HEAT KERNEL ANALYSIS ON INFINITE-DIMENSIONAL GROUPS … · 2010-07-21 · SEGAL-BARGMANN TRANSFORM...

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HEAT KERNEL ANALYSIS ON INFINITE-DIMENSIONAL GROUPS Masha Gordina University of Connecticut http://www.math.uconn.edu/~gordina 6th Cornell Probability Summer School July 2010

Transcript of HEAT KERNEL ANALYSIS ON INFINITE-DIMENSIONAL GROUPS … · 2010-07-21 · SEGAL-BARGMANN TRANSFORM...

Page 1: HEAT KERNEL ANALYSIS ON INFINITE-DIMENSIONAL GROUPS … · 2010-07-21 · SEGAL-BARGMANN TRANSFORM AND TAYLOR MAP IN RN AND CN † R. H. Cameron, Some examples of Fourier{Wiener transforms

HEAT KERNEL ANALYSIS ONINFINITE-DIMENSIONAL GROUPS

Masha Gordina

University of Connecticut

http://www.math.uconn.edu/~gordina

6th Cornell Probability Summer School

July 2010

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SEGAL-BARGMANN TRANSFORM AND TAYLOR MAPIN RN AND CN

• R. H. Cameron, Some examples of Fourier–Wiener transforms of analyticfunctionals, Duke Math.J. 12 (1945), 485–488.

• R. H. Cameron, and W. T. Martin, Fourier–Wiener transforms of analyticfunctionals, Duke Math. J. 12 (1945), 489–507.

• R. H. Cameron, and W. T. Martin, Fourier–Wiener transforms of function-als belonging to L2 over the space C, Duke Math. J. 14 (1947), 99–107.

• V. Bargmann, On a Hilbert space of analytic functions and an associatedintegral transform, Part I, Communications on Pure and Applied Mathemat-ics 14 (1961), 187–214.

• L. Gross, P. Malliavin, Hall’s transform and the Segal-Bargmann map, inIto’s stochastic calculus and probability theory, Springer, 1996, 73–116.

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〈u, v〉 =n∑

j=1ujvj u, v ∈ Cn, bilinear inner product

z = x − iy z = x + iy ∈ Cn, x, y ∈ Rn

〈u, v〉 the usual inner product in Cn

|z|2 = 〈z, z〉 the usual norm on Cn

pt (x) =1

(2πt)n/2e−|x|2

2t , x ∈ Rn

µt (z) =1

(2πt)ne−|z|2

t , z ∈ Cn

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The heat kernels, pt and µt, are the fundamental solutions to the heatequations in Rn and Cn with the Laplacians

∆ =n∑

j=1

∂2

∂x2j

∆C =n∑

j=1

(∂2

∂x2j

+∂2

∂y2j

)

z = (z1, ..., zn) ∈ Cn, zj = xj + iyj, xj, yj ∈ R.

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Namely, pt is the solution to

∂u (t, x)

∂t− ∆u (t, x) = 0, x ∈ Rn, 0 < t < ∞,

u (0, x) = δ (x) ,

where δ (x) is the Dirac delta function.

The general solution to the heat equation on Rn is given by a convolutionwith pt with the initial value u (0, x) = f (x)

u (t, x) =

Rnpt (x − y) f (y) dy = e−|x|2

2t

Rnf (y) e

〈x,y〉t pt (y) dy

(e

t∆2 f

)(x) = (pt ∗ f) (x) , f ∈ L2 (Rn, dx)

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H (Cn) the space of holomorphic functions on Cn

HL2 (Cn, µt (z) dz) H (Cn)⋂

L2 (Cn, µt (z) dz)

Theorem (Bargmann) Let t > 0 and 1 < p < ∞. For any functionf ∈ Lp (Rn, pt (x) dx) the convolution pt ∗ f has a unique analyticcontinuation, Stf , to Cn given by

(Stf) (z) = e−〈z,z〉2t

Rnf (u) e−〈z,u〉

t pt (u) du, z ∈ Cn.

