HEAT KERNEL ANALYSIS ON INFINITE-DIMENSIONAL HEISENBERG GROUPS

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1 HEAT KERNEL ANALYSIS ON INFINITE-DIMENSIONAL HEISENBERG GROUPS BRUE K. DRIVER AND MARIA GORDINA Abstract. We introduce a class of non-commutative Heisenberg like infinite dimensional Lie groups based on an abstract Wiener space. The Ricci curvature tensor for these groups is computed and shown to be bounded. Brownian motion and the corresponding heat kernel measures, {ν t} t>, are also studied. We show that these heat kernel measures admit: 1 Gaussian like upper bounds, ameron-martin type quasi-invariance results, 3 good L p bounds on the corresponding Radon-Nykodim derivatives, 4 integration by parts formulas, and 5 logarithmic Sobolev inequalities. The last three results heavily rely on the boundedness of the Ricci tensor. ontents 1. Introduction 1.1. A finite-dimensional paradigm 1.. Summary of results 3. Abstract Wiener Space Preliminaries 5 3. Infinite-Dimensional Heisenberg Type Groups Length and distance estimates Norm estimates Examples Finite Dimensional Projections and ylinder Functions Brownian Motion and Heat Kernel Measures 4.1. A quadratic integral Brownian Motion on G ω Finite Dimensional Approximations 5 5. Path space quasi-invariance 3 6. Heat Kernel Quasi-Invariance The Ricci urvature on Heisenberg type groups Examples revisited Heat Inequalities Infinite-dimensional Radon-Nikodym derivative estimates Logarithmic Sobolev Inequality Future directions 47 Date: September 9, 8 File:RealHeisenberg5 8.tex Mathematics Subject lassification. Primary; 35K5,43A15 Secondary; 58G3. Key words and phrases. Heisenberg group, heat kernel, quasi-invariance, logarithmic Sobolev inequality. This research was supported in part by NSF Grant DMS-5468 and the Miller Institute at the University of alifornia, at Berkeley. Research was supported in part by NSF Grant DMS

2 DRIVER AND GORDINA Appendix A. Wiener Space Results 48 Appendix B. The Ricci tensor on a Lie group 5 Appendix. Proof of Theorem Proof of Theorem References Introduction The aim of this paper is to construct and study properties of heat kernel measures on certain infinite-dimensional Heisenberg groups. In this paper the Heisenberg groups will be constructed from a skew symmetric form on an abstract Wiener space. A typical example of such a group is the Heisenberg group of a symplectic vector space. Before describing our results let us recall some typical heat kernel results for finite-dimensional Riemannian manifolds A finite-dimensional paradigm. Let M, g be a complete connected n dimensional Riemannian manifold n <, g be the Laplace Beltrami operator acting on M, and Ric denote the associated Ricci tensor. Recall see for example Strichartz [48], Dodziuk [15] and Davies [1] that the closure,, of c M is self-adjoint on L M, dv, where dv gdx 1... dx n is the Riemann volume measure on M. Moreover, the semi-group P t : e t / has a symmetric positive integral heat kernel, p t x, y, such that M p tx, ydv y 1 for all x M and 1.1 P t fx : e t / f x p t x, yfydv y for all f L M. M Theorem 1. summarizes some of the results that we would like to extend to our infinite-dimensional Heisenberg group setting. Notation 1.1. If µ is a probability measure on a measure space Ω, F and f L 1 µ L 1 Ω, F, µ, we will often write µf for the integral, Ω fdµ. Theorem 1.. Beyond the assumptions above, let us further assume that Ric ki for some k R. Then 1. 1 p t x, y is a smooth function. The Ricci curvature assumption is not needed here. M p t x, y dv y 1, see for example Davies [1, Theorem 5..6 ]. 3 Given a point o M, let dν t x : p t o, x dv x for all t >. Then {ν t } t> may be characterized as the unique family of probability measures such that the function t ν t f : M fdν t is continuously differentiable, d dt ν tf 1 ν t f, and lim t ν t f fo for all f B M, the bounded functions on M. 4 There exist constants, c c K, n, T and K, n, T, such that, 1.3 pt, x, y V x, t/ exp c d x, y t,

3 HEAT KERNELS ON INFINITE-DIMENSIONAL HEISENBERG GROUPS 3 for all x, y M and t, T ], where d x, y is the Riemannian distance from x to y and V x, r is the volume of the r ball centered at x. 5 The heat kernel measure, ν T, for any T > satisfies the following logarithmic Sobolev inequality; 1.4 ν T f log f k 1 1 e kt v T f + ν T f log ν T f, for f c M. These results are fairly standard. For item 3. see [17, Theorem.6 ], for Eq. 1.3 see for example Theorems 5.6.4, 5.6.6, and in Saloff-oste [47] and for more detailed bounds see [39, 1, 46, 13, 6]. The logarithmic Sobolev inequality in Eq. 1.4 generalizes Gross [8] original Logarithmic Sobolev inequality valid for M R n and is due in this generality to D. Bakry and M. Ledoux, see [, 3, 38]. Also see [3, 1, 5, 51, 1] and Driver and Lohrenz [, Theorem.9] for the case of interest here, namely when M is a uni-modular Lie group with a left invariant Riemannian metric. When passing to infinite-dimensional Riemannian manifolds we will no longer have available the Riemannian volume measure. Because of this problem, we will take item 3. of Theorem 1. as our definition of the heat kernel measure. The heat kernel upper bound in Eq. 1.3 also does not make sense in infinite dimensions. However, the following consequence almost does: there exists c T > such that 1.5 M exp c d o, x t dν t x < for all < t T. In fact Eq. 1.5 will not hold in infinite dimensions either. It will be necessary to replace the distance function, d, by a weaker distance function as happens in Fernique s theorem for Gaussian measure spaces. With these results as background we are now ready to summarize the results of this paper. 1.. Summary of results. Let us describe the setting informally, for precise definitions see Sections and 3. Let W, H, µ be an abstract Wiener space, be a finite-dimensional inner product space, and ω : W W be a continuous skew symmetric bilinear quadratic form on W. The set g W can be equipped with a Lie bracket by setting [A, a, B, b], ω A, B. As in the case for the Heisenberg group of a symplectic vector space, the Lie algebra g W can be given the group structure by defining w 1, c 1 w, c w 1 + w, c 1 + c + 1 ω ω 1, w. The set W with the group structure will be denoted by G or G ω. The Lie subalgebra g M H is called the ameron-martin subalgebra, and g M equipped with the same group multiplication denoted by G M and called the ameron- Martin subgroup. We equip G M with the left invariant Riemannian metric which agrees with the natural Hilbert inner product, A, a, B, b gm : A, B H + a, b, on g M Te G M. In Section 3 we give several examples of this abstract setting including the standard finite-dimensional Heisenberg group.

