On Functions Which are Superharmonic for Markov Processes

Abstract

Let $X = (x_t ,\zeta ,\mathcal{M}_t ,{\bf P}_x )$ be a standard Markov process on a locally compact separable Hausdorff space $(E,\mathcal{O})$. An almost Borel measurable function $f(x):E \to ( { - \infty , + \infty } ]$ is called superharmonic if it satisfies the following conditions: a) it is intrinsically continuous; b) ${\bf M}_x f(x(\tau _G )) \leqq f(x)$ for any $x \in E$ and any open set G with a compact closure, where $\tau _G $ is the hitting time for the set $E - G$.The main results are stated in Theorems 1 and 2. In these theorems S denotes the set of $x \in E$ for which $x_t $ coincides with x (${\bf P}_x $ almost surely) during a positive random time interval $[0,\delta ]$; the symbol $\mathcal{U}$ denotes any open base of $\mathcal{O}$, and $\mathcal{V}$ is the class of all sets U of the type $U \in \mathcal{U}$ or $U = V - S$, where $V \in \mathcal{U}$.Theorem 1.A non-negative almost Borel function$f(x)$, $x \in E$, is superharmonic if and only if it is intrinsically continuous and\[ {\bf ...