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.
Highlights
Introduction and Statement of the ResultsThe goal of this work is to develop a probabilistic approach for studying the trace at the boundary of positive solutions to the semilinear parabolic equation Ou ont 1/2Au- U2 (1)This approach has been inspired by our previous work [14] and the recent paper of Dynkin and Kuznetsov [7], which both dealt with the trace at the boundary for related semilinear elliptic partial differential equations
Our main probabilistic tool is the path-valued process called the Browniau snake, whose connections with semilinear partial differential equations have been investigated in several recent papers [10, 13, 14]
Since the Brownian snake is closely related to the super-Brownian motion, part of these connections can be viewed as a reformulation of Dynkin’s important work on the relation between superprocesses and partial differential equations [3,4,5]
Summary
This approach has been inspired by our previous work [14] and the recent paper of Dynkin and Kuznetsov [7], which both dealt with the trace at the boundary for related semilinear elliptic partial differential equations. When d 1, it follows from well-known results (see e.g., Sugitani [18], Theorem 1) that the measures X are absolutely continuous, and, more precisely, we may write Xt(dY Yt(Y)dy where the process (Yt(y);t > 0, y ) is jointly continuous. With this notation at hand, we can state our last result, which is analogous to the main result of [14].
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