4 4 DRIVER AND GORDINA The main objects of our study are a Brownian motion in G and the corresponding heat kernel measure defined in Section 4. Namely, let {B t, B t} t be a Brownian motion on g with variance determined by ] E [ B s, B s, A, a gm B t, B t,, c gm Re A, a,, c gm min s, t for all s, t [,, A, H and a, c. Then the Brownian motion on G is the continuous G valued process defined by g t B t, B t + 1 t ω B τ, db τ. For T > the heat kernel measure on G is ν T Law g T. It is shown in orollary 4.5 that {ν t } t> satisfies item 3. of Theorem 1. with o, G ω. Theorem 4.16 gives heat kernel measure bounds that may be viewed as a noncommutative version of Fernique s theorem for Gω. In light of Theorem 3.1 this result is also analogous to the integrated Gaussian upper bound in Eq In Theorem 5. we prove quasi-invariance for the path space measure associated to the Brownian motion, g, on G with respect to multiplication on the left by finite energy paths in the ameron-martin subgroup G M. In light of the results in Malliavin [4] it is surprising that Theorem 5. holds. Theorem 5. is then used to prove quasi-invariance of the heat kernel measures with respect to both right and left multiplication Theorem 6.1 and orollary 6., as well as integration by parts formulae on the path space and for the heat kernel measures, see orollaries These results can be interpreted as the first steps towards proving ν t is a strictly positive smooth measure. In this infinite-dimensional it is natural to interpret quasi-invariance and integration by parts formulae as properties of the smoothness of the heat kernel measure, see [17, Theorem 3.3] for example. In Section 7 we compute the Ricci curvature and check that not only it is bounded from below see Proposition 7., but also that the Ricci curvature of certain finitedimensional approximations are bounded from below with constants independent of the approximation. Based on results in [18], these bounds allow us to give another proof of the quasi-invariance result for ν t and at the same time to get L p estimates on the corresponding Radon-Nikodym derivatives, see Theorem 8.1. These estimates are crucial for the heat kernel analysis on the spaces of holomorphic functions which is the subject of our paper [19]. In Theorem 8.3 we show that an analogue of the logarithmic Sobolev inequality in Eq. 1.4 holds in our setting as well. In Section 9 we give a list of open questions and further possible developments of the results of this paper. We expect our methods to be applicable to a much larger class of infinite-dimensional nilpotent groups. Finally, we refer to papers of H. Airault, P. Malliavin, D. Bell, Y. Inahama concerning quasi-invariance, integration by parts formulae and the logarithmic Sobolev inequality on certain infinite-dimensional curved spaces [1, 4, 6, 5, 3].. Abstract Wiener Space Preliminaries Suppose that X is a real separable Banach space and B X is the Borel σ algebra on X.

5 HEAT KERNELS ON INFINITE-DIMENSIONAL HEISENBERG GROUPS 5 Definition.1. A measure µ on X, B X is called a mean zero, non-degenerate Gaussian measure provided that its characteristic functional is given by.1 ˆµu : e iux dµ x e 1 qu,u for all u X, X where q q µ : X X R is a quadratic form such that qu, v qv, u and qu qu, u with equality iff u, i.e. q is a real inner product on X. In what follows we frequently make use of the fact that. p : x p X dµ x < for all 1 p <. X This is a consequence of Skorohod s inequality see for example [36, Theorem 3.].3 e λ x X dµ x < for all λ < ; X or the even stronger Fernique s inequality see for example [8, Theorem.8.5] or [36, Theorem 3.1].4 e δ x X dµ x < for some δ >. X Lemma.. If u, v X, then.5 u x v x dµ x q u, v and.6 qu, v u X v X. X Proof. Let u µ : µ u 1 denote the law of u under µ. Then by Equation.1, and hence,.7 u µ dx X 1 πq u, u e 1 qu,u x dx u x dµ x q µ u, u q u, u. Polarizing this identity gives Equation.5 which along with Equation. implies Equation.6. The next theorem summarizes some well known properties of Gaussian measures that we will use freely below. Theorem.3. Let µ be a Gaussian measure on a real separable Banach space, X. For x X let ux.8 x H : sup u X \{} qu, u and define the ameron-martin subspace, H X, by.9 H {h X : h H < }. Then 1 H is a dense subspace of X;

6 6 DRIVER AND GORDINA there exists a unique inner product,, H on H such that h H h, h for all h H. Moreover, with this inner product H is a separable Hilbert space. 3 For any h H.1 h X h H, where is as in.. 4 If {e j } is an orthonormal basis for H, then for any u, v H.11 q u, v u, v H u e j v e j. The proof of this standard theorem is relegated to Appendix A see Theorem A.1. Remark.4. It follows from Equation.1 that any u X restricted to H is in H. Therefore we have.1 u dµ q u, u u H u e j, X where {e j } is an orthonormal basis for H. More generally, if ϕ is a linear bounded map from X to, where is a real Hilbert space, then.13 ϕ H : ϕ e j ϕ x dµ x <. To prove Equation.13, let {f j } be an orthonormal basis for. Then ϕ x dµ x ϕ x, f j dµ x ϕ x, f j dµ x X X ϕ, f j k1 H X k1 X ϕ e k, f j ϕ e k, f j ϕ e k ϕ H. A simple consequence of Eq..13 is that.14 ϕ H ϕ X x X dµ x ϕ X. X k1 3. Infinite-Dimensional Heisenberg Type Groups Throughout the rest of this paper X, H, µ will denote a real abstract Wiener space, i.e. X is a real separable Banach space, H is a real separable Hilbert space densely embedded into X, and µ is a Gaussian measure on X, B X such that Equation.1 holds with q u, u : u H, u H H. Following the discussion in [35] and [3] we will say that a possibly infinite dimensional Lie algebra, g, is of Heisenberg type if : [g, g] is contained in the center of g. If g is of Heisenberg type and W is a complementary subspace to

7 HEAT KERNELS ON INFINITE-DIMENSIONAL HEISENBERG GROUPS 7 in g, we may define a bilinear map, ω : W W, by ω w, w [w, w ] for all w, w W. Then for ξ i : w i + c i W g, i 1, we have [ξ 1, ξ ] [w 1 + c 1, w + c ] + ω w 1, w. If we now suppose G is a finite-dimensional Lie group with Lie algebra g, then by the Baker-ampbell-Dynkin-Hausdorff formula e ξ 1 e ξ e ξ 1+ξ + 1 [ξ 1,ξ ] e w 1+w +c 1 +c + 1 ωw 1,w. In particular, we may introduce a group structure on g by defining w 1 + c 1 w + c w 1 + w + c 1 + c + 1 ω w 1, w. With this as motivation, we are now going to introduce a class of Heisenberg type Lie groups based on the following data. Notation 3.1. Let W, H, µ be an abstract Wiener space, be a finite-dimensional inner product space, and ω : W W be a continuous skew symmetric bilinear quadratic form on W. Further let 3.1 ω : sup { ω w 1, w : w 1, w W with w 1 W w W 1}. be the uniform norm on ω which is finite by the assumed continuity of ω. We now define g : W which is a Banach space in the norm 3. w, c g : w W + c. We further define g M : H which is a Hilbert space relative to the inner product 3.3 A, a, B, b gm : A, B H + a, b. The associated Hilbertian norm on g M is given by 3.4 A, a gm : A H + a. It is easily checked that defining 3.5 [w 1, c 1, w, c ] :, ω w 1, w for all w 1, c 1, w, c g makes g into a Lie algebra such that g M is Lie subalgebra of g. Note that this definition implies that [g, g] is contained in the center of g. It is also easy to verify that we may make g into a group using the multiplication rule 3.6 w 1, c 1 w, c w 1 + w, c 1 + c + 1 ω w 1, w. The latter equation may be more simply expressed as 3.7 g 1 g g 1 + g + 1 [g 1, g ], where g i w i, c i, i 1,. As sets G and g are the same. The identity in G is e, and the inverse is given by g 1 g for all g w, c G. Let us observe that {} is in the center of both G and g and for h in the center of G, g h g + h. In particular, since [g, h] {} it follows that k [g, h] k + [g, h] for all k, g, h G.

