Probabilistic representations of solutions of elliptic boundary value problem and non-symmetric semigroups

In this paper, we use a probabilistic approach to show that there exists a unique, bounded continuous solution to the Dirichlet boundary value problem for a general class of second order non-symmetric elliptic operators L   with singular coefficients, which does not necessarily have the maximum principle. The theory of Dirichlet forms and heat kernel estimates play a crucial role in our approach. A probabilistic representation of the non-symmetric semigroup {Tt}t≥0{Tt}t≥0 generated by L is also given.