HEAT KERNEL ANALYSIS ON INFINITE-DIMENSIONAL GROUPS. Masha Gordina. University of Connecticut.

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1 HEAT KERNEL ANALYSIS ON INFINITE-DIMENSIONAL GROUPS Masha Gordina University of Connecticut 6th Cornell Probability Summer School July 2010

2 SEGAL-BARGMANN TRANSFORM AND TAYLOR MAP IN R N AND C N R. H. Cameron, Some examples of Fourier Wiener transforms of analytic functionals, Duke Math.J. 12 (1945), R. H. Cameron, and W. T. Martin, Fourier Wiener transforms of analytic functionals, Duke Math. J. 12 (1945), R. H. Cameron, and W. T. Martin, Fourier Wiener transforms of functionals belonging to L 2 over the space C, Duke Math. J. 14 (1947), V. Bargmann, On a Hilbert space of analytic functions and an associated integral transform, Part I, Communications on Pure and Applied Mathematics 14 (1961), L. Gross, P. Malliavin, Hall s transform and the Segal-Bargmann map, in Itô s stochastic calculus and probability theory, Springer, 1996,

3 u, v = n j=1 u j v j u, v C n, bilinear inner product z = x iy z = x + iy C n, x, y R n u, v z 2 = z, z the usual inner product in C n the usual norm on C n p t (x) = 1 e n/2 (2πt) x 2 2t, x R n µ t (z) = 1 (2πt) n e z 2 t, z C n

4 The heat kernels, p t and µ t, are the fundamental solutions to the heat equations in R n and C n with the Laplacians = n j=1 2 x 2 j C = n j=1 ( 2 x 2 j + 2 y 2 j ) z = (z 1,..., z n ) C n, z j = x j + iy j, x j, y j R.

5 Namely, p t is the solution to u (t, x) t u (t, x) = 0, x R n, 0 < t <, u (0, x) = δ (x), where δ (x) is the Dirac delta function. The general solution to the heat equation on R n is given by a convolution with p t with the initial value u (0, x) = f (x) u (t, x) = x 2 2t R p R n t (x y) f (y) dy = e f (y) e x,y t n ) (x) = (p t f) (x), f L 2 (R n, dx) ( e t 2 f p t (y) dy

6 H (C n ) HL 2 (C n, µ t (z) dz) the space of holomorphic functions on C n H (C n ) L 2 (C n, µ t (z) dz) Theorem (Bargmann) Let t > 0 and 1 < p <. For any function f L p (R n, p t (x) dx) the convolution p t f has a unique analytic continuation, S t f, to C n given by (S t f) (z) = e z,z 2t z,u f (u) e t p t (u) du, z C n. Rn Moreover, S t is a unitary operator from L 2 (R n, p t (x) dx) onto the space HL 2 (C n, µ t (z) dz).

7 Plan of a proof. Step 1. (S t f) (z) H and (S t f) (x) = (p t f) (x) for any x R n Step 2. (Polynomial basis over C n ) For a multiindex α = (k 1,..., k n ) of nonnegative integers let n z α = z k j, α = α! = j=1 n j=1 n j=1 k j k j!

8 Then the set {z α } forms an orthogonal basis of HL 2 (C n, µ t (z) dz). If f H (C n ) and has the pointwise convergent power series f(z) = α c α z α, c α C, then C n f (z) 2 µ t dz = α c α 2 t α α! The first series is convergent in L 2 (C n, µ t ) if either side of the second identity is finite. Step 3.(Bargmann s pointwise bounds) If f HL 2 (C n, µ t ), then f(z) 2 e z 2 t f 2 L 2 (C n,µ t ), z Cn.

9 Remark (Monotonicity of L 2 norm.) Let 0 k < n. Denote by µ (k) t heat kernel density for C k. Then the f (z) 2 µ (k) C k t dz C n f (z) 2 µ (n) t dz, f H (C n ). Step 4. (Total sets) Let f a (u) = e a,u, a C n, u R n, g a (z) = e a,z, a C n, z C n. For any 1 < p < the sets {f a : a R n } and {f a : a ir n } are each total in L p (R n, p t dx), that is, the linear span of the set is dense in L p (R n, p t dx). The set {g a : a R n } is total in HL 2 (C n, µ t dz).