8 8 DRIVER AND GORDINA Definition 3.. When we want to emphasize the group structure on g we denote g by G or G ω. Similarly, when we view g M as a subgroup of G it will be denoted by G M and will be called the ameron Martin subgroup. Lemma 3.3. The Banach space topologies on g and g M make G and G M into topological groups. Proof. Since g 1 g, the map g g 1 is continuous in the g and g M topologies. Since g 1, g g 1 + g and g 1, g [g 1, g ] are continuous in both the g and g M topologies, it follows from Equation 3.7 that g 1, g g 1 g is continuous as well. For later purposes it is useful to observe, by Equations 3.5 and 3.7, that 3.8 g 1 g g g 1 g + g g + 1 ω g 1 g g g for any g 1, g G. Notation 3.4. To each g G, let l g : G G and r g : G G denote left and right multiplication by g respectively. Notation 3.5 Linear differentials. Suppose f : G is a Frechét smooth function. For g G and h, k g let f g h : h f g d f g + th dt and f g h k : h k f g. Here and in the sequel a prime on a symbol will be used denote its derivative or differential. As G is a vector space, to each g G we can associate the tangent space as in the following notation to G at g, T g G, which is naturally isomorphic to G. Notation 3.6. For v, g G, let v g T g G denote the tangent vector satisfying, v g f f g v for all Frechét smooth functions, f : G. We will write g and g M for T e G and T e G M respectively. Of course as sets we may view g and g M as G and G M respectively. For h g, let h be the left invariant vector field on G such that h g h when g e. More precisely if σ t G is any smooth curve such that σ e and σ h e.g. σ t th, then 3.9 h g lg h : d g σ t. dt As usual we view h as a first order differential operator acting on smooth functions, f : G, by 3.1 hf g d f g σ t. dt Proposition 3.7. Let f : G be a smooth function, h A, a g and g w, c G. Then 3.11 h g : lg h A, a + 1 ω w, A for all g w, c G g

9 HEAT KERNELS ON INFINITE-DIMENSIONAL HEISENBERG GROUPS 9 and in particular using Notation A, af g f g A, a + 1 ω w, A. Furthermore, if h A, a, k B, b, and then 3.13 h kf k hf [h, k]f. In other words, the Lie algebra structure on g induced by the Lie algebra structure on the left invariant vector fields on G is the same as the Lie algebra structure defined in Eq Proof. Since th t A, a is a curve in G passing through the identity at t, we have d h g [g th] d [w, c ta, a] dt dt d [ w + ta, c + ta + t ] dt ω w, A A, a + 1 ω w, A. So by the chain rule, 3.14 where hf g f g h g and hence h kf g d [f g th dt k ] g th f g h g k g + f g d g th, dt k d d g th B, a + 1 dt k dt ω w + ta, B, 1 ω A, B. Since f g is symmetric, it now follows by subtracting Equation 3.14 with h and k interchanged from itself that h kf k hf g f g, ω A, B f g [h, k] [h, k]f g as desired. Lemma 3.8. The one parameter group in G, e th, determined by h A, a g, is given by 3.15 e th th t A, a. Proof. Letting w t, c t : e th, according to Equation 3.11 we have that d w t, c t A, a + 1 dt ω w t, A with w and c. The solution to this differential equation is easily seen to be given by Equation 3.15.

10 1 DRIVER AND GORDINA 3.1. Length and distance estimates. Notation 3.9. Let T > and M 1 denote the collection of 1 -paths, g : [, T ] G M. The length of g is defined as 3.16 l GM g T l g 1 s g s gm ds. As usual, the Riemannian distance between x, y G M is defined as d GM x, y inf { l GM g : g 1 M such that g x and g T y }. It will also be convenient to define y : d GM e, y for all y G M. The value of T > used in defining d M is irrelevant since the length functional is reparametrization invariant. Let 3.17 : sup { ω h, k : h H k H 1} ω <. The inequality in Eq is a consequence of Eq..1 and the definition of ω in Eq Proposition 3.1. Let ε : 1/ where is as in Eq Then for all x, y G M, 3.18 d GM x, y 1 + x y gm gm y x gm and in particular, x d GM e, x x gm. Moreover, there exists K < such that if x, y G M with d GM x, y < ε/ 1/, then 3.19 y x gm K 1 + x gm y gm d GM x, y. As a consequence of Eqs and 3.19 we see that the topology on G M induced by d GM is the same as the Hilbert topology induced by gm. Remark The equivalence of these two topologies in an infinite-dimensional setting has been addressed in [4] in the case of Hilbert-Schmidt groups of operators. Proof. For notational simplicity, let T 1. If g s w s, a s is a path in M 1 for s 1, then by Equation 3.11 l g 1 s g s w s, a s 1 3. ωws, w s g s 1 [g s, g s] and we may write Equation 3.16 more explicitly as l GM g g s 1 [g s, g s] ds. gm If we now apply Equation 3.1 to g s x + s y x for s 1, we see that 1 d GM x, y l GM g y x 1 [x + s y x, y x] ds gm y x 1 [x, y x] 1 + gm x y x gm gm.

