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

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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).

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  • Poisson kernel
  • Green’s kernel
  • Harnack inequality
  • 3-G inequality
  • Super-Brownian motion
  • Recursive moment formulae
  • X-harmonic function

Mathematics Subject Classifications (2010)

  • 60J45
  • 60J68