Heat Kernel Analysis on Infinite-Dimensional Heisenberg Groups
aa r X i v : . [ m a t h . P R ] M a y HEAT KERNEL ANALYSIS ON INFINITE-DIMENSIONALHEISENBERG GROUPS
BRUCE K. DRIVER † AND MARIA GORDINA ∗ Abstract.
We introduce a class of non-commutative Heisenberg like infinitedimensional Lie groups based on an abstract Wiener space. The Ricci curva-ture tensor for these groups is computed and shown to be bounded. Brownianmotion and the corresponding heat kernel measures, { ν t } t> , are also stud-ied. We show that these heat kernel measures admit: 1) Gaussian like upperbounds, 2) Cameron-Martin type quasi-invariance results, 3) good L p – boundson the corresponding Radon-Nykodim derivatives, 4) integration by parts for-mulas, and 5) logarithmic Sobolev inequalities. The last three results heavilyrely on the boundedness of the Ricci tensor. Contents
1. Introduction 21.1. A finite-dimensional paradigm 21.2. Summary of results 32. Abstract Wiener Space Preliminaries 53. Infinite-Dimensional Heisenberg Type Groups 73.1. Length and distance estimates 103.2. Norm estimates 123.3. Examples 143.4. Finite Dimensional Projections and Cylinder Functions 184. Brownian Motion and Heat Kernel Measures 204.1. A quadratic integral 214.2. Brownian Motion on G ( ω ) 234.3. Finite Dimensional Approximations 255. Path space quasi-invariance 326. Heat Kernel Quasi-Invariance 387. The Ricci Curvature on Heisenberg type groups 407.1. Examples revisited 438. Heat Inequalities 458.1. Infinite-dimensional Radon-Nikodym derivative estimates 458.2. Logarithmic Sobolev Inequality 469. Future directions 47 Date : October 30, 2018
File:Driver˙Gordina˙JFA˙05˙2008.tex .1991
Mathematics Subject Classification.
Primary; 35K05,43A15 Secondary; 58G32.
Key words and phrases.
Heisenberg group, heat kernel, quasi-invariance, logarithmic Sobolevinequality. † This research was supported in part by NSF Grant DMS-0504608 and the Miller Institute atthe University of California, at Berkeley. ∗ Research was supported in part by NSF Grant DMS-0706784.
Appendix A. Wiener Space Results 48Appendix B. The Ricci tensor on a Lie group 50Appendix C. Proof of Theorem 3.12 51C.1. Proof of Theorem 3.12. 53References 541.
Introduction
Both authors have been greatly influenced by Professor Malliavin and his workover the years. In particular this paper is partially an attempt to better understandMalliavin’s paper, [40]. It is with great pleasure to us that this article appears(assuming it is accepted) in this special edition of JFA dedicated to Professor PaulMalliavin.The aim of this paper is to construct and study properties of heat kernel measureson certain infinite-dimensional Heisenberg groups. In this paper the Heisenberggroups will be constructed from a skew symmetric form on an abstract Wienerspace. A typical example of such a group is the Heisenberg group of a symplecticvector space. Before describing our results let us recall some typical heat kernelresults for finite-dimensional Riemannian manifolds.1.1.
A finite-dimensional paradigm.
Let (
M, g ) be a complete connected n – dimensional Riemannian manifold ( n < ∞ ) , ∆ = ∆ g be the Laplace Beltramioperator acting on C ( M ) , and Ric denote the associated Ricci tensor. Recall (seefor example Strichartz [48], Dodziuk [15] and Davies [12]) that the closure, ¯∆ , of∆ | C ∞ c ( M ) is self-adjoint on L ( M, dV ) , where dV = √ gdx . . . dx n is the Riemannvolume measure on M. Moreover, the semi-group P t := e t ¯∆ / has a symmetricpositive integral (heat) kernel, p t ( x, y ) , such that R M p t ( x, y ) dV ( y ) x ∈ M and(1.1) P t f ( x ) := (cid:16) e t ¯∆ / f (cid:17) ( x ) = Z M p t ( x, y ) f ( y ) dV ( y ) for all f ∈ L ( M ) . Theorem 1.2 summarizes some of the results that we would like to extend to ourinfinite-dimensional Heisenberg group setting.
Notation 1.1. If µ is a probability measure on a measure space (Ω , F ) and f ∈ L ( µ ) = L (Ω , F , µ ) , we will often write µ ( f ) for the integral, R Ω f dµ. Theorem 1.2.
Beyond the assumptions above, let us further assume that
Ric > kI for some k ∈ R . Then (1) p t ( x, y ) is a smooth function. (The Ricci curvature assumption is notneeded here.) (2) R M p t ( x, y ) dV ( y ) = 1 , (see for example Davies [12, Theorem 5.2.6 ] ). (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 measuressuch that the function t → ν t ( f ) := R M f dν t is continuously differentiable, (1.2) ddt ν t ( f ) = 12 ν t (∆ f ) , and lim t ↓ ν t ( f ) = f ( o ) for all f ∈ BC ( M ) , the bounded C –functions on M. EAT KERNELS ON INFINITE-DIMENSIONAL HEISNEBERG GROUPS 3 (4)
There exist constants, c = c ( K, n, T ) and C = C ( K, n, T ) , such that, (1.3) p ( t, x, y ) CV (cid:16) x, p t/ (cid:17) exp (cid:18) − c d ( x, y ) t (cid:19) , for all x, y ∈ M and t ∈ (0 , T ] , where d ( x, y ) is the Riemannian distancefrom 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 loga-rithmic Sobolev inequality; (1.4) ν T ( f log f ) k − (cid:0) − e − kT (cid:1) v T (cid:0) |∇ f | (cid:1) + ν T ( f ) log ν T (cid:0) f (cid:1) , for f ∈ C ∞ c ( M ) . These results are fairly standard. For item 3. see [17, Theorem 2.6 ], for Eq.(1.3) see for example Theorems 5.6.4, 5.6.6, and 5.4.12 in Sallof-Coste [47] and formore detailed bounds see [39, 12, 46, 13, 26]). The logarithmic Sobolev inequalityin Eq. (1.4) generalizes Gross’ [28] original Logarithmic Sobolev inequality validfor M = R n and is due in this generality to D. Bakry and M. Ledoux, see [2, 3, 38].Also see [30, 10, 52, 51, 21] and Driver and Lohrenz [22, Theorem 2.9] for the caseof interest here, namely when M is a uni-modular Lie group with a left invariantRiemannian metric.When passing to infinite-dimensional Riemannian manifolds we will no longerhave available the Riemannian volume measure. Because of this problem, we willtake item 3. of Theorem 1.2 as our definition of the heat kernel measure. The heatkernel upper bound in Eq. (1.3) also does not make sense in infinite dimensions.However, the following consequence almost does: there exists c ( T ) > Z M exp (cid:18) c d ( o, x ) t (cid:19) dν t ( x ) < ∞ for all 0 < t ≤ T. In fact Eq. (1.5) will not hold in infinite dimensions either. It will be necessaryto replace the distance function, d, by a weaker distance function as happens inFernique’s theorem for Gaussian measure spaces. With these results as backgroundwe are now ready to summarize the results of this paper.1.2. Summary of results.
Let us describe the setting informally, for precise def-initions see Sections 2 and 3. Let (
W, H, µ ) be an abstract Wiener space, C be afinite-dimensional inner product space, and ω : W × W → C be a continuous skewsymmetric bilinear quadratic form on W. The set g = W × C can be equipped witha Lie bracket by setting [( A, a ) , ( B, b )] = (0 , ω ( A, B )) . As in the case for the Heisenberg group of a symplectic vector space, the Lie algebra g = W × C can be given the group structure by defining( w , c ) · ( w , c ) = (cid:18) w + w , c + c + 12 ω ( ω , w ) (cid:19) . The set W × C with the group structure will be denoted by G or G ( ω ). The Lie sub-algebra g CM = H × C is called the Cameron-Martin subalgebra, and g CM equipped DRIVER AND GORDINA with the same group multiplication denoted by G CM and called the Cameron-Martin subgroup. We equip G CM with the left invariant Riemannian metric whichagrees with the natural Hilbert inner product, h ( A, a ) , ( B, b ) i g CM := h A, B i H + h a, b i C , on g CM ∼ = T e G CM . In Section 3 we give several examples of this abstract settingincluding the standard finite-dimensional Heisenberg group.The main objects of our study are a Brownian motion in G and the correspondingheat kernel measure defined in Section 4. Namely, let { ( B ( t ) , B ( t )) } t > be aBrownian motion on g with variance determined by E h h ( B ( s ) , B ( s )) , ( A, a ) i g CM · h ( B ( t ) , B ( t )) , ( C, c ) i g CM i = Re h ( A, a ) , ( C, c ) i g CM min ( s, t )for all s, t ∈ [0 , ∞ ) , A, C ∈ H ∗ and a, c ∈ C . Then the Brownian motion on G isthe continuous G –valued process defined by g ( t ) = (cid:18) B ( t ) , B ( t ) + 12 Z t ω ( B ( τ ) , dB ( τ )) (cid:19) . For
T > G is ν T = Law ( g ( T )) . It is shown inCorollary 4.5 that { ν t } t> satisfies item 3. of Theorem 1.2 with o = (0 , ∈ G ( ω ) . Theorem 4.16 gives heat kernel measure bounds that may be viewed as a non-commutative version of Fernique’s theorem for G ( ω ). In light of Theorem 3.12 thisresult is also analogous to the integrated integrated Gaussian upper bound in Eq.(1.5).In Theorem 5.2 we prove quasi-invariance for the path space measure associatedto the Brownian motion, g, on G with respect to multiplication on the left by finiteenergy paths in the Cameron-Martin subgroup G CM . (In light of the results inMalliavin [40] it is surprising that Theorem 5.2 holds.) Theorem 5.2 is then usedto prove quasi-invariance of the heat kernel measures with respect to both rightand left multiplication (Theorem 6.1 and Corollary 6.2), as well as integration byparts formulae on the path space and for the heat kernel measures, see Corollaries5.6 – 6.5. 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 naturalto interpret quasi-invariance and integration by parts formulae as properties assmoothness 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 boundedfrom below (see Proposition 7.2), but also that the Ricci curvature of certain finite-dimensional “approximations” are bounded from below with constants independentof the approximation. Based on results in [18], these bounds allow us to giveanother 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 holomorphicfunctions which is the subject of our paper [19]. In Theorem 8.3 we show thatanalogue of the logarithmic Sobolev inequality in Eq. (1.4) holds in our setting aswell.In Section 9 we give a list of open questions and further possible developmentsof the results of this paper. We expect our methods to be applicable to a muchlarger class of infinite-dimensional nilpotent groups. EAT KERNELS ON INFINITE-DIMENSIONAL HEISNEBERG GROUPS 5
Finally, we refer to papers of H. Airault, P. Malliavin, D. Bell, Y. Inahama con-cerning quasi-invariance, integration by parts formulae and the logarithmic Sobolevinequality on certain infinite-dimensional curved spaces ([1, 4, 6, 5, 32]).2.
Abstract Wiener Space Preliminaries
Suppose that X is a real separable Banach space and B X is the Borel σ –algebraon X . Definition 2.1.
A measure µ on ( X, B X ) is called a (mean zero, non-degenerate) Gaussian measure provided that its characteristic functional is given by(2.1) ˆ µ ( u ) := Z X e iu ( x ) dµ ( x ) = e − q ( u,u ) for all u ∈ X ∗ , where q = q µ : X ∗ × X ∗ → R is a quadratic form such that q ( u, v ) = q ( v, u ) and q ( u ) = q ( u, u ) > u = 0 , i.e. q is a real inner product on X ∗ . In what follows we frequently make use of the fact that(2.2) C p := Z X k x k pX dµ ( x ) < ∞ for all 1 p < ∞ . This is a consequence of Skorohod’s inequality (see for example [36, Theorem 3.2])(2.3) Z X e λ k x k X dµ ( x ) < ∞ for all λ < ∞ ;or the even stronger Fernique’s inequality (see for example [8, Theorem 2.8.5] or[36, Theorem 3.1])(2.4) Z X e δ k x k X dµ ( x ) < ∞ for some δ > . Lemma 2.2. If u, v ∈ X ∗ , then (2.5) Z X u ( x ) v ( x ) dµ ( x ) = q ( u, v ) and (2.6) | q ( u, v ) | C k u k X ∗ k v k X ∗ . Proof.
Let u ∗ µ := µ ◦ u − denote the law of u under µ. Then by Equation (2.1),( u ∗ µ ) ( dx ) = 1 p πq ( u, u ) e − q ( u,u ) x dx and hence,(2.7) Z X u ( x ) dµ ( x ) = q µ ( u, u ) = q ( u, u ) . Polarizing this identity gives Equation (2.5) which along with Equation (2.2) impliesEquation (2.6). (cid:3)
The next theorem summarizes some well known properties of Gaussian measuresthat we will use freely below.
DRIVER AND GORDINA
Theorem 2.3.
Let µ be a Gaussian measure on a real separable Banach space, X. For x ∈ X let (2.8) k x k H := sup u ∈ X ∗ \{ } | u ( x ) | p q ( u, u ) and define the Cameron-Martin subspace, H ⊂ X, by (2.9) H = { h ∈ X : k h k H < ∞} . Then (1) H is a dense subspace of X ; (2) there exists a unique inner product, h· , ·i H on H such that k h k H = h h, h i for all h ∈ H. Moreover, with this inner product H is a separable Hilbertspace. (3) For any h ∈ H (2.10) k h k X p C k h k H , where C is as in (2.2) . (4) If { e j } ∞ j =1 is an orthonormal basis for H, then for any u, v ∈ H ∗ (2.11) q ( u, v ) = h u, v i H ∗ = ∞ X j =1 u ( e j ) v ( e j ) . The proof of this standard theorem is relegated to Appendix A – see TheoremA.1.
Remark 2.4.
It follows from Equation (2.10) that any u ∈ X ∗ restricted to H isin H ∗ . Therefore we have(2.12) Z X u dµ = q ( u, u ) = k u k H ∗ = ∞ X j =1 | u ( e j ) | , where { e j } ∞ j =1 is an orthonormal bases for H. More generally, if ϕ is a linear boundedmap from W to C , where C is a real Hilbert space, then(2.13) k ϕ k H ∗ ⊗ C =: ∞ X j =1 k ϕ ( e j ) k C = Z X k ϕ ( x ) k C dµ ( x ) < ∞ . To prove Equation (2.13), let { f j } ∞ j =1 be an orthonormal basis for C . Then Z X k ϕ ( x ) k C dµ ( x ) = Z X ∞ X j =1 (cid:12)(cid:12) h ϕ ( x ) , f j i C (cid:12)(cid:12) dµ ( x ) = ∞ X j =1 Z X (cid:12)(cid:12) h ϕ ( x ) , f j i C (cid:12)(cid:12) dµ ( x )= ∞ X j =1 (cid:13)(cid:13) h ϕ ( · ) , f j i C (cid:13)(cid:13) H ∗ = ∞ X j =1 ∞ X k =1 (cid:12)(cid:12) h ϕ ( e k ) , f j i C (cid:12)(cid:12) = ∞ X k =1 ∞ X j =1 (cid:12)(cid:12) h ϕ ( e k ) , f j i C (cid:12)(cid:12) = ∞ X k =1 k ϕ ( e k ) k C = k ϕ k H ∗ ⊗ C . A simple consequence of Eq. (2.14) is that(2.14) k ϕ k H ∗ ⊗ C k ϕ k W ∗ ⊗ C Z X k x k W dµ ( x ) = C k ϕ k W ∗ ⊗ C . EAT KERNELS ON INFINITE-DIMENSIONAL HEISNEBERG GROUPS 7 Infinite-Dimensional Heisenberg Type Groups
Throughout the rest of this paper (
X, H, µ ) will denote a real abstract Wienerspace , i.e. X is a real separable Banach space, H is a real separable Hilbert spacedensely embedded into X , and µ is a Gaussian measure on ( X, B X ) such thatEquation (2.1) holds with q ( u, u ) := h u | H , u | H i H ∗ . Following the discussion in [35] and [23] we will say that a (possibly infinite–dimensional) Lie algebra, g , is of Heisenberg type if C := [ g , g ] is contained inthe center of g . If g is of Heisenberg type and W is a complementary subspace to C in g , we may define a bilinear map, ω : W × W → C , by ω ( w, w ′ ) = [ w, w ′ ] forall w, w ′ ∈ C . Then for ξ i := w i + c i ∈ W ⊕ C = g , i = 1 , ξ , ξ ] = [ w + c , w + c ] = 0 + ω ( w , w ) . If we now suppose G is a finite-dimensional Lie group with Lie algebra g , then bythe Baker-Campbell-Dynkin-Hausdorff formula e ξ e ξ = e ξ + ξ + [ ξ ,ξ ] = e w + w + c + c + ω ( w ,w ) . In particular, we may introduce a group structure on g by defining( w + c ) · ( w + c ) = w + w + c + c + 12 ω ( ω , w ) . With this as motivation, we are now going to introduce a class of Heisenberg typeLie groups based on the following data.
Notation 3.1.
Let ( W, H, µ ) be an abstract Wiener space, C be a finite-dimensionalinner product space, and ω : W × W → C be a continuous skew symmetric bilinearquadratic form on W. Further let (3.1) k ω k := sup {k ω ( w , w ) k C : w , w ∈ W with k w k W = k w k W = 1 } . be the uniform norm on ω which is finite by the assumed continuity of ω. We now define g := W × C which is a Banach space in the norm(3.2) k ( w, c ) k g := k w k W + k c k C . We further define g CM := H × C which is a Hilbert space relative to the productinner product(3.3) h ( A, a ) , ( B, b ) i g CM := h A, B i H + h a, b i C . The associated Hilbertian norm on g CM is given by(3.4) k ( A, a ) k g CM := q k A k H + k a k C . It is easily checked that defining(3.5) [( w , c ) , ( w , c )] := (0 , ω ( w , w ))for all ( w , c ) , ( w , c ) ∈ g makes g into a Lie algebra such that g CM is Lie sub-algebra of g . Note that this definition implies that C = [ g , g ] is contained in thecenter of g . It is also easy to verify that we may make g into a group using themultiplication rule(3.6) ( w , c ) · ( w , c ) = (cid:18) w + w , c + c + 12 ω ( w , w ) (cid:19) DRIVER AND GORDINA
The latter equation may be more simply expressed as(3.7) g g = g + g + 12 [ g , g ] , where g i = ( w i , c i ), i = 1 ,
2. As sets G and g are the same.The identity in G is e = (0 ,
0) and the inverse is given by g − = − g for all g = ( w, c ) ∈ G. Let us observe that { } × C is in the center of both G and g andfor h in the center of G, g · h = g + h . In particular, since [ g, h ] ∈ { } × C it followsthat k · [ g, h ] = k + [ g, h ] for all k, g, h ∈ G. Definition 3.2.
