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In this paper, we use a probabilistic approach to show that there exists aunique, bounded continuous solution to the Dirichlet boundary value problem fora general class of second order non-symmetric elliptic operators $L$ withsingular coefficients, which does not necessarily have the maximum principle.The theory of Dirichlet forms and heat kernel estimates play a crucial role inour approach. A probabilistic representation of the non-symmetric semigroup$\{T_t\}_{t\ge 0}$ generated by $L$ is also given.

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