10 Step 5. (Transform on exponentials) (S t f a ) (z) = e z,u +t a 2 2, (S t f a, S t f b ) L 2 (C n,µ t dz) = (f a, f b ) L 2 (R n,p t dx), a, b Cn.

11 Fock space and the Taylor map α a multilinear form on C n... C n {e j } n j=1 an orthonormal basis of C n α 2 Mult k (C n,c) = α ( ) e j1,..., e jk 2 e j m {e j} n j=1 Mult k (C n, C) k-linear forms on C n, Mult 0 (C n, C) = C Sym k (C n, C) symmetric forms in Mult k (C n, C) α 2 t = k=0 t k k! α k 2 Mult k (C n,c), α k Mult k (C n, C)

12 T t (C n ) the full Fock space of α = α 2 t <. k=0 α k, α k Mult k (C n, C), F t (C n ) the bosonic Fock space of α T t (C n ) such that α k Sym k (C n, C) H (C n ) the space of holomorphic functions f : C n C. D h f(z) d ds t=0 f (z + th) the directional derivative of f H (C n ) D h1 D hk f (z) is a symmetric k linear form on C n which is linear in each variable separately. It is denoted by D k f (z) as a linear form on C n... C n.

13 ( ) D k f (z) (h 1,..., h k ) = ( D h1...d hk f ) (z), z, h 1,..., h k C n D 0 f (z) = f (z) k=0 D k f (z) is a symmetric linear form on C n (1 D) 1 z f = k=0 D k f (z) Sym (C n, C)

14 If (1 D) 1 z f 2 t = k=0 t k k! Dk f (z) 2 <, then (1 D) 1 z f F t (C n ).

15 Theorem (Bargmann-Krée) For any f HL 2 (C n, dµ t ) (1 D) 1 0 f F t (C n ) Moreover, the map (1 D) 1 0 : HL 2 (C n, µ t (z)) F t (C n ) is a surjective isometry. Plan of a proof. The isometry: (1 D) 1 0 f 2 t = C n f (z) 2 dµ t (z), f H (C n )

16 Surjectivity: let α F t (C n ) define Then f (z) = k=0 α k (z,..., z). k! f (z) 2 k=0 α k (z,..., z) k! 2 k=0 α k z k k! 2 k=0 t k α k 2 k! k=0 z k t k k! α 2 t e z 2 t, z C n

17 and (1 D) 1 0 f = α.

18 SUMMING UP ground state transformation L 2 (R n, dx) L 2 (R n, dp t ) f f ϕ S t HL 2 (C n, dµ t ) B T F t dp t = p t dx, dµ t (z) = µ t (z)dz F t is a bosonic Fock space, where C n is a state space of a single particle. HL 2 (C n, µ t ) ={square integrable holomorphic functions}

19 B :H k1 (x j1 )... H kn (x jn ) x k 1 j 1 s... s x k n j n S t :H k1 (x j1 )... H kn (x jn ) z k 1 j 1... z k n j n H k (x) is an Hermite polynomial. T :z k 1 j 1... z k n j n x k 1 j 1 s... s x k n j n

20 FINITE-DIMENSIONAL MATRIX LIE GROUPS G L 2 (G, dg) a finite-dimensional compact matrix Lie group with the identity I the space of square integrable functions on G with respect to right invariant Haar measure dg g Lie algebra of G with the inner product, derivatives t=o f D ξ f (g) = ξf (g) = dt d g G, f C (G) {ξ i } d i=1 an orthonormal basis of the Lie algebra g ( ge tξ), ξ g, a Laplacian d j=1 ξ j 2 f

21 P t the heat semigroup e t on L 2 (G, dg) heat kernel p t a left convolution kernel on G such that P t f (g) = f p t (g) = f (gk) p t (k) dk, for all f C (G). G

22 The heat kernel measure p t (g) dg can be described also as the distribution in t of the process g t satisfying the Stratonovich stochastic differential equation δg t = g t δw t, g 0 = I or Itô s stochastic differential equation dg t = g t dw t g t d 1 ξ 2 i dt, g 0 = I W t the Brownian motion on the Lie algebra g with the identity operator as its covariance W t = d b t i ξ i i=1

