Global heat kernel estimates for symmetric jump processes

Authors:
Zhen-Qing Chen, Panki Kim and Takashi Kumagai

Journal:
Trans. Amer. Math. Soc. **363** (2011), 5021-5055

MSC (2010):
Primary 60J75, 60J35; Secondary 31C25, 31B05

Published electronically:
March 10, 2011

Corrigendum:
Trans. Amer. Math. Soc. 367 (2015), 7515.

MathSciNet review:
2806700

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Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, we study sharp heat kernel estimates for a large class of symmetric jump-type processes in for all . A prototype of the processes under consideration are symmetric jump processes on with jumping intensity

**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**, 10.1090/S0002-9947-08-04544-3**2.**Martin T. Barlow, Richard F. Bass, and Takashi Kumagai,*Parabolic Harnack inequality and heat kernel estimates for random walks with long range jumps*, Math. Z.**261**(2009), no. 2, 297–320. MR**2457301**, 10.1007/s00209-008-0326-5**3.**Martin T. Barlow, Alexander Grigor′yan, 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**, 10.1515/CRELLE.2009.005**4.**Richard F. Bass and David A. Levin,*Transition probabilities for symmetric jump processes*, Trans. Amer. Math. Soc.**354**(2002), no. 7, 2933–2953 (electronic). MR**1895210**, 10.1090/S0002-9947-02-02998-7**5.**Luis A. Caffarelli, Sandro Salsa, and Luis Silvestre,*Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian*, Invent. Math.**171**(2008), no. 2, 425–461. MR**2367025**, 10.1007/s00222-007-0086-6**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), no. 2, suppl., 245–287 (English, with French summary). MR**898496****7.**René Carmona, Wen Chen Masters, and Barry Simon,*Relativistic Schrödinger operators: asymptotic behavior of the eigenfunctions*, J. Funct. Anal.**91**(1990), no. 1, 117–142. MR**1054115**, 10.1016/0022-1236(90)90049-Q**8.**P. Carr, H. Geman, D. Madan and M. Yor. The Fine Structure of Asset Returns: An Empirical Investigation,*Journal of Business*,**75(2)**(2002), 305-332.**9.**Zhen-Qing Chen,*Gaugeability and conditional gaugeability*, Trans. Amer. Math. Soc.**354**(2002), no. 11, 4639–4679 (electronic). MR**1926893**, 10.1090/S0002-9947-02-03059-3**10.**Zhen-Qing Chen, Panki Kim, and Takashi Kumagai,*Weighted Poincaré inequality and heat kernel estimates for finite range jump processes*, Math. Ann.**342**(2008), no. 4, 833–883. MR**2443765**, 10.1007/s00208-008-0258-8**11.**Zhen-Qing Chen, Panki Kim, and Takashi Kumagai,*On heat kernel estimates and parabolic Harnack inequality for jump processes on metric measure spaces*, Acta Math. Sin. (Engl. Ser.)**25**(2009), no. 7, 1067–1086. MR**2524930**, 10.1007/s10114-009-8576-7**12.**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**, 10.1016/S0304-4149(03)00105-4**13.**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**, 10.1007/s00440-007-0070-5**14.**Zhen-Qing Chen and Renming Song,*Drift transforms and Green function estimates for discontinuous processes*, J. Funct. Anal.**201**(2003), no. 1, 262–281. MR**1986161**, 10.1016/S0022-1236(03)00087-9**15.**Z.-Q. Chen and K. Tsuchida. Large deviation, compact embedding and differentiability of spectral functions. In preparation.**16.**Masatoshi Fukushima, Yōichi Ōshima, and Masayoshi Takeda,*Dirichlet forms and symmetric Markov processes*, de Gruyter Studies in Mathematics, vol. 19, Walter de Gruyter & Co., Berlin, 1994. MR**1303354****17.**I. Koponen, Analytic approach to the problem of convergence of truncated Levy flights towards the Gaussian stochastic process,*Physical Review E*,**52**(1995), 1197-1199.**18.**Elliott H. Lieb and Horng-Tzer Yau,*The stability and instability of relativistic matter*, Comm. Math. Phys.**118**(1988), no. 2, 177–213. MR**956165****19.**Rosario N. Mantegna and H. Eugene Stanley,*Stochastic process with ultraslow convergence to a Gaussian: the truncated Lévy flight*, Phys. Rev. Lett.**73**(1994), no. 22, 2946–2949. MR**1303317**, 10.1103/PhysRevLett.73.2946**20.**J. Masamune and T. Uemura. Conservation property of symmetric jump processes. To appear in*Ann. Inst. Henri Poincaré Probab. Stat*.**21.**A. Matacz. Financial modeling and option theory with the truncated Lévy process.*Int. J. Theor. Appl. Finance***3(1)**(2000), 143-160.**22.**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****23.**Jan Rosiński,*Tempering stable processes*, Stochastic Process. Appl.**117**(2007), no. 6, 677–707. MR**2327834**, 10.1016/j.spa.2006.10.003**24.**Masayoshi Takeda and Kaneharu Tsuchida,*Differentiability of spectral functions for symmetric 𝛼-stable processes*, Trans. Amer. Math. Soc.**359**(2007), no. 8, 4031–4054 (electronic). MR**2302522**, 10.1090/S0002-9947-07-04149-9**25.**Kaneharu Tsuchida,*Differentiability of spectral functions for relativistic 𝛼-stable processes with application to large deviations*, Potential Anal.**28**(2008), no. 1, 17–33. MR**2366397**, 10.1007/s11118-007-9065-1

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Additional Information

**Zhen-Qing Chen**

Affiliation:
Department of Mathematics, University of Washington, Seattle, Washington 98195

Email:
zchen@math.washington.edu

**Panki Kim**

Affiliation:
Department of Mathematical Science, Seoul National University, Seoul 151-747, South Korea

Email:
pkim@snu.ac.kr

**Takashi Kumagai**

Affiliation:
Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan

Email:
kumagai@kurims.kyoto-u.ac.jp

DOI:
https://doi.org/10.1090/S0002-9947-2011-05408-5

Keywords:
Dirichlet form,
jump process,
jumping kernel,
parabolic Harnack inequality,
heat kernel estimates

Received by editor(s):
March 23, 2010

Published electronically:
March 10, 2011

Additional Notes:
The first author’s research was partially supported by NSF Grant DMS-0906743.

The second author’s research was supported by a National Research Foundation of Korea Grant funded by the Korean Government (2009-0087117)

The second and the third authors’ research was partially supported by the Global COE program at the Department of Mathematics, Faculty of Science, Kyoto University.

Article copyright:
© Copyright 2011
Zhen-Qing Chen, Panki Kim, and Takashi Kumagai