Brownian trajectory is a regular lateral boundary for the heat equation

摘要

The one-dimensional heat equation in the domain x > w(t), t > 0, is considered. Here wt is a trajectory of Brownian motion. For almost any trajectory, it is proved that if the boundary data are continuous, then the solution is continuous in the closure of the domain. The proof is based on Davis's law of square root for Brownian motion or on its weaker version, which is obtained by using the theory of stochastic partial differential equations. [References: 15]