Differentiability of stochastic flow of reflected Brownian motions

TL;DR

This paper proves that the stochastic flow of reflected Brownian motions in smooth multidimensional domains is differentiable with respect to initial conditions, using excursion theory and Skorokhod equation analysis.

Contribution

It establishes the differentiability of reflected Brownian motion flows and provides a representation of the derivative as a multiplicative functional.

Findings

Proves differentiability of stochastic flow in multidimensional domains

Derives a representation of the derivative as a multiplicative functional

Uses excursion theory and Skorokhod equation analysis

Abstract

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.