A remark on the existence of a diffusion process with non-local boundary conditions

Abstract

It is known that a diffusion process on a domain D with smooth boundary is determined by a pair of analytical data (A, L), where A is a second order differential operator of elliptic type and L is a Wentzell's boundary condition which consists of the sum of a second order differential operator and non-local terms. Here we shall afford a concrete example in this framework. We discuss the case where the boundary condition L possesses two non-local terms, one corresponds to the Cauchy process on the boundary and the other to a stable process of order β∈(0,1)∩Q having inward jumps from the boundary. We shall show analytically the existence and uniqueness of the Feller semigroup, and hence the diffusion