11 HEAT KERNELS ON INFINITE-DIMENSIONAL HEISENBERG GROUPS 11 As we may interchange the roles of x and y in this inequality, the proof of Equation 3.18 is complete. Let { } B ε : x g M : x gm ε, y B ε, and g : [, 1] G M be a 1 path such that g, e and g 1 y. Further let T [, 1] be the first time that g exits B ε with the convention that T 1 if g [, 1] B ε. Then from Equation 3.1 l GM g l GM g [,T ] T [ g s gm 1 ] [g s, g s] gm ds 3. 1 ε T g s gm ds 1 g T g M 1 y g M. Optimizing Equation 3. over g implies y d GM e, y 1 y g M for all y B ε. 1 ε g T gm If in the above argument y was not in B ε, then the path g would have had to exit B ε and we could conclude that l GM g g T gm / ε/ and therefore that d GM e, y ε/. Hence we have shown that y d GM e, y 1 min ε, y gm for all y G M. Now suppose that x, y G M and without loss of generality that x gm y gm. Using the left invariance of d GM, it follows that 3.3 d GM x, y d GM e, x 1 y 1 min ε, x 1 y gm. If we further suppose that d GM x, y < ε, we may conclude from Equation 3.3 that y x 1 [x, y] x 1 y gm d GM x, y. gm If we write x A, a and y B, b, it follows that B A H + b a 1 ω A, B and therefore B A H d GM x, y and b a b a 1 ω A, B + 1 ω A, B d GM x, y + 1 ω A, B A 4d G M x, y d GM x, y + A H B A H d GM x, y 1 + A H d GM x, y 1 + x. gm

12 1 DRIVER AND GORDINA ombining these results shows that if d GM x, y < ε then y x g M 4d G M x, y x gm from which Equation 3.19 easily follows. We are most interested in the case where {ω A, B : A, B H} is a total subset of, i.e. span {ω A, B : A, B H}. In this case it turns out that straight line paths are bad approximations to the geodesics joining e G M to points x G M far away from e. For points x G M distant from e it is better to use horizontal paths instead which leads to the following distance estimates. Theorem 3.1. Suppose that {ω A, B : A, B H} is a total subset of. Then there exists ω < such that 3.4 d M e, A, a ω A H + a for all A, a g M. Moreover, for any ε > there exists γ ε > such that and 3.5 γ ε A H + a d M e, A, a if d M e, A, a ε. Thus away from any neighborhood of the identity, d M e, A, a is comparable to A H + a. Since this theorem is not central to the rest of the paper we will relegate its proof to Appendix. The main point of Theorem 3.1 is to explain why Theorem 4.16 is an infinite dimensional analogue of the integrated Gaussian heat kernel bound in Eq Norm estimates. Notation Suppose H and are real complex Hilbert spaces, L : H is a bounded operator, ω : H H is a continuous complex bilinear form, and {e j } is an orthonormal basis for H. The Hilbert Schmidt norms of L and ω are defined by 3.6 L H : Le j, and 3.7 ω ω H H : ω e i, e j. i, It is easy to verify directly that these definitions are basis independent. Also see Equation 3.9 below. Proposition Suppose that W, H, µ is a real abstract Wiener space, ω : W W is as in Notation 3.1, and {e j } is an orthonormal basis for H. Then 3.8 ω w, H ω w W for all w W

13 HEAT KERNELS ON INFINITE-DIMENSIONAL HEISENBERG GROUPS 13 and 3.9 ω W W where is as in Equation.. ω w, w dµ w dµ w ω <, Proof. From Equation.13, ω w, H ω w, w dµ w W ω w W w W dµ w ω w W. W Similarly, viewing w ω w, as a continuous linear map from W to H it follows from Eqs..13 and.14, that ω h ω h, H H ω w, H dµ w W ω w W dµ w ω. W Remark The Lie bracket on g M has the following continuity property, [A, a, B, b] gm A, a gm B, b gm where ω as in Eq This is a consequence of the following simple estimates [A, a, B, b] gm, ω A, B gm ω A, B A H B H A, a gm B, b gm. This continuity property of the Lie bracket is often used to prove that the exponential map is a local diffeomorphism e.g. see [4] in the case of infinite-dimensional matrix groups. In the Heisenberg group setting the exponential map is a global diffeomorphism as follows from Lemma 3.8, where we have not used continuity of the Lie bracket. Lemma Suppose that H is a real Hilbert space, is a real finite-dimensional inner product space, and l : H is a continuous linear map. Then for any orthonormal basis {e j } of H the series 3.3 l e j l e j and 3.31 l e j e j H are convergent and independent of the basis. 1 1 If we were to allow to be an infinite-dimensional Hilbert space here, we would have to assume that l is Hilbert Schmidt. When dim <, l : H is Hilbert Schmidt iff it is bounded.

14 14 DRIVER AND GORDINA Proof. If {f i } dim i1 is an orthonormal basis for, then l e j l e j l e j dim i1 f i, l e j dim i1 f i, l H < which shows that the sum in Equation 3.3 is absolutely convergent and that l is Hilbert Schmidt. Similarly, since {l e j e j } is an orthogonal set in H and l e j e j H l e j <, the sum in Equation 3.31 is convergent as well. Recall that if H and K are two real Hilbert spaces then the Hilbert space tensor product, H K, is unitarily equivalent to the space of Hilbert Schmidt operators, HS H, K, from H to K. Under this identification, h k H K corresponds to the operator still denoted by h k in HS H, K defined by; H h h, h H k K. Using this identification we have that for all c ; l e j l e j c l e j l e j, c l e j e j, l c l e j, l c e j ll c and l e j e j c e j l e j, c e j e j, l c l c, which clearly shows that Equations 3.3 and3.31 are basis independent Examples. Here we describe several examples including finite-dimensional Heisenberg groups. As we mentioned earlier a typical example of such a group is the Heisenberg group of a symplectic vector space. For each of the examples presented we will explicitly compute the norm ω of the form ω as defined in Equation 3.7. In Section 7 we will also explicitly compute the Ricci tensor for each of the examples introduced here. To describe some of the examples below, it is convenient to use complex Banach and Hilbert spaces. However, for the purposes of this paper the complex structure on these spaces should be forgotten. In doing so we will use the following notation. If X is a complex vector space, let X Re denote X thought of as a real vector space. If H,, H is a complex Hilbert space, we define, HRe Re, H in which case H Re,, HRe becomes a real Hilbert space. Before going to the examples, let us record the relationship between the complex and real Hilbert-Schmidt norms of Notation 3.13.

15 HEAT KERNELS ON INFINITE-DIMENSIONAL HEISENBERG GROUPS 15 Lemma Suppose H and are complex Hilbert spaces, L : H and ω : H H are as in Notation 3.13, and c. Then 3.3 L HRe L Re H, 3.33 ω,, c Re ω,, c HRe H, H H Re and 3.34 ω, H Re H Re Re 4 ω, H H. Proof. A straightforward proof. Example 3.18 Finite-dimensional real Heisenberg group. Let R, W H n Re R n, and ω w, z : Im w, z be the standard symplectic form on R n, where w, z w z is the usual inner product on n. Any element of the group H n R : G ω can be written as g z, c, where z n and c R. As above, the Lie algebra, h n R, of Hn R is, as a set, equal to Hn R itself. If {e j} n is an orthonormal basis for R n then {e j, ie j } n is an orthonormal basis for H and real Hilbert-Schmidt norm of ω is given by 3.35 ω H H n n [Im εe j, δe k ] δ j,k n. j,k1 ε,δ {1,i} j,k1 Example 3.19 Finite-dimensional complex Heisenberg group. Suppose that W H n n,, and ω : W W is defined by ω w 1, w, z 1, z w 1 z w z 1. Any element of the group H n : G ω can be written as g z 1, z, c, where z 1, z n and c. As above, the Lie algebra, h n, of Hn is, as a set, equal to H n itself. In this case {e j,,, e j } n is a complex orthonormal basis for H. The complex Hilbert-Schmidt norm of the symplectic form ω is given by n ω H H ω e j,,, e j n. Example 3.. Let K,, be a complex Hilbert space and Q be a strictly positive trace class operator on K. For h, k K, let h, k Q : h, Qk and h Q : h, h Q. Also let K Q,, Q denote the Hilbert space completion of K, Q. Analogous to Example 3.18, let H : K Re, W : K Q Re, and ω : W W R : be defined by ω w, z Im w, z Q for all w, z W. Then G ω W R is a real group and W, H determines an abstract Wiener space see for example [36, Exercise 17, p. 59] and [8, Example 3.9.7]. Let {e j } be an orthonormal basis for K so that {e j, ie j } is an orthonormal basis for H, Re, K. Then the real Hilbert-Schmidt norm of ω is given by ω [ H H Im εe j, Qδe k ] j,k1 ε,δ {1,i}