When we want to emphasize the group structure on g we denote g by G or G ( ω ). Similarly, when we view g CM as a subgroup of G it will be denotedby G CM and will be called the Cameron–Martin subgroup.
Lemma 3.3.
The Banach space topologies on g and g CM make G and G CM intotopological groups.Proof. Since g − = − g, the map g g − is continuous in the g and g CM topologies.Since ( g , g ) g + g and ( g , g ) [ g , g ] are continuous in both the g and g CM topologies, it follows from Equation (3.7) that ( g , g ) g · g is continuousas well. (cid:3) For later purposes it is useful to observe, by Equations (3.5) and (3.7), that(3.8) k g g k g k g k g + k g k g + 12 k ω k k g k g k g k g for any g , g ∈ G. Notation 3.4.
To each g ∈ G, let l g : G → G and r g : G → G denote left andright multiplication by g respectively. Notation 3.5 (Linear differentials) . Suppose f : G → C is a Frech´et smoothfunction. For g ∈ G and h, k ∈ g let f ′ ( g ) h := ∂ h f ( g ) = ddt (cid:12)(cid:12)(cid:12) f ( g + th ) 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 ordifferential. As G is a vector space, to each g ∈ G we can associate the tangent space (as inthe 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 Frech´et smooth functions, f : G → C . We will write g and g CM for T e G and T e G CM respectively. Of course as setswe may view g and g CM as G and G CM respectively. For h ∈ g , let ˜ h be the leftinvariant 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 (0) = e and ˙ σ (0) = h (e.g. σ ( t ) = th ) , then(3.9) ˜ h ( g ) = l g ∗ h := ddt (cid:12)(cid:12)(cid:12) g · σ ( t ) . EAT KERNELS ON INFINITE-DIMENSIONAL HEISNEBERG GROUPS 9
As usual we view ˜ h as a first order differential operator acting on smooth functions, f : G → C , by(3.10) (cid:16) ˜ hf (cid:17) ( g ) = ddt (cid:12)(cid:12)(cid:12) f ( g · σ ( t )) . Proposition 3.7.
Let f : G → C be a smooth function, h = ( A, a ) ∈ g and g = ( w, c ) ∈ G. Then (3.11) e h ( g ) := l g ∗ h = (cid:18) A, a + 12 ω ( w, A ) (cid:19) g for all g = ( w, c ) ∈ G and in particular using Notation 3.6 (3.12) ^ ( A, a ) f ( g ) = f ′ ( g ) (cid:18) A, a + 12 ω ( w, A ) (cid:19) . Furthermore, if h = ( A, a ) , k = ( B, b ) , and then (3.13) (cid:16) ˜ h ˜ kf − ˜ k ˜ hf (cid:17) = ] [ h, k ] f. In other words, the Lie algebra structure on g induced by the Lie algebra structureon the left invariant vector fields on G is the same as the Lie algebra structuredefined in Eq. (3.5) .Proof. Since th = t ( A, a ) is a curve in G passing through the identity at t = 0 , wehave e h ( g ) = ddt (cid:12)(cid:12)(cid:12) [ g · ( th )] = ddt (cid:12)(cid:12)(cid:12) [( w, c ) · t ( A, a )]= ddt (cid:12)(cid:12)(cid:12) (cid:20)(cid:18) w + tA, c + tδ + t ω ( w, A ) (cid:19)(cid:21) = (cid:18) A, a + 12 ω ( w, A ) (cid:19) . So by the chain rule, (cid:16) ˜ hf (cid:17) ( g ) = f ′ ( g ) ˜ h ( g ) and hence (cid:16) ˜ h ˜ kf (cid:17) ( g ) = ddt (cid:12)(cid:12)(cid:12) h f ′ ( g · th ) ˜ k ( g · th ) i = f ′′ ( g ) (cid:16) ˜ h ( g ) ⊗ ˜ k ( g ) (cid:17) + f ′ ( g ) ddt (cid:12)(cid:12)(cid:12) ˜ k ( g · th ) , (3.14)where ddt (cid:12)(cid:12)(cid:12) ˜ k ( g · th ) = ddt (cid:12)(cid:12)(cid:12) (cid:18) B, a + 12 ω ( w + tA, B ) (cid:19) = (cid:18) , ω ( A, B ) (cid:19) . Since f ′′ ( g ) is symmetric, it now follows by subtracting Equation (3.14) with h and k interchanged from itself that (cid:16) ˜ h ˜ kf − ˜ k ˜ hf (cid:17) ( g ) = f ′ ( g ) (0 , ω ( A, B )) = f ′ ( g ) [ h, k ] = (cid:16) ] [ h, k ] f (cid:17) ( g )as desired. (cid:3) Lemma 3.8.
The one parameter group in
G, e th , determined by h = ( A, a ) ∈ g , isgiven by (3.15) e th = th = t ( A, δ ) . Proof.
Letting ( w ( t ) , c ( t )) := e th , according to Equation (3.11) we have that ddt ( w ( t ) , c ( t )) = (cid:18) A, a + 12 ω ( w ( t ) , A ) (cid:19) with w (0) = 0 and c (0) = 0 . The solution to this differential equation is easily seen to be given by Equation(3.15). (cid:3)
Length and distance estimates.Notation 3.9.
Let
T > and C CM denote the collection of C -paths, g : [0 , T ] → G CM . The length of g is defined as (3.16) ℓ G CM ( g ) = Z T (cid:13)(cid:13) l g − ( s ) ∗ g ′ ( s ) (cid:13)(cid:13) g CM ds. As usual, the Riemannian distance between x, y ∈ G CM is defined as d G CM ( x, y ) = inf (cid:8) ℓ G CM ( g ) : g ∈ C CM such that g (0) = x and g ( T ) = y (cid:9) . It will also be convenient to define | y | := d G CM ( e , y ) for all y ∈ G CM . (Thevalue of T > used in defining d C CM is irrelevant since the length functionalis reparametrization invariant.) Let(3.17) C := sup {k ω ( h, k ) k C : k h k H = k k k H = 1 } C k ω k < ∞ . The inequality in Eq. (3.17) is a consequence of Eq. (2.10) and the definition of k ω k in Eq. (3.1). Proposition 3.10.
Let ε := 1 /C where C is as in Eq. (3.17). Then for all x, y ∈ G CM , (3.18) d G CM ( x, y ) (cid:18) C k x k g CM ∧ k y k g CM (cid:19) k y − x k g CM and in particular, | x | = d G CM ( e , x ) k x k g CM . Moreover, there exists
K < ∞ suchthat if x, y ∈ G CM with d G CM ( x, y ) < ε/ / C, then (3.19) k y − x k g CM K (cid:16) k x k g CM ∧ k y k g CM (cid:17) d G CM ( x, y ) . As a consequence of Eqs. (3.18) and (3.19) we see that the topology on G CM induced by d G CM is the same as the Hilbert topology induced by k·k g CM . Remark 3.11.
The equivalence of these two topologies in an infinite-dimensionalsetting has been addressed in [24] 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 C CM for 0 s , then by Equation (3.11) l g − ( s ) ∗ g ′ ( s ) = (cid:18) w ′ ( s ) , a ′ ( s ) − ω ( w ( s ) , w ′ ( s )) (cid:19) (3.20) = g ′ ( s ) −
12 [ g ( s ) , g ′ ( s )]and we may write Equation (3.16) more explicitly as(3.21) ℓ G CM ( g ) = Z (cid:13)(cid:13)(cid:13)(cid:13) g ′ ( s ) −
12 [ g ( s ) , g ′ ( s )] (cid:13)(cid:13)(cid:13)(cid:13) g CM ds. EAT KERNELS ON INFINITE-DIMENSIONAL HEISNEBERG GROUPS 11
If we now apply Equation (3.21) to g ( s ) = x + s ( y − x ) for 0 s , we see that d G CM ( x, y ) ℓ G CM ( g ) = Z (cid:13)(cid:13)(cid:13)(cid:13) ( y − x ) −
12 [ x + s ( y − x ) , ( y − x )] (cid:13)(cid:13)(cid:13)(cid:13) g CM ds = (cid:13)(cid:13)(cid:13)(cid:13) ( y − x ) −
12 [ x, ( y − x )] (cid:13)(cid:13)(cid:13)(cid:13) g CM (cid:18) C k x k g CM (cid:19) k y − x k g CM . As we may interchange the roles of x and y in this inequality, the proof of Equation(3.18) is complete.Let B ε := n x ∈ g CM : k x k g CM ε o ,y ∈ B ε , and g : [0 , → G CM be a C –path such that g (0) = (0 ,
0) = e and g (1) = y. Further let T ∈ [0 ,
1] be the first time that g exits B ε with the conventionthat T = 1 if g ([0 , ⊂ B ε . Then from Equation (3.21) ℓ G CM ( g ) > ℓ G CM (cid:0) g | [0 ,T ] (cid:1) > Z T (cid:20) k g ′ ( s ) k g CM − k [ g ( s ) , g ′ ( s )] k g CM (cid:21) ds > (cid:18) − C ε (cid:19) · Z T k g ′ ( s ) k g CM ds > (cid:18) − C ε (cid:19) · k g ( T ) k g CM > k g ( T ) k g CM > k y k g CM . (3.22)Optimizing Equation (3.22) over g implies | y | = d G CM ( e , y ) > k y k g CM for all y ∈ B ε . 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 G CM ( g ) > k g ( T ) k g CM / ε/ d G CM ( e , y ) > ε/ . Hence we have shown that | y | = d G CM ( e , y ) >
12 min (cid:16) ε, k y k g CM (cid:17) for all y ∈ G CM . Now suppose that x, y ∈ G CM and (without loss of generality) that k x k g CM k y k g CM . Using the left invariance of d G CM , it follows that(3.23) d G CM ( x, y ) = d G CM (cid:0) e , x − y (cid:1) >
12 min (cid:16) ε, (cid:13)(cid:13) x − y (cid:13)(cid:13) g CM (cid:17) . If we further suppose that d G CM ( x, y ) < ε , we may conclude from Equation (3.23)that (cid:13)(cid:13)(cid:13)(cid:13) y − x −
12 [ x, y ] (cid:13)(cid:13)(cid:13)(cid:13) g CM = (cid:13)(cid:13) x − y (cid:13)(cid:13) g CM d G CM ( x, y ) . If we write x = ( A, a ) and y = ( B, b ) , it follows that k B − A k H + (cid:13)(cid:13)(cid:13)(cid:13) b − a − ω ( A, B ) (cid:13)(cid:13)(cid:13)(cid:13) C d G CM ( x, y )and therefore k B − A k H d G CM ( x, y ) and k b − a k C (cid:13)(cid:13)(cid:13)(cid:13) b − a − ω ( A, B ) (cid:13)(cid:13)(cid:13)(cid:13) C + (cid:13)(cid:13)(cid:13)(cid:13) ω ( A, B ) (cid:13)(cid:13)(cid:13)(cid:13) C d G CM ( x, y ) + 12 k ω ( A, B − A ) k C d G CM ( x, y ) + C k A k H k B − A k H d G CM ( x, y ) (cid:18) C k A k H (cid:19) d G CM ( x, y ) (cid:18) C k x k g CM (cid:19) . Combining these results shows that if d G CM ( x, y ) < ε then k y − x k g CM d G CM ( x, y ) (cid:18) C k x k g CM (cid:19) ! from which Equation (3.19) easily follows. (cid:3) We are most interested in the case where { ω ( A, B ) :
A, B ∈ H } is a total subsetof C , i.e. span { ω ( A, B ) :
A, B ∈ H } = C . In this case it turns out that straightline paths are bad approximations to the geodesics joining e ∈ G CM to points x ∈ G CM far away from e . For points x ∈ G CM distant from e it is better to use“horizontal” paths instead which leads to the following distance estimates. Theorem 3.12.
Suppose that { ω ( A, B ) :
A, B ∈ H } is a total subset of C . Thenthere exists C ( ω ) < ∞ such that (3.24) d CM ( e , ( A, a )) C ( ω ) (cid:18) k A k H + q k a k C (cid:19) for all ( A, a ) ∈ g CM . Moreover, for any ε > there exists γ ( ε ) > such that and (3.25) γ ( ε ) (cid:18) k A k H + q k a k C (cid:19) d CM ( e , ( A, a )) if d CM ( e , ( A, a )) ≥ ε . Thus away from any neighborhood of the identity, d CM ( e , ( A, a )) is comparable to k A k H + p k a k C . Since this theorem is not central to the rest of the paper we will relegate its proofto Appendix C. The main point of Theorem 3.12 is to explain why Theorem 4.16is an infinite dimensional analogue of the integrated Gaussian heat kernel bound inEq. (1.5).3.2.
Norm estimates.Notation 3.13.
Suppose H and C are real (complex) Hilbert spaces, L : H → C is a bounded operator, ω : H × H → C is a continuous (complex) bilinear form,and { e j } ∞ j =1 is an orthonormal basis for H. The Hilbert–Schmidt norms of L and ω are defined by (3.26) k L k H ∗ ⊗ C := ∞ X j =1 k Le j k C , and (3.27) k ω k = k ω k H ∗ ⊗ H ∗ ⊗ C := ∞ X i,j =1 k ω ( e i , e j ) k C . It is easy to verify directly that these definitions are basis independent. Also seeEquation (3.29) below.
EAT KERNELS ON INFINITE-DIMENSIONAL HEISNEBERG GROUPS 13
Proposition 3.14.
Suppose that ( W, H, µ ) is a real abstract Wiener space, ω : W × W → C is as in Notation 3.1, and { e j } ∞ j =1 is an orthonormal basis for H. Then (3.28) k ω ( w, · ) k H ∗ ⊗ C C k ω k k w k W for all w ∈ W and (3.29) k ω k = Z W × W k ω ( w, w ′ ) k C dµ ( w ) dµ ( w ′ ) k ω k C < ∞ , where C is as in Equation (2.2) .Proof. From Equation (2.13), k ω ( w, · ) k H ∗ ⊗ C = Z W k ω ( w, w ′ ) k C dµ ( w ′ ) k ω k k w k W Z W k w ′ k W dµ ( w ′ ) = C k ω k k w k W . Similarly, viewing w → ω ( w, · ) as a continuous linear map from W to H ∗ ⊗ C itfollows from Eqs. (2.13) and (2.14), that k ω k = k h ω ( h, · ) k H ∗ ⊗ ( H ∗ ⊗ C ) = Z W k ω ( w, · ) k H ∗ ⊗ C dµ ( w ) Z W C k ω k k w k W dµ ( w ) = C k ω k . (cid:3) Remark 3.15.
The Lie bracket on g CM has the following continuity property, k [( A, a ) , ( B, b )] k g CM C k ( A, a ) k g CM k ( B, b ) k g CM where C ≤ k ω k as in Eq. (3.17). This is a consequence of the following simpleestimates k [( A, a ) , ( B, b )] k g CM = k (0 , ω ( A, B )) k g CM = k ω ( A, B ) k C C k A k H k B k H C k ( A, a ) k g CM k ( B, b ) k g CM . This continuity property of the Lie bracket is often used to prove that the exponen-tial map is a local diffeomorphism (e.g. see [24] in the case of infinite-dimensionalmatrix groups). In the Heisenberg group setting the exponential map is a globaldiffeomorphism as follows from Lemma 3.8, where we have not used continuity ofthe Lie bracket.
Lemma 3.16.
Suppose that H is a real Hilbert space, C is a real finite-dimensionalinner product space, and ℓ : H → C is a continuous linear map. Then for anyorthonormal basis { e j } ∞ j =1 of H the series (3.30) ∞ X j =1 ℓ ( e j ) ⊗ ℓ ( e j ) ∈ C ⊗ C and (3.31) ∞ X j =1 ℓ ( e j ) ⊗ e j ∈ C ⊗ H are convergent and independent of the basis. Proof. If { f i } dim C i =1 is an orthonormal basis for C , then ∞ X j =1 k ℓ ( e j ) ⊗ ℓ ( e j ) k C ⊗ C = ∞ X j =1 k ℓ ( e j ) k C = dim C X i =1 ∞ X j =1 ( f i , ℓ ( e j )) = dim C X i =1 k ( f i , ℓ ( · )) k H ∗ < ∞ which shows that the sum in Equation (3.30) is absolutely convergent and that ℓ is Hilbert–Schmidt. Similarly, since { ℓ ( e j ) ⊗ e j } ∞ j =1 is an orthogonal set in C ⊗ H and ∞ X j =1 k ℓ ( e j ) ⊗ e j k C ⊗ H = ∞ X j =1 k ℓ ( e j ) k C < ∞ , 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 tensorproduct, 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 correspondsto 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 ∈ C ; ∞ X j =1 ℓ ( e j ) ⊗ ℓ ( e j ) c = ∞ X j =1 ℓ ( e j ) h ℓ ( e j ) , c i C = ∞ X j =1 ℓ ( e j ) h e j , ℓ ∗ c i C = ℓ ∞ X j =1 h e j , ℓ ∗ c i C e j = ℓℓ ∗ c and ∞ X j =1 ℓ ( e j ) ⊗ e j c = ∞ X j =1 e j h ℓ ( e j ) , c i C = ∞ X j =1 e j h e j , ℓ ∗ c i C = ℓ ∗ c, which clearly shows that Equations (3.30) and(3.31) are basis–independent. (cid:3) Examples.
Here we describe several examples including finite-dimensionalHeisenberg groups. As we mentioned earlier a typical example of such a groupis the Heisenberg group of a symplectic vector space. For each of the examplespresented we will explicitly compute the norm k ω k of the form ω as defined inEquation (3.27). In Section 7 we will also explicitly compute the Ricci tensor foreach of the examples introduced here.To describe some of the examples below, it is convenient to use complex Banachand Hilbert spaces. However, for the purposes of this paper the complex structureon 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· , ·i H ) is a complex Hilbert space, we define h· , ·i H Re = Re h· , ·i H in whichcase (cid:0) H Re , h· , ·i H Re (cid:1) becomes a real Hilbert space. Before going to the examples, If we were to allow C to be an infinite-dimensional Hilbert space here, we would have toassume that ℓ is Hilbert–Schmidt. When dim C < ∞ , ℓ : H → C is Hilbert–Schmidt iff it isbounded. EAT KERNELS ON INFINITE-DIMENSIONAL HEISNEBERG GROUPS 15 let us record the relationship between the complex and real Hilbert Schmidt normsof Notation 3.13.
