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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Spatial asymptotics at infinity for heat kernels of integro-differential operators


Authors: Kamil Kaleta and Paweł Sztonyk
Journal: Trans. Amer. Math. Soc. 371 (2019), 6627-6663
MSC (2010): Primary 47D03, 60J35, 35A08; Secondary 60G51, 60E07, 35S10
DOI: https://doi.org/10.1090/tran/7538
Published electronically: October 24, 2018
MathSciNet review: 3937340
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Abstract: We study a spatial asymptotic behavior at infinity of kernels $ p_t(x)$ for convolution semigroups of nonlocal pseudo-differential operators. We give general and sharp sufficient conditions under which the limits

$\displaystyle \lim _{r \to \infty } \frac {p_t(r\theta -y)}{t \, \nu (r\theta )}, \quad t \in T, \ \ \theta \in E, \ \ y \in \mathbb{R}^d$    

exist and can be effectively computed. Here $ \nu $ is the corresponding Lévy density, $ T \subset (0,\infty )$ is a bounded time-set, and $ E$ is a subset of the unit sphere in $ \mathbb{R}^d$, $ d \geq 1$. Our results are local on the unit sphere. They apply to a wide class of convolution semigroups, including those corresponding to highly asymmetric (finite and infinite) Lévy measures. Key examples include fairly general families of stable, tempered stable, jump-diffusion, and compound Poisson semigroups. A main emphasis is put on the semigroups with Lévy measures that are exponentially localized at infinity, for which our assumptions and results are strongly related to the existence of the multidimensional exponential moments. Here a key example is the evolution semigroup corresponding to the so-called quasi-relativistic Hamiltonian $ \sqrt {-\Delta +m^2} - m$, $ m>0$. As a byproduct, we also obtain sharp two-sided estimates of the kernels $ p_t$ in generalized cones, away from the origin.

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Additional Information

Kamil Kaleta
Affiliation: Faculty of Pure and Applied Mathematics, Wrocław University of Science and Technology, Wybrzeże Wyspiańskiego 27, 50-370 Wrocław, Poland; and Institut für mathematische Stochastik, Fachrichtung Mathematik, Technische Universität Dresden, 01062 Dresden, Germany
Email: kamil.kaleta@pwr.edu.pl

Paweł Sztonyk
Affiliation: Faculty of Pure and Applied Mathematics, Wrocław University of Science and Technology, Wybrzeże Wyspiańskiego 27, 50-370 Wrocław, Poland
Email: pawel.sztonyk@pwr.edu.pl

DOI: https://doi.org/10.1090/tran/7538
Keywords: Convolution semigroup, L\'evy measure, L\'evy process, relativistic process, tempered process, relativistic Hamiltonian, convolution of measures, transition density, heat kernel, asymptotics, subexponential decay, exponential decay, exponential moment, light tail
Received by editor(s): June 19, 2017
Received by editor(s) in revised form: February 8, 2018
Published electronically: October 24, 2018
Additional Notes: Research was supported in part by the National Science Centre, Poland, grant no. 2015/17/B/ST1/01233, and by the Alexander von Humboldt Foundation, Germany. We also thank the Alexander von Humboldt Foundation for funding our research stays at Institut für Mathematische Stochastik, TU Dresden, Germany, during which parts of this paper were written. We are very grateful to our host, Professor René L. Schilling, for his kind hospitality and numerous discussions during these stays.
Article copyright: © Copyright 2018 American Mathematical Society