# Moment Densities of Super Brownian Motion, and a Harnack Estimate for a Class of X-harmonic Functions

Research paper by Thomas S. Salisbury, A. Deniz Sezer

Indexed on: 29 Jun '14Published on: 29 Jun '14Published in: Potential Analysis

#### Abstract

This paper features a comparison inequality for the densities of the moment measures of super-Brownian motion. These densities are defined recursively for each n≥1 in terms of the Poisson and Green’s kernels, hence can be analyzed using the techniques of classical potential theory. When n=1, the moment density is equal to the Poisson kernel, and the comparison is simply the classical inequality of Harnack. For n>1 we find that the constant in the comparison inequality grows at most exponentially with n. We apply this to a class of X-harmonic functions Hν of super-Brownian motion, introduced by Dynkin. We show that for a.e. Hν in this class, $$H^{\nu }(\mu )<\infty$$ for every μ.