Lemma 3.17.
Suppose H and C are complex Hilbert spaces, L : H → C and ω : H × H → C are as in Notation 3.13, and c ∈ C . Then k L k H ∗ Re ⊗ C Re = 2 k L k H ∗ ⊗ C , (3.32) (cid:13)(cid:13) h ω ( · , · ) , c i C Re (cid:13)(cid:13) H ∗ Re ⊗ H ∗ Re = 2 kh ω ( · , · ) , c i C k H ∗ ⊗ H ∗ , (3.33) and (3.34) k ω ( · , · ) k H ∗ Re ⊗ H ∗ Re ⊗ C Re = 4 k ω ( · , · ) k H ∗ ⊗ H ∗ ⊗ C . Proof.
A straightforward proof. (cid:3)
Example 3.18 (Finite-dimensional real Heisenberg group) . Let C = R , W = H =( C n ) Re ∼ = R n , and ω ( w, z ) := Im h w, z i be the standard symplectic form on R n , where h w, z i = w · ¯ z is the usual inner product on C n . Any element of the group H n R := G ( ω ) can be written as g = ( z, c ) , where z ∈ C n and c ∈ R . As above, the Liealgebra, h n R , of H n R is, as a set, equal to H n R itself. If { e j } nj =1 is an orthonormal basisfor R n then { e j , ie j } nj =1 is an orthonormal basis for H and (real) Hilbert Schmidtnorm of ω is given by (3.35) k ω k H ∗ ⊗ H ∗ = n X j,k =1 X ε,δ ∈{ ,i } [Im h εe j , δe k i ] = n X j,k =1 δ j,k = 2 n. Example 3.19 (Finite-dimensional complex Heisenberg group) . Suppose that W = H = C n × C n , C = C , and ω : W × W → C is defined by ω (( w , w ) , ( z , z )) = w · z − w · z . Any element of the group H n C := G ( ω ) can be written as g = ( z , z , c ) , where z , z ∈ C n and c ∈ C . As above, the Lie algebra, h n C , of H n C is, as a set, equal to H n C itself. In this case { ( e j , , (0 , e j ) } nj =1 is a complex orthonormal basis for H. The (complex) Hilbert-Schmidt norm of the symplectic form ω is given by k ω k H ∗ ⊗ H ∗ = 2 n X j =1 | ω (( e j , , (0 , e j )) | = 2 n. Example 3.20.
Let ( K, h· , ·i ) be a complex Hilbert space and Q be a strictlypositive trace class operator on K. For h, k ∈ K, let h h, k i Q := h h, Qk i and k h k Q := q h h, h i Q . Also let (cid:16) K Q , h· , ·i Q (cid:17) denote the Hilbert space completionof (cid:16) K, k·k Q (cid:17) . Analogous to Example 3.18, let H := K Re , W := ( K Q ) Re , and ω : W × W → R =: C be defined by ω ( w, z ) = Im h w, z i Q for all w, z ∈ W. Then G ( ω ) = W × R is a real group and ( W, H ) determines an abstract Wienerspace (see for example [36, Exercise 17, p. 59] and [8, Example 3.9.7] )). Let { e j } ∞ j =1 be an orthonormal basis for K so that { e j , ie j } ∞ j =1 is an orthonormal basisfor ( H, Re h· , ·i K ) . Then the real Hilbert-Schmidt norm of ω is given by k ω k H ∗ ⊗ H ∗ = ∞ X j,k =1 X ε,δ ∈{ ,i } (cid:2) Im h εe j , Qδe k i (cid:3) = 2 ∞ X j,k =1 (cid:2) Im h e j , Qe k i + Re h e j , Qe k i (cid:3) = 2 ∞ X j,k =1 |h e j , Qe k i| = 2 ∞ X k =1 k Qe k k = 2 k Q k HS = 2 tr Q . (3.36) Example 3.21.
Again let ( K, h· , ·i ) , Q, and (cid:16) K Q , h· , ·i Q (cid:17) be as in the previousexample. 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 → C be defined by ω (( w , w ) , ( z , z )) = h w , ¯ z i Q − h w , ¯ z i Q , which is skew symmetric because the conjugation commutes with Q. If { e j } ∞ j =1 is anorthonormal basis for K, then { ¯ e j } ∞ j =1 is also an orthonormal basis for K (becausethe conjugation is isometric) and { ( e j , , (0 , e j ) } ∞ j =1 is a orthonormal basis for H. Hence, the (complex) Hilbert-Schmidt norm of ω is given by k ω k H ∗ ⊗ H ∗ = ∞ X j,k =1 (cid:0) | ω (( e j , , (0 , e k )) | + | ω ((0 , e k ) , ( e j , | (cid:1) = 2 ∞ X j,k =1 |h e j , Q ¯ e k i| = 2 ∞ X k =1 k Q ¯ e k k = 2 k Q k HS = 2 tr Q . (3.37) Example 3.22.
Suppose that ( V, h· , ·i V ) is a d –dimensional F –inner product space( F = R or C ) , C is a finite-dimensional F – inner product space, α : V × V → C isan anti-symmetric bilinear form on V, and { q j } ∞ j =1 is a sequence of positive numberssuch that P ∞ j =1 q j < ∞ . Let W = v ∈ V N : ∞ X j =1 q j k v j k V < ∞ and H = v ∈ V N : ∞ X j =1 k v j k V < ∞ ⊂ W, each of which are Hilbert spaces when equipped with the inner products h v, w i W := ∞ X j =1 q j h v j , w j i V and h v, w i H := ∞ X j =1 h v j , w j i V respectively. Further let ω : W × W → C be defined by ω ( v, w ) = ∞ X j =1 q j α ( v j , w j ) . EAT KERNELS ON INFINITE-DIMENSIONAL HEISNEBERG 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 ) | = ∞ X j =1 q j | α ( v j , w j ) | k α k ∞ X j =1 q j k v j k V k w j k V k α k k v k W k w k W , we have k ω k k α k . For v ∈ V, let v ( j ) := (0 , . . . , , v, , , . . . ) ∈ H wherethe v is put in the j th – position. If { u a } da =1 is an orthonormal basis for V, then { u a ( j ) : a = 1 , . . . , d } ∞ j =1 is an orthonormal basis for H. Therefore, kh ω ( · , · ) , c i C k H ∗ ⊗ H ∗ = ∞ X j,k =1 d X a,b =1 h ω ( u a ( j ) , u b ( k )) , c i C = ∞ X j =1 d X a,b =1 q j h α ( u a , u b ) , c i C = ∞ X j =1 q j kh α ( · , · ) , c i C k V ∗ ⊗ V ∗ for all c ∈ C . (3.38) Example 3.23.
Let ( V, h· , ·i , α ) be as in Example 3.22 with F = C ,W = { σ ∈ C ([0 , , V ) : σ (0) = 0 } and H be the associated Cameron – Martin space, H := H ( V ) = (cid:26) h ∈ W : Z k h ′ ( s ) k V ds < ∞ (cid:27) , wherein R k h ′ ( s ) k V ds := ∞ if h is not absolutely continuous. Further let η be acomplex measure on [0 , and ω ( σ , σ ) := Z α ( σ ( s ) , σ ( s )) dη ( s ) for all σ , σ ∈ W. Then ( W, H, ω ) satisfies all of the assumptions in Notation 3.1. Let { u a } da =1 be theorthonormal basis of V , { l j } ∞ j =1 be the orthonormal basis of H ( R ) , then { l j u a : a =1 , , . . . , d } ∞ j =1 is an orthonormal basis of H and (see [22, Lemma 3.8] ) (3.39) ∞ X j =1 l j ( s ) l j ( t ) = s ∧ t for all s, t ∈ [0 , . If we let λ be the total variation of η, then dη = ρdλ , where ρ = dηdλ . Hence if d ¯ η ( t ) := ¯ ρ ( t ) dλ ( t ) and c ∈ C , then kh ω ( · , · ) , c i C k = ∞ X j,k =1 d X a,b =1 (cid:12)(cid:12) h ω ( l j u a , l k u b ) , c i C (cid:12)(cid:12) = ∞ X j,k =1 d X a,b =1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z [0 , l j ( s ) l k ( s ) ρ ( s ) dλ ( s ) h α ( u a , u b ) , c i C (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)
28 DRIVER AND GORDINA = kh α ( · , · ) , c i C k · ∞ X j,k =1 Z [0 , l j ( s ) l k ( s ) l j ( t ) l k ( t ) ρ ( s ) ¯ ρ ( t ) dλ ( s ) dλ ( t )= kh α ( · , · ) , c i C k · Z [0 , ( s ∧ t ) ρ ( s ) ¯ ρ ( t ) dλ ( s ) dλ ( t )= kh α ( · , · ) , c i C k · Z [0 , ( s ∧ t ) dη ( s ) d ¯ η ( t ) , (3.40) wherein we have used Equation (3.39) in the fourth equality above. Summing thisequation over c in an orthonormal basis for C shows (3.41) k ω k = k α k · Z [0 , ( s ∧ t ) dη ( s ) d ¯ η ( t ) . Finite Dimensional Projections and Cylinder 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 i H ∈ Ran( i ∗ ) ⊂ H ∗ } or in other words, h ∈ H is in H ∗ iff h· , h i H ∈ H ∗ extends to a continuous linearfunctional on W. (We will continue to denote the continuous extension of h· , h i H to W by h· , h i H . ) Because H is a dense subspace of W, i ∗ is injective and because i is injective, i ∗ has a dense range. Since h
7→ h· , h i H as a map from H to H ∗ is aconjugate linear isometric isomorphism, it follows from the above comments that H ∗ ∋ h → h· , h i H ∈ W ∗ is a conjugate linear isomorphism too, and that H ∗ is adense subspace of H .Now suppose that P : H → H is a finite rank orthogonal projection such that P H ⊂ H ∗ . Let { e j } nj =1 be an orthonormal basis for P H and ℓ j = h· , e j i H ∈ W ∗ . Then we may extend P to a (unique) continuous operator from W → H (stilldenoted by P ) by letting(3.42) P w := n X j =1 h k, e j i H e j = n X j =1 ℓ j ( w ) e j for all w ∈ W. For all w ∈ W we have, k P w k H C ( P ) k P w k W and k P w k W n X i =1 kh· , e i i H k W k e i k W ! k w k W and therefore there exists C < ∞ such that(3.43) k P w k H C k w k W for all w ∈ W. Notation 3.24.
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 theform given in Equation (3.42) . Further, let G P := P W × C (a subgroup of G CM ) and π = π P : G → G P be defined by π P ( w, c ) := ( P w, c ) . EAT KERNELS ON INFINITE-DIMENSIONAL HEISNEBERG GROUPS 19
Remark 3.25.
The reader should be aware that π P : G → G P ⊂ G CM is not (forgeneral 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 ′ ) = 12 (0 , 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 ) + (0 , c ) = ( w, b ) · (0 , c ) for all w ∈ W and b, c ∈ C , we may also writeEquation 3.44 as(3.46) π P [( w, c ) · ( w ′ , c ′ )] = π P ( w, c ) · π P ( w ′ , c ′ ) · Γ P ( w, w ′ ) . Definition 3.26.
A function f : G → C is said to be a ( smooth) cylinderfunction if it may be written as f = F ◦ π P for some P ∈ Proj ( W ) and some(smooth) function F : G P → C . Notation 3.27.
For g = ( w, c ) ∈ G, let γ ( g ) and χ ( g ) be the elements of g CM ⊗ g CM defined by γ ( g ) := ∞ X j =1 (0 , ω ( w, e j )) ⊗ ( e j , and χ ( g ) := ∞ X j =1 (0 , ω ( w, e j )) ⊗ (0 , ω ( w, e j )) where { e j } ∞ j =1 is any orthonormal basis for H. Both γ and χ are well defined becauseof Lemma 3.16. Notation 3.28 (Left differentials) . Suppose f : G → C is a smooth cylinderfunction. For g ∈ G and h, h , ..., h n ∈ g , n ∈ N let (cid:0) D f (cid:1) ( g ) = f ( g ) and ( D n f ) ( g ) ( h ⊗ ... ⊗ h n ) = ˜ h ... ˜ h n f ( g ) , (3.47) where ˜ hf is given as in Equation (3.10) or Equation (3.12) . We will write Df for D f. Proposition 3.29.
Let { e j } ∞ j =1 and { f ℓ } dℓ =1 be orthonormal bases for H and C respectively. Then for any smooth cylinder function, f : G → C , (3.48) Lf ( g ) := ∞ X j =1 (cid:20) ^ ( e j , f (cid:21) ( g ) + d X ℓ =1 (cid:20) ^ (0 , f ℓ ) f (cid:21) ( g ) is well defined. Moreover, if f = F ◦ π P , ∂ h is as in Notation 3.5 for all h ∈ g CM , (3.49) ∆ H f ( g ) := ∞ X j =1 ∂ e j , f ( g ) = (∆ P H F ) ( P w, c ) and (3.50) ∆ C f ( g ) := d X ℓ =1 h ∂ ,f ℓ ) f i ( g ) = (∆ C F ) ( P w, c ) , then (3.51) Lf ( g ) = (∆ H f + ∆ C f ) ( g ) + f ′′ ( g ) (cid:18) γ ( g ) + 14 χ ( g ) (cid:19) . Proof.
The proof of the second equality in Eq. (3.49) is straightforward and willbe left to the reader. Recall from Equation (3.12) that(3.52) ^ ( e j , f ( g ) = f ′ ( g ) (cid:18) e j , ω ( w, e j ) (cid:19) . Applying ^ ( e j ,
0) to both sides of Equation (3.52) gives ^ ( e j , f ( g ) = f ′′ ( g ) (cid:18)(cid:18) e j , ω ( w, e j ) (cid:19) ⊗ (cid:18) e j , ω ( w, e j ) (cid:19)(cid:19) (3.53) = f ′′ ( g ) (( e j , ⊗ ( e j , f ′′ ( g ) ((0 , ω ( w, e j )) ⊗ ( e j , f ′′ ( g ) ((0 , ω ( w, e j )) ⊗ (0 , ω ( w, e j )))(3.54)wherein we have used, ∂ e j ω ( · , e j ) = ω ( e j , e j ) = 0 . Summing Equation (3.54) on j shows, ∞ X j =1 (cid:20) ^ ( e j , f (cid:21) ( g ) = ∞ X j =1 f ′′ ( g ) (( e j , ⊗ ( e j , f ′′ ( g ) (cid:18) γ ( g ) + 14 χ ( g ) (cid:19) = ∞ X j =1 ∂ e j , f ( g ) + f ′′ ( g ) (cid:18) γ ( g ) + 14 χ ( g ) (cid:19) . 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 C appearing inEquation (3.48). (cid:3) Brownian Motion and Heat Kernel Measures
For the Hilbert space stochastic calculus background needed for this section, seeM´etivier [41]. For the background on Itˆo integral relative to an abstract Wienerspace–valued Brownian motion, see Kuo [36, pages 188-207] (especially Theorem5.1), Kusuoka and Stroock [37, p. 5], and the appendix in [16].Suppose now that ( B ( t ) , B ( t )) is a smooth curve in g CM with ( B (0) , B (0)) =(0 ,
0) and consider solving, for g ( t ) = ( w ( t ) , c ( t )) ∈ G CM , the differential equation(4.1) ( ˙ w ( t ) , ˙ c ( t )) = ˙ g ( t ) = l g ( t ) ∗ (cid:16) ˙ B ( t ) , ˙ B ( t ) (cid:17) with g (0) = (0 , . By Equation (3.11), it follows that( ˙ w ( t ) , ˙ c ( t )) = (cid:18) ˙ B ( t ) , ˙ B ( t ) + 12 ω (cid:16) w ( t ) , ˙ B ( t ) (cid:17)(cid:19) and therefore the solution to Equation (4.1) is given by(4.2) g ( t ) = ( w ( t ) , c ( t )) = (cid:18) B ( t ) , B ( t ) + 12 Z t ω (cid:16) B ( τ ) , ˙ B ( τ ) (cid:17) dτ (cid:19) . Below in subsection 4.2, we will replace B and B by Brownian motions and usethis to define a Brownian motion on G. EAT KERNELS ON INFINITE-DIMENSIONAL HEISNEBERG GROUPS 21
A quadratic integral.
Let { ( B ( t ) , B ( t )) } t > be a Brownian motion on g with variance determined by E h h ( B ( s ) , B ( s )) , ( A, a ) i g CM h ( B ( t ) , B ( t )) , ( C, c ) i g CM i = Re h ( A, a ) , ( C, c ) i g CM min ( s, t )for all s, t ∈ [0 , ∞ ) , A, C ∈ H ∗ and a, c ∈ C . Also let { e j } ∞ j =1 ⊂ H ∗ be an orthonor-mal basis for H. For n ∈ N , define P n ∈ Proj ( W ) as in Notation 3.24, i.e.(4.3) P n ( w ) = n X j =1 h w, e j i H e j = n X j =1 ℓ j ( w ) e j for all w ∈ W. Proposition 4.1.