Moreover, St is a unitary operator from L2 (Rn, pt (x) dx) onto the spaceHL2 (Cn, µt (z) dz).

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Plan of a proof.

Step 1. (Stf) (z) ∈ H and (Stf) (x) = (pt ∗ f) (x) for any x ∈ Rn

Step 2. (Polynomial basis over Cn) For a multiindex α = (k1, ..., kn) ofnonnegative integers let

zα =n∏

j=1

zkj,

|α| =n∑

j=1

kj

α! =n∏

j=1

kj!

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Then the set zα forms an orthogonal basis of HL2 (Cn, µt (z) dz). Iff ∈ H (Cn) and has the pointwise convergent power series

f(z) =∑α

cαzα, cα ∈ C,

then

Cn|f (z) |2µtdz =

∑α

|cα|2t|α|α!

The first series is convergent in L2 (Cn, µt) if either side of the secondidentity is finite.

Step 3.(Bargmann’s pointwise bounds) If f ∈ HL2 (Cn, µt), then

|f(z)|2 6 e|z|2

t ‖f‖2L2(Cn,µt)

, z ∈ Cn.

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Remark (Monotonicity of L2 norm.) Let 0 6 k < n. Denote by µ(k)t the

heat kernel density for Ck. Then

Ck|f (z) |2µ(k)

t dz 6∫

Cn|f (z) |2µ(n)

t dz, f ∈ H (Cn) .

Step 4. (Total sets) Let

fa (u) = e〈a,u〉, a ∈ Cn, u ∈ Rn,

ga (z) = e〈a,z〉, a ∈ Cn, z ∈ Cn.

For any 1 < p < ∞ the sets fa : a ∈ Rn and fa : a ∈ iRn areeach total in Lp (Rn, ptdx), that is, the linear span of the set is dense inLp (Rn, ptdx). The set ga : a ∈ Rn is total in HL2 (Cn, µtdz).

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Step 5. (Transform on exponentials)

(Stfa) (z) = e〈z,u〉+t|a|22 ,

(Stfa, Stfb)L2(Cn,µtdz) = (fa, fb)L2(Rn,ptdx) , a, b ∈ Cn.

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Fock space and the Taylor map

α a multilinear form on Cn × ... × Cn

ejnj=1 an orthonormal basis of Cn

‖α‖2Multk(Cn,C) =

ejm∈ejnj=1

|α (ej1, ..., ejk

) |2

Multk (Cn,C) k-linear forms on Cn, Mult0 (Cn,C) = C

Symk (Cn,C) symmetric forms in Multk (Cn,C)

‖α‖2t=

∞∑

k=0

tk

k!‖αk‖2

Multk(Cn,C), αk ∈ Multk (Cn,C)

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Tt (Cn) the full Fock space of α =∞∑

k=0αk, αk ∈ Multk (Cn,C),

‖α‖2t < ∞.

Ft (Cn) the bosonic Fock space of α ∈ Tt (Cn) such that αk ∈Symk (Cn,C)

H (Cn) the space of holomorphic functions f : Cn → C.

Dhf(z) dds |t=0 f (z + th) the directional derivative of f ∈ H (Cn)

Dh1· · · Dhk

f (z) is a symmetric k–linear form on Cn which is linear

in each variable separately. It is denoted by Dkf (z) as a linear form onCn × ... × Cn.

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(Dkf (z)

)(h1, ..., hk) =

(Dh1

...Dhkf

)(z) , z, h1, ..., hk ∈ Cn

D0f (z) = f (z)

∞∑

k=0

Dkf (z) is a symmetric linear form on Cn

(1 − D)−1z f =

∞∑

k=0

Dkf (z) ∈ Sym (Cn,C)

Page 14: HEAT KERNEL ANALYSIS ON INFINITE-DIMENSIONAL GROUPS … · 2010-07-21 · SEGAL-BARGMANN TRANSFORM AND TAYLOR MAP IN RN AND CN † R. H. Cameron, Some examples of Fourier{Wiener transforms

If

‖ (1 − D)−1z f‖2

t =∞∑

k=0

tk

k!|Dkf (z) |2 < ∞,

then (1 − D)−1z f ∈ Ft (Cn).

