Maria Gordina

Holomorphic functions and the heat kernel measure on an infinite dimensional complex orthogonal group

The heat kernel measure μt is constructed on an infinite dimensional complex group using a diffusion in a Hilbert space. Then it is proved that holomorphic polynomials on the group are square integrable with respect to the heat kernel measure. The closure of these polynomials, H L2(S OH S, μt), is one of two spaces of holomorphic functions we consider. The second space, H L2(S O(∞)), consists of functions which are holomorphic on an analog of the Cameron–Martin subspace for the group. It is proved that there is an isometry from the first space to the second one.The main theorem is that an infinite dimensional nonlinear analog of the Taylor expansion defines an isometry from H L2(S O(∞)) into the Hilbert space associated with a Lie algebra of the infinite dimensional group. This is an extension to infinite dimensions of an isometry of B. Driver and L. Gross for complex Lie groups.All the results of this paper are formulated for one concrete group, the Hilbert–Schmidt complex orthogonal group, though our methods can be applied in more general situations.

Quasi-invariance for heat kernel measures on sub-Riemannian infinite-dimensional Heisenberg groups

We study heat kernel measures on sub-Riemannian infinite- dimensional Heisenberg-like Lie groups. In particular, we show that Cameron-Martin type quasi-invariance results hold in this subelliptic setting and give L p -estimates for the Radon-Nikodym derivatives. The main ingredient in our proof is a generalized curvature-dimension esti- mate which holds on approximating finite-dimensional projection groups. Such estimates were first introduced by Baudoin and Garofalo in (4).

A note on local controllability on Lie groups

In this work we study linear control systems as addressed by Cardetti and Mittenhuber, 2005. In particular, in this note we present the explicit form of the subgroup where the system is locally controllable.

HEAT KERNEL ANALYSIS ON INFINITE DIMENSIONAL GROUPS

The paper gives an overview of our previous results concerning heat kernel measures in infinite dimensions. We give a history of the subject first, and then describe the construction of heat kernel measure for a class of infinite-dimensional groups. The main tool we use is the theory of stochastic differential equations in infinite dimensions. We provide examples of groups to which our results can be applied. The case of finite-dimensional matrix groups is included as a particular case. 1. Motivation and history. In this paper we review our results in [7], [8], [9], [11] and show how they fit into a broader picture. In order to see the challenges this study presents, we first review what is known in finite dimensions and for the flat infinite-dimensional case. Our main results concern analogues of heat kernel (Gaussian) measures on infinitedimensional groups. 1.1. Finite-dimensional noncommutative case. The initial motivation for our work was the following result for finite-dimensional Lie groups. It has a long history which we address later in this section. Let G be a finite-dimensional connected complex Lie group with a Lie algebra g. The Lie algebra is a complex Hilbert space with a norm | · |, and we denote by {ξi} i=1 an orthonormal basis of g as a real vector space. By identifying g with left-invariant vector fields at the identity e we can define the derivative ∂if(g) = d dt ∣∣∣∣ t=0 f(gei), g ∈ G, and the Laplacian ∆f = ∑ ∂ i f. The heat kernel measure μt has the heat kernel as its density with respect to a Haar measure dx. The heat kernel is the fundamental solution to the heat equation ∂μt(x) ∂t = 14∆μt(x), t > 0, x ∈ G, μt(x)dx weakly −−−−−→ t→0 δe(dx). Research supported by the NSF Grant DMS-0306468. Date: December 14, 2004.

Quasi-invariance for the pinned Brownian motion on a Lie group

We give a new proof of the well-known fact that the pinned Wiener measure on a Lie group is quasi-invariant under right multiplication by finite energy paths. The main technique we use is the time reversal. This approach is different from what B. Driver used to prove quasi-invariance for the pinned Brownian motion on a compact Riemannian manifold.

Equivalence of the Brownian and Energy Representations

We consider two unitary representations of infinite-dimensional groups of smooth paths with values in a compact Lie group. The first representation is induced by the quasi-invariance of the Wiener measure, and the second representation is the energy representation. We define these representations and their basic properties, and then we prove that these representations are unitarily equivalent. Bibliography: 28 titles.

Left-invariant geometries on SU(2) are uniformly doubling

A classical aspect of Riemannian geometry is the study of estimates that hold uniformly over some class of metrics. The best known examples are eigenvalue bounds under curvature assumptions. In this paper, we study the family of all left-invariant geometries on SU(2). We show that left-invariant geometries on SU(2) are uniformly doubling and give a detailed estimate of the volume of balls that is valid for any of these geometries and any radius. We discuss a number of consequences concerning the spectrum of the associated Laplacians and the corresponding heat kernels.

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

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 cannot 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.