For each n, let M nt := R t ω ( B ( τ ) , dP n B ( τ )) . Then (1) { M nt } t > is an L –martingale and there exists an L –martingale, { M t } t > with values in C such that (4.4) lim n →∞ E (cid:20) max t T k M t − M nt k C (cid:21) = 0 for all T < ∞ . (2) The quadratic variation of M is given by; (4.5) h M i t = Z t k ω ( B ( τ ) , · ) k H ∗ ⊗ C dτ. (3) The square integrable martingale, M t , is well defined independent of thechoice of the orthonormal basis, { e j } ∞ j =1 and hence will be denoted by R t ω ( B ( τ ) , dB ( τ )) . (4) For each p ∈ [1 , ∞ ) , { M t } t > is L p –integrable and there exists c p < ∞ suchthat E (cid:18) sup t T k M t k p C (cid:19) 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 M Pt := R t ω ( B ( τ ) , dP B ( τ )) . Let
P, Q ∈ Proj ( W )and choose an orthonormal basis, { v l } Nl =1 for Ran ( P ) + Ran ( Q ) . We then have E (cid:20)(cid:13)(cid:13)(cid:13) M PT − M QT (cid:13)(cid:13)(cid:13) C (cid:21) = E Z T N X l =1 k ω ( B ( τ ) , ( P − Q ) v l ) k C dτ = E Z T ∞ X l =1 k ω ( B ( τ ) , ( P − Q ) e l ) k C dτ (4.6) = Z T ∞ X l =1 ∞ X k =1 k ω ( e k , ( P − Q ) e l ) k C τ dτ = T ∞ X l =1 ∞ X k =1 k ω ( e k , ( P − Q ) e l ) k C . (4.7)Taking P = P n and Q = P m with m n in Eq. (4.7) allows us to conclude that E h k M nT − M mT k C i = T n X j = m +1 ∞ X l =1 k ω ( e l , e j ) k C → m, n → ∞ because k ω k < ∞ by Proposition 3.14. Since the space of continuous L –martingales on [0 , T ] is complete in the norm, N → E k N T k C and, by Doob’smaximal inequality ([34, Proposition 7.16]), there exists c < ∞ such that E (cid:20) max t T k N t k p C (cid:21) c E k N T k p C , it follows that there exists a square integrable C –valued martingale, { M t } t > , suchthat Eq. (4.4) holds.2. Since the quadratic variation of M n is given by h M n i t = Z t k ω ( B ( τ ) , dP n B ( τ )) k C = Z t n X l =1 k ω ( B ( τ ) , e l ) k C dτ and E [ |h M i t − h M n i t | ] q E [ h M − M n i t ] · E [ h M + M n i t ]= q E k M t − M nt k C · E k M t + M nt k C → n → ∞ , Eq. (4.5) easily follows.3. Suppose now that (cid:8) e ′ j (cid:9) ∞ j =1 ⊂ 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 (cid:13)(cid:13)(cid:13) M P n T − M P ′ n T (cid:13)(cid:13)(cid:13) C = T ∞ X l =1 ∞ X k =1 k ω ( e k , ( P n − P ′ n ) e l ) k C . Since ∞ X l =1 ∞ X k =1 k ω ( e k , P ′ n e l ) − ω ( e k , e l ) k C = ∞ X l =1 ∞ X k =1 k ω ( e k , ( P ′ n − I ) e l ) k C = ∞ X l =1 ∞ X k =1 k ω ( e ′ k , ( P ′ n − I ) e ′ l ) k C = ∞ X l = n +1 ∞ X k =1 k ω ( e ′ k , e ′ l ) k C → n → ∞ and similarly but more easily, P ∞ l =1 P ∞ k =1 k ω ( e k , P n e l ) − ω ( e k , e l ) k C → n →∞ , we may pass to the limit in Eq. (4.8) to learn that lim n →∞ E (cid:13)(cid:13)(cid:13) M P n T − M P ′ n T (cid:13)(cid:13)(cid:13) C =0 .
4. By Jensen’s inequality Z T k ω ( B ( s ) , · ) k H ∗ ⊗ C ds ! p/ = T p/ Z T k ω ( B ( s ) , · ) k H ∗ ⊗ C dsT ! p/ T p/ Z T k ω ( B ( s ) , · ) k pH ∗ ⊗ C dsT = T p − Z T k ω ( B ( s ) , · ) k pH ∗ ⊗ C ds. EAT KERNELS ON INFINITE-DIMENSIONAL HEISNEBERG GROUPS 23
Combining this estimate with Equation (3.28) and then applying either Skoro-hod’s or Fernique’s inequality (see Equations (2.3) or (2.4)) shows E h h M i p/ T i T p − Z T E k ω ( B ( s ) , · ) k pH ∗ ⊗ C ds T p − Z T C p/ k ω k p k B ( s ) k pW ds T p − C p/ k ω k p Z W k y k pW dµ ( y ) Z T s p/ ds = T p − C p/ k ω k p C p T p/ p/ c ′ p T p . (4.9)As a consequence of the Burkholder-Davis-Gundy inequality (see for example [49,Corollary 6.3.1a on p.344], [45, Appendix A.2], or [41, p. 212] and [34, Theorem17.7] for the real case), for any p > c ′′ p < ∞ such that E (cid:18) sup t T k M t k C (cid:19) p c ′′ p E h h M i p/ T i = c ′′ p c ′ p T p =: c p T p . (cid:3) Brownian Motion on G ( ω ) . Motivated by Eq. (4.2) we have the followingdefinition.
Definition 4.2.
Let ( B ( t ) , B ( t )) be a g – valued Brownian motion as in subsec-tion 4.1. A Brownian motion on G is the continuous G –valued process definedby(4.10) g ( t ) = (cid:18) B ( t ) , B ( t ) + 12 Z t ω ( B ( τ ) , dB ( τ )) (cid:19) . Further, for
T > , let ν T = Law ( g ( T )) be a probability measure on G. We referto ν T as the time T heat kernel measure on G. Remark 4.3.
An alert reader may complain that we should use the Stratonovichintegral in Eq. (4.10) rather than the Itˆo integral. However, these two integrals areequal since ω is a skew symmetric form Z t ω ( B ( τ ) , ◦ dB ( τ )) = Z t ω ( B ( τ ) , dB ( τ )) + 12 Z t ω ( dB ( τ ) , dB ( τ ))= Z t ω ( B ( τ ) , dB ( τ )) . Theorem 4.4 (The generator of g ( t )) . The generator of g ( t ) is the operator L defined in Proposition 3.29. More precisely, if f : G → C is a smooth cylinderfunction, then (4.11) d [ f ( g ( t ))] = f ′ ( g ( t )) dg ( t ) + 12 Lf ( g ( t ) dt where L is given in Proposition 3.29, f ′ is defined as in Notation 3.5 and dg ( t ) = (cid:18) dB ( t ) , dB ( t ) + 12 ω ( B ( t ) , dB ( t )) (cid:19) . Proof.
Let us begin by observing that dg ( t ) ⊗ dg ( t ) = (cid:18) dB ( t ) , dB ( t ) + 12 ω ( B ( t ) , dB ( t )) (cid:19) ⊗ = (cid:20)(cid:18) dB ( t ) , ω ( B ( t ) , dB ( t )) (cid:19) + (0 , dB ( t )) (cid:21) ⊗ = ∞ X j =1 (cid:18) e j , ω ( B ( t ) , e j ) (cid:19) ⊗ dt + d X ℓ =1 (0 , f ℓ ) ⊗ dt (4.12)where { f ℓ } dℓ =1 is an orthonormal basis for C and { e j } ∞ j =1 is an orthonormal basisfor H. Hence, as a consequence of Itˆo’s formula, we have d [ f ( g ( t ))] = f ′ ( g ( t )) dg ( t ) + 12 f ′′ ( g ( t )) ( dg ( t ) ⊗ dg ( t ))= f ′ ( g ( t )) dg ( t )+ 12 f ′′ ( g ( t )) ∞ X j =1 (cid:18) e j , ω ( B ( t ) , e j ) (cid:19) ⊗ dt + 12 f ′′ ( g ( t )) d X ℓ =1 (0 , f ℓ ) ⊗ dt = f ′ ( g ( t )) dg ( t ) + 12 ∞ X j =1 (cid:18) ^ ( e j , f (cid:19) ( g ( t )) dt + 12 d X ℓ =1 (cid:18) ^ (0 , f ℓ ) f (cid:19) ( g ( t )) dt = f ′ ( g ( t )) ( dg ( t )) + 12 Lf ( g ( t )) dt. (cid:3) For the next corollary, let P ∈ Proj ( W ) as in Equation (3.42), F ∈ C ( P H × C , C ) , and f = F ◦ π P : G → C be a cylinder function where P ∈ Proj ( W ) . We will further suppose there exist 0 < K, p < ∞ such that(4.13) | F ( h, c ) | + k F ′ ( h, c ) k + k F ′′ ( h, c ) k K (1 + k h k H + k c k C ) p for any h ∈ P H and c ∈ C . Further let { f ℓ } dℓ =1 be an orthonormal basis for C andextend { e j } nj =1 to an orthonormal basis, { e j } ∞ j =1 , for H. Corollary 4.5. If f : G → C is a cylinder function as above, then (4.14) E [ f ( g ( T ))] = f ( e ) + 12 Z T E [( Lf ) ( g ( t ))] dt, i.e. (4.15) ν T ( f ) = f ( e ) + 12 Z T ν t ( Lf ) dt. In other words, ν t weakly solves the heat equation ∂ t ν t = 12 Lν t with lim t ↓ ν t = δ e . EAT KERNELS ON INFINITE-DIMENSIONAL HEISNEBERG GROUPS 25
Proof.
Integrating Equation (4.11) shows(4.16) f ( g ( T )) = f ( e ) + N T + 12 Z T Lf ( g ( τ )) dτ where N t := Z t f ′ ( g ( τ )) dg ( τ ) = Z t f ′ ( g ( τ )) (cid:18) dB ( τ ) , dB ( τ ) + 12 dM τ (cid:19) and M t = R t ω ( B ( τ ) , dB ( τ )) . Using Eqs. (4.12) and (4.13) there exists C = C ( P, k ω k ) < ∞ such that d h N i t := | dN t | = h f ′ ( g t ) ⊗ f ′ ( g t ) , dg t ⊗ dg t i = ∞ X j =1 (cid:12)(cid:12)(cid:12)(cid:12) f ′ ( g ( t )) (cid:18) e j , ω ( B ( t ) , e j ) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) dt + d X ℓ =1 | f ′ ( g ( t )) (0 , f ℓ ) | dt = n X j =1 (cid:12)(cid:12)(cid:12)(cid:12) f ′ ( g ( t )) (cid:18) e j , ω ( B ( t ) , e j ) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) dt + d X ℓ =1 | f ′ ( g ( t )) (0 , f ℓ ) | dt C ( P, k ω k ) (1 + k P B ( t ) k H + k B ( t ) k C ) p (cid:16) k B ( t ) k W + 1 (cid:17) dt C (1 + k B ( t ) k W + k B ( t ) k C ) p +2 dt, wherein we have used Equation (3.43) for the last inequality. From this inequalityand either of Equations (2.3) or (2.4), we find E [ h N i T ] C Z T E (1 + k B ( t ) k W + k B ( t ) k C ) p +2 dt < ∞ and hence that N t is a square integrable martingale. Therefore we may take theexpectation of Equation (4.16) which implies Equation (4.14). (cid:3) Finite Dimensional Approximations.Proposition 4.6.
Let { P n } ∞ n =1 ⊂ 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 (cid:20) max t T k B ( t ) − B n ( t ) k pW (cid:21) = 0 for all p ∈ [1 , ∞ ) , and (4.19) lim n →∞ max t T k B ( t ) − B n ( t ) k W = 0 a.s.Proof. Let { w k } ∞ k =1 ⊂ W be a countable dense set and for each k ∈ N , choose ϕ k ∈ W ∗ such that k ϕ k k W ∗ = 1 and ϕ k ( w k ) = k w k k W . We then have, k w k W = sup k | ϕ k ( w ) | = sup Re ϕ k ( w ) for all w ∈ W. By [8, Theorem 3.5.7] with A = I , if ε n ( t ) := B ( t ) − B n ( t ) , then(4.20) lim n →∞ E k ε n ( T ) k pW = 0 for all p ∈ [1 , ∞ ) . Since { ϕ k ( ε n ( t )) } t > is (up to a multiplicative factor) a standard Brownian mo-tion, {| ϕ k ( ε n ( t )) |} t > is a submartingale for each k ∈ N and therefore so is {k ε n ( t ) k = sup k | ϕ k ( ε n ( t )) |} t > . Hence, according to Doob’s inequality, for each p ∈ [1 , ∞ ) there exists C p < ∞ such that(4.21) E (cid:12)(cid:12)(cid:12)(cid:12) max t T k ε n ( t ) k W (cid:12)(cid:12)(cid:12)(cid:12) p C p E k ε n ( T ) k pW . Combining Equation (4.21) with Equation (4.20) proves Equation (4.18). Equation(4.19) now follows from Equation (4.18) and [11, Proposition 2.11]. To apply thisproposition, let E be the Banach space, C ([0 , T ] , W ) equipped with the sup-norm,and let ξ k := ℓ k ( B ( · )) e k ∈ E for all k ∈ N . (cid:3) 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 ) := (cid:18) P B ( t ) , B ( t ) + 12 Z t ω ( P B ( τ ) , P dB ( τ )) (cid:19) . Then (4.22) g ( t ) = g P ( t ) (cid:18) QB ( t ) , Z t [2 ω ( QB ( τ ) , P dB ( τ )) + ω ( QB ( τ ) , QdB ( τ ))] (cid:19) , and (4.23) g P ( t ) − π P ( g ( t )) = 12 (cid:18) , Z t [ ω ( B ( τ ) , dB ( τ )) − ω ( P B ( τ ) , P dB ( τ ))] (cid:19) . Also, if { P n } ∞ n =1 ⊂ Proj ( W ) are as in Eq. (4.3) and (4.24) g n ( t ) = g P n ( t ) = (cid:18) P n B ( t ) , B ( t ) + 12 Z t ω ( P n B ( τ ) , dP n B ( τ )) (cid:19) , then (4.25) lim n →∞ E (cid:20) max t T k g ( t ) − g n ( t ) k p g (cid:21) = 0 for all p < ∞ . Proof.
A simple computation shows l g P ( t ) − ∗ ◦ dg P ( t ) = (cid:18) dP B ( t ) , dB ( t ) + ω ( P B ( t ) , P dB ( t ))+ ω ( − P B ( t ) , P dB ( t )) (cid:19) = ( dP B ( t ) , dB ( t )) = d ( P B ( t ) , B ( t )) . Hence it follows that g P solves the stochastic differential equation, dg P ( t ) = l g P ( t ) ∗ ◦ d ( P B ( t ) , B ( t )) with g P (0) = 0and therefore g P is a G P –valued Brownian motion. The proof of the equalities inEquations (4.22) and (4.23) follows by elementary manipulations which are left tothe reader.In light of Equation (4.18) of Proposition 4.6, to prove the last assertion we mustshow(4.26) lim n →∞ E (cid:20) max t T | M t ( n ) | p (cid:21) = 0 , EAT KERNELS ON INFINITE-DIMENSIONAL HEISNEBERG GROUPS 27 where M t ( n ) is the local martingale defined by M t ( n ) := Z t [ ω ( B ( τ ) , dB ( τ )) − ω ( B n ( τ ) , dB n ( τ ))] . Since h M ( n ) i T = ∞ X j =1 Z T k ω ( B ( τ ) , e j ) k C dτ + n X j =1 Z T k ω ( B n ( τ ) , e j ) k C dτ − n X j =1 Z T h ω ( B ( τ ) , e j ) , ω ( B n ( τ ) , e j ) i C dτ and 2 T E [ h M ( n ) i T ] = ∞ X j,k =1 k ω ( e k , e j ) k C − n X k =1 n X j =1 k ω ( e k , e j ) k C → n → ∞ , it follows by the Burkholder-Davis-Gundy inequalities that M ( n ) is amartingale and Equation (4.26) holds for p = 2 and hence for p ∈ [1 , . By Doob’s maximal inequality ([34, Proposition 7.16]), to prove Equation (4.26)for p > , it suffices to show lim n →∞ E [ | M T ( n ) | p ] = 0 . However, M T ( n ) has Itˆo’schaos expansion terminating at degree two and hence by a theorem of Nelson (see[44, Lemma 2 on p. 415] and [43, pp. 216-217]) for each j ∈ N there exists c j < ∞ such that E h M jT ( n ) i c j (cid:2) E M T ( n ) (cid:3) j . (This result also follows from Nelson’shypercontractivity for the Ornstein-Uhlenbeck operator.) This clearly suffices tocomplete the proof of the theorem. (cid:3) Lemma 4.8.
For all P ∈ Proj ( W ) and t > , let ν Pt := Law ( g P ( t )) . Then ν Pt ( dx ) = p Pt ( e, x ) dx, where dx is the Riemannian volume measure (equal to aHaar measure) p Pt ( x, y ) is the heat kernel on G P . Proof.
An application of Corollary 4.5 with G replaced by G P implies that ν Pt =Law ( g P ( t )) is a weak solution to the heat equation on G P . The result now followsas an application of [17, Theorem 2.6 ]. (cid:3)
Corollary 4.9.
For any
T > , the heat kernel measure, ν T , is invariant underthe inversion map, g g − for any g ∈ G .Proof. It is well known (see for example [20, Proposition 3.1]) that heat kernelmeasures based at the identity of a finite-dimensional Lie group are invariant underinversion. Now suppose that f : G → R is a bounded continuous function. Bypassing 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, E f (cid:16) g ( T ) − (cid:17) = lim n →∞ E f (cid:16) g n ( T ) − (cid:17) = lim n →∞ E f ( g n ( T )) = E f ( g ( T )) . This completes the proof because ν T is the law of g ( T ) . (cid:3) We are now going to give exponential bounds which are much stronger than themoment estimates in Equation (4.9) of Proposition 4.1. Before doing so we need torecall the following result of Cameron–Martin and Kac, [9, 33].
Lemma 4.10 (Cameron–Martin and Kac) . Let { b s } s > be a one dimensional Brow-nian motion. Then for any T > and λ ∈ [0 , π T );(4.27) E " exp λ Z T b s ds ! = [cos ( λT )] − / < ∞ . Proof.
When T = 1 , simply follow the proof of [31, Equation (6.9) on p. 472] with λ replaced by − λ . For general
T > , by a change of variables and a Brownianmotion scaling we have Z T b s ds = T Z b tT dt d = T Z b t dt. Therefore, E " exp λ Z T b s ds ! = E (cid:20) exp (cid:18) λ T Z b s ds (cid:19)(cid:21) (4.28) = cos − / (cid:16) √ λ T (cid:17) provided that λ ∈ [0 , π T ) . (cid:3) Remark 4.11.
For our purposes below, all we really need later from Lemma 4.10is the qualitative statement that for λT > E " exp λ Z T b s ds ! = 1 + λ T O (cid:0) λ T (cid:1) . Instead of using Lemma 4.10 we can derive this statement as an easy consequenceof the scaling identity in Equation (4.28) along with the analyticity (use Fernique’stheorem) of the function, F ( z ) := E (cid:20) exp (cid:18) z Z b s ds (cid:19)(cid:21) for | z | small. Proposition 4.12. If { N t } t > is a continuous local martingale such that N = 0 . Then (4.30) E e | N t | q E (cid:2) e h N i t (cid:3) . Proof.
