Heat kernel analysis on infinite-dimensional Heisenberg groups

Probab. Theory Relat. Fields

2009

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We introduce a class of non-commutative, complex, infinite-dimensional Heisenberg like Lie groups based on an abstract Wiener space. The holomorphic functions which are also square integrable with respect to a heat kernel measure μ on these groups are studied. In particular, we establish a unitary equivalence between the square integrable holomorphic functions and a certain completion of the universal enveloping algebra of the "Lie algebra" of this class of groups. Using quasi-invariance of the heat kernel measure, we also construct a skeleton map which characterizes globally defined functions from the L 2 (ν)-closure of holomorphic polynomials by their values on the Cameron-Martin subgroup.

Section: Proposition 44 (Generator Ofmentioning

confidence: 58%

“…As was shown in [4,Lemma 3.3], these Banach space topologies on W × C and H × C make G and G C M into topological groups.…”

Section: Notation 33 Let G C M Denote H × C When Viewed As a Lie Submentioning

confidence: 82%

“…In this subsection, we are going to recall one of the key theorems from [4]. We first need a little more notation.…”

Section: Heat Kernel Quasi-invariance Propertiesmentioning

confidence: 99%

Section: Heat Kernel Quasi-invariance Propertiesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Square integrable holomorphic functions on infinite-dimensional Heisenberg type groups

Driver

Probab. Theory Relat. Fields

2009

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An Application of a Functional Inequality to Quasi-Invariance in Infinite Dimensions

Convexity and Concentration

2017

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One way to interpret smoothness of a measure in infinite dimensions is quasi-invariance of the measure under a class of transformations. Usually such settings lack a reference measure such as the Lebesgue or Haar measure, and therefore we can not use smoothness of a density with respect to such a measure. We describe how a functional inequality can be used to prove quasi-invariance results in several settings. In particular, this gives a different proof of the classical Cameron-Martin (Girsanov) theorem for an abstract Wiener space. In addition, we revisit several more geometric examples, even though the main abstract result concerns quasi-invariance of a measure under a group action on a measure space.1991 Mathematics Subject Classification. Primary 58G32 58J35; Secondary 22E65 22E30 22E45 58J65 60B15 60H05.1 2 GORDINA moreover we do not refer to the extensive literature on the subject, as it is beyond the scope of our paper.We start by describing an abstract setting of how finite-dimensional approximations can be used to prove such a quasi-invariance. In [9] this method was applied to projective and inductive limits of finite-dimensional Lie groups acting on themselves by left or right multiplication. In that setting a functional inequality (integrated Harnack inequality) on the finite-dimensional approximations leads to a quasi-invariance theorem on the infinite-dimensional group space. Similar methods were used in the elliptic setting on infinite-dimensional Heisenberg-like groups in [8], and on semi-infinite Lie groups in [14]. Note that the assumptions we make below in Section 3 have been verified in these settings, including the sub-elliptic case for infinite-dimensional Heisenberg group in [1]. Even though the integrated Harnack inequality we use in these situations have a distinctly geometric flavor, we show in this paper that it does not have to be.The paper is organized as follows. The general setting is described in Section 2 and 3, where Theorem 3.2 is the main result. One of the ingredients for this result is quasi-invariance for finite-dimensional approximations which is described in Section 3. We review the connection between an integrated Harnack inequality and Wang's Harnack inequality in Section 4. Finally, Section 5 gives several examples of how one can use Theorem 3.2. We describe in detail the case of an abstract Wiener space, where the group in question is identified with the Cameron-Martin subspace acting by translation on the Wiener space. In addition we discuss elliptic (Riemannian) and sub-elliptic (sub-Riemannian) infinite-dimensional groups which are examples of a subgroup acting on the group by multiplication.Acknowledgement. The author is grateful to Sasha Teplyaev and Tom Laetsch for useful discussions and helpful comments.

A subelliptic Taylor isomorphism on infinite-dimensional Heisenberg groups

Probab. Theory Relat. Fields

Melcher

2011

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Abstract. Let G denote an infinite-dimensional Heisenberg-like group, which is a class of infinite-dimensional step 2 stratified Lie groups. We consider holomorphic functions on G that are square integrable with respect to a heat kernel measure which is formally subelliptic, in the sense that all appropriate finite dimensional projections are smooth measures. We prove a unitary equivalence between a subclass of these square integrable holomorphic functions and a certain completion of the universal enveloping algebra of the "Cameron-Martin" Lie subalgebra. The isomorphism defining the equivalence is given as a composition of restriction and Taylor maps.