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Theorem (Bargmann-Kree) For any f ∈ HL2 (Cn, dµt)

(1 − D)−10 f ∈ Ft (Cn)

Moreover, the map (1 − D)−10 : HL2 (Cn, µt (z)) → Ft (Cn) is a

surjective isometry.

Plan of a proof.

The isometry:

‖ (1 − D)−10 f‖2

t =

Cn|f (z) |2dµt (z) , f ∈ H (Cn)

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Surjectivity: let α ∈ Ft (Cn) define

f (z) =∞∑

k=0

αk (z, ..., z)

k!.

Then

|f (z) |2 6

∞∑

k=0

|αk (z, ..., z) |k!

2

6

∞∑

k=0

|αk||z|kk!

2

6∞∑

k=0

tk|αk|2k!

∞∑

k=0

|z|ktkk!

6 ‖α‖2te

|z|2t , z ∈ Cn

Page 17: HEAT KERNEL ANALYSIS ON INFINITE-DIMENSIONAL GROUPS … · 2010-07-21 · SEGAL-BARGMANN TRANSFORM AND TAYLOR MAP IN RN AND CN † R. H. Cameron, Some examples of Fourier{Wiener transforms

and

(1 − D)−10 f = α.

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SUMMING UP

ground state transformation

St

T

BL2(Rn, dx) L2(Rn, dpt)

HL2(Cn, dµt)

Ft

----PPPPPPPPPPPPPPPq

PPPPPPPPPPPPPPPq

PPPPPPPPPPPPPPPq

PPPPPPPPPPPPPPPq

PPPPPPPPPPPPPPPq

³³³³³³³³³³³³³1

³³³³³³³³³³³³³1

³³³³³³³³³³³³³1

³³³³³³³³³³³³³1

³³³³³³³³³³³³³1

????

f fϕ----

dpt = ptdx, dµt(z) = µt(z)dz

Ft is a bosonic Fock space, where Cn is a state space of a single particle.

HL2(Cn, µt) =square integrable holomorphic functions

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B :Hk1(xj1) · ... · Hkn

(xjn) 7→ x

k1j1

⊗s ... ⊗s xknjn

St :Hk1(xj1) · ... · Hkn

(xjn) 7→ z

k1j1

· ... · zknjn

Hk(x) is an Hermite polynomial.

T :zk1j1

· ... · zknjn

7→ xk1j1

⊗s ... ⊗s xknjn

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FINITE-DIMENSIONAL MATRIX LIE GROUPS

G a finite-dimensional compact matrix Lie group with theidentity I

L2 (G, dg) the space of square–integrable functions on G with re-spect to right invariant Haar measure dg

g Lie algebra of G with the inner product 〈·, ·〉

derivatives Dξf (g) = ξf (g) = ddt

∣∣∣t=of(getξ

), ξ ∈ g,

g ∈ G, f ∈ C∞ (G)

ξidi=1 an orthonormal basis of the Lie algebra g

∆ a Laplaciand∑

j=1ξj

2f

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Pt the heat semigroup et∆ on L2 (G, dg)

heat kernel pt a left convolution kernel on G such that

Ptf (g) = f ∗ pt (g) =

Gf (gk) pt (k) dk, for all f ∈ C∞ (G) .

