1,k2∑
l1,...,ln∈Zd
l1+...+ln=k1
(∏ni=1 ϕli −
∏ni=1 ψli)
2 (t− t
)
(∏ni=1 λli) (
∑ni=1 λli)
≤ n! 2 δk1,k2∑
l1,...,ln∈Zd
l1+...+ln=k1
(∏ni=1 ϕli −
∏ni=1 ψli)
2 (t− t
)
(∏ni=1 (λli)
(1+1/n))
(2.2.75)
for all k1, k2 ∈ Zd and all t, t ∈ R with t ≤ t. Next note that Corollary 59 ensures that
supk1∈Zd
∑
l1,...,ln∈Zd
l1+...+ln=k1
(λk1)γ
(∏ni=1 (λli)
(1+1/n))
<∞ (2.2.76)
for all γ ∈(0, 1 + 1
n + (n− 1)min(1 + 1
n − d2 , 0) )
and therefore, we obtain that
∑
k1∈Zd
∑
l1,...,ln∈Zd
l1+...+ln=k1
(λk1 )2β
(∏ni=1 (λli)
(1+1/n)) <∞ (2.2.77)
for all β ∈(−∞, 12 + 1
2n + (n− 1)min(12 + 1
2n − d4 , 0)− d
4
). Combining this, (2.2.75) and dominated
88
convergence implies for every β ∈(−∞, 12 + 1
2n − d4 + (n− 1)min
(12 + 1
2n − d4 , 0) )
that
supt,t∈R,t<t
∑
k1,k2∈Zd
∣
∣
∣
∣
E
[
⟨
gk1 ,(
V ϕt,t
)n−
(
V ψt,t
)n⟩
H
⟨
gk2 ,(
V ϕt,t
)n−
(
V ψt,t
)n⟩
H
]∣
∣
∣
∣
(t−t)·(λk1λk2 )−β → 0 (2.2.78)
as (Φ0,≤1)2 ∋ (ϕ, ψ) → (1, 1). In the next step observe that Definition (2.2.14) implies that
E
⟨
g−k1 ,[
(V ϕt,t1
)n − (V ψt,t1
)n]
−[
(V ϕt,t2
)n − (V ψt,t2
)n]⟩
H
·⟨
gk2 ,[
(V ϕt,t1
)n − (V ψt,t1
)n]
−[
(V ϕt,t2
)n − (V ψt,t2
)n]⟩
H
= E
[⟨
g−k1 , (V ϕt1,t2)n − (V ψt1,t2)n⟩
H
⟨
gk2 , (V ϕt1,t2)n − (V ψt1,t2)n⟩
H
]
(2.2.79)
for all k1, k2 ∈ Zd, ϕ, ψ ∈ Φ0,≤1 and all t1, t2 ∈ R with t ≤ t1 ≤ t2. Combining this with (2.2.78) shows
for every β ∈(−∞, 12 + 1
2n − d4 + (n− 1)min
(12 + 1
2n − d4 , 0) )
that
supt∈R,
t1,t2∈[t,∞),t1 6=t2
∑
k1,k2∈Zd
∣
∣
∣
∣
∣
∣
∣
E
⟨
g−k1,
[
(Vϕ
t,t1)n − (V
ψ
t,t1)n
]
−
[
(Vϕ
t,t2)n − (V
ψ
t,t2)n
]⟩
H
·
⟨
gk2,
[
(Vϕ
t,t1)n − (V
ψ
t,t1)n
]
−
[
(Vϕ
t,t2)n − (V
ψ
t,t2)n
]⟩
H
∣
∣
∣
∣
∣
∣
∣
(λk1λk2 )−β |t1−t2|
→ 0 (2.2.80)
as (Φ0,≤1)2 ∋ (ϕ, ψ) → (1, 1). Combining (2.2.78) and (2.2.80) with Lemma 52 completes the proof of
Proposition 70.
Proposition 70 shows convergence of averaged Wick powers under the assumption that n, d ∈
2, 3, . . . with n+1n−1 >
d2 . Lemma 72 below, in particular, proves that averaged Wick powers fail to
converge if n, d ∈ 2, 3, . . . with n+1n−1 ≤ d
2 . In the proof of Lemma 72 the following lemma is used.
Lemma 71. Assume the setting of Subsection 2.2.1 and let n, d ∈ 2, 3, . . . with (n+1)(n−1) ≤ d
2 . Then
∑
l1,...,ln∈Zd
l1+...+ln=v
1
(∏ni=1 λli)(λv+
∑ni=1 λli)
= ∞ for all v ∈ Zd.
89
Proof of Lemma 71. Note that
∑
l1,...,ln∈Zd
l1+...+ln=v
1
(∏ni=1 λli) (λv +
∑ni=1 λli)
≥∑
l1,...,ln∈Zd
l1+...+ln=v
1
(λv +∑n
i=1 λli)(n+1)
≥∑
l1,...,ln∈Zd
l1+...+ln=v
1(
λv + λ(v−l1−...−ln) +∑n−1i=1 λli
)(n+1)
≥ 1
(n+ 1)(n+1)
∑
l1,...,ln∈Zd
l1+...+ln=v
1(
λv +∑n−1i=1 λli
)(n+1)
≥ 1
(2λv (n+ 1))(n+1)
∑
l1,...,ln∈Zd
l1+...+ln=v
1(∑n−1
i=1 λli
)(n+1)
=1
(2λv (n+ 1))(n+1)
∑
k∈Zd(n−1)
1(
1 + ‖k‖2Rd(n−1)
)(n+1)
= ∞
(2.2.81)
for all v ∈ Zd. The proof of Lemma 71 is thus completed.
Lemma 72 (Divergence of averaged Wick powers). Assume the setting of Subsection 2.2.1, let n, d ∈
2, 3, . . . with (n+1)(n−1) ≤ d
2 and let C0, C1, . . . , Cn−1 : Φ0 → R be arbitrary functions. Then it holds for
every v ∈ Zd and every t0, t ∈ R with t0 < t that
E
∣∣∣∣∣
⟨
gv,
ˆ t
t0
(
(V ϕs )n −
n−1∑
k=0
Ck(ϕ) · (V ϕs )k
)
ds
⟩
H
∣∣∣∣∣
2
→ ∞ as Φ0 ∋ ϕ→ 1. (2.2.82)
Proof of Lemma 72. Throughout this proof let C0, C1, . . . Cn : Φ0 → R be the unique functions satis-
fying C0(0) = −C0(0), C1(0) = −C1(0), . . . , Cn−1(0) = −Cn−1(0), Cn(0) = 1 and
xn −n−1∑
k=0
Ck(ϕ) · xk =
n∑
k=0
Ck(ϕ) ·
∑
v∈Zd
(ϕv)2
λv
k2
·Hk
x
√∑
v∈Zd(ϕv)2
λv
(2.2.83)
90
for all x ∈ R, ϕ ∈ Φ0\0 and all t ∈ R (cf. (2.2.60)). Then Lemma 68 and Lemma 69 imply that
E
∣∣∣∣∣
⟨
gv,
ˆ t
t0
(
(V ϕs )n −n−1∑
k=0
Ck(ϕ) · (V ϕs )k
)
ds
⟩
H
∣∣∣∣∣
2
= E
∣∣∣∣∣
n∑
k=0
⟨
gv,
ˆ t
t0
Ck(ϕ)(: (V ϕs )k :
)ds
⟩
H
∣∣∣∣∣
2
=n∑
k,l=0
Ck(ϕ) · Cl(ϕ) · E[⟨gv, (V ϕt0,t)k
⟩
H
⟨gv, (V ϕt0,t)l
⟩
H
]
=
n∑
k=0
∣∣∣Ck(ϕ)
∣∣∣
2
E
[∣∣∣
⟨gv, (V ϕt0,t)k
⟩
H
∣∣∣
2]
≥∣∣∣Cn(ϕ)
∣∣∣
2
E
[∣∣∣
⟨gv, (V ϕt0,t)n
⟩
H
∣∣∣
2]
= n! (2π)2d
∑
l1,...,ln∈Zd
l1+...+ln=v
n∏
i=1
(ϕli)2
λli
ˆ t
t0
ˆ t
t0
e−(∑ni=1 λli)|s1−s2| ds2 ds1
≥ n! (2π)2d
∑
l1,...,ln∈Zd
l1+...+ln=v
(n∏
i=1
(ϕli)2
λli
)(t− t0)
(
1− e−(t−t0)
2
)
(∑n
i=1 λli)
≥ n! (2π)2d
∑
l1,...,ln∈Zd
l1+...+ln=v
(∏ni=1 (ϕli)
2)
(t− t0)(
1− e−(t−t0)
2
)
(∏ni=1 λli) (λv +
∑ni=1 λli)
(2.2.84)
for all v ∈ Zd, t0, t ∈ R with t0 ≤ t and all ϕ ∈ Φ0. Combining this with Lemma 71 completes the
proof of Lemma 72.
2.2.6 Convolutional Wick powers of Ornstein-Uhlenbeck processes
Lemma 73 (Correlation of convolutional Wick powers of V ϕ, ϕ ∈ Φ0, in Fourier space). Assume the
setting of Subsection 2.2.1. Then
1
(2π)2d
E[⟨gk1 , •(V ϕ1
t1 )n1•⟩
H
⟨gk2 , •(V ϕ2
t2 )n2•⟩
H
]
=
n1! δn1,n2 δk1,k2∑
l1,...,ln1∈Zd
l1+...+ln1=k1
[(
∏n1i=1 ϕ
(1)li
ϕ(2)li
)
(∏n1i=1 λli)
·´ t1−∞´ t2−∞ e−λk1 (t1−s1+t2−s2)−(
∑n1i=1 λli)|s1−s2| ds2 ds1
] : n1n2 6= 0
δn1,n2 δk1,k2 δk1,0 : n1n2 = 0
(2.2.85)
for all t1, t2 ∈ R, k1, k2 ∈ Zd, ϕ(1), ϕ(2) ∈ Φ0 and all n1, n2 ∈ N0.
91
Proof of Lemma 73. Combining the identity
1
(2π)2d
E
[⟨
gk1 , •(V ϕ
(1)
t1
)n1•⟩
H
⟨
gk2 , •(V ϕ
(2)
t2
)n2•⟩
H
]
=
ˆ t1
−∞
ˆ t2
−∞e−λk1 (t1−s1) e−λk2 (t2−s2)
E
[⟨
gk1 , :(V ϕ
(1)
s1
)n1:⟩
H
⟨
gk2 , :(V ϕ
(2)
s2
)n2:⟩
H
]
(2π)2dds1 ds2
(2.2.86)
for all ϕ(1), ϕ(2) ∈ Φ0, k1, k2 ∈ Zd, n1, n2 ∈ N0 and all t1, t2 ∈ R with t1 ≤ t2 with Lemma 64 completes
the proof of Lemma 73.
Lemma 74 (Time integrals for convolutional Wick powers). Assume the setting of Subsection 2.2.1.
Then
ˆ t1
−∞
ˆ t2
−∞e−a(t1−s1+t2−s2)−b|s1−s2| ds2 ds1 =
e−a(t2−t1)
a (a+ b)+
(e−b(t2−t1)−e−a(t2−t1))(a−b)(a+b) : a 6= b
(t2−t1)e−a(t2−t1)
(a+b) : a = b
(2.2.87)
andˆ t
−∞
ˆ t
−∞e−a(2t−s1−s2)−b|s1−s2| ds2 ds1 =
1
a (a+ b)(2.2.88)
for all a, b ∈ (0,∞) and all t, t1, t2 ∈ R with t1 ≤ t2.
Proof of Lemma 74. First of all, note that
ˆ t1
−∞
ˆ t2
−∞1(u1,u2)∈R2 : u2≤u1(s1, s2) · e−a(t1−s1+t2−s2)−b|s1−s2| ds2 ds1
=
ˆ t1
−∞
ˆ t1
−∞1(u1,u2)∈R2 : u2≤u1(s1, s2) · e−a(t1−s1+t2−s2)−b|s1−s2| ds2 ds1
=
ˆ t1
−∞
ˆ s1
−∞e−a(t1−s1+t2−s2)−b(s1−s2) ds2 ds1
=
´ t1−∞ e−a(t1+t2−2s1) ds1
(a+ b)=e−a(t2−t1) − e−a(t1+t2−2t0)
2a (a+ b),
(2.2.89)
92
that
ˆ t1
−∞
ˆ t2
−∞1(u1,u2)∈R2 : u1≤u2≤t1(s1, s2) · e−a(t1−s1+t2−s2)−b|s1−s2| ds2 ds1
=
ˆ t1
−∞
ˆ t1
−∞1(u1,u2)∈R2 : u1≤u2(s1, s2) · e−a(t1−s1+t2−s2)−b|s1−s2| ds2 ds1
=
ˆ t1
−∞
ˆ t1
−∞1(u1,u2)∈R2 : u2≤u1(s1, s2) · e−a(t1−s1+t2−s2)−b|s1−s2| ds2 ds1
=e−a(t2−t1) − e−a(t1+t2−2t0)
2a (a+ b)
(2.2.90)
and that
ˆ t1
−∞
ˆ t2
−∞1(u1,u2)∈R2 : u1≤t1≤u2(s1, s2) · e−a(t1−s1+t2−s2)−b|s1−s2| ds2 ds1
=
ˆ t2
t1
ˆ t1
−∞e−a(t1−s1+t2−s2)−b(s2−s1) ds1 ds2
=1
(a+ b)
ˆ t2
t1
(
e−a(t2−s2)−b(s2−t1))
ds2 =
e−b(t2−t1)−e−a(t2−t1)
(a−b)(a+b) : a 6= b
(t2−t1)e−a(t2−t1)
(a+b) : a = b
(2.2.91)
for all t1, t2 ∈ R with t1 ≤ t2. Combining (2.2.89)–(2.2.91) proves that
ˆ t1
−∞
ˆ t2
−∞e−a(t1−s1+t2−s2)−b|s2−s1| ds2 ds1 =
e−a(t2−t1)
a (a+ b)+
e−a(t2−t1)−e−b(t2−t1)
(a−b)(a+b) : a 6= b
(t2−t1)e−a(t2−t1)
(a+b) : a = b
(2.2.92)
for all t1, t2 ∈ R with t1 ≤ t2. The proof of Lemma 74 is thus completed.
The next proposition proves convergence of convolutional Wick powers under the assumption that
n, d ∈ 2, 3, . . . with n+1n−1 >
d2 . Its proof uses Lemma 73, Lemma 74 and Lemma 52.
Proposition 75 (Convergence of convolutional Wick powers). Assume the setting of Subsection 2.2.1
and let n, d ∈ 2, 3, . . . with n+1n−1 >
d2 . Then there exists an up to indistinguishability unique stochastic
process
•(V )n• : R× Ω → ∩β∈(−∞,2+n(2−d)
2 )CβP([0, 2π]d,R) (2.2.93)
with continuous sample paths which satisfies for every T, p ∈ (0,∞) and every α ∈ (0, 12 ), β ∈ R with
93
2α+ β < 2 + n(2−d)2 that
‖•(V ϕ)n • − • (V )n•‖Lp(Ω;Cα([−T,T ],CβP([0,2π]d,R))) → 0 as Φ0,≤1 ∋ ϕ→ 1. (2.2.94)
Proof of Proposition 75. Lemma 73 and Lemma 74 imply
1
(2π)2d
E
[⟨
gk1 , •(V ϕt)n • − •
(V ψt)n•⟩
H
⟨
gk2 , •(V ϕt)n • − •
(V ψt)n•⟩
H
]
= n! δk1,k2∑
l1,...,ln∈Zd
l1+...+ln=k1
∏ni=1
[ϕli ]2
λli− 2
∏ni=1
ϕliψliλli
+∏ni=1
[ψli ]2
λli
·´ t
−∞´ t
−∞ e−λk1 (2t−s1−s2)−(∑ni=1 λli)|s2−s1| ds2 ds1
= n! δk1,k2∑
l1,...,ln∈Zd
l1+...+ln=k1
(∏ni=1 ϕli −
∏ni=1 ψli)
2
(∏ni=1 λli)λk1 (λk1 +
∑ni=1 λli)
≤ n! δk1,k2
(λk1 )max(4−d,1)
∑
l1,...,ln∈Zd
l1+...+ln=k1
(∏ni=1 ϕli −
∏ni=1 ψli)
2
(∏ni=1 (λli)
(1+min(d−2,1)n )
)
(2.2.95)
for all k1, k2 ∈ Zd, t ∈ R and all ϕ, ψ ∈ Φ0. Next note that Corollary 59 ensures that
supk1∈Zd
∑
l1,...,ln∈Zd
l1+...+ln=k1
(λk1)γ
(∏ni=1 (λli)
(1+min(d−2,1)n )
)
<∞ (2.2.96)
for all γ ∈(−∞, d2 +min(d− 2, 1) + n
(1− d
2
) ). Therefore, we obtain that
∑
k1∈Zd
∑
l1,...,ln∈Zd
l1+...+ln=k1
(λk1)(2β−max(4−d,1))
(∏ni=1 (λli)
(1+min(d−2,1)n )
) <∞ (2.2.97)
for all β ∈(−∞, 4−(d−2)n
4
). Combining this with (2.2.95) and dominated convergence shows for every
β ∈(−∞, 1− (d−2)n
4
)that
supt∈R
∑
k1,k2∈Zd
∣
∣
∣E
[
〈gk1 ,•(V ϕt )n•−•(V ψt )n•〉
H〈gk2 ,•(V ϕt )n•−•(V ψt )
n•〉H
]∣
∣
∣
(λk1λk2 )−β → 0 as (Φ0,≤1)
2 ∋ (ϕ, ψ) → (1, 1).
(2.2.98)
94
In the next step let hk1,l1,...,ln : R2 → R, k1, l1, . . . , ln ∈ Zd, be functions defined through
hk1,l1,...,ln(t1, t2) :=
ˆ t1
−∞
ˆ t2
−∞e−λk1 (t1−s1+t2−s2)−(
∑ni=1 λli)|s1−s2| ds2 ds1 (2.2.99)
for all t1, t2 ∈ R and all k1, l1, . . . , ln ∈ Zd. Then observe that Lemma 74 implies that
hk1,l1,...,ln(t1, t1)− 2hk1,l1,...,ln(t1, t2) + hk1,l1,...,ln(t2, t2)
=2(1− e−λk1 (t2−t1)
)
λk1 (λk1 +∑ni=1 λli)
− 2 ·
e−(∑ni=1 λli)(t2−t1)−e−λk1 (t2−t1)
(λk1−∑
ni=1 λli)(λk1+
∑
ni=1 λli)
: λk1 6=∑ni=1 λli
(t2−t1)e−λk1 (t2−t1)
(λk1+[∑
ni=1 λli ])
: λk1 =∑ni=1 λli
≤ 2(1− e−λk1 (t2−t1)
)
λk1 (λk1 +∑ni=1 λli)
≤ 2 (λk1 )2α
(t2 − t1)2α
λk1 (λk1 +∑n
i=1 λli)=
2 (λk1 )(2α−1)
(t2 − t1)2α
(λk1 +∑n
i=1 λli)
(2.2.100)
for all k1, l1, . . . , ln ∈ Zd, α ∈ [0, 12 ] and all t1, t2 ∈ R with t1 ≤ t2. Lemma 73 hence shows that
1
(2π)2d
E
⟨
g−k1 ,[
•(V ϕt1 )n • − • (V ψt1 )n•]
−[
•(V ϕt2 )n • − • (V ψt2 )n•]⟩
H
·⟨
gk2 ,[
•(V ϕt1 )n • − • (V ψt1 )n•]
−[
•(V ϕt2 )n • − • (V ψt2 )n•]⟩
H
= n! δk1,k2∑
l1,...,ln∈Zd
l1+...+ln=k1
(
∏ni=1(ϕli)
2−2∏ni=1 ϕliψli+
∏ni=1(ψli)
2)
(hk1,l1,...,ln (t1,t1)−2hk1,l1,...,ln (t1,t2)+hk1,l1,...,ln(t2,t2))
(∏
ni=1 λli)
= n! δk1,k2∑
l1,...,ln∈Zd
l1+...+ln=k1
(∏ni=1 ϕli−
∏ni=1 ψli)
2(hk1,l1,...,ln(t1,t1)−2hk1,l1,...,ln (t1,t2)+hk1,l1,...,ln (t2,t2))(∏ni=1 λli)
≤ (n+ 1)! δk1,k2
∑
l1,...,ln∈Zd
l1+...+ln=k1
(∏ni=1 ϕli −
∏ni=1 ψli)
2(λk1 )
(2α−1)
(∏ni=1 λli) (λk1 +
∑ni=1 λli)
(t2 − t1)
2α
≤ (n+ 1)! δk1,k2
(λk1)(max(4−d,1)−2α)
∑
l1,...,ln∈Zd
l1+...+ln=k1
(∏ni=1 ϕli −
∏ni=1 ψli)
2
(∏ni=1 (λli)
(1+min(d−2,1)n )
)
(t2 − t1)
2α
(2.2.101)
for all k1, k2 ∈ Zd, ϕ, ψ ∈ Φ0,≤1, α ∈ [0, 12 ] and all t1, t2 ∈ R with t1 ≤ t2. In addition, Corollary 59
95
ensures that
supk1∈Zd
∑
l1,...,ln∈Zd
l1+...+ln=k1
(λk1)γ
(∏ni=1 (λli)
(1+min(d−2,1)n )
)
<∞ (2.2.102)
for all γ ∈(−∞, d2 +min(d− 2, 1) + n
(1− d
2
) ). Therefore, we obtain that
∑
k1∈Zd
∑
l1,...,ln∈Zd
l1+...+ln=k1
(λk1 )(2α+2β−max(4−d,1))
(∏ni=1 (λli)
(1+min(d−2,1)n )
) <∞ (2.2.103)
for all α, β ∈ R with α + β < 1 − (d−2)n4 . Combining this with (2.2.101) and dominated convergence
implies for every α ∈ [0, 12 ], β ∈ R with α+ β < 1− (d−2)n4 that
supt1,t2∈R
t1 6=t2
∑
k1,k2∈Zd
∣
∣
∣
∣
∣
∣
∣
E
⟨
g−k1,
[
•(Vϕt1
)n • − • (V
ψt1
)n•
]
−
[
•(Vϕt2
)n • − • (V
ψt2
)n•
]⟩
H
·
⟨
gk2,
[
•(Vϕt1
)n • − • (V
ψt1
)n•
]
−
[
•(Vϕt2
)n • − • (V
ψt2
)n•
]⟩
H
∣
∣
∣
∣
∣
∣
∣
(λk1λk2 )−β |t1−t2|2α
→ 0 as (Φ0,≤1)
2 ∋ (ϕ, ψ) → (1, 1).