23 where b t i are real-valued Brownian motions mutually independent on a probability space (Ω, F, P ). P t f (I) = E [f (g t )] Itô s calculus as the box calculus dt dt = 0, dt db t = 0, db t db t = dt dw t dw t = d db t i ξ i d db t i ξ i = d ξ 2 i dt. i=1 i=1 i=1 Itô s product formula for B, C : M n (C) L (M n (C)), F, G : M n (C) M n (C)

24 then dx t = B (X t ) dw t + F (X t ) dt dy t = C (Y t ) dw t + G (Y t ) dt d (X t Y t ) = dx t Y t + X t dy t + dx t dy t = B (X t ) dw t Y t + F (X t ) Y t dt + X t C (Y t ) dw t + X t G (Y t ) dt + B (X t ) dw t C (Y t ) dw t

25 G = SU (2) g = su (2) { A = { A = ( ) a b c d ( ) α β γ δ } : A A = I, det A = 1 } : A + A = 0, tr A = 0 inner product on su (2) A, B = tr B A U 1 AU, U 1 BU = A, B, A, B su (2), U SU (2) o.n.b. ξ j = i 2 σ j, j = 1, 2, 3 Pauli matrices σ 1 = ( ), σ 2 = ( 0 i i 0 ), σ 3 = ( )

26 For any U SU (2) U = U (ϕ, θ, ψ) = cos θ 2 ei(ϕ+ψ) 2 i sin θ 2 ei(ϕ ψ) 2 i sin θ 2 e i(ϕ ψ) 2 cos θ 2 e i(ϕ+ψ) 2 Euler angles 0 ϕ < 2π, 0 θ π, 2π ψ < 2π Theorem (Laplacian on SU (2)) = ξ ξ2 2 + ξ3 2 = 2 2 θ sin 2 θ 2 ϕ 2 4 cos θ sin 2 θ sin 2 θ 2 ψ + 2cos θ sin θ θ 2 ϕ ψ

27 Let b t 1, bt 2 and bt 3 be three real-valued independent Brownian motions. Then the Brownian motion on SU (2) is a solution to δg t = g t (δb t 1 ξ 1 + δb t 2 ξ 2 + δb t 3 ξ 3 ), g 0 = ( ) since dg t = g t (db t 1 ξ 1 + db t 2 ξ 2 + db t 3 ξ 3 ) 3 2 g tdt, g 0 = ( ) σ σ2 2 + σ2 3 = 3I. Theorem. The process g t SU (2) with probability 1. Its generator is the Laplacian described above.

28 Proof. 1. g t is unitary. First write an SDE for g t since W t = W t. By Itô s product formula dg t = dw tg t 1 2 g t dt, g 0 = ( ) d ( g t g t ) = dgt g t + g tdg t + dg t dg t = g t dw t g t 3 2 g tg t dt g tdw t g t 3 2 g tg t dt + dg t dg t = g t gt dt + g tgt dt = 0,

29 since dw t dw t = d j=1 ξ 2 j dt = 3Idt. 2. det g t = 1. Denote g t = ( At B t C t D t ) W t = ( iα t β t + iγ t β t + iγ t iα t ) where α t, β t, γ t are independent real-valued Brownian motions.

30 da t = ia t dα t B t dβ t + ib t dγ t 3 2 A tdt db t = ib t dα t + A t dβ t + ia t dγ t 3 2 B tdt dc t = ic t dα t D t dβ t + id t dγ t 3 2 C tdt dd t = id t dα t + C t dβ t + ic t dγ t 3 2 D tdt By Itô s product formula d det g t = D t da t +A t dd t C t db t B t dc t +da t dd t db t dc t Thus d det g t = 0 with g 0 = 1.

