Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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ON ESTIMATES OF POISSON KERNELS FOR SYMMETRIC LÉVY PROCESSES

  • Kang, Jaehoon (Department of Mathematical Sciences Seoul National University) ;
  • Kim, Panki (Department of Mathematical Sciences and Research Institute of Mathematics Seoul National University)
  • Received : 2012.10.09
  • Published : 2013.09.01
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Abstract

In this paper, using elementary calculus only, we give a simple proof that Green function estimates imply the sharp two-sided pointwise estimates for Poisson kernels for subordinate Brownian motions. In particular, by combining the recent result of Kim and Mimica [5], our result provides the sharp two-sided estimates for Poisson kernels for a large class of subordinate Brownian motions including geometric stable processes.

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References

  1. K. Bogdan, T. Byczkowski, T. Kulczycki, M. Ryznar, R. Song, and Z. Vondracek, Potential analysis of stable processes and its extensions, Lecture Notes in Mathematics, 1980. Springer-Verlag, Berlin, 2009.
  2. Z.-Q. Chen and R. Song, Estimates on Green functions and Poisson kernels for symmetric stable processes, Math. Ann. 312 (1998), no. 3, 465-601. https://doi.org/10.1007/s002080050232
  3. N. Ikeda and S. Watanabe, On some relations between the harmonic measure and the Levy measure for a certain class of Markov processes, J. Math. Kyoto Univ. 2 (1962), 79-95. https://doi.org/10.1215/kjm/1250524975
  4. P. Kim and A. Mimica, Harnack inequalities for subordinate Brownian motions, Electron. J. Probab. 17 (2012), no. 37, 23 pp.
  5. P. Kim and A. Mimica, Green function estimates for subordinate Brownian motions: stable and beyond, Trans. Amer. Math. Soc., to appear.
  6. P. Kim, R. Song, and Z. Vondracek, Boundary Harnack principle for subordinate Brownian motion, Stochastic Process. Appl. 119 (2009), no. 5, 1601-1631. https://doi.org/10.1016/j.spa.2008.08.003
  7. P. Kim, R. Song, and Z. Vondracek, Potential theory of subordinated Brownian motions revisited, Stochastic analysis and applications to finance, essays in honour of Jia-an Yan, Interdisciplinary Mathematical Sciences 13, pp. 243-290, World Scientific, 2012.
  8. P. Kim, R. Song, and Z. Vondracek, Two-sided Green function estimates for killed subordinate Brownian motions, Proc. Lond. Math. Soc. (3) 104 (2012), no. 5, 927-958. https://doi.org/10.1112/plms/pdr050
  9. P. Kim, R. Song, and Z. Vondracek, Global uniform boundary Harnack principle with explicit decay rate and its application, Preprint, 2012.
  10. T. Kulczycki, Properties of Green function of symmetric stable processes, Probab. Math. Statist. 17 (1997), no. 2, Acta Univ. Wratislav. No. 2029, 339-364.

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