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The heat kernel measure pt (g) dg can be described also as the distribu-tion in t of the process gt satisfying the Stratonovich stochastic differentialequation

δgt = gtδWt, g0 = I

or Ito’s stochastic differential equation

dgt = gtdWt +1

2gt

d∑

1

ξ2i dt, g0 = I

Wt the Brownian motion on the Lie algebra g with theidentity operator as its covariance

Wt =d∑

i=1

btiξi

Page 23: HEAT KERNEL ANALYSIS ON INFINITE-DIMENSIONAL GROUPS … · 2010-07-21 · SEGAL-BARGMANN TRANSFORM AND TAYLOR MAP IN RN AND CN † R. H. Cameron, Some examples of Fourier{Wiener transforms

where bti are real-valued Brownian motions mutually independent on a prob-

ability space (Ω, F, P ).

Ptf (I) = E [f (gt)]

Ito’s calculus as the box calculus

dt · dt = 0, dt · dBt = 0, dBt · dBt = dt

dWt · dWt =

d∑

i=1

dbtiξi

·

d∑

i=1

dbtiξi

=

d∑

i=1

ξ2i

dt.

Ito’s product formula for B, C : Mn (C) → L (Mn (C)), F, G :Mn (C) → Mn (C)

Page 24: HEAT KERNEL ANALYSIS ON INFINITE-DIMENSIONAL GROUPS … · 2010-07-21 · SEGAL-BARGMANN TRANSFORM AND TAYLOR MAP IN RN AND CN † R. H. Cameron, Some examples of Fourier{Wiener transforms

dXt = B (Xt) dWt + F (Xt) dt

dYt = C (Yt) dWt + G (Yt) dt

then

d (XtYt) = dXtYt + XtdYt + dXt · dYt =

B (Xt) dWtYt + F (Xt) Ytdt + XtC (Yt) dWt + XtG (Yt) dt

+ B (Xt) dWt · C (Yt) dWt

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G = SU (2)

A =

(a bc d

): A∗A = I, det A = 1

g = su (2)

A =

(α βγ δ

): A∗ + A = 0, tr A = 0

inner product on su (2) 〈A, B〉 = tr B∗A

〈U−1AU, U−1BU〉 = 〈A, B〉, A, B ∈ su (2) , U ∈ SU (2)

o.n.b. ξj = i√2σj, j = 1, 2, 3

Pauli matrices

σ1 =

(0 11 0

), σ2 =

(0 −ii 0

), σ3 =

(1 00 −1

)

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For any U ∈ SU (2)

U = U (ϕ, θ, ψ) =

cos θ

2 ei(ϕ+ψ)

2 i sin θ2 ei

(ϕ−ψ)2

i sin θ2 e−i

(ϕ−ψ)2 cos θ

2 e−i(ϕ+ψ)

2

Euler angles 0 6 ϕ < 2π, 0 6 θ 6 π, −2π 6 ψ < 2π

Theorem (Laplacian on SU (2))

∆ = ξ12+ ξ2

2+ ξ3

2

= 2∂2

∂θ2+

2

sin2 θ

∂2

∂ϕ2− 4

cos θ

sin2 θ

∂2

∂ϕ∂ψ

+2

sin2 θ

∂2

∂2ψ+ 2

cos θ

sin θ

∂θ

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Let bt1, bt

2 and bt3 be three real-valued independent Brownian motions. Then

the Brownian motion on SU (2) is a solution to

δgt = gt

(δbt

1ξ1 + δbt2ξ2 + δbt

3ξ3

), g0 =

(1 00 1

)

dgt = gt

(dbt

1ξ1 + dbt2ξ2 + dbt

3ξ3

)− 3

2gtdt, g0 =

(1 00 1

)

since

σ21 + σ2

2 + σ23 = 3I.

Theorem. The process gt ∈ SU (2) with probability 1. Its generator is theLaplacian described above.

Page 28: HEAT KERNEL ANALYSIS ON INFINITE-DIMENSIONAL GROUPS … · 2010-07-21 · SEGAL-BARGMANN TRANSFORM AND TAYLOR MAP IN RN AND CN † R. H. Cameron, Some examples of Fourier{Wiener transforms

Proof.

