Unbounded perturbations of two-dimensional diffusion processes with nonlocal boundary conditions

TL;DR

This paper investigates conditions under which unbounded perturbations of elliptic operators with nonlocal boundary conditions generate Feller semigroups in bounded planar regions, extending the theory of multidimensional diffusion processes.

Contribution

It establishes sufficient conditions for unbounded elliptic operator perturbations with nonlocal boundary conditions to generate Feller semigroups, including cases with measures intersecting the boundary.

Findings

Provided criteria for the generator property of nonlocal operators

Extended the class of perturbations allowing Feller semigroup generation

Analyzed measures with support intersecting the boundary

Abstract

The existence of Feller semigroups arising in the theory of multidimensional diffusion processes is studied. Unbounded perturbations of elliptic operators (in particular, integro-differential operators) are considered in plane bounded regions. Their domain of definition is given by a nonlocal boundary condition involving the integral over the closure of the region with respect to a nonnegative Borel measure. The support of the measure may intersect with the boundary, and the measure need not be small. We formulate sufficient conditions on unbounded perturbations of elliptic operator and on the Borel measure in the nonlocal boundary condition which guarantee that the corresponding nonlocal operator is a generator of a Feller semigroup.