On Boundary Conditions For Multidimensional Diffusion Processes

Abstract

The problem considered in the paper is as follows: given an elliptical operator $\mathfrak{A}$ in a closed bounded region K, the most general boundary conditions are sought, which restrict $\mathfrak{A}$ to an infinite-simal operator of a Markov process in K. This problem is solved for the case when K is a circle or a sphere and only for processes invariant by rotations. In the general case when a process is given, boundary conditions are found, which are satisfied by all smooth functions in the domain of the infinitesimal operator of the process; however, it is not known whether this domain can be constructed from the boundary conditions.