(2.2.104)
Combining (2.2.98) and (2.2.104) with Lemma 52 completes the proof of Proposition 75.
Proposition 75 shows convergence of convolutional Wick powers under the assumption that n, d ∈
2, 3, . . . with n+1n−1 >
d2 . In the case n, d ∈ 2, 3, . . . with n+1
n−1 ≤ d2 , convolutional Wick powers fail
to converge. This is the subject of the next lemma.
Lemma 76 (Divergence of convolutional Wick powers). Assume the setting of Subsection 2.2.1, let
n, d ∈ 2, 3, . . . with n+1n−1 ≤ d
2 and let C0, C1, . . . , Cn−1 : Φ0 → R be arbitrary functions. Then it holds
for every v ∈ Zd and every t ∈ R that
E
∣∣∣∣∣
⟨
gv,
ˆ t
−∞eA(t−s)
(
(V ϕs )n −
n−1∑
k=0
Ck(ϕ) · (V ϕs )k
)
ds
⟩
H
∣∣∣∣∣
2
→ ∞ as Φ0 ∋ ϕ→ 1. (2.2.105)
Proof of Lemma 76. Throughout this proof let C0, C1, . . . Cn : Φ0 → R be the unique functions satis-
fying C0(0) = −C0(0), C1(0) = −C1(0), . . . , Cn−1(0) = −Cn−1(0), Cn(0) = 1 and
xn −n−1∑
k=0
Ck(ϕ) · xk =
n∑
k=0
Ck(ϕ) ·
∑
v∈Zd
(ϕv)2
λv
k2
·Hk
x
√∑
v∈Zd(ϕv)2
λv
(2.2.106)
96
for all x ∈ R, ϕ ∈ Φ0\0 and all t ∈ R (cf. (2.2.60)). Then Lemma 73 implies that
E
∣∣∣∣∣
⟨
gv,
ˆ t
−∞eA(t−s)
(
(V ϕs )n −n−1∑
k=0
Ck(ϕ) · (V ϕs )k
)
ds
⟩
H
∣∣∣∣∣
2
= E
∣∣∣∣∣
n∑
k=0
⟨
gv,
ˆ t
−∞eA(t−s)
[
Ck(ϕ)(: (V ϕt )k :
)]
ds
⟩
H
∣∣∣∣∣
2
=n∑
k,l=0
Ck(ϕ) · Cl(ϕ) · E[
〈gv, •(V ϕt )k•〉H⟨gv, •(V ϕt )l•
⟩
H
]
=
n∑
k=0
∣∣∣Ck(ϕ)
∣∣∣
2
E[∣∣⟨gv, •(V ϕt )k•
⟩
H
∣∣2]
≥∣∣∣Cn(ϕ)
∣∣∣
2
E[
|〈gv, •(V ϕt )n•〉H |2]
=n! (2π)2d
λv
∑
l1,...,ln∈Zd
l1+...+ln=v
(∏ni=1 (ϕli)
2)
(∏ni=1 λli) (λv +
∑ni=1 λli)
(2.2.107)
for all t ∈ R, v ∈ Zd and all ϕ ∈ Φ0. Combining this with Lemma 71 completes the proof of
Lemma 76.
2.2.7 Summary
The following table briefly summarizes the results of Proposition 65, Proposition 70 and Proposition 75
and of Lemma 67, Lemma 72 and Lemma 76. Recall that the main arguments for the results from
Propositions 65, 70 and 75 presented in the table are certain summability properities; see (2.2.49) and
(2.2.53) in the case of Wick powers, (2.2.75) and (2.2.77) in the case of averaged Wick powers and
(2.2.95) and (2.2.97) in the case of convolutional Wick powers. In the table ε ∈ (0,∞) is an arbitrarily
small positive real number, CαP is an abbreviation for CαP([0, 2π]d,R) where α ∈ R and d ∈ N and
the expressions WP, AWP and CWP are abbreviations for Wick powers, averaged Wick powers and
convolutional Wick powers respectively.
97
......
......
......
n = 5
WP:
C−εP
AWP:
C1/5−εP
CWP:
C2−εP
No WP
No AWP
No CWP
No WP
No AWP
No CWP
No WP
No AWP
No CWP
No WP
No AWP
No CWP
. . .
n = 4
WP:
C−εP
AWP:
C1/4−εP
CWP:
C2−εP
No WP
AWP:
C−1−εP
CWP:
C−εP
No WP
No AWP
No CWP
No WP
No AWP
No CWP
No WP
No AWP
No CWP
. . .
n = 3
WP:
C−εP
AWP:
C1/3−εP
CWP:
C2−εP
No WP
AWP:
C−1/2−εP
CWP:
C1/2−εP
No WP
No AWP
No CWP
No WP
No AWP
No CWP
No WP
No AWP
No CWP
. . .
n = 2
WP:
C−εP
AWP:
C1/2−εP
CWP:
C2−εP
WP:
C−1−εP
AWP:
C−εP
CWP:
C1−εP
No WP
AWP:
C−1−εP
CWP:
C−εP
No WP
AWP:
C−2−εP
CWP:
C−1−εP
No WP
No AWP
No CWP
. . .
d = 2 d = 3 d = 4 d = 5 d = 6 . . .
98
2.3 Stochastic partial differential equations (SPDEs)
2.3.1 Local existence and uniqueness of mild solutions of deterministic
nonautonomous partial differential equations
This subsection investigates local existence and uniqueness questions for mild solutions of deterministic
nonautonomous evolution equations of the form
∂
∂tx(t) = Ax(t) +
n∑
i=1
Fi(t, x(t)) (2.3.1)
on a real Banach space (U, ‖·‖U ) for t ∈ [t0, T ] where t0, T ∈ R are real numbers with t0 < T , where
A : D(A) ⊂ U → U is a negative generator of a strongly continuous analytic semigroup, where n ∈ N
is a natural number and where F1, . . . , Fn are suitable functions that are locally Lipschitz continuous
on appropriate spaces.
To investigate these questions, we impose the following setting. Throughout this subsection, let
(U, ‖·‖U ) be a real Banach space, let A : D(A) ⊂ U → U be a negative generator of a strongly
continuous analytic semigroup on U and let(Ur, ‖·‖Ur
):= (D((−A)r), ‖(−A)r(·)‖U ) for all r ∈ R.
Next define
‖F‖Cnα,β,γ,δ([t0,T ]) :=
supt∈[t0,T ]
n∑
i=1
[
‖Fi(t, 0)‖Uαi + supx,y∈Umax(βi,γi)
x 6=y
‖Fi(t, x)− Fi(t, y)‖Uαi(1 + ‖x‖δiUβi + ‖y‖δiUβi
)‖x− y‖Uγi
]
∈ [0,∞](2.3.2)
for all F = (F1, . . . , Fn) ∈ C([t0, T ] × Umax(β1,γ1), Uα1) × . . . × C([t0, T ] × Umax(βn,γn), Uαn), α =
(α1, . . . , αn), β = (β1, . . . , βn), γ = (γ1, . . . , γn) ∈ Rn, δ = (δ1, . . . , δn) ∈ [0,∞)n, t0 ∈ (−∞, T ), T ∈ R
and all n ∈ N. Furthermore, define
Cnα,β,γ,δ([t0, T ]) :=
F ∈(
C([t0, T ]× Umax(β1,γ1), Uα1)× . . .
× C([t0, T ]× Umax(βn,γn), Uαn))
: ‖F‖Cnα,β,γ,δ([t0,T ]) <∞
(2.3.3)
for all α = (α1, . . . , αn), β = (β1, . . . , βn), γ = (γ1, . . . , γn) ∈ Rn, δ = (δ1, . . . , δn) ∈ [0,∞)n, t0 ∈
(−∞, T ), T ∈ R and all n ∈ N. Observe that the pairs(Cnα,β,γ,δ([t0, T ]), ‖·‖Cnα,β,γ,δ([t0,T ])
)for α, β, γ ∈
Rn, δ ∈ [0,∞)n, t0 ∈ (−∞, T ), T ∈ R and n ∈ N are normed real vector spaces. In the next step
99
define
Cnα,β,γ,δ([t0,∞)) :=
(F1, . . . , Fn) ∈(
C([t0,∞)× Umax(β1,γ1), Uα1)× . . .
× C([t0,∞)× Umax(βn,γn), Uαn))
:(
∀T ∈ (t0,∞) :
‖(F1|[t0,T ]×Umax(β1,γ1), . . . , Fn|[t0,T ]×Umax(βn,γn)
)‖Cnα,β,γ,δ([t0,T ]) <∞)
(2.3.4)
for all α = (α1, . . . , αn), β = (β1, . . . , βn), γ = (γ1, . . . , γn) ∈ Rn, δ = (δ1, . . . , δn) ∈ [0,∞)n, t0 ∈ R
and all n ∈ N. Moreover, we equip Cnα,β,γ,δ([t0,∞)) with the metric dCnα,β,γ,δ([t0,∞)) : Cnα,β,γ,δ([t0,∞))×
Cnα,β,γ,δ([t0,∞)) → [0,∞) defined through
dCnα,β,γ,δ([t0,∞))(F,G) :=
∞∑
k=1
1
2kmin
(
1,∥∥((F1 −G1)|[t0,t0+k]×Umax(β1,γ1)
, . . . ,
(Fn −Gn)|[t0,t0+k]×Umax(βn,γn)
)∥∥Cnα,β,γ,δ([t0,t0+k])
)
(2.3.5)
for all F = (F1, . . . , Fn), G = (G1, . . . , Gn) ∈ Cnα,β,γ,δ([t0,∞)), α = (α1, . . . , αn), β = (β1, . . . , βn),
γ = (γ1, . . . , γn) ∈ Rn, δ = (δ1, . . . , δn) ∈ [0,∞)n, t0 ∈ R and all n ∈ N. Finally, note that the triangle
inequality and the definition of ‖F‖Cnα,β,γ,δ([t0,T ]) imply that
‖Fi(t, x)‖Uαi ≤ ‖Fi(t, x)− Fi(t, 0)‖Uαi + ‖Fi(t, 0)‖Uαi
≤[
supy∈Umax(βi,γi)\0
‖Fi(t,y)−Fi(t,0)‖Uαi(1+‖y‖δiUβi
)‖y‖Uγi
+ ‖Fi(t, 0)‖Uαi
]
(1 + ‖x‖δiUβi
)(1 + ‖x‖Uγi
)
≤ ‖F‖Cnα,β,γ,δ([t0,T ])
(1 + ‖x‖δiUβi
) (1 + ‖x‖Uγi
)
(2.3.6)
for all t ∈ [t0, T ], x ∈ Umax(βi,γi), i ∈ 1, 2, . . . , n, F = (F1, . . . , Fn) ∈ Cnα,β,γ,δ([t0, T ]), α =
(α1, . . . , αn), β = (β1, . . . , βn), γ = (γ1, . . . , γn) ∈ Rn, δ = (δ1, . . . , δn) ∈ [0,∞)n, t0 ∈ (−∞, T ),
T ∈ R and all n ∈ N.
Lemma 77 (Local existence and uniqueness of mild solutions). Assume the setting in the begin-
ning of Subsection 2.3.1, let r0, t0 ∈ R, T ∈ (t0,∞), v ∈ Ur0 , n ∈ N, α = (α1, . . . , αn) ∈ Rn, β =
(β1, . . . , βn), γ = (γ1, . . . , γn) ∈ [r0,∞)n, δ = (δ1, . . . , δn) ∈ [0,∞)n, r1 ∈ [max(β1, . . . , βn, γ1, . . . , γn), 1+
min(α1, . . . , αn)) with maxi∈1,...,n[γi − min(αi, r0) + δi(βi − r0)] < 1 and let F = (F1, . . . , Fn) ∈
Cnα,β,γ,δ([t0, T ]). Then there exist a real number τ ∈ (t0, T ] such that there exists a unique continuous
function x : [t0, τ ] → Ur0 satisfying x|(t0,τ ] ∈ C((t0, τ ], Ur1), sups∈(t0,τ ] (s− t0)(r1−r0) ‖x(s)‖Ur1 < ∞
100
and x(t) = eA(t−t0) v +∑n
i=1
´ t
t0eA(t−s) Fi(s, x(s)) ds for all t ∈ [t0, τ ].
Observe that all integrals appearing in Lemma 77 are well-defined. Indeed, under the assumptions
of Lemma 77 it holds that if τ ∈ (t0, T ] and if x : [t0, τ ] → Ur0 is a continuous function which satisfies
x|(t0,τ ] ∈ C((t0, τ ], Ur1) and sups∈(t0,τ ] (s− t0)(r1−r0) ‖x(s)‖Ur1 < ∞, then (2.3.6) and interpolation
(see, e.g., Theorem 37.6 in Sell & You [SY02]) imply that
ˆ t
t0
∥∥eA(t−s) Fi(s, x(s))
∥∥Ur1
ds ≤ˆ t
t0
‖eA(t−s)‖L(Uαi ,Ur1) ‖Fi(s, x(s))‖Uαids
≤ ‖F‖Cnα,β,γ,δ([t0,T ])
[
sups∈(0,T−t0]
‖eAs‖L(Uαi,Ur1 )
smin(αi−r1,0)
]ˆ t
t0
(1 + ‖x(s)‖δiUβi
) (1 + ‖x(s)‖Uγi
)
(t− s)max(r1−αi,0) ds
≤ ‖F‖Cnα,β,γ,δ([t0,T ])
[
sups∈(0,T−t0]
‖eAs‖L(Uαi,Ur1 )
smin(αi−r1,0)
][
sups∈(t0,τ ]
(1 + ‖x(s)‖Uγi )(s− t0)
(r0−γi)
]
·[
sups∈(t0,τ ]
(1 + ‖x(s)‖δiUβi )(s− t0)
δi(r0−βi)
]ˆ t
t0
1
(t− s)max(r1−αi,0) (s− t0)
(γi−r0+δi(βi−r0)) ds <∞
(2.3.7)
for all t ∈ [t0, τ ] and all i ∈ 1, 2, . . . , n where we used r1 < 1 + minj∈1,...,n αj ≤ 1 + αi and
γi − r0 + δi(βi − r0) < 1 for all i ∈ 1, 2, . . . , n in the last line of (2.3.7). We now present the proof
of Lemma 77.
