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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, Hν(μ)<∞$H^{\nu }(\mu )<\infty $ for every μ.

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