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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

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


Author: R. K. Getoor
Journal: Trans. Amer. Math. Soc. 354 (2002), 901-924
MSC (2000): Primary 60J40; Secondary 60J25, 60J45, 31C05
Published electronically: October 4, 2001
MathSciNet review: 1867363
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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.


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Additional Information

R. K. Getoor
Affiliation: Department of Mathematics, University of California, San Diego, La Jolla, California 92093-0112

DOI: https://doi.org/10.1090/S0002-9947-01-02931-2
Keywords: Markov processes, harmonic functions, Schr\"odinger operators, Poisson representation, Dirichlet problem
Received by editor(s): October 26, 2000
Received by editor(s) in revised form: April 11, 2001
Published electronically: October 4, 2001
Article copyright: © Copyright 2001 American Mathematical Society