Proof of Lemma 77. Lemma 77 follows from an application of the Banach fixed point theorem. For
this several preparations are needed. First, let κ ∈ [0,∞) be a real number defined through
κ :=
[
2 + r1 − r0 + T + ‖F‖Cnα,β,γ,δ([t0,T ]) +
n∑
i=1
δi
](4+|r0|+|r1|+maxi∈1,...,n|αi|)
+
1∑
j=0
n∑
i=1
[1
min(1 + αi − rj , 1)+B(1+min(αi−rj ,0),1+r0−γi+δi(r0−βi))
]
+ maxj∈0,1
maxθ∈r0,r1,α1,...,αn
supt∈(t0,T ]
[
(t− t0)max(rj−θ,0) ∥∥eA(t−t0)
∥∥L(Uθ,Urj )
]
+ maxθ∈β1,...,βn∪γ1,...,γn
supv∈Ur1v 6=0
1 +
‖v‖Uθ‖v‖
(θ−r0)
(r1−r0)
Ur1‖v‖
(r1−θ)(r1−r0)
Ur0
(1+∑ni=1 δi)
<∞
(2.3.8)
where B : (0,∞)2 → (0,∞) is the Beta function defined through B(x,y) :=´ 1
0 (1− s)(x−1)
s(y−1) ds for
all x, y ∈ (0,∞). Observe that the quantity κ is indeed finite; see, e.g., Theorems 37.5 and 37.6 in Sell
101
& You [SY02]. Next define real vector spaces E[t0,τ ] ⊂ C([t0, τ ], Ur0), τ ∈ (t0, T ], through
E[t0,τ ] :=
x ∈ C([t0, τ ], Ur0) :
x|(t0,τ ] ∈ C((t0, τ ], Ur1) and
supt∈(t0,τ ] (t− t0)(r1−r0) ‖x(t)‖Ur1 <∞
(2.3.9)
for all τ ∈ (t0, T ], define norms ‖·‖E[t0,τ]: E[t0,τ ] → [0,∞), τ ∈ (t0, T ], through
‖x‖E[t0,τ]:=
1∑
j=0
[
supt∈(t0,τ ]
[
(t− t0)(rj−r0) ‖x(t)‖Urj
]]
(2.3.10)
for all τ ∈ (t0, T ], define sets E[t0,τ ],v ⊂ E[t0,τ ], τ ∈ (t0, T ], v ∈ Ur0 , through
E[t0,τ ],v :=
x ∈ E[t0,τ ] : ‖x‖E[t0,τ]≤ κ7
(1 + ‖v‖Ur0
)
(2.3.11)
for all τ ∈ (t0, T ] and all v ∈ Ur0 and define mappings Φ[t0,τ ],v : E[t0,τ ] → E[t0,τ ], τ ∈ (t0, T ], v ∈ Ur0 ,
through
(Φ[t0,τ ],vx)(t) := eA(t−t0) v +n∑
i=1
ˆ t
t0
eA(t−s) Fi(s, x(s)) ds (2.3.12)
for all t ∈ [t0, τ ], x ∈ E[t0,τ ], τ ∈ (t0, T ] and all v ∈ Ur0 . Note that (2.3.7) ensures that the mappings
Φ[t0,τ ],v, τ ∈ (t0, T ], v ∈ Ur0 , are well-defined. We now establish a few estimates for the mappings
Φ[t0,τ ],v, τ ∈ (t0, T ], v ∈ Ur0 . First, observe that
∥∥(Φ[t0,τ ],v0)(t)
∥∥Urj
≤∥∥eA(t−t0)v
∥∥Urj
+n∑
i=1
ˆ t
t0
∥∥eA(t−s)∥∥
L(Uαi ,Urj )
∥∥Fi(s, 0)
∥∥Uαi
ds
≤∥∥eA(t−t0)
∥∥L(Ur0 ,Urj )
‖v‖Ur0 + κ2n∑
i=1
ˆ t
t0
(t− s)min(αi−rj ,0) ds
≤ κ (t− t0)(r0−rj) ‖v‖Ur0 + κ2
n∑
i=1
(t− t0)min(1+αi−rj,1)
min(1 + αi − rj , 1)
≤ κ5 (t− t0)(r0−rj) (1 + ‖v‖Ur0
)
(2.3.13)
for all j ∈ 0, 1, t ∈ (t0, τ ], τ ∈ (t0, T ], v ∈ Ur0 and hence
∥∥Φ[t0,τ ],v(0)
∥∥E[t0,τ]
≤ κ6(1 + ‖v‖Ur0
)(2.3.14)
102
for all τ ∈ (t0, T ], v ∈ Ur0 . In the next step observe that
∥∥(Φ[t0,τ ],vx)(t) − (Φ[t0,τ ],vy)(t)
∥∥Urj
≤n∑
i=1
ˆ t
t0
∥∥eA(t−s)[Fi(s, x(s)) − Fi(s, y(s))
]∥∥Urj
ds
≤n∑
i=1
ˆ t
t0
∥∥eA(t−s)∥∥
L(Uαi ,Urj )
∥∥Fi(s, x(s))− Fi(s, y(s))
∥∥Uαi
ds
≤ κn∑
i=1
ˆ t
t0
(t− s)min(αi−rj ,0) ∥∥Fi(s, x(s)) − Fi(s, y(s))∥∥Uαi
ds
≤ κ2n∑
i=1
ˆ t
t0
(t− s)min(αi−rj,0)
[
1 + ‖x(s)‖δiUβi + ‖y(s)‖δiUβi]
‖x(s)− y(s)‖Uγi ds
≤ κ3n∑
i=1
ˆ t
t0
(t− s)min(αi−rj,0) ‖x(s)− y(s)‖
(γi−r0)
(r1−r0)
Ur1‖x(s)− y(s)‖
(r1−γi)
(r1−r0)
Ur0
·[
1 + ‖x(s)‖(βi−r0)δi(r1−r0)
Ur1‖x(s)‖
(r1−βi)δi(r1−r0)
Ur0+ ‖y(s)‖
(βi−r0)δi(r1−r0)
Ur1‖y(s)‖
(r1−βi)δi(r1−r0)
Ur0
]
ds
(2.3.15)
and therefore
(t− t0)(rj−r0) ∥∥(Φ[t0,τ ],vx)(t) − (Φ[t0,τ ],vy)(t)
∥∥Urj
≤ κ3 (t− t0)(rj−r0) ‖x− y‖E[t0,t]
n∑
i=1
ˆ t
t0
(t− s)min(αi−rj ,0)
·(
(s− t0)(r0−γi) + (s− t0)
(r0−γi+δi(r0−βi))[
‖x‖δiE[t0,t]+ ‖y‖δiE[t0,t]
])
ds
≤ κ5 (t− t0)(rj−r0)
[
1 + ‖x‖E[t0,t]+ ‖y‖E[t0,t]
](1+∑ni=1 δi) ‖x− y‖E[t0,t]
·n∑
i=1
ˆ (t−t0)
0
(t− t0 − s)min(αi−rj ,0) s(r0−γi+δi(r0−βi)) ds
= κ5[
1 + ‖x‖E[t0,t]+ ‖y‖E[t0,t]
](1+∑ni=1 δi) ‖x− y‖E[t0,t]
·n∑
i=1
(t− t0)(1+min(αi,rj)−γi+δi(r0−βi))B(1+min(αi−rj ,0),1+r0−γi+δi(r0−βi))
(2.3.16)
103
and hence
(t− t0)(rj−r0) ∥∥(Φ[t0,τ ],vx)(t) − (Φ[t0,τ ],vy)(t)
∥∥Urj
≤ κ6 (t− t0)mini∈1,...,n[1−(γi−min(αi,rj)+δi(βi−r0))]
[
1 + ‖x‖E[t0,t]+ ‖y‖E[t0,t]
](1+∑ni=1 δi)
· ‖x− y‖E[t0,t]
[n∑
i=1
B(1+min(αi−rj ,0),1+r0−γi+δi(r0−βi))
]
≤ κ7 (t− t0)[1−maxi∈1,...,n(γi−min(αi,rj)+δi(βi−r0))]
[
1 + ‖x‖E[t0,t]+ ‖y‖E[t0,t]
](1+∑ni=1 δi)
· ‖x− y‖E[t0,t]
(2.3.17)
for all j ∈ 0, 1, t ∈ (t0, τ ], x, y ∈ E[t0,τ ], τ ∈ (t0, T ], v ∈ Ur0 . Hence, we get
∥∥Φ[t0,τ ],v(x)− Φ[t0,τ ],v(y)
∥∥E[t0,τ]
≤ κ8 (τ − t0)[1−maxi∈1,...,n(γi−min(αi,r0)+δi(βi−r0))]
·[
1 + ‖x‖E[t0,τ]+ ‖y‖E[t0,τ]
]κ
‖x− y‖E[t0,τ]
(2.3.18)
for all x, y ∈ E[t0,τ ], v ∈ Ur0 , τ ∈ (t0, T ]. Combining (2.3.14) and (2.3.18) results in
∥∥Φ[t0,τ ],v(x)
∥∥E[t0,τ]
≤∥∥Φ[t0,τ ],v(x) − Φ[t0,τ ],v(0)
∥∥E[t0,τ]
+∥∥Φ[t0,τ ],v(0)
∥∥E[t0,τ]
≤ κ8 (τ − t0)[1−maxi∈1,...,n(γi−min(αi,r0)+δi(βi−r0))]
[
1 + ‖x‖E[t0,τ]
]κ
‖x‖E[t0,τ]
+ κ6(1 + ‖v‖Ur0
)
(2.3.19)
for all x ∈ E[t0,τ ], v ∈ Ur0 , τ ∈ (t0, T ]. The assumption
maxi∈1,...,n
[γi −min(αi, r0) + δi (βi − r0)] < 1 (2.3.20)
together with inequalities (2.3.18) and (2.3.19) implies that there exists a mapping ρ : Ur0 → (t0, T ]
such that
∥∥Φ[t0,ρ(v)],v(x)
∥∥E[t0,ρ(v)]
≤ 1 + κ6(1 + ‖v‖Ur0
),
∥∥Φ[t0,ρ(v)],v(x) − Φ[t0,ρ(v)],v(y)
∥∥E[t0,ρ(v)]
≤ 1
2‖x− y‖E[t0,ρ(v)]
(2.3.21)
for all x, y ∈ E[t0,ρ(v)],v, v ∈ Ur0 . This ensures that Φ[t0,ρ(v)],v
(E[t0,ρ(v)],v
)⊂ E[t0,ρ(v)],v for all v ∈ Ur0 .
The Banach fixed point theorem hence proves that there exist unique functions xv ∈ E[t0,ρ(v)],v, v ∈ Ur0 ,
104
such that Φ[t0,ρ(v)],v(xv) = xv for all v ∈ Ur0 . This completes the proof of Lemma 77.
Lemma 77 shows, under suitable assumptions, that there exists a unique local mild solution of
(2.3.1). This solution can be extended to a maximal interval of definition. This is the subject of
the next corollary. It follows directly from Lemma 77 and a standard argument from the ordinary
differential equations literature and its proof is therefore omitted.
Corollary 78 (Maximal mild solutions). Assume the setting in the beginning of Subsection 2.3.1, let
r0, t0 ∈ R, T ∈ (t0,∞), v ∈ Ur0 , n ∈ N, α = (α1, . . . , αn) ∈ Rn, β = (β1, . . . , βn), γ = (γ1, . . . , γn) ∈
[r0,∞)n, δ = (δ1, . . . , δn) ∈ [0,∞)n, r1 ∈ [max(β1, . . . , βn, γ1, . . . , γn), 1 + min(α1, . . . , αn)) with
maxi∈1,...,n[γi − min(αi, r0) + δi(βi − r0)] < 1 and let F = (F1, . . . , Fn) ∈ Cnα,β,γ,δ([t0, T ]). Then
there exist a unique real number τ ∈ (t0, T ] and a unique continuous function x : [t0, τ) → Ur0 satisfy-
ing x|(t0,τ) ∈ C((t0, τ), Ur1), sups∈(t0,t] (s− t0)(r1−r0) ‖x(s)‖Ur1 <∞, limsրτ
[1
(T−s) +‖x(s)‖Ur1]= ∞
and x(t) = eA(t−t0) v +∑n
i=1
´ t
t0eA(t−s) Fi(s, x(s)) ds for all t ∈ (t0, τ).
The next result shows, under suitable assumptions, that the unique maximal mild solution of (2.3.1)
enjoys a bit more regularity than the regularity asserted in Corollary 78.
Corollary 79 (More regularity for maximal mild solutions). Assume the setting in the beginning
of Subsection 2.3.1, let r0, t0 ∈ R, T ∈ (t0,∞), v ∈ Ur0 , n ∈ N, α = (α1, . . . , αn) ∈ Rn, β =
(β1, . . . , βn), γ = (γ1, . . . , γn) ∈ [r0,∞)n, δ = (δ1, . . . , δn) ∈ [0,∞)n with max(β1, . . . , βn, γ1, . . . , γn) <
1 + min(α1, . . . , αn) and maxi∈1,...,n[γi −min(αi, r0) + δi(βi − r0)] < 1 and let F = (F1, . . . , Fn) ∈
Cnα,β,γ,δ([t0, T ]). Then there exist a unique real number τ ∈ (t0, T ] and a unique continuous func-
tion x : [t0, τ) → Ur0 satisfying x|(t0,τ) ∈ C((t0, τ), Ur1), sups∈(t0,t] (s− t0)(r1−r0) ‖x(s)‖Ur1 < ∞,
limsրτ
[‖x(s)‖Umax(β1,...,βn,γ1,...,γn)
+ 1(T−s)
]= ∞ and x(t) = eA(t−t0) v+
∑ni=1
´ t
t0eA(t−s) Fi(s, x(s)) ds
for all t ∈ (t0, τ) and all r1 ∈ [r0, 1 + min(α1, . . . , αn)).
Proof of Corollary 79. First of all, Corollary 78 implies that there exists a unique real number τ ∈
(t0, T ] and a unique continuous function x : [t0, τ) → Ur0 satisfying x|(t0,τ) ∈ C((t0, τ), Umax(β1,...,βn,γ1,...,γn)),
limsրτ
[1
(T−s) + ‖x(s)‖Umax(β1,...,βn,γ1,...,γn)
]= ∞ and
sups∈(t0,t]
(s− t0)(max(β1,...,βn,γ1,...,γn)−r0) ‖x(s)‖Umax(β1,...,βn,γ1,...,γn)
<∞ (2.3.22)
and
x(t) = eA(t−t0) v +n∑
i=1
ˆ t
t0
eA(t−s) Fi(s, x(s)) ds (2.3.23)
105
for all t ∈ (t0, τ). Next we observe similar as in (2.3.7) that (2.3.6) and interpolation (see, e.g.,
Theorem 37.6 in Sell & You [SY02]) imply that
ˆ t
t0
∥∥eA(t−s) Fi(s, x(s))
∥∥Ur1
ds ≤ˆ t
t0
‖eA(t−s)‖L(Uαi ,Ur1 ) ‖Fi(s, x(s))‖Uαi ds
≤ ‖F‖Cnα,β,γ,δ([t0,T ])
[
sups∈(0,T−t0]
‖eAs‖L(Uαi,Ur1 )
smin(αi−r1,0)
]ˆ t
t0
(1 + ‖x(s)‖δiUβi
) (1 + ‖x(s)‖Uγi
)
(t− s)max(r1−αi,0) ds
≤ ‖F‖Cnα,β,γ,δ([t0,T ])
[
sups∈(0,T−t0]
‖eAs‖L(Uαi,Ur1 )
smin(αi−r1,0)
] [
sups∈(t0,t]
(1 + ‖x(s)‖Uγi )(s− t0)
(r0−γi)
]
·[
sups∈(t0,t]
(1 + ‖x(s)‖δiUβi )(s− t0)
δi(r0−βi)
]ˆ t
t0
1
(t− s)max(r1−αi,0) (s− t0)(γi−r0+δi(βi−r0)) ds <∞
(2.3.24)
for all t ∈ [t0, τ), i ∈ 1, 2, . . . , n and all r1 ∈ (−∞, 1 + min(α1, . . . , αn)) where we used γi − r0 +
δi(βi − r0) < 1 for all i ∈ 1, 2, . . . , n in the last line of (2.3.24). This proves that x(t) ∈ Ur1 for all
t ∈ (t0, τ) and all r1 ∈ (−∞, 1 + min(α1, . . . , αn)) and that
sups∈(t0,t]
(s− t0)(r1−r0) ‖x(s)‖Ur1 <∞ (2.3.25)
for all t ∈ (t0, τ) and all r1 ∈ [r0, 1+min(α1, . . . , αn)). Applying Lemma 77 then proves that x|(t0,τ) ∈
C((t0, τ), Ur1) for all r1 ∈ [r0, 1 + min(α1, . . . , αn)). This completes the proof of Lemma 79.
We now present and prove the main result of this subsection. It shows, under suitable assumptions,
that the unique local mild solutions of (2.3.1) depend continuously in an appropriate sense on the
possibly nonlinear vector fields in (2.3.1).
Theorem 80 (Continuous dependence on the data on bounded time intervals). Assume the set-
ting in the beginning of Subsection 2.3.1 and let r0 ∈ R, n ∈ N, α = (α1, . . . , αn) ∈ Rn, β =
(β1, . . . , βn), γ = (γ1, . . . , γn) ∈ [r0,∞)n, δ = (δ1, . . . , δn) ∈ [0,∞)n with max(β1, . . . , βn, γ1, . . . , γn) <
1+min(α1, . . . , αn) and maxi∈1,...,n[γi−min(αi, r0)+(βi−r0)δi
]< 1. Then there exist unique lower
semicontinuous functions τ t0,T : Cnα,β,γ,δ([t0, T ]) × Ur0 → (t0, T ], t0, T ∈ R with t0 < T , and unique
functions xt0,T : Cnα,β,γ,δ([t0, T ]) ×Ur0 → ∪s∈(t0,T ]C([t0, s), Ur0), t0, T ∈ R with t0 < T , which satisfy
xt0,TF,v ∈ C([t0, τt0,TF,v ), Ur0), x
t0,TF,v |
(t0,τt0,T
F,v )∈ C((t0, τ
t0,TF,v ), Ur1), sups∈(t0,t](s− t0)
(r1−r0) ‖xt0,TF,v (s)‖Ur1 <
∞ and
limsրτ
t0,T
F,v
[1
(T−s) + ‖xt0,TF,v (s)‖Umax(β1,...,βn,γ1,...,γn)
]
= ∞ (2.3.26)
106
and
xt0,TF,v (t) = eA(t−t0) v +n∑
i=1
ˆ t
t0
eA(t−s) Fi(s, xt0,TF,v (s)) ds (2.3.27)
for all t ∈ (t0, τt0,TF,v ), v ∈ Ur0 , r1 ∈ [r0, 1 + min(α1, . . . , αn)), F = (F1, . . . , Fn) ∈ Cnα,β,γ,δ([t0, T ]) and
all t0, T ∈ R with t0 < T . In addition, it holds for every t0, T ∈ R with t0 < T , every t ∈ (t0, T ] and
every r1 ∈ [r0, 1 + min(α1, . . . , αn)) that the function
Cnα,β,γ,δ([t0, T ])× Ur0 ∋ (F, v) 7→
xt0,TF,v (t) : t < τF,v
∞ : t ≥ τF,v
∈ Ur1 ∪ ∞ (2.3.28)
is Borel measurable. Moreover, it holds that
limN→∞
sups∈(t0,t]
(s− t0)(r1−r0) ‖xt0,TF1,v1
(s)− xt0,TFN ,vN(s)‖Ur1
+ ‖xt0,TF1,v1(s)− xt0,TFN ,vN
(s)‖Ur0
= 0 (2.3.29)
for all t ∈ (t0, τF,v), r1 ∈ [r0, 1 + min(α1, . . . , αn)), (vN )N∈N ⊂ Ur0 , (FN )N∈N ⊂ Cnα,β,γ,δ([t0, T ]) with
limN→∞ ‖F1 − FN‖Cnα,β,γ,δ([t0,T ]) = limN→∞ ‖v1 − vN‖Ur0 = 0 and all t0, T ∈ R with t0 < T .
Proof of Theorem 80. First of all, observe that Corollary 79 ensures that there exist unique functions
τ t0,T : Cnα,β,γ,δ([t0, T ]) × Ur0 → (t0, T ], t0, T ∈ R with t0 < T , and xt0,T : Cnα,β,γ,δ([t0, T ]) × Ur0 →
∪s∈(t0,T ]C([t0, s), Ur0), t0, T ∈ R with t0 < T , satisfying xt0,TF,v ∈ C([t0, τt0,TF,v ), Ur0), x
t0,TF,v |
(t0,τt0,T
F,v )∈
C((t0, τt0,TF,v ), Ur1), sups∈(t0,t] (s− t0)
(r1−r0) ‖xt0,TF,v (s)‖Ur1 <∞ and
limsրτF,v
[1
(T − s)+ ‖xt0,TF,v (s)‖Umax(β1,...,βn,γ1,...,γn)
]
= ∞ (2.3.30)
and
xt0,TF,v (t) = eA(t−t0) v +n∑
i=1
ˆ t
t0
eA(t−s) Fi(s, xt0,TF,v (s)) ds (2.3.31)
for all t ∈ (t0, τt0,TF,v ), v ∈ Ur0 , F = (F1, . . . , Fn) ∈ Cnα,β,γ,δ([t0, T ]), t0, T ∈ R with t0 < T and all
r1 ∈ [r0, 1 + min(α1, . . . , αn)). It thus remains to prove that τ t0,T , t0, T ∈ R with t0 < T , are lower
semicontinuous and that (2.3.28) and (2.3.29) are fulfilled.
