Advertisement

Mean Value Inequalities for Jump Processes

  • Zhen-Qing Chen
  • Takashi KumagaiEmail author
  • Jian Wang
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 229)

Abstract

Parabolic Harnack inequalities are one of the most important inequalities in analysis and PDEs, partly because they imply Hölder regularity of the solutions of heat equations. Mean value inequalities play an important role in deriving parabolic Harnack inequalities. In this paper, we first survey the recent results obtained in Chen et al. (Stability of heat kernel estimates for symmetric non-local Dirichlet forms, 2016, [15]; Stability of parabolic Harnack inequalities for symmetric non-local Dirichlet forms, 2016, [16]) on the study of stability of heat kernel estimates and parabolic Harnack inequalities for symmetric jump processes on general metric measure spaces. We then establish the \(L^p\)-mean value inequalities for all \(p\in (0, 2]\) for these processes.

Keywords

Symmetric jump process Heat kernel estimate Harnack inequality Stability Mean value inequality 

MSC 2010

Primary 31B05 60J35 60J75 Secondary 31C25 35K08 60J45 

References

  1. 1.
    Andres, S., Barlow, M.T.: Energy inequalities for cutoff-functions and some applications. J. Reine Angew. Math. 699, 183–215 (2015)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Aronson, D.G.: Bounds on the fundamental solution of a parabolic equation. Bull. Am. Math. Soc. 73, 890–1896 (1967)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Barlow, M.T., Bass, R.F.: Stability of parabolic Harnack inequalities. Trans. Am. Math. Soc. 356, 1501–1533 (2003)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Barlow, M.T., Bass, R.F., Kumagai, T.: Stability of parabolic Harnack inequalities on metric measure spaces. J. Math. Soc. Jpn. 58, 485–519 (2006)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Barlow, M.T., Bass, R.F., Kumagai, T.: Parabolic Harnack inequality and heat kernel estimates for random walks with long range jumps. Math. Z. 261, 297–320 (2009)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Barlow, M.T., Grigor’yan, A., Kumagai, T.: On the equivalence of parabolic Harnack inequalities and heat kernel estimates. J. Math. Soc. Jpn. 64, 1091–1146 (2012)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Björn, A., Björn, J.: Nonlinear Potential Theory on Metric Spaces. EMS Tracts in Mathematics, vol. 17. European Mathematical Society, Zurich (2011)CrossRefGoogle Scholar
  8. 8.
    Castro, A.D., Kuusi, T., Palatucci, G.: Nonlocal Harnack inequalities. J. Funct. Anal. 267, 1807–1836 (2014)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Castro, A.D., Kuusi, T., Palatucci, G.: Local behavior of fractional \(p\)-minimizers. Annales de l’Institut Henri Poincaré (C) Non Linear Anal. 33, 1279–1299 (2016)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Chen, Z.-Q.: On notions of harmonicity. Proc. Am. Math. Soc. 137, 3497–3510 (2009)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Chen, Z.-Q., Fukushima, M.: Symmetric Markov Processes, Time Change, and Boundary Theory. Princeton University Press, Princeton (2012)zbMATHGoogle Scholar
  12. 12.
    Chen, Z.-Q., Kumagai, T.: Heat kernel estimates for stable-like processes on \(d\)-sets. Stoch. Process. Appl. 108, 27–62 (2003)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Chen, Z.-Q., Kumagai, T.: Heat kernel estimates for jump processes of mixed types on metric measure spaces. Probab. Theory Relat. Fields 140, 277–317 (2008)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Chen, Z.-Q., Kuwae, K.: On subhamonicity for symmetric Markov processes. J. Math. Soc. Jpn. 64, 1181–1209 (2012)CrossRefGoogle Scholar
  15. 15.
    Chen, Z.-Q., Kumagai, T., Wang, J.: Stability of heat kernel estimates for symmetric non-local Dirichlet forms. Preprint 2016. arXiv:1604.04035
  16. 16.
    Chen, Z.-Q., Kumagai, T., Wang, J.: Stability of parabolic Harnack inequalities for symmetric non-local Dirichlet forms. Preprint 2016. arXiv:1609.07594
  17. 17.
    Delmotte, T.: Parabolic Harnack inequality and estimates of Markov chains on graphs. Rev. Math. Iberoam. 15, 181–232 (1999)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Fukushima, M., Oshima, Y., Takeda, M.: Dirichlet Forms and Symmetric Markov Processes, 2 rev. and ext. edn. de Gruyter, Berlin (2011)Google Scholar
  19. 19.
    Giaquinta, M., Giusti, E.: On the regularity of the minima of variational integrals. Acta Math. 148, 31–46 (1982)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Reprint of the 1998 edition. Classics in Mathematics, xiv+517 pp. Springer, Berlin (2001)Google Scholar
  21. 21.
    Grigor’yan, A.: The heat equation on noncompact Riemannian manifolds. (in Russian) Matem. Sbornik. 182, 55-87 (1991); (English transl.) Math. USSR. Sbornik 72, 47–77 (1992)Google Scholar
  22. 22.
    Grigor’yan, A., Hu, J.: Upper bounds of heat kernels on doubling spaces. Mosco Math. J. 14, 505–563 (2014)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Grigor’yan, A., Hu, E., Hu, J.: Two-sided estimates of heat kernels of jump type Dirichlet forms. Preprint 2016. https://www.math.uni-bielefeld.de/~grigor/gcap.pdf
  24. 24.
    Grigor’yan, A., Hu, J., Lau, K.-S.: Generalized capacity, Harnack inequality and heat kernels on metric spaces. J. Math. Soc. Jpn. 67, 1485–1549 (2015)CrossRefGoogle Scholar
  25. 25.
    Kuusi, T., Mingione, G., Sire, Y.: Nonlocal equations with measure data. Commun. Math. Phys. 337, 1317–1368 (2015)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Li, P., Yau, S.-T.: On the parabolic kernel of the Schrödinger operator. Acta Math. 156(3–4), 153–201 (1986)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Ma, Z.-M., Röckner, M.: Introduction to the Theory of (Non-Symmetric) Dirichlet Forms. Springer, New York (1992)CrossRefGoogle Scholar
  28. 28.
    Moser, J.: On Harnack’s inequality for parabolic differential equations. Commun. Pure Appl. Math. 17, 101–134 (1964)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Murugan, M., Saloff-Coste, L.: Heat kernel estimates for anomalous heavy-tailed random walks. Preprint 2015. arXiv:1512.02361
  30. 30.
    Saloff-Coste, L.: A note on Poincaré, Sobolev, and Harnack inequalities. Int. Math. Res. Notices 2, 27–38 (1992)CrossRefGoogle Scholar
  31. 31.
    Saloff-Coste, L.: Aspects of Sobolev-type Inequalities. London Mathematical Society Lecture Notes, vol. 289. Cambridge University Press, Cambridge (2002)zbMATHGoogle Scholar
  32. 32.
    Sturm, K.-T.: Analysis on local Dirichlet spaces -III. The parabolic Harnack inequality. J. Math. Pures Appl. 75, 273–297 (1996)MathSciNetzbMATHGoogle Scholar
  33. 33.
    Torchinsky, A.: Real-variable Methods in Harmonic Analysis. Academic Press, London (1986)zbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.University of WashingtonSeattleUSA
  2. 2.RIMSKyoto UniversityKyotoJapan
  3. 3.Fujian Normal UniversityFuzhouPeople’s Republic of China

Personalised recommendations