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

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 H ν of super-Brownian motion, introduced by Dynkin. We show that for a.e. H ν in this class, H^ν(μ )<∞ for every μ .