On distribution density of the first exit point of a diffusion process with break from a small circle neighborhood of its initial point

A two-dimensional homogeneous diffusion process with break is considered. A distribution of the first exit point of such a process from an arbitrary neighborhood of zero as a function of the initial point of the process is determined by an elliptic partial differential equation of second order with constant coefficients and corresponds to the solution of the Dirichlet problem for this equation. A connection of this Dirichlet problem with a distribution density of the first exit point of the process from a small circle neighborhood of zero is proved. In terms of this asymptotic the necessary and sufficient conditions are proved for a function of the initial point of the process to satisfy a partial view of the elliptical partial differential equation of second order, which corresponds to a standard Wiener process with drift and break.