Mathematical Research Letters

Volume 13 (2006)

Number 1

On extreme X-harmonic functions

Pages: 59 – 69



E. B. Dynkin (Cornell University)


All positive harmonic functions in an arbitrary domain $E$ of a Euclidean space can be decomposed in a unique way into extreme functions. The latter can be obtained by a passage to the limit from $k^y(x)=\frac{g(x,y)}{g(a,y)}$ where $g(x,y)$ is the Green function of the Laplacian and $a$ is a fixed point of $E$. Our goal is to get similar results 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 $\M_c(E)$ 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. Extreme $X$- harmonic functions play the same role as their classical counterpart. We describe a limit process for getting these functions. Instead of the ratio $\frac{g(x,y)}{g(a,y)}$ 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.

Published 1 January 2006