CONTINUITY AND ESTIMATES OF THE LIOUVILLE HEAT KERNEL...

CONTINUITY AND ESTIMATES OF THE LIOUVILLE HEAT KERNEL WITHAPPLICATIONS TO SPECTRAL DIMENSIONS

SEBASTIAN ANDRES AND NAOTAKA KAJINO

ABSTRACT. The Liouville Brownian motion (LBM), recently introduced by Garban,Rhodes and Vargas and in a weaker form also by Berestycki, is a diffusion pro-cess evolving in a planar random geometry induced by the Liouville measure Mγ ,formally written as Mγ(dz) = eγX(z)−γ2E[X(z)2]/2dz, γ ∈ (0, 2), for a (massive)Gaussian free field X. It is an Mγ -symmetric diffusion defined as the time changeof the two-dimensional Brownian motion by the positive continuous additive func-tional with Revuz measure Mγ .

In this paper we provide a detailed analysis of the heat kernel pt(x, y) of theLBM. Specifically, we prove its joint continuity, a locally uniform sub-Gaussian up-per bound of the form pt(x, y) ≤ C1t

−1 log(t−1) exp(−C2(|x − y|β/t)

1β−1

)for

small t for each β > (γ + 2)2/2 and an on-diagonal lower bound of the formpt(x, x) ≥ C3t

−1(log(t−1)

)−η for small t for Mγ -a.e. x with some concrete con-stant η > 0. As applications, we also show that the pointwise spectral dimensionequals 2Mγ -a.e. and that the global spectral dimension is 2 as well.

1. INTRODUCTION

One of the main mathematical issues in the theory of 2d-Liouville quantum grav-ity is to construct a random geometry on a two-dimensional manifold (say R2

equipped with the Euclidian metric dx2) which can be formally described by a Rie-mannian metric tensor of the form

eγX(x) dx2, (1.1)

where X is a massive Gaussian free field on R2 on a probability space (Ω,A,P) andγ ∈ (0, 2) is a parameter. The study of Liouville quantum gravity is mainly motivatedby the so-called KPZ-formula (for Knizhnik, Polyakov and Zamolodchikov), whichrelates some geometric quantities in a number of models in statistical physics totheir formulation in a setup governed by this random geometry. In this context, bythe KPZ relation the parameter γ can be expressed in terms of the central chargeof the underlying model. We refer to [10] and to the survey article [13] for moredetails on this topic.

Date: January 29, 2015.2010 Mathematics Subject Classification. Primary: 60J35, 60J55, 60J60, 60K37; Secondary:

31C25, 60J45, 60G15.Key words and phrases. Liouville quantum gravity, Gaussian multiplicative chaos, Liouville Brown-

ian motion, heat kernel, spectral dimension.S.A. was partially supported by the CRC 1060 “The Mathematics of Emergent Effects”, Bonn.

1

2 SEBASTIAN ANDRES AND NAOTAKA KAJINO

However, to give rigorous sense to the expression (1.1) is a highly non-trivialproblem. Namely, as the correlation function of the Gaussian free field X exhibitsshort scale logarithmically divergent behaviour, the field X is not a function butonly a random distribution. In other words, the underlying geometry is too roughto make sense in the classical Riemannian framework, so some regularisation isrequired. While it is not clear how to execute a regularisation procedure on the levelof the metric, the method is quite performing in order to construct the associatedvolume form. More precisely, using the theory of Gaussian multiplicative chaosestablished by Kahane in [19] (see also [24]), by a certain cutoff procedure onecan define the associated volume measure Mγ for γ ∈ (0, 2), called the Liouvillemeasure. It can be interpreted as given by

Mγ(A) =

∫AeγX(z)−γ

2

2 E[X(z)2]dz,

but this expression for Mγ is only very formal, for Mγ is known to be singular w.r.t.the Lebesgue measure by a result [19, (141)] by Kahane (see also [24, Theorems4.1 and 4.2]). Recently, in [15] Garban, Rhodes and Vargas have constructed thenatural diffusion process (Bt)t≥0 associated with (1.1), which they call the LiouvilleBrownian motion (LBM). Similar results have been simultaneously obtained in [5].On a formal level, B is the solution of the SDE

dBt = e−γ2X(Bt)+

γ2

4 E[X(Bt)2]dBt,

where B is a standard Brownian motion on R2 independent of X. In view of theDambis-Dubins-Schwarz theorem this SDE representation suggests to define theLBM B as a time change of the planar Brownian motion B. This has been rigorouslycarried out in [15], and then by general theory the LBM turns out to be symmetricw.r.t. the Liouville measure Mγ . In the companion paper [16] Garban, Rhodes andVargas also identified the Dirichlet form associated with B and they showed thatthe transition semigroup is absolutely continuous w.r.t. Mγ , meaning that the Liou-ville heat kernel pt(x, y) exists. Moreover, they observed that the intrinsic metric dBgenerated by that Dirichlet form is identically zero, which indicates that

limt↓0

t log pt(x, y) = −dB(x, y)2

2= 0, x, y ∈ R2,

and therefore some non-Gaussian heat kernel behaviour is expected. This degen-eracy of the intrinsic metric is known to occur typically for diffusions on fractals,whose heat kernels indeed satisfy the so-called sub-Gaussian estimates; see e.g. thesurvey articles [3, 21] and references therein.

In this paper we continue the analysis of the Liouville heat kernel, which hasbeen initiated simultaneously and independently in [22]. As our first main resultwe obtain the continuity of the heat kernel.

CONTINUITY AND ESTIMATES OF THE LIOUVILLE HEAT KERNEL 3

Theorem 1.1. Let γ ∈ (0, 2). Then P-a.s. the Liouville heat kernel pt(x, y) is (0,∞)-valued and jointly continuous on (0,∞)×R2×R2, and in particular the Liouville Brow-nian motion B is irreducible. Moreover, the associated transition semigroup (Pt)t>0

defined by

Ptf(x) := Ex[f(Bt)] =

∫R2

pt(x, y)f(y)Mγ(dy)

is strong Feller, i.e. Ptf is continuous for any bounded Borel measurable f : R2 → R.

As mentioned above, due to the degeneracy of the intrinsic metric we expectanomalous off-diagonal bounds for the heat kernel to hold. Indeed, we obtain thefollowing sub-Gaussian upper bound. We denote by | · | the Euclidean norm on R2.

Theorem 1.2. Let γ ∈ (0, 2). Then P-a.s. the following holds. For any boundedU ⊂ R2 and any β > (γ + 2)2/2, there exist Ci = Ci(X, γ, U, β) > 0, i = 1, 2, suchthat

pt(x, y) = pt(y, x) ≤ C1 t−1 log(t−1) exp

(− C2

(|x− y|β ∧ 1

t

)1/(β−1))(1.2)

for all t ∈ (0, 12 ], x ∈ R2 and y ∈ U .

Since β > (γ+2)2/2 > 2, the off-diagonal part exp(−C2((|x−y|β∧1)/t)1/(β−1)

)of

the bound (1.2) indicates that the process diffuses slower than the two-dimensionalBrownian motion, which is why such a bound is called sub-Gaussian. We do notexpect that the lower bound (γ + 2)2/2 for the exponent β is best possible. Unfor-tunately, Theorem 1.2 alone does not even exclude the possibility that β could betaken arbitrarily close to 2, which in the case of the two-dimensional torus has beenin fact disproved in a recent result [22, Theorem 5.1] by Maillard, Rhodes, Vargasand Zeitouni showing that β satisfying (1.2) for small t must be at least 2 + γ2/4.

From the conformal invariance of the planar Brownian motion B it is natural toexpect that the LBM B as a time change of B admits two-dimensional behaviour, aswas observed by physicists in [1] and in a weak form proved in [25] (see Remark1.5 below). The on-diagonal part t−1 log(t−1) in (1.2) shows a sharp upper bound inthis spirit except for a logarithmic correction, and we will also prove the followingon-diagonal lower bound valid for Mγ-a.e. x ∈ R2, which matches (1.2) besidesanother logarithmic correction.

Theorem 1.3. Let γ ∈ (0, 2). Then, P-a.s., for Mγ-a.e. x ∈ R2 there exist C3 =

C3(X, γ, |x|) > 0 and t0(x) = t0(X, γ, x) ∈ (0, 12 ] such that

pt(x, x) ≥ C3 t−1(

log(t−1))−η

, ∀t ∈ (0, t0(x)], (1.3)

for some explicit constant η > 0 (for instance η = 34 is enough).

Combining the on-diagonal estimates in Theorems 1.2 and 1.3 we can immedi-ately identify the pointwise spectral dimension as 2.


Corollary 1.4. Let γ ∈ (0, 2). Then, P-a.s., for Mγ-a.e. x ∈ R2,

limt↓0

2 log pt(x, x)

− log t= 2.

Essentially from Theorem 1.2 and Corollary 1.4 we shall deduce that the globalspectral dimension, that is the growth order of the Dirichlet eigenvalues of thegenerator on bounded domains, is also 2; see Subsection 6.2 for details.

Remark 1.5. In [25, Theorem 3.6] the following result on the spectral dimensionhas been proved: Almost surely w.r.t. P, for any α > 0 and for all x ∈ R2,

limy→x

∫ ∞0

e−λttαpt(x, y) dt <∞, ∀λ > 0, (1.4)

and

limy→x

∫ ∞0

e−λtpt(x, y) dt =∞, ∀λ > 0. (1.5)

In [25] the left hand sides were interpreted as the integrals w.r.t. t of the on-diagonalheat kernel pt(x, x), which was needed due to the lack of the knowledge of thecontinuity of pt(x, y). By Theorem 1.1 this interpretation can be made rigorousnow, and moreover, (1.4) follows immediately from Theorem 1.2. On the otherhand, (1.5) is actually an easy consequence of the Dirichlet form theory. Indeed, by[12, Exercises 2.2.2 and 4.2.2]

∫∞0 e−λtpt(x, x) dt is equal to the reciprocal of the

λ-order capacity of the singleton x w.r.t. the LBM, and this capacity is zero by [12,Lemma 6.2.4 (i)] and the fact that the same holds for the planar Brownian motion.

The proofs of our main results above are mainly based on the moment estimatesfor the Liouville measure Mγ by [19, 26] and those for the exit times of the LBMB from balls by [15], together with the general fact from time change theory thatthe Green operator of the LBM has exactly the same integral kernel as that of theplanar Brownian motion (see (2.3) below). To turn those moment estimates intoP-almost sure statements, we need some Borel-Cantelli arguments that cannot pro-vide us with uniform control on various random constants over unbounded sets.For this reason we can expect the estimate (1.2) to hold only locally uniformly, sothat in Theorem 1.2 we cannot drop the dependence of the constants C1, C2 onU or the cutoff of |x − y| at 1 in the exponential. Also to remove the logarithmiccorrections in (1.2) and (1.3) and the restriction to Mγ-a.e. points in Theorem 1.3and Corollary 1.4 one would need to have good uniform control on the ratios of theMγ-measures of concentric balls with different radii, which we cannot hope for inview of [6, Appendix A].

The LBM can also be constructed on other domains like the torus, the sphereor planar domains D ⊂ R2 equipped with a log-correlated Gaussian field like the(massive or massless) Gaussian free field (cf. [15, Section 2.9]). In fact, Theo-rem 1.1 has been simultaneously and independently obtained in [22] for the LBMon the torus, where thanks to the boundedness of the space one can utilise the


eigenfunction expansion of the heat kernel to prove its continuity and the strongFeller property of the semigroup. On the other hand, in our case of R2 the Liouvilleheat kernel pt(x, y) does not admit such an eigenfunction expansion and the proofof its continuity and the strong Feller property requires some additional arguments.Therefore, although the proofs of our results should directly transfer to the otherdomains mentioned above, we have decided to work on the plane R2 in this paperfor the sake of simplicity and in order to stress that our methods also apply to thecase of unbounded domains.

