Leonard Gross’s Work in Infinite-dimensional Analysis and Heat Kernel Analysis

This paper describes a certain part of Leonard Gross’s work in infinite-dimensional analysis, connected to the Gross Ergodicity Theorem. I then look at way in which Gross’s work helped to create a new subject within (mostly) finite-dimensional analysis, a subject which may be called “harmonic analysis with respect to heat kernel measure.” This subject transfers to Lie groups certain constructions on Rn that involves a Gaussian measure. On the Lie group, the role of the Gaussian measure is played by a heat kernel measure. 1. The Gross Ergodicity Theorem and its Consequences The purpose of this article is to outline one part of Leonard Gross’s work in infinite-dimensional analysis and to show how that work lead to the development of a new discipline within (mostly) finite-dimensional analysis, a discipline which may be called “harmonic analysis with respect to heat kernel measure,” or “heat kernel analysis” more briefly. The story begins with what is now called the Gross Ergodicity Theorem, established in [21]. The result may be described as follows. Let K be a connected compact Lie group equipped with a bi-invariant Riemannian metric. Let W (K) denote the continuous path group, i.e., the group of continuous maps of [0, 1] into K sending 0 to the identity e in K. We consider on W (K) the Wiener measure ρ. Now let L(K) denote the finite-energy loop group, i.e., the group of maps of [0, 1] into K sending both 0 and 1 to e and having one distributional derivative in L. Then the left action of L(K) on W (K) leaves the Wiener measure quasi-invariant. Theorem 1.1. Given f ∈ L(W (K), ρ), suppose that for all l ∈ L(K), f(l · g) = f(g) for almost every g in W (K). Then there exists a measurable function φ on K such that f(g) = φ(g(1)) for almost every g in W (K). That is to say, if a function on the path group is invariant under the left action of the loop group, then that function depends only on the endpoint of the path. The difficulty in this theorem comes in the mismatch between the levels of smoothness: The paths in W (K) have to be continuous rather than finite energy, because the Wiener measure does not live on finite-energy paths (they form a set of measure zero). Meanwhile, the loops in L(K) have to be finite energy rather 2000 Mathematics Subject Classification. Primary 60H30.