1. gt is unitary. First write an SDE for g∗t

dg∗t = −dWtg

∗t − 1

2g∗t dt, g∗

0 =

(1 00 1

)

since W ∗t = −Wt.

By Ito’s product formula

d(gtg

∗t

)= dgtg

∗t + gtdg∗

t + dgt · dg∗t

= gtdWtg∗t − 3

2gtg

∗t dt − gtdWtg

∗t − 3

2gtg

∗t dt + dgt · dg∗

t

= −gtg∗t dt + gtg

∗t dt = 0,

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since dWt · dW ∗t =

d∑j=1

ξ2jdt = −3Idt.

2. det gt = 1. Denote

gt =

(At BtCt Dt

)

Wt =

(iαt βt + iγt

−βt + iγt −iαt

)

where αt, βt, γt are independent real-valued Brownian motions.

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dAt = iAtdαt − Btdβt + iBtdγt − 3

2Atdt

dBt = −iBtdαt + Atdβt + iAtdγt − 3

2Btdt

dCt = iCtdαt − Dtdβt + iDtdγt − 3

2Ctdt

dDt = −iDtdαt + Ctdβt + iCtdγt − 3

2Dtdt

By Ito’s product formula

d det gt = DtdAt+AtdDt−CtdBt−BtdCt+dAt·dDt−dBt·dCt

Thus d det gt = 0 with g0 = 1.

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Theorem (B. Driver, L. Gross) Let G be a connected complex Lie group,and g its Lie algebra. Suppose that f is in H (G). Define the kth Taylorcoefficient of f at I to be the element Dkf (I) in g⊗k determined by

〈Dkf (I) , ξ1 ⊗ ... ⊗ ξk〉 = ξ1...ξkf (I) , ξ1, ..., ξk ∈ g

with D0f (I) = f (I). Then the map (1 − D)−1I : f 7→

∞∑k=0

Dkf (I)

from H (G) into T ′ (g) is a unitary operator from HL2 (G, µt) onto Ft.

• B. Driver, On the Kakutani–Ito–Segal–Gross and the Segal–Bargmann–Hall isomorphisms, J. of Funct. Anal. 133 (1995), 69–128.

• B. Driver, L. Gross, Hilbert spaces of holomorphic functions on complexLie groups, New trends in stochastic analysis (Charingworth, 1994), 76–106,1997, in Proceedings of the 1994 Taniguchi Symposium.

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HEAT KERNEL ANALYSIS ONHILBERT-SCHMIDT GROUPS

What happens on infinite-dimensional curved spaces

• What is a heat kernel (Gaussian) measure?

No Haar measure, no heat kernel, only a heat kernel measure

• Lie algebra and Laplacian

Different choices of a norm on a Lie algebra give different Lie algebrasm

the Lie algebra determines directions of differentiationm

the Lie algebra and a norm on it determines a Laplacian and the Wienerprocess.

• Cameron-Martin subspace and holomorphic skeletons

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RESULTS FOR SOME INFINITE-DIMENSIONAL

CURVED SPACES.

Loop groups: heat kernel measures, the Cameron-Martin subgroup, Ricciis bounded from below, quasi-invariance of the heat kernel measures (S. Aida,B. Driver, S. Fang, P. Malliavin, I. Shigekawa ...)

Path spaces and groups: heat kernel measures, the Cameron-Martinsubspace, Taylor map (S. Aida, M. Cecil, B. Driver, S. Fang, E. Hsu, O.Enchev–D. Stroock)

Diff(S1), Diff(S1)/S1: heat kernel measures, the Cameron-Martinsubgroup, Ricci is bounded from below, quasi-invariance of the heat kernelmeasures (H. Airault, M. Bowick, A. A. Kirillov, G., P. Lescot, P. Malliavin,S.Rajeev, M. Wu, D. V. Yur’ev, B Zumino...)