In [22] Maillard, Rhodes, Vargas and Zeitouni have also obtained upper andlower estimates of the Liouville heat kernel on the torus. Their heat kernel upperbound in [22, Theorem 4.2] involves an on-diagonal part of the form Ct−(1+δ)

for any δ > 0 and an off-diagonal part of the form exp(− C(|x − y|β/t)1/(β−1)

)for any β > β0(γ), where β0(γ) is a constant larger than our lower bound (γ +

2)2/2 on the exponent β and satisfies limγ↑2 β0(γ) = ∞. Thus Theorem 1.2 givesa better estimate, and we prove it by self-contained, purely analytic argumentswhile the proof in [22] relies on (1.4), whose proof in [25] is technically involved.Concerning lower bounds, an on-diagonal lower bound as in Theorem 1.3 is nottreated in [22]. On the other hand, their off-diagonal lower bound [22, Theorem5.1], which implies the bound β ≥ 2 + γ2/4 for any such exponent β as in (1.2) (inthe case of the torus) as mentioned above after Theorem 1.2, is not covered by ourresults.

The rest of the paper is organised as follows. In Section 2 we recall the con-struction of the LBM [15] and we introduce the precise setup. In Section 3 weprove preliminary estimates on the volume decay of the Liouville measure and onthe exit times from balls needed in the proofs. In Section 4 we show that the resol-vent operators of the killed LBM have the strong Feller property, which is needed inSection 5, where we prove Theorems 1.1 and 1.2. In Subsection 5.1 we show thecontinuity of the Dirichlet heat kernel associated with the LBM killed upon exitinga bounded domain by using its eigenfunction expansion, and in Subsection 5.2 wethen deduce the continuity of the heat kernel and the strong Feller property on un-bounded domains, as well as Theorem 1.2, using a recent result in [18]. Finally,in Section 6 we show the on-diagonal lower bound in Theorem 1.3 and therebyidentify the pointwise and global spectral dimensions as 2.

Throughout the paper, we write C for random positive constants depending onthe realisation of the field X, which may change on each appearance, whereas thenumbered random positive constants Ci will be kept the same. Analogously, non-random positive constants will be denoted by c or ci, respectively. We denote by | · |the Euclidean norm on R2 and by B(x,R) = y ∈ R2 : |x− y| < R, x ∈ R2, R > 0,open Euclidean balls in R2 and for abbreviation we set BR := B(0, R). Finally, forU ⊆ R2 we write ‖f‖∞ := supx∈U |f(x)| for bounded f : U → R.


2. LIOUVILLE BROWNIAN MOTION

2.1. Massive Gaussian free field. Consider a massive Gaussian free field X on thewhole plane R2, i.e. a Gaussian Hilbert space associated with the Sobolev spaceH1

m

defined as the closure of C∞c (R2) w.r.t. the inner product

〈f, g〉m := m2

∫R2

f(x) g(x) dx+

∫R2

∇f(x) · ∇g(x) dx,

where the quantity m > 0 is called the mass. More precisely, (〈X, f〉m)f∈H1m

is afamily of centered Gaussian random variables on a probability space (Ω,A,P) with

E[〈X, f〉m 〈X, g〉m

]= 2π 〈f, g〉m.

In other words, the covariance function is given by the massive Green functionassociated with the operator m2 −∆, which can be written as

g(m)(x, y) =

∫ ∞0

e−m2

2u− |x−y|

2

2udu

2u=

∫ ∞1

k(m)(u(x− y))

udu,

where

k(m)(z) =1

2

∫ ∞0

e−m2

2v|z|2− v

2 dv.

Following [15] we now introduce an n-regularised version of X. To that aim let(cn)n≥1 be an unbounded increasing sequence with c1 = 1 and let (Yn)n≥1 be afamily of independent centered continuous Gaussian fields on R2 defined also on(Ω,A,P) with covariance

E[Yn(x)Yn(y)

]=

∫ cn+1

cn

k(m)(u(x− y))

udu =: kn(x, y). (2.1)

Then, the n-regularised field is defined as

Xn(x) =n∑k=1

Yk(x),

the associated random measure Mn = Mn,γ is given by

Mn(dx) = exp(γXn(x)− γ2

2 E[Xn(x)2])dx.

Here the factor γ ≥ 0 is a parameter. By the classical theory of Gaussian multiplica-tive chaos, established in Kahane’s seminal work [19] (see also [24]), P-a.s. thefamily (Mn)n converges vaguely to a limiting Radon measure M = Mγ called theLiouville measure, which has full support if γ ∈ [0, 2) and is identically zero if γ ≥ 2.Throughout the paper, we will assume from now on that γ ∈ (0, 2) is fixed.


2.2. Definition of Liouville Brownian Motion. The Liouville Brownian motionhas been constructed by Garban, Rhodes and Vargas in [15] as the canonical diffu-sion process under the geometry induced by the field X and the measure M . Moreprecisely, they have constructed a positive continuous additive functional Ftt≥0

of the planar Brownian motion B naturally associated with the measure M andthey have defined the LBM as Bt = BF−1

t. In this subsection we briefly recall the

construction.Let Ω′ := C([0,∞),R2) and G be the Borel σ-field on Ω′. Further, let Pxx∈R2 be

the family of probability measures on (Ω′,G) such that under each Px the coordinateprocess (Bt)t≥0 is a two-dimensional Brownian motion starting at x. We denote byG0t = σ(Bs; s ≤ t), t ≥ 0, G0

∞ = σ(Bs; s < ∞), and by Gtt∈[0,∞] the minimumcompleted admissible filtration for B as defined e.g. in [12, Section A.2]. Moreover,let θtt≥0 be the family of shift mappings on Ω′, i.e. Bt+s = Bt θs, s, t ≥ 0. Finally,we write qt(x, y) = (2πt)−1 exp(−|x−y|2/2t), t ≥ 0, x, y ∈ R2 for the Gaussian heatkernel on R2.

Definition 2.1. i) A process A = (At)t≥0 is a positive additive functional (PCAF) of Bin the strict sense, if At is a Gt-measurable, [0,∞]-valued random variable for everyt ≥ 0 and if there exists a set Λ ∈ G∞, called a defining set for A, such that

a) for all x ∈ R2, Px[Λ] = 1,b) for all t ≥ 0, θt(Λ) ⊂ Λ,c) for all ω ∈ Λ, s, t ≥ 0,

At+s(ω) = At(ω) +As θt(ω),

and t 7→ At(ω) is a [0,∞)-valued continuous function with A0(ω) = 0.

ii) Two such functionals A1 and A2 are called equivalent if Px[A1t = A2

t ] = 1 forall t > 0, x ∈ R2, or equivalently, there exists a defining set Λ ∈ G∞ for both A1 andA2 such that A1

t (ω) = A2t (ω) for all t ≥ 0, ω ∈ Λ.

iii) For any such A a Borel measure µA on R2 satisfying∫R2

f(y)µA(dy) = limt↓0

1

t

∫R2

Ex

[ ∫ t

0f(Bs) dAs

]dx,

for any non-negative Borel function f is called the Revuz measure of A, which existsuniquely by general theory (see e.g. [7, Theorem A.3.5]).

For every n ∈ N let now Fnt : Ω× Ω′ → [0,∞) be defined as

Fnt :=

∫ t

0exp(γXn(Bs)− γ2

2 E[Xn(Bs)2])ds, t ≥ 0,

which is strictly increasing. Note that for every n the functional Fn is a PCAF of Bin the strict sense with defining set Ω′ and Revuz measure Mn.

Theorem 2.2 ([15, Theorem 2.7]). Almost surely w.r.t. P the following hold:i) There exists a unique PCAF F whose Revuz measure is M .

ii) For all x ∈ R2, Px-a.s., F is strictly increasing and satisfies limt→∞ Ft =∞.


iii) For all x ∈ R2, Fn converges to F in Px-probability in C([0,∞),R), equippedwith the topology of uniform convergence on compact sets.

The process (B, Pxx∈R2), P-a.s. defined by Bt := BF−1t

, t ≥ 0, is called the (massive)Liouville Brownian motion (LBM).

Thanks to Theorem 2.2, we can apply the general theory of time changes ofMarkov processes to have the following properties of the LBM: first, it is a recurrentdiffusion on R2 by [12, Theorems A.2.12 and 6.2.3]. Furthermore by [12, Theorem6.2.1 (i)] (see also [15, Theorem 2.18]), the LBM is M -symmetric, i.e. its transitionsemigroup (Pt) given by

Pt(x,A) := Ex[Bt ∈ A]

for t ∈ (0,∞), x ∈ R2 and a Borel set A ⊂ R2, satisfies∫R2

Ptf · g dM =

∫R2

f · Ptg dM

for all Borel measurable functions f, g : R2 → [0,∞]. Here the Borel measurabil-ity of Pt(·, A) can be deduced from [15, Corollary 2.20] (or from Proposition 2.4below).

Remark 2.3. It is claimed in [15, Corollary 2.20] that (Pt) is a Feller semigroup,meaning that Pt preserves the space of bounded continuous functions. Note thatthis is different from the notion of a Feller semigroup as for instance in [7, 12],i.e. a strongly continuous Markov semigroup on the space of continuous functionsvanishing at infinity. It is not known whether (Pt) is a Feller semigroup in the lattersense.

It is natural to expect that the LBM can be constructed in such a way that it de-pends measurably on the randomness of the field X. However, this measurabilitydoes not seem obvious from the construction in [15], since there the existence of thePCAF F has been deduced from some general theory on the Revuz correspondencefor P-a.e. fixed realisation of M . To overcome this issue, in the following proposi-tion we show the pathwise convergence of Fn towards F which also ensures themeasurability of F and B w.r.t. A⊗ G0

∞. The proof is given in Appendix A.

Proposition 2.4. There exists a set Λ ∈ A⊗ G0∞ such that the following hold:

i) For P-a.e. ω ∈ Ω, Px[Λω] = 1 for any x ∈ R2, where Λω := ω′ ∈ Ω′ :

(ω, ω′) ∈ Λ.ii) For every (ω, ω′) ∈ Λ the following limits exist in R for all 0 < s < t:

Fs,t(ω, ω′) := lim

n→∞(Fnt (ω, ω′)− Fns (ω, ω′)),

Ft(ω, ω′) := lim

s′↓0Fs′,t(ω, ω

′).

Moreover, with F0(ω, ω′) := 0, F·(ω, ω′) : [0,∞) → [0,∞) is continuous,strictly increasing and satisfies limt→∞ Ft(ω, ω

′) =∞.


iii) Let t ≥ 0 and set Ft := t on Λc. Then Ft is A⊗ G0∞-measurable.

iv) For P-a.e. ω ∈ Ω, the process (Ft(ω, ·))t≥0 is a PCAF of B in the strict sensewith defining set Λω.

The previous proposition implies easily that F indeed has the Revuz measure M .More strongly, we have the following proposition valid for any starting point x ∈ R2,which we prove for later use in a slightly more general setting in Appendix B.

Proposition 2.5. Almost surely w.r.t. P the following holds. For any x ∈ R2 and anyBorel functions η : [0,∞)→ [0,∞] and f : R2 → [0,∞],

Ex

[ ∫ ∞0

η(t) f(Bt) dFt

]=

∫ ∞0

∫R2

η(t) f(y) qt(x, y)M(dy) dt,

and in particular, for any t > 0,∫R2

f(y)M(dy) =1

t

∫R2

Ex

[ ∫ t

0f(Bs) dFs

]dx.