16 16 DRIVER AND GORDINA 3.36 j,k1 [ Im e j, Qe k + Re e j, Qe k ] Qe k Q HS tr Q. k1 Example 3.1. Again let K,,, Q, and K Q,, Q j,k1 e j, Qe k be as in the previous example. Let us further assume that K is equipped with a conjugation, k k, which is isometric and commutes with Q. Analogously to Example 3.19, let W : K Q K Q, H K K, and let ω : W W be defined by ω w 1, w, z 1, z w 1, z Q w, z 1 Q, which is skew symmetric because the conjugation commutes with Q. If {e j } is an orthonormal basis for K, then {ē j } is also an orthonormal basis for K because the conjugation is isometric and {e j,,, e j } is a orthonormal basis for H. Hence, the complex Hilbert-Schmidt norm of ω is given by 3.37 ω H H ω ej,,, e k + ω, e k, e j, j,k1 e j, Qē k Qē k Q HS tr Q. j,k1 k1 Example 3.. Suppose that V,, V is a d dimensional F inner product space F R or, is a finite-dimensional F inner product space, α : V V is an anti-symmetric bilinear form on V, and {q j } is a sequence of positive numbers such that q j <. Let W v V N : q j v j V < and H v V N : v j V < W, each of which are Hilbert spaces when equipped with the inner products v, w W : q j v j, w j V v, w H : v j, w j V and respectively. Further let ω : W W be defined by ω v, w q j α v j, w j.

17 HEAT KERNELS ON INFINITE-DIMENSIONAL HEISENBERG GROUPS 17 Then W Re, H Re is an abstract Wiener space see for example [36, Exercise 17, p. 59] and [8, Example 3.9.7] and, since ω v, w q j α v j, w j α q j v j V w j V α v W w W, we have ω α. For v V, let v j :,...,, v,,,... H where the v is put in the j th position. If {u a } d a1 is an orthonormal basis for V, then {u a j : a 1,..., d} is an orthonormal basis for H. Therefore, 3.38 d ω,, c H H ω u a j, u b k, c j,k1 a,b1 d qj α u a, u b, c a,b1 q j α,, c V V for all c. Example 3.3. Let V,,, α be as in Example 3. with F, W {σ [, 1], V : σ } and H be the associated ameron Martin space, { 1 } H : H V h W : h s V ds <, wherein 1 h s V ds : if h is not absolutely continuous. Further let η be a complex measure on [, 1] and ω σ 1, σ : 1 α σ 1 s, σ s dη s for all σ 1, σ W. Then W, H, ω satisfies all of the assumptions in Notation 3.1. Let {u a } d a1 be the orthonormal basis of V, {l j } be the orthonormal basis of HR, then {l ju a : a 1,,..., d} is an orthonormal basis of H and see [, Lemma 3.8] 3.39 l j s l j t s t for all s, t [, 1]. If we let λ be the total variation of η, then dη ρdλ, where ρ dη dλ. Hence if d η t : ρ t dλ t and c, then ω,, c d ω lj u a, l k u b, c j,k1 a,b1 d j,k1 a,b1 [,1] l j s l k s ρ s dλ s α u a, u b, c

18 18 DRIVER AND GORDINA α,, c l j s l k s l j t l k t ρ s ρ t dλ s dλ t j,k1 [,1] α,, c s t ρ s ρ t dλ s dλ t [,1] 3.4 α,, c s t dη s d η t, [,1] wherein we have used Equation 3.39 in the fourth equality above. Summing this equation over c in an orthonormal basis for shows 3.41 ω α s t dη s d η t. [,1] 3.4. Finite Dimensional Projections and ylinder Functions. Let i : H W be the inclusion map, and i : W H be its transpose, i.e. i l : l i for all l W. Also let H : {h H :, h H Rani H } or in other words, h H is in H iff, h H H extends to a continuous linear functional on W. We will continue to denote the continuous extension of, h H to W by, h H. Because H is a dense subspace of W, i is injective and because i is injective, i has a dense range. Since h, h H as a map from H to H is a conjugate linear isometric isomorphism, it follows from the above comments that H h, h H W is a conjugate linear isomorphism too, and that H is a dense subspace of H. Now suppose that P : H H is a finite rank orthogonal projection such that P H H. Let {e j } n be an orthonormal basis for P H and l j, e j H W. Then we may extend P to a unique continuous operator from W H still denoted by P by letting 3.4 P w : n k, e j H e j n l j w e j for all w W. For all w W we have, P w H P P w W and n P w W, e i H W e i W w W i1 and therefore there exists < such that 3.43 P w H w W for all w W. Notation 3.4. Let Proj W denote the collection of finite rank projections on W such that P W H and P H : H H is an orthogonal projection, i.e. P has the form given in Equation 3.4. Further, let G P : P W a subgroup of G M and be defined by π P w, c : P w, c. π π P : G G P

19 HEAT KERNELS ON INFINITE-DIMENSIONAL HEISENBERG GROUPS 19 Remark 3.5. The reader should be aware that π P : G G P G M is not for general ω and P Proj W a group homomorphism. In fact we have, 3.44 π P [w, c w, c ] π P w, c π P w, c Γ P w, w, where 3.45 Γ P w, w 1, ω w, w ω P w, P w. So unless ω is supported on the range of P, π P is not a group homomorphism. Since, w, b +, c w, b, c for all w W and b, c, we may also write Equation 3.44 as 3.46 π P [w, c w, c ] π P w, c π P w, c Γ P w, w. Definition 3.6. A function f : G is said to be a smooth cylinder function if it may be written as f F π P for some P Proj W and some smooth function F : G P. Notation 3.7. For g w, c G, let γ g and χ g be the elements of g M g M defined by γ g :, ω w, e j e j, and χ g :, ω w, e j, ω w, e j where {e j } is any orthonormal basis for H. Both γ and χ are well defined because of Lemma Notation 3.8 Left differentials. Suppose f : G is a smooth cylinder function. For g G and h, h 1,..., h n g, n N let D f g f g and 3.47 D n f g h 1... h n h 1... h n f g, where hf is given as in Equation 3.1 or Equation 3.1. We will write Df for D 1 f. Proposition 3.9. Let {e j } and {f l} d l1 be orthonormal bases for H and respectively. Then for any smooth cylinder function, f : G, [ ] 3.48 Lf g : e j, d [ ] f g +, f l f g is well defined. Moreover, if f F π P, h is as in Notation 3.5 for all h g M, 3.49 H f g : e j, f g P HF P w, c and 3.5 f g : d l1 l1 [,fl f ] g F P w, c,

