Nearly hyperharmonic functions are infima of excessive functions

TL;DR

This paper proves that nearly hyperharmonic functions on a Hunt process space are precisely the infima of excessive functions, providing a simple, complete proof with novel reductions and measure-theoretic insights.

Contribution

It offers a concise, comprehensive proof that nearly hyperharmonic functions are infima of excessive functions, with new reductions and measure integration results.

Findings

Nearly hyperharmonic functions are infima of excessive functions.

The proof involves a reduction to a case with finite expected visits.

Integral infima hold for all finite measures, not just those with finite integrals.

Abstract

Let $\mathfrak X$ be a Hunt process on a locally compact space $X$ such that the set $\mathcal E_{\mathfrak X}$ of its Borel measurable excessive functions separates points, every function in $\mathcal E_{\mathfrak X}$ is the supremum of its continuous minorants in $\mathcal E_{\mathfrak X}$ and there are strictly positive continuous functions $v,w\in\mathcal E_{\mathfrak X}$ such that $v/w$ vanishes at infinity. A numerical function $u\ge 0$ on $X$ is said to be nearly hyperharmonic, if $\int^\ast u\circ X_{\tau_V}\,dP^x\le u(x)$ for all $x\in X$ and relatively compact open neighborhoods $V$ of $x$, where $\tau_V$ denotes the exit time of $V$. For every such function $u$, its lower semicontinous regularization $\hat u$ is excessive. The main purpose of the paper is to give a short, complete and understandable proof for the statement that every Borel measurable nearly hyperharmonic function on $X$ is the infimum of its majorants in $E_{\mathfrak X}$. The major novelties of our approach are the following: 1. A quick reduction to the special case, where starting at $x\in X$ with $u(x)<\infty$ the expected number of times the process $\mathfrak X$ visits the set of points $y\in X$, where $\hat u(y):=\liminf_{z\to y} u(z)<u(y)$, is finite. 2. The statement that the integral $\int u\,d\mu$ is the infimum of all integrals $\int w\,d\mu$, $w\in E_{\mathfrak X}$ and $w\ge u$, not only for measures $\mu$ satisfying $\int w\,d\mu<\infty$ for some excessive majorant $w$ of $u$, but also for all finite measures. At the end, the measurability assumption on $u$ is weakened considerably.