Science in China Series A: Mathematics

, Volume 52, Issue 7, pp 1423–1445

Symmetric jump processes and their heat kernel estimates



We survey the recent development of the DeGiorgi-Nash-Moser-Aronson type theory for a class of symmetric jump processes (or equivalently, a class of symmetric integro-differential operators). We focus on the sharp two-sided estimates for the transition density functions (or heat kernels) of the processes, a priori Hölder estimate and parabolic Harnack inequalities for their parabolic functions. In contrast to the second order elliptic differential operator case, the methods to establish these properties for symmetric integro-differential operators are mainly probabilistic.


symmetric jump process diffusion with jumps pseudo-differential operator Dirichlet form a prior Hölder estimates parabolic Harnack inequality global and Dirichlet heat kernel estimates Lévy system 


60J35 47G30 60J45 31C05 31C25 60J75 


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  1. 1.
    Stroock D W. Diffusion semigroup corresponding to uniformly elliptic divergence form operator. Lecture Notes in Math, 1321: 316–347 (1988)CrossRefMathSciNetGoogle Scholar
  2. 2.
    Bertoin J. Lévy Processes. Cambridge: Cambridge University Press, 1996MATHGoogle Scholar
  3. 3.
    Janicki A, Weron A. Simulation and Chaotic Behavior of α-Stable Processes. New York: Dekker, 1994Google Scholar
  4. 4.
    Klafter J, Shlesinger M F, Zumofen G. Beyond Brownian motion. Physics Today, 49: 33–39 (1996)CrossRefGoogle Scholar
  5. 5.
    Samorodnitsky G, Taqqu M S. Stable Non-Gaussian Random Processes. New York-London: Chapman & Hall, 1994MATHGoogle Scholar
  6. 6.
    Caffarelli L A, Salsa S, Silvestre Luis. Regularity estimates for the solution and the free boundary to theobstacle problem for the fractional Laplacian. Invent Math, 171(1): 425–461 (2008)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Silvestre L. Hölder estimates for solutions of integro differential equations like the fractional Laplace. Indiana Univ Math J, 55: 1155–1174 (2006)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Blumenthal R M, Getoor R K. Some theorems on stable processes. Trans Amer Math Soc, 95: 263–273 (1960)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Kolokoltsov V. Symmetric stable laws and stable-like jump-diffusions. Proc London Math Soc, 80: 725–768 (2000)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Bass R F, Levin D A. Transition probabilities for symmetric jump processes. Trans Amer Math Soc, 354: 2933–2953 (2002)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Chen Z-Q, Kumagai T. Heat kernel estimates for stable-like processes on d-sets. Stochastic Process Appl, 108: 27–62 (2003)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Chen Z-Q, Kumagai T. Heat kernel estimates for jump processes of mixed types on metric measure spaces. Probab Theory Related Fields, 140: 277–317 (2008)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Chen Z-Q, Kim P, Kumagai T. Weighted Poincaré inequality and heat kernel estimates for finite range jump processes. Math Ann, 342: 833–883 (2008)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Hurst S R, Platen E, Rachev S T. Option pricing for a logstable asset price model. Math Comput Modelling, 29: 105–119 (1999)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Matacz A. Financial modeling and option theory with the truncated Lévy process. Int J Theor Appl Finance, 3(1): 143–160 (2000)MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Barlow M T, Bass R F, Chen Z-Q, et al. Non-local Dirichlet forms and symmetric jump processes. Trans Amer Math Soc, 361: 1963–1999 (2009)MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Chen Z-Q, Kumagai T. A priori Hölder estimate, parabolic Harnack inequality and heat kernel estimates for diffusions with jumps. Revista Mathemática Iberoamericana, to appear (2009)Google Scholar
  18. 18.
    Zhang Q S. The boundary behavior of heat kernels of Dirichlet Laplacians. J Differential Equations, 182: 416–430 (2002)MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Chen Z-Q, Kim P, Song R. Heat kernel estimates for Dirichlet fractional Laplacian. J European Math Soc, to appear (2009)Google Scholar
  20. 20.
    Chen Z-Q, Kim P, Song R. Two-sided heat kernel estimates for censored stable-like processes. Probab Theory Related Fields, to appear (2009)Google Scholar
  21. 21.
    Chen Z-Q, Kim P, Song R. Sharp heat kernel estimates for relativistic stable processes. Preprint, 2009Google Scholar
  22. 22.
    Bass R F. SDEs with jumps. Notes for Cornell Summer School 2007.
  23. 23.
    Chen Z-Q. Multidimensional symmetric stable processes. Korean J Comput Appl Math, 6: 227–266 (1999)MATHMathSciNetGoogle Scholar
  24. 24.
    Benjamini I, Chen Z-Q, Rohde S. Boundary trace of reflecting Brownian motions. Probab Theory Relat Fields, 129: 1–17 (2004)MATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Bass R F, Levin D A. Harnack inequalities for jump processes. Potential Anal, 17: 375–388 (2002)MATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Chen Z-Q, Song R. Drift transforms and Green function estimates for discontinuous processes. J Funct Anal, 201: 262–281 (2003)MATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Herbst I W, Sloan D. Perturbation of translation invariant positive preserving semigroups on L 2(ℝN). Trans Amer Math Soc, 236: 325–360 (1978)MATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Meyer P A. Renaissance, recollements, mélanges, ralentissement de processus de Markov. Ann Inst Fourier, 25: 464–497 (1975)Google Scholar
  29. 29.
    Ikeda N, Nagasawa M, Watanabe S. A construction of Markov process by piecing out. Proc Japan Acad Ser A Math Sci, 42: 370–375 (1966)MATHMathSciNetGoogle Scholar
  30. 30.
    Barlow M T, Grigor’yan A, Kumagai T. Heat kernel upper bounds for jump processes and the first exit time. J Reine Angew Math, 626: 135–157 (2009)MATHMathSciNetGoogle Scholar
  31. 31.
    Song R, Vondraĉek Z. Parabolic Harnack inequality for the mixture of Brownian motion and stable process. Tohoku Math J, 59: 1–19 (2007)MATHCrossRefMathSciNetGoogle Scholar
  32. 32.
    Carlen E A, Kusuoka S, Stroock D W. Upper bounds for symmetric Markov transition functions. Ann Inst Heri Poincaré-Probab Statist, 23: 245–287 (1987)MathSciNetGoogle Scholar
  33. 33.
    Chen Z-Q, Song R. Estimates on Green functions and Poisson kernels of symmetric stable processes. Math Ann, 312: 465–601 (1998)MATHCrossRefMathSciNetGoogle Scholar
  34. 34.
    Kulczycki T. Properties of Green function of symmetric stable processes. Probab Math Stat, 17: 381–406 (1997)MathSciNetGoogle Scholar
  35. 35.
    Chen Z-Q, Tokle J. Global heat kernel estimates for fractional Laplacians in un bounded open sets. Preprint, 2009Google Scholar
  36. 36.
    Bogdan K, Burdzy K, Chen Z Q. Censored stable processes. Probab Theory Related Fields, 127: 89–152 (2003)MATHCrossRefMathSciNetGoogle Scholar
  37. 37.
    Chen Z-Q, Kim P. Green function estimate for censored stable processes. Probab Theory Related Fields, 124: 595–610 (2002)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Science in China Press and Springer-Verlag GmbH 2009

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of WashingtonSeattleUSA
  2. 2.Department of MathematicsBeijing Institute of TechnologyBeijingChina

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