2.3. The Liouville Dirichlet form. By virtue of Propositions 2.4 and 2.5, we canapply the general theory of Dirichlet forms to obtain an explicit description of theDirichlet form associated with the LBM, as it has been done in [15, 16].

Denote by H1(R2) the standard Sobolev space, that is

H1(R2) = f ∈ L2(R2, dx) : ∇f ∈ L2(R2, dx),

on which we define the form

E(f, g) =1

2

∫R2

∇f · ∇g dx. (2.2)

Recall that (E , H1(R2)) is the Dirichlet form of the planar Brownian motion B. ByH1e (R2) we denote the extended Dirichlet space, that is the set of dx-equivalence

classes of Borel measurable functions f on R2 such that limn→∞ fn = f ∈ R dx-a.e.for some (fn)n ⊂ H1(R2) satisfying limk,l→∞ E(fk−fl, fk−fl) = 0. By [7, Theorem2.2.13] we have the following identification of H1

e (R2):

H1e (R2) = f ∈ L2

loc(R2, dx) : ∇f ∈ L2(R2, dx).

The capacity of a set A ⊂ R2 is defined by

Cap(A) = infB⊂R2openA⊂B

inff∈H1(R2)

f |B≥1 dx-a.e.

E(f, f) +

∫R2

f2 dx

A set A ⊂ R2 is called polar if Cap(A) = 0. We call a function f quasi-continuousif for any ε > 0 there exists an open U ⊂ R2 with Cap(U) < ε such that f |R2\Uis real-valued and continuous. By [12, Theorem 2.1.7] any f ∈ H1

e (R2) admitsa quasi-continuous dx-version f , which is unique up to polar sets by [12, Lemma2.1.4].

Then, as the Liouville measure M is a Radon measure on R2 and does not chargepolar sets by [15, Theorem 2.2] (or by Propositions 2.4, 2.5 and [7, Theorem 4.1.1(i)]), the Dirichlet form (E ,F) of the LBM B is a strongly local, regular Dirichlet


form on L2(R2,M) which takes on the following explicit form by [12, Theorem6.2.1]: The domain is given by

F =u ∈ L2(R2,M) : u = f M -a.e. for some f ∈ H1

e (R2),

which can be identified with f ∈ H1e (R2) : f ∈ L2(R2,M) by [12, Lemma 6.2.1],

and for f, g ∈ F the form E(f, g) is given by (2.2).

2.4. The killed Liouville Brownian motion. Let U be a non-empty open subsetof R2 and let U ∪ ∂U be its one-point compactification. We denote by TU :=

infs ≥ 0 : Bs 6∈ U the exit time of the Brownian motion B from U and byτU := infs ≥ 0 : Bs 6∈ U that of the LBM B. Since by definition Bt = BF−1

t, t ≥ 0,

and F is a homeomorphism on [0,∞), we have τU = FTU . Let now BU and BUdenote the Brownian motion and the LBM, respectively, killed upon exiting U . Thatis, they are diffusions on U defined by

BUt :=

Bt if t < TU ,

∂U if t ≥ TU ,BUt :=

Bt if t < τU ,

∂U if t ≥ τU .

Then, for t, λ ∈ (0,∞) and a Borel function f : U → R we write

PUt f(x) := Ex[f(BUt )

]and RUλ f(x) := Ex

[ ∫ τU

0e−λtf(Bt) dt

]for the semigroup and resolvent operators associated with the killed LBM, respec-tively. If U is bounded, as a time change of BU the killed LBM BU has the sameintegral kernel for its Green operator GU as BU (cf. Proposition B.1), namely forany non-negative Borel function f ,

GUf(x) := Ex

[ ∫ τU

0f(Bt) dt

]= Ex

[ ∫ TU

0f(Bt) dFt

]=

∫UgU (x, y)f(y)M(dy).

(2.3)

Here gU denotes the Euclidian Green kernel given by

gU (x, y) =

∫ ∞0

qUt (x, y) dt, x, y ∈ R2,

for the jointly continuous transition density qUt of BU : qUt (x, y)dy = Px[BUt ∈ dy]

and qUt (x, y) = 0 if (x, y) 6∈ U × U . Finally, we recall (see e.g. [12, Example 1.5.1])that the Green function gB(x0,R) over a ball B(x0, R) is of the form

gB(x0,R)(x, y) =1

πlog

1

|x− y|+ Ψx0,R(x, y), (2.4)

for some bounded continuous function Ψx0,R on B(x0, R)×B(x0, R).


3. PRELIMINARY ESTIMATES

3.1. Volume decay estimates. For our analysis of the Liouville heat kernel somegood control on the volume of small balls under the Liouville measure is needed.An upper estimate has already been established in [15], so in the next lemma weprovide a lower bound. The argument is based on some bounds on the negativemoments of the mass of small balls. Such bounds have been proven in [26] inthe case where the multiplicative chaos is associated with a family of Gaussianfields whose covariance function is contained a certain class of convolution kernels.Since it is not clear to the authors whether the cut-off procedure producing theapproximating measures Mn is covered by the results in [26] we give a comparisonargument in Lemma C.1.

Lemma 3.1. Let α1 := 12(γ + 2)2 and α2 := (2 − γ)2/2. Then, P-a.s., for any ε > 0

and any R ≥ 1, there exist Ci = Ci(X, γ,R, ε) > 0, i = 4, 5, such that

C4 rα1+ε ≤M(B(x, r)) ≤ C5 r

α2−ε, ∀x ∈ BR, r ∈ (0, 1]. (3.1)

Proof. By the monotonicity of (3.1) in ε it suffices to show (3.1) P-a.s. for each ε.The upper bound is proven in [15, Theorem 2.2]. We show the lower bound in thesame manner. Note that α1 = (2 + ξ(q))/q for q := 2/γ where ξ(q) := 2q+ q(1+q)

2 γ2.Let R ≥ 1 be fixed and, setting rn := 2−nR for n ∈ Z, we define

ΛR,n :=(

k2nR,

l2nR

)| k, l ∈ Z, −2n ≤ k, l ≤ 2n

⊂ [−R,R]2. (3.2)

Then, by Cebysev’s inequality

P[

infx∈ΛR,n

M(B(x, rn)) ≤ 2−n(α1+ε)]

= P[

supx∈ΛR,n

M(B(x, rn))−q ≥ 2n(α1+ε)q]≤∑

x∈ΛR,n

P[M(B(x, rn))−q ≥ 2n(α1+ε)q

]≤ 2−n(α1+ε)q

∑x∈ΛR,n

E[M(B(x, rn))−q

]= 2−n(α1+ε)q 22n E

[M(B(0, rn))−q

]= c 2n(2+ξ(q)−(α1+ε)q) = c 2−εqn

for some constant c = c(γ,R) > 0, where we used the spatial stationarity of the lawof M and Lemma C.1. In particular,

∑∞n=1 P

[infx∈ΛR,n B(x, rn) ≤ 2−n(α1+ε)

]<∞,

so that by the Borel-Cantelli lemma P-a.s. for someC = C(X, γ,R) > 0 we have thatM(B(x, rn)) ≥ C2−n(α1+ε) for all x ∈ λR,n for all n ≥ 1. Since for every y ∈ BR andr ∈ (0, 1] we have B(y, r) ⊃ B(x, rn) for some x ∈ ΛR,n with n satisfying 2−n r,the claim follows.

3.2. Exit time estimates. In this subsection we provide some lower estimates onthe exit times from balls, which are needed in the proof of Theorems 1.1 and 1.2.More precisely, we establish estimates on the negative moments and the tail be-haviour at zero of these exit times.


Proposition 3.2. For any q > 0, P-a.s., for any κ > 2+ξ(q) with ξ(q) := 2q+ q(1+q)2 γ2

and any R ≥ 1, there exists a random constant C6 = C6(X, γ,R, q, κ) > 0 such that

Ex[τ−qB(x,r)

]≤ C6 r

−κ, ∀r ∈ (0, 1], x ∈ BR. (3.3)

Proof. Since (3.3) is weaker for larger κ and smaller R, it suffices to show (3.3)P-a.s. for each κ and R. First we note that from [15, Proposition 2.12] and Fatou’slemma, we get for all q > 0 and for all x ∈ R2,

EEx[τ−qB(x,r)

]≤ c r−ξ(q), ∀r ∈ (0, 1], (3.4)

for some c = c(γ, q) > 0. As in the proof of Lemma 3.1 above let rn := 2−nR

and ΛR,n be defined as in (3.2) for any n ∈ Z. In the sequel we write Eµ forthe expectation operator associated with the law of a Brownian motion with initialdistribution µ. Let x ∈ R2 and let µx,rn = Px[BτB(x,rn)

∈ ·] be the distribution of theLBM upon exiting B(x, rn). Since B(z, rn) ⊂ B(x, 2rn) for any z ∈ ∂B(x, rn), wehave τB(z,rn) ≤ τB(x,2rn), Pz-a.s. Hence, using (3.4) we get

EEz[τ−qB(x,2rn)

]≤ EEz

[τ−qB(z,rn)

]≤ c r−ξ(q)n ,

provided that n is large enough so that rn ≤ 1. The µx,rn(dz)-integral of this be-comes

EEµx,rn[τ−qB(x,2rn)

]≤ c r−ξ(q)n .

Let now κ > 2 + ξ(q) and n0 satisfying rn0 ≤ 1 be fixed. Then, for all n ≥ n0 weobtain by Cebysev’s inequality,

P[

supx∈ΛR,n+1

Eµx,rn[τ−qB(x,2rn)

]≥ r−κn

]≤ rκn

∑x∈ΛR,n+1

EEµx,rn[τ−qB(x,2rn)

]≤ c

2n(κ−ξ(q)−2)

for some c = c(R, q, κ) > 0. Hence, by our choice of κ,∑n≥n0

P[

supx∈ΛR,n+1


]≥ r−κn

]<∞

and we apply the Borel-Cantelli lemma to obtain that P-a.s. for all n with rn ≤ 1

and for all x ∈ ΛR,n+1,


]≤ C r−κn (3.5)

for some random constant C = C(X, γ,R, q, κ) > 0.Now for any r ∈ (0, 1] we choose n ≥ n0 such that rn ≤ 2

5r < 2rn. For all y ∈ BR,by construction there exists an x ∈ ΛR,n+1 such that |x − y| ≤ 1

2rn. Furthermore,note thatB(x, rn) ⊂ B(x, 2rn) ⊂ B(y, r) and therefore τB(x,rn) < τB(x,2rn) ≤ τB(y,r),


Py-a.s. In particular, by using the strong Markov property

Ey[τ−qB(y,r)

]≤ Ey

[τ−qB(x,2rn)

]≤ Ey

[(τB(x,rn) + τB(x,2rn) θτB(x,rn)

)−q]≤ Ey

[(τB(x,2rn) θτB(x,rn)

)−q]= EyEBτB(x,rn)

[τ−qB(x,2rn)

]= Eµyx,rn

[τ−qB(x,2rn)

],

where µyx,rn := Py[BτB(x,rn)∈ ·]. Since the exit laws of the LBM and the planar

Brownian motion coincide, i.e. µyx,rn = Py[BTB(x,rn)∈ ·], the exact formula for

the distribution of a Brownian motion upon exiting balls (see e.g. [4, TheoremII.1.17]) implies that µyx,rn ≤ cµx,rn for some explicit constant c > 0. (This can beregarded as an application of the scale-invariant elliptic Harnack inequality.) Thus,Ey[τ−qB(y,r)

]≤ cEµx,rn [τ−qB(x,2rn)] and the claim follows from (3.5).

