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

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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References

  1. 1.

    Cranston, M., Fabes, E., Zhao, Z.: Conditional Gauge and Potential Theory for the Schrodinger Operator. Trans. Am. Math. Soc. 307 (1), 171–194 (1988)

    MathSciNet  MATH  Google Scholar 

  2. 2.

    Doob, J. L.: Classical Potential Theory and its Probabilistic Counterpart. Springer, New York (1984)

    Google Scholar 

  3. 3.

    Dynkin, E. B.: Diffusions, superdiffusions and partial differential equations. Colloquium Publications 50. Amer. Math. Soc., Providence. (2002)

  4. 4.

    Dynkin, E. B.: Harmonic functions and exit boundary of superdiffusion. J. Funct. Anal. 206, 33–68 (2004)

    MathSciNet  Article  MATH  Google Scholar 

  5. 5.

    Dynkin, E. B.: Superdiffusions and positive solutions of nonlinear partial differential equations. University Lecture Series 34. Amer. Math. Soc., Providence. (2004)

  6. 6.

    Dynkin, E. B.: A note on X-harmonic functions. Illinois J. Math. 50, 1–4 (2006)

    MathSciNet  Google Scholar 

  7. 7.

    Dynkin, E. B.: On extreme X-harmonic functions. Math. Res. Lett. 13, 59–69 (2006)

    MathSciNet  Article  MATH  Google Scholar 

  8. 8.

    McConnell, T. R.: A conformal inequality related to the conditional gauge theorem. Trans. Amer. Math. Soc. 318, 721–733 (1990)

    MathSciNet  Article  MATH  Google Scholar 

  9. 9.

    Salisbury, T. S., Sezer, A. D.: Conditioning super-Brownian motion on its boundary statistics, and fragmentation. Ann. Probab. 41, 3617–3657 (2013)

    MathSciNet  Article  MATH  Google Scholar 

  10. 10.

    Salisbury, T. S., Verzani, J.: On the conditioned exit measures of super Brownian motion. Probab. Theory Relat. Fields 115, 237–285 (1999)

    MathSciNet  Article  MATH  Google Scholar 

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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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Keywords

  • 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