Symmetric jump processes : localization, heat kernels and convergence
Annales de l'I.H.P. Probabilités et statistiques, Tome 46 (2010) no. 1, pp. 59-71.

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.

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.

DOI : 10.1214/08-AIHP201
Classification : 60J35, 60J75, 45K05
Mots-clés : symmetric jump processes, Dirichlet forms, heat kernels, Harnack inequalities, weak convergence, non-local operators 
@article{AIHPB_2010__46_1_59_0,
     author = {Bass, Richard F. and Kassmann, Moritz and Kumagai, Takashi},
     title = {Symmetric jump processes : localization, heat kernels and convergence},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     pages = {59--71},
     publisher = {Gauthier-Villars},
     volume = {46},
     number = {1},
     year = {2010},
     doi = {10.1214/08-AIHP201},
     mrnumber = {2641770},
     zbl = {1201.60078},
     language = {en},
     url = {http://www.numdam.org/articles/10.1214/08-AIHP201/}
}
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
SP  - 59
EP  - 71
VL  - 46
IS  - 1
PB  - Gauthier-Villars
UR  - http://www.numdam.org/articles/10.1214/08-AIHP201/
DO  - 10.1214/08-AIHP201
LA  - en
ID  - AIHPB_2010__46_1_59_0
ER  - 
%0 Journal Article
%A Bass, Richard F.
%A Kassmann, Moritz
%A Kumagai, Takashi
%T Symmetric jump processes : localization, heat kernels and convergence
%J Annales de l'I.H.P. Probabilités et statistiques
%D 2010
%P 59-71
%V 46
%N 1
%I Gauthier-Villars
%U http://www.numdam.org/articles/10.1214/08-AIHP201/
%R 10.1214/08-AIHP201
%G en
%F AIHPB_2010__46_1_59_0
Bass, Richard F.; Kassmann, Moritz; Kumagai, Takashi. Symmetric jump processes : localization, heat kernels and convergence. Annales de l'I.H.P. Probabilités et statistiques, Tome 46 (2010) no. 1, pp. 59-71. doi : 10.1214/08-AIHP201. http://www.numdam.org/articles/10.1214/08-AIHP201/

[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. | Numdam | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[5] R. F. Bass and T. Kumagai. Symmetric Markov chains on ℤd with unbounded range. Trans. Amer. Math. Soc. 360 (2008) 2041-2075. | MR | Zbl

[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. | Numdam | MR | Zbl

[7] Z. Q. Chen and T. Kumagai. Heat kernel estimates for stable-like processes on d-sets. Stochastic Process. Appl. 108 (2003) 27-62. | MR | Zbl

[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. | MR | Zbl

[9] R. Husseini and M. Kassmann. Markov chain approximations for symmetric jump processes. Potential Anal. 27 (2007) 353-380. | MR | Zbl

[10] D. W. Stroock and W. Zheng. Markov chain approximations to symmetric diffusions. Ann. Inst. H. Poincaré Probab. Statist. 33 (1997) 619-649. | Numdam | MR | Zbl

[11] P. Sztonyk. Regularity of harmonic functions for anisotropic fractional Laplacian. Math. Nachr. To appear. | MR | Zbl

Cité par Sources :