Abstract
Let J be the Lévy density of a symmetric Lévy process in \(\mathbb {R}^{d}\) with its Lévy exponent satisfying a weak lower scaling condition at infinity. Consider the non-symmetric and non-local operator
where κ(x, z) is a Borel function on \(\mathbb {R}^{d}\times \mathbb {R}^{d}\) satisfying 0 < κ 0 ≤ κ(x, z) ≤ κ 1, κ(x, z) = κ(x,−z) and |κ(x, z) − κ(y, z)|≤ κ 2|x − y|β for some β ∈ (0, 1]. We construct the heat kernel p κ(t, x, y) of \(\mathcal {L}^{\kappa }\), establish its upper bound as well as its fractional derivative and gradient estimates. Under an additional weak upper scaling condition at infinity, we also establish a lower bound for the heat kernel p κ.
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Acknowledgements
We are grateful to Xicheng Zhang for several valuable comments, in particular for suggesting the improvement of the gradient estimate (1.17). We also thank Karol Szczypkowski for pointing out some mistakes in an earlier version of this paper and Jaehoon Lee for reading the manuscript and giving helpful comments.
Panki Kim was supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government(MSIP) (No. 2016R1E1A1A01941893)
Renming Song supported in part by a grant from the Simons Foundation (208236)
Zoran Vondraček was supported in part by the Croatian Science Foundation under the project 3526.
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Kim, P., Song, R. & Vondraček, Z. Heat Kernels of Non-symmetric Jump Processes: Beyond the Stable Case. Potential Anal 49, 37–90 (2018). https://doi.org/10.1007/s11118-017-9648-4
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DOI: https://doi.org/10.1007/s11118-017-9648-4
Keywords
- Heat kernel estimates
- Subordinate Brownian motion
- Symmetric Lévy process
- Non-symmetric operator
- Non-symmetric Markov process