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 $n\ge 1$ 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 $H^{\nu }$ of super-Brownian motion, introduced by Dynkin. We show that for a.e. $H^{\nu }$ in this class, $H^{\nu }(\mu )<\mathrm{\infty }$ for every $\mu$.