Mathematische Annalen

, Volume 342, Issue 4, pp 833–883 | Cite as

Weighted Poincaré inequality and heat kernel estimates for finite range jump processes

  • Zhen-Qing Chen
  • Panki KimEmail author
  • Takashi Kumagai


It is well-known that there is a deep interplay between analysis and probability theory. For example, for a Markovian infinitesimal generator \({\mathcal{L}}\) , the transition density function p(t, x, y) of the Markov process associated with \({\mathcal{L}}\) (if it exists) is the fundamental solution (or heat kernel) of \({\mathcal{L}}\) . A fundamental problem in analysis and in probability theory is to obtain sharp estimates of p(t, x, y). In this paper, we consider a class of non-local (integro-differential) operators \({\mathcal{L}}\) on \({\mathbb{R}^d}\) of the form
$$\mathcal{L}u(x) = \lim\limits_{{\varepsilon \downarrow 0}} \int\limits_{\{y\in \mathbb {R}^d: \, |y-x| > \varepsilon\}} (u(y)-u(x)) J(x, y) dy,$$
where \({\displaystyle J(x, y)= \frac{c (x, y)}{|x-y|^{d+\alpha}} {\bf 1}_{\{|x-y| \leq \kappa\}}}\) for some constant \({\kappa > 0}\) and a measurable symmetric function c(x, y) that is bounded between two positive constants. Associated with such a non-local operator \({\mathcal{L}}\) is an \({\mathbb{R}^d}\) -valued symmetric jump process of finite range with jumping kernel J(x, y). We establish sharp two-sided heat kernel estimate and derive parabolic Harnack principle for them. Along the way, some new heat kernel estimates are obtained for more general finite range jump processes that were studied in (Barlow et al. in Trans Am Math Soc, 2008). One of our key tools is a new form of weighted Poincaré inequality of fractional order, which corresponds to the one established by Jerison in (Duke Math J 53(2):503–523, 1986) for differential operators. Using Meyer’s construction of adding new jumps, we also obtain various a priori estimates such as Hölder continuity estimates for parabolic functions of jump processes (not necessarily of finite range) where only a very mild integrability condition is assumed for large jumps. To establish these results, we employ methods from both probability theory and analysis extensively.

Mathematics Subject Classification (2000)

