Brownian motion with boundary diffusion.

摘要

Diffusion in a bounded domain can be described by a second order elliptic operator with Wentzell's boundary condition. We consider a Brownian motion with boundary diffusive condition.;We first prove a weak existence and uniqueness theorem for Brownian motion with boundary diffusion in bounded Lipschitz domain. This is done through showing that Dirichlet forms corresponding to the process is regular. Skorohod representation is also also computed, and the process is shown to be a semi-martingale in piecewise smooth domain. Boundary Diffusive Boundary Harnack Principle for positive harmonic functions is stated, and proved using probabilistic method. Finally, we prove a version of Invariance Principle that gives us a sufficient condition for a sequence of Markov chains to converge to Brownian motion with diffusive boundary conditions.