## Illinois Journal of Mathematics

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

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

#### 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