Primary 60J75 60J35 Secondary 31C25 31C05 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Barlow, M.T., Bass, R.F., Chen, Z.-Q., Kassmann, M.: Non-local Dirichlet forms and symmetric jump processes. Trans. Am. Math. Soc. (to appear)Google Scholar
  2. 2.
    Barlow, M.T., Bass, R.F., Kumagai, T.: Parabolic Harnack inequality and heat kernel estimates for random walks with long range jumps. Math. Z. (to appear)Google Scholar
  3. 3.
    Barlow, M.T., Grigor’yan, A., Kumagai, T.: Heat kernel upper bounds for jump processes. J. Reine Angew. Math. (to appear)Google Scholar
  4. 4.
    Bass R.F., Levin D.A.: Transition probabilities for symmetric jump processes. Trans. Am. Math. Soc. 354, 2933–2953 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Bertoin J.: Lévy Processes. Cambridge University Press, Cambridge (1996)zbMATHGoogle Scholar
  6. 6.
    Blumenthal R.M., Getoor R.K.: Markov Processes and Potential Theory. Academic Press, Reading (1968)zbMATHGoogle Scholar
  7. 7.
    Caffarelli L.A., Salsa S., Silvestre L.: Regularity estimates for the solution and the free boundary to the obstacle problem for the fractional Laplacian. Invent. Math. 171(1), 425–461 (2008)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Carlen E.A., Kusuoka S., Stroock D.W.: Upper bounds for symmetric Markov transition functions. Ann. Inst. H. Poincaré Prob. Stat. 23, 245–287 (1987)MathSciNetGoogle Scholar
  9. 9.
    Chen Z.-Q.: Gaugeability and conditional gaugeability. Trans. Am. Math. Soc. 354, 4639–4679 (2002)zbMATHCrossRefGoogle Scholar
  10. 10.
    Chen, Z.-Q., Kim, P., Kumagai, T.: Notes on heat kernel estimates and parabolic Harnack inequality for jump processes, in preparationGoogle Scholar
  11. 11.
    Chen Z.-Q., Kumagai T.: Heat kernel estimates for stable-like processes on d-sets. Stoch. Process Appl. 108, 27–62 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Chen Z.-Q., Kumagai T.: Heat kernel estimates for jump processes of mixed types on metric measure spaces. Prob. Theory Relat. Fields 140, 277–317 (2008)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Chen, Z.-Q., Rohde, S.: SLE driven by symmetric stable processes. Preprint (2007)Google Scholar
  14. 14.
    Fabes E.B., Stroock D.W.: A new proof of Moser’s parabolic Harnack inequality using the old ideas of Nash. Arch. Rational Mech. Anal. 96(4), 327–338 (1986)zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Fukushima M., Oshima Y., Takeda M.: Dirichlet forms and symmetric Markov processes. de Gruyter, Berlin (1994)zbMATHGoogle Scholar
  16. 16.
    Gilbarg D., Trudinger N.S.: Elliptic Partial Differential Equations of Second Order, 2nd edn. Springer, Heidelberg (1983)zbMATHGoogle Scholar
  17. 17.
    Grzywny, T., Ryznar, M.: Estimates of Green function for some perturbations of fractional Laplacian. Illinois J. Math. (to appear)Google Scholar
  18. 18.
    Hu J., Kumagai T.: Nash-type inequalities and heat kernels for non-local Dirichlet forms. Kyushu J. Math. 60, 245–265 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Hurst S.R., Platen E., Rachev S.T.: Option pricing for a logstable asset price model. Math. Comput. Model. 29, 105–119 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Ikeda N., Watanabe S.: Stochastic Differential Equations and Diffusion Processes, 2nd edn. North-Holland Publishing Co., Amsterdam (1989)zbMATHGoogle Scholar
  21. 21.
    Janicki A., Weron A.: Simulation and Chaotic Behavior of α-Stable Processes. Dekker, New York (1994)Google Scholar
  22. 22.
    Jerison D.: The Poincaré inequality for vector fields satisfying Hörmander’s condition. Duke Math. J. 53(2), 503–523 (1986)zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Kassmann, M.: A priori estimates for integro-differential operators with measurable kernels. Calc. Var. Partial Diff. Equat. (to appear)Google Scholar
  24. 24.
    Kim P., Song R.: Potential theory of truncated stable processes. Math. Z. 256(1), 139–173 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Kim P., Song R.: Boundary behavior of harmonic functions for truncated stable processes. J. Theor. Prob. 21, 287–321 (2008)zbMATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Klafter J., Shlesinger M.F., Zumofen G.: Beyond Brownian motion. Phys. Today 49, 33–39 (1996)CrossRefGoogle Scholar
  27. 27.
    Kolokoltsov V.: Symmetric stable laws and stable-like jump-diffusions. Proc. Lond. Math. Soc. 80, 725–768 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Matacz A.: Financial modeling and option theory with the truncated Lévy process. Int. J. Theor. Appl. Finance 3(1), 143–160 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    Saloff-Coste L.: Aspects of Sobolev-type Inequalities. Cambridge University Press, Cambridge (2002)zbMATHGoogle Scholar
  30. 30.
    Saloff-Coste L., Stroock D.W.: Opérateurs uniformément sous-elliptiques sur les groupes de Lie. J. Funct. Anal. 98(1), 97–121 (1991)zbMATHCrossRefMathSciNetGoogle Scholar
  31. 31.
    Silvestre L.: Hölder estimates for solutions of integro differential equations like the fractional Laplace. Indiana Univ. Math. J. 55, 1155–1174 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  32. 32.
    Takeda M., Tsuchida K.: Criticality of generalized Schrödinger operators and differentiability of spectral functions. Adv. Stud. Pure Math. 41, 333–350 (2004)MathSciNetGoogle Scholar
  33. 33.
    Takeda M., Tsuchida K.: Differentiability of spectral functions for symmetric α-stable processes. Trans. Am. Math. Soc. 359, 4031–4054 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  34. 34.
    Tsuchida K.: Differentiability of spectral functions for relativistic α-stable processes with application to large deviations. Potential Anal. 28(1), 17–33 (2008)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of WashingtonSeattleUSA
  2. 2.Department of MathematicsSeoul National UniversitySeoulSouth Korea
  3. 3.Department of Mathematics, Faculty of ScienceKyoto UniversityKyotoJapan

Personalised recommendations