Finely $\mu$-harmonic functions of a Markov process

Abstract:

Let $X$ be a Borel right process and $m$ a fixed excessive measure. Given a finely open nearly Borel set $G$ we define an operator $\Lambda _G$ which we regard as an extension of the restriction to $G$ of the generator of $X$. It maps functions on $E$ to (locally) signed measures on $G$ not charging $m$-semipolars. Given a locally smooth signed measure $\mu$ we define $h$ to be (finely) $\mu$-harmonic on $G$ provided $(\Lambda _G + \mu ) h = 0$ on $G$ and denote the class of such $h$ by $\mathcal H^\mu _f (G)$. Under mild conditions on $X$ we show that $h \in \mathcal H^\mu _f (G)$ is equivalent to a local “Poisson” representation of $h$. We characterize $\mathcal H^\mu _f (G)$ by an analog of the mean value property under secondary assumptions. We obtain global Poisson type representations and study the Dirichlet problem for elements of $\mathcal H^\mu _f (G)$ under suitable finiteness hypotheses. The results take their nicest form when specialized to Hunt processes.