Proposition 3.3. Almost surely w.r.t. P, for any β > α1 and any R ≥ 1 there existrandom constants Ci = Ci(X, γ,R, β) > 0, i = 7, 8, such that

Px[τB(x,r) ≤ t] ≤ C7 exp(−C8(rβ/t)1/(β−1)

), ∀x ∈ BR, r ∈ (0, 1], t > 0.

Proof. Let β > α1 and R ≥ 1. By [18, Theorem 7.2] (see also [17, Theorem 9.1]) itis enough to show that there exist ε ∈ (0, 1) and δ > 0 such that Px

[τB(x,r) ≤ t

]≤ ε

for all x ∈ BR, r ∈ (0, 1] and t ∈ (0, δrβ]. Indeed, set q := 2/γ and let ε ∈ (0, 1),δ := (ε/C3)1/q and κ := (α1 + β)q/2, so that κ > α1q = 2 + ξ(q) by β > α1 =

(2 + ξ(q))/q. Then, by Cebysev’s inequality and Proposition 3.2 we have

Px[τB(x,r) ≤ δrβ

]= Px

[(τB(x,r))

−q ≥ (δrβ)−q]≤ C3 δ

q rβq−κ ≤ ε,

proving the claim.

4. STRONG FELLER PROPERTY OF THE RESOLVENTS

In this section we prove that the resolvent operator of the killed LBM BU has thestrong Feller property. We will mainly follow the arguments in [16, Theorem 2.4],where the corresponding Feller property of the original LMB B is established. Theessential ingredients are a coupling lemma and the following lemma.

Lemma 4.1 ([14, Lemma 2.19]). Almost surely w.r.t. P, for all R > 0,

limT→0

supn≥1

supx∈BR

Ex[FnT ] = 0.

Proposition 4.2. Almost surely w.r.t. P, for any non-empty open set U ⊆ R2 and forany λ > 0 the resolvent operator RUλ is strong Feller, i.e. it maps Borel measurablebounded functions on U to continuous bounded functions on U .

Proof. Let f : U → R be Borel measurable and bounded and recall that TU =

infs ≥ 0 : Bs 6∈ U denotes the exit time of the Brownian motion B from U . First


note that since τU = FTU , the resolvent operator of BU can be written as

RUλ f(x) = Ex

[ ∫ τU

0e−λtf(Bt) dt

]= Ex

[ ∫ TU

0e−λFtf(Bt) dFt

]= Ex

[ ∫ TU∧ε

0e−λFtf(Bt) dFt

]+ Ex

[ ∫ TU

TU∧εe−λFtf(Bt) dFt

]=: Nε(x) +RU,ελ f(x), (4.1)

for any ε > 0. From Lemma 4.1 we deduce that Nε(x) ≤ ‖f‖∞Ex[Fε] convergestowards zero uniformly over compact subsets of R2 as ε → 0. Hence, it is enoughto show that RU,ελ f is continuous. By the Markov property of the Brownian motionwe obtain that

RU,ελ f(x) = Ex

[1lTU>ε

∫ TU

εe−λFtf(Bt) dFt

]= Ex

[1lTU>εEBε

[ ∫ TU

0e−λFtf(Bt) dFt

]e−λFε

]= Ex

[1lTU>εe

−λFεRUλ f(Bε)].

Now for any two points x, y ∈ R2 we use the coupling lemma [15, Lemma 2.9],which allows to construct a couple (Bx, By) consisting of two Brownian motionsBx and By starting at x and y, respectively, with the property that both processescoincide after a stopping time τxy. We denote by Px,y the law of (Bx, By) and byEx,y the corresponding expectation. Then, we obtain∣∣RU,ελ f(x)−RU,ελ f(y)

∣∣=∣∣∣Ex,y[1lTxU>εe−λFxε RUλ f(Bx

ε )]− Ex,y

[1lT yU>ε

e−λFyε RUλ f(By

ε )]∣∣∣

≤∣∣∣Ex,y[1lTxU>εe−λFxε (RUλ f(Bx

ε )−RUλ f(Byε ))]∣∣∣

+∣∣∣Ex,y[(1lTxU>εe−λFxε − 1lT yU>ε

e−λFyε)RUλ f(By

ε )]∣∣∣ . (4.2)

Note that on the event τxy ≤ ε ∩ T xU > ε ∩ T yU > ε we have RUλ f(Bxε ) =

RUλ f(Byε ). Thus, the first term in (4.2) can be estimated from above by

Ex,y

[1lTxU>ε

∣∣RUλ f(Bxε )−RUλ f(By

ε )∣∣] ≤ 2λ−1‖f‖∞Ex,y

[1lτxy>ε∪T yU≤ε

]≤ 2λ−1‖f‖∞

(Px,y[τxy > ε] + Px,y[T

yU ≤ ε]

),

where we used the trivial bound ‖RUλ f‖∞ ≤ λ−1‖f‖∞. Since for some r > 0 smallenough,

Px,y[T yU ≤ ε

]≤ Px,y

[T yB(y,r) ≤ ε

]= Py

[TB(y,r) ≤ ε

]≤ 2e−r

2/4ε,

the second term converges to zero uniformly over y in a compact set as ε → 0

(see e.g. [4, Proposition I.4.8]). On the other hand, for any ε > 0 the first termconverges to zero as |x− y| → 0 (cf. again [15, Lemma 2.9]).


The second term in (4.2) can be estimated from above by

λ−1‖f‖∞Ex,y[∣∣1lTxU>ε − 1lT yU>ε

∣∣ e−λFxε + 1lT yU>ε∣∣e−λFxε − e−λF yε ∣∣]

≤ λ−1‖f‖∞(Px,y [T xU ≤ ε] + Px,y

[T yU ≤ ε

]+ Ex,y

[∣∣e−λFxε − e−λF yε ∣∣]) .As above, the first two terms converge to zero uniformly over compact sets as ε→ 0.Moreover, by the same arguments as in [16, pp. 10–11],

lim|x−y|→0

Ex,y

[∣∣e−λFxε − e−λF yε ∣∣] = 0, (4.3)

which then finishes the proof. For the sake of completeness we repeat the argumenthere. For any ε > 0,

Ex,y

[∣∣e−λFxε − e−λF yε ∣∣] ≤ 2Px,y[ε ≤ τxy] + Ex,y

[∣∣e−λFxε − e−λF yε ∣∣ 1lε>τxy]= 2Px,y[ε ≤ τxy] + Ex,y

[∣∣∣e−λ(Fxτxy+Fxε −Fxτxy)− e−λ

(F yτxy+F yε −F yτxy

)∣∣∣ 1lε>τxy] .Since on the event ε > τxy, we have F xε − F xτxy = F yε − F yτxy , we further obtainthat

Ex,y

[∣∣e−λFxε − e−λF yε ∣∣] ≤ 2Px,y [ε ≤ τxy] + Ex,y

[∣∣e−λFxτxy − e−λF yτxy ∣∣ 1lε>τxy]≤ 2Px,y [ε ≤ τxy] + Ex,y

[min

(2, λ|F xτxy − F

yτxy |)]

≤ 2Px,y[ε ≤ τxy] + Ex,y[min

(2, λF xδ + λF yδ

)]+ 2Px,y[δ < τxy],

for any arbitrary δ > 0. Thus, using the coupling lemma [15, Lemma 2.9] as above,

lim supy→x

Ex,y

[∣∣e−λFxε − e−λF yε ∣∣] ≤ Ex [min(2, 2λFδ

)].

Finally, since limδ→0Ex[Fδ] = 0 uniformly over x in a compact subset of R2 (seeagain Lemma 4.1), we obtain (4.3).

5. CONTINUITY AND UPPER BOUNDS OF THE HEAT KERNELS

Throughout Sections 5 and 6 we fix any environment ω ∈ Ω such that all theconclusions of Proposition 2.4 iv), Lemma 3.1, Propositions 3.3, 4.2 and B.1 hold.

The purpose of this section is to prove Theorem 5.1 below on the continuity ofthe heat kernels as well as Theorem 1.2. Recall that F equipped with the norm‖f‖2F := E(f, f) + ‖f‖2L2(R2,M) is a Hilbert space. For any open set U ⊂ R2, wedefine FU to be the closure in (F , ‖ · ‖F ) of the set of all functions in F that arecompactly supported in U . Then, it is well known that (E ,FU ) is the Dirichlet formassociated with the killed Liouville Brownian motion BU and that it is regular onL2(U,M) (see e.g. [12, Theorems 4.4.2 and 4.4.3]). The corresponding generatoron L2(U,M) is denoted by LU , its domain by D(LU ), and the associated semigroupand resolvent operators by (TUt )t>0 and (GUλ )λ>0, respectively.

Theorem 5.1. For any non-empty open set U ⊆ R2 the following hold:


i) There exists a (unique) jointly continuous function pU = pUt (x, y) : (0,∞) ×U × U → [0,∞) such that for all (t, x) ∈ (0,∞) × U , Px[BUt ∈ dy] =

pUt (x, y)M(dy), which we refer to as the Dirichlet Liouville heat kernel on U .ii) The semigroup operator PUt is strong Feller, i.e. it maps Borel measurable

bounded functions on U to continuous bounded functions on U .iii) If U is connected, pUt (x, y) ∈ (0,∞) for any (t, x, y) ∈ (0,∞)×U ×U , and in

particular, BU is irreducible.

From now on we will write pt(·, ·) instead of pR2

t (·, ·) and call it the (global)Liouville heat kernel. Note that Theorem 1.1 follows directly from Theorem 5.1 bychoosing U = R2.

5.1. The heat kernel on bounded open sets. In this subsection we will proveTheorem 5.1 for a fixed non-empty, bounded, open U ⊂ R2. The case of unboundedopen sets will be treated later in Subsection 5.2. We denote by ‖f‖p the Lp(U,M)-norm, p ≥ 1, and by 〈·, ·〉 the L2(U,M)-inner product. Let R > 0 be such thatU ⊂ BR.

Proposition 5.2 (Faber-Krahn-type inequality). There exists C9 = C9(X, γ,R) > 0

such that the smallest eigenvalue λ1(U) of −LU satisfies

λ1(U) ≥ C9

M(U) log(2 + 1

M(U)

) .Proof. First we recall that λ1(U) can be expressed by the variational formula

λ1(U)−1 = sup 〈GUf, f〉, f ≥ 0, ‖f‖2 = 1 (5.1)

by [12, Theorems 1.5.4 and 4.2.6], since gU ∈ L2(U × U,M ×M) by (2.4) and acalculation similar to (B.3). Setting ν := πα2/2 = (1− γ

2 )2, as U ⊂ BR we have forany f ≥ 0 with ‖f‖2 = 1,

〈GUf, f〉 ≤ 〈GBRf, f〉 =

∫U

∫U

exp(νgBR(x, y)

)M(dy)M(dx) (5.2)

+

∫U

∫U

f(x) f(y)

νlog(

1 +f(x) f(y)

ν

)M(dy)M(dx).

Here we used the fact that ab ≤ a log(1 + a) + eb for any a, b ≥ 0, which can beeasily verified, with a = f(x) f(y)

ν and b = νgBR(x, y). Recall the Green function overBR can be represented as in (2.4), which implies∫U

∫U

exp(νgBR(x, y)

)M(dy)M(dx) ≤ cM(U)2 +

∫U

∫U

1

|x− y|ν/πM(dy)M(dx).


