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

 

Some theorems on Feller processes: Transience, local times and ultracontractivity


Authors: René L. Schilling and Jian Wang
Journal: Trans. Amer. Math. Soc. 365 (2013), 3255-3286
MSC (2010): Primary 60J25, 60J75, 35S05
Published electronically: August 21, 2012
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Abstract: We present sufficient conditions for the transience and the existence of local times of a Feller process, and the ultracontractivity of the associated Feller semigroup; these conditions are sharp for Lévy processes. The proof uses a local symmetrization technique and a uniform upper bound for the characteristic function of a Feller process. As a by-product, we obtain for stable-like processes (in the sense of R. Bass) on $ \mathbb{R}^d$ with smooth variable index $ \alpha (x)\in (0,2)$ a transience criterion in terms of the exponent $ \alpha (x)$; if $ d=1$ and $ \inf _{x\in \mathbb{R}} \alpha (x)\in (1,2)$, then the stable-like process has local times.


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Additional Information

René L. Schilling
Affiliation: Institut für Mathematische Stochastik, TU Dresden, 01062 Dresden, Germany
Email: rene.schilling@tu-dresden.de

Jian Wang
Affiliation: School of Mathematics and Computer Science, Fujian Normal University, 350007, Fuzhou, People’s Republic of China
Email: jianwang@fjnu.edu.cn

DOI: http://dx.doi.org/10.1090/S0002-9947-2012-05738-2
PII: S 0002-9947(2012)05738-2
Keywords: Feller process, characteristic function, symbol, (local) symmetrization, stable-like process, ultracontractivity, transience, local time
Received by editor(s): August 16, 2011
Received by editor(s) in revised form: October 31, 2011
Published electronically: August 21, 2012
Article copyright: © Copyright 2012 American Mathematical Society