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

摘要

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.