SettingDn := B(x, 21−nR)\B(x, 2−nR) and using Lemma 3.1 with ε ∈ (0, α2−ν/π)

we further obtain∫U

∫U

1

|x− y|ν/πM(dy)M(dx) ≤

∫U

∫B(x,2R)

1

|x− y|ν/πM(dy)M(dx)

≤∞∑n=0

∫U

∫Dn

(2−nR)−ν/πM(B(x, 21−nR)

)M(dx) ≤ CM(U)

for some C = C(X, γ,R) > 0. Denoting MU = M(· ∩U)/M(U), the second term in(5.2) becomes

M(U)2

∫U

∫U

f(x) f(y)

νlog(

1 +f(x) f(y)

ν

)MU (dy) MU (dx).

Set H(s) = s2 and I(s) := s log(1 + s), s ≥ 0. Then, the function H I−1 is convexand we apply Jensen’s inequality to obtain

H I−1

(∫U

∫U

f(x) f(y)

νlog(

1 +f(x) f(y)

ν

)MU (dy) MU (dx)

)≤∫U

∫U

(f(x) f(y)

ν

)2

MU (dy) MU (dx) =1

ν2M(U)2,

where we used that ‖f‖2 = 1. Hence,∫U

∫U

f(x) f(y)

νlog(

1 +f(x) f(y)

ν

)MU (dy) MU (dx)

≤ I H−1( 1

ν2M(U)2

)=

1

νM(U)log(

1 +1

νM(U)

).

Finally, choosing the constants large enough, we combine the above considerationsto conclude that

〈GUf, f〉 ≤CM(U) +1

νM(U) log

(1 +

1

νM(U)

)≤ CM(U) log

(2 +

1

M(U)

),

and by the variational formula for λ1(U) in (5.1) the claim follows.

In the next proposition we derive a Nash-type inequality from the above Faber-Krahn inequality, which implies the ultracontractivity of the semigroup (TUt ) andon-diagonal estimate on (TUt ) of the same form as stated for pt(·, ·) in Theorem 1.2.

Proposition 5.3. There exists a constant C10 = C10(X, γ,R) such that

‖TUt ‖L1→L∞ ≤ C10 t−1 log(t−1), ∀t ∈ (0, 1

2 ]. (5.3)

Here ‖A‖L1→L∞ denotes the operator norm of a bounded linear operator A mappingfrom L1(U,M) to L∞(U,M).


Proof. Since U ⊂ BR and therefore ‖TUt ‖L1→L∞ ≤ ‖TBRt ‖L1→L∞ it is enough to

show (5.3) for (TBRt ). Recall that λ1(U) can also be expressed by the variationalformula

λ1(U) = infE(f, f)

‖f‖22: f ∈ FU , f 6= 0

.

Hence, the Faber-Krahn inequality in Proposition 5.2 can be rewritten as

‖f‖22 ≤ C ψ(M(U)) E(f, f), ∀f ∈ FU , (5.4)

where ψ(s) := s log(2 + 1s ). Next we will verify that

‖f‖22 ≤ C ψ(M(supp(f))

)E(f, f), ∀f ∈ FBR , (5.5)

where supp(f) denotes the M -essential support of f in BR. First, for f ∈ FBRwith supp(f) compact, (5.5) follows by choosing a decreasing sequence (Un)n≥1 ofopen subsets of BR with

⋂n Un = supp(f), applying (5.4) with U = Un and letting

n → ∞. Next, for general f ∈ FBR , as |f | ∈ FBR and E(|f |, |f |) ≤ E(f, f) wemay assume f ≥ 0. Let (fn)n ⊂ FBR be a sequence with supp(fn) compact and‖fn − f‖F → 0, where by [12, Theorem 1.4.2 (v)] we may assume that fn ≥ 0 forall n. Then, since f ∧ fn ∈ FBR , supp(f ∧ fn) is a compact subset of supp(f) and

‖f − f ∧ fn‖F = ‖ (f − fn)+ ‖F ≤ ‖f − fn‖F → 0,

we conclude (5.5) for all f ∈ FBR by letting n→∞ in (5.5) for f ∧ fn.Now, since ψ is strictly increasing on [0,∞), by [2, Proposition 10.3] (5.5) implies

that

‖f‖22 ≤ C ψ(C/‖f‖22

)E(f, f) for all f ∈ FBR with 0 < ‖f‖1 ≤ 1.

In particular, for such f we have θ(‖f‖22) ≤ E(f, f) with θ(s) := C s2(log(2+Cs))−1

and by [8, Proposition II.1] we obtain that

‖TBRt ‖L1→L∞ ≤ m(t), t > 0, (5.6)

for a decreasing C1-bijection m on (0,∞) satisfying

θ(m(t)) = −m′(t), limt↓0

m(t) =∞. (5.7)

Set Φ(s) :=∫∞s θ(u)−1du. Then, m(t) = Φ−1(t) is the solution of (5.7). Moreover,

for any s ∈ (0,∞),

Φ(s) =

∫ ∞s

C

u2log(2 + Cu) du ≤ C

slog(2 + Cs) =: Ψ(s).

Note that Ψ is strictly decreasing on (0,∞). In particular, Ψ−1(t) ≥ Φ−1(t) for allt ∈ (0,∞). Finally, since for t small enough Ψ(t−1 log(t−1)) ≤ Ct we conclude that

m(Ct) = Φ−1(Ct) ≤ Ψ−1(Ct) ≤ C t−1 log(t−1),

and the claim follows from (5.6).


Now we prove Theorem 5.1 for bounded open sets U . Given the ultracontractiv-ity of (TUt ) in Proposition 5.3 and the strong Feller property in Proposition 4.2, ageneral result in [9] provides the existence of a continuous kernel pUt (·, ·) for (TUt ).Then, we still have to identify this kernel as the transition density of BU .

Proof of Theorem 5.1 for bounded U . We divide the proof of i) into several steps.Step 1: In the first step we show the existence of pUt (x, y) and its continuity.

Let (λn)n≥1 be the eigenvalues of −LU in increasing order (repeated accordingto multiplicity) and let (ϕn)n≥1 ⊂ D(LU ) be a complete orthonormal system ofL2(U,M) such that −LUϕn = λnϕn for each n ≥ 1. Then, by the ultracontractivityof the semigroup

(TUt)t>0

established in Proposition 5.3, that is TUt (L2(U,M)) ⊂L∞(U,M) and TUt : L2(U,M) → L∞(U,M) is a bounded linear operator for anyt > 0, we may choose a bounded Borel measurable version of ϕn for each n. Further,by the strong Feller property of the resolvents in Proposition 4.2 we have RUµϕn ∈Cb(U) for any λ > 0, and since

RUλ ϕn = GUλϕn = (λ+ λn)−1ϕn M -a.e., (5.8)

there exists a continuous version of ϕn, which we still denote by ϕn. Then by [9,Theorem 2.1.4], the series

pUt (x, y) :=

∞∑n=1

e−λntϕn(x)ϕn(y) (5.9)

converges uniformly on [s0,∞)×U ×U for all s0 > 0, from which the continuity ofpUt (x, y) follows, and this defines an integral kernel for TUt , namely for each t > 0

and f ∈ L2(U,M),

TUt f(x) =

∫UpUt (x, y)f(y)M(dy), for M -a.e. x ∈ U .

Step 2: In this step we show that RUλ is absolutely continuous w.r.t. the Liouvillemeasure, i.e. we establish the existence of the resolvent kernel. For any Borel set Awith M(A) = 0 we have RUλ 1lA(x) = GUλ 1lA(x) = 0 for M -a.e. x ∈ U . Since M hasfull support, RUλ 1lA(x) = 0 on a dense subset of U and since x 7→ RUλ 1lA(x) is contin-uous by the strong Feller property in Proposition 4.2, it is idendically zero, and theabsolute continuity follows. Therefore, there exists a resolvent kernel rUλ (·, ·) suchthat for all bounded Borel functions f ,

RUλ f(x) =

∫UrUλ (x, y) f(y)M(dy).

Step 3: Next we will show that for any x ∈ U ,∫ ∞0

e−λt(∫

UpUt (x, y) f(y)M(dy)

)dt =

∫ ∞0

e−λtEx[f(BUt )] dt, (5.10)

for all λ > 0 and all non-negative bounded Borel functions f . Recall that PUt f(x) =

Ex[f(BUt )] denotes the transition semigroup of BU . Then for any ε > 0, since


PUε = TUε M -a.e., by the absolute continuity of RUλ w.r.t. M we have∫ ∞ε

e−λtPUt f(x) dt = e−λεRUλ (PUε f)(x) = e−λεRUλ (TUε f)(x)

= e−λεRUλ

( ∞∑n=1

e−λnε〈ϕn, f〉ϕn

)(x) =

∞∑n=1

e−(λ+λn)ε 1

λ+ λn〈ϕn, f〉ϕn(x),

where we also used (5.8) and the uniform convergence of the series in (5.9). Settingaεn =

∫∞ε e−(λ+λn)tdt, we further get∫ ∞

εe−λtPUt f(x) dt =

∞∑n=1

aεn ϕn(x)〈ϕn, f〉 = limN→∞

N∑n=1

∫ ∞ε

e−(λ+λn)tϕn(x)〈ϕn, f〉 dt

=

∫ ∞ε

( ∞∑n=1

e−λntϕn(x)〈ϕn, f〉)e−λtdt =

∫ ∞ε

(∫UpUt (x, y) f(y)M(dy)

)e−λt dt,

and by taking the limit as ε ↓ 0 we obtain (5.10).Step 4: Finally, we now prove that Px[BUt ∈ dy] = pUt (x, y)M(dy) for all (t, x) ∈

(0,∞)×U . By the uniqueness of Laplace transforms for positive measures on [0,∞)

(see e.g. [11, Section XIII.1 Theorem 1a]), we get for all non-negative boundedBorel functions f ,∫

UpUt (x, y) f(y)M(dy) = Ex

[f(BUt )

], for a.e. t ∈ [0,∞). (5.11)

If in addition f is continuous, the continuity of pU established in Step 1 implies that(5.11) holds for all t > 0. Finally a density argument gives the claim. The proof ofi) is now complete.

For ii), since pUt is bounded by [9, Theorem 2.1.4], the claim is immediate fromthe continuity of pUt and the fact that M(U) < ∞. Finally, iii) follows from [20,Proposition A.3 (2)].

5.2. The heat kernel on unbounded open sets. The proof of Theorem 5.1 forunbounded U is based on the following lemma, which essentially contains Theo-rem 1.2 already.

Lemma 5.4. For R ≥ 1 and β > α1, there exist Ci = Ci(X, γ,R, β), i = 11, 12, suchthat for any non-empty open bounded U ⊂ R2,

pUt (x, y) = pUt (y, x) ≤ C11 t−1 log(t−1) exp

(− C12

(|x− y|β ∧ 1

t

)1/(β−1))for all t ∈ (0, 1

2 ], x ∈ R2 and y ∈ BR, where we extend the kernel pU (·, ·) to a functionon (0,∞)× R2 × R2 by setting pUt (x, y) := 0 for (x, y) ∈ (U × U)c for any t > 0.

Proof. Since for every n ∈ N we have the on-diagonal bounds on pBn in Proposi-tion 5.3, given the exit time estimates in Proposition 3.3, the result follows from[18, Theorem 1.1].


Remark 5.5. The constants appearing in the upper bound in Lemma 5.4 do notdepend on the set U . Therefore, for any R ≥ 1 there exists C13 = C13(X, γ,R),also not depending on U , such that pU1/2(x, y) ≤ C13 for all x ∈ R2 and y ∈ BR. Inparticular, by the semigroup property we have for all t ∈ (1

2 ,∞) and such x and y,

pUt (x, y) =

∫R2

pUt−1/2(x, z) pU1/2(z, y)M(dz) ≤ C13

∫R2

pUt−1/2(x, z)M(dz) ≤ C13.