20 DRIVER AND GORDINA then 3.51 Lf g H f + f g + f g γ g + 14 χ g. Proof. The proof of the second equality in Eq is straightforward and will be left to the reader. Recall from Equation 3.1 that 3.5 ej, f g f g e j, 1 ω w, e j. Applying e j, to both sides of Equation 3.5 gives e j, f g f g e j, 1 ω w, e j e j, ω w, e j f g e j, e j, + f g, ω w, e j e j, f g, ω w, e j, ω w, e j wherein we have used, ej ω, e j ω e j, e j. Summing Equation 3.54 on j shows, [ ] e j, f g f g e j, e j, + f g γ g + 14 χ g e j, f g + f g γ g + 14 χ g. The formula in Equation 3.51 for Lf is now easily verified and this shows that Lf is independent of the choice of orthonormal bases for H and appearing in Equation Brownian Motion and Heat Kernel Measures For the Hilbert space stochastic calculus background needed for this section, see Métivier [41]. For the background on Itô integral relative to an abstract Wiener space valued Brownian motion, see Kuo [36, pages 188-7] especially Theorem 5.1, Kusuoka and Stroock [37, p. 5], and the appendix in [16]. Suppose now that B t, B t is a smooth curve in g M with B, B, and consider solving, for g t w t, c t G M, the differential equation 4.1 ẇ t, ċ t ġ t l gt Ḃ t, Ḃ t with g,. By Equation 3.11, it follows that ẇ t, ċ t Ḃ t, Ḃ t + 1 w ω t, Ḃ t and therefore the solution to Equation 4.1 is given by 4. g t w t, c t B t, B t + 1 t ω B τ, Ḃ τ dτ. Below in subsection 4., we will replace B and B by Brownian motions and use this to define a Brownian motion on G.

21 HEAT KERNELS ON INFINITE-DIMENSIONAL HEISENBERG GROUPS A quadratic integral. Let {B t, B t} t be a Brownian motion on g with variance determined by ] E [ B s, B s, A, a gm B t, B t,, c gm Re A, a,, c gm min s, t for all s, t [,, A, H and a, c. Also let {e j } H be an orthonormal basis for H. For n N, define P n Proj W as in Notation 3.4, i.e. n n 4.3 P n w w, e j H e j l j w e j for all w W. Proposition 4.1. For each n, let M n t : t ω B τ, dp nb τ. Then 1 {Mt n } t is an L martingale and there exists an L martingale, {M t } t with values in such that [ ] 4.4 lim E max M t Mt n n t T for all T <. The quadratic variation of M is given by; 4.5 M t t ω B τ, H dτ. 3 The square integrable martingale, M t, is well defined independent of the choice of the orthonormal basis, {e j } and hence will be denoted by t ω B τ, db τ. 4 For each p [1,, {M t } t is L p integrable and there exists c p < such that E sup t T M t p c p T p < for all T <. This estimate will be considerably generalized in Proposition 4.13 below. Proof. 1. For P Proj W let Mt P : t ω B τ, dp B τ. Let P, Q Proj W and choose an orthonormal basis, {v l } N l1 for Ran P + Ran Q. We then have [ M ] P E T M Q T T N E ω B τ, P Q v l dτ l1 T E ω B τ, P Q e l dτ l1 T ω e k, P Q e l τdτ T l1 k1 l1 k1 ω e k, P Q e l. Taking P P n and Q P m with m n in Eq. 4.7 allows us to conclude that [ ] E MT n MT m T n ω e l, e j as m, n jm+1 l1

22 DRIVER AND GORDINA because ω < by Proposition Since the space of continuous L martingales on [, T ] is complete in the norm, N E N T and, by Doob s maximal inequality [34, Proposition 7.16], there exists c < such that [ ] E max N t p t T ce N T p, it follows that there exists a square integrable valued martingale, {M t } t, such that Eq. 4.4 holds.. Since the quadratic variation of M n is given by t t n M n t ω B τ, dp n B τ ω B τ, e l dτ and E [ M t M n t ] E [ M M n t ] E [ M + M n t ] E M t Mt n E M t + Mt n as n, Eq. 4.5 easily follows. 3. Suppose now that { e j} H is another orthonormal basis for H and P n : W H are the corresponding orthogonal projections. Taking P P n and P P n in Eq. 4.7 gives, 4.8 E M P n Since l1 k1 T M P n T T l1 k1 l1 ω e k, P n P n e l. ω e k, P ne l ω e k, e l ω e k, P n I e l l1 k1 l1 k1 ω e k, P n I e l ln+1 k1 ω e k, e l as n and similarly but more easily, l1 k1 ω e k, P n e l ω e k, e l as n, we may pass to the limit in Eq. 4.8 to learn that lim n E M P n T M P n T. 4. By Jensen s inequality p/ T p/ T ω B s, H ds T p/ ω B s, ds H T T T p/ ω B s, p H ds T T T p 1 ω B s, p H ds.

23 HEAT KERNELS ON INFINITE-DIMENSIONAL HEISENBERG GROUPS 3 ombining this estimate with Equation 3.8 and then applying either Skorohod s or Fernique s inequality see Equations.3 or.4 shows [ ] T E T p 1 E ω B s, p H ds 4.9 M p/ T T T p 1 p/ ω p B s p W ds T T p 1 p/ ω p y p W dµ y s p/ ds W T p/+1 T p 1 p/ ω p p p/ + 1 c pt p. As a consequence of the Burkholder-Davis-Gundy inequality see for example [49, orollary 6.3.1a on p.344], [45, Appendix A.], or [41, p. 1] and [34, Theorem 17.7] for the real case, for any p there exists c p < such that p [ ] E sup M t c pe M p/ T c pc pt p : c p T p. t T 4.. Brownian Motion on G ω. Motivated by Eq. 4. we have the following definition. Definition 4.. Let B t, B t be a g valued Brownian motion as in subsection 4.1. A Brownian motion on G is the continuous G valued process defined by 4.1 g t B t, B t + 1 t ω B τ, db τ. Further, for T >, let ν T Law g T be a probability measure on G. We refer to ν T as the time T heat kernel measure on G. Remark 4.3. An alert reader may complain that we should use the Stratonovich integral in Eq. 4.1 rather than the Itô integral. However, these two integrals are equal since ω is a skew symmetric form t ω B τ, db τ t t ω B τ, db τ + 1 ω B τ, db τ. t ω db τ, db τ Theorem 4.4 The generator of g t. The generator of g t is the operator L defined in Proposition 3.9. More precisely, if f : G is a smooth cylinder function, then 4.11 d [f g t] f g t dg t + 1 Lfg t dt where L is given in Proposition 3.9, f is defined as in Notation 3.5 and dg t db t, db t + 1 ω B t, db t.