By Itˆo’s formula, we know that Z t := e N t −h N i t / = e N t − h N i t is a non-negative local martingale. If { σ n } ∞ n =1 is a localizing sequence of stoppingtimes for Z, then, by Fatou’s lemma, E [ Z t |B s ] lim inf n →∞ E [ Z σ n t |B s ] = lim inf n →∞ Z σ n s = Z s . This shows that Z is a supermartingale and in particular that E [ Z t ] E Z = 1 . By the Cauchy-Schwarz inequality we find E (cid:2) e N t (cid:3) = E h e N t −h N i t e h N i t i q E (cid:2) e N t − h N i t (cid:3) · E (cid:2) e h N i t (cid:3) = q E [ Z t ] · E (cid:2) e h N i t (cid:3) q E (cid:2) e h N i t (cid:3) (4.31) EAT KERNELS ON INFINITE-DIMENSIONAL HEISNEBERG GROUPS 29
Applying this inequality with N replaced by − N and using e | x | e x + e − x easilygive Equation (4.30). (cid:3) Proposition 4.13.
Let n ∈ N , T > , d = dim R C , (4.32) γ := sup ∞ X j =1 (cid:12)(cid:12) h ω ( h, e j ) , c i C (cid:12)(cid:12) : k h k H = k c k C = 1 k ω k < ∞ and for P ∈ Proj ( W ) let B P ( t ) := P B ( t ) . Then for all (4.33) 0 λ < π dT √ γ , (4.34) sup P ∈ Proj( W ) E (cid:20) exp (cid:18) λ (cid:13)(cid:13)(cid:13)(cid:13)Z t ω ( B P ( τ ) , dB P ( τ )) (cid:13)(cid:13)(cid:13)(cid:13) C (cid:19)(cid:21) < ∞ and (4.35) E (cid:20) exp (cid:18) λ (cid:13)(cid:13)(cid:13)(cid:13)Z t ω ( B ( τ ) , dB ( τ )) (cid:13)(cid:13)(cid:13)(cid:13) C (cid:19)(cid:21) < ∞ . Proof.
Equation (4.35) follows by choosing { P n } ∞ n =1 ⊂ Proj ( W ) as in Eq. (4.3)and then using Fatou’s lemma in conjunction with the estimate in Equation (4.34).So we need only to concentrate on proving Equation (4.34).Fix a P ∈ Proj ( W ) as in Equation (3.42) and let M Pt := Z t ω ( B P ( τ ) , dB P ( τ )) . If { f ℓ } dℓ =1 is an orthonormal basis for C , then (cid:13)(cid:13) M Pt (cid:13)(cid:13) C d X ℓ =1 (cid:12)(cid:12)(cid:10) M Pt , f ℓ (cid:11) C (cid:12)(cid:12) , and it follows by H¨older’s inequality and the martingale estimate in Proposition4.12 that E h e λ k M Pt k C i E h e λ P dℓ =1 |h M Pt ,f ℓ i C | i d Y ℓ =1 (cid:16) E h e λd |h M Pt ,f ℓ i C | i(cid:17) /d d Y ℓ =1 r E h e λ d h h M P · ,f ℓ i C i t i! /d = 2 d Y ℓ =1 (cid:16) E h e λ d hh M P · ,f ℓ i C i t i(cid:17) / d . (4.36)We will now evaluate each term in the product in Eq. (4.36). So let c := f ℓ and N t := (cid:10) M Pt , c (cid:11) C , and Q P : H → H and Q : H → H be the unique non-negativesymmetric operators such that, for all h ∈ H, n X j =1 (cid:12)(cid:12) h ω ( P h, e j ) , c i C (cid:12)(cid:12) = h Q P h, h i H for all h ∈ H and ∞ X j =1 (cid:12)(cid:12) h ω ( h, e j ) , c i C (cid:12)(cid:12) = h Qh, h i H for all h ∈ H. Also let { q l ( P ) } ∞ l =1 be the eigenvalues listed in decreasing order (counted withmultiplicities) for Q P and observe that(4.37) q ( P ) = sup h =0 h Q P h, h ik h k H sup h =0 h QP h, P h ik h k H sup h =0 h Qh, h ik h k H γ. With this notation, the quadratic variation of N is given by(4.38) h N i T = Z T n X j =1 (cid:12)(cid:12) h ω ( B P ( t ) , e j ) , c i C (cid:12)(cid:12) dt = Z T h Q P B P ( t ) , B P ( t ) i H dt. Moreover, by expanding B P ( τ ) in an orthonormal basis of eigenvectors of Q P | P H it follows that(4.39) h N i T = n X l =1 q l ( P ) Z T b l ( τ ) dτ where { b l } nl =1 is a sequence of independent Brownian motions. Hence it followsthat E h e λ d hh M P · ,f ℓ i C i T i = E h e λ d h N i T i = n Y l =1 E " exp λ d q l ( P ) Z T b l ( τ ) dτ ! . (4.40)If Eq. (4.33) holds then (using Eq. (4.37))2 λd p q ( P ) = p λ d q ( P ) λd √ γ < π/ T and we may apply Lemma 4.10 to find E " exp λ d q l ( P ) Z T b l ( τ ) dτ ! = 1 r cos (cid:16) λd p q l ( P ) T (cid:17) = exp (cid:18) −
12 ln cos (cid:16) λd p q l ( P ) T (cid:17)(cid:19) . (4.41)Moreover, a simple calculus exercise shows for any k ∈ (0 , π/
2) there exists c ( k ) < ∞ such that − ln cos ( x ) c ( k ) x for 0 x k. Taking k = 2 λd √ γT we mayapply this estimate to Eq. (4.41) and combine the result with Eq. (4.40) to find E h e λ d hh M P · ,f ℓ i C i T i n Y l =1 exp (cid:0) c ( k ) 4 λ d T q l ( P ) (cid:1) = exp (cid:0) c ( k ) 4 λ d T tr ( Q P ) (cid:1) . Since Q P P QP, we havetr Q P tr Q = ∞ X l =1 h Qe l , e l i H = ∞ X j,l =1 (cid:12)(cid:12) h ω ( e l , e j ) , c i C (cid:12)(cid:12) = kh ω ( · , · ) , c i C k < ∞ . Combining the last two equations (recalling that c = f ℓ ) then shows,(4.42) E h e λ d hh M P · ,f ℓ i C i T i exp (cid:16) c ( k ) 4 λ d T kh ω ( · , · ) , f ℓ i C k (cid:17) . EAT KERNELS ON INFINITE-DIMENSIONAL HEISNEBERG GROUPS 31
Using this estimate back in Eq. (4.36) gives, E h e λ k M Pt k C i c ( k ) 2 λ dT d X ℓ =1 kh ω ( · , · ) , f ℓ i C k ! (4.43) = 2 exp (cid:16) c ( k ) 2 λ dT k ω k (cid:17) which completes the proof as this last estimate is independent of P ∈ Proj ( W ) . (cid:3) Proposition 4.14.
Suppose that ν and µ are Gaussian measures on W such q ν ( f ) := ν (cid:0) f (cid:1) q µ ( f ) := µ (cid:0) f (cid:1) for all f ∈ W ∗ R . If g : [0 , ∞ ) → [0 , ∞ ) is anon-negative, non-decreasing, C – function, then Z W g ( k w k ) dν ( w ) Z W g ( k w k ) dµ ( w ) . Proof.
Theorem 3.3.6 in [8, p. 107] 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 : k w k < t } is convex and balanced, it follows that µ ( B t ) ν ( B t ) or equiv-alently that 1 − ν ( B t ) − µ ( B t ) for all t > . Since Z W g ( k w k ) dν ( w ) = Z W (cid:20) g (0) + Z ∞ t k w k g ′ ( t ) dt (cid:21) dν ( w )= g (0) + Z ∞ (cid:18) g ′ ( t ) Z W t k w k dν ( w ) (cid:19) dt = g (0) + Z ∞ g ′ ( t ) [1 − ν ( B t )] dt (4.44)with the same formula holding when ν is replaced by µ, it follows that Z W g ( k w k ) dν ( w ) = g (0) + Z ∞ g ′ ( t ) [1 − ν ( B t )] dt g (0) + Z ∞ dtg ′ ( t ) [1 − µ ( B t )] = Z W g ( k w k ) dµ ( w ) . (cid:3) Definition 4.15.
Let ρ : G → [0 , ∞ ) be defined as ρ ( w, c ) := k w k W + k c k C . In analogy to Gross’ theory of measurable semi-norms (see e.g. Definition 5 in[27]) in the abstract Wiener space setting and in light of Theorem 3.12, we view ρ as a “measurable” extension of d G CM Theorem 4.16 (Integrated Gaussian heat kernel bounds) . There exists a δ > such that for all ε ∈ (0 , δ ) , T > , p ∈ [1 , ∞ ) , (4.45) sup P ∈ Proj( W ) E h e εT ρ ( g P ( T )) i < ∞ and Z G e εT ρ ( g ) dν T ( g ) < ∞ whenever ε < δ. Proof.
Let ε ′ := ε/T. For P ∈ Proj ( W ) ,ρ ( g P ( T )) k B P ( T ) k W + k B ( T ) k C + 12 k N P ( T ) k C , where N P ( T ) := R T ω ( B P ( t ) , dB P ( t )) and therefore, E h e ε ′ ρ ( g P ( T )) i E h e ε ′ [ k B P ( T ) k W + k N P ( T ) k C ] i · E h e ε ′ k B ( T ) k C i . Moreover, by H¨older’s inequality we have, E h e ε ′ ρ ( g P ( T )) i E h e ε ′ k B ( T ) k C i r E h e ε ′ k B P ( T ) k W i · E (cid:2) e ε ′ k N P ( T ) k C (cid:3) E h e ε ′ k B ( T ) k C i s E h e ε ′ k B ( T ) k W i · sup P ′ ∈ Proj( W ) E (cid:2) e ε ′ k N P ′ ( T ) k C (cid:3) . wherein we have made use of Proposition 4.14 to conclude that E h e ε ′ k B P ( T ) k W i E h e ε ′ k B ( T ) k W i = E h e ε ′ T k B (1) k W i which is finite by Fernique’s theorem provided 2 ε = 2 ε ′ T < δ ′ for some δ ′ > . Similarly by Proposition 4.13,sup P ′ ∈ Proj( W ) E h e ε ′ k N P ′ ( T ) k C i < ∞ provided ε = ε ′ T < π √ γ . The assertion in Equation (4.45) now follows from theseobservations and the fact that E h e ε ′ k B ( T ) k C i < ∞ for all ε ′ > . (cid:3) Path space quasi-invariance
Notation 5.1.
Let W T ( G ) denote the collection of continuous paths, g : [0 , T ] → G such that g (0) = e . Moreover, if V is a separable Hilbert space, let H T ( V ) denote the collection of absolutely continuous functions (see [14, pages 106-107] ), h : [0 , T ] → V such that h (0) = 0 and k h k H T ( V ) := Z T (cid:13)(cid:13)(cid:13) ˙ h ( t ) (cid:13)(cid:13)(cid:13) V dt ! / < ∞ . By polarization, we endow H T ( V ) with the inner product h h, k i H T ( V ) = Z T D ˙ h ( t ) , ˙ k ( t ) E V dt. Theorem 5.2 (Path space quasi-invariance) . Suppose
T > , k ( · ) = ( A ( · ) , a ( · )) ∈H T ( g CM ) (thought of as a finite energy path in G CM ) , and g ( · ) is the G –valuedBrownian motion in Equation (4.10) . Then over the finite time interval, [0 , T ] , the laws of k · g and g are equivalent, i.e. they are mutually absolutely continuousrelative to one another. More precisely, if F : W T ( G ) → [0 , ∞ ] is a measurablefunction, then (5.1) E [ F ( k · g )] = E h ˜ Z k ( B, B ) F ( g ) i , where (5.2)˜ Z k ( B, B ) := exp R T D ˙ A ( t ) , dB ( t ) E H − R T (cid:13)(cid:13)(cid:13) ˙ A ( t ) (cid:13)(cid:13)(cid:13) H dt + R T D ˙ a ( t ) + ω (cid:16) A ( t ) − B ( t ) , ˙ A ( t ) (cid:17) , dB ( t ) E C − R T (cid:13)(cid:13)(cid:13) ˙ a ( t ) + ω (cid:16) A ( t ) − B ( t ) , ˙ A ( t ) (cid:17)(cid:13)(cid:13)(cid:13) C dt . EAT KERNELS ON INFINITE-DIMENSIONAL HEISNEBERG GROUPS 33
Moreover, Equation (5.2) is valid for all measurable functions, F : W T ( G ) → C such that E [ | F ( k · g ) | ] = E h ˜ Z k ( B, B ) | F ( g ) | i < ∞ . Proof.
The Cameron–Martin theorem states (see for example, [36, Theorem 1.2 onp. 113]) that(5.3) E [ F ( B, B )] = E [ Z k ( B, B ) F (( B, B ) − k )] , where(5.4) Z k ( B, B ) := exp R T hD ˙ A ( t ) , dB ( t ) E H + h ˙ a ( t ) , dB ( t ) i C i − R T (cid:20)(cid:13)(cid:13)(cid:13) ˙ A ( t ) (cid:13)(cid:13)(cid:13) H + k ˙ a ( t ) k C (cid:21) dt . Since ( k · g ) ( t ) = (cid:18) B ( t ) + A ( t ) , B ( t ) + a ( t ) + 12 Z t ω ( B ( τ ) , dB ( τ )) + 12 ω ( A ( t ) , B ( t )) (cid:19) (5.5)is mapped to (cid:18) B ( t ) , B ( t ) + 12 Z t ω (( B − A ) ( τ ) , d ( B − A ) ( τ )) + 12 ω ( A ( t ) , ( B − A ) ( t )) (cid:19) under the transformation B → B − A and B → B − a, we may conclude fromEquation (5.3) that(5.6) E [ F ( k · g )] = E [ Z k ( B, B ) F ( B, B + c )] , where c ( t ) = 12 Z t ω (( B − A ) ( τ ) , d ( B − A ) ( τ )) + 12 ω ( A ( t ) , ( B − A ) ( t )) . By taking the differential of c, one easily shows that c ( t ) = 12 Z t ω ( B ( τ ) , dB ( τ )) + u B ( t ) , where(5.7) u B ( t ) := 12 Z t ω (cid:16) A ( τ ) − B ( τ ) , ˙ A ( τ ) (cid:17) dτ. Hence Equation (5.6) may be rewritten as(5.8) E [ F ( k · g )] = E (cid:20) Z k ( B, B ) F (cid:18) B, B + u B + 12 Z · ω ( B ( t ) , dB ( t )) (cid:19)(cid:21) . Freezing the integration over B (i.e. using Fubini’s theorem) we may use theCameron-Martin theorem one more time to make the transformation, B → B − u B . Doing so gives E [ F ( k · g )] = E (cid:20) ˜ Z k ( B, B ) F (cid:18)(cid:18) B, B + 12 Z · ω ( B ( t ) , dB ( t )) (cid:19)(cid:19)(cid:21) = E h ˜ Z k ( B, B ) F ( g ) i , (5.9) where(5.10)˜ Z k ( B, B ) := Z k ( B, B − u B ) exp Z T h ˙ u B ( t ) , dB ( t ) i C − Z T k ˙ u B ( t ) k C dt ! . A little algebra shows that ˜ Z k ( B, B ) defined in Equation (5.10) may be expressedas in Equation (5.2). (cid:3) Remark 5.3.
The above proof fails if we try to use it to prove the right quasi-invariance on the path space measure, i.e. that g · k has a law which is absolutelycontinuous to that of g. In this case( g · k ) ( t ) = (cid:18) B ( t ) + A ( t ) , B ( t ) + a ( t ) + 12 Z t ω ( B ( τ ) , dB ( τ )) − ω ( A ( t ) , B ( t )) (cid:19) and then making the transformation, B → B − A and B → B − a gives E [ F ( g · k )] = E [ Z k ( B, B ) F ( B, B + c )]where c ( t ) = 12 Z t ω ( B ( τ ) , dB ( τ )) + u B ( t )and u B ( t ) = 12 Z t [ ω ( A, dA ) − ω ( A, dB )] . The argument breaks down at this point since u B is no longer absolutely continuousin t. Hence we can no longer use the Cameron – Martin theorem to translate awaythe u B term. Proposition 5.4.
There exists a δ > and a function C ( p, u ) ∈ (0 , ∞ ] , for < p < ∞ and u < ∞ , which is non-decreasing in each of its variables, C ( p, u ) < ∞ whenever (5.11) p (cid:16) p δ/u (cid:17) , and, (5.12) E h ˜ Z k ( B, B ) p i C (cid:16) p, k k k H T ( g CM ) (cid:17) for all k ∈ H T ( g CM ) . Proof.