Lemma 5.6. For any increasing sequence (Un) of open subsets of R2 with⋃n Un = R2,

limn→∞

Px[τUn < t

]= 0,

uniformly in (t, x) over compact subsets of [0,∞)× R2.

Proof. It suffices to prove the uniform convergence in (t, x) over [0, T ]×BR for anyT,R ∈ (0,∞). By monotonicity we may assume t = T . Then, for any x ∈ BR and nsuch that B2R ⊂ Un we obtain by the strong Markov property

Px[τUn < T

]= Px

[τB2R

≤ τUn < T]

= Px[τB2R

≤ τB2R+ τUn θτB2R

< T]

≤ Px[τUn θτB2R

< T]

= Ex

[PBτB2R

[τUn < T

]]= Pµx0,2R

[τUn < T

],

where µx0,2R := Px[BτB2R∈ ·] as in the proof of Proposition 3.2 above. Arguing

precisely as there, from an explicit formula for the exit distribution of a Brownianmotion (see [4, Theorem II.1.17]) we get that µx0,2R ≤ cµ0,2R for some explicit c > 0

with µ0,2R = µ00,2R. Thus Px

[τUn < T

]≤ cPµ0,2R

[τUn < T

], which converges to zero

as n → ∞ by the dominated convergence theorem since the trajectory of B|[0,T ] isbounded and therefore contained in Un for n large enough.

Proof of Theorem 5.1 for unbounded U . i) Let R ≥ 1 and let f : R2 → [0,∞) bebounded and Borel measurable with f |BcR = 0. Note that for any l > k ≥ R + 1 wehave Bk ⊂ Bl and therefore τBk < τBl , Px-a.s., for all x ∈ Bk. Then, by f |∂Bk = 0

and the strong Markov property [18, Proposition 3.4] we obtain for any t > 0 andx ∈ Bk,

PBlt f(x) = PBkt f(x) + Ex[1lτBk<t<τBl

f(Bt)]

= PBkt f(x) + Ex

[1lτBk<t

PBlt−τBkf(BτBk )

].

Recall that

PBlt−τBkf(BτBk ) =

∫BR

f(y) pBlt−τBk(BτBk , y)M(dy).

Now, note that by Remark 5.5,

supn

sup[ 12,∞)×R2×R2

pBn· (·, ·) ≤ C13 <∞.

Further, as k ≥ R+ 1 we have dist(BR, Bck) ≥ 1 and therefore Lemma 5.4 gives

supn,k

sup(0, 1

2)×∂Bk×BR

pBn· (·, ·) ≤ C14 <∞


for some C14 = C14(X, γ,R). Hence,

PBlt f(x) ≤ PBkt f(x) + (C13 ∨ C14)Px[τBk < t

] ∫BR

f(y)M(dy),

and since f ≥ 0 with f |BcR = 0 is arbitrary we obtain

pBlt (x, y) ≤ pBkt (x, y) + (C13 ∨ C14)Px[τBk < t

], (5.12)

for all x ∈ Bk, y ∈ BR and t ∈ (0,∞); recall that by the results in the last subsectionfor all k, l ∈ N the Dirichlet heat kernels pBl and pBk are continuous. Thus itfollows from (5.12) and Lemma 5.6 that the limit pt(x, y) := limn p

Bnt (x, y) exists

and is continuous on (0,∞) × R2 × BR. Since R ≥ 1 is arbitrary and the relationPx[Bt ∈ dy] = pt(x, y)M(dy) can be obtained from that for bounded U by monotoneconvergence, statement i) follows for the global heat kernel pt(x, y), i.e. for the caseU = R2. For general unbounded, open U ⊂ R2, statement i) follows by similararguments from the fact that for any k, l ∈ N with k < l,

0 ≤ pU∩Blt (x, y)− pU∩Bkt (x, y) ≤ pBlt (x, y)− pBkt (x, y), x, y ∈ R2, t > 0.

In order to see the second inequality, notice that if (x, y) ∈ (Bk×Bk)c this inequalityholds trivially, and for (x, y) ∈ Bk × Bk, by the continuity of the Dirichlet heatkernels,(pBlt − p

Bkt − p

U∩Blt + pU∩Bkt

)(x, y) = lim

r↓0

Px[Bt ∈ B(y, r), τU ∨ τBk ≤ t < τBl

]M((B(y, r)

) ≥ 0.

ii) Let x ∈ U and t, ε > 0. Since∫R2

pt(x, y)M(dy) = Px[Bt ∈ R2] = 1

by lims→∞ Fs =∞, Px-a.s., we can choose n ∈ N such that x ∈ Bn and∫Bn

pt(x, y)M(dy) > 1− ε.

Then, by the continuity of pt there exists r > 0 such that B(x, r) ⊂ U and∫Bn

pt(z, y)M(dy) > 1− ε, ∀z ∈ B(x, r).

Hence, for such z,∫U\Bn

pUt (z, y)M(dy) ≤∫Bcn

pt(z, y)M(dy) < ε.

Now, for any bounded Borel function f : U → R and z ∈ B(x, r), writing

PUt f(z) =

∫U\Bn

f(y) pUt (z, y)M(dy) +

∫U∩Bn

f(y) pUt (z, y)M(dy),


we obtain∣∣PUt f(x)− PUt f(z)∣∣

≤ 2 ‖f‖∞ε+∣∣∣ ∫

U∩BnpUt (x, y)f(y)M(dy)−

∫U∩Bn

pUt (z, y)f(y)M(dy)∣∣∣

≤ (2‖f‖∞ + 1) ε,

provided |x − z| is sufficiently small, which proves the strong Feller property. Inthe last step we used the fact that, since 0 ≤ pUt ≤ pt on B(x, r) × (Bn ∩ U) wherept is bounded and pUt is continuous, the function z 7→

∫U∩Bn p

Ut (z, y)f(y)M(dy) is

continuous on B(x, r) by the dominated convergence theorem.iii) Since U is connected, for any x, y ∈ U there exists a connected bounded open

set V ⊂ U with x, y ∈ V and by the corresponding result for bounded sets we havepUt (x, y) ≥ pVt (x, y) > 0 for any t > 0.

Proof of Theorem 1.2. This is immediate from Lemma 5.4, since pt = limn pBnt as

shown in the above proof.

6. ON-DIAGONAL LOWER BOUNDS AND SPECTRAL DIMENSIONS

In this section we prove the on-diagonal lower bound in Theorem 1.3. Indeed, wewill show a more general result that also covers the Dirichlet Liouville heat kernel.Recall that we have fixed an environment ω ∈ Ω as declared at the beginning ofSection 5.

Theorem 6.1. For M -a.e. x ∈ R2 the following holds. For any open set U ⊆ R2

containing x there exist C15 = C15(X, γ, |x|) > 0 and t0(x) = t0(X, γ, x, U) ∈ (0, 12 ]

such that

pUt (x, x) ≥ C15 t−1(

log(t−1))−η

, ∀t ∈ (0, t0(x)], (6.1)

for some explicit constant η > 0 (for instance η = 34 is enough).

In particular, Theorem 6.1 immediately implies Theorem 1.3 by choosing U =

R2. Furthermore, we can directly deduce the pointwise spectral dimension of theDirichlet Liouville heat kernel.

Corollary 6.2. For M -a.e. x ∈ R2, for any open set U ⊆ R2 containing x,

limt↓0

2 log pUt (x, x)

− log t= 2. (6.2)

Proof. This is immediate from the lower bound in Theorem 6.1 and, since pUt (x, x) ≤pt(x, x), from the on-diagonal part of the upper bound in Theorem 1.2.


6.1. Proof of Theorem 6.1. In order to show Theorem 6.1 we need further mo-ment and tail estimates on the exit times from balls. First, we recall the representa-tion of the expected exit time in terms of the Green kernel.

Lemma 6.3. For any non-empty open set U ⊆ R2 and any x ∈ U ,

Ex[τU ] =

∫UgU (x, y)M(dy).

Proof. This follows immediately from Proposition B.1.

Lemma 6.4. For any R ≥ 1 there exists C16 = C16(X, γ,R) > 0 such that

Ex[τB(x,r)] ≤ C16M(B(x, r)) log 1r , ∀r ∈ (0, 1

2 ], x ∈ BR.

Proof. Recall that the Green function over BR+1 can be written as in (2.4) Then, asB(x, r) ⊂ BR+1 and therefore gB(x,r) ≤ gBR+1

, we have

Ex[τB(x,r)] =

∫B(x,r)

gB(x,r)(x, y)M(dy)

≤∫B(x,r)

(1

πlog

1

|x− y|+ Ψ0,R+1(x, y)

)M(dy).

Setting Dn := B(x, 21−nr) \B(x, 2−nr), n ≥ 1, we obtain further that

Ex[τB(x,r)] ≤ cM(B(x, r)) + c

∞∑n=1

(n+ log 1r )M(Dn),

with c = c(R) > 0. On the other hand, by Lemma 3.1, for any ε > 0,

(n+ log 1r )M(B(x, 21−nr)

)M(B(x, r)

) ≤ C n 2−n(α2−ε)rα2−α1−2ε ≤ C 2−n(α2−ε)/2,

provided n ≥ c log 1r with c = c(γ) > 0. Hence,

Ex[τB(x,r)] ≤cM(B(x, r)) + C∑

n≥c log 1r

2−n(α2−ε)/2M(B(x, r))

+ c log1

r

∑n<c log 1

r

M(Dn),

which implies the claim.

Lemma 6.5. There exists a constant c1 > 0 such that

Ex[τB(x,r)

]≥ c1M

(B(x, r/2)

), ∀r > 0, x ∈ R2.

Proof. This follows from the following scale invariance of the Green kernel:

gB(x,r)(x, y) = gB(0,r)(0, y − x) = gB(0,λr)

(0, λ(y − x)

)


for any x, y ∈ R2, r > 0 and λ > 0 (cf. e.g. [12, Example 1.5.1]). Indeed, using thisrelation we obtain

Ex[τB(x,r)] =

∫B(x,r)

gB(x,r)(x, y)M(dy) ≥∫B(x,r/2)

gB(0,1)

(0, y−xr

)M(dy)

≥ cM(B(x, r/2)

),

which is the claim.

Proposition 6.6. For any R ≥ 1 there exists C17 = C17(X, γ,R) > 0 such that

Px[τB(x,r) ≤ t

]≤ 1− C17

M(B(x, r/2)

)M(B(x, 3r)

)log 1

r

,

for all x ∈ BR, r ∈ (0, 12 ] and 0 < t ≤ 1

2Ex[τB(x,r)

].

Proof. Obviously, τB(x,r) ≤ t + 1lτB(x,r)>t(τB(x,r) − t) for any t > 0, so using theMarkov property we get

Ex[τB(x,r)

]≤ t+ Ex

[1lτB(x,r)>t(τB(x,r) − t)

]= t+ Ex

[1lτB(x,r)>tEBt

[τB(x,r)

]]≤ t+ Px

[τB(x,r) > t

]sup

y∈B(x,r)Ey[τB(x,r)

].

For t ≤ 12Ex[τB(x,r)] this implies that

Px[τB(x,r) ≤ t

]≤ 1 +

t− Ex[τB(x,r)]

supy∈B(x,r)Ey[τB(x,r)

] ≤ 1− 12

Ex[τB(x,r)]

supy∈B(x,r)Ey[τB(x,r)

] .Finally, note that by Lemma 6.4

supy∈B(x,r)

Ey[τB(x,r)] ≤ supy∈B(x,r)

Ey[τB(y,2r)

]≤ C log 1

r supy∈B(x,r)

M(B(y, 2r))

≤ CM(B(x, 3r)) log 1r ,

and the claim follows Lemma 6.5.

We are now in the position to show an on-diagonal lower bound on the Dirichletheat kernel.