For this let r1 ∈ [max(β1, . . . , βn, δ1, . . . , δn), 1+min(α1, . . . , αn)) be an arbitrary real number and
107
let κ[t0,T ] ∈ [0,∞), t0, T ∈ R with t0 < T , be real numbers defined through
κ[t0,T ] :=1∑
j=0
n∑
i=1
[1
(1 + αi − rj)+B(1+min(αi−rj ,0),1+r0−γi+δi(r0−βi))
]
+
[
2 + n+ r1 − r0 + |T − t0|+n∑
i=1
δi
](4+|r0|+|r1|+maxi∈1,...,n|αi|)
+ maxj∈0,1
maxθ∈r0,r1,α1,...,αn
supt∈(t0,T ]
[
(t− t0)max(rj−θ,0) ‖eA(t−t0)‖L(Uθ,Urj )
]
+ maxθ∈β1,...,βn∪γ1,...,γn
supv∈Ur1v 6=0
1 +
‖v‖Uθ‖v‖
(θ−r0)
(r1−r0)
Ur1‖v‖
(r1−θ)
(r1−r0)
Ur0
+‖v‖Ur0‖v‖Ur1
(1+∑ni=1 δi)
<∞
(2.3.32)
for all t0, T ∈ R with t0 < T where B : (0,∞)2 → (0,∞) is the Beta function defined through
B(x,y) :=´ 1
0(1− s)
(x−1)s(y−1) ds for all x, y ∈ (0,∞). Then observe that
∥∥xt0,TF,v (t)− xt0,T
F ,v(t)∥∥Urj
≤∥∥eA(t−t0)∥∥
L(Urk ,Urj )‖v − v‖Urk
+
n∑
i=1
ˆ t
t0
∥∥eA(t−s)∥∥
L(Uαi ,Urj )
∥∥Fi(s, x
t0,TF,v (s))− Fi(s, x
t0,T
F ,v(s))
∥∥Uαi
ds
≤ κ[t0,T ] (t− t0)min(rk−rj ,0) ‖v − v‖Urk
+
n∑
i=1
ˆ t
t0
κ[t0,T ] (t− t0)min(αi−rj ,0) ‖Fi(s, xt0,TF,v (s))− Fi(s, x
t0,TF,v (s))‖Uαi ds
+
n∑
i=1
ˆ t
t0
κ[t0,T ] (t− t0)min(αi−rj ,0) ‖Fi(s, xt0,TF,v (s))− Fi(s, x
t0,T
F ,v(s))‖Uαi ds
(2.3.33)
and inequality (2.3.6) therefore implies that
∥∥xt0,TF,v (t)− xt0,T
F ,v(t)∥∥Urj
≤ κ[t0,T ] (t− t0)min(rk−rj,0) ‖v − v‖Urk
+ κ[t0,T ] ‖F − F‖Cnα,β,γ,δ([t0,T ])
·n∑
i=1
ˆ t
t0
(t− s)min(αi−rj ,0) (1 + ‖xt0,TF,v (s)‖δiβi
) (1 + ‖xt0,TF,v (s)‖Uγi
)ds
+ κ[t0,T ] ‖F‖Cnα,β,γ,δ
([t0,T ])
·n∑
i=1
ˆ t
t0
(t− s)min(αi−rj ,0)
(
1 + ‖xt0,TF,v (s)‖δiUβi + ‖xt0,TF ,v
(s)‖δiUβi)
· ‖xt0,TF,v (s)− xt0,TF ,v
(s)‖Uγi ds
(2.3.34)
108
and the definition of κ[t0,T ] hence shows that
∥∥xt0,TF,v (t)− xt0,T
F ,v(t)∥∥Urj
≤ κ[t0,T ] (t− t0)min(rk−rj,0) ‖v − v‖Urk
+[κ[t0,T ]
]3 ‖F − F‖Cnα,β,γ,δ
([t0,T ])
n∑
i=1
ˆ t
t0
(t− s)min(αi−rj ,0)
·[
1 + ‖xt0,TF,v (s)‖(r1−βi)δi(r1−r0)
Ur0‖xt0,TF,v (s)‖
(βi−r0)δi(r1−r0)
Ur1
] [
1 + ‖xt0,TF,v (s)‖(r1−γi)
(r1−r0)
Ur0‖xt0,TF,v (s)‖
(γi−r0)
(r1−r0)
Ur1
]
ds
+[κ[t0,T ]
]3 ‖F‖Cnα,β,γ,δ([t0,T ])
n∑
i=1
ˆ t
t0
(t− s)min(αi−rj ,0)
·[
1 + ‖xt0,TF,v (s)‖(r1−βi)δi(r1−r0)
Ur0‖xt0,TF,v (s)‖
(βi−r0)δi(r1−r0)
Ur1+ ‖xt0,T
F ,v(s)‖
(r1−βi)δi(r1−r0)
Ur0‖xt0,T
F ,v(s)‖
(βi−r0)δi(r1−r0)
Ur1
]
· ‖xt0,TF,v (s)− xt0,TF ,v
(s)‖(r1−γi)
(r1−r0)
Ur0‖xt0,TF,v (s)− xt0,T
F ,v(s)‖
(γi−r0)
(r1−r0)
Ur1ds
(2.3.35)
for all j, k ∈ 0, 1, t ∈ (t0, τt0,TF,v ) ∩ (t0, τ
t0,T
F ,v), v, v ∈ Ur0 , F, F ∈ Cnα,β,γ,δ([t0, T ]) and all t0, T ∈ R with
t0 < T . This, in particular, implies that
∥∥xt0,TF,v (t)− xt0,T
F ,v(t)∥∥Ur1
≤ κ[t0,T ] ‖v − v‖Ur1
+[κ[t0,T ]
]5 ‖F − F‖Cnα,β,γ,δ([t0,T ])
n∑
i=1
ˆ t
t0
(t− s)min(αi−r1,0)
·[
1 + ‖xt0,TF,v (s)‖δiUr1] [
1 + ‖xt0,TF,v (s)‖Ur1]
ds
+[κ[t0,T ]
]5 ‖F‖Cnα,β,γ,δ([t0,T ])
n∑
i=1
ˆ t
t0
(t− s)min(αi−r1,0)
·[
1 + ‖xt0,TF,v (s)‖δiUr1 + ‖xt0,TF ,v
(s)‖δiUr1]
‖xt0,TF,v (s)− xt0,TF ,v
(s)‖Ur1 ds
(2.3.36)
and the estimates(1+|x|δi
)(1+|x|
)≤ κ (1 + |x|)(2+
∑nj=1 δj) and
(1+|x|δi+|y|δi
)≤ κ (1 + |x|+ |y|)(1+
∑nj=1 δj)
109
for all x, y ∈ R and all i ∈ 1, . . . , n hence give
∥∥xt0,TF,v (t)− xt0,T
F ,v(t)∥∥Ur1
≤ κ[t0,T ] ‖v − v‖Ur1
+[κ[t0,T ]
]6 ‖F − F‖Cnα,β,γ,δ([t0,T ])
[
1 + sups∈[t0,t]
‖xt0,TF,v (s)‖Ur1
](2+∑ni=1 δi)
·n∑
i=1
ˆ t
t0
(t− s)min(αi−r1,0) ds
+[κ[t0,T ]
]6‖F‖Cnα,β,γ,δ([t0,T ])
[
1 + sups∈[t0,t]
‖xt0,TF,v (s)‖Ur1 + sups∈[t0,t]
‖xt0,TF ,v
(s)‖Ur1
](1+∑ni=1 δi)
·n∑
i=1
ˆ t
t0
(t− s)min(αi−r1,0) ‖xt0,TF,v (s)− xt0,T
F ,v(s)‖Ur1 ds
(2.3.37)
for all t ∈ (t0, τt0,TF,v ) ∩ (t0, τ
t0,T
F ,v), v, v ∈ Ur1 , F, F ∈ Cnα,β,γ,δ([t0, T ]) and all t0, T ∈ R with t0 < T .
Therefore, we obtain that
∥∥xt0,TF,v (t)− xt0,T
F ,v(t)∥∥Ur1
≤ κ[t0,T ] ‖v − v‖Ur1
+[κ[t0,T ]
]8 ‖F − F‖Cnα,β,γ,δ([t0,T ])
[
1 + sups∈[t0,t]
‖xt0,TF,v (s)‖Ur1
](2+∑ni=1 δi)
·ˆ t
t0
(t− s)min(α1−r1,...,αn−r1,0) ds
+[κ[t0,T ]
]8 ‖F‖Cnα,β,γ,δ([t0,T ])
[
1 + sups∈[t0,t]
‖xt0,TF,v (s)‖Ur1 + sups∈[t0,t]
‖xt0,TF ,v
(s)‖Ur1
](1+∑ni=1 δi)
·ˆ t
t0
(t− s)min(α1−r1,...,αn−r1,0) ‖xt0,TF,v (s)− xt0,T
F ,v(s)‖Ur1 ds
(2.3.38)
and hence
∥∥xt0,TF,v (t)− xt0,T
F ,v(t)∥∥Ur1
≤[κ[t0,T ]
]10[
‖v − v‖Ur1 + ‖F − F‖Cnα,β,γ,δ
([t0,T ])
][
1 + sups∈[t0,t]
‖xt0,TF,v (s)‖Ur1
](2+∑ni=1 δi)
+[κ[t0,T ]
]8 ‖F‖Cnα,β,γ,δ([t0,T ])
[
1 + sups∈[t0,t]
‖xt0,TF,v (s)‖Ur1 + sups∈[t0,t]
‖xt0,TF ,v
(s)‖Ur1
](1+∑ni=1 δi)
·ˆ t
t0
(t− s)min(α1−r1,...,αn−r1,0) ‖xt0,TF,v (s)− xt0,T
F ,v(s)‖Ur1 ds
(2.3.39)
for all t ∈ (t0, τt0,TF,v ) ∩ (t0, τ
t0,T
F ,v), v, v ∈ Ur1 , F, F ∈ Cnα,β,γ,δ([t0, T ]) and all t0, T ∈ R with t0 < T . A
110
generalization of Gronwall’s lemma (see Lemma 7.1.1 in Henry [Hen81]) therefore implies
sups∈[t0,t]
∥∥xt0,TF,v (s)− xt0,T
F ,v(s)∥∥Ur1
≤ Emin(α1−r1,...,αn−r1,0)
[
[κ[t0,T ]
]9 ‖F‖Cnα,β,γ,δ
([t0,T ])
·[
1 + sups∈[t0,t]
‖xt0,TF,v (s)‖Ur1 + sups∈[t0,t]
‖xt0,TF ,v
(s)‖Ur1](1+
∑ni=1 δi)
]
[κ[t0,T ]
]10
·[
‖v − v‖Ur1 + ‖F − F‖Cnα,β,γ,δ([t0,T ])
] [
1 + sups∈[t0,t]
‖xt0,TF,v (s)‖Ur1](2+
∑ni=1 δi)
(2.3.40)
for all t ∈ (t0, τt0,TF,v )∩ (t0, τ
t0,T
F ,v), v, v ∈ Ur1 , F, F ∈ Cnα,β,γ,δ([t0, T ]) and all t0, T ∈ R with t0 < T where
Er : [0,∞) → [0,∞), r ∈ (−1, 0], is a family of functions defined through Er(x) :=∑∞
n=0(x·Γ(r+1))n
Γ(n(r+1)+1)
for all x ∈ [0,∞) and all r ∈ (−1, 0]. As in (2.3.11) and (2.3.12), we now define sets E[t0,T ], t0, T ∈ R
with t0 < T , and functions ‖·‖E[t0,T ]: E[t0,T ] → [0,∞), t0, T ∈ R with t0 < T , by
E[t0,T ] :=
y ∈ C([t0, T ], Ur0) :
y|(t0,T ] ∈ C((t0, T ], Ur1) and
supt∈(t0,T ] (t− t0)(r1−r0) ‖y(t)‖Ur1 <∞
(2.3.41)
for all t0, T ∈ R with t0 < T and by ‖y‖E[t0,T ]:=∑1
j=0 supt∈(t0,τ ] (t− t0)(rj−r0) ‖y(t)‖Urj for all
y ∈ E[t0,T ], t0, T ∈ R with t0 < T . Then we get from (2.3.35) that
∥∥xt0,TF,v (t)− xt0,T
F ,v(t)∥∥Urj
≤ κ[t0,T ] (t− t0)(r0−rj) ‖v − v‖Ur0
+[κ[t0,T ]
]4 ‖F − F‖Cnα,β,γ,δ([t0,T ])
n∑
i=1
ˆ t
t0
(t− s)min(αi−rj ,0) (s− t0)
(r0−γi+δi(r0−βi))
·[
1 + ‖xt0,TF,v (s)‖(r1−βi)δi(r1−r0)
Ur0(s− t0)
(βi−r0)δi ‖xt0,TF,v (s)‖(βi−r0)δi(r1−r0)
Ur1
]
·[
1 + ‖xt0,TF,v (s)‖(r1−γi)
(r1−r0)
Ur0(s− t0)
(γi−r0) ‖xt0,TF,v (s)‖(γi−r0)
(r1−r0)
Ur1
]
ds
+[κ[t0,T ]
]4 ‖F‖Cnα,β,γ,δ([t0,T ])
n∑
i=1
ˆ t
t0
(t− s)min(αi−rj ,0) (s− t0)
(r0−γi+δi(r0−βi))
·[
1 + ‖xt0,TF,v (s)‖(r1−βi)δi(r1−r0)
Ur0(s− t0)
(βi−r0)δi ‖xt0,TF,v (s)‖(βi−r0)δi(r1−r0)
Ur1
+ ‖xt0,TF ,v
(s)‖(r1−βi)δi(r1−r0)
Ur0(s− t0)
(βi−r0)δi ‖xt0,TF ,v
(s)‖(βi−r0)δi(r1−r0)
Ur1
]
· ‖xt0,TF,v (s)− xt0,TF ,v
(s)‖(r1−γi)
(r1−r0)
Ur0(s− t0)
(γi−r0) ‖xt0,TF,v (s)− xt0,TF ,v
(s)‖(γi−r0)
(r1−r0)
Ur1ds
(2.3.42)
111
and therefore
∥∥xt0,TF,v (t)− xt0,T
F ,v(t)∥∥Urj
≤ κ[t0,T ] (t− t0)(r0−rj) ‖v − v‖Ur0
+[κ[t0,T ]
]4 ‖F − F‖Cnα,β,γ,δ([t0,T ])
n∑
i=1
ˆ t
t0
(t− s)min(αi−rj ,0) (s− t0)
(r0−γi+δi(r0−βi)) ds
·[
1 + ‖xt0,TF,v |[t0,t]‖δiE[t0,t]
] [
1 + ‖xt0,TF,v |[t0,t]‖E[t0,t]
]
+[κ[t0,T ]
]4 ‖F‖Cnα,β,γ,δ([t0,T ])
n∑
i=1
ˆ t
t0
(t− s)min(αi−rj ,0) (s− t0)
(r0−γi+δi(r0−βi)) ds
·[
1 + ‖xt0,TF,v |[t0,t]‖δiE[t0,t]+ ‖xt0,T
F ,v|[t0,t]‖δiE[t0,t]
]
‖(xt0,TF,v − xt0,TF ,v
)|[t0,t]‖E[t0,t]
(2.3.43)
and the estimates(1+|x|δi
)(1+|x|
)≤ κ (1 + |x|)(2+
∑nj=1 δj) and
(1+|x|δi+|y|δi
)≤ κ (1 + |x|+ |y|)(1+
∑nj=1 δj)
for all x, y ∈ R and all i ∈ 1, . . . , n hence show that
∥∥xt0,TF,v (t)− xt0,T
F ,v(t)∥∥Urj
≤ κ[t0,T ] (t− t0)(r0−rj) ‖v − v‖Ur0
+[κ[t0,T ]
]5 ‖F − F‖Cnα,β,γ,δ([t0,T ])
[
1 + ‖xt0,TF,v |[t0,t]‖E[t0,t]
](2+∑ni=1 δi)
·n∑
i=1
ˆ t
t0
(t− s)min(αi−rj ,0) (s− t0)(r0−γi+δi(r0−βi)) ds
+[κ[t0,T ]
]5 ‖F‖Cnα,β,γ,δ([t0,T ])
[
1 + ‖xt0,TF,v |[t0,t]‖E[t0,t]+ ‖xt0,T
F ,v|[t0,t]‖E[t0,t]
](1+∑ni=1 δi)
· ‖(xt0,TF,v − xt0,TF ,v
)|[t0,t]‖E[t0,t]
n∑
i=1
ˆ t
t0
(t− s)min(αi−rj ,0) (s− t0)
(r0−γi+δi(r0−βi)) ds
(2.3.44)
for all j ∈ 0, 1, t ∈ (t0, τt0,TF,v ) ∩ (t0, τ
t0,T
F ,v), v, v ∈ Ur0 , F, F ∈ Cnα,β,γ,δ([t0, T ]) and all t0, T ∈ R with
t0 < T . The estimate
n∑
i=1
ˆ t
t0
(t− s)min(αi−rj ,0) (s− t0)
(r0−γi+δi(r0−βi)) ds
=
n∑
i=1
(t− t0)(1+min(αi−rj ,0)+r0−γi+δi(r0−βi))B(1+min(αi−rj ,0),1+r0−γi+δi(r0−βi))
≤ κ[t0,T ] (t− t0)[r0−rj+mini∈1,...,n(1+min(αi,r0)−γi+δi(r0−βi))]
·n∑
i=1
B(1+min(αi−rj ,0),1+r0−γi+δi(r0−βi))
≤[κ[t0,T ]
]2(t− t0)
[r0−rj+mini∈1,...,n(1+min(αi,r0)−γi+δi(r0−βi))]
(2.3.45)
112
for all j ∈ 0, 1, t ∈ (t0, T ] and all t0, T ∈ R with t0 < T therefore proves that
(t− t0)(rj−r0) ∥∥xt0,TF,v (t)− xt0,T
F ,v(t)∥∥Urj
≤ κ[t0,T ] ‖v − v‖Ur0
+[κ[t0,T ]
]7 ‖F − F‖Cnα,β,γ,δ([t0,T ])
[
1 + ‖xt0,TF,v |[t0,t]‖E[t0,t]
](2+∑ni=1 δi)
· (t− t0)mini∈1,...,n(1+min(αi,r0)−γi+δi(r0−βi))
+[κ[t0,T ]
]7 ‖F‖Cnα,β,γ,δ([t0,T ])
[
1 + ‖xt0,TF,v |[t0,t]‖E[t0,t]+ ‖xt0,T
F ,v|[t0,t]‖E[t0,t]
](1+∑ni=1 δi)
· ‖(xt0,TF,v − xt0,TF ,v
)|[t0,t]‖E[t0,t](t− t0)
mini∈1,...,n(1+min(αi,r0)−γi+δi(r0−βi))
(2.3.46)
for all j ∈ 0, 1, t ∈ (t0, τt0,TF,v ) ∩ (t0, τ
t0,T
F ,v), v, v ∈ Ur0 , F, F ∈ Cnα,β,γ,δ([t0, T ]) and all t0, T ∈ R with
t0 < T . Hence, we obtain
‖(xt0,TF,v − xt0,TF ,v
)|[t0,t]‖E[t0,t]≤[κ[t0,T ]
]2 ‖v − v‖Ur0+[κ[t0,T ]
]8 ‖F − F‖Cnα,β,γ,δ([t0,T ])
[
1 + ‖xt0,TF,v |[t0,t]‖E[t0,t]
](2+∑ni=1 δi)
· (t− t0)mini∈1,...,n(1+min(αi,r0)−γi+δi(r0−βi))
+[κ[t0,T ]
]8 ‖F‖Cnα,β,γ,δ([t0,T ])
[
1 + ‖xt0,TF,v |[t0,t]‖E[t0,t]+ ‖xt0,T
F ,v|[t0,t]‖E[t0,t]
](1+∑ni=1 δi)
· ‖(xt0,TF,v − xt0,TF ,v
)|[t0,t]‖E[t0,t](t− t0)
mini∈1,...,n(1+min(αi,r0)−γi+δi(r0−βi))
(2.3.47)
for all t ∈ (t0, τt0,TF,v ) ∩ (t0, τ
t0,T
F ,v), v, v ∈ Ur0 , F, F ∈ Cnα,β,γ,δ([t0, T ]) and all t0, T ∈ R with t0 < T .
Rearranging finally results in
‖(xt0,TF,v − xt0,TF ,v
)|[t0,t]‖E[t0,t]
[
1− (t− t0)mini∈1,...,n(1+min(αi,r0)−γi+δi(r0−βi))
·[κ[t0,T ]
]8 ‖F‖Cnα,β,γ,δ([t0,T ])
[
1 + ‖xt0,TF,v |[t0,t]‖E[t0,t]+ ‖xt0,T
F ,v|[t0,t]‖E[t0,t]
](1+∑ni=1 δi)
]
≤[κ[t0,T ]
]9[
‖F − F‖Cnα,β,γ,δ([t0,T ]) + ‖v − v‖Ur0] [
1 + ‖xt0,TF,v |[t0,t]‖E[t0,t]
](2+∑ni=1 δi)
(2.3.48)
for all t ∈ (t0, τt0,TF,v ) ∩ (t0, τ
t0,T
F ,v), v, v ∈ Ur0 , F, F ∈ Cnα,β,γ,δ([t0, T ]) and all t0, T ∈ R with t0 < T .
