Short Communications
On Functions Which are Superharmonic for a Markov Process
M. G. Šur Moscow
Abstract:
Let
$X=(x_t,\zeta,\mathcal{M}_t,{\mathbf P}_x)$ 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)
${\mathbf M}_x f(x(\tau_G))\leqq f(x)$ for any
$x\in E$ and any open set
$G$ with a compact closure, where
$\tau_G$ is the hitting time for the set
$E\setminus G$.
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
$x$ (
${\mathbf P}_x$ 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
$U=V\setminus S$, where
$V\in\mathcal{U}$.
Theorem 1.
{\it A non-negative almost Borel function
$f(x)$,
$x\in E$, is superharmonic if and only if it is intrinsically continuous and
$$
M_x f\left({x\left({\tau_U}\right)}\right)\leqq f(x)
$$
for any
$x\in E$ and any
$U\in\mathcal{V}$.}
Theorem 2.
{\it A non-negative function
$f(x)$,
$x\in E$, which is semicontinuous from below is superharmonic if and only if it satisfies the condition
$(*)$ for any
$x\in E$ and any
$U\in\mathcal{U}$.}
Received: 22.06.1963
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