Nearly Hyperharmonic Functions are Infima of Excessive Functions

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 $\int ^*u\circ X_{\tau _V}\,\text {d}P^x\le u(x)$ for every $x\in X$ and every relatively compact open neighborhood V of x, where $\tau _V$ denotes the exit time of V. For every such function u, its lower semicontinuous regularization $\hat{u}$ is excessive. The main purpose of the paper is to give a short, complete and understandable proof for the statement that $u=\inf \{w\in {\mathcal {E}}_{{\mathfrak {X}}}:w\ge u\}$ for every Borel measurable nearly hyperharmonic function on X. Principal 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 $\hat{u}(y):=\liminf _{z\rightarrow y} u(z)<u(y)$, is finite.

• 2. The consequent use of (only) the strong Markov property.

• 3. The proof of the equality $\int u\,\text {d}\mu =\inf \{\int w\,\text {d}\mu :w\in \mathcal {E}_{\mathfrak {X}},\,w\ge u\}$ not only for measures $\mu$ satisfying $\int w\,\text {d}\mu <\infty$ for some excessive majorant w of u, but also for all finite measures.

At the end, the measurability assumption on u is weakened considerably.