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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Cranston, M., Fabes, E., Zhao, Z.: Conditional Gauge and Potential Theory for the Schrodinger Operator. Trans. Am. Math. Soc. 307 (1), 171–194 (1988)
Doob, J. L.: Classical Potential Theory and its Probabilistic Counterpart. Springer, New York (1984)
Dynkin, E. B.: Diffusions, superdiffusions and partial differential equations. Colloquium Publications 50. Amer. Math. Soc., Providence. (2002)
Dynkin, E. B.: Harmonic functions and exit boundary of superdiffusion. J. Funct. Anal. 206, 33–68 (2004)
Dynkin, E. B.: Superdiffusions and positive solutions of nonlinear partial differential equations. University Lecture Series 34. Amer. Math. Soc., Providence. (2004)
Dynkin, E. B.: A note on X-harmonic functions. Illinois J. Math. 50, 1–4 (2006)
Dynkin, E. B.: On extreme X-harmonic functions. Math. Res. Lett. 13, 59–69 (2006)
McConnell, T. R.: A conformal inequality related to the conditional gauge theorem. Trans. Amer. Math. Soc. 318, 721–733 (1990)
Salisbury, T. S., Sezer, A. D.: Conditioning super-Brownian motion on its boundary statistics, and fragmentation. Ann. Probab. 41, 3617–3657 (2013)
Salisbury, T. S., Verzani, J.: On the conditioned exit measures of super Brownian motion. Probab. Theory Relat. Fields 115, 237–285 (1999)
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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
Keywords
- Poisson kernel
- Green’s kernel
- Harnack inequality
- 3-G inequality
- Super-Brownian motion
- Recursive moment formulae
- X-harmonic function