The heat operator in infinite dimensions

TL;DR

This paper investigates the properties of the heat operator associated with the Laplacian in infinite-dimensional Gaussian spaces, establishing its contraction behavior across various L^p spaces with explicit conditions.

Contribution

It proves that the heat operator acts as a contraction between specific L^p spaces in infinite-dimensional Wiener spaces, providing two elementary proofs of this result.

Findings

Heat operator is a contraction from L^2(B,μ_s) to L^2(B,μ_{s-t}) for t<s.

More generally, it is a contraction from L^p(B,μ_s) to L^q(B,μ_{s-t}) under certain conditions.

The paper provides two elementary proofs of the contraction property.

Abstract

Let (H,B) be an abstract Wiener space and let \mu_{s} be the Gaussian measure on B with variance s. Let \Delta be the Laplacian (*not* the number operator), that is, a sum of squares of derivatives associated to an orthonormal basis of H. I will show that the heat operator \exp(t\Delta/2) is a contraction operator from L^2(B,\mu_{s} to L^2(B,\mu_{s-t}), for all t<s. More generally, the heat operator is a contraction from L^p(B,\mu_{s}) to L^q(B,\mu_{s-t}) for t<s, provided that p and q satisfy (p-1)/(q-1) \leq s/(s-t). I give two proofs of this result, both very elementary.