Nearly hyperharmonic functions are infima of excessive functions
A talk in the Oberseminar Geometric Analysis series by
Wolfhard Hansen from Bielefeld
| 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
\begin{equation*}
\int^\ast u\circ X_{\tau_V}\,dP^x\le u(x)
\end{equation*}
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 semicontinous regularization
$\hat u$ is excessive.
The talk will present a short and complete proof for the statement
that
\begin{equation*}
u=\inf \{w\in \mathcal E_{\mathfrak X}\colon w\ge u\}
\end{equation*}
for every Borel measurable nearly hyperharmonic function $u$ on $X$.
The main novelty is a quick reduction to the very special case,
where
starting in $\{ u <\infty\}$ the process $\mathfrak X$ hits the
set $\{ û < u \}$ almost surely only
finitely many times. |