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Probabilistic representations of solutions to the heat equation

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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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Authors and Affiliations

  1. Indian Statistical Institute, R.V. College Post, 560 059, Bangalore, India

    B. Rajeev & S. Thangavelu

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  1. B. Rajeev
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  2. S. Thangavelu
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Correspondence to B. Rajeev.

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Rajeev, B., Thangavelu, S. Probabilistic representations of solutions to the heat equation. Proc. Indian Acad. Sci. (Math. Sci.) 113, 321–332 (2003). https://doi.org/10.1007/BF02829609

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  • DOI: https://doi.org/10.1007/BF02829609

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