Strong Feller Continuity of Feller Processes and Semigroups

TL;DR

This paper explores two equivalent characterizations of the strong Feller property for Feller processes, generalizing existing results and linking process properties with harmonic function continuity.

Contribution

It introduces new criteria based on local absolute continuity and Orlicz-ultracontractivity, providing a more natural framework for analyzing strong Feller continuity.

Findings

Established equivalence of two characterizations of strong Feller property.

Generalized conditions for strong Feller continuity beyond previous results.

Connected process regularity with harmonic function continuity.

Abstract

We study two equivalent characterizations of the strong Feller property for a Markov process and of the associated sub-Markovian semigroup. One is described in terms of locally uniform absolute continuity, whereas the other uses local Orlicz-ultracontractivity. These criteria generalize many existing results on strong Feller continuity and seem to be more natural for Feller processes. By establishing the estimates of the first exit time from balls, we also investigate the continuity of harmonic functions for Feller processes which enjoy the strong Feller property.