Probabilistic representations of solutions to the heat equation

2003/10/17 by B. Rajeev, S. Thangavelu · 1 citation
Mathematics · #math.PR #math.AP

paper · pdf

published as Proc. Indian Acad. Sci. (Math. Sci.), Vol. 113, No. 3, August 2003, pp. 321-332 · 12 pages

arxiv created 2003/10/17 · arxiv updated 2009/12/01

Abstract

In this paper we provide a new (probabilistic) proof of a classical result in partial differential equations, viz. if ϕ is a tempered distribution, then the solution of the heat equation for the Laplacian, with initial condition ϕ, is given by the convolution of ϕ with the heat kernel (Gaussian density). Our results also extend the probabilistic representation of solutions of the heat equation to initial conditions that are arbitrary tempered distributions.

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• Products and Convolutions in Hermite-Sobolev spaces

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