For the purposes of this proof, let E B and E B denote the expectation rela-tive to B and B respectively, so that by Fubini’s theorem E = E B E B = E B E B , We may write ˜ Z k ( B, B ) as˜ Z k ( B, B ) := ζ ( B ) exp Z T h ˙ a ( t ) + ˙ u B ( t ) , dB ( t ) i C ! where ζ ( B ) := exp Z T D ˙ A ( t ) , dB ( t ) E H − Z T (cid:13)(cid:13)(cid:13) ˙ A ( t ) (cid:13)(cid:13)(cid:13) H dt − Z T k ˙ a ( t ) + ˙ u B ( t ) k C dt ! and u B ( t ) is as in Equation (5.7). Hence it follows that, E B h ˜ Z k ( B, B ) p i = ζ p ( B ) E B " exp p Z T h ˙ a ( t ) + ˙ u B ( t ) , dB ( t ) i C ! EAT KERNELS ON INFINITE-DIMENSIONAL HEISNEBERG GROUPS 35 = ζ p ( B ) exp p Z T k ˙ a ( t ) + ˙ u B ( t ) k C dt ! = U V (5.13)where U := exp p Z T D ˙ A ( t ) , dB ( t ) E H − Z T (cid:13)(cid:13)(cid:13) ˙ A ( t ) (cid:13)(cid:13)(cid:13) H dt !! and V = exp p − p Z T k ˙ a ( t ) + ˙ u B ( t ) k C dt ! . Note that when p = 1 , Equation (5.13) becomes E B h ˜ Z k ( B, B ) i = exp Z T D ˙ A ( t ) , dB ( t ) E H − Z T (cid:13)(cid:13)(cid:13) ˙ A ( t ) (cid:13)(cid:13)(cid:13) H dt ! , from which it easily follows that E h ˜ Z k ( B, B ) i = E B E B h ˜ Z k ( B, B ) i = 1 . Now suppose that p > . By the Cauchy – Schwarz inequality, E h ˜ Z k ( B, B ) p i = E B [ U V ] (cid:0) E B (cid:2) V (cid:3)(cid:1) / (cid:0) E B (cid:2) U (cid:3)(cid:1) / . Because E U = exp − p Z T (cid:13)(cid:13)(cid:13) ˙ A ( t ) (cid:13)(cid:13)(cid:13) H dt ! E " exp p Z T D ˙ A ( t ) , dB ( t ) E H ! = exp (cid:16) p k A k H T ( H ) (cid:17) exp (cid:16) p k k k H T ( g CM ) (cid:17) < ∞ , we have reduced the problem to estimating E V . By elementary estimates we have k ˙ u B ( t ) k C = 14 (cid:13)(cid:13)(cid:13) ω (cid:16) A ( t ) − B ( t ) , ˙ A ( t ) (cid:17)(cid:13)(cid:13)(cid:13) C k ω k k A ( t ) − B ( t ) k W (cid:13)(cid:13)(cid:13) ˙ A ( t ) (cid:13)(cid:13)(cid:13) W k ω k (cid:13)(cid:13)(cid:13) ˙ A ( t ) (cid:13)(cid:13)(cid:13) W (cid:16) k A ( t ) k W + 4 k B ( t ) k W (cid:17) and hence k ˙ a ( t ) + ˙ u B ( t ) k k ˙ a ( t ) k C + 2 k ˙ u B ( t ) k C = 2 k ˙ a ( t ) k C + k ω k (cid:13)(cid:13)(cid:13) ˙ A ( t ) (cid:13)(cid:13)(cid:13) W (cid:18) k A ( t ) k W + 4 sup t T k B ( t ) k W (cid:19) . (5.14)By Equation (2.10) there exits c < ∞ such that k·k W c k·k H . Since k A ( t ) k H Z T (cid:13)(cid:13)(cid:13) ˙ A ( τ ) (cid:13)(cid:13)(cid:13) H dτ √ T k A k H T ( H ) , we find V C exp (cid:18) (cid:0) p − p (cid:1) c k ω k k A k H T ( H ) sup t T k B ( t ) k W (cid:19) where C = exp (cid:16)(cid:0) p − p (cid:1) (cid:16) k a k H T ( C ) + c T k ω k k A k H T ( H ) (cid:17)(cid:17) C ′ (cid:16) p, k k k H T ( g CM ) (cid:17) < ∞ . Now by Fernique’s theorem as in Equation 2.4 there exists δ ′ > M := E (cid:20) exp (cid:18) δ ′ sup t T k B ( t ) k W (cid:19)(cid:21) < ∞ and hence it follows that E V C ′ (cid:16) p, k k k H T ( g CM ) (cid:17) · M < ∞ provided4 (cid:0) p − p (cid:1) c k ω k k A k H T ( H ) (cid:0) p − p (cid:1) c k ω k k k k H T ( g CM ) δ ′ . The latter condition holds provided p q δ/ k k k H T ( g CM ) δ := (cid:16) c k ω k (cid:17) − δ ′ > . (cid:3) Definition 5.5.
We will say that a function, F : W T ( G ) → R ( W T ( G ) as inNotation 5.1) is polynomially bounded if there exist constants K, M < ∞ suchthat(5.15) | F ( g ) | K t ∈ [0 ,T ] k g ( t ) k G ! M for all g ∈ W T ( G ) . Given a finite energy path, k ( t ) = ( A ( t ) , a ( t )) ∈ g CM , we say that F is right k differentiable if dds (cid:12)(cid:12)(cid:12) F (( sk ) · g ) =: (cid:16) ˆ kF (cid:17) ( g )exists for all g ∈ W T ( G ) . Corollary 5.6 (Path space integration by parts) . Let k ( · ) = ( A ( · ) , a ( · )) ∈H T ( g CM ) and F : W T ( G ) → R be a k – differentiable function such that F and ˆ kF are polynomial bounded functions on W T ( G ) . Then (5.16) E h(cid:16) ˆ kF (cid:17) ( g ) i = E [ F ( g ) z k ] where (5.17) z k := Z T hD ˙ A ( t ) , dB ( t ) E H + D ˙ a ( t ) − ω (cid:16) B ( t ) , ˙ A ( t ) (cid:17) , dB ( t ) E C i . Moreover, E | z k | p < ∞ for all p ∈ [1 , ∞ ) . Proof.
From Theorem 5.2, we have that for any s ∈ R (5.18) E [ F (( sk ) · g )] = E h ˜ Z sk ( B, B ) F ( g ) i . Formally differentiating this identity at s = 0 and interchanging the derivativeswith the expectations immediately leads to Equation (5.16). To make this rigorouswe need only to verify that derivative interchanges are permissible. From Equations(3.8) and (5.15), there exists C ( k ) < ∞ such thatsup | s | (cid:12)(cid:12)(cid:12)(cid:12) dds F (( sk ) · g ) (cid:12)(cid:12)(cid:12)(cid:12) = sup | s | (cid:12)(cid:12)(cid:12)(cid:16) ˆ kF (cid:17) (( sk ) · g ) (cid:12)(cid:12)(cid:12) EAT KERNELS ON INFINITE-DIMENSIONAL HEISNEBERG GROUPS 37 K sup | s | t ∈ [0 ,T ] k [ sk ( t )] · g ( t ) k G ! M C ( k ) t ∈ [0 ,T ] k g ( t ) k G ! M wherein the last expression is integrable by Fernique’s theorem and the momentestimate in Proposition 4.1. Therefore, dds | E [ F (( sk ) · g )] = E (cid:20) dds | F (( sk ) · g ) (cid:21) = E h(cid:16) ˆ kF (cid:17) ( g ) i . To see that we may also differentiate the right side of Equation (5.18), observethat ˜ Z sk ( B, B ) = exp (cid:0) sz k + s β + s γ + s κ (cid:1) where β = − Z T (cid:13)(cid:13)(cid:13) ˙ A ( t ) (cid:13)(cid:13)(cid:13) H dt + 12 Z T D ω (cid:16) A ( t ) , ˙ A ( t ) (cid:17) , dB ( t ) E C − Z T (cid:13)(cid:13)(cid:13) ˙ a ( t ) − ω (cid:16) B ( t ) , ˙ A ( t ) (cid:17)(cid:13)(cid:13)(cid:13) C dt,γ = − Z T Re D ˙ a ( t ) − ω (cid:16) B ( t ) , ˙ A ( t ) (cid:17) , ω (cid:16) A ( t ) , ˙ A ( t ) (cid:17)E C dt, and κ = − Z T (cid:13)(cid:13)(cid:13) ω (cid:16) A ( t ) , ˙ A ( t ) (cid:17)(cid:13)(cid:13)(cid:13) C dt. Using Fernique’s theorem again and estimates similar to those used in the proof ofProposition 5.4, one shows for any p ∈ [1 , ∞ ) that there exists s ( p ) > E " sup | s | s ( p ) (cid:12)(cid:12)(cid:12)(cid:12) dds ˜ Z sk ( B, B ) (cid:12)(cid:12)(cid:12)(cid:12) p < ∞ . Therefore we may differentiate past the expectation to find dds | E h F ( g ) ˜ Z sk ( B, B ) i = E (cid:20) F ( g ) dds | ˜ Z sk ( B, B ) (cid:21) = E [ F ( g ) z k ] . The fact that z k has finite moments of all orders follows by the martingale argumentsalong with Nelson’s theorem as described in the proof of Lemma 4.7. Alternatively,observe that R T D ˙ A ( t ) , dB ( t ) E H is Gaussian and hence has finite moments of allorders. If we let M t := R t D ˙ a − ω (cid:16) B, ˙ A (cid:17) , dB E C , then M is a martingale suchthat h M i T = Z T (cid:13)(cid:13)(cid:13) ˙ a ( t ) − ω (cid:16) B ( t ) , ˙ A ( t ) (cid:17)(cid:13)(cid:13)(cid:13) C dt C (cid:18) t T k B ( t ) k W (cid:19) . So by Fernique’s theorem, E [ h M i pT ] < ∞ for all p < ∞ and hence by theBurkholder-Davis-Gundy inequalities, E | M T | p < ∞ for all 1 p < ∞ . (cid:3) Heat Kernel Quasi-Invariance
In this section we will use the results of Section 5 to prove both quasi-invarianceof the heat kernel measures, { ν T } T > , relative to left and right translations byelements of G CM . Theorem 6.1 (Left quasi-invariance of the heat kernel measure) . Let
T > and ( A, a ) ∈ G CM . Then ( A, a ) · g ( T ) and g ( T ) have equivalent laws. More precisely,if f : G → [0 , ∞ ] is a measurable function, then (6.1) E [ f (( A, a ) · g ( T ))] = E (cid:2) f ( g ( T )) ¯ Z k ( g ( T ) (cid:3) , where (6.2) ¯ Z k ( g ( T )) = E (cid:2) ζ ( A,a ) ( B, B ) (cid:12)(cid:12) σ ( g ( T )) (cid:3) and ln ζ ( A,a ) ( B, B ) := 1 T h A, B ( T ) i H − k A k H T + 1 T Z T h a − ω ( B ( t ) , A ) , dB ( t ) i C − T Z T k a − ω ( B ( t ) , A ) k C dt. (6.3) Proof.
An application of Theorem 5.2 with F ( g ) := f ( g ( T )) and k ( t ) := tT ( A, a )implies E [ f (( A, a ) · g ( T ))] = E [ F ( k · g )] = E h ˜ Z k ( B, B ) · F ( g ) i = E h ˜ Z k ( B, B ) f ( g ( T )) i , (6.4)where after a little manipulation one shows, ˜ Z k ( B, B ) = ζ ( A,a ) ( B, B ) . By con-ditioning on σ ( g ( T )) we can also write Equation (6.4) as in Equation (6.1). (cid:3) Corollary 6.2 (Right quasi-invariance of the heat kernel measure) . The heat kernelmeasure, ν T , is also quasi-invariant under right translations, and (6.5) dν T ◦ r − k dν T ( g ) = ¯ Z k − (cid:0) g − (cid:1) . where ¯ Z k = dν T ◦ l − k /dν T is as in Theorem 6.1.Proof. Recall from Corollary 4.9 that ν T is invariant under the inversion map, g → g − . From this observation and Theorem 6.1 it follows that ν T is also quasi-invariant under right translations of elements of G CM . In more detail, if k ∈ G CM and f : G → R is a bounded measurable function, then Z G f ( g · k ) dν T ( g ) = Z G f (cid:0) g − · k (cid:1) dν T ( g ) = Z G f (cid:16)(cid:0) k − g (cid:1) − (cid:17) dν T ( g )= Z G f (cid:0) g − (cid:1) ¯ Z k − ( g ) dν T ( g ) = Z G f ( g ) ¯ Z k − (cid:0) g − (cid:1) dν T ( g ) . Equation (6.5) is a consequence of this identity. (cid:3)
Just like in the case abstract Wiener spaces we have the following strong con-verses of Theorem 6.1 and Corollary 6.2.
EAT KERNELS ON INFINITE-DIMENSIONAL HEISNEBERG GROUPS 39
Proposition 6.3.
Suppose that k ∈ G \ G CM and T > , then ν T ◦ l − k and ν T aresingular and ν T ◦ r − k and ν T are singular.Proof. Let k = ( A, a ) ∈ G \ G CM with a ∈ C and A ∈ W \ H. Given a measurablesubset, V ⊂ W, we have ν T ( V × C ) = P ( B ( T ) ∈ V ) =: µ T ( V ) , where µ T is Wiener measure on W with variance T. It is well known (see e.g.Corollary 2.5.3 in [8]) that if A ∈ W \ H that µ T ( · − A ) is singular relative to µ T ( · ) , i.e. we may partition W into two disjoint measurable sets, W and W suchthat µ T ( W ) = 1 = µ T ( W − A ) . A simple computation shows for any V ⊂ W that l − k ( V × C ) = r − k ( V × C ) = ( V − A ) × C . Thus if we define G i := W i × C for i = 0 , , we have that G is the disjoint union of G and G and ν T ( G ) = µ T ( W ) = 1 while ν T (cid:0) r − k ( G ) (cid:1) = ν T (cid:0) l − k ( G ) (cid:1) = ν T (( W − A ) × C ) = µ T ( W − A ) = 1 . (cid:3) Corollary 6.4 (Right heat kernel integration by parts) . Let k := ( A, a ) ∈ g CM andsuppose that f : G → C is a smooth function such that f and ˆ kf are polynomiallybounded. Then E h(cid:16) ˆ kf (cid:17) ( g ( T )) i = E [ f ( g ( T )) z k ] where ˆ kf ( g ) := dds (cid:12)(cid:12)(cid:12) f (( sk ) g ) and z k := 1 T " h A, B ( T ) i H + h a, B ( T ) i C − Z T h ω ( B ( t ) , A ) , dB ( t ) i C . Moreover, with ν T := Law ( g ( T )) , the above formula gives, Z G (cid:16) ˆ kf (cid:17) dν T ( g ) = Z G f ( g ) ¯ z k ( g ) dν T ( g ) , where (6.6) ¯ z k ( g ( T )) := E ( z k | σ ( g ( T ))) . Proof.
This is a special case of Corollary 5.6, with k ( t ) := tT ( A, a ) and F ( g ) := f ( g ( T )) . (cid:3) Corollary 6.5 (Left heat kernel integration by parts) . Let k := ( A, a ) ∈ g CM andsuppose that f : G → C is a smooth function such that f and ˜ kf are polynomiallybounded. Then Z G (cid:16) ˜ kf (cid:17) dν T ( g ) = Z G f ( g ) ¯ z lk ( g ) dν T ( g ) , where ˜ kf ( g ) := dds (cid:12)(cid:12)(cid:12) f ( g ( sk )) and (6.7) ¯ z lk ( g ) = − ¯ z k (cid:0) g − (cid:1) where ¯ z k is defined in Equation (6.6) . Proof.
Let u ( g ) := f (cid:0) g − (cid:1) so that f ( g ) = u (cid:0) g − (cid:1) . Then (cid:16) ˜ kf (cid:17) ( g ) = dds | f ( g · ( sk )) = dds | u (cid:0) ( − sk ) · g − (cid:1) = − (cid:16) ˆ ku (cid:17) (cid:0) g − (cid:1) . Therefore by Corollary 6.4 and two uses of Corollary 4.9 we find Z G (cid:16) ˜ kf (cid:17) dν T ( g ) = − Z G (cid:16) ˆ ku (cid:17) (cid:0) g − (cid:1) dν T ( g ) = − Z G (cid:16) ˆ ku (cid:17) ( g ) dν T ( g )= − Z G u ( g ) ¯ z k ( g ) dν T ( g ) = − Z G f (cid:0) g − (cid:1) ¯ z k ( g ) dν T ( g )= − Z G f ( g ) ¯ z k (cid:0) g − (cid:1) dν T ( g ) . (cid:3) Definition 6.6. A cylinder polynomial is a cylinder function, f = F ◦ π P : G → C , where P ∈ Proj ( W ) and F is a real or complex polynomial function on P H × C . Corollary 6.7 (Closability of the Dirichlet Form) . Given real–valued cylindricalpolynomials, u, v on G, let E T ( u, v ) := Z G h grad u, grad v i H dν T , where grad u : G → g CM is the gradient of u defined by h grad u, k i g CM = ˜ ku for all k ∈ g CM . Then E T is closable and its closure, E T , is a Dirichlet form on Re L ( G, ν T ) . Proof.
The closability of E T is equivalent to the closability of the gradient operator,grad : L ( ν T ) → L ( ν T ) ⊗ g CM , with the domain, D (grad) , being the space ofcylinder polynomials on G. To check the latter statement it suffices to show thatgrad has a densely defined adjoint which is easily accomplished. Indeed, if k ∈ g CM and u and v are cylinder polynomials, then h grad u, v · k i L ( ν T ) ⊗ g CM = Z G ˜ ku · v dν T = Z G h ˜ k ( u · v ) − u · ˜ kv i dν T = D u, − ˜ kv + ¯ z lk v E L ( ν T ) , wherein we have used the product rule in the second equality and Corollary 6.5for the third. This shows that v · k is contained in the domain of grad ∗ andgrad ∗ ( v · k ) = − ˜ kv + ¯ z lk v, where z lk is as in Eq. (6.7). This completes the proofsince linear combination of functions of the form v · k with k ∈ g CM and v being acylinder polynomial is dense in L ( ν T ) ⊗ g CM . (cid:3) The Ricci Curvature on Heisenberg type groups
In this section we compute the Ricci curvature for G ( ω ) and its finite-dimensionalapproximations. This information will be used in Section 8 to prove a logarithmicSobolev inequality for ν T and to get detailed L p –bounds on the Radon-Nikodymderivatives of ν T under translations by elements from G CM . EAT KERNELS ON INFINITE-DIMENSIONAL HEISNEBERG GROUPS 41
Notation 7.1.
Let ( W, H, ω ) be as in Notation 3.1, P ∈ Proj ( W ) , and G P = P W × C ⊂ G CM as in Notation 3.24. We equip G P with the left invariant Rie-mannian metric induced from restriction of the (real part of the) inner product on g CM = H × C to Lie ( G P ) = P H × C . Further, let
Ric P denote the associated Riccitensor at the identity in G P . Proposition 7.2. If ( W, H, ω, P ) as in Notation 7.1, P ∈ Proj ( W ) is as in Eq.(3.42), and ( A, a ) ∈ P H × C , then (cid:10) Ric P ( A, a ) , ( A, a ) (cid:11) H × C = 14 n X j,k =1 (cid:12)(cid:12) h ω ( e k , e j ) , a i C (cid:12)(cid:12) − n X k =1 k ω ( A, e k ) k C (7.1) = 14 kh ω ( · , · ) , a i C k P H ) ∗ ⊗ ( P H ) ∗ − k ω ( A, · ) k P H ) ∗ ⊗ C . (7.2) Proof.
We are going to compute Ric P using the formula in Equation (B.3) of Ap-pendix B. If { f ℓ } dim C ℓ =1 is an orthonormal basis for C , then(7.3) n X k =1 (cid:13)(cid:13) ad ( e k , ( A, a ) (cid:13)(cid:13) H × C + dim C X ℓ =1 (cid:13)(cid:13) ad (0 ,f ℓ ) ( A, a ) (cid:13)(cid:13) H × C = n X k =1 k ω ( e k , A ) k C . If (
B, b ) ∈ P H × C , then ad ∗ ( B,b ) ( A, a ) = n X j =1 D ad ∗ ( B,b ) ( A, a ) , ( e j , E g CM ( e j , dim C X ℓ =1 D ad ∗ ( B,b ) ( A, a ) , (0 , f ℓ ) E g CM (0 , f ℓ )= n X j =1 h ( A, a ) , [( B, b ) , ( e j , i g CM ( e j , dim C X ℓ =1 h ( A, a ) , [( B, b ) , (0 , f ℓ )] i g CM (0 , f ℓ )= n X j =1 h ( A, a ) , (0 , ω ( B, e j )) i g CM ( e j ,
0) = n X j =1 ( a, ω ( B, e j )) C ( e j , . This then immediately implies(7.4) n X k =1 (cid:13)(cid:13)(cid:13) ad ∗ ( e k , ( A, a ) (cid:13)(cid:13)(cid:13) g CM + dim C X ℓ =1 (cid:13)(cid:13)(cid:13) ad ∗ (0 ,f ℓ ) ( A, a ) (cid:13)(cid:13)(cid:13) g CM = n X k =1 n X j =1 h a, ω ( e k , e j ) i C . Using Equations (7.3) and (7.4) with the formula for the Ricci tensor in Equation(B.3) of Appendix B implies Equation (7.1). (cid:3)
Corollary 7.3.
