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.