31 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 Taylor coefficient of f at I to be the element D k f (I) in g k determined by D k f (I), ξ 1... ξ k = ξ 1... ξ k f (I), ξ 1,..., ξ k g with D 0 f (I) = f (I). Then the map (1 D) 1 I : f k=0 D k f (I) from H (G) into T (g) is a unitary operator from HL 2 (G, µ t ) onto F t. B. Driver, On the Kakutani Itô Segal Gross and the Segal Bargmann Hall isomorphisms, J. of Funct. Anal. 133 (1995), B. Driver, L. Gross, Hilbert spaces of holomorphic functions on complex Lie groups, New trends in stochastic analysis (Charingworth, 1994), , 1997, in Proceedings of the 1994 Taniguchi Symposium.

32 HEAT KERNEL ANALYSIS ON HILBERT-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 algebras the Lie algebra determines directions of differentiation the Lie algebra and a norm on it determines a Laplacian and the Wiener process. Cameron-Martin subspace and holomorphic skeletons

33 RESULTS FOR SOME INFINITE-DIMENSIONAL CURVED SPACES. Loop groups: heat kernel measures, the Cameron-Martin subgroup, Ricci is 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-Martin subspace, Taylor map (S. Aida, M. Cecil, B. Driver, S. Fang, E. Hsu, O. Enchev D. Stroock) Diff(S 1 ), Diff(S 1 )/S 1 : heat kernel measures, the Cameron-Martin subgroup, Ricci is bounded from below, quasi-invariance of the heat kernel measures (H. Airault, M. Bowick, A. A. Kirillov, G., P. Lescot, P. Malliavin, S.Rajeev, M. Wu, D. V. Yur ev, B Zumino...)

34 Hilbert-Schmidt groups: heat kernel measures, the Taylor map, holomorphic skeletons, a Cameron-Martin subgroup, Ricci =, Diff(S 1 )/S 1 and Sp HS (G. in PA 00, JFA 00, 05) infinite-dimensional nilpotent groups: heat kernel measures, quasiinvariance of the heat kernel measures, the Taylor map, holomorphic skeletons, a Cameron-Martin subgroup, Ricci is bounded from below, log Sobolev inequality (Driver, Gordina JFA 09, PTRF, JDG, T. Melcher JFA 09)

35 HILBERT-SCHMIDT GROUPS: INFINITE MATRIX GROUPS B(H) bounded linear operators on a complex Hilbert space H. G=GL(H) Q HS invertible elements of B(H). a bounded linear symmetric nonnegative operator on HS. Hilbert-Schmidt operators on H with the inner product (A, B) HS = T rb A. g = g CM HS an infinite-dimensional Lie algebra with a Hermitian inner product (, ), A g = Q 1/2 A HS.

36 G CM GL(H)Cameron-Martin group {x GL(H), d(x, I)< } d(x, y) the Riemannian distance induced by d(x, y) = inf g(0)=x g(1)=y 1 0 g(s) 1 g (s) g ds

37 HS as infinite matrices HS= matrices {a ij } such that i,j a ij 2 <. e ij = i j , Qe ij=λ ij e ij, λ ij ξ ij = λ ij e ij. Q is a trace class operator i,j λ ij <. For example, λ ij = r i+j, 0<r<1.

38 (i) The Hilbert-Schmidt general group GL HS =GL(H) (I + HS), Lie algebra gl HS =HS, gcm = Q 1/2 HS. (ii) The Hilbert-Schmidt orthogonal group SO HS is the connected component of {B : B I HS, B T B=BB T =I}. Lie algebra so HS ={A : A HS, A T = A}, gcm = Q 1/2 so HS.

39 (iii) The Hilbert-Schmidt symplectic group Sp HS ={X : X I HS, ( ) 0 I J=. I 0 X T JX=J}, where Lie algebra sp HS = {X : X HS, X T J +JX=0}, g CM = Q 1/2 sp HS.

40 STOCHASTIC DIFFERENTIAL EQUATIONS ON HS, HEAT KERNEL MEASURES (G., JFA 2000) W t W i t a Brownian motion in HS with the covariance operator Q, that is, W t = Wt i ξ i, i=1 one-dimensional independent real Brownian motions {ξ j } j=1an orthonormal basis of g as a real space T = 1 2 ξj 2 j=1

41 Theorem (G.) Suppose that Q is a trace-class operator. Then [SDEs] the stochastic differential equations dg t = T G t dt + dw t G t, G 0 = X, dz t = Z t T dt Z t dw t, Z 0 = Y have unique solutions in HS. [Inverse]. The solutions of these SDEs with G 0 = Z 0 = I satisfy Z t G t = I with probability 1 for any t > 0.