For P ∈ Proj ( W ) as in (3.42) , let (7.5) k P ( ω ) := −
12 sup n k ω ( · , A ) k P H ) ∗ ⊗ C : A ∈ P H, k A k P H = 1 o . Also let (7.6) k ( ω ) := −
12 sup k A k H =1 k ω ( · , A ) k H ∗ ⊗ C > − k ω k > −∞ . Then k P ( ω ) is the largest constant k ∈ R such that (7.7) (cid:10) Ric P ( A, a ) , ( A, a ) (cid:11) P H × C > k k ( A, a ) k P H × C for all ( A, a ) ∈ P H × C and k ( ω ) is the largest constant k ∈ R such that Equation (7.7) holds uniformlyfor all P ∈ Proj ( W ) . Proof.
Let us observe that by Equation (7.1) (cid:10)
Ric P ( A, a ) , ( A, a ) (cid:11) P H × C k ( A, a ) k P H × C > (cid:10) Ric P ( A, , ( A, (cid:11) P H × C k ( A, k P H × C the optimal lower bound, k P ( ω ) , for Ric p is determined by k P ( ω ) = inf A ∈ P H \{ } (cid:10) Ric P ( A, , ( A, (cid:11) P H × C k ( A, k P H × C = inf A ∈ P H \{ } − k ω ( · , A ) k P H ) ∗ ⊗ C k A k P H ! which is equivalent to Equation (7.5). It is now a simple matter to check that k ( ω ) = inf P ∈ Proj( W ) k P ( ω ) which is the content of the last assertion of the theorem. (cid:3) In revisiting the examples from Section 3.3 we will have a number of cases where H and C are complex Hilbert spaces and ω : H × H → C will be a complex bilinearform. In these cases it will be convenient to express the Ricci curvature in terms ofthese complex structures. Proposition 7.4.
Suppose that H and C are complex Hilbert spaces, ω : H × H → C is complex bi-linear, and P : H → H is a finite rank (complex linear) orthogonalprojection. We make G P = P H × C into a Lie group using the group law inEquation (3.6) . Letting and endow G P with the left invariant Riemannian metricwhich agrees with h· , ·i [ g P ] Re := Re h· , ·i g P on g P = P H × C at the identity in G P . Then for all ( A, a ) ∈ g P , (cid:10) Ric P ( A, a ) , ( A, a ) (cid:11) [ g P ] Re = 12 kh ω ( · , · ) , a i C k P H ) ∗ ⊗ ( P H ) ∗ − k ω ( A, · ) k P H ) ∗ ⊗ C (7.8) = 12 n X j,k =1 (cid:12)(cid:12) h ω ( e k , e j ) , a i C (cid:12)(cid:12) − n X k =1 k ω ( A, e k ) k C , (7.9) where { e j } nj =1 is any orthonormal basis for P H.
Proof.
Applying Equation (7.2) with
P H, C , and g P being replaced by ( P H ) Re , C Re , and [ g P ] Re implies (cid:10) Ric P ( A, a ) , ( A, a ) (cid:11) [ g P ] Re = 14 (cid:13)(cid:13) h ω ( · , · ) , a i C Re (cid:13)(cid:13) P H ) ∗ Re ⊗ ( P H ) ∗ Re − k ω ( A, · ) k P H ) ∗ Re ⊗ C Re . EAT KERNELS ON INFINITE-DIMENSIONAL HEISNEBERG GROUPS 43
However, by Lemma 3.17 we also know that (cid:13)(cid:13) h ω ( · , · ) , a i C Re (cid:13)(cid:13) P H ) ∗ Re ⊗ ( P H ) ∗ Re = 2 kh ω ( · , · ) , a i C k P H ) ∗ ⊗ ( P H ) ∗ and k ω ( A, · ) k P H ) ∗ Re ⊗ C Re = 2 k ω ( A, · ) k P H ) ∗ ⊗ C which completes the proof of Equation (7.8). (cid:3) Remark 7.5.
By letting n → ∞ in Propositions 7.2 and 7.4, it is reasonable tointerpret the Ricci tensor on G CM to be determined by h Ric (
A, a ) , ( A, a ) i [ g CM ] Re = α F (cid:18) kh a, ω ( · , · ) i C k H ∗ ⊗ H ∗ − k ω ( · , A ) k H ∗ ⊗ C (cid:19) (7.10) = α F ∞ X j,k =1 (cid:12)(cid:12) h a, ω ( e k , e j ) i C (cid:12)(cid:12) − ∞ X k =1 k ω ( e k , A ) k C , (7.11)where { e j } ∞ j =1 is an orthonormal basis for H, F is either R or C and α F is one ortwo respectively. Moreover if C = F , then Equation (7.10) may be written as(7.12) h Ric (
A, a ) , ( A, a ) i [ g CM ] Re = α F (cid:18) k ω ( · , · ) k H ∗ ⊗ H ∗ · | a | − k ω ( · , A ) k H ∗ (cid:19) . Examples revisited.
Using Equation (7.10), it is straight forward to com-pute the Ricci tensor on G for each of the Examples 3.18 – 3.23. Lemma 7.6.
The Ricci tensor for G CM associated to each of the structures intro-duced in Examples 3.18 and 3.19 are given (respectively) by (7.13) h Ric ( z, c ) , ( z, c ) i h n R = nc − k z k C n for all ( z, c ) ∈ C n × R , and (7.14) h Ric ( z, c ) , ( z, c ) i [ h n C ] Re = n | c | − k z k C n for all ( z, c ) ∈ C n × C . Proof.
We omit the proof of this lemma as it can be deduced from the next propo-sition by taking Q = I. (cid:3) Proposition 7.7.
The Ricci tensor for G CM associated to each of the structuresintroduced in Examples 3.20 and 3.21 are given (respectively) by (7.15) h Ric ( h, c ) , ( h, c ) i g CM = 12 h c tr Q − k Qh k H i for all ( h, c ) ∈ H × R , and (7.16) h Ric ( k , k , c ) , ( k , k , c ) i [ g CM ] Re = | c | tr Q − k Qk k K − k Qk k K for all ( k , k , c ) ∈ K × K × R . Proof.
We start with the proof of Equation (7.15). In this case, k ω ( · , h ) k H ∗ Re = ∞ X j =1 h ω ( e j , A ) + ω ( ie j , A ) i = ∞ X j =1 (cid:20)(cid:16) Im h h, e j i Q (cid:17) + (cid:16) Im h h, ie j i Q (cid:17) (cid:21) = ∞ X j =1 (cid:20)(cid:16) Im h h, e j i Q (cid:17) + (cid:16) Re h h, e j i Q (cid:17) (cid:21) = ∞ X j =1 (cid:12)(cid:12)(cid:12) h h, e j i Q (cid:12)(cid:12)(cid:12) = k Qh k H and from Equation (3.36) k ω k = 2 tr (cid:0) Q (cid:1) . Using these results in Equation (7.12)with F = R gives Equation (7.15) with F = C and H = K × K, Equation (7.16)follows from Equation (7.12) with F = C , Equation (3.36), and the following iden-tity; k ω (( k , k ) , · ) k H ∗ = ∞ X j =1 (cid:16) | ω (( k , k ) , ( e j , | + | ω (( k , k ) , (0 , e j )) | (cid:17) = ∞ X j =1 |h k , Q ¯ e j i| + ∞ X j =1 |h k , Q ¯ e j i| = k Qk k K + k Qk k K . (7.17) (cid:3) Proposition 7.8.
The Ricci tensor for G CM associated to the structure introducedin Example 3.22 is given by (7.18) h Ric ( v, c ) , ( v, c ) i [ g CM ] Re = ∞ X j =1 q j h Ric α ( v j , c ) , ( v j , c ) i V Re × C Re ∀ ( v, c ) ∈ H × F , where Ric α denotes the Ricci tensor on G ( α ) := V × C as is defined by Equation (7.19) below.Proof. Using Equation (3.38) along with the identity, k ω ( · , v ) k H ∗ ⊗ C = ∞ X j =1 d X a =1 k ω ( u a ( j ) , v ) k C = ∞ X j =1 d X a =1 q j k α ( u a , v j ) k C = ∞ X j =1 q j k α ( · , v j ) k V ∗ ⊗ C , in Equation (7.10) shows h Ric ( v, c ) , ( v, c ) i [ g CM ] Re = α F ∞ X j =1 q j (cid:18) kh α ( · , · ) , c i C k − k α ( · , v j ) k V ∗ ⊗ C (cid:19) . Moreover, by a completely analogous finite-dimensional application of Equation(7.10), we have(7.19) h Ric α ( v j , c ) , ( v j , c ) i V Re × C Re = α F (cid:18) kh α ( · , · ) , c i C k − k α ( · , v j ) k V ∗ ⊗ C (cid:19) . Combining these two identities completes the proof. (cid:3)
EAT KERNELS ON INFINITE-DIMENSIONAL HEISNEBERG GROUPS 45
Proposition 7.9.
Let α : V × V → C be as in Example 3.23. For v ∈ V, let α v : V → C be defined by α v w = α ( v, w ) and α ∗ v : C → V be its adjoint. TheRicci tensor for G CM associated to the structure introduced in Example 3.23 is thengiven by h Ric ( h, c ) , ( h, c ) i [ g CM ] Re = 12 "Z [0 , ( s ∧ t ) dη ( s ) d ¯ η ( t ) · kh c, α ( · , · ) i C k V ∗ ⊗ V ∗ (7.20) − Z [0 , ( s ∧ t ) tr (cid:16) α ∗ h ( t ) α h ( s ) (cid:17) dη ( s ) d ¯ η ( t ) . Proof.
In this example we have k ω ( h, · ) k H ∗ ⊗ C = ∞ X j =1 d X a =1 k ω ( h, l j u a ) k C = ∞ X j =1 d X a =1 (cid:13)(cid:13)(cid:13)(cid:13)Z α ( h ( s ) , l j ( s ) u a ) dη ( s ) (cid:13)(cid:13)(cid:13)(cid:13) C = ∞ X j =1 d X a =1 (cid:28)Z α ( h ( s ) , l j ( s ) u a ) dη ( s ) , Z α ( h ( t ) , l j ( t ) u a ) dη ( t ) (cid:29) C = Z dη ( s ) Z d ¯ η ( t ) ( s ∧ t ) d X a =1 (cid:10) α h ( s ) u a , α h ( t ) u a (cid:11) C = Z [0 , s ∧ t h tr (cid:16) α ∗ h ( t ) α h ( s ) (cid:17)i dη ( s ) d ¯ η ( t ) . Using this identity along with Equation (3.40) in Equation (7.10) with α F = α C = 2implies Equation (7.20). (cid:3) Heat Inequalities
Infinite-dimensional Radon-Nikodym derivative estimates.
Recallfrom Theorem 6.1 and Corollary 6.2, we have already shown that ν T ◦ l − h and ν T ◦ r − h are absolutely continuous to ν T for all h ∈ G CM and T > . These re-sults were based on the path space quasi-invariance formula given Theorem 5.2.However, in light of the results in Malliavin [40] it is surprising that Theorem 5.2holds at all and we do not expect it to extend to many other situations. Therefore,it is instructive to give an independent proof of Theorem 6.1 and Corollary 6.2which will work for a much larger class of examples. The alternative proof have theadded advantage of giving detailed size estimates on the resulting Randon-Nikodymderivatives.
Theorem 8.1.
For all h ∈ G CM and T > , ν T ◦ l − h and ν T ◦ r − h are abso-lutely continuous to ν T . Let Z lh := d ( ν T ◦ l − h ) dν T and Z rh := d ( ν T ◦ r − h ) dν T be the respectiveRandon-Nikodym derivatives, k ( ω ) is given in Equation (7.6) , and c ( t ) := te t − for all t ∈ R with the convention that c (0) = 1 . Then for all p < ∞ , Z lh and Z rh are both in L p ( ν T ) and satisfy the estimate (8.1) k Z ∗ h k L p ( ν T ) exp (cid:18) c ( k ( ω ) T ) ( p − T d G CM ( e , h ) (cid:19) , where ∗ = l or ∗ = r. Proof.
The proof of this theorem is an application Theorem 7.3 and Corollary 7.4in [18] on quasi-invariance of the heat kernel measures for inductive limits of finite-dimensional Lie groups. In applying these results the reader should take: G = G CM , A = Proj ( W ) , s P := π P , ν P = Law ( g P ( T )) , and ν = ν T = Law ( g ( T )) . We now verify that the hypotheses [18, Theorem 7.3] are satisfied. These assump-tions include a densness condition on the inductive limit group, consistency of theheat kernel measures on finite-dimensional Lie groups, uniform bound on the Riccicurvature, and finally that the length of a path in the inductive limit group can beapproximated by the lengths of paths in finite-dimensional groups.(1) By Proposition 3.10, ∪ P ∈ Proj( W ) G P is a dense subgroup of G CM . (2) From Lemma 4.7, for any { P n } ∞ n =1 ⊂ Proj ( W ) with P n | H ↑ I H and f ∈ BC ( G, R ) (the bounded continuous maps from G to R ) , we have Z G f dν = lim n →∞ Z G Pn ( f ◦ i P ) dν P n . (3) Corollary 7.3 shows that Ric P > k ( ω ) g P for all P ∈ Proj ( W ) . (4) Lastly we have to verify that for any P ∈ Proj ( W ) , and k ∈ C ([0 , , G CM ) with k (0) = e, there exists an increasing sequence, { P n } ∞ n =1 ⊂ Proj ( W ) such that P ⊂ P n , P n ↑ I on H, and(8.2) ℓ G CM ( k ) = lim n →∞ ℓ G Pn ( π n ◦ k ) , where π n := π P n and ℓ G CM ( k ) is the length of k (see Notation 3.9 with T =1). However, with k ( t ) = ( A ( t ) , c ( t )) , using the dominated convergencetheorem applied to the identity (see Equation (3.21)); ℓ G Pn ( π n ◦ k ) = Z (cid:13)(cid:13)(cid:13)(cid:13) π n ˙ k ( t ) − h π n k ( t ) , π n ˙ k ( t ) i(cid:13)(cid:13)(cid:13)(cid:13) g CM dt = Z s(cid:13)(cid:13)(cid:13) P n ˙ A ( t ) (cid:13)(cid:13)(cid:13) H + (cid:13)(cid:13)(cid:13)(cid:13) ˙ c ( t ) − ω (cid:16) P n A ( t ) , P n ˙ A ( t ) (cid:17)(cid:13)(cid:13)(cid:13)(cid:13) C dt shows Equation (8.2) holds for any such choice of P n | H ↑ I H with P ⊂ P n ∈ Proj ( W ) . (cid:3) Remark 8.2.
In the case of infinite-dimensional matrix groups three out of fourassumptions hold as has been shown in [24]. The condition that fails is the uniformbounds on the Ricci curvature which is one of the main results in [25].8.2.
Logarithmic Sobolev Inequality.Theorem 8.3.
Let ( E T , D ( E T )) be the closed Dirichlet form in Corollary 6.7 and k ( ω ) be as in Equation (7.6) . Then for all real-valued f ∈ D ( E T ) , the following EAT KERNELS ON INFINITE-DIMENSIONAL HEISNEBERG GROUPS 47 logarithmic Sobolev inequality holds (8.3) Z G (cid:0) f ln f (cid:1) dν T − e − k ( ω ) T k ( ω ) E T ( f, f ) + Z G f dν T · ln Z G f dν T , where ν T = Law ( g ( T )) is the heat kernel measure on G as in Definition 4.2.Proof. Let f : G → R be a cylinder polynomial as in Definition 6.6. Following themethod of Bakry and Ledoux applied to G P (see [22, Theorem 2.9] for the caseneeded here) shows E (cid:2)(cid:0) f log f (cid:1) ( g P ( T )) (cid:3) − e − k P ( ω ) T k P ( ω ) E (cid:13)(cid:13)(cid:0) grad P f (cid:1) ( g P ( T )) (cid:13)(cid:13) G P + E (cid:2) f ( g P ( T )) (cid:3) log E (cid:2) f ( g P ( T )) (cid:3) (8.4)where k P ( ω ) is as in Equation (7.6). Since the function, x → x − (1 − e − x ) , isdecreasing and k ( ω ) k P ( ω ) for all P ∈ Proj ( W ) , Equation (8.4) also holds with k P ( ω ) replaced by k ( ω ) . With this observation along with Lemma 4.7, we maypass to the limit at P ↑ I in Equation (8.4) to find E (cid:2)(cid:0) f log f (cid:1) ( g ( T )) (cid:3) − e − k ( ω ) T k ( ω ) E | grad f ( g ( T )) | + E (cid:2) f ( g ( T )) (cid:3) log E (cid:2) f ( g ( T )) (cid:3) . This is equivalent to Equation (8.3) when f is a cylinder polynomial. The result forgeneral f ∈ D ( E T ) then holds by a standard (and elementary) limiting argument– see the end of Example 2.7 in [29]. (cid:3) Future directions
In this last section we wish to speculate on a number of ways that the results inthis paper might be extended.(1) It would be interesting to see what happens if we set B to be identicallyzero so that g ( t ) in Equation (4.2) becomes(9.1) g ( t ) = (cid:18) B ( t ) , Z t ω (cid:16) B ( τ ) , ˙ B ( τ ) (cid:17) dτ (cid:19) . The generator now is L = P ∞ k =1 ^ ( e k , and if ω ( g CM × g CM ) is a totalsubset of C , L would satisfy H¨ormander’s condition for hypoellipticity. Ifdim H were finite, it would follow that the heat kernel measure, ν T , is asmooth positive measure and hence quasi-invariant. When dim H is infinitewe do not know if ν T is still quasi-invariant. Certainly both proofs whichwere given above when B was not zero now break down.(2) It should be possible to remove the restriction on C being finite-dimensional, i.e. we expect much of what we have done to go throughwhen C is a separable Hilbert space. In doing so one would have to modifythe finite-dimensional approximations used in the theory to truncate C aswell.(3) It should be possible to widen the class of admissible ω s substantially. Theidea is to assume that ω is only defined from H × H → C such that k ω k < ∞ . Under this relaxed assumption, we will no longer have a group structure on G := W × C . Nevertheless, with a little work one can stillmake sense of Brownian motion process defined in Definition 4.2 by letting(9.2) Z t ω ( B ( τ ) , dB ( τ )) := L – lim n →∞ Z t ω ( P n B ( τ ) , dP n B ( τ )) . In fact, using Nelson’s hypercontractivity and the fact that Z t ω ( P n B ( τ ) , dP n B ( τ ))is in the second homogeneous chaos subspace, the convergence in Equation(9.2) is in L p for all p ∈ [1 , ∞ ) . In this setting we expect the path spacequasi-invariance results of Section 5 to remain valid. Similarly, as the lowerbound on the Ricci curvature only depends on ω | H × H , we expect the re-sults of Section 8.1 to go through as well. As a consequence, G should carrya measurable left and right actions by element of G CM and these actionsshould leave the heat kernel measures (end point distributions of the Brow-nian motion on G ) quasi-invariant. One might call the resulting structurea quasigroup . Unfortunately, this term has already been used in abstractalgebra.(4) We also expect that level of non-commutativity of G may be increased.To be more precise, under suitable hypotheses, it should be possible tohandle more general graded nilpotent Lie groups. However, when the levelof nilpotency of G is increased, there will likely be trouble with the pathspace quasi-invariance in section 5. Nevertheless, the methods of Section8.1 should survive and therefore we still expect the heat kernel measure tobe quasi-invariant. Acknowledgement.