Proposition 6.7. For any R ≥ 1 there exist c2 > 0 and C18 = C18(X, γ,R) > 0 suchthat

pB(x,r)t (x, x) ≥ C18

(M(B(x, r/2)

)M(B(x, 3r)

)log 1

r

)2

M(B(x, r)

)−1,

for all x ∈ BR, r ∈ (0, 12 ] and 0 < t ≤ c2M

(B(x, r/2)

).


Proof. First note that for every t ≥ 0 by the Cauchy-Schwarz inequality, the symme-try of the Dirichlet kernel pB(x,r)

t (x, y) and the Chapman-Kolmogorov equation,

Px[τB(x,r) > t

]2= Px

[Bt ∈ B(x, r), τB(x,r) > t

]2=(∫

B(x,r)pB(x,r)t (x, y)M(dy)

)2

≤M(B(x, r)

) ∫B(x,r)

(pB(x,r)t (x, y)

)2M(dy) = M

(B(x, r)

)pB(x,r)2t (x, x).

On the other hand, since Ex[τB(x,r)

]≥ c1M

(B(x, r/2)

)by Lemma 6.5, we have by

Proposition 6.6 for t ≤ 12c1M

(B(x, r/2)

),

Px[τB(x,r) > t

]≥ C

M(B(x, r/2)

)M(B(x, 3r)

)log 1

r

,

which gives the result.

Corollary 6.8. Let c3 > 0. Then the conclusion of Theorem 6.1 holds for any x ∈ R2

that admits r0 = r0(x) ∈ (0, 12 ] such that

M(B(x, 2r)

)≤ c3

(log 1

r

)4M(B(x, r)

), ∀r ∈ (0, r0]. (6.3)

Proof. For any x ∈ U satisfying (6.3) let r1 = r1(x) < r0/2 be such that B(x, r1) ⊂U . Further, for t ≤ c2M(B(x, r1/2)) choose r = r(t) ∈ (0, r1) such that

c2M(B(x, r/4)

)≤ t ≤ c2M

(B(x, r/2)

).

Note that by Lemma 3.1, log r(t) log t for small t. Then by Proposition 6.7,

t pUt (x, x) ≥ t pB(x,r0)t (x, x) ≥ t pB(x,r)

t (x, x)

≥ CM(B(x, r/4)

)M(B(x, r)

) (M(B(x, r/2)

)M(B(x, 3r)

) )2 (log 1

r

)−2. (6.4)

Now, by using (6.3) we have

M(B(x, r/4)

)M(B(x, r)

) =M(B(x, r/4)

)M(B(x, r/2)

) M(B(x, r/2))

M(B(x, r)

) ≥ c ( log 1r

)−8,

and

M(B(x, r/2)

)M(B(x, 3r)

) ≥ M(B(x, r/2)

)M(B(x, 4r)

) ≥ c ( log 1r

)−12.

Combining these esimates with (6.4) gives the result.

Theorem 6.1 follows now from Corollary 6.8 and the following result.

Proposition 6.9. Let µ be a Borel measure on R2 with µ(B(x, r)

)∈ (0,∞) for all x ∈

R2 and r > 0. Then for some c3 > 0 for µ-a.e. x ∈ R2 there exists r0 = r0(x) ∈ (0, 12 ]

such that

µ(B(x, 2r)

)≤ c3

(log 1

r

)4µ(B(x, r)

), ∀r ∈ (0, r0].


Proof. Fix an arbitrary x0 ∈ R2. Set Bx0 = B(x0, 1), rk := 2−k, k ∈ Z, and µx0 :=

µ(· ∩Bx0) and let

An :=x ∈ Bx0 |µ

(B(x, rn−1)

)≥ n2µ

(B(x, rn)

), n ∈ N.

Further let Λn :=x0 + ( k

2n ,l

2n ) | k, l ∈ Z, −2n ≤ k, l ≤ 2n

. Then, since Bx0 ⊂⋃x∈Λn+1

B(x, rn+1),∫Bx0

µ(B(y, rn−1)

)µ(B(y, rn)

) dµx0(y) ≤∑

x∈Λn+1

∫B(x,rn+1)

µ(B(y, rn−1)

)µ(B(y, rn)

) µx0(dy)

≤∑

x∈Λn+1

∫B(x,rn+1)

µ(B(x, rn−2)

)µ(B(x, rn+1)

) µx0(dy) ≤∑

x∈Λn+1

µ(B(x, rn−2)

)≤ c µ(B(x0, 4))

for some c > 0. By Cebysev’s inequality this implies∞∑n=1

µx0(An) ≤ c µ(B(x0, 4))

∞∑n=1

n−2 <∞,

and by the Borel-Cantelli lemma for µ-a.e. x ∈ Bx0 there exists n0(x) ∈ N such that

µ(B(x, rn−1)

)≤ n2µ

(B(x, rn)

), ∀n ≥ n0(x). (6.5)

Now let x ∈ Bx0 satisfy (6.5), let r ∈ (0, rn0(x)) and let n ≥ n0(x) be such thatrn+1 ≤ r < rn. Then by applying (6.5) twice,

µ(B(x, 2r)

)≤ µ

(B(x, rn−1)

)≤ n2(n+ 1)2µ

(B(x, rn+1)

)≤ 4n4µ

(B(x, r)

)with n < (log 1

r )/ log 2. Finally, since x0 is arbitrary, the claim follows.

6.2. Global spectral dimension. Let U ⊂ R2 be non-empty, open and bounded.Then, as above in Section 5.1, let (λn(U))n≥1 be the eigenvalues of −LU in increas-ing order (repeated according to multiplicity). Then, setting

ZU (t) :=

∫UpUt (x, x)M(dx) =

∞∑n=1

e−λn(U)t,

we obtain the following global spectral dimension from Theorem 6.2.

Corollary 6.10. For any non-empty, open and bounded U ⊂ R2,

limt↓0

2 logZU (t)

− log t= 2. (6.6)

Proof. As pUt (x, x) ≤ pt(x, x), the on-diagonal part of the upper bound in Theo-rem 1.2 immediately implies that

lim supt↓0

2 logZU (t)

− log t≤ 2,

so it remains to prove the lower bound. By Theorem 6.1 we have limt↓0 pUt (x, x) =

∞ for M -a.e. x ∈ U . In particular, there exists t0 ∈ (0, 1) such that M(A) > 0 for


A := x ∈ U : pUt0(x, x) ≥ 1. Noting that ZU (t) ≥∫A p

Ut (x, x)M(dx), we obtain by

Jensen’s inequality

logZU (t) ≥ logM(A) + log

∫ApUt (x, x)

M(dx)

M(A)

≥ logM(A) +

∫A

log pUt (x, x)M(dx)

M(A).

Then, since t 7→ pUt (x, x) is decreasing, log pUt (x, x) ≥ 0 for all x ∈ A and t ∈ (0, t0],so we may use Fatou’s lemma and Corollary 6.2 to conclude that

lim inft↓0

logZU (t)

− log t≥∫A

lim inft↓0

log pUt (x, x)

− log t

M(dx)

M(A)= 1,

which proves the lower bound.

Remark 6.11. It is not clear to the authors whether the counterpart of (6.6) for theeigenvalue counting function NU (λ) := #n ∈ N : λn(U) ≤ λ also holds.

APPENDIX A. PROOF OF PROPOSITION 2.4

The proof will be based on the following results proved in [15].

Theorem A.1. For each x ∈ R2, P× Px-a.s. the following hold:

i) For all t ≥ 0, Ft := limn→∞ Fnt exists in R.

ii) The mapping [0,∞) 3 t 7→ Ft ∈ [0,∞) is continuous, strictly increasing andsatisfies F0 = 0 and limt→∞ Ft =∞.

Proof. See [15, Lemma 2.8 and Proof of Theorem 2.7].

We start with a preparatory lemma.

Lemma A.2. Almost surely w.r.t. P, for all x ∈ R2,

limT→0

lim infn→∞

FnT = 0, Px-a.s.

Proof. For all x ∈ R2, by Fatou’s lemma

Ex[

lim infn→∞

FnT]≤ lim inf

n→∞Ex[FnT ] <∞,

so using the dominated convergence theorem the claim follows immediately fromLemma 4.1.

For all t ≥ 0 we denote by Λt the set of all (ω, ω′) ∈ Ω× Ω′ such that:

i) For all u ≥ t, Ft,u(ω, ω′) := limn→∞ Fnu (ω, ω′)− Fnt (ω, ω′) exists in R.

ii) The mapping [t,∞) 3 u 7→ Ft,u(ω, ω′) ∈ [0,∞) is continuous, strictly in-creasing and satisfies Ft,t(ω, ω′) = 0 and limu→∞ Ft,u(ω, ω′) =∞.


We will write Λωt = ω′ ∈ Ω′ : (ω, ω′) ∈ Λt. Note that Λωt = θ−1t (Λω0 ), t ≥ 0, since

Fn is a PCAF of B for every n. Further, we have Λt ∈ A⊗G0∞, since all the defining

properties of Λt can be rephrased in terms of the values of Fns for dyadic rationals by virtue of the monotonicity of Fn. Finally, recall that by Theorem A.1 we haveP× Px[Λ0] = 1 for all x ∈ R2.

Lemma A.3. For P-a.e. ω, Px[Λωt ] = 1 for all t > 0 and x ∈ R2.

Proof. Fix any probability measure µ on R2 with full support. Since EPx[Λω0 ] = 1 forall x ∈ R2, using Fubini’s theorem integration w.r.t. µ gives that EPµ[Λω0 ] = 1 withPµ :=

∫R2 Px µ(dx). In particular, for P-a.e. ω, Pµ[(Λω0 )c] = 0 and thus

Py[(Λω0 )c] = 0, for dy-a.e. y ∈ R2. (A.1)

Now, for every such ω, every x ∈ R2 and t > 0 we use the Markov property to obtain

Px[Λωt ] = Px[θ−1t (Λω0 )

]= Ex

[1lΛω0 θt

]= Ex

[PBt [Λ

ω0 ]]

=1

2πt

∫R2

Py[Λω0 ] e−|y−x|

2/2t dy = 1,

where we used (A.1) in the last step.

Proof of Proposition 2.4. Let

Λ :=⋂q∈Q+

Λq ∩

(ω, ω′) ∈ Ω× Ω′ : limT→0

lim infn→∞

FnT (ω, ω′) = 0,

where Q+ denotes the set of all positive q ∈ Q. Then, i) follows immediately fromLemma A.2 and Lemma A.3.

Let now (ω, ω′) ∈ Λ be arbitrary. Then, ω′ ∈ Λωq for each q ∈ Q+, so for all s ≥ q

the limit Fq,s exists in R and s 7→ Fq,s(ω, ω′) continuous and strictly increasing and

and lims→∞ Fq,s(ω, ω′) =∞. Thus, for all 0 < s < t the limit

Fs,t(ω, ω′) := lim

n→∞Fnt (ω, ω′)− Fns (ω, ω′)

exists in R. Hence, for any T > 0 and 0 < s < t ≤ T ,

Ft,T (ω, ω′)− Fs,T (ω, ω′) = limn→∞

(Fnt (ω, ω′)− Fns (ω, ω′)

)≤ lim inf

n→∞FnT (ω, ω′).

But since limT→0 lim infn FnT (ω, ω′) = 0, this implies that (Fs,T (ω, ω′))s∈[0,T ] is a

Cauchy sequence as s ↓ 0, so there exists

FT (ω, ω′) = lims↓0

Fs,T (ω, ω′).

