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2006 A note on $X$-harmonic functions
E. B. Dynkin
Illinois J. Math. 50(1-4): 385-394 (2006). DOI: 10.1215/ijm/1258059479


The Martin boundary theory allows one to describe all positive harmonic functions in an arbitrary domain $E$ of a Euclidean space starting from the functions $k^y(x)=\sfrac{g(x,y)}{g(a,y)}$, where $g(x,y)$ is the Green function of the Laplacian and $a$ is a fixed point of $E$. In two previous papers a similar theory was developed for a class of positive functions on a space of measures. These functions are associated with a superdiffusion $X$ and we call them $X$-harmonic. Denote by $\M_c(E)$ the set of all finite measures $\mu$ supported by compact subsets of $E$. $X$-harmonic functions are functions on $\M_c(E)$ characterized by a mean value property formulated in terms of exit measures of a superdiffusion. Instead of the ratio $\sfrac{g(x,y)}{g(a,y)}$ we use a Radon-Nikodym derivative of the probability distribution of an exit measure of $X$ with respect to the probability distribution of another such measure. The goal of the present note is to find an expression for this derivative.


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E. B. Dynkin. "A note on $X$-harmonic functions." Illinois J. Math. 50 (1-4) 385 - 394, 2006.


Published: 2006
First available in Project Euclid: 12 November 2009

zbMATH: 1109.60064
MathSciNet: MR2247833
Digital Object Identifier: 10.1215/ijm/1258059479

Primary: 60J50
Secondary: 31C05, 60J45

Rights: Copyright © 2006 University of Illinois at Urbana-Champaign


Vol.50 • No. 1-4 • 2006
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