Heat Kernel Analysis on Infinite Dimensional Groups

Abstract

Abstract. 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. 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 {ξ i } 2n 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 derivativeand the Laplacian ∆f = ∂ 2 i f. The heat kernel measure µ t 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

A Brownian Motion on the Diffeomorphism Group of the Circle

“…1.1 by using the theory of stochastic differential equations in Hilbert spaces as developed by G. DaPrato and J. Zabczyk in [2]. Using this method, M. Gordina [4][5][6] and M. Wu [8] constructed a Brownian motion in several Hilbert-Schmidt groups. The construction relied on the fact that these Hilbert-Schmidt groups are Hilbert Lie groups.…”

Diffeomorphisms of the circle and Brownian motions on an infinite-dimensional symplectic group

“…To prove this theorem we will use Theorem 7.4 from the book by G. DaPrato and J. Zabczyk [3] as it has been done in [6,7]. It is enough to check 1.…”

“…For such matrix groups, the method of [6,7] can be used to construct a Brownian motion living in the group. This construction relies on the fact that these groups can be embedded into a larger Hilbert space of Hilbert-Schmidt operators.…”

“…We will use the fact that Proof. The proof is adapted from papers by M. Gordina [6,7]. Let Y t be the solution to Equation (6.5) and Y …”