Conservation property of symmetric jump processes
Masamune, Jun ; Uemura, Toshihiro
Annales de l'I.H.P. Probabilités et statistiques, Tome 47 (2011), p. 650-662 / Harvested from Numdam
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Motivés par les récents développements dans la théorie des processus de sauts, nous étudions leur propriété de conservation. Nous montrons qu'un processus de saut est conservatif sous certaines conditions sur la croissance du volume de l'espace sous-tendant et sur le taux de saut du processus. Nous donnons des exemples de processus satisfaisant ces conditions.

Motivated by the recent development in the theory of jump processes, we investigate its conservation property. We will show that a jump process is conservative under certain conditions for the volume-growth of the underlying space and the jump rate of the process. We will also present examples of jump processes which satisfy these conditions.

Publié le : 2011-01-01
DOI : https://doi.org/10.1214/09-AIHP368
Classification:  60J75,  31C25,  35R09 
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     author = {Masamune, Jun and Uemura, Toshihiro},
     title = {Conservation property of symmetric jump processes},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     volume = {47},
     year = {2011},
     pages = {650-662},
     doi = {10.1214/09-AIHP368},
     mrnumber = {2841069},
     zbl = {1230.60090},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIHPB_2011__47_3_650_0}
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Masamune, Jun; Uemura, Toshihiro. Conservation property of symmetric jump processes. Annales de l'I.H.P. Probabilités et statistiques, Tome 47 (2011) pp. 650-662. doi : 10.1214/09-AIHP368. http://gdmltest.u-ga.fr/item/AIHPB_2011__47_3_650_0/

[1] N. Aronszajn and K. T. Smith. Theory of Bessel potentials I. Ann. Inst. Fourier (Grenoble) 11 (1961) 385-475. | Numdam | MR 143935 | Zbl 0102.32401

[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 2465826 | Zbl 1166.60045

[3] M. F. Chen. From Markov Chains to Non-Equilibrium Particle Systems, 2nd edition. World Scientific, River Edge, NJ, 2004. | MR 2091955 | Zbl 0753.60055

[4] Z. Q. Chen, P. Kim and T. Kumagai. Weighted Poincaré inequality and heat kernel estimates for finite range jump processes. Math. Ann. 342 (2008) 833-883. | MR 2443765 | Zbl 1156.60069

[5] 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 2357678 | Zbl 1131.60076

[6] E. B. Davies. The heat kernel bounds, conservation of probability and the Feller property. Festschrift on the occasion of the 70th birthday of Shmuel Agmon. J. Anal. Math. 58 (1992) 99-119. | MR 1226938 | Zbl 0808.58041

[7] S. W. Hawking and G. F. R. Ellis. The Large Scale Structure of Space-Time. Cambridge Monographs on Mathematical Physics 1. Cambridge Univ. Press, London, 1973. | MR 424186 | Zbl 0265.53054

[8] M. Fukushima, Y. Oshima and M. Takeda. Dirichlet Forms and Symmetric Markov Processes. de Gruyter Studies in Mathematics 19. Walter de Gruyter, Berlin, 1994. | MR 1303354 | Zbl 0838.31001

[9] M. P. Gaffney. The conservation property of the heat equation on Riemannian manifolds. Comm. Pure Appl. Math. 12 (1959) 1-11. | MR 102097 | Zbl 0102.09202

[10] A. Grigor'Yan. Stochastically complete manifolds. Dokl. Akad. Nauk SSSR 290 (1986) 534-537 (Russian). | MR 860324 | Zbl 0632.58041

[11] K. Ichihara. Explosion problems for symmetric diffusion processes. Trans. Amer. Math. Soc. 298 (1986) 515-536. | MR 860378 | Zbl 0612.60068

[12] Y. Isozaki and T. Uemura. A family of symmetric stable-like processes and its global path properties. Probab. Math. Statist. 24 (2004) 145-164. | MR 2148226 | Zbl 1063.60119

[13] N. Jacob. Pseudo differential operators and Markov processes, Vol. 1-3. Imperial Colledge Press, London, 2001. | MR 1873235 | Zbl 1076.60003

[14] L. Karp and P. Li. The heat equation on complete Riemannian manifold. Preprint, 1982.

[15] S. Karlin and J. L. Mcgregor. The differential equations of birth-and-death processes, and the Stieltjes moment problem. Trans. Amer. Math. Soc. 85 (1957) 489-546. | MR 91566 | Zbl 0091.13801

[16] J. Masamune. Analysis of the Laplacian of an incomplete manifold with almost polar boundary. Rend. Mat. Appl. (7) 25 (2005) 109-126. | MR 2142127 | Zbl 1086.58014

[17] J. Masamune and T. Uemura. Lp-Liouville property for nonlocal operators. Preprint, 2008.

[18] Y. Oshima. On conservativeness and recurrence criteria for Markov processes. Potential Anal. 1 (1992) 115-131. | MR 1245880 | Zbl 1081.60545

[19] R. L. Schilling. Conservativeness of semigroups generated by pseudo-differential operators. Potential Anal. 9 (1998) 91-104. | MR 1644108 | Zbl 0917.60066

[20] R. L. Schilling and T. Uemura. On the Feller property of Dirichlet forms generated by pseudo-differential operator. Tohoku Math. J. (2) 59 (2007) 401-422. | MR 2365348 | Zbl 1141.31006

[21] K. T. Sturm. Analysis on local Dirichlet spaces. I. Recurrence, conservativeness and Lp-Liouville properties. J. Reine Angew. Math. 456 (1994) 173-196. | MR 1301456 | Zbl 0806.53041

[22] M. Takeda. On the conservativeness of the Brownian motion on a Riemannian manifold. Bull. London Math. Soc. 23 (1991) 86-88. | MR 1111541 | Zbl 0748.60070

[23] T. Uemura. On some path properties of symmetric stable-like processes for one dimension. Potential Anal. 16 (2002) 79-91. | MR 1880349 | Zbl 0998.31005

[24] T. Uemura. On symmetric stable-like processes: Some path properties and generators. J. Theoret. Probab. 17 (2004) 541-555. | MR 2091550 | Zbl 1067.60020