Invariant, super and quasi-martingale functions of a Markov process

TL;DR

This paper characterizes the space of excessive functions of a Markov process through quasimartingale properties, unifies various function classes, and explores applications to invariant measures and semi-Dirichlet forms.

Contribution

It provides a unifying framework linking excessive, harmonic, and martingale functions, clarifying their relationships and applications in Markov process theory.

Findings

The space of excessive functions equals functions that are quasimartingales when composed with the process.

In the conservative case, harmonic, co-harmonic, invariant, co-invariant, and martingale functions coincide.

A two-step method for establishing the existence of invariant probability measures is proposed.

Abstract

We identify the linear space spanned by the real-valued excessive functions of a Markov process with the set of those functions which are quasimartingales when we compose them with the process. Applications to semi-Dirichlet forms are given. We provide a unifying result which clarifies the relations between harmonic, co-harmonic, invariant, co-invariant, martingale and co-martingale functions, showing that in the conservative case they are all the same. Finally, using the co-excessive functions, we present a two-step approach to the existence of invariant probability measures.