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https://www.ias.ac.in/article/fulltext/pmsc/113/03/0321-0332

Brownian motion; heat equation; translation operators; infinite dimensional stochastic differential equations

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.

B Rajeev¹ S Thangavelu¹

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

Published

August 2003

Manuscript received

16 November 2002