We now use (2.3.40) and (2.3.48) to prove (2.3.29). For this let t0, T ∈ R be real numbers with t0 <
T , let ε ∈ (0, 1] be a real number defined through ε := mini∈1,...,n (1 + min(αi, r0)− γi + δi(r0 − βi))
and let (vN )N∈N ⊂ Ur0 and FN = (FN,1, . . . , FN,n) ∈ Cnα,β,γ,δ([t0, T ]) , N ∈ N, be sequences with
limN→∞ ‖v1 − vN‖Ur0 = limN→∞ ‖F1 − FN‖Cnα,β,γ,δ([t0,T ]) = 0 and ‖F1 − FN‖Cnα,β,γ,δ([t0,T ]) ≤ 1 for all
113
N ∈ N. Then observe that (2.3.48) ensures that
∥∥(xt0,TF1,v1
− xt0,TFN ,vN)|[t0,t]
∥∥E[t0,t]
[
1−[κ[t0,T ]
]8(t− t0)
ε[
2 + ‖xt0,TF1,v1|[t0,t]‖E[t0,t]
+ ‖xt0,TFN ,vN|[t0,t]‖E[t0,t]
+ ‖F1‖Cnα,β,γ,δ([t0,T ])
](2+∑ni=1 δi)
]
≤[κ[t0,T ]
]9
·[
‖F1 − FN‖Cnα,β,γ,δ([t0,T ]) + ‖v1 − vN‖Ur0] [
1 + ‖xt0,TF1,v1|[t0,t]‖E[t0,t]
](2+∑ni=1 δi)
(2.3.49)
for all t ∈ (t0, τt0,TF1,v1
) ∩ (t0, τt0,TFN ,vN
) and all N ∈ N. This implies that
‖(xt0,TF1,v1− xt0,TFN ,vN
)|[t0,t]‖E[t0,t]
·[
1−[κ[t0,T ]
]8(t− t0)
ε[
4 + 2 ‖xt0,TF1,v1|[t0,t]‖E[t0,t]
+ ‖F1‖Cnα,β,γ,δ([t0,T ])
](2+∑ni=1 δi)
]
≤[κ[t0,T ]
]9[
‖F1 − FN‖Cnα,β,γ,δ([t0,T ]) + ‖v1 − vN‖Ur0] [
1 + ‖xt0,TF1,v1|[t0,t]‖E[t0,t]
](2+∑ni=1 δi)
(2.3.50)
for all t ∈s ∈ (t0, τ
t0,TF1,v1
) ∩ (t0, τt0,TFN ,vN
) : ‖xt0,TFN ,vN|[t0,t]‖E[t0,s]
≤ 2 + ‖xt0,TF1,v1|[t0,t]‖E[t0,s]
and all N ∈ N.
In the next step let t ∈ (t0, τt0,TF1,v1
) and N ∈ N be real numbers with the property that
[κ[t0,T ]
]8(t− t0)
ε[
4 + 2 ‖xt0,TF1,v1|[t0,t]‖E[t0,t]
+ ‖F1‖Cnα,β,γ,δ([t0,T ])
](2+∑ni=1 δi) ≤ 1
2 (2.3.51)
for all t ∈ (t0, t] and with the property that
[κ[t0,T ]
]9[
‖F1 − FN‖Cnα,β,γ,δ([t0,T ]) + ‖v1 − vN‖Ur0]
·[
1 + ‖xt0,TF1,v1|[t0,t]‖E[t0,t]
](2+∑ni=1 δi) ≤ 1
2
(2.3.52)
for all N ∈ N, N + 1, . . . =: N. Then we obtain from (2.3.50) that
‖(xt0,TF1,v1− xt0,TFN ,vN
)|[t0,t]‖E[t0,t]≤ 2
[κ[t0,T ]
]9
·[
‖F1 − FN‖Cnα,β,γ,δ([t0,T ]) + ‖v1 − vN‖Ur0] [
1 + ‖xt0,TF1,v1|[t0,t]‖E[t0,t]
](2+∑ni=1 δi) ≤ 1
(2.3.53)
for all t ∈s ∈ (t0, t]∩ (t0, τ
t0,TFN ,vN
) : ‖xt0,TFN ,vN|[t0,t]‖E[t0,s]
≤ 2+ ‖xt0,TF1,v1|[t0,t]‖E[t0,s]
and all N ∈ N. This
implies that
‖(xt0,TF1,v1− xt0,TFN ,vN
)|[t0,t]‖E[t0,t]≤ 2
[κ[t0,T ]
]9
·[
‖F1 − FN‖Cnα,β,γ,δ([t0,T ]) + ‖v1 − vN‖Ur0] [
1 + ‖xt0,TF1,v1|[t0,t]‖E[t0,t]
](2+∑ni=1 δi)
(2.3.54)
114
for all N ∈ N and we hence get
limN→∞
‖(xt0,TF1,v1− xt0,TFN ,vN
)|[t0,t]‖E[t0,t]= 0. (2.3.55)
In the next step we define vN ∈ Ur1 , N ∈ N∪1, through vN := xt0,TFN ,vN(t) for all N ∈ N∪1 and we
define FN ∈ Cnα,β,γ,δ([t, T ]), N ∈ N∪1, through FN := (FN,1|[t,T ], . . . , FN,n|[t,T ]) for all N ∈ N∪1.
Note that (vN )N∈N∪1 is well-defined since t < τ t0,TFN ,vNfor all N ∈ N ∪ 1. Furthermore, we obtain
from (2.3.40) that
sups∈[t,t]
∥∥xt,T
F1,v1(s)− xt,T
FN ,vN(s)∥∥Ur1
≤ Emin(α1−r1,...,αn−r1,0)
[
[κ[t,T ]]9 ‖FN‖Cnα,β,γ,δ([t,T ])
·[
1 + sups∈[t,t]
‖xt,TF1,v1
(s)‖Ur1 + sups∈[t,t]
‖xt,TFN ,vN
(s)‖Ur1](1+
∑ni=1 δi)
]
[κ[t,T ]]10
·[
‖v1 − vN‖Ur1 + ‖F1 − FN‖Cnα,β,γ,δ([t,T ])
][
1 + sups∈[t,t]
‖xt,TF1,v1
(s)‖Ur1
](2+∑ni=1 δi)
(2.3.56)
for all t ∈ (t, τ t,TF1,v1
) ∩ (t, τ t,TFN ,vN
) and all N ∈ N. This implies that
sups∈[t,t]
∥∥xt,T
F1,v1(s)− xt,T
FN ,vN(s)∥∥Ur1
≤ [κ[t0,T ]]10 · Emin(α1−r1,...,αn−r1,0)
[
[κ[t0,T ]]9
·[
2 + sups∈[t,t]
‖xt,TF1,v1
(s)‖Ur1 + sups∈[t,t]
‖xt,TFN ,vN
(s)‖Ur1 + ‖F1‖Cnα,β,γ,δ
([t0,T ])
](2+∑ni=1 δi)
]
·[
‖v1 − vN‖Ur1 + ‖F1 − FN‖Cnα,β,γ,δ([t0,T ])
][
1 +‖xt0,TF1,v1
|[t0,t]‖E[t0,t]
(t− t0)(r1−r0)
](2+∑ni=1 δi)
(2.3.57)
and therefore
sups∈[t,t]
∥∥xt0,TF1,v1
(s)− xt0,TFN ,vN(s)∥∥Ur1
≤ [κ[t0,T ]]10 ·Emin(α1−r1,...,αn−r1,0)
[
[κ[t0,T ]]9
·[
4 + 2 sups∈[t,t]
‖xt0,TF1,v1(s)‖Ur1 + ‖F1‖Cnα,β,γ,δ([t0,T ])
](2+∑ni=1 δi)
]
·[
‖v1 − vN‖Ur1 + ‖F1 − FN‖Cnα,β,γ,δ([t0,T ])
][
1 +‖xt0,TF1,v1
|[t0,t]‖E[t0,t]
(t− t0)(r1−r0)
](2+∑ni=1 δi)
(2.3.58)
for all t ∈s ∈ (t, τ t0,TF1,v1
) ∩ (t, τ t0,TFN ,vN) : supu∈[t,s] ‖xt0,TFN ,vN
(u)‖Ur1 ≤ 2+ supu∈[t,s] ‖xt0,TF1,v1(u)‖Ur1
and
all N ∈ N. In the next step we observe that (2.3.55) proves that there exists a non-decreasing family
115
Nt ∈ N, t ∈ (t, τ t0,TF1,v1), of natural numbers such that
Emin(α1−r1,...,αn−r1,0)
[[
4 + 2 sups∈[t,t]
‖xt0,TF1,v1(s)‖Ur1 + ‖F1‖Cnα,β,γ,δ([t0,T ])
](2+∑ni=1 δi)
· [κ[t0,T ]]9
]
[κ[t0,T ]]10[
‖v1 − vN‖Ur1 + ‖F1 − FN‖Cnα,β,γ,δ([t0,T ])
]
·[
1 +‖xt0,TF1,v1
|[t0,t]‖E[t0,t]
(t− t0)(r1−r0)
](2+∑ni=1 δi)
≤ 1
(2.3.59)
for all N ∈ Nt, Nt + 1, . . . and all t ∈ (t, τ t0,TF1,v1). Combining this with (2.3.58) results in
sups∈[t,t]
‖xt0,TF1,v1(s)− xt0,TFN ,vN
(s)‖Ur1 ≤ Emin(α1−r1,...,αn−r1,0)
[
[κ[t0,T ]]9
·[
4 + 2 sups∈[t,t]
‖xt0,TF1,v1(s)‖Ur1 + ‖F1‖Cnα,β,γ,δ([t0,T ])
](2+∑ni=1 δi)
]
[κ[t0,T ]]10
·[
‖v1 − vN‖Ur1 + ‖F1 − FN‖Cnα,β,γ,δ([t0,T ])
][
1 +‖xt0,TF1,v1
|[t0,t]‖E[t0,t]
(t− t0)(r1−r0)
](2+∑ni=1 δi)
(2.3.60)
for all N ∈ Nt, Nt + 1, . . . and all t ∈ (t, τ t0,TF1,v1). Inequality (2.3.60) implies that τ t0,T is lower
semicontinuous and combining (2.3.60) with (2.3.54) proves that
limN→∞
sups∈(t0,t]
(s− t0)(r1−r0) ‖xt0,TF1,v1
(s)− xt0,TFN ,vN(s)‖Ur1
+ ‖xt0,TF1,v1(s)− xt0,TFN ,vN
(s)‖Ur0
= 0 (2.3.61)
for all t ∈ (t0, τF1,v1). Interpolation (see, e.g., Theorem 37.6 in Sell & You [SY02]) hence implies that
(2.3.29) is fulfilled. Since every lower semicontinuous function is Borel measurable, we obtain that τ t0,T
is Borel measurable. Therefore, we get for every t ∈ [t0, T ] that the sets (F, v) ∈ Cnα,β,γ,δ([t0, T ]) ×
Ur0 : τt0,TF,v > t and (F, v) ∈ Cnα,β,γ,δ([t0, T ]) × Ur0 : τ
t0,TF,v ≤ t are Borel measurable subsets of
Cnα,β,γ,δ([t0, T ])× Ur0 and (2.3.29) implies for every t ∈ (t0, T ] and every r ∈ [r0, r1] that the mapping
(F, v) ∈ Cnα,β,γ,δ([t0, T ]) × Ur0 : τt0,TF,v > t ∋ (F, v) 7→ xF,v(t) ∈ Ur is continuous and, in particular,
Borel measurable. These two facts imply (2.3.28) and this completes the proof of Theorem 80.
Theorem 80 investigates solutions of (2.3.1) on a bounded time interval. The next corollary extends
this result to unbounded time intervals.
Corollary 81 (Continuous dependence on the data on unbounded time intervals). Assume the set-
116
ting in the beginning of Subsection 2.3.1 and let t0, r0 ∈ R, n ∈ N, α = (α1, . . . , αn) ∈ Rn, β =
(β1, . . . , βn), γ = (γ1, . . . , γn) ∈ [r0,∞)n, δ = (δ1, . . . , δn) ∈ [0,∞)n with max(β1, . . . , βn, γ1, . . . , γn) <
1 + min(α1, . . . , αn) and
maxi∈1,...,n
[γi −min(αi, r0) + δi(βi − r0)
]< 1. (2.3.62)
Then there exist a unique lower semicontinuous function τ : Cnα,β,γ,δ([t0,∞)) × Ur0 → (t0,∞] and a
unique function x : Cnα,β,γ,δ([t0,∞))×Ur0 → ∪s∈(t0,∞]C([t0, s), Ur0) satisfying xF,v ∈ C([t0, τF,v), Ur0),
xF,v|(t0,τF,v) ∈ C((t0, τF,v), Ur1), sups∈(t0,t](s− t0)(r1−r0) ‖xF,v(s)‖Ur1 <∞ and
limsրτF,v
[τF,v + ‖xF,v(s)‖Umax(β1,...,βn,γ1,...,γn)
]= ∞ (2.3.63)
and
xF,v(t) = eA(t−t0) v +n∑
i=1
ˆ t
t0
eA(t−s) Fi(s, xF,v(s)) ds (2.3.64)
for all t ∈ (t0, τF,v), v ∈ Ur0 , r1 ∈ [r0, 1+min(α1, . . . , αn)) and all F = (F1, . . . , Fn) ∈ Cnα,β,γ,δ([t0,∞)).
In addition, it holds for every t ∈ (t0,∞) and every r1 ∈ [r0, 1 + min(α1, . . . , αn)) that the function
Cnα,β,γ,δ([t0,∞))× Ur0 ∋ (F, v) 7→
xF,v(t) : t < τF,v
∞ : t ≥ τF,v
∈ Ur1 ∪ ∞ (Measurability property)
is Borel measurable. Moreover, it holds that
limN→∞
sups∈(t0,t]
(s− t0)(r1−r0) ‖xF1,v1(s)− xFN ,vN (s)‖Ur1
+ ‖xF1,v1(s)− xFN ,vN (s)‖Ur0
= 0 (Continuity property)
for all t ∈ (t0, τF1,v1), r1 ∈ [r0, 1 + min(α1, . . . , αn)), (vN )N∈N ⊂ Ur0 , (FN )N∈N ⊂ Cnα,β,γ,δ([t0,∞))
with limN→∞ dCnα,β,γ,δ([t0,∞))(F1, FN ) = limN→∞ ‖v1 − vN‖Ur0 = 0.
Corollary 81 follows immediately from Theorem 80 and its proof is therefore omitted.
117
2.3.2 SPDEs with space-time white noise and polynomial nonlinearities in
two space dimensions
The aim of this subsection is to prove local existence and uniqueness of mild solutions of SPDEs in
two space dimensions with polynomial nonlinearities of the form
dXt =[Xt + κn(t) : (Xt)
n : + . . .+ κ2(t) : (Xt)2 : +κ1(t)Xt + κ0(t)
]dt+ dWt (2.3.65)
for t ∈ [0,∞) with periodic boundary conditions on (0, 2π)2 where n ∈ N is an arbitrary natural
number, where κ0, κ1, . . . , κn ∈ C([0,∞),R) are arbitrary continuous functions, where (Wt)t≥0 is
a cylindrical I-Wiener process and where : (Xt)2 :, . . . , : (Xt)
n : are suitable renormalizations of
(Xt)2, . . . , (Xt)
n for t ∈ [0,∞). The precise result is formulated in the following theorem.
Theorem 82 (Polynomial nonlinearities in two space dimensions). Let (Ω,F ,P) be a probability space,
let n ∈ N, t0 ∈ R, κ0, κ1, . . . , κn ∈ C([t0,∞),R), η ∈ (− 2n , 0), let V = : (V )1 :, : (V )2 :, . . . , : (V )n :
: [t0,∞)×Ω → ∩r∈(−∞,0) CrP([0, 2π]2,R) be stochastic processes with continuous sample paths given by
Propositions 65 and 66 and let ξ : Ω → CηP([0, 2π]2,R) be a random variable. Then there exists a unique
random variable τ : Ω → (t0,∞] and a unique stochastic process X : [t0,∞)×Ω → CηP([0, 2π]2,R)∪∞
such that for every ω ∈ Ω it holds that Xt(ω) = ∞ for all t ∈ [τ(ω),∞), that
(Xs(ω))s∈[t0,τ(ω)) ∈ C([t0, τ(ω)), CηP ([0, 2π]2,R)
), (2.3.66)
(Xs(ω)− Vs(ω))s∈(t0,∞) ∈ C((t0,∞),∩ν∈(0,2) [CνP([0, 2π]2,R) ∪ ∞]
), (2.3.67)
sups∈(t0,t]
(s− t0)(r−η)
2 ‖Xs(ω)− Vs(ω)‖CrP([0,2π]2,R) <∞ (2.3.68)
for all r ∈ [η, 2) and all t ∈ (t0, τ(ω)) and that
Xt(ω) = eA2(t−t0) ξ(ω) + Vt(ω)− eA2(t−t0) Vt0(ω) +
ˆ t
t0
eA2(t−s)(
κ0(t) + (κ1(t) + 1)Xt(ω)
+n∑
w=2
κw(t)
[
(Xt(ω)− Vt(ω))w +
w−1∑
k=0
w
k
(Xt(ω)− Vt(ω))
k(
: (Vt)(w−k) :
)
(ω)
])
ds
(2.3.69)
for all t ∈ [t0, τ(ω)). In that sense, the stochastic process X is a local mild solution of the SPDE (2.3.65).
118
Let us briefly compare Proposition 4.4 in Da Prato & Debussche [DPD03] with Theorem 82 above.
In the setting of Theorem 82 we note that
ˆ t
t0
‖Xs(ω)− Vs(ω)‖pCrP([0,2π]2,R) ds
≤[
sups∈(t0,t]
(s− t0)p(r−η)
2 ‖Xs(ω)− Vs(ω)‖pCrP([0,2π]2,R)
]ˆ t
t0
(s− t0)p(η−r)
2 ds
=
[
sups∈(t0,t]
(s− t0)p(r−η)
2 ‖Xs(ω)− Vs(ω)‖pCrP([0,2π]2,R)
]
(t− t0)(1+ p(η−r)
2 )(
1 + p(η−r)2
) <∞
(2.3.70)
and hence
(Xs(ω)− Vs(ω))s∈[t0,t]∈ C
([t0, t]; CηP([0, 2π]2,R)
)∩ Lp
([t0, t]; CrP([0, 2π]2,R)
)(2.3.71)
for all t ∈ (t0, τ(ω)), ω ∈ Ω, r ∈ [η, 2p + η) and all p ∈ (0,∞). Equation (2.3.71) implies the regularity
statement in Proposition 4.4 in Da Prato & Debussche [DPD03] and this demonstrates that Theorem 82
above implies Proposition 4.4 in Da Prato & Debussche [DPD03].
Proof of Theorem 82. We show Theorem 82 through an application of Corollary 81. For this applica-
tion define (U, ‖·‖U ) :=(C0P([0, 2π]
2,R), ‖·‖C0P([0,2π]2,R)
)and
(Ur, ‖·‖Ur
):= (D((−A2)
r), ‖(−A2)r(·)‖U )
for all r ∈ R. Moreover, define r0 := η2 ∈ (− 1
n , 0) and let ε ∈ (0,min(12 ,1n + r0)) be a real number.
Observe that this ensures that n (ε− r0) < 1. Next define α := −ε, β := ε, γ := ε, δ := n− 1 and let
Fω : [t0,∞)× Umax(β,γ) → Uα, ω ∈ Ω, be functions defined through
Fω(t, y) = κ0(t) + (κ1(t) + 1) (y + Vt(ω)) +
n∑
w=2
κw(t)
yw +
w−1∑
k=0
w
k
yk
(
: (Vt)(w−k) :
)
(ω)
(2.3.72)
for all y ∈ Umax(β,γ), t ∈ [t0,∞), ω ∈ Ω. Then note for every ω ∈ Ω that Fω ∈ C1α,β,γ,δ([t0,∞)); see
(2.3.4) for the definition of C1α,β,γ,δ([t0,∞)). Next observe that
[max(β, γ), 1 + α
)=[ε, 1− ε
)6= ∅ (2.3.73)
119
and
γ −min(α, r0) + (β − r0)δ = ε−min(−ε, r0) + (ε− r0) (n− 1)
= ε− r0 + (ε− r0) (n− 1) = n (ε− r0) < 1.
