Distribution density of the first exit point of a two-dimensional diffusion process from a circle neighborhood of its initial point: the inhomogeneous case

A two-dimensional diffusion process is considered. The distribution of the first exit point of such a process from an arbitrary domain of its values is determined, as a function of the initial point of the process, by an elliptic second-order differential equation, and corresponds to the solution of the Dirichlet problem for this equation (the case of nonconstant coefficients). We examine the distribution density of the first exit point of the process from the small circular neighborhood of its initial point and study its relation to the Dirichlet problem. In terms of this asymptotics, we prove two theorems, which provide sufficient conditions and necessary conditions for the distribution of the first exit point, as a function of the initial point of the process, to satisfy a certain second-order elliptic differential equation corresponding to the standard Wiener process with drift and break. The removable second-order terms of the expansion in powers of the radius of the small circular neighborhood of the initial point of the process are identified. In terms of the removable terms, these two theorems are combined as a single theorem giving a necessary and sufficient condition for correspondence to this Wiener process.