## Illinois Journal of Mathematics

### A note on $X$-harmonic functions

E. B. Dynkin

#### 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

#### 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