On notions of harmonicity

TL;DR

This paper explores the equivalence between analytic and probabilistic definitions of harmonic functions within symmetric Hunt processes on locally compact metric spaces, extending to more general processes and infinite-dimensional spaces.

Contribution

It establishes the equivalence of harmonicity notions for symmetric Hunt processes and discusses extensions to broader classes of processes and spaces.

Findings

Proves the equivalence of analytic and probabilistic harmonicity.

Extends results to symmetric right processes on Lusin spaces.

Addresses harmonicity in infinite-dimensional spaces.

Abstract

In this paper, we address the equivalence of the analytic and probabilistic notions of harmonicity in the context of general symmetric Hunt processes on locally compact separable metric spaces. Extensions to general symmetric right processes on Lusin spaces including infinite dimensional spaces are mentioned at the end of this paper.