Heat Kernel Analysis on Infinite-dimensional Groups

نویسندگان

  • Masha Gordina
  • R. H. Cameron
  • W. T. Martin
چکیده

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.

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تاریخ انتشار 2010