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

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 μ(y, dη), y ∈ ∂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. © 2008 MAIK Nauka.