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Hilbert-Schmidt groups: heat kernel measures, the Taylor map, holo-morphic skeletons, a Cameron-Martin subgroup, Ricci = −∞, Diff(S1)/S1

and SpHS (G. in PA ’00, JFA ’00, ’05)

infinite-dimensional nilpotent groups: heat kernel measures, quasi-invariance of the heat kernel measures, the Taylor map, holomorphic skele-tons, a Cameron-Martin subgroup, Ricci is bounded from below, log Sobolevinequality (Driver, Gordina JFA’09, PTRF, JDG, T. Melcher JFA’09)

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HILBERT-SCHMIDT GROUPS:INFINITE MATRIX GROUPS

B(H) bounded linear operators on a complex Hilbert space H .

G=GL(H) invertible elements of B(H).

Q a bounded linear symmetric nonnegative operator on HS.

HS Hilbert-Schmidt operators on H with the inner product(A,B)HS = TrB∗A.

g = gCM ⊆ HS an infinite-dimensional Lie algebra with a Hermitian innerproduct (·, ·), |A|g = |Q−1/2A|HS.

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GCM ⊆ GL(H)Cameron-Martin group x∈GL(H), d(x, I)< ∞

d(x, y) the Riemannian distance induced by | · |

d(x, y) = infg(0)=xg(1)=y

1∫0

|g(s)−1g′(s)|gds

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HS as infinite matrices

HS= matrices aij such that∑i,j

|aij|2 < ∞.

eij=

j

i

. . . . . . . .

. . . 1 . . .

. . . . . . . .

. . . . . . . .

, Qeij=λijeij, λij > 0.

ξij=√

λijeij.

Q is a trace class operator ⇐⇒ ∑i,j

λij < ∞.

For example, λij = ri+j, 0<r<1.

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(i) The Hilbert-Schmidt general group

GLHS=GL(H)⋂

(I + HS),

Lie algebra glHS=HS, gCM = Q1/2HS.

(ii) The Hilbert-Schmidt orthogonal group SOHS is the connectedcomponent of

B : B − I ∈ HS, BT B=BBT=I.

Lie algebra soHS=A : A ∈ HS, AT= − A,

gCM = Q1/2soHS.

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(iii) The Hilbert-Schmidt symplectic group

SpHS=X : X − I ∈ HS, XT JX=J, where

J=

(0 −II 0

).

Lie algebra spHS= X :X ∈ HS, XT J +JX=0,

gCM = Q1/2spHS.

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STOCHASTIC DIFFERENTIAL EQUATIONS ON HS,HEAT KERNEL MEASURES (G., JFA 2000)

Wt a Brownian motion in HS with the covariance operator Q, thatis,

Wt=∞∑

i=1

W it ξi,

W it one-dimensional independent real Brownian motions

ξj∞j=1an orthonormal basis of g as a real space

T = 12

∞∑j=1

ξ2j

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Theorem (G.)

Suppose that Q is a trace-class operator. Then

• [SDEs] the stochastic differential equations

dGt = TGtdt + dWtGt, G0 = X,

dZt = ZtTdt − ZtdWt, Z0 = Y

have unique solutions in HS.

•[Inverse]. The solutions of these SDEs with G0 = Z0 = I satisfy

ZtGt = I with probability 1 for any t > 0.

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• [Kolmogorov’s backward equation] The function v(t, X) = Ptϕ(X) isa unique solution to the parabolic type equation

∂tv(t, X)=1

2

∞∑

j=1

D2v(t, X)(ξjX ⊗ξjX)+(TX, Dv(t, X))HS.

Kolmogorov’s backward equation=the group heat equation.

Definition. The heat kernel measure µt on HS is the transitionprobability of the stochastic process Gt, that is, µt(A) = P (Gt ∈ A).

Open question: quasi-invariance of the heat kernel measures.

