The heat operator in infinite dimensions

Let (H,B) be an abstract Wiener space and let μ_{s} be the Gaussian measure on B with variance s. Let Δbe 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(tΔ/2) is a contraction operator from L^2(B,μ_{s} to L^2(B,μ_{s-t}), for all t<s. More generally, the heat operator is a contraction from L^p(B,μ_{s}) to L^q(B,μ_{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.

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