24 4 DRIVER AND GORDINA Proof. Let us begin by observing that dg t dg t db t, db t + 1 ω B t, db t 4.1 [db t, 1 ] ω B t, db t +, db t e j, 1 ω B t, e j dt + d, f l dt l1 where {f l } d l1 is an orthonormal basis for and {e j} for H. Hence, as a consequence of Itô s formula, we have is an orthonormal basis d [f g t] f g t dg t + 1 f g t dg t dg t f g t dg t + 1 f g t e j, 1 ω B t, e j dt + 1 f g t f g t dg t d l1, f l f g t dt f g t dg t + 1 Lf g t dt. e j, f g t dt d, f l dt For the next corollary, let P Proj W as in Equation 3.4, F P H,, and f F π P : G be a cylinder function where P Proj W. We will further suppose there exist < K, p < such that 4.13 F h, c + F h, c + F h, c K 1 + h H + c p for any h P H and c. Further let {f l } d l1 be an orthonormal basis for and extend {e j } n to an orthonormal basis, {e j}, for H. orollary 4.5. If f : G is a cylinder function as above, then 4.14 E [f g T ] f e + 1 i.e ν T f f e + 1 T T In other words, ν t weakly solves the heat equation E [Lf g t] dt, ν t Lf dt. t ν t 1 Lν t with lim t ν t δ e. l1

25 HEAT KERNELS ON INFINITE-DIMENSIONAL HEISENBERG GROUPS 5 Proof. Integrating Equation 4.11 shows 4.16 f g T f e + N T + 1 where N t : t f g τ dg τ t T Lf g τ dτ f g τ db τ, db τ + 1 dm τ and M t t ω B τ, db τ. Using Eqs. 4.1 and 4.13 there exists P, ω < such that d N t : dn t f g t f g t, dg t dg t f g t e j, 1 ω B t, e j dt + n f g t e j, 1 ω B t, e j dt + d f g t, f l dt l1 d f g t, f l dt 1 P, ω 1 + P B t H + B t p B t W + 1 dt 1 + B t W + B t p+ dt, wherein we have used Equation 3.43 for the last inequality. From this inequality and either of Equations.3 or.4, we find E [ N T ] T l1 E 1 + B t W + B t p+ dt < and hence that N t is a square integrable martingale. Therefore we may take the expectation of Equation 4.16 which implies Equation Finite Dimensional Approximations. Proposition 4.6. Let {P n } n1 Proj W be as in Eq. 4.3 and 4.17 B n t : P n B t P n H H W. Then 4.18 lim n E and [ ] max B t B n t p t T W for all p [1,, 4.19 lim n max t T B t B n t W a.s. Proof. Let {w k } k1 W be a countable dense set and for each k N, choose ϕ k W such that ϕ k W 1 and ϕ k w k w k W. We then have, w W sup ϕ k w sup Re ϕ k w for all w W. k By [8, Theorem 3.5.7] with A I, if ε n t : B t B n t, then 4. lim E ε n T p n W for all p [1,. Since {ϕ k ε n t} t is up to a multiplicative factor a standard Brownian motion, { ϕ k ε n t } t is a submartingale for each k N and therefore so is

26 6 DRIVER AND GORDINA { ε n t sup k ϕ k ε n t } t. Hence, according to Doob s inequality, for each p [1, there exists p < such that p 4.1 E max ε n t t T W p E ε n T p W. ombining Equation 4.1 with Equation 4. proves Equation Equation 4.19 now follows from Equation 4.18 and [11, Proposition.11]. To apply this proposition, let E be the Banach space, [, T ], W equipped with the sup-norm, and let ξ k : l k B e k E for all k N. Lemma 4.7 Finite Dimensional Approximations to g t. For P Proj W, Q : I W P, let g P t be the Brownian motion on G P defined by g P t : P B t, B t + 1 t ω P B τ, P db τ. Then 4. and g t g P t QB t, 1 t 4.3 g P t 1 π P g t 1 [ω QB τ, P db τ + ω QB τ, QdB τ],, t [ω B τ, db τ ω P B τ, P db τ]. Also, if {P n } n1 Proj W are as in Eq. 4.3 and 4.4 g n t g Pn t P n B t, B t + 1 t ω P n B τ, dp n B τ, then 4.5 lim n E for all 1 p <. [ ] max g t g n t p t T g Proof. A simple computation shows dp Bt, l gp t 1 dg db t + 1 P t ω P Bt, P dbt ω P Bt, P dbt + 1 dp Bt, db t d P Bt, B t. Hence it follows that g P solves the stochastic differential equation, dg P t l gp t d P Bt, B t with g P and therefore g P is a G P valued Brownian motion. The proof of the equalities in Equations 4. and 4.3 follows by elementary manipulations which are left to the reader. In light of Equation 4.18 of Proposition 4.6, to prove the last assertion we must show [ ] 4.6 lim n E max M t n p t T,

27 HEAT KERNELS ON INFINITE-DIMENSIONAL HEISENBERG GROUPS 7 where M t n is the local martingale defined by Since and M n T M t n : n T t T T E [ M n T ] [ω B τ, db τ ω B n τ, db n τ]. ω B τ, e j dτ + n T ω B τ, e j, ω B n τ, e j dτ j,k1 ω e k, e j n k1 ω B n τ, e j dτ n ω e k, e j as n, it follows by the Burkholder-Davis-Gundy inequalities that M n is a martingale and Equation 4.6 holds for p and hence for p [1, ]. By Doob s maximal inequality [34, Proposition 7.16], to prove Equation 4.6 for p, it suffices to show lim n E [ M T n p ]. However, M T n has Itô s chaos expansion terminating at degree two and hence by a theorem of Nelson see [44, Lemma [ on p. 415] ] and [43, pp ] for each j N there exists c j < such that E M j T n [ c j EM T n ] j. This result also follows from Nelson s hypercontractivity for the Ornstein-Uhlenbeck operator. This clearly suffices to complete the proof of the theorem. Lemma 4.8. For all P Proj W and t >, let νt P : Law g P t. Then νt P dx p P t e, x dx, where dx is the Riemannian volume measure equal to a Haar measure p P t x, y is the heat kernel on G P. Proof. An application of orollary 4.5 with G replaced by G P implies that νt P Law g P t is a weak solution to the heat equation on G P. The result now follows as an application of [17, Theorem.6 ]. orollary 4.9. For any T >, the heat kernel measure, ν T, is invariant under the inversion map, g g 1 for any g G. Proof. It is well known see for example [, Proposition 3.1] that heat kernel measures based at the identity of a finite-dimensional Lie group are invariant under inversion. Now suppose that f : G R is a bounded continuous function. By passing to a subsequence if necessary, we may assume that the sequence of G valued random variables, {g n T } t, in Lemma 4.7 converges almost surely to g T. Therefore by the dominated convergence theorem, Ef g T 1 lim Ef g n T 1 lim Ef g n T Ef g T. n n This completes the proof because ν T is the law of g T. We are now going to give exponential bounds which are much stronger than the moment estimates in Equation 4.9 of Proposition 4.1. Before doing so we need to recall the following result of ameron Martin and Kac, [9, 33].