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CAMERON-MARTIN GROUP AND ISOMETRIES

HL2(µt) the closure in L2(µt) of holomorphic polynomials HP on HS.

Theorem (G.) [Holomorphic skeletons]

For any f ∈ HL2(µt) there is a holomorphic function f on GCM suchthat for any x ∈ GCM and pm ∈ HP

if pmL2(µt)−−−−→ f, then pm(x) −→ f(x).

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This skeleton f is given by the formula

f(x)=∞∑

k=0

06s16...6sk61

(DkIf)(c(s1) ⊗ ... ⊗ c(sk))d~s,

where Dkf is the kth derivative of function f , and c(s) = g(s)−1g′(s)for any smooth path g(s) from I to x.

A larger space of holomorphic functions:

Ht (GCM) holomorphic functions on GCM such that

‖f‖t,∞ = limn→∞ ‖f‖t,n= lim

n→∞

Gn

|f(z)|2dµnt (z) < ∞.

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Theorem(G.) [Pointwise estimates]

For any f ∈ Ht(GCM), g ∈ GCM , 0 < s < t

|(Dkf)(g)|2(g∗)⊗k 6 k!

sk‖f‖2t,∞ exp

(d2(g,I)

t−s

).

Theorem (G.) Q : HS → HS (or soHS or spHS).

• If Q is trace class, then the heat kernel measure lives in GLHS (or SOHS

or SpHS ), and Ht(GCM) is an infinite dimensional Hilbert space.

• If the covariance operator Q is the identity operator, then Ht(GCM)contains only constant functions.

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Theorem(G.) [ISOMETRIES]

• The skeleton map is an isometry from HL2(µt) to Ht(GCM) (therestriction map on holomorphic polynomials HP extends to an isometrybetween the spaces HL2(µt) and Ht(GCM)).

• If Q is a trace class operator, then HL2(µt) is an infinite-dimensionalHilbert space.

• The Taylor map f 7→∞∑

k=0Dkf(I) is an isometry from Ht(GCM) to

the Fock space Ft(g), a subspace of the dual of the tensor algebra of g

with the norm

|α|2t=∞∑

k=0

tk

k!|αk|2.

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HP

inclusionmap

−−−→ HL2(µt)

skeleton“restriction”

map−−−→ Ht(GCM)

Taylormap

−−−→ Ft(g)

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Ft(g) a Hilbert space in the dual of the universal enveloping algebraof the Lie algebra g

T (g) the tensor algebra over g

J the two-sided ideal in T (g) generated byξ ⊗ η − η ⊗ ξ − [ξ, η]; ξ, η ∈ g.

T ′(g) =∞∑

k=0(g⊗k)∗, the algebraic dual of the tensor algebra T (g)

.

J0 the annihilator of J in the dual space T ′(g).

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‖α‖2t=

∞∑k=0

tk

k!|αk|2(g⊗k)∗, α=

∞∑k=0

αk, αk ∈ (g⊗k)∗, k=0, 1, 2, ...

The generalized bosonic Fock space is Ft(g) = α ∈ J0 : ‖α‖2t < ∞

This is the space of Taylor coefficients of functions from Ht(GCM).

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CAMERON-MARTIN GROUP AND EXPONENTIAL MAP

Definition. The Cameron-Martin group is

GCM=x ∈ GL(H), d(x, I) < ∞, where d is the Riemanniandistance induced by | · |:

d(x, y)= infg(0)=xg(1)=y

1∫

0

|g(s)−1g′(s)|gds

Finite dimensional approximations:

gn ascending finite dimensional Lie subalgerbas of g,

Gn Lie groups with Lie algebras

Assumption: all gn are invariant subspaces of Q.

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Theorem (G.) If |[X, Y ]| 6 c|X||Y | then

1. g =⋃

gn, and the exponential map is a local diffeomorphism from g

to GCM .

2.⋃

Gn is dense in GCM in the Riemannian distance induced by | · |.