Moreover, since 0 ≤ FT (ω, ω′) ≤ lim infn FnT (ω, ω′) → 0 as T → 0, we obtain

limT→0 FT (ω, ω′) = 0. Thus, setting F0(ω, ω′) = 0, the mapping [0,∞) 3 T 7→FT (ω, ω′) is continuous and it is also stricly increasing. To see the latter, note that


for all 0 < s < t, choosing q ∈ Q+ such that q ≤ s, we have

Ft(ω, ω′)− Fs(ω, ω′) = lim

ε↓0

(Fε,t(ω, ω

′)− Fε,s(ω, ω′))

= limn→∞

(Fnt (ω, ω′)− Fns (ω, ω′)

)= Fq,t(ω, ω

′)− Fq,s(ω, ω′) > 0,

where we used in the last step that the function u 7→ Fq,u(ω, ω′) is strictly increas-ing on [q,∞) since (ω, ω′) ∈ Λq. Finally, since for any T > 0 and every q ∈ Q+

with q ≤ T we have FT (ω, ω′) ≥ Fq,T (ω, ω′) and (ω, ω′) ∈ Λq we obtain thatlimT→∞ FT (ω, ω′) =∞. This completes the proof of ii).

Statement iii) is clear, so it remains to show iv). Let ω ∈ Ω satisfy the property instatement i). First, for any t > 0, Λω ∈ G0 ⊂ Gt by i), and the Gt-measurability ofFs,t(ω, ·)|Λω implies that of Ft(ω, ·). Next we show that θt(Λω) ⊂ Λω for all t ≥ 0.Since

lim infn→∞

FnT θt = limn→∞

FnT+t − Fnt = Ft,t+T → 0, as T → 0,

it suffices to verify that θt(Λω) ⊂ Λωq for all q ∈ Q+. But this follows from the factthat for any q ∈ Q+ and u ≥ q,

Fnu θt − Fnq θt = (Fnu+t − Fnt )− (Fnq+t − Fnt ) = Fnu+t − Fnq+t,

converges to Fq+t,u+t having the required properties. Finally, a similar argumentshows that Fs+t = Ft + Fs θt, s, t ≥ 0. Indeed, for t > 0 we have

Fs+t − Ft = limr↓0

Fr,s+t − Fr,t = limr↓0

limn→∞

[Fns+t − Fnr − Fnt + Fnr

]= lim

n→∞

[Fns+t − Fnt

]= lim

n→∞Fns θt = Fs θt.

Therefore Ft(ω, ·) is a PCAF of B in the strict sense with defining set Λω.

APPENDIX B. THE REVUZ CORRESPONDENCE BETWEEN M AND F

Proposition B.1. Almost surely w.r.t. P the following holds. For any non-empty, openset U ⊆ R2, for all x ∈ R2 and all Borel measurable functions η : [0,∞)→ [0,∞] andf : U → [0,∞],

Ex

[ ∫ TU

0η(t) f(BU

t ) dFt

]=

∫ ∞0

∫Uη(t) f(y) qUt (x, y)M(dy) dt, (B.1)

where qUt denotes the Gaussian Dirichlet heat kernel, i.e. qUt (x, y)dy = Px[BUt ∈ dy].

Lemma B.2. Almost surely w.r.t. P the following holds. For all non-empty open U ⊆R2, all x ∈ U , t > 0, and all measurable, bounded f : U → [0,∞) with compactsupport contained in U ,

∫ t∧TU0 f(BU

s ) dFns n is uniformly Px-integrable.

Proof. We shall prove that

supn≥1

Ex

[(∫ t∧TU

0f(BU

s ) dFns

)2 ]<∞. (B.2)


For any bounded Borel function h : U → R, using the Markov property and the factthat qUt (x, y) ≤ qt(x, y) for all t > 0, x, y ∈ U ,

Ex

[(∫ t∧TU

0h(BU

s ) ds

)2 ]=

∫U

∫Uh(y)h(z)

∫ t

0

∫ t

sqs(x, y)qu−s(y, z) du ds dz dy.

Further, note that∫ t

0

∫ t

sqs(x, y)qu−s(y, z) du ds ≤

∫ t

0qs(x, y) ds

∫ t

0qu(y, z) du

≤ 1

2π2

∫ t/(y−x)2

0s−1e−

12sds

∫ t/(z−y)2

0u−1e−

12udu

≤ 1

2π2

(c+ log+

t

(y − x)2

)(c+ log+

t

(z − y)2

),

where log+ = log(· ∨ 1). Let now R > 0 be such that supp(f) ⊂ BR. Then, choosingh(y) = f(y) exp

(γXn(y)− γ2

2 E[Xn(y)]),

Ex

[(∫ t

0f(BU

s ) dFns

)2 ]= Ex

[(∫ t

0h(BU

s ) ds

)2 ]≤ 1

2π2

∫U

∫Uh(y)h(z)

(c+ log+

t

(y − x)2

)(c+ log+

t

(z − y)2

)dz dy

≤ 1

2π2‖f‖∞

∫BR

∫BR

(c+ log+

t

(y − x)2

)(c+ log+

t

(z − y)2

)Mn(dz)Mn(dy).

Setting Dk = B(y, 21−nR) \B(y, 2−nR), y ∈ BR, k ∈ N, by [15, Theorem 2.2]∫BR

(c+ log+

t

(z − y)2

)Mn(dz) ≤ cM(BR) +

∞∑k=0

∫Dk

log+

t

(2−nR)2Mn(dz)

≤ cM(BR) + C∞∑k=1

(21−kR)α2−ε <∞, (B.3)

uniformly in n for some constant C = C(X, γ,R, t) and any ε ∈ (0, α2). Sincex ∈ BR the term

∫BR

(c+log+

t(y−x)2

)Mn(dy) can be treated similarly and therefore

(B.2) follows.

Lemma B.3. Almost surely w.r.t. P, for any non-empty open set U ⊆ R2 and all x ∈ U ,∫ TU

01lx(B

Us ) dFs = 0, Px-a.s.

Proof. For δ > 0 let fδ be the continuous function on U with the property thatfδ(x) = 1, fδ ≡ 0 on B(x, δ)c and decaying linearly on B(x, δ). Then, a similarcomputation as in the proof of Lemma B.2 above gives that for any t > 0,

Ex

[ ∫ t∧TU

0fδ(B

Us ) dFns

]≤ 1

2π

∫B(x,δ)

(c+ log+

t

(y − x)2

)Mn(dy) ≤ C δα2−ε,


uniformly in n for some constant C = C(X, γ, t) and any ε small enough. ByLemma B.2 taking the limit n→∞ yields Ex

[ ∫ t∧TU0 fδ(B

Us ) dFs

]≤ C δα2−ε. Finally,

since t > 0 is arbitrary, we let δ → 0 to get the claim by monotone convergence.

Proof of Proposition B.1. By a monotone class argument it suffices to consider con-tinuous functions η and f with compact supports in (0,∞) and R2, respectively.First note that by Fubini’s theorem we have for every n ∈ N,

Ex

[ ∫ TU

0η(t) f(BU

t ) dFnt

]=

∫ ∞0

∫Uη(t) f(y) qUt (x, y)Mn(dy), (B.4)

and we need to show that taking limits on both sides of (B.4) we obtain (B.1).Since dFn converges vaguely to dF on (0,∞) the convergence of the left handside of (B.4) follows from the uniform Px-integrability of

∫ TU0 η(t) f(BU

t ) dFnt n inLemma B.2.

For the convergence of the right hand side of (B.4), note first that we may replacef in (B.1) by f1lU\x by Lemma B.3 and the fact that M(x) = 0. Then, we caneasily construct a sequence of continuous functions (fk)k having compact supportsin U \x such that pointwise fk ↑ f1lU\x as k →∞. Thus, it is enough to considerthe case x 6∈ supp(f). But in this case the function y 7→

∫∞0 η(t) f(y) qUt (x, y) dt is

continuous with compact support, and the desired convergence follows from thevague convergence of Mn to M .

APPENDIX C. NEGATIVE MOMENTS OF THE LIOUVILLE MEASURE

Lemma C.1. Let q > 0 and set ξ(q) := 2q + q(1+q)2 γ2. Then there exists c4 = c4(γ, q)

such that for any r ∈ (0, 1],

supn≥1

E[Mn(B(0, r))−q

]≤ c4r

−ξ(q).

We remark that the law of Mn is spatially stationary, so the same statement ap-plies to any ball B(x0, r), x0 ∈ R2 and r ∈ (0, 1].

Proof. Recall that Xn(x) =∑n

j=1 Yj , where for each j, Yj is a centered Gaussianfield with covariance kernel kj defined in (2.1). By construction we have

∑∞j=1 kj =

g(m) and g(m) can be written as

g(m)(x, y) = log+

1

|x− y|+ Ψm(x, y),

for some bounded continuous function Ψm. In particular, for a sufficiently large T ,∞∑j=1

kj(x, y) = g(m)(x, y) ≤ log+

T

|x− y|:= k0(x, y), if |x− y| < 2.

Hence, for all n ≥ 1 and ε > 0 there exists δ0 > 0 such thatn∑j=1

kj(x, y) ≤ ε+ ϑδ ∗ k0(x, y), ∀|x− y| < 2, δ < δ0,


where ϑ(x) := 5π

(1− |x|1/2

)+and ϑδ := 1

δ2ϑ( ·δ ). Then, ϑ is positive definite (see

[23]) and∫R2 ϑ(x) dx = 1. For all δ > 0 let

(X0δ (x)

)x∈R2 be the centered Gaussian

field with covariance

E[X0δ (x)X0

δ (y)]

= ϑδ ∗ k0(y − x)

and by [26, Theorem 2.1] the associated random measures

M0δ (dx) = exp

(X0δ (x)− 1

2E[X0δ (x)

])dx

converge vaguely to some measureM0 as δ tends to zero. Further, for ε > 0 let Zε ∼N (0, ε) be independent of (X0

δ (x))x∈R2 . By Kahane’s convexity inequality (see [24,Theorem 2.1] or [19]) we have for all bounded convex functions F : [0,∞) → R,all collections of points (yi)1≤i≤N ⊂ B(0, 1) and all non-negative weights (pi)1≤i≤N ,N ∈ N,

E[F( N∑i=1

pi eXn(yi)−

12E[Xn(yi)

2])] ≤ E[F( N∑i=1

pi eX0δ (yi)+Zε−

12E[X0

δ (yi)2]− ε2

)].

By choosing appropriate points yi and weights pi we deduce that for all r ∈ (0, 1],

E[F( ∫

B(0,r)eXn(y)−1

2E[Xn(y)2]dy)]≤ E

[F( ∫

B(0,r)eX

0δ (y)+Zε−1

2E[X0δ (y)2]− ε2dy

)],

which can be rewritten as

E[F(Mn

(B(0, r)

))]≤ E

[F(eZε−

ε2M0

δ

(B(0, r)

))]Taking the limit δ → 0 and ε→ 0 afterwards, we obtain

E[F(Mn

(B(0, r)

))]≤ E

[F(M0(B(0, r)

))].

Finally, we choose F (t) = λq−1e−λt for λ > 0 and, since 1Γ(q)

∫∞0 λq−1e−λt dλ = t−q,

integration over λ yields

E[Mn(B(0, r))−q

]≤ E

[M0(B(0, r))−q

].

The statement now follows from [26, Proposition 3.7].

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2010.

RHEINISCHE FRIEDRICH-WILHELMS UNIVERSITAT BONN

Current address: Endenicher Allee 60, 53115 BonnE-mail address: [email protected]

DEPARTMENT OF MATHEMATICS, GRADUATE SCHOOL OF SCIENCE, KOBE UNIVERSITY

Current address: Rokkodai-cho 1-1, Nada-ku, Kobe 657-8501, Japan.E-mail address: [email protected]

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