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
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.
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.