Journal of Applied Mathematics and Stochastic Analysis
Volume 9 (1996), Issue 4, Pages 399-414

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

Copyright © 1996 Jean-François Le Gall. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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.