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Pseudo-differential operators and Markov semigroups on compact Lie groups

Applebaum, D. (2011) Pseudo-differential operators and Markov semigroups on compact Lie groups. Journal of Mathematical Analysis and Applications, 384 (2). pp. 331-348. ISSN 0022-247X

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Abstract

We extend the Ruzhansky-Turunen theory of pseudo-differential operators on compact Lie groups into a tool that can be used to investigate group-valued Markov processes in the spirit of the work in Euclidean spaces of N. Jacob and collaborators. Feller semigroups, their generators and resolvents are exhibited as pseudo-differential operators and the symbols of the operators forming the semigroup are expressed in terms of the Fourier transform of the transition kernel. The symbols are explicitly computed for some examples including the Feller processes associated to stochastic flows arising from solutions of stochastic differential equations on the group driven by Levy processes. We study a family of Levy-type linear operators on general Lie groups that are pseudo-differential operators when the group is compact and find conditions for them to give rise to symmetric Dirichlet forms. (C) 2011 Elsevier Inc. All rights reserved.

Item Type: Article
Copyright, Publisher and Additional Information: © 2011 Elsevier. This is an author produced version of a paper subsequently published in Journal of Mathematical Analysis and Applications. Uploaded in accordance with the publisher's self-archiving policy.
Keywords: Feller semigroup; Pseudo-differential operator; Symbol; Fourier transform; Peter-Weyl theorem; Lie group; Lie algebra; Convolution semigroup; Courrege-Hunt operator; Sobolev space; Dirichlet form; Beurling-Deny representation
Institution: The University of Sheffield
Academic Units: The University of Sheffield > Faculty of Science (Sheffield) > School of Mathematics and Statistics (Sheffield)
Depositing User: Miss Anthea Tucker
Date Deposited: 16 Sep 2011 10:54
Last Modified: 08 Feb 2013 17:34
Published Version: http://dx.doi.org/10.1016/j.jmaa.2011.05.067
Status: Published
Publisher: Elsevier
Refereed: Yes
Identification Number: 10.1016/j.jmaa.2011.05.067
URI: http://eprints.whiterose.ac.uk/id/eprint/43276

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