28 8 DRIVER AND GORDINA Lemma 4.1 ameron Martin and Kac. Let {b s } s be a one dimensional Brownian motion. Then for any T > and λ [, π T [ ; λ ] T 4.7 E exp b sds [cos λt ] 1/ <. Proof. When T 1, simply follow the proof of [31, Equation 6.9 on p. 47] with λ replaced by λ. For general T >, by a change of variables and a Brownian motion scaling we have T 1 1 b sds T b tt dt d T b t dt. Therefore, 4.8 E [ exp λ provided that λ [, π T. T b sds ] [ λ T E exp cos 1/ λ T 1 ] b sds Remark For our purposes below, all we really need later from Lemma 4.1 is the qualitative statement that for λt > sufficiently small [ λ ] T 4.9 E exp b sds 1 + λ T + O λ 4 T 4. 4 Instead of using Lemma 4.1 we can derive this statement as an easy consequence of the scaling identity in Equation 4.8 along with the analyticity use Fernique s theorem of the function, F z : E [ exp z 1 ] b sds for z small. Proposition 4.1. If {N t } t is a continuous local martingale such that N. Then 4.3 Ee Nt E [ e t] N. Proof. By Itô s formula, we know that Z t : e N t N t / e N t N t is a non-negative local martingale. If {σ n } n1 is a localizing sequence of stopping times for Z, then, by Fatou s lemma, E [Z t B s ] lim inf E n [Zσ n t B s ] lim inf n Zσ n s Z s. This shows that Z is a supermartingale and in particular that E [Z t ] EZ 1. By the auchy-schwarz inequality we find E [ e Nt] ] E [e Nt N t e N t E [ ] [ ] e N t N t E e N t E [Z t ] E [ ] e N t E [ ] 4.31 e N t

29 HEAT KERNELS ON INFINITE-DIMENSIONAL HEISENBERG GROUPS 9 Applying this inequality with N replaced by N and using e x e x +e x easily give Equation 4.3. Proposition Let n N, T >, d dim R, 4.3 γ : sup ω h, e j, c : h H c 1 ω < and for P Proj W let B P t : P B t. Then for all π 4.33 λ < 4dT γ, [ 4.34 sup E exp λ P ProjW and [ 4.35 E exp λ t t ] ω B P τ, db P τ < ] ω B τ, db τ <. Proof. Equation 4.35 follows by choosing {P n } n1 Proj W as in Eq. 4.3 and then using Fatou s lemma in conjunction with the estimate in Equation So we need only to concentrate on proving Equation Fix a P Proj W as in Equation 3.4 and let M P t : t ω B P τ, db P τ. If {f l } d l1 is an orthonormal basis for, then d M t P Mt P, f l l1 and it follows by Hölder s inequality and the martingale estimate in Proposition 4.1 that ] E [e λ M P t E [e λ P d l1 M P t,f l ] d E [e λd M P t,f l ] 1/d 4.36 l1, d ] E [e 1/d λ d M P,f l t l1 d l1 [ ] E e λ d M P 1/d,f l t. We will now evaluate each term in the product in Eq So let c : f l and N t : Mt P, c, and Q P : H H and Q : H H be the unique non-negative symmetric operators such that, for all h H, n ω P h, e j, c QP h, h H for all h H

30 3 DRIVER AND GORDINA and ω h, e j, c Qh, h H for all h H. Also let {q l P } l1 be the eigenvalues listed in decreasing order counted with multiplicities for Q P and observe that Q P h, h QP h, P h Qh, h 4.37 q 1 P sup h h sup H h h sup H h h γ. H With this notation, the quadratic variation of N is given by T n T 4.38 N T ω B P t, e j, c dt Q P B P t, B P t H dt. Moreover, by expanding B P τ in an orthonormal basis of eigenvectors of Q P P H it follows that n T 4.39 N T q l P b l τ dτ l1 where {b l } n l1 is a sequence of independent Brownian motions. Hence it follows that ] ] E [e λ d M P,f l T E [e λ d N T [ n ] T 4.4 E exp λ d q l P b l τ dτ. l1 If Eq holds then using Eq λd q 1 P 4λ d q 1 P λd γ < π/t and we may apply Lemma 4.1 to find [ ] E 4.41 exp λ d q l P T b l τ dτ 1 cos λd q l P T exp 1 ln cos λd q l P T. Moreover, a simple calculus exercise shows for any k, π/ there exists c k < such that 1 ln cos x c k x for x k. Taking k λd γt we may apply this estimate to Eq and combine the result with Eq. 4.4 to find ] E [e λ d M P,f l n T exp c k 4λ d T q l P exp c k 4λ d T tr Q P. l1 Since Q P P QP, we have tr Q P tr Q Qe l, e l H l1 j,l1 ω e l, e j, c ω,, c <. ombining the last two equations recalling that c f l then shows, ] 4.4 E [e λ d M P,f l T exp c k 4λ d T ω,, f l.

31 HEAT KERNELS ON INFINITE-DIMENSIONAL HEISENBERG GROUPS 31 Using this estimate back in Eq gives, ] E [e λ M P t 4.43 exp c k λ dt exp c k λ dt ω d ω,, f l which completes the proof as this last estimate is independent of P Proj W. Proposition Suppose that ν and µ are Gaussian measures on W such q ν f : ν f q µ f : µ f for all f WR. If g : [, [, is a non-negative, non-decreasing, 1 function, then g w dν w g w dµ w. W Proof. Theorem in [8, p. 17] states that if q ν q µ then µ A ν A for every Borel set A which is convex and balanced. In particular, since B t : {w W : w < t} is convex and balanced, it follows that µ B t ν B t or equivalently that 1 ν B t 1 µ B t for all t. Since [ ] g w dν w g + 1 t w g t dt dν w W W g + g t 1 t w dν w dt 4.44 g + W l1 W g t [1 ν B t ] dt with the same formula holding when ν is replaced by µ, it follows that g w dν w g + g t [1 ν B t ] dt W g + dtg t [1 µ B t ] g w dµ w. Definition Let ρ : G [, be defined as ρ w, c : w W + c. In analogy to Gross theory of measurable semi-norms see e.g. Definition 5 in [7] in the abstract Wiener space setting and in light of Theorem 3.1, we view ρ as a measurable extension of d GM. Theorem 4.16 Integrated Gaussian heat kernel bounds. There exists a δ > such that for all ε, δ, T >, p [1,, [ ] 4.45 sup E e ε T ρ g P T < and e ε T ρ g dν T g < P ProjW whenever ε < δ. Proof. Let ε : ε/t. For P Proj W, ρ g P T B P T W + B T + 1 N P T, G W