(2.3.74)
We can thus apply Corollary 81 to obtain the existence of a unique lower semicontinuous function
ρ : C1α,β,γ,δ([t0, T ])×Ur0 → (t0,∞] and to obtain the existence of a unique function y : C1
α,β,γ,δ([t0,∞))×
Ur0 → ∪s∈(t0,∞] C([t0, s), Ur0) which satisfy yG,v ∈ C([t0, ρG,v), Ur0), yG,v|(t0,ρG,v) ∈ C((t0, ρG,v), Ur1)
and
sups∈(t0,t]
(s− t0)(r1−r0) ‖yG,v(s)‖Ur1 <∞ = lim
sրρG,v
[ρG,V + ‖yG,v(s)‖Uε
](2.3.75)
and
yG,v(t) = eA2(t−t0) v +
ˆ t
t0
eA2(t−s)G(s, yG,v(s)) ds (2.3.76)
for all t ∈ (t0, ρG,v), v ∈ Ur0 , r1 ∈[η2 , 1− ε
), G ∈ C1
α,β,γ,δ([t0, T ]) and all T ∈ (0,∞). Next we define
functions τ : Ω → (t0,∞] and X : [t0,∞) × Ω → Ur0 ∪ ∞ through τ(ω) := ρFω, ξ(ω)−Vt0 (ω) for all
ω ∈ Ω and through
Xt(ω) :=
yFω, ξ(ω)−Vt0 (ω)(t) + Vt(ω) : t < τ(ω)
∞ : t ≥ τ(ω)
(2.3.77)
for all t ∈ [t0,∞) and all ω ∈ Ω. This definition together with (2.3.76) ensures that
Xt(ω)− Vt(ω) = eA2(t−t0)(ξ(ω)− Vt0(ω))+
ˆ t
t0
eA2(t−s) Fω(s,Xt(ω)− Vt(ω)
)ds (2.3.78)
for all t ∈ (t0, τ(ω) and all ω ∈ Ω. Combining this with (2.3.72) proves that X fulfills (2.3.69). In the
next step we note that
B(
C([t0,∞),
[C−ε/2P ([0, 2π]2,R)
]×n))
= σC([t0,∞),[C−ε/2
P ([0,2π]2,R)]×n)
(
C([t0,∞),
[C−ε/2P ([0, 2π]2,R)
]×n)
∋ f 7→ f(t) ∈[C−ε/2P ([0, 2π]2,R)
]×n: t ∈ [t0,∞)
)
.
(2.3.79)
120
This implies that the mapping
Ω ∋ ω 7→(Vt(ω), (:(Vt)
2:)(ω), . . . , (:(Vt)n:)(ω)
)
t∈[t0,∞)∈ C
([t0,∞),
[C−ε/2P ([0, 2π]2,R)
]×n)(2.3.80)
is F/B(C([t0,∞),
[C−ε/2P ([0, 2π]2,R)
]×n))-measurable. This ensures that the mapping
Ω ∋ ω 7→(Fω, ξ(ω)
)∈ C1
α,β,γ,δ([t0,∞))× Ur0 (2.3.81)
is F/B(C1α,β,γ,δ([t0,∞))×Ur0
)-measurable. Combining this with Corollary 81 proves that τ is a random
variable and that X is a stochastic process (see (Measurability property) in Corollary 81 for details).
Since ε ∈ (0,min(12 ,1n + r0)) was arbitrary, the proof of Theorem 82 is completed.
2.3.3 SPDEs with space-time white noise and quadratic nonlinearities in
three space dimensions
The aim of this subsection is to prove local existence and uniqueness of mild solutions of SPDEs in
three space dimensions with quadratic nonlinearities of the form
dXt =[Xt + κ2(t) : (Xt)
2 : +κ1(t)Xt + κ0(t)]dt+ dWt (2.3.82)
for t ∈ [0,∞) with periodic boundary conditions on (0, 2π)3 where κ0, κ1, κ2 ∈ C([0,∞),R) are ar-
bitrary continuous functions, where (Wt)t≥0 is a cylindrical I-Wiener process and where : (Xt)2 : is
a suitable renormalization of (Xt)2 for t ∈ [0,∞). The precise result is formulated in the following
theorem.
Theorem 83 (Quadratic nonlinearities in three space dimensions). Let (Ω,F ,P) be a probability space,
let t0 ∈ R, κ0, κ1, κ2 ∈ C([t0,∞),R), η ∈ (−1,− 12 ), let V : [t0,∞) × Ω → ∩r∈(−∞,−1/2)CrP([0, 2π]3,R)
and :(V )2 : : [t0,∞) × Ω → ∩r∈(−∞,−1)CrP([0, 2π]3,R) be stochastic processes with continuous sample
paths given by Propositions 65 and 66 and let ξ : Ω → CηP([0, 2π]3,R) be a random variable. Then there
exists a unique random variable τ : Ω → (t0,∞] and a unique stochastic process X : [t0,∞) × Ω →
CηP([0, 2π]3,R) ∪ ∞ such that for every ω ∈ Ω it holds that Xt(ω) = ∞ for all t ∈ [τ(ω),∞), that
(Xs(ω))s∈[t0,τ(ω)) ∈ C([t0, τ(ω)), CηP ([0, 2π]3,R)), (2.3.83)
121
(Xs(ω)− Vs(ω))s∈(t0,∞) ∈ C((t0,∞),∩ν∈( 1
2 ,1)[CνP([0, 2π]3,R) ∪ ∞]
), (2.3.84)
sups∈(t0,t]
(s− t0)(r−η)
2 ‖Xs(ω)− Vs(ω)‖CrP([0,2π]3,R) <∞ (2.3.85)
for all r ∈ [η, 1) and all t ∈ (t0, τ(ω)) and that
Xt(ω) = eA3(t−t0) ξ(ω) + Vt(ω)− eA3(t−t0) V0(ω) +
ˆ t
t0
eA3(t−s)[
κ2(t)(
(Xt(ω)− Vt(ω))2
+ 2 (Xt(ω)− Vt(ω)) Vt(ω) +(: (Vt)
2 :)(ω))
+ (κ1(t) + 1)Xt(ω) + κ0(t)
]
ds (2.3.86)
for all t ∈ [t0, τ(ω)). In that sense, the stochastic process X is a local mild solution of the SPDE (3.2.16).
Proof of Theorem 83. We show Theorem 83 through an application of Corollary 81. For this applica-
tion define (U, ‖·‖U ) :=(C0P([0, 2π]
3,R), ‖·‖C0P([0,2π]3,R)
)and
(Ur, ‖·‖Ur
):= (D((−A3)
r), ‖(−A3)r(·)‖U )
for all r ∈ R. Moreover, define r0 := η2 ∈ (− 1
2 ,− 14 ) and let ε ∈ (0, 14 + r0
2 ) be a real number. Observe
that this ensures that 2ε− r0 <12 and that ε < 1
8 . Next define α := − 12 − ε, β := − 1
4 − ε2 , γ := 1
4 + ε
and δ := 1 and let Fω : [t0,∞)× Umax(β,γ) → Uαi , ω ∈ Ω, be functions defined through
Fω(t, y) := κ2(t)(y2 + 2Vt(ω) y +
(: (Vt)
2 :)(ω))+ (κ1(t) + 1) (y + Vt(ω)) + κ0(t) (2.3.87)
for all y ∈ Umax(β,γ), t ∈ [t0,∞), ω ∈ Ω. Then note for every ω ∈ Ω that Fω ∈ C1α,β,γ,δ([t0,∞)); see
(2.3.4) for the definition of C1α,β,γ,δ([t0,∞)). Next observe that
[max(β, γ), 1 + α
)=[14 + ε, 12 − ε
)6= ∅ (2.3.88)
and
γ −min(α, r0) + (β − r0)δ = γ + β − α− r0
= ε2 + 1
2 + ε− r0 ≤ 12 + 2ε− r0 < 1.
(2.3.89)
We can thus apply Theorem 80 to obtain the existence of a unique lower semicontinuous function
ρ : C1α,β,γ([t0,∞))×Ur0 → (t0,∞] and to obtain the existence of a unique function y : C1
α,β,γ([t0,∞))×
Ur0 → ∪s∈(t0,∞] C([t0, s), Ur0) which satisfy yG,v ∈ C([t0, ρG,v), Ur0), yG,v|(t0,ρG,v) ∈ C((t0, ρG,v), Ur1)
and
sups∈(t0,t]
(s− t0)(r1−r0) ‖yG,v(s)‖Ur1 <∞ = lim
sրρG,v
[
ρG,v + ‖yG,v(s)‖U 14+ε
]
(2.3.90)
122
and
yG,v(t) = eA3(t−t0) v +
ˆ t
t0
eA3(t−s)G(s, yG,v(s)) ds (2.3.91)
for all t ∈ (t0, ρG,v), v ∈ Ur0 , r1 ∈[η2 ,
12 − ε
)and all G ∈ C1
α,β,γ,δ([t0,∞)). Next we define functions
τ : Ω → (t0,∞] and X : [t0,∞) × Ω → Ur0 ∪ ∞ through τ(ω) := ρFω , ξ(ω)−Vt0 (ω) for all ω ∈ Ω and
through
Xt(ω) :=
yFω, ξ(ω)−Vt0 (ω)(t) + Vt(ω) : t < τ(ω)
∞ : t ≥ τ(ω)
(2.3.92)
for all t ∈ [t0,∞) and all ω ∈ Ω. This definition together with (2.3.91) ensures that
Xt(ω)− Vt(ω) = eA3(t−t0)(ξ(ω)− Vt0 (ω))+
ˆ t
t0
eA3(t−s) Fω(s,Xt(ω)− Vt(ω)) ds (2.3.93)
for all t ∈ (t0, τ(ω)) and all ω ∈ Ω. Combining this with (2.3.87) proves that X fulfills (2.3.86). In the
next step we note that
B(
C([t0,∞), C−(1+ε)/2
P ([0, 2π]3,R)× C−(2+ε)/2P ([0, 2π]3,R)
))
= σC([t0,∞),C−(1+ε)/2
P ([0,2π]3,R)×C−(2+ε)/2P ([0,2π]3,R))
(
C([t0,∞), C−(1+ε)/2
P ([0, 2π]3,R)× C−(2+ε)/2P ([0, 2π]3,R)
)
∋ f 7→ f(t) ∈ C−(1+ε)/2P ([0, 2π]3,R)× C−(2+ε)/2
P ([0, 2π]3,R) : t ∈ [t0,∞))
.
(2.3.94)
This implies that the mapping
Ω ∋ ω 7→(Vt(ω), (: (Vt)
2 :)(ω))
t∈[t0,∞)∈ C
([t0,∞), C−(1+ε)/2
P ([0, 2π]3,R)× C−(2+ε)/2P ([0, 2π]3,R)
)
(2.3.95)
is F/B(C([t0,∞), C−(1+ε)/2
P ([0, 2π]3,R) × C−(2+ε)/2P ([0, 2π]3,R))
)-measurable and this shows that the
mapping
Ω ∋ ω 7→(Fω, ξ(ω)− Vt0(ω)
)∈ C1
α,β,γ,δ([t0,∞))× Ur0 (2.3.96)
is F/B(C1α,β,γ,δ([t0,∞))×Ur0
)-measurable. Combining this with Corollary 81 proves that τ is a random
variable and that X is a stochastic process (see (Measurability property) in Corollary 81 for details).
Since ε ∈ (0, 14 + r02 ) was arbitrary, the proof of Theorem 83 is completed.
123
Chapter 3
Exact renormalization group study of
the shear flow
3.1 Introduction
In the paper [AM90] Avellaneda and Majda proposed the following model of shear flow along y-axis:
∂T δ
∂t+ vδ(x, t)
∂T δ
∂y=
1
2ν0∆T
δ T δ∣∣t=0
= T0(δx, δy) (3.1.1)
as a simplified model for the advection-diffusion of a passive scalar, where the velocity field v has mean
zero Gaussian distribution and
⟨
|vδ(k)|2⟩
=√2π1δ≤|k|≤1 |k|1−ǫ (steady case) (3.1.2)
⟨|vδ(k, ω)|2
⟩=
√2π1δ≤|k|≤1|k|1−ǫ
|k|zω2 + |k|2z (unsteady case) (3.1.3)
and v is the Fourier transform of v, and 1δ≤|k|≤1 is a cutoff function with the infrared cutoff being
δ > 0 and ultraviolet cutoff being 1, ν0 is a positive constants, −∞ < ǫ < 4, z ≥ 0 and T0 is a given
function. Note that vδ doesn’t depend on t in the steady case. The authors of [AM90] identifies three
regimes in ǫ in the steady case and five regimes in (ǫ, z) in the unsteady case, and different (space-time)
124
scaling behaviors and effective equations for the averaged solution in the limit
T (x, y, t) = limδ→0
⟨
T δ(x
δ,y
δ,t
δα)
⟩
(3.1.4)
in different regimes, which is summarized in the following table. Notice that in the table ρ2(δ) is equal
to the δα in our paper.
In another paper [AM92], Avellaneda and Majda also applied the Yakhot-Orszag’s (also Forster-
Nelson-Stephen’s) approximate RG method developed in [FNS77, OY99, YO86] to the unsteady case,
and found three regimes in (ǫ, z), which except for the mean field regime and a hyperscaling regime
II (allowing the diffusivity perhaps being different numerically), fail to match the other regimes in
the exact result. The following two pictures compares the different regimes with the left one the
exact result and the right one the approximate RG result. This approximate RG method involves
assumptions that are hard to be justified and drastic truncations which lead to the wrong results. In
particular, the approximate RG method truncates the effect equations at every scale so that they only
have one dynamical parameter: the coefficient in front of ∂2yyT , while in Section 3.4, we show that the
exact effect equations at every scale should be parametrized by kernels.
125
In this paper we show that at least for this simple model, exact RG method, inspired by [Pol84,
WK74] is capable to recover precisely all the regimes with correct scalings. The method in [Pol84,
WK74] was used to study the (Euclidean) quantum field theories, which typically consist of a Gaussian
field φ on Rd with distribution formally written as e−12
´
(∂φ(x))2ddx, and a functional in φ that is written
in a form eV(φ), for instance V(φ) = − 14
´
φ(x)4ddx. An untraviolet cutoff Λ0 is needed to make sense
of the functional, and then the RG scheme is to decompose φ into slow and high fluctuation parts
φ(k) = φ<(k) + φ>(k) separated by a scale k ∼ e−lΛ0 and average out eV(φ) w.r.t. φ>, denoted by⟨eV(φ)
⟩
φ>, followed by a rescaling k → e−lk. Polchinski [Pol84] is able to write down an exact evolution
equation which describes the dynamics of⟨eV(φ)
⟩
φ>as l grows. Following this approach, we regard
T (x, y, t) as a functional of the Gaussian field v for each x, y, t, and exploit the special structures of
the Polchinski equation in our case of shear flow.
We will consider the Fourier transform of T in y, i.e. T (x, ξ, t), which satisfies
∂T
∂t+ iv(x, t)ξT = −1
2ν0ξ
2T +1
2ν0∂
2xT (3.1.5)
This form will allow us to apply the Feynman-Kac formula
T (x, ξ, t) = E
[
e−ν02 ξ
2te−iξ´ t0v(x+
√ν0Bs,t−s)dsT
∣∣t=0
(x+√ν0Bt, ξ)
]
(3.1.6)
where B is a standard Brownian motion in R starting from origin, which is independent of v. We will
always write E for expectation over B and 〈−〉 for expectation over v. Here and after we omit the
subscripts δ on T and v for simplicity of notations, so their dependence on the infrared cutoff will be
126
implicit.
Finally we remark that the applicability of the RG method to general passive scalar problem would
be very interesting. For previous mathematical works see [BK91, SZ06].
3.2 Steady case
3.2.1 The Polchinski equation
First of all, we decompose the Gaussian random field into low and high fluctuation parts:
v(x) = v<(x) + v>(x) (3.2.1)
where⟨
|v>(k)|2⟩
=√2π1e−l≤|k|≤1 |k|1−ǫ (3.2.2)
⟨
|v<(k)|2⟩
=√2π1δ≤|k|≤e−l |k|1−ǫ (3.2.3)
Let Tl(x, ξ, t; v<) =⟨
T (x, ξ, t)⟩
v>be the average of T (x, ξ, t) over v>.
Proposition 84. Tl(x, ξ, t; v<) satisfies the Polchinski equation ([Pol84])
∂Tl(x, ξ, t; v<)
∂l=
ˆ ˆ
∂
∂lCl(x
′ − y′)δ2Tl(x, ξ, t; v<)
δv<(x′)δv<(y′)dx′dy′ (3.2.4)
where the right hand side involves functional derivatives of Tl w.r.t. v<, and
Cl(x′ − y′) =
ˆ 1
e−lei(x
′−y′)k |k|1−ǫ dk (3.2.5)
and the initial condition at l = 0 is
T0(x, ξ, t; v) = T (x, ξ, t) = E
[
e−ν02 ξ
2te−iξ´ t0v(x+
√ν0Bs)dsT
∣∣t=0
(x+√ν0Bt, ξ)
]
(3.2.6)
Proof. Let µl be the (mean zero) Gaussian density of v>. It’s covariance is Cl and when l = 0 it
concentrates on v = 0. Therefore µl is the fundamental solution of a heat equation with ∂∂lCl(x
′ − y′)
as Laplacian coefficient. By µl(v>) = µl(−v>), we have⟨
T (x, ξ, t)⟩
v>= µl ∗ T (x, ξ, t) where ∗ is the
convolution. Therefore⟨
T (x, ξ, t)⟩
v>solves the same heat equation with initial data at l = 0 given
127
by T (x, ξ, t).
Following the procedure of Polchinski [Pol84], let’s assume that Tl(x, ξ, t; v<) has the following
form:
Tl(x, ξ, t; v<) = E
[
e∑
n≥0 Un(x,ξ,t;v<,B,l)T∣∣t=0
(x +√ν0Bt, ξ)
]
(3.2.7)
where U0 is independent of v<, U1 is linear in v<, etc. and Un is n-th order in v<.
In fact, following the proof of Prop 84, e∑
n≥0 Un(x,ξ,t;v<,B) also satisfies the Polchinski equation
(3.2.4), with initial condition at l = 0 given by
e−ν02 ξ
2te−iξ´
t0v(x+
√ν0Bs)ds (3.2.8)
Therefore it’s easy to check that
∂∑
n≥0 Un
∂l=
ˆ ˆ
∂
∂lCl(x
′ − y′)
δ2∑
n≥0 Un
δv<(x′)δv<(y′)+δ∑
n≥0 Un
δv<(x′)
δ∑
n≥0 Un
δv<(y′)
dx′dy′ (3.2.9)
By comparing the orders in v<, we obtain a system of equations
∂Un∂l
=
ˆ ˆ
∂
∂lCl(x
′ − y′)
δ2Un+2
δv<(x′)δv<(y′)+
∑
p+q=n+2
δUpδv<(x′)
δUqδv<(y′)
dx′dy′ (3.2.10)
Because when l = 0, Un = 0 for n ≥ 2, and by inspection of the above equation ∂∂lUn = 0 for n ≥ 1,
we conclude that Un = 0 for n ≥ 2 and
U1(x, ξ, t; v<, B, l) = −iξˆ t
0
v<(x+√ν0Bs)ds (3.2.11)
for all l ≥ 0. Therefore
Tl(x, ξ, t; v<) = E
[
eU0(x,ξ,t;v<,B,l)−iξ´ t0v(x+
√ν0Bs)dsT
∣∣t=0
(x+√ν0Bt, ξ)
]
(3.2.12)
where
∂U0
∂l= −ξ2
ˆ ˆ
∂
∂lCl(x
′ − y′)δ´ t
0 v<(x+√ν0Bs)ds
δv<(x′)
δ´ t
0 v<(x+√ν0Bs)ds
δv<(y′)dx′dy′ (3.2.13)
128
i.e.
∂U0
∂l= −ξ2
ˆ t
0
ˆ t
0
∂
∂lCl(
√ν0(Bs −Bs′))dsds
′ (3.2.14)
The RG procedure consists of the above integration out of v> followed by rescaling. Formally,
we can view the distribution of Tl(x, ξ, t) as superpositions (over Brownian paths) of quantum field
theoretic measures:
exp
−1
2
ˆ
(v<(x
′)(−i∂)ǫ−1v<(x′))dx′ + U1(l) + U0(l)
(3.2.15)
with
1
2
ˆ
(v(x′)(−i∂)ǫ−1v(x′)
)dx′ =
ˆ e−l
δ
|k|ǫ−1|v<(k)|2dk (3.2.16)
With rescaling x→ elx, t→ eαlt, we determine a scaling exponent for v so that the quadratic term is
preserved:
v<(elx) → e(ǫ/2−1)lv(x) (3.2.17)
Define
V1(x, ξ, t; v,B, l) = U1(elx, e−lξ, eαlt; e(ǫ/2−1)lv<(e
−l·), B, l) (3.2.18)
V0(x, ξ, t;B, l) = U0(elx, e−lξ, eαlt;B, l) +
ν02e(α−2)lξ2t (3.2.19)
Note that we have separated out the initial condition − ν02 ξ
2t from U0, as a matter of convenience. By
(3.2.11) (3.2.14) we have our RG flow with rescaling incorporated:
∂V1
∂l = (α+ ǫ2 − 2)V1 + (α2 − 1)
´
δV1
δBsBsds
∂V0
∂l = (2α− 2)V0 − e(2α−2)lξ2´ t
0
´ t
0∂∂l
[
Cl((Bs −Bs′)e
α2 l)]
dsds′(3.2.20)
3.2.2 Fixed points for the steady case
Before we discuss fixed points for different regimes, let’s make a general remark concerning the infrared
cutoff δ. The infrared cutoff δ is a technical issue to make sense of the SPDE, but it prevents the
RG to flow forever l → ∞, namely the RG stops at scale l = − log δ. However, as a dynamic system,
(3.2.20) indeed exists as l → ∞. Therefore, suppose that (V1(l), V0(l)) → (V ⋆1 , V⋆0 ) as l → ∞ where
(V ⋆1 , V⋆0 ) is a fixed point, we will have (under certain suitable norm) ‖(V1(l), V0(l))− (V ⋆1 , V
⋆0 )‖ < a(l)
129
where a(l) → 0 as l → ∞. In particular,
‖(V1(− log δ), V0(− log δ))− (V ⋆1 , V⋆0 )‖ < a(− log δ) (3.2.21)
Once we have this bound, we will then let δ → 0 (see (3.1.4)).
