On the existence of a Feller semigroup with atomic measure in nonlocal boundary condition

Abstract

The existence of Feller semigroups arising in the theory of multidimensional diffusion processes is studied. An elliptic operator of second order is considered on a plane bounded region $G$. Its domain of definition consists of continuous functions satisfying a nonlocal condition on the boundary of the region. In general, the nonlocal term is an integral of a function over the closure of the region $G$ with respect to a nonnegative Borel measure $\mu(y,d\eta)$, $y\in\partial G$. It is proved that the operator is a generator of a Feller semigroup in the case where the measure is atomic. The smallness of the measure is not assumed.