Symmetric jump processes : localization, heat kernels and convergence

Richard F. Bass; Moritz Kassmann; Takashi Kumagai

Annales de l'I.H.P. Probabilités et statistiques (2010)

  • Volume: 46, Issue: 1, page 59-71
  • ISSN: 0246-0203

Abstract

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We consider symmetric processes of pure jump type. We prove local estimates on the probability of exiting balls, the Hölder continuity of harmonic functions and of heat kernels, and convergence of a sequence of such processes.

How to cite

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Bass, Richard F., Kassmann, Moritz, and Kumagai, Takashi. "Symmetric jump processes : localization, heat kernels and convergence." Annales de l'I.H.P. Probabilités et statistiques 46.1 (2010): 59-71. <http://eudml.org/doc/242929>.

@article{Bass2010,
abstract = {We consider symmetric processes of pure jump type. We prove local estimates on the probability of exiting balls, the Hölder continuity of harmonic functions and of heat kernels, and convergence of a sequence of such processes.},
author = {Bass, Richard F., Kassmann, Moritz, Kumagai, Takashi},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {symmetric jump processes; Dirichlet forms; heat kernels; Harnack inequalities; weak convergence; non-local operators},
language = {eng},
number = {1},
pages = {59-71},
publisher = {Gauthier-Villars},
title = {Symmetric jump processes : localization, heat kernels and convergence},
url = {http://eudml.org/doc/242929},
volume = {46},
year = {2010},
}

TY - JOUR
AU - Bass, Richard F.
AU - Kassmann, Moritz
AU - Kumagai, Takashi
TI - Symmetric jump processes : localization, heat kernels and convergence
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2010
PB - Gauthier-Villars
VL - 46
IS - 1
SP - 59
EP - 71
AB - We consider symmetric processes of pure jump type. We prove local estimates on the probability of exiting balls, the Hölder continuity of harmonic functions and of heat kernels, and convergence of a sequence of such processes.
LA - eng
KW - symmetric jump processes; Dirichlet forms; heat kernels; Harnack inequalities; weak convergence; non-local operators
UR - http://eudml.org/doc/242929
ER -

References

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  1. [1] M. T. Barlow and R. F. Bass. The construction of Brownian motion on the Sierpinski carpet. Ann. Inst. H. Poincaré Probab. Statist. 25 (1989) 225–257. Zbl0691.60070MR1023950
  2. [2] M. T. Barlow, R. F. Bass, Z.-Q. Chen and M. Kassmann. Non-local Dirichlet forms and symmetric jump processes. Trans. Amer. Math. Soc. 361 (2009) 1963–1999. Zbl1166.60045MR2465826
  3. [3] M. T. Barlow, A. Grigor’yan and T. Kumagai. Heat kernel upper bounds for jump processes and the first exit time. J. Reine Angew. Math. To appear. Zbl1158.60039MR2492992
  4. [4] R. F. Bass and M. Kassmann. Hölder continuity of harmonic functions with respect to operators of variable order. Comm. Partial Diferential Equations 30 (2005) 1249–1259. Zbl1087.45004MR2180302
  5. [5] R. F. Bass and T. Kumagai. Symmetric Markov chains on ℤd with unbounded range. Trans. Amer. Math. Soc. 360 (2008) 2041–2075. Zbl1130.60076MR2366974
  6. [6] E. A. Carlen, S. Kusuoka and D. W. Stroock. Upper bounds for symmetric Markov transition functions. Ann. Inst. H. Poincaré Probab. Statist. 23 (1987) 245–287. Zbl0634.60066MR898496
  7. [7] Z. Q. Chen and T. Kumagai. Heat kernel estimates for stable-like processes on d-sets. Stochastic Process. Appl. 108 (2003) 27–62. Zbl1075.60556MR2008600
  8. [8] Z. Q. Chen and T. Kumagai. Heat kernel estimates for jump processes of mixed types on metric measure spaces. Probab. Theory Related Fields 140 (2008) 277–317. Zbl1131.60076MR2357678
  9. [9] R. Husseini and M. Kassmann. Markov chain approximations for symmetric jump processes. Potential Anal. 27 (2007) 353–380. Zbl1128.60071MR2353972
  10. [10] D. W. Stroock and W. Zheng. Markov chain approximations to symmetric diffusions. Ann. Inst. H. Poincaré Probab. Statist. 33 (1997) 619–649. Zbl0885.60065MR1473568
  11. [11] P. Sztonyk. Regularity of harmonic functions for anisotropic fractional Laplacian. Math. Nachr. To appear. Zbl1194.47044MR2604123

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