Mean field regime ǫ < 0
With diffusive scaling, i.e. α = 2, we see that
∂V1∂l
=ǫ
2V1 (3.2.22)
and this together with ǫ < 0 implies V1 → 0.
We immediately see that in the mean field regime, the fixed points are
e−12
´
(v(x′)(−i∂)ǫ−1v(x′))dx′+V0 (3.2.23)
for all dimensionless function V0, for instance V0 = Dξ2t, D ∈ R. Indeed, by (3.2.10) and dimension-
lessness of V0 we know these are the only fixed points restricted to the space V0, V1, V≥2 = 0.
These discussions mean that there’re many fixed points. Now we come to the question of which
specific fixed point (i.e. V0 =?) RG goes to starting from U1, U0. Solving equation (3.2.14) with
U0(l = 0) = − ν02 ξ
2t,
U0(l) = −ν02ξ2t− ξ2
ˆ t
0
ˆ t
0
Cl(√ν0(Bs −Bs′))dsds
′ (3.2.24)
and therefore
V0(l) = −e−2lξ2ˆ e2lt
0
ˆ e2lt
0
Cl(√ν0(Bs −Bs′))dsds
′ (3.2.25)
As argued in [AM90], using ergodicity arguments, the last term in the above equation goes to
2
πν0tξ2ˆ 1
0
|k|−1−ǫdk = − 2tξ2
πν0ǫ(3.2.26)
as l → ∞. In fact, following [AM90], one can construct a process X so that X ′′> = − 2
ν0v>, and
by Ito’s formula, ξ´ t
0v>(
√ν0Bs)ds =
√ν0ξ´ t
0X ′>(
√ν0Bs)dBs + R>(t) where it can be shown that
R>(t) upon rescaling goes to 0 as l → ∞. The Ito integral is a martingale with quadratic varia-
tion ν0ξ2´ t
0X ′>(
√ν0Bs)
2ds, which upon rescaling and by ergodicity theorem goes to ν0ξ2t⟨X ′(0)2
⟩=
130
2πν0
tξ2´ 1
0|k|−1−ǫdk. Finally we obtain the conclusion by observing that the correlation of ξ
´ t
0v>(
√ν0Bs)ds
is given by the last term of (3.2.24).
Therefore, the effective equation is
∂T
∂t=
1
2ν0∆T − 2
πν0ǫTyy (3.2.27)
Fixed points with non-vanishing V ⋆1 : the regime 2 < ǫ < 4
We have seen that in the mean field regime the fixed point has V ⋆1 = 0. We now look for a scaling
so that fixed points have linear terms in v (i.e. V ⋆1 6= 0), for ǫ > 0. By (3.2.20),∂V ⋆1∂l = 0 can be
guaranteed by
α = 2− ǫ
2(3.2.28)
and
(α
2− 1)
ˆ
δV ⋆1δBs
Bsds = 0 (3.2.29)
Since α2 − 1 6= 0,
´ δV ⋆1δBs
Bsds = 0, which implies that V ⋆1 doesn’t depend on B. In fact
V ⋆1 = −iξˆ t
0
v(x)ds = −iξtv(x) (3.2.30)
Now with V ⋆1 already found, we identify the constant term in the fixed point, namely V ⋆0 . By
(3.2.20) with B = 0
∂V0∂l
= (2α− 2)V0 −1
πe(2α−2)lξ2
ˆ t
0
ˆ t
0
∂
∂l
ˆ 1
e−l|k|1−ǫ dkdsds′ (3.2.31)
To find the fixed point, the right hand side being equal to 0 implies that
(2α− 2)V ⋆0 − 1
πe(2α−2)lξ2
ˆ t
0
ˆ t
0
e−l(2−ǫ)dsds′ = 0 (3.2.32)
Using the scaling we found above 2α− 2 = 2− ǫ,
(2α− 2)V ⋆0 − 1
πξ2t2 = 0 (3.2.33)
131
namely,
V ⋆0 =1
π
ξ2t2
2− ǫ= − 1
πξ2t2ˆ ∞
1
|k|1−ǫ dk (3.2.34)
where the last equality holds if ǫ > 2 by direct calculation, which is written into this integral form for
the reader to compare it with the result in [AM90]. In fact we can see that for 0 < ǫ < 2, 1πξ2t2
2−ǫ is
not the fixed point that the RG flow will converge to, starting from our initial condition V1(0), V0(0),
as discussed in the following remark. Finally, the term − ν02 ξ
2t in U0 goes to zero under the scaling
α = 2− ǫ2 .
Remark 85. We give more explanations about the fixed point here, by looking at the effects of inte-
gration and of rescaling separately. We would like to see that − 1π ξ
2t2´∞1 |k|1−ǫ dk is unchanged under
integration out e−l < |k| < 1 followed by rescaling. Indeed, integration out e−l < |k| < 1 gives an
increment
− 1
πξ2t2ˆ ∞
1
|k|1−ǫ dk → − 1
πξ2t2ˆ ∞
1
|k|1−ǫ dk − 1
πξ2t2ˆ 1
e−l|k|1−ǫ dk = − 1
πξ2t2ˆ ∞
e−l|k|1−ǫ dk
(3.2.35)
and rescaling it back to unit cutoff, together with a change of variable k → e−lk, gives
− 1
πe(2α−2)lξ2t2
ˆ ∞
1
|k|1−ǫ e(ǫ−2)ldk = − 1
πξ2t2ˆ ∞
1
|k|1−ǫ dk (3.2.36)
In the case 0 < ǫ < 2, solving (3.2.13)(3.2.14) we obtain
U0(l) = U0(0)−1
πξ2ˆ t
0
ˆ t
0
ˆ 1
e−l|k|1−ǫ eik(Bs−Bs′)dkdsds′ (3.2.37)
with U0(0) = − ν02 ξ
2t. Upon rescaling with scaling exponents described above, the Brownian motion
B damps out and V0(l) → − 1π ξ
2t2´∞1
|k|1−ǫ dk = −∞ for 0 < ǫ < 2.
Unlike mean field regime where the constant term in v of the fixed point is arbitrary dimensionless
quantity, here the linear term determines uniquely the constant term in v of the fixed point. Indeed: in
mean field regime, which fixed point it converges depends on initial condition (ν0), but in hyperscaling
regime 2 < ǫ < 4, initial condition (ν0) doesn’t affect the infrared behavior.
As mentioned in the beginning of this section, though there appears an infrared cutoff δ, we can
still talk about fixed point V ⋆0 of the dynamic flow of V0 independently. To find the effective equation
132
(see (3.1.4)), we should first take an IR cutoff δ for v, and run the RG flow until e−l = δ, where
V0(e−l = δ) ≈ − 1
πξ2t2ˆ el
1
|k|1−ǫ dk (3.2.38)
which is very close to V ⋆0 , and ≈ means up to a very weak correction by the Brownian motion. Then
we take δ → 0.
Nonlocal regime
In nonlocal regime 0 < ǫ < 2, an interesting fixed point can’t have linear term: indeed, if it had linear
term, the discussions for regime 2 ≤ ǫ < 4 would all apply but according to Remark 85 the RG would
converge to V ⋆0 = ∞ if 0 < ǫ < 2 (the integral is ultravoilet divergent).
We solve the equation (3.2.14) for U0
U0(l) = U0(0)−1
πξ2t2ˆ 1
0
ˆ 1
0
ˆ 1
e−lei(Bs−Bs′ )
√tk |k|1−ǫ dkdsds′ (3.2.39)
with U0(0) = − ν02 ξ
2t. The last term under rescaling x→ elx, t→ eαlt becomes
− 1
πξ2t2e2(α−1)l
ˆ 1
0
ˆ 1
0
ˆ 1
e−lei(Bs−Bs′)e
αl/2√tk |k|1−ǫ dkdsds′ (3.2.40)
For this to converge to a nontrivial fixed point as l → ∞, we have to change variable for k. We look
for a nonlocal fixed point, i.e. V ⋆0 (B) which depends on B. Let k → e−αl/2t−1/2k
− 1
πe(α+αǫ/2−2)lξ2t1+ǫ/2
ˆ 1
0
ˆ 1
0
ˆ eαl/2√t
e(α/2−1)l√t
ei(Bs−Bs′)k |k|1−ǫ dkdsds′ (3.2.41)
Choosing α = 21+ǫ/2 , we obtain a nontrivial fixed point as l → ∞
V ⋆0 = − 1
πξ2t1+ǫ/2
ˆ 1
0
ˆ 1
0
ˆ
ei(Bs−Bs′)k |k|1−ǫ dkdsds′ (3.2.42)
Notice that U0(0) = − ν02 ξ
2t vanishes under rescaling.
133
3.3 Unsteady case
3.3.1 The Polchinski equation
As in the steady case, we decompose the Gaussian field
v(x, t) = v<(x, t) + v>(x, t) (3.3.1)
where⟨
|v>(k, ω)|2⟩
=√2π1e−l≤|k|≤1|k|1−ǫ
|k|z|k|2z + ω2
(3.3.2)
⟨
|v<(k, ω)|2⟩
=√2π1δ≤|k|≤e−l |k|1−ǫ
|k|z|k|2z + ω2
(3.3.3)
Repeating the proof of Prop 84, the Polchinski equation for Tl(x, ξ, t, v<) is now modified as
∂Tl(x, ξ, t; v<)
∂l=
ˆ ˆ
∂
∂lCl(x
′ − y′, t′ − r′)δ2Tl(x, ξ, t; v<)
δv<(x′, t′)δv<(y′, r′)dx′dy′dt′dr′ (3.3.4)
where
Cl(x′ − y′, t′ − r′) =
ˆ ∞
−∞
ˆ 1
e−lei(x
′−y′)k+i(t′−r′)ω |k|1−ǫ |k|z|k|2z + ω2
dkdω (3.3.5)
and the initial condition at l = 0 is
T0(x, ξ, t; v) = T (x, ξ, t) = E
[
e−ν02 ξ
2te−iξ´
t0v(x+
√ν0Bs,t−s)dsT
∣∣t=0
(x+√ν0Bt, ξ)
]
(3.3.6)
Following the procedure in the steady case, we write the system of equations for Un, and find that
Un = 0 for n ≥ 2 and
U1 = −iξˆ t
0
v(x+√ν0Bs, t− s)ds (3.3.7)
for all l ≥ 0. The RG flow for U0 is
∂U0
∂l= −ξ2
ˆ ˆ ˆ ˆ
∂
∂lCl(x
′ − y′, t′ − r′)
δ´ t
0 v<(x+√ν0Bs, t− s)ds
δv<(x′, t′)
δ´ t
0 v<(x+√ν0Bs, t− s)ds
δv<(y′, r′)dx′dy′dt′dr′
(3.3.8)
i.e.
∂U0
∂l= −ξ2
ˆ t
0
ˆ t
0
∂
∂lCl(Bs − Bs′ , s− s′)dsds′ (3.3.9)
134
In order to find the rescaling exponent for v we identify the free propagator in the unsteady case:
ˆ ˆ
v(−i∂x)ǫ−1−z((−i∂x)2z + (−i∂t)2)vdxdt =ˆ ˆ
δ≤|k|≤1
|k|ǫ−1−z(|k|2z + ω2)|v|2dkdω (3.3.10)
There are two terms; we will discuss different cases in which different term dominates. In the integration
step, both terms together plays the role of free propagator, while in the rescaling step, since in general
the two terms have different dimensions, we can only rescale so that only one of them is invariant
and the other one damps out. For this reason we define rescaled functions with scaling exponent of v
implicit:
V1(x, ξ, t; v,B) = U1(elx, e−lξ, eαlt; e[v]lv<(e
−l·, e−αlt), B) (3.3.11)
V0(x, ξ, t;B) = U0(elx, e−lξ, eαlt;B) +
ν02e(α−2)lξ2t (3.3.12)
We have our RG flow with rescaling incorporated:
∂V1
∂l = (α− 1 + [v])V1 + (α2 − 1)´
∂V1
∂BsBsds
∂V0
∂l = (2α− 2)V0 − e(2α−2)lξ2´ t
0
´ t
0∂∂l
[
Cl((Bs −Bs′)e
α2 l, s− s′
)]
dsds′(3.3.13)
with initial condition at l = 0 givin by
V1(l = 0) = −iξˆ t
0
v(x+√ν0Bs, t− s)ds V0(l = 0) = 0 (3.3.14)
3.3.2 Fixed points for the unsteady case
Regime I (mean field) ǫ < 0, z ≥ 2 ∪ ǫ < 2− z, 0 < z < 2
With diffusive scaling, i.e. α = 2, we see that
∂V1∂l
= (1 + [v])V1 (3.3.15)
We discuss two cases separately. In the case z ≥ 2, ω2 dominates k2z + ω2, so [v] is determined by
requiring thatˆ ˆ
v(−i∂x)ǫ−1−z(−i∂t)2vdxdt (3.3.16)
135
is invariant, which implies
[v] =ǫ− z
2(3.3.17)
It’s easy to see 1 + [v] < 0 if ǫ < 0, so by (3.3.15) V1 → 0.
In the case 0 < z < 2, k2z dominates k2z + ω2, so [v] is determined by requiring that
ˆ ˆ
v(−i∂x)ǫ−1−z(−i∂x)2zvdxdt (3.3.18)
is invariant, which implies
[v] =ǫ+ z
2− 2 (3.3.19)
We still have 1 + [v] < 0 if ǫ < 2− z, so by (3.3.15) V1 → 0.
As in the steady case, all dimensionless V0 are fixed points. To study the question of which
specific fixed point RG converges to starting from V1(l = 0), V0(l = 0), we solve equation (3.3.9) with
U0(l = 0) = − ν02 ξ
2t,
U0(l) = −ν02ξ2t− ξ2
ˆ t
0
ˆ t
0
Cl(√ν0(Bs −Bs′), s− s′)dsds′ (3.3.20)
and therefore
V0(l) = −e−2lξ2ˆ e2lt
0
ˆ e2lt
0
Cl(√ν0(Bs −Bs′), s− s′)dsds′ (3.3.21)
As argued in [AM90], using ergodicity arguments, the right hand side of the above equation goes to
−D(ǫ, z) = − 2
πtξ2ˆ 1
0
(ν02|k|2 + |k|z)−1|k|1−ǫdk (3.3.22)
as l → ∞.
Therefore, the effective equation is
∂T
∂t=
1
2ν0∆T +D(ǫ, z)Tyy (3.3.23)
136
Regime II: 2− z < ǫ < 4− 2z
We look for fixed points with nonzero linear terms. Suppose ∂2zx dominates ∂2zx + ∂2t . With scaling
x→ elx, t→ eαt, we scale v so that the quadratic term
ˆ ˆ
(v<(x
′)∂ǫ−1−zx ∂2zx v<(x
′))dx′dt (3.3.24)
is preserved, i.e. [v] = (ǫ+ z − α− 2)l/2:
v<(elx, eαlt) → e(ǫ+z−α−2)l/2v(x, t) (3.3.25)
By (3.3.13),∂V ⋆1∂l = 0 can be guaranteed by α = [v] + 1 i.e.
α = 4− ǫ− z (3.3.26)
and
(α
2− 1)
ˆ
∂V ⋆1∂Bs
Bsds = 0 (3.3.27)
It’s easy to check that α2 − 1 6= 0, so
´ ∂V ⋆1∂Bs
Bsds = 0, which implies that V ⋆1 doesn’t depend on B. In
fact
V ⋆1 = −iξˆ t
0
v(x, t− s)ds (3.3.28)
Now with V ⋆1 already found, we identify the constant term in the fixed point, namely V ⋆0 . By
(3.3.13) with B = 0
∂V0∂l
= (2α− 2)V0 −1
πe(2α−2)lξ2
ˆ t
0
ˆ t
0
∂
∂l
ˆ ∞
−∞
ˆ 1
e−l|k|1−ǫ eiω(s−s′)eαl |k|z
ω2 + |k|2z dkdωdsds′ (3.3.29)
By straightforward calculations,
G(k, t; l) :=
ˆ t
0
ˆ t
0
ˆ ∞
−∞eiω(s−s
′)eαl |k|zω2 + |k|2z dωdsds
′
= t2[
1
|k|zteαl −1
(|k|zteαl)2 (1− e−|k|zteαl)
] (3.3.30)
137
We have |k|zt−1eαlG(k, t; l) → 1 as l → ∞. Replacing G(k, t; l) with t|k|zeαl ,
∂V0∂l
= (2α− 2)V0 −1
πe(2α−2)lξ2t
∂
∂l
ˆ 1
e−l|k|1−ǫ−z e−αldk (3.3.31)
The right hand side being equal to 0 implies that
(2α− 2)V ⋆0 − 1
πe(2α−2)lξ2te−l(2−ǫ−z)e−αl − 1
πe(2α−2)lξ2t
ˆ 1
e−l|k|1−ǫ−z (−α)e−αldk = 0 (3.3.32)
Observe that the third term divided by (−α) solves (3.3.31), therefore we obtain the fixed point
equation
(2α− 2)V ⋆0 − 1
πe(2α−2)lξ2te−l(2−ǫ−z)e−αl − αV ⋆0 = 0 (3.3.33)
Using the scaling we found above α = 4− ǫ− z,
(α − 2)V ⋆0 − 1
πξ2t = 0 (3.3.34)
namely,
V ⋆0 = − 1
π
ξ2t
ǫ+ z − 2= − 1
πξ2t
ˆ ∞
1
|k|1−ǫ−z dk (3.3.35)
where the last equality is written into this integral form for the reader to compare it with the result
in [AM90]. Finally, the term − ν02 ξ
2t in U0 goes to zero.
Regime III: 4− 2z < ǫ < 4, z < 2 ∪ 2 < ǫ < 4, z ≥ 2
Next, suppose ∂2t dominates ∂2zx + ∂2t . i.e. α < z. Observe that kz
k2z+ω2 → kze−lz
k2ze−lz+ω2e−2αl converges
to δ(ω) as l → ∞. Namely, in the fixed point, the Gaussian field v(k, ω) = 0 unless ω = 0. Let
v(k) =´
v(k, t)dt. The fixed point has the form
e−´
(v(x′)∂ǫ−1x v(x′))dx′−iξtv(x)ds+U⋆0 (3.3.36)
and U⋆0 is thus the same with that of hyperscaling regime for steady case
U⋆0 = ξ2t2ˆ
|k|1−ǫ ψ0(|k|)dk (3.3.37)
138
and´ (v(x′)∂ǫ−1
x v(x′))dx′ and iξtv(x)ds being both marginal gives, which is the same with that of
hyperscaling regime for steady case, α = 2− ǫ/2.
Regime IV: 4− 2z < ǫ < 2, 1 < z < 2
Solving the equation (3.3.9) for U0
U0(l) = U0(0)−1
πξ2t2ˆ 1
0
ˆ 1
0
ˆ ∞
−∞
ˆ 1
e−lei(Bs−Bs′)
√tk+iω(s−s′)t |k|1−ǫ |k|z
ω2 + |k|2z dkdωdsds′ (3.3.38)
with U0(0) = − ν02 ξ
2t. The last term under rescaling x→ elx, t→ eαlt becomes
− 1
πξ2t2+
ǫ−2z e((2−
2−ǫz )α−2)l
ˆ 1
0
ˆ 1
0
ˆ ∞
−∞
ˆ 1
e−lei(Bs−Bs′)e
αl/2√tk+iω(s−s′)teαl |k|1−ǫ |k|z
ω2 + |k|2z dkdωdsds′
(3.3.39)
We choose α so that (2 − 2−ǫz )α− 2 = 0 i.e.
