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

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

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Authors and Affiliations

  1. York University, Toronto, Ontario, Canada

    Thomas S. Salisbury

  2. University of Calgary, Calgary, Alberta, Canada

    A. Deniz Sezer

Authors
  1. Thomas S. Salisbury
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  2. A. Deniz Sezer
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Correspondence to A. Deniz Sezer.

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Both authors are supported in part by NSERC.

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Salisbury, T.S., Sezer, A.D. Moment Densities of Super Brownian Motion, and a Harnack Estimate for a Class of X-harmonic Functions. Potential Anal 41, 1347–1358 (2014). https://doi.org/10.1007/s11118-014-9420-y

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  • DOI: https://doi.org/10.1007/s11118-014-9420-y

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