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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

   

 

Global heat kernel estimates for symmetric jump processes


Authors: Zhen-Qing Chen, Panki Kim and Takashi Kumagai
Journal: Trans. Amer. Math. Soc. 363 (2011), 5021-5055
MSC (2010): Primary 60J75, 60J35; Secondary 31C25, 31B05
Published electronically: March 10, 2011
MathSciNet review: 2806700
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Abstract: In this paper, we study sharp heat kernel estimates for a large class of symmetric jump-type processes in $ \mathbb{R}^d$ for all $ t>0$. A prototype of the processes under consideration are symmetric jump processes on $ \mathbb{R}^d$ with jumping intensity

$\displaystyle \frac{1}{\Phi(\vert x-y\vert)} \int_{[\alpha_1, \alpha_2]} \frac{c(\alpha, x,y)} {\vert x-y\vert^{d+\alpha}} \nu(d\alpha) , $

where $ \nu$ is a probability measure on $ [\alpha_1, \alpha_2] \subset (0, 2)$, $ \Phi$ is an increasing function on $ [ 0, \infty )$ with $ c_1e^{c_2r^{\beta}} \le \Phi(r) \le c_3 e^{c_4r^{\beta}}$ with $ \beta \in (0,\infty)$, and $ c(\alpha , x, y)$ is a jointly measurable function that is bounded between two positive constants and is symmetric in $ (x, y)$. They include, in particular, mixed relativistic symmetric stable processes on $ \mathbb{R}^d$ with different masses. We also establish the parabolic Harnack principle.


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Additional Information

Zhen-Qing Chen
Affiliation: Department of Mathematics, University of Washington, Seattle, Washington 98195
Email: zchen@math.washington.edu

Panki Kim
Affiliation: Department of Mathematical Science, Seoul National University, Seoul 151-747, South Korea
Email: pkim@snu.ac.kr

Takashi Kumagai
Affiliation: Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan
Email: kumagai@kurims.kyoto-u.ac.jp

DOI: http://dx.doi.org/10.1090/S0002-9947-2011-05408-5
PII: S 0002-9947(2011)05408-5
Keywords: Dirichlet form, jump process, jumping kernel, parabolic Harnack inequality, heat kernel estimates
Received by editor(s): March 23, 2010
Published electronically: March 10, 2011
Additional Notes: The first author’s research was partially supported by NSF Grant DMS-0906743.
The second author’s research was supported by a National Research Foundation of Korea Grant funded by the Korean Government (2009-0087117)
The second and the third authors’ research was partially supported by the Global COE program at the Department of Mathematics, Faculty of Science, Kyoto University.
Article copyright: © Copyright 2011 Zhen-Qing Chen, Panki Kim, and Takashi Kumagai