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The heat operator in infinite dimensions

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 Added by Brian C. Hall
 Publication date 2007
  fields Physics
and research's language is English
 Authors Brian C. Hall




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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(tDelta/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.



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