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Parabolic Harnack Inequality Implies the Existence of Jump Kernel

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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 jumping 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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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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Authors and Affiliations

  1. Department of Mathematical Sciences, Tsinghua University, Beijing, 100084, China

    Guanhua Liu

  2. Department of Mathematics, University of British Columbia, Vancouver, BC V6T 1Z2, Canada

    Guanhua Liu & Mathav Murugan

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  1. Guanhua Liu
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  2. Mathav Murugan
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Correspondence to Mathav Murugan.

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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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