Conservation property of symmetric jump processes
Annales de l'I.H.P. Probabilités et statistiques, Tome 47 (2011) no. 3, pp. 650-662.

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.

DOI : 10.1214/09-AIHP368
Classification : 60J75, 31C25, 35R09
Mots-clés : conservation property, symmetric Dirichlet forms with jumps, derivation property 
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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) no. 3, pp. 650-662. doi : 10.1214/09-AIHP368. http://www.numdam.org/articles/10.1214/09-AIHP368/

[1] N. Aronszajn and K. T. Smith. Theory of Bessel potentials I. Ann. Inst. Fourier (Grenoble) 11 (1961) 385-475. | 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. F. Chen. From Markov Chains to Non-Equilibrium Particle Systems, 2nd edition. World Scientific, River Edge, NJ, 2004. | MR | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

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

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

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

[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 | Zbl

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

[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 | Zbl

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

[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 | Zbl

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

[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 | Zbl

[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 | Zbl

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

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

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

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