HEAT KERNEL ANALYSIS ONINFINITE-DIMENSIONAL GROUPS
Masha Gordina
University of Connecticut
http://www.math.uconn.edu/~gordina
6th Cornell Probability Summer School
July 2010
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.
〈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
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.
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)
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).
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!
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.
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).
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.
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)
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.
(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)
If
‖ (1 − D)−1z f‖2
t =∞∑
k=0
tk
k!|Dkf (z) |2 < ∞,
then (1 − D)−1z f ∈ Ft (Cn).
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)
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
and
(1 − D)−10 f = α.
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
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
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
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) .
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
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)
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
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
)
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
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.
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,
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.
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.
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.
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
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...)
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)
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.
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
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.
(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.
(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.
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
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.
• [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.
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).
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) < ∞.
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.
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.
HP
inclusionmap
−−−→ HL2(µt)
skeleton“restriction”
map−−−→ Ht(GCM)
Taylormap
−−−→ Ft(g)
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).
‖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).
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.
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 | · |.