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

TL;DR

This paper proves the existence of Feller semigroups generated by elliptic operators with nonlocal boundary conditions involving atomic measures, expanding understanding of diffusion processes with nonlocal interactions.

Contribution

It establishes that such operators generate Feller semigroups without requiring the measure to be small, specifically when the measure is atomic.

Findings

Feller semigroup existence is proven for operators with atomic measures.

No smallness condition on the measure is necessary.

The results apply to multidimensional diffusion processes with nonlocal boundary conditions.

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.