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.
"Differentiability of Stochastic Flow of Reflected Brownian Motions." Electron. J. Probab. 14 2182 - 2240, 2009. https://doi.org/10.1214/EJP.v14-700