Diffusion processes associated with elliptic operators with measure-valued coefficients

Abstract

In this paper, we give a survey on diffusion processes associated with elliptic operators with measure-valued coefficients. We first introduce the existence and uniqueness of Brownian motion and reflected Brownian with measure-valued drifts as well as the Gaussian type estimates of the transition functions. We then study the Dirichlet and Neumann boundary value problems for the associated generators of the diffusion processes and obtain the probabilistic representations for the solutions. we also present the small time asymptotics and large deviations for the Brownian motions with drifts.