The Heat Equation and Reflected Brownian Motion in Time-Dependent Domains

Abstract

The paper is concerned with reflecting Brownian motion (RBM) in domains with deterministic moving boundaries, also known as "non-cylindrical domains," and its connections with partial differential equations. Construction is given for reflecting Brownian motion in C3-smooth time-dependent domains in the n-dimensional Euclidean space Rn. Various sample path properties of RBM, its two-sided transition density functions estimate and the probabilistic representation of solutions for the corresponding partial differential equations are obtained. Furthermore, the onedimensional case is throughly studied, with the assumptions on the smoothness of the boundary drastically relaxed.