Doubly Feller property of Brownian motions with Robin boundary condition

TL;DR

This paper investigates the properties of Hunt processes associated with Sobolev spaces under Robin boundary conditions on unbounded Lipschitz domains, establishing their doubly Feller nature and conditions for semigroup compactness.

Contribution

It proves the doubly Feller property for semigroups of Hunt processes with Robin boundary conditions on unbounded Lipschitz domains, and provides criteria for their compactness.

Findings

Semigroups are doubly Feller under Robin boundary conditions.

Provides a condition for the semigroup's compactness.

Extends understanding of boundary behavior in stochastic processes.

Abstract

In this paper, we consider first order Sobolev spaces with Robin boundary condition on unbounded Lipschitz domains. Hunt processes are associated with these spaces. We prove that the semigroup of these processes are doubly Feller. As a corollary, we provide a condition for semigroups generated by these processes being compact.