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

 

Non-local Dirichlet forms and symmetric jump processes


Authors: Martin T. Barlow, Richard F. Bass, Zhen-Qing Chen and Moritz Kassmann
Journal: Trans. Amer. Math. Soc. 361 (2009), 1963-1999
MSC (2000): Primary 60J35; Secondary 60J75, 45K05, 31B05
Published electronically: October 23, 2008
MathSciNet review: 2465826
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Abstract: We consider the non-local symmetric Dirichlet form $ (\mathcal{E}, \mathcal{F})$ given by

$\displaystyle \mathcal{E} (f,f)=\int\limits_{\mathbb{R}^d} \int\limits_{\mathbb{R}^d} (f(y)-f(x))^2 J(x,y) \, dx\, dy $

with $ \mathcal{F}$ the closure with respect to $ \mathcal{E}_1$ of the set of $ C^1$ functions on $ \mathbb{R}^d$ with compact support, where $ \mathcal{E}_1 (f, f):=\mathcal{E} (f, f)+\int_{\mathbb{R}^d} f(x)^2 dx$, and where the jump kernel $ J$ satisfies

$\displaystyle \kappa_1\vert y-x\vert^{-d-\alpha} \leq J(x,y) \leq \kappa_2\vert y-x\vert^{-d-\beta} $

for $ 0<\alpha< \beta <2, \, \vert x-y\vert<1$. This assumption allows the corresponding jump process to have jump intensities whose sizes depend on the position of the process and the direction of the jump. We prove upper and lower estimates on the heat kernel. We construct a strong Markov process corresponding to $ (\mathcal{E}, \mathcal{F})$. We prove a parabolic Harnack inequality for non-negative functions that solve the heat equation with respect to $ \mathcal{E}$. Finally we construct an example where the corresponding harmonic functions need not be continuous.


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Additional Information

Martin T. Barlow
Affiliation: Department of Mathematics, University of British Columbia, Vancouver, British Columbia, Canada V6T 1Z2
Email: barlow@math.ubc.ca

Richard F. Bass
Affiliation: Department of Mathematics, University of Connecticut, Storrs, Connecticut 06269-3009
Email: bass@math.uconn.edu

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

Moritz Kassmann
Affiliation: Institut für Angewandte Mathematik, Universität Bonn, Beringstrasse 6, D-53115 Bonn, Germany
Email: kassmann@iam.uni-bonn.de

DOI: http://dx.doi.org/10.1090/S0002-9947-08-04544-3
PII: S 0002-9947(08)04544-3
Keywords: Jump processes, symmetric processes, integro-differential operators, Harnack inequality, Dirichlet forms, heat kernel, harmonic, parabolic.
Received by editor(s): September 29, 2006
Received by editor(s) in revised form: May 4, 2007
Published electronically: October 23, 2008
Additional Notes: The research of the first author was partially supported by NSERC (Canada)
The research of the second author was partially supported by NSF grant DMS-0601783.
The research of the third author was partially supported by NSF grant DMS-0600206.
The research of the fourth author was partially supported by DFG (Germany) through Sonderforschungsbereich 611.
Article copyright: © Copyright 2008 Martin T. Barlow, Richard F. Bass, Zhen-Qing Chen, and Moritz Kassmann