Illinois Journal of Mathematics

A note on $X$-harmonic functions

E. B. Dynkin

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Abstract

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.

Article information

Source
Illinois J. Math., Volume 50, Number 1-4 (2006), 385-394.

Dates
First available in Project Euclid: 12 November 2009

Permanent link to this document
https://projecteuclid.org/euclid.ijm/1258059479

Digital Object Identifier
doi:10.1215/ijm/1258059479

Mathematical Reviews number (MathSciNet)
MR2247833

Zentralblatt MATH identifier
1109.60064

Subjects
Primary: 60J50: Boundary theory
Secondary: 31C05: Harmonic, subharmonic, superharmonic functions 60J45: Probabilistic potential theory [See also 31Cxx, 31D05]

Citation

Dynkin, E. B. A note on $X$-harmonic functions. Illinois J. Math. 50 (2006), no. 1-4, 385--394. doi:10.1215/ijm/1258059479. https://projecteuclid.org/euclid.ijm/1258059479


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