Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Symmetric jump processes: Localization, heat kernels and convergence

Richard F. Bass, Moritz Kassmann, and Takashi Kumagai

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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.

Résumé

Nous considérons des processus symétriques purement discontinus. Nous obtenons des estimations locales pour les probabilités de sortie d’une boule, la continuité hölderienne des fonctions harmoniques et des noyaux de la chaleur, et la convergence d’un suite de tels processus.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 46, Number 1 (2010), 59-71.

Dates
First available in Project Euclid: 1 March 2010

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1267454108

Digital Object Identifier
doi:10.1214/08-AIHP201

Mathematical Reviews number (MathSciNet)
MR2641770

Zentralblatt MATH identifier
1201.60078

Subjects
Primary: 60J35: Transition functions, generators and resolvents [See also 47D03, 47D07]
Secondary: 60J75: Jump processes 45K05: Integro-partial differential equations [See also 34K30, 35R09, 35R10, 47G20]

Keywords
Symmetric jump processes Dirichlet forms Heat kernels Harnack inequalities Weak convergence Non-local operators

Citation

Bass, Richard F.; Kassmann, Moritz; Kumagai, Takashi. Symmetric jump processes: Localization, heat kernels and convergence. Ann. Inst. H. Poincaré Probab. Statist. 46 (2010), no. 1, 59--71. doi:10.1214/08-AIHP201. https://projecteuclid.org/euclid.aihp/1267454108


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References

  • [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.
  • [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.
  • [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.
  • [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.
  • [5] R. F. Bass and T. Kumagai. Symmetric Markov chains on ℤd with unbounded range. Trans. Amer. Math. Soc. 360 (2008) 2041–2075.
  • [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.
  • [7] Z. Q. Chen and T. Kumagai. Heat kernel estimates for stable-like processes on d-sets. Stochastic Process. Appl. 108 (2003) 27–62.
  • [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.
  • [9] R. Husseini and M. Kassmann. Markov chain approximations for symmetric jump processes. Potential Anal. 27 (2007) 353–380.
  • [10] D. W. Stroock and W. Zheng. Markov chain approximations to symmetric diffusions. Ann. Inst. H. Poincaré Probab. Statist. 33 (1997) 619–649.
  • [11] P. Sztonyk. Regularity of harmonic functions for anisotropic fractional Laplacian. Math. Nachr. To appear.