The heat operator in infinite dimensions

Original · EN

Let (H,B) be an abstract Wiener space and let μₛ 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 (tΔ/2) is a contraction operator from L²(B,μₛ to L²(B,μₛ₋ₜ), for all t<s. More generally, the heat operator is a contraction from Lᵖ(B,μₛ) to Lq(B,μₛ₋ₜ) for t<s, provided that p and q satisfy (p-1)/(q-1) ≤ s/(s-t). I give two proofs of this result, both very elementary.