α =2z
2z + ǫ− 2(3.3.40)
By the same calculations of (3.3.30),
ˆ 1
0
ˆ 1
0
ˆ ∞
−∞eiω(s−s
′)teαl |k|zω2 + |k|2z dωdsds
′
=1
|k|zteαl −1
(|k|zteαl)2 (1 − e−|k|zteαl)
(3.3.41)
Changing variable k → e−αl/zt−1/zk, we have
V0(l) = − 1
πξ2t2e2(α−1)l
ˆ eαl/zt1/z
e(α/z−1)lt1/zei(Bs−Bs′ )e
αl/2e−αl/zt1/2−1/zk |k|1−ǫ g(|k|z)dk (3.3.42)
where g(k) = 1k − 1
k2 (1− e−k). Using z < 2, the Brownian motion term goes to 0 as l → ∞. Therefore
V0(l) → − 1
πξ2t2+
ǫ−2z
ˆ ∞
0
|k|1−ǫ g(|k|z)dk = V ⋆0 (3.3.43)
Notice that U0(0) = − ν02 ξ
2t vanishes under rescaling.
Regime V (nonlocal regime): 0 < ǫ < 2, z > 2
This regime is treated essentially the same with the nonlocal regime of steady case.
139
The solution for U0 is
U0(l) = U0(0)−1
πξ2t2ˆ 1
0
ˆ 1
0
ˆ 1
e−lei(Bs−Bs′)
√tk |k|1−ǫ e−|k|z|s−s′|tdkdsds′ (3.3.44)
with U0(0) = − ν02 ξ
2t. The last term under rescaling x → elx, t → eαlt and a change of variable for
k → e−αl/2t−1/2k becomes
− 1
πe(α+αǫ/2−2)lξ2t1+ǫ/2
ˆ 1
0
ˆ 1
0
ˆ eαl/2√t
e(α/2−1)l√t
ei(Bs−Bs′ )k |k|1−ǫ e−|k|z|s−s′|t1−z/2e(1−z/2)αldkdsds′
(3.3.45)
Since z > 2, 1− z/2 → 0. Choosing α = 21+ǫ/2 , we obtain a nontrivial fixed point as l → ∞
V ⋆0 = − 1
πξ2t1+ǫ/2
ˆ 1
0
ˆ 1
0
ˆ
ei(Bs−Bs′)k |k|1−ǫ dkdsds′ (3.3.46)
Notice that U0(0) = − ν02 ξ
2t vanishes under rescaling.
3.4 Effective SPDE
In this section we derive the effective SPDEs for Tl at arbitrary scale l. In particular we will see that
the effective SPDEs contain kernels that are in very general forms, not restricted to heat kernels which
would make the effective SPDEs local equations as assumed in the Yakhot-Orszag type calculations
[AM92]. Recall our notation
Tl(x, ξ, t; v<) =⟨
T (x, ξ, t)⟩
v>(3.4.1)
Proposition 86. Tl(x, ξ, t; v<) satisfies the following SPDE:
∂tTl + iv<(x, t)ξTl =ν02∂2xTl −
1
2ν0ξ
2Tl +
ˆ
R
Kl(x, x, ξ, t)T∣∣t=0
(x, ξ)dx (3.4.2)
where the kernel of the integral operator Kl is a superposition of kernels over the ensemble of Brownian
bridge paths Bs : s ∈ [0, T ], B(0) = x, B(t) = x.
Kl(x, x, ξ, t) =1√
2πν0te−
(x−x)2
2ν0t E
[
e−ν02 ξ
2te−iξ´
t0v<(Bs)ds
e−ξ2
2
´ t0
´ t0R(Bs−Bs′ ,s−s′)dsds′ξ2
ˆ t
0
R(Bs − x)ds
] (3.4.3)
140
and
R(x, t) =1
π
ˆ ∞
−∞
ˆ 1
e−leixk+itω |k|1−ǫ |k|z
ω2 + |k|2z dkdω (3.4.4)
Proof. By the Feynman-Kac representation,
T (x, ξ, t) = E
[
e−ν02 ξ
2te−iξ´
t0v(x+
√ν0Bs,t−s)dsT
∣∣t=0
(x+√ν0Bt, ξ)
]
(3.4.5)
where E is the expectation over Brownian motion B. We average out v> and obtain
Tl(x, ξ, t; v<)
=E
[
e−ν02 ξ
2teU1(x,ξ,t)e−ξ2
2
´
t0
´
t0 〈v>(
√ν0(Bs−Bs′),s−s′)v>(0,0)〉dsds′ T
∣∣t=0
(x+√ν0Bt, ξ)
]
=E
[
e−ν02 ξ
2teU1(x,ξ,t)e−ξ2
2
´
t0
´
t0Rl(
√ν0(Bs−Bs′),s−s′)dsds′ T
∣∣t=0
(x+√ν0Bt, ξ)
]
(3.4.6)
where U1(x, ξ, t) = −iξ´ t
0v<(x+
√ν0Bs, t− s)ds, and
Rl(√ν0(Bs −Bs′), s− s′) =
1
π
ˆ ∞
−∞
ˆ 1
e−lei
√ν0(Bs−Bs′)k+i(s−s′)ω |k|1−ǫ |k|z
ω2 + |k|2z dkdω (3.4.7)
Now we derive the effective PDE for Tl. Since the generator of Bt is ∂2x,
ν02∂2xTl = lim
r→0
1
r
E[Tl(x+
√ν0Br, ξ, t)− Tl(x, ξ, t)
]
= limr→0
1
r
E
[
e−ν02 ξ
2te−iξ´ t0v<(x+
√ν0Br+
√ν0Bs,t−s)ds
e−ξ2
2
´
t0
´
t0R(
√ν0(Bs−Bs′ ),s−s′)dsds′ T
∣∣t=0
(x+√ν0Br +
√ν0Bt, ξ)
]
− [r = 0]
(3.4.8)
where the term [r = 0] means the same as the first term except r = 0. Because Br+Bt ∼ Br+t in law,
ν02∂2xTl = lim
r→0
1
r
E
[
e−ν02 ξ
2te−iξ´ t+rr
v<(x+√ν0Bs,t−s)ds
e−ξ2
2
´
t+rr
´
t+rr
R(√ν0(Bs−Bs′))dsds′ T
∣∣t=0
(x+√ν0Bt+r, ξ)
]
− [r = 0]
= limr→0
1
r
E
[
e−ν02 ξ
2(t+r)e−iξ´ t+r0
v<(x+√ν0Bs,t−s)ds
(
eν02 ξ
2r+iξ´ r0v<(x+
√ν0Bs,t−s)ds − 1 + 1
)
e−ξ2
2
´
t+r0
´
t+r0
R(√ν0(Bs−Bs′),s−s′)dsds′
(
eξ2
2
´
ΓR(
√ν0(Bs−Bs′ ),s−s′)dsds′ − 1 + 1
)
T∣∣t=0
(x+√ν0Bt+r, ξ)
]
− [r = 0]
(3.4.9)
141
where
Γ = (s, s′) ∈ [0, t+ r]2\[0, t]2 (3.4.10)
Thereforeν02∂2xTl = ∂tTl +
(1
2ν0ξ
2 + iv<(x, t)ξ
)
Tl
+ E
[
e−ν02 ξ
2te−iξ´ t0v<(x+
√ν0Bs,t−s)dse−
ξ2
2
´ t0
´ t0R(
√ν0(Bs−Bs′ ),s−s′)dsds′
limr→0
1
r
(ξ2
2
ˆ
Γ
R(√ν0(Bs −Bs′), s− s′)dsds′
)
T∣∣t=0
(x+√ν0Bt, ξ)
]
(3.4.11)
The limit of 1r
´
Γas r → 0 with Γ being the infinitesimally thin region defined above is the line integral
times 2, thus
ν02∂2xTl = ∂tTl +
(1
2ν0ξ
2 + iv<(x, t)ξ
)
Tl
+ E
[
e−ν02 ξ
2te−iξ´ t0v<(x+
√ν0Bs,t−s)dse−
ξ2
2
´ t0
´ t0R(
√ν0(Bs−Bs′ ),s−s′)dsds′
ξ2ˆ t
0
R(√ν0(Bs −Bt), s− t)ds · T
∣∣t=0
(x+√ν0Bt, ξ)
]
=∂tTl +
(1
2ν0ξ
2 + iv<(x)ξ
)
Tl +
ˆ
R
Kl(x, x, ξ, t)T∣∣t=0
(x, ξ)dx
(3.4.12)
where
Kl(x, x, ξ, t) =1√
2πν0te−
(x−x)2
2ν0t E
[
e−ν02 ξ
2te−iξ´
t0v<(Bs,t−s)ds
e−ξ2
2
´
t0
´
t0R(Bs−Bs′ ,s−s′)dsds′ξ2
ˆ t
0
R(Bs − x)ds
] (3.4.13)
and B is a Brownian bridge on [0, t] with variance ν0 and B(0) = x, B(t) = x.
142
Bibliography
[AM90] Marco Avellaneda and Andrew J. Majda. Mathematical models with exact renormalization
for turbulent transport. Comm. Math. Phys., 131(2):381–429, 1990.
[AM92] Marco Avellaneda and Andrew J. Majda. Approximate and exact renormalization theories
for a model for turbulent transport. Phys. Fluids A, 4(1):41–57, 1992.
[AR91] Sergio Albeverio and Michael Röckner. Stochastic differential equations in infinite dimen-
sions: solutions via Dirichlet forms. Probab. Theory Relat. Fields, 89:347–386, 1991.
[Bał83] Tadeusz Bałaban. Ultraviolet stability in field theory. The ϕ43 model. In Scaling and self-
similarity in physics (Bures-sur-Yvette, 1981/1982), volume 7 of Progr. Phys., pages 297–
319. Birkhäuser Boston, Boston, MA, 1983.
[Bau12] Roland Bauerschmidt. A simple method for finite range decomposition of quadratic forms
and gaussian fields. arXiv preprint arXiv:1206.2212, 2012.
[BCM88] V. S. Borkar, R. T. Chari, and S. K. Mitter. Stochastic quantization of field theory in finite
and infinite volume. Journal of functional analysis, 81(1):184ￜ206, 1988.
[BDH95] D. Brydges, J. Dimock, and T. R. Hurd. The short distance behavior of (φ4)3. Comm.
Math. Phys., 172(1):143–186, 1995.
[BDH98] D. Brydges, J. Dimock, and T. R. Hurd. A non-Gaussian fixed point for φ4 in 4−ǫ dimensions.
Comm. Math. Phys., 198(1):111–156, 1998.
[BGM04] David C. Brydges, G. Guadagni, and P. K. Mitter. Finite range decomposition of Gaussian
processes. J. Statist. Phys., 115(1-2):415–449, 2004.
143
[BK91] J. Bricmont and A. Kupiainen. Random walks in asymmetric random environments. Comm.
Math. Phys., 142(2):345–420, 1991.
[BMS03] D. C. Brydges, P. K. Mitter, and B. Scoppola. Critical (Φ4)3,ǫ. Comm. Math. Phys., 240(1-
2):281–327, 2003.
[Bry09] David C. Brydges. Lectures on the renormalisation group. In Statistical mechanics, vol-
ume 16 of IAS/Park City Math. Ser., pages 7–93. Amer. Math. Soc., Providence, RI, 2009.
[BS10] David Brydges and Gordon Slade. Renormalisation group analysis of weakly self-avoiding
walk in dimensions four and higher. In Proceedings of the International Congress of Mathe-
maticians. Volume IV, pages 2232–2257, New Delhi, 2010. Hindustan Book Agency.
[BT06] David Brydges and Anna Talarczyk. Finite range decompositions of positive-definite func-
tions. J. Funct. Anal., 236(2):682–711, 2006.
[BY90] David Brydges and Horng-Tzer Yau. Grad φ perturbations of massless Gaussian fields.
Comm. Math. Phys., 129(2):351–392, 1990.
[CS11] Joseph Conlon and Thomas Spencer. Strong convergence to the homogenized limit of elliptic
equations with random coefficients. arXiv preprint arXiv:1101.4914, 2011.
[DD05] T. Delmotte and J.-D. Deuschel. On estimating the derivatives of symmetric diffusions in
stationary random environment, with applications to ∇φ interface model. Probab. Theory
Related Fields, 133(3):358–390, 2005.
[Del99] Thierry Delmotte. Parabolic Harnack inequality and estimates of Markov chains on graphs.
Rev. Mat. Iberoamericana, 15(1):181–232, 1999.
[Die] H.W. Diehl. Field-theoretic approach to critical behaviour at surfaces. Phase transitions
and critical phenomena Volume 10.
[Dim09] J. Dimock. Infinite volume limit for the dipole gas. J. Stat. Phys., 135(3):393–427, 2009.
[Dim11] Jonathan Dimock. The renormalization group according to balaban - i. small fields. arXiv
preprint arXiv:1108.1335, 2011.
[DPD03] G. Da Prato and A. Debussche. Strong solutions to the stochastic quantization equations.
Ann. Probab., 31(4):1900–1916, 2003.
144
[DT07] Giuseppe Da Prato and Luciano Tubaro. Wick powers in stochastic PDEs: an introduction.
Technical Report UTM 711, Matematica, University of Trento, page 39 pages, 2007. Available
online at http://eprints.biblio.unitn.it/1189/1/UTM711.pdf.
[DZ92] Giuseppe Da Prato and Jerzy Zabczyk. Stochastic equations in infinite dimensions, vol-
ume 44 of Encyclopedia of Mathematics and its Applications. Cambridge University Press,
Cambridge, 1992.
[DZ96] G. Da Prato and J. Zabczyk. Ergodicity for infinite-dimensional systems, volume 229 of
London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge,
1996.
[EJS13] Weinan E, Arnulf Jentzen, and Hao Shen. Renormalized powers of Ornstein-Uhlenbeck
processes and well-posedness of stochastic Ginzburg-Landau equations. arXiv preprint
arXiv:1302.5930, 2013.
[Fal12] Pierluigi Falco. Kosterlitz-Thouless transition line for the two dimensional Coulomb gas.
Comm. Math. Phys., 312(2):559–609, 2012.
[FNS77] Dieter Forster, David R. Nelson, and Michael J. Stephen. Large-distance and long-time
properties of a randomly stirred fluid. Phys. Rev. A (3), 16(2):732–749, 1977.
[FP78] Jürg Fröhlich and Yong Moon Park. Correlation inequalities and the thermodynamic limit
for classical and quantum continuous systems. Comm. Math. Phys., 59(3):235–266, 1978.
[FS81a] Jürg Fröhlich and Thomas Spencer. Kosterlitz-Thouless transition in the two-dimensional
plane rotator and Coulomb gas. Phys. Rev. Lett., 46(15):1006–1009, 1981.
[FS81b] Jürg Fröhlich and Thomas Spencer. The Kosterlitz-Thouless transition in two-dimensional
abelian spin systems and the Coulomb gas. Comm. Math. Phys., 81(4):527–602, 1981.
[FS81c] Jürg Fröhlich and Thomas Spencer. On the statistical mechanics of classical Coulomb and
dipole gases. J. Statist. Phys., 24(4):617–701, 1981.
[Gal85] Giovanni Gallavotti. Renormalization theory and ultraviolet stability for scalar fields via
renormalization group methods. Rev. Modern Phys., 57(2):471–562, 1985.
[GG96] D. Gatarek and B. Goldys. Existence,uniqueness and ergodicity for the stochastic quantiza-
tion equation. Studia Math., 119:179–193, 1996.
145
[GJ87] J. Glimm and A. Jaffe. Quantum physics: a functional integral point of view. Springer-Verlag
New York, 1987.
[GK80] K. Gaw‘edzki and A. Kupiainen. A rigorous block spin approach to massless lattice theories.
Comm. Math. Phys., 77(1):31–64, 1980.
[GK83] K. Gaw‘edzki and A. Kupiainen. Block spin renormalization group for dipole gas and (∇ϕ)4.
Ann. Physics, 147(1):198–243, 1983.
[Hai] Martin Hairer. Solving the KPZ equation. To appear in Ann. of Math. (2), page 103 pages.
[Hen81] Daniel Henry. Geometric theory of semilinear parabolic equations, volume 840 of Lecture
Notes in Mathematics. Springer-Verlag, Berlin, 1981.
[HRW12] Martin Hairer, Marc D. Ryser, and Hendrik Weber. Triviality of the 2D stochastic Allen-
Cahn equation. Electron. J. Probab., 17(39):1–14, 2012.
[JLM85] G. Jona-Lasinio and P. K. Mitter. On the stochastic quantization of field theory. Commu-
nications in Mathematical Physics, 101(3):409ￜ436, 1985.
[Kum10] T. Kumagai. Random walks on disordered media and their scaling limits. Notes of St. Flour
lectures, available at http://www.kurims.kyoto-u.ac.jp/ kumagai/, 2010.
[Law91] Gregory F. Lawler. Intersections of random walks. Probability and its Applications.
Birkhäuser Boston Inc., Boston, MA, 1991.
[LL10] Gregory F. Lawler and Vlada Limic. Random walk: a modern introduction, volume 123
of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge,
2010.
[LR98] Vitali Liskevich and Michael Röckner. Strong uniqueness for certain infinite-dimensional
Dirichlet operators and applications to stochastic quantization. Ann. Scuola Norm. Sup.
Pisa Cl. Sci. (4), 27(1):69–91 (1999), 1998.
[Lun09] Alessandra Lunardi. Interpolation theory. Appunti. Scuola Normale Superiore di Pisa (Nuova
Serie). [Lecture Notes. Scuola Normale Superiore di Pisa (New Series)]. Edizioni della Nor-
male, Pisa, second edition, 2009.
146
[MR99] R. Mikulevicius and B. L. Rozovskii. Martingale problems for stochastic PDEï¿œs. Stochastic
partial differential equations: six perspectives, 64:243ￜ326, 1999.
[Nee92] J. Neerven. The adjoint of a semigroup of linear operators. Springer-Verlag Berlin and New
York, 1992.
[OY99] Steven A. Orszag and Victor Yakhot. Analysis of the ǫ-expansion in turbulence theory:
approximate renormalization group for diffusion of a passive scalar in a random velocity
field. J. Sci. Comput., 14(2):147–195, 1999.
[Pol84] Joseph Polchinski. Renormalization and effective lagrangians. Nuclear Physics B,
231(2):269–295, 1984.
[PS95] Michael E. Peskin and Daniel V. Schroeder. An introduction to quantum field theory.
Addison-Wesley Publishing Company Advanced Book Program, Reading, MA, 1995. Edited
and with a foreword by David Pines.
[PW81] G. Parisi and Y.S. Wu. Perturbation theory without gauge fixing. Sci.Sinica, 24(4):483–496,
1981.
[Sim79] B. Simon. Functional integration and quantum physics, volume 86. Academic Press, 1979.
[SY02] George R. Sell and Yuncheng You. Dynamics of evolutionary equations, volume 143 of
Applied Mathematical Sciences. Springer-Verlag, New York, 2002.
[SZ06] Alain-Sol Sznitman and Ofer Zeitouni. An invariance principle for isotropic diffusions in
random environment. Invent. Math., 164(3):455–567, 2006.
[Wal86] John B. Walsh. An introduction to stochastic partial differential equations, volume 1180 of
Lecture Notes in Mathematics. Springer, Berlin, 1986.
[WK74] Kenneth G Wilson and John Kogut. The renormalization group and the ǫ expansion. Physics
Reports, 12(2):75–199, 1974.
[YO86] Victor Yakhot and Steven A. Orszag. Renormalization group analysis of turbulence. I. Basic
theory. J. Sci. Comput., 1(1):3–51, 1986.
147
![Renormalization Theory for Interacting Crumpled Manifolds · Renormalization Group equation to predict a scaling behavior. Beyond the one-loop calculations of [16][17][18][19] for](https://static.fdocuments.in/doc/165x107/5f0f6a717e708231d4440cf8/renormalization-theory-for-interacting-crumpled-manifolds-renormalization-group.jpg)
