A probabilistic representation of the solution of the directional derivative problem

Let $\mathbf A$ be аn elliptic differential operator of the second order in a domain $D$ of an $N$-dimentional Euclidean space; $l$ be a smooth vector field on the boundary. A probabilistic representation for the solution of the boundary value problem $Au=0$, $\partial u/dl|_{\partial D}=f$ is given in terms of the local time on the boundary. The central limit theorem is proved for a functional of the type of the local time on the boundary.