Journal of Applied Mathematics and Stochastic Analysis 
Volume 9 (1996), Issue 4, Pages 399-414
doi:10.1155/S1048953396000354

A probabilistic approach to the trace at the boundary for solutions of a semilinear parabolic partial differential equation

Jean-François Le Gall

 Université Paris VI, Laboratoire de Probabilités, 4, Place Jussieu, Paris Cedex 05 75252, France
Received 1 August 1996; Revised 1 October 1996

Abstract

We use the path-valued process called the “Brownian snake” to investigate the trace at the boundary of nonnegative solutions of a semilinear parabolic partial differential equation. In particular, we characterize possible traces and in dimension one we prove that nonnegative solutions are in one-to-one correspondence with their traces at the origin. We also provide probabilistic representations for various classes of solutions.

This article is dedicated to the memory of Roland L. Dobrushin.