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Escape rate of symmetric jump-diffusion processes


Author: Yuichi Shiozawa
Journal: Trans. Amer. Math. Soc. 368 (2016), 7645-7680
MSC (2010): Primary 31C25; Secondary 60J75
DOI: https://doi.org/10.1090/tran6681
Published electronically: February 10, 2016
MathSciNet review: 3546778
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Abstract: We study the escape rate of symmetric jump-diffusion processes generated by regular Dirichlet forms. We derive an upper bound of the escape rate by using the volume growth of the underlying measure and the growth of the canonical coefficient. Our result allows the (sub-)exponential volume growth and the unboundedness of the canonical coefficient.


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  • [1] Martin T. Barlow, Richard F. Bass, Zhen-Qing Chen, and Moritz Kassmann, Non-local Dirichlet forms and symmetric jump processes, Trans. Amer. Math. Soc. 361 (2009), no. 4, 1963-1999. MR 2465826 (2010e:60163), https://doi.org/10.1090/S0002-9947-08-04544-3
  • [2] Martin T. Barlow, Alexander Grigoryan, and Takashi Kumagai, Heat kernel upper bounds for jump processes and the first exit time, J. Reine Angew. Math. 626 (2009), 135-157. MR 2492992 (2009m:58077), https://doi.org/10.1515/CRELLE.2009.005
  • [3] Richard F. Bass, Adding and subtracting jumps from Markov processes, Trans. Amer. Math. Soc. 255 (1979), 363-376. MR 542886 (81b:60070), https://doi.org/10.2307/1998181
  • [4] Zhen-Qing Chen, Symmetric jump processes and their heat kernel estimates, Sci. China Ser. A 52 (2009), no. 7, 1423-1445. MR 2520585 (2010i:60220), https://doi.org/10.1007/s11425-009-0100-0
  • [5] Zhen-Qing Chen and Takashi Kumagai, Heat kernel estimates for stable-like processes on $ d$-sets, Stochastic Process. Appl. 108 (2003), no. 1, 27-62. MR 2008600 (2005d:60135), https://doi.org/10.1016/S0304-4149(03)00105-4
  • [6] Zhen-Qing Chen and Takashi Kumagai, Heat kernel estimates for jump processes of mixed types on metric measure spaces, Probab. Theory Related Fields 140 (2008), no. 1-2, 277-317. MR 2357678 (2009e:60186), https://doi.org/10.1007/s00440-007-0070-5
  • [7] E. B. Davies, Heat kernel bounds, conservation of probability and the Feller property, J. Anal. Math. 58 (1992), 99-119. Festschrift on the occasion of the 70th birthday of Shmuel Agmon. MR 1226938 (94e:58136), https://doi.org/10.1007/BF02790359
  • [8] Matthew Folz, Volume growth and stochastic completeness of graphs, Trans. Amer. Math. Soc. 366 (2014), no. 4, 2089-2119. MR 3152724, https://doi.org/10.1090/S0002-9947-2013-05930-2
  • [9] Masatoshi Fukushima, Yoichi Oshima, and Masayoshi Takeda, Dirichlet forms and symmetric Markov processes, Second revised and extended edition, de Gruyter Studies in Mathematics, vol. 19, Walter de Gruyter & Co., Berlin, 2011. MR 2778606 (2011k:60249)
  • [10] Matthew P. Gaffney, The conservation property of the heat equation on Riemannian manifolds, Comm. Pure Appl. Math. 12 (1959), 1-11. MR 0102097 (21 #892)
  • [11] A. A. Grigoryan, Stochastically complete manifolds, Dokl. Akad. Nauk SSSR 290 (1986), no. 3, 534-537 (Russian). MR 860324 (88a:58209)
  • [12] Alexander Grigoryan, Integral maximum principle and its applications, Proc. Roy. Soc. Edinburgh Sect. A 124 (1994), no. 2, 353-362. MR 1273753 (95c:35045), https://doi.org/10.1017/S0308210500028511
  • [13] Alexander Grigoryan, Escape rate of Brownian motion on Riemannian manifolds, Appl. Anal. 71 (1999), no. 1-4, 63-89. MR 1690091 (2000c:58065), https://doi.org/10.1080/00036819908840705
  • [14] Alexander Grigor'yan, Heat kernel and analysis on manifolds, AMS/IP Studies in Advanced Mathematics, vol. 47, American Mathematical Society, Providence, RI; International Press, Boston, MA, 2009. MR 2569498 (2011e:58041)
  • [15] Alexander Grigoryan and Elton Hsu, Volume growth and escape rate of Brownian motion on a Cartan-Hadamard manifold, Sobolev spaces in mathematics. II, Int. Math. Ser. (N. Y.), vol. 9, Springer, New York, 2009, pp. 209-225. MR 2484627 (2010f:58045), https://doi.org/10.1007/978-0-387-85650-6_10
  • [16] Alexander Grigor'yan, Jiaxin Hu, and Ka-Sing Lau, Estimates of heat kernels for non-local regular Dirichlet forms, Trans. Amer. Math. Soc. 366 (2014), no. 12, 6397-6441. MR 3267014, https://doi.org/10.1090/S0002-9947-2014-06034-0
  • [17] Alexander Grigor'yan, Xueping Huang, and Jun Masamune, On stochastic completeness of jump processes, Math. Z. 271 (2012), no. 3-4, 1211-1239. MR 2945605, https://doi.org/10.1007/s00209-011-0911-x
  • [18] Alexander Grigor'yan and Mark Kelbert, On Hardy-Littlewood inequality for Brownian motion on Riemannian manifolds, J. London Math. Soc. (2) 62 (2000), no. 2, 625-639. MR 1783649 (2001k:58075), https://doi.org/10.1112/S002461070000123X
