Abstract
We prove that the parabolic Harnack inequality implies the existence of jump kernel for symmetric pure jump process. This allows us to remove a technical assumption on the jum** measure in the recent characterization of the parabolic Harnack inequality for pure jump processes by Chen, Kumagai and Wang. The key ingredients of our proof are the Lévy system formula and estimates on the heat kernel.
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Barlow, M.T., Bass, R.F.: Stability of parabolic Harnack inequalities. Trans. Amer. Math. Soc. 356(4), 1501–1533 (2004)
Barlow, M.T., Bass, R.F., Kumagai, T.: Stability of parabolic Harnack inequalities on metric measure spaces. J. Math. Soc. Japan 58(2), 485–519 (2006)
Barlow, M.T., Bass, R.F., Chen, Z.-Q., Kassmann, M.: Non-local Dirichlet forms and symmetric jump processes. Trans. Amer. Math. Soc. 361(4), 1963–1999 (2009)
Bass, M.T., Barlow R.F., Kumagai, T.: Parabolic Harnack inequality and heat kernel estimates for random walks with long range jumps. Math. Z. 261 (2), 297–320 (2009)
Barlow, M.T., Grigor’yan, A., Kumagai, T.: On the equivalence of parabolic Harnack inequalities and heat kernel estimates. J. Math. Soc. Japan 64 (4), 1091–1146 (2012)
Bass, R.F., Chen, Z.-Q.: Regularity of harmonic functions for a class of singular stable-like processes. Math. Z. 266, 489–503 (2010)
Chen, Z.-Q.: Private communication (2020)
Chen, Z.-Q., Fukushima, M.: Symmetric Markov Processes, Time Change, and Boundary Theory. Princeton University Press, Princeton (2012)
Chen, Z.-Q., Kumagai, T., Wang, J.: Stability of parabolic Harnack inequalities for symmetric non-local Dirichlet forms. J. Eur. Math. Soc. (JEMS) (to appear) ar**v:1703.09385
Chen, Z.-Q., Kumagai, T., Wang, J.: Mean value inequalities for jump processes. stochastic partial differential equations and related fields, In Honor of Michael röckner, SPDERF, Bielefeld, pp. 421–437, Springer Proc. in Math and Stat., 229 (2018)
Chen, Z.-Q., Kumagai, T., Wang, J.: Heat kernel estimates and parabolic Harnack inequalities for symmetric Dirichlet forms, Adv. Math. (to appear) ar**v:1908.07650v1
Delmotte, T.: Parabolic Harnack inequality and estimates of Markov chains on graphs. Rev. Math. Iberoamericana 15, 181–232 (1999)
Felsinger, M., Kassmann, M.: Local regularity for parabolic nonlocal operators. Comm. Part. Diff. Eq. 38(9), 1539–1573 (2013)
Fukushima, M., Oshima, Y., Takeda, M.1: Dirichlet forms and symmetric Markov processes. Berlin, 2nd ed. (2011)
Grigor’yan, A.: The heat equation on noncompact Riemannian manifolds. (in Russian). Matem. Sbornik. 182, 55–87 (1991). (English transl.) Math. USSR Sbornik 72 (1992), 47–77
Grigor’yan, A., Hu, J., Lau, K.-S.: Generalized capacity, Harnack inequality and heat kernels of Dirichlet forms on metric spaces. J. Math. Soc. Japan 67, 1485–1549 (2015)
Hebisch, W., Saloff-Coste, L.: On the relation between elliptic and parabolic Harnack inequalities. Ann. Inst. Fourier (Grenoble) 51(5), 1437–1481 (2001)
Heinonen, J.: Lectures on Analysis on Metric Spaces, Universitext. Springer-Verlag, New York (2001). x + 140 pp
Kassmann, M.: Harnack inequalities: an introduction. Bound. Value Probl., Art. ID 81415, 21 pp (2007)
Kassmann, M., Kim, K.-Y., Kumagai, T.: Heat kernel bounds for nonlocal operators with singular kernels, (preprint) ar**v:1910.04242 (2019)
Saloff-Coste, L.: A note on Poincaré, Sobolev, and Harnack inequalities. Inter. Math. Res. Notices 2, 27–38 (1992)
Sturm, K.-T.: Analysis on local Dirichlet spaces III. The parabolic Harnack inequality. J. Math. Pures. Appl. 75(9), 273–297 (1996)
Xu, F.: A class of singular symmetric Markov processes. Potential Anal. 38(1), 207–232 (2013)
Acknowledgments
We are grateful to Zhen-Qing Chen for providing us an alternate proof of Lemma 2.4 that avoids the use of near diagonal lower bound. We thank Jian Wang for helpful comments on a previous draft and in particular for the example in Remark 2.5. We thank the anonymous referee for a careful reading of the paper, for helpful suggestions in general and especially concerning Remark 2.6.
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Research partially supported by China Scholarship Council.
Research partially supported by NSERC and the Canada research chairs program.
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Liu, G., Murugan, M. Parabolic Harnack Inequality Implies the Existence of Jump Kernel. Potential Anal 57, 155–166 (2022). https://doi.org/10.1007/s11118-021-09909-0
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DOI: https://doi.org/10.1007/s11118-021-09909-0