The Heat Equation and Reflected Brownian Motion in Time-dependent Domains by Krzysztof Burdzy,1

Heat Equation and Reflected Brownian Motion in Time Dependent Domains

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

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 C-smooth time-dependent domains in the n-dimensional Euclidean space R. Various sample path properties of RBM, its two-sided transiti...

Ju n 20 07 On pathwise uniqueness for reflecting Brownian motion in C 1 + γ domains ∗ Richard F . Bass and Krzysztof Burdzy

Pathwise uniqueness holds for the Skorokhod stochastic differential equation in C 1+γ-domains in R d for γ > 1/2 and d ≥ 3.

Differentiability of Stochastic Flow of Reflected Brownian Motions

We prove that a stochastic flow of reflected Brownian motions in a smooth multidimensional domain is differentiable with respect to its initial position. The derivative is a linear map represented by a multiplicative functional for reflected Brownian motion. The method of proof is based on excursion theory and analysis of the deterministic Skorokhod equation.

Discrete Approximations to Reflected Brownian Motion∗

In this paper we investigate three discrete or semi-discrete approximation schemes for reflected Brownian motion on bounded Euclidean domains. For a class of bounded domains D in R that includes all bounded Lipschitz domains and the von Koch snowflake domain, we show that the laws of both discrete and continuous time simple random walks on D ∩ 2Z moving at the rate 2 with stationary initial dis...