The first author would like to thank the Berkeley mathe-matics department and the Miller Institute for Basic Research in Science for theirsupport of this project in its latter stages.
Appendix A. Wiener Space Results
The well known material presented in this Appendix may be (mostly) found inthe books [36] and [8]. In particular, the following theorem is based in part onLemma 2.4.1 on p. 59 of [8], and Theorem 3.9.6 on p. 138 [8].
Theorem A.1.
Let ( X, B X , µ ) be a Gaussian measure space as in Definition 2.1.Then (1) ( H, k·k H ) is a normed space such that (A.1) k h k X p C k h k H for all h ∈ H, where C is as in (2.2) . (2) Let K be the closure of X ∗ in Re L ( µ ) and for f ∈ K let ιf := h f := Z X xf ( x ) dµ ( x ) ∈ X, where the integral is to be interpreted as a Bochner integral. Then ι ( K ) = H and ι : K → H is an isometric isomorphism of real Banach spaces. Since K is a real Hilbert space it follows that k·k H is a Hilbertian norm on H. EAT KERNELS ON INFINITE-DIMENSIONAL HEISNEBERG GROUPS 49 (3) H is a separable Hilbert space and (A.2) ( ιu, h ) H = u ( h ) for all u ∈ X ∗ and h ∈ H. (4) The Cameron-Martin space, H, is dense in X. (5) The quadratic form q may be computed as (A.3) q ( u, v ) = ∞ X i =1 u ( e i ) v ( e i ) where { e i } ∞ i =1 is any orthonormal basis for H .Notice that by Item 1. H i ֒ → X is continuous and hence so is X ∗ i tr ֒ → H ∗ ∼ = H =( · , · ) H ∗ . Equation (A.3) asserts that q = ( · , · ) H ∗ (cid:12)(cid:12) X ∗ × X ∗ . Proof.
1. Using Equation (2.6) we find k h k X = sup u ∈ X ∗ \{ } | u ( h ) |k u k X ∗ sup u ∈ X ∗ \{ } | u ( h ) | p q ( u, u ) /C p C k h k H , and hence if k h k H = 0 then k h k X = 0 and so h = 0 . If h, k ∈ H, then for all u ∈ X ∗ , | u ( h ) | k h k H p q ( u ) and | u ( k ) | k k k H p q ( u ) so that | u ( h + k ) | | u ( h ) | + | u ( k ) | ( k h k H + k k k H ) p q ( u ) . This shows h + k ∈ H and k h + k k H k h k H + k k k H . Similarly, if λ ∈ R and h ∈ H, then λh ∈ H and k λh k H = | λ | k h k H . Therefore H is a subspace of W and( H, k·k H ) is a normed space.2. For f ∈ K and u ∈ X ∗ (A.4) u ( ιf ) = u Z X xf ( x ) dµ ( x ) = Z X u ( x ) f ( x ) dµ ( x )and hence | u ( ιf ) | k u k L ( µ ) k f k L ( µ ) = p q ( u ) k f k K which shows that ιf ∈ H and k ιf k H k f k K . Moreover, by choosing u n ∈ X ∗ suchthat L ( µ ) − lim n →∞ u n = f, we findlim n →∞ | u n ( ιf ) | p q ( u n ) = lim n →∞ (cid:12)(cid:12)(cid:12)(cid:12)R X u n ( x ) f ( x ) dµ ( x ) (cid:12)(cid:12)(cid:12)(cid:12) k u n k L ( µ ) = k f k L ( µ ) k f k L ( µ ) = k f k L ( µ ) from which it follows k ιf k H = k f k K . So we have shown that ι : K → H isan isometry. Let us now show that ι ( K ) = H. Given h ∈ H and u ∈ X ∗ letˆ h ( u ) = u ( h ) . Since (cid:12)(cid:12)(cid:12) ˆ h ( u ) (cid:12)(cid:12)(cid:12) | u ( h ) | p q ( u ) k h k H = k u k L ( µ ) k h k H = k u k K k h k H the functional ˆ h extends continuously to K. We will continue to denote this exten-sion by ˆ h ∈ K ∗ . Since K is a Hilbert space, there exists f ∈ K such thatˆ h ( u ) = Z X f ( x ) u ( x ) dµ ( x ) for all u ∈ X ∗ (and in fact all u ∈ K ) . Thus we have, for all u ∈ X ∗ , that u ( h ) = Z X u ( x ) f ( x ) dµ ( x ) = u (cid:18)Z X xf ( x ) dµ ( x ) (cid:19) = u ( ιf ) . From this equation we conclude that h = ιf and hence ι ( K ) = H. H is a separable since it is unitarily equivalent to K ⊂ L ( X, B , µ ) and L ( X, B , µ ) is separable. Suppose that u ∈ X ∗ , f ∈ K and h = ιf ∈ H. Then( ιu, h ) H = ( ιu, ιf ) H = ( u, f ) K = Z X u ( x ) f ( x ) dµ ( x ) = u (cid:18)Z X xf ( x ) dµ ( x ) (cid:19) = u ( ιf ) = u ( h ) .
4. For sake of contradiction, if H ⊂ X were not dense, then, by the Hahn–Banachtheorem, there would exist u ∈ X ∗ \{ } such that u ( H ) = 0 . However from Equation(A.2), we would then have q ( u, u ) = ( ιu, ιu ) H = u ( ιu ) = 0 . Because we have assumed that q to be an inner product on X ∗ , u must be zerocontrary to u being in X ∗ \ { } .
5. Let { e i } ∞ i =1 be an orthonormal basis for H, then for u, v ∈ X ∗ ,q ( u, v ) = ( u, v ) K = ( ιu, ιv ) H = ∞ X i =1 ( ιu, e i ) H ( e i , ιv ) H = ∞ X i =1 u ( e i ) v ( e i )wherein the last equality we have again used Equation (A.2). (cid:3) Appendix B. The Ricci tensor on a Lie group
In this appendix we recall a formula for the Ricci tensor relative to a left invariantRiemannian metric, h· , ·i , on any finite-dimensional Lie Group, G. Let ∇ be theLevi-Civita covariant derivative on T G, for any X ∈ g let ˜ X ( g ) = l g ∗ X be the leftinvariant vector field on G such that ˜ X ( e ) = X. , and for X, Y ∈ g , let D X Y := ∇ X ˜ Y ∈ g . Since ∇ ˜ X ˜ Y is a left invariant vector field and (cid:16) ∇ ˜ X ˜ Y (cid:17) ( e ) = ∇ X ˜ Y = D X Y, we have the identity; ∇ ˜ X ˜ Y = ^ D X Y .
Similarly the Ricci curvature tensor, Ric , (and more generally the full curvature tensor) is invariant under left translations,i.e. Ric g = l g − ∗ Ric e l g ∗ for all g ∈ G. Hence it suffices to compute the Ricci tensorat e ∈ G. We will abuse notation and simply write Ric for Ric e . Proposition B.1 (The Ricci tensor on G ) . Continuing the notation above, for all
X, Y ∈ g we have (B.1) D X Y := 12 ([ X, Y ] − ad ∗ X Y − ad ∗ Y X ) ∈ g , where ad ∗ X denotes the adjoint of ad X relative to h· , ·i e . We also have, (B.2) h Ric
X, X i = tr (cid:0) ad ad ∗ X X (cid:1) −
12 tr (cid:0) ad X (cid:1) + 14 X Y ∈ Γ | ad ∗ Y X | − X Y ∈ Γ | ad Y X | , EAT KERNELS ON INFINITE-DIMENSIONAL HEISNEBERG GROUPS 51 where Γ ⊂ g is any orthonormal basis for g . In particular if g is nilpotentthen tr (cid:0) ad ad ∗ X X (cid:1) = 0 and tr (cid:0) ad X (cid:1) = 0 and therefore Equation (B.2) reduces to (B.3) h Ric
X, X i = 14 X Y ∈ Γ | ad ∗ Y X | − X Y ∈ Γ | ad Y X | > − X Y ∈ Γ | ad Y X | . These results may be found in [7], see Lemma 7.27, Theorem 7.30, and Corollary7.33 for the computations of the Levi -Civita covariant derivative, the curvaturetensor, and the Ricci curvature tensor respectively.
Appendix C. Proof of Theorem 3.12
Before giving the proof of Theorem 3.12 it will be necessary to develop the notionCarnot-Carath´eodory distance function, δ, in this infinite dimensional context. Notation C.1.
Let
T > and C HCM denote the horizontal elements in C CM , where g ∈ C CM is horizontal iff l g ( s ) − ∗ g ′ ( s ) ∈ H × { } for all s. We then define, δ ( x, y ) = inf (cid:8) ℓ G CM ( g ) : g ∈ C HCM such that g (0) = x and g ( T ) = y (cid:9) with the infimum of the empty set is taken to be infinite. Observe that δ ( x, y ) > d CM ( x, y ) for all x, y ∈ G CM . The following theoremdescribes the behavior of δ. Theorem C.2. If { ω ( A, B ) :
A, B ∈ H } is a total subset of C , then there exists c ∈ (0 , such that (C.1) c (cid:18) k A k H + q k a k C (cid:19) δ ( e , ( A, a )) c − (cid:18) k A k H + q k a k C (cid:19) for all ( A, a ) ∈ g CM . Proof.
Our proof will be modeled on the standard proof of this result in the finitedimensional context, see for example [50, 42]. The only thing we must be carefulof is to avoid using any compactness arguments.For any left invariant metric, d, (e.g. d = δ or d = d CM ) on G CM we have(C.2) d ( e , xy ) d ( e , x ) + d ( x, xy ) = d ( e , x ) + d ( e , y ) ∀ x, y ∈ G CM . Given any path g = ( w, c ) ∈ C CM joining e to ( A, a ) , we have from Eq. (3.20)that ℓ G CM ( g ) = Z q k w ′ ( s ) k H + k c ′ ( s ) − ω ( w ( s ) , w ′ ( s )) / k C ds > Z k w ′ ( s ) k H ds > k A k H from which it follows that(C.3) δ ( e , ( A, a )) > d CM ( e , ( A, > k A k H . Since the path g ( t ) = ( tA,
0) is horizontal and k A k H = ℓ G CM ( g ) > δ ( e , ( A, > d CM ( e , ( A, > k A k H it follows that(C.4) δ ( e , ( A, d ( e , ( A, k A k H for all A ∈ H. Given
A, B ∈ H, let ξ ( t ) = A cos t + B sin t for 0 t π and g ( t ) = (cid:18) ξ ( t ) − A, Z t ω (cid:16) ξ ( τ ) − A, ˙ ξ ( τ ) (cid:17) dτ (cid:19) so that l g ( t ) − ∗ ˙ g ( t ) = ( ξ ( t ) , , g (0) = e , and g (2 π ) = (cid:18) , Z π ω (cid:16) ξ ( τ ) , ˙ ξ ( τ ) (cid:17) dτ (cid:19) = (cid:18) , Z π ω ( A, B ) dτ (cid:19) = (0 , πω ( A, B )) . From this one horizontal curve we may conclude that δ ( e , (0 , πω ( A, B ))) ℓ G CM ( g ) = Z π k− A sin t + B cos t k H dt π ( k A k H + k B k H ) . (C.5)Choose { A ℓ , B ℓ } dℓ =1 ⊂ H such that { πω ( A ℓ , B ℓ ) } dℓ =1 is a basis for C . Let (cid:8) ε ℓ (cid:9) dℓ =1 be the corresponding dual basis. Hence for any a ∈ C we have δ ( e , (0 , a )) = δ e , d Y ℓ =1 (cid:0) , ε ℓ ( a ) πω ( A ℓ , B ℓ ) (cid:1)! d X ℓ =1 δ (cid:0) e , (cid:0) , ε ℓ ( a ) πω ( A ℓ , B ℓ ) (cid:1)(cid:1) = d X ℓ =1 δ (cid:18) e , (cid:18) , πω (cid:18) sgn( ε ℓ ( a )) q | ε ℓ ( a ) | A ℓ , q | ε ℓ ( a ) | B ℓ (cid:19)(cid:19)(cid:19) π d X ℓ =1 (cid:18)(cid:13)(cid:13)(cid:13)(cid:13)q | ε ℓ ( a ) | A ℓ (cid:13)(cid:13)(cid:13)(cid:13) H + (cid:13)(cid:13)(cid:13)(cid:13)q | ε ℓ ( a ) | B ℓ (cid:13)(cid:13)(cid:13)(cid:13) H (cid:19) , wherein we have used Eq. (C.2) for the first inequality and Eq. (C.5) for the secondinequality. It now follows by simple estimates that(C.6) δ ( e , (0 , a )) C d X ℓ =1 q | ε ℓ ( a ) | C vuut d X ℓ =1 | ε ℓ ( a ) | C ( ω ) q k a k C . for some constants C C C ( ω ) < ∞ . Combining Eqs. (C.2), (C.4), and (C.6)gives, δ ( e , ( A, a )) = δ ( e , ( A,
0) (0 , a )) δ ( e , ( A, δ ( e , (0 , a )) k A k H + C ( ω ) q k a k C . (C.7)To prove the analogous lower bound we will make use of the dilation homomor-phisms defined for each λ > ϕ λ ( w, c ) = (cid:0) λw, λ c (cid:1) for all ( w, c ) ∈ g CM = G CM . One easily verifies that ϕ λ is both a Lie algebra homomorphism on g CM and a grouphomomorphism on G CM . Using the homomorphism property it follows that l ϕ λ ( g ( t )) − ∗ ddt ϕ λ ( g ( t )) = ϕ λ (cid:16) l g ( t ) − ∗ ˙ g ( t ) (cid:17) EAT KERNELS ON INFINITE-DIMENSIONAL HEISNEBERG GROUPS 53 and consequently; if g is any horizontal curve, then ϕ λ ◦ g is again horizontal and ℓ G CM ( ϕ λ ◦ g ) = λℓ G CM ( g ) . From these observations we may conclude that(C.8) δ ( ϕ λ ( x ) , ϕ λ ( y )) = λδ ( x, y ) for all x, y ∈ G CM . By Proposition 3.10, we know there exists ε >
K < ∞ such that(C.9) Kδ ( e , x ) ≥ Kd G CM ( e , x ) ≥ k x k g CM whenever k x k g CM ≤ ε. For arbitrary x = ( A, a ) ∈ G CM , choose λ > ε = k ϕ λ ( x ) k = λ k A k H + λ k a k C , i.e. λ = q k A k H + 4 k a k C ε − k A k H k a k C . It then follows from Eqs. (C.8) and (C.9) that λKδ ( e , x ) = Kδ ( e , ϕ λ ( x )) ≥ ε, i.e. δ ( e , x ) ≥ ε K λ = 2 ε K k a k C q k A k H + 4 k a k C ε − k A k H = 2 ε k a k C K k A k H r k a k C ε k A k H − . (C.10)Since √ x − ≤ min ( x/ , √ x ) we have1 √ x − ≥ max (cid:18) x , √ x (cid:19) ≥ x + 12 √ x . Using this estimate with x = 4 k a k C k A k − H ε in Eq. (C.10) shows δ ( e , x ) ≥ ε k a k C K k A k H k A k H k a k C ε + k A k H ε k a k C ! = 12 K (cid:16) k A k H + ε k a k C (cid:17) , which implies the lower bound in Eq. (C.1). (cid:3) We are now ready to give the proof of Theorem 3.12C.1.
Proof of Theorem 3.12.
Proof.
The first assertion in Eq. (3.24) of Theorem 3.12 follows from TheoremC.2 and the previously observed fact that d CM δ. To prove Eq. (3.25), let ε < ε/ ε > d CM ( e , x ) ε then k x k g CM Kd CM ( e , x ) Kε . So if x = ( A, a ) , we have k A k H Kε and k a k C Kε and hence by Theorem C.2, δ ( e , x ) c − (cid:0) Kε + √ Kε (cid:1) . This implies that(C.11) M ( ε ) := sup { δ ( e , x ) : x ∋ d CM ( e , x ) ≤ ε } c − (cid:16) Kε + p Kε (cid:17) < ∞ . Now suppose that x ∈ G CM with d CM ( e , x ) > ε . Choose a curve, g ∈ C CM suchthat g (0) = e , g (1) = x, and ℓ G CM ( g ) d CM ( e , x ) + ε / . Also choose ε ∈ ( ε / , ε ] such that ℓ G CM ( g ) = nε with n ∈ N and let 0 = t < t < t < · · · 1] such that ℓ G CM (cid:0) g | [ t i − ,t i ] (cid:1) = ε for i = 1 , , . . . , n. If x i := g ( t i ) for i = 0 , . . . , n, then ε > ε = ℓ G CM (cid:0) g | [ t i − ,t i ] (cid:1) > d CM ( x i − , x i ) and therefore from Eq. 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