On Stochastic Representation for Solutions of the Dirichlet Problem for Elliptic Equations in Divergence Form

We provide a stochastic representation of weak solutions of the Dirichlet problem for general second-order uniformly elliptic linear equations in divergence form. The representation is given in terms of solutions of some generalized linear backward stochastic differential equation associated with the problem and may be considered as a generalization of the Feynman-Kac formula.