  • [19] Elton P. Hsu and Guangnan Qin, Volume growth and escape rate of Brownian motion on a complete Riemannian manifold, Ann. Probab. 38 (2010), no. 4, 1570-1582. MR 2663637 (2011e:58060), https://doi.org/10.1214/09-AOP519
  • [20] Xueping Huang, Escape rate of Markov chains on infinite graphs, J. Theoret. Probab. 27 (2014), no. 2, 634-682. MR 3195830, https://doi.org/10.1007/s10959-012-0456-x
  • [21] Xueping Huang, A note on the volume growth criterion for stochastic completeness of weighted graphs, Potential Anal. 40 (2014), no. 2, 117-142. MR 3152158, https://doi.org/10.1007/s11118-013-9342-0
  • [22] Xueping Huang and Yuichi Shiozawa, Upper escape rate of Markov chains on weighted graphs, Stochastic Process. Appl. 124 (2014), no. 1, 317-347. MR 3131296, https://doi.org/10.1016/j.spa.2013.08.004
  • [23] Kanji Ichihara, Explosion problems for symmetric diffusion processes, Trans. Amer. Math. Soc. 298 (1986), no. 2, 515-536. MR 860378 (88b:60178), https://doi.org/10.2307/2000634
  • [24] Nobuyuki Ikeda, Masao Nagasawa, and Shinzo Watanabe, A construction of Markov processes by piecing out, Proc. Japan Acad. 42 (1966), 370-375. MR 0202197 (34 #2070)
  • [25] A. Khintchine, Zwei Sätze über stochastische Prozesse mit stabilen Verteilungen, Rec. Math. [Mat. Sbornik] N.S. 3 (45) (1938), 577-584.
  • [26] Kazuhiro Kuwae, Functional calculus for Dirichlet forms, Osaka J. Math. 35 (1998), no. 3, 683-715. MR 1648400 (99h:31014)
  • [27] Jun Masamune and Toshihiro Uemura, Conservation property of symmetric jump processes, Ann. Inst. Henri Poincaré Probab. Stat. 47 (2011), no. 3, 650-662 (English, with English and French summaries). MR 2841069 (2012f:60291), https://doi.org/10.1214/09-AIHP368
  • [28] Jun Masamune, Toshihiro Uemura, and Jian Wang, On the conservativeness and the recurrence of symmetric jump-diffusions, J. Funct. Anal. 263 (2012), no. 12, 3984-4008. MR 2990064, https://doi.org/10.1016/j.jfa.2012.09.014
  • [29] P. A. Meyer, Renaissance, recollements, mélanges, ralentissement de processus de Markov, Ann. Inst. Fourier (Grenoble) 25 (1975), no. 3-4, xxiii, 465-497 (French, with English summary). Collection of articles dedicated to Marcel Brelot on the occasion of his 70th birthday. MR 0415784 (54 #3862)
  • [30] Yōichi Ōshima, On conservativeness and recurrence criteria for Markov processes, Potential Anal. 1 (1992), no. 2, 115-131. MR 1245880 (94k:60118), https://doi.org/10.1007/BF01789234
  • [31] S. Ouyang, Volume growth, comparison theorem and escape rate of diffusion process, preprint.
  • [32] René L. Schilling, Conservativeness and extensions of Feller semigroups, Positivity 2 (1998), no. 3, 239-256. MR 1653474 (2000c:47085), https://doi.org/10.1023/A:1009748105208
  • [33] Yuichi Shiozawa, Conservation property of symmetric jump-diffusion processes, Forum Math. 27 (2015), no. 1, 519-548. MR 3334071, https://doi.org/10.1515/forum-2012-0043
  • [34] Yuichi Shiozawa and Toshihiro Uemura, Explosion of jump-type symmetric Dirichlet forms on $ \mathbb{R}^d$, J. Theoret. Probab. 27 (2014), no. 2, 404-432. MR 3195820, https://doi.org/10.1007/s10959-012-0424-5
  • [35] Karl-Theodor Sturm, Analysis on local Dirichlet spaces. I. Recurrence, conservativeness and $ L^p$-Liouville properties, J. Reine Angew. Math. 456 (1994), 173-196. MR 1301456 (95i:31003), https://doi.org/10.1515/crll.1994.456.173
  • [36] Masayoshi Takeda, On a martingale method for symmetric diffusion processes and its applications, Osaka J. Math. 26 (1989), no. 3, 605-623. MR 1021434 (91d:60193)
  • [37] Masayoshi Takeda, On the conservativeness of the Brownian motion on a Riemannian manifold, Bull. London Math. Soc. 23 (1991), no. 1, 86-88. MR 1111541 (92d:58220), https://doi.org/10.1112/blms/23.1.86
  • [38] Jian Wang, Stability of Markov processes generated by Lévy-type operators, Chinese Ann. Math. Ser. A 32 (2011), no. 1, 33-50 (Chinese, with English and Chinese summaries); English transl., Chinese J. Contemp. Math. 32 (2011), no. 1, 33-52. MR 2663819 (2012e:60198)

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Additional Information

Yuichi Shiozawa
Affiliation: Department of Environmental and Mathematical Sciences, Graduate School of Natural Science and Technology, Okayama University, Okayama 700-8530, Japan
Email: shiozawa@ems.okayama-u.ac.jp

DOI: https://doi.org/10.1090/tran6681
Received by editor(s): October 16, 2013
Received by editor(s) in revised form: October 6, 2014
Published electronically: February 10, 2016
Additional Notes: The author was supported in part by the Grant-in-Aid for Young Scientists (B) 23740078.
Article copyright: © Copyright 2016 American Mathematical Society

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