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• W1655008047 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." @default.
• W1655008047 created "2016-06-24" @default.
• W1655008047 creator A5033238483 @default.
• W1655008047 date "2001-10-04" @default.
• W1655008047 modified "2023-09-25" @default.
• W1655008047 title "Finely $mu $-harmonic functions of a Markov process" @default.
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• W1655008047 doi "https://doi.org/10.1090/s0002-9947-01-02931-2" @default.
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