42 [Kolmogorov s backward equation] The function v(t, X) = P t ϕ(x) is a unique solution to the parabolic type equation t v(t, X)=1 2 D 2 v(t, X)(ξ j X ξ j X) + (T X, Dv(t, X)) HS. j=1 Kolmogorov s backward equation=the group heat equation. Definition. The heat kernel measure µ t on HS is the transition probability of the stochastic process G t, that is, µ t (A) = P (G t A). Open question: quasi-invariance of the heat kernel measures.

43 CAMERON-MARTIN GROUP AND ISOMETRIES HL 2 (µ t ) the closure in L 2 (µ t ) of holomorphic polynomials HP on HS. Theorem (G.) [Holomorphic skeletons] For any f HL 2 (µ t ) there is a holomorphic function f on G CM such that for any x G CM and p m HP if p m L 2 (µ t ) f, then p m (x) f(x).

44 This skeleton f is given by the formula f(x)= (DI k f)(c(s 1)... c(s k ))d s, k=0 0 s 1... s k 1 where D k f is the kth derivative of function f, and c(s) = g(s) 1 g (s) for any smooth path g(s) from I to x. A larger space of holomorphic functions: H t (G CM ) f t, = holomorphic functions on G CM such that lim n f t,n= lim n G n f(z) 2 dµ n t (z) <.

45 Theorem(G.) [Pointwise estimates] For any f H t (G CM ), g G CM, 0 < s < t ( ) (D k f)(g) 2 k! (g ) k s k f 2 t, exp d 2 (g,i) t s. Theorem (G.) Q : HS HS (or so HS or sp HS ). If Q is trace class, then the heat kernel measure lives in GL HS (or SO HS or Sp HS ), and H t (G CM ) is an infinite dimensional Hilbert space. If the covariance operator Q is the identity operator, then H t (G CM ) contains only constant functions.

46 Theorem(G.) [ISOMETRIES] The skeleton map is an isometry from HL 2 (µ t ) to H t (G CM ) (the restriction map on holomorphic polynomials HP extends to an isometry between the spaces HL 2 (µ t ) and H t (G CM )). If Q is a trace class operator, then HL 2 (µ t ) is an infinite-dimensional Hilbert space. The Taylor map f k=0 D k f(i) is an isometry from H t (G CM ) to the Fock space F t (g), a subspace of the dual of the tensor algebra of g with the norm α 2 t = k=0 t k k! α k 2.

47 HP inclusion map HL 2 (µ t ) skeleton restriction map H t (G CM ) Taylor map F t (g)

48 F t (g) T (g) a Hilbert space in the dual of the universal enveloping algebra of the Lie algebra g the tensor algebra over g J the two-sided ideal in T (g) generated by {ξ η η ξ [ξ, η]; ξ, η g}. T (g) J 0 =. k=0 (g k ), the algebraic dual of the tensor algebra T (g) the annihilator of J in the dual space T (g).

49 α 2 t = k=0 t k k! α k 2 (g k ), α= k=0 α k, α k (g k ), k=0, 1, 2,... The generalized bosonic Fock space is F t (g) = {α J 0 : α 2 t < } This is the space of Taylor coefficients of functions from H t (G CM ).

50 CAMERON-MARTIN GROUP AND EXPONENTIAL MAP Definition. The Cameron-Martin group is G CM ={x GL(H), d(x, I) < }, where d is the Riemannian distance induced by : d(x, y)= inf g(0)=x g(1)=y 1 Finite dimensional approximations: 0 g(s) 1 g (s) g ds g n ascending finite dimensional Lie subalgerbas of g, G n Lie groups with Lie algebras Assumption: all g n are invariant subspaces of Q.

51 Theorem (G.) If [X, Y ] c X Y then 1. g = g n, and the exponential map is a local diffeomorphism from g to G CM. 2. G n is dense in G CM in the Riemannian distance induced by.