# Quasimartingales associated to Markov processes

## TL;DR

This paper characterizes functions of Markov processes whose evaluations are quasimartingales, showing they are differences of excessive functions, and explores their stability under various process transformations.

## Contribution

It provides a complete characterization of quasimartingale functions for Markov processes and extends Fukushima's semimartingale results to semi-Dirichlet forms.

## Key findings

- u(X) is a quasimartingale iff u is the difference of two finite excessive functions
- Quasimartingale property is preserved under killing, time change, and subordination
- Extension of semimartingale characterization to semi-Dirichlet forms

## Abstract

For a fixed right process $X$ we investigate those functions $u$ for which $u(X)$ is a quasimartingale. We prove that $u(X)$ is a quasimartingale if and only if $u$ is the dif- ference of two finite excessive functions. In particular, we show that the quasimartingale nature of $u$ is preserved under killing, time change, or Bochner subordination. The study relies on an analytic reformulation of the quasimartingale property for $u(X)$ in terms of a certain variation of $u$ with respect to the transition function of the process. We provide sufficient conditions under which $u(X)$ is a quasimartingale, and finally, we extend to the case of semi-Dirichlet forms a semimartingale characterization of such functionals for symmetric Markov processes, due to Fukushima.