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
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 L\'{e}vy processes. We study a family of L\'{e}vy-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.
Metadata
Item Type: | Article |
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Authors/Creators: |
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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. Article available under the terms of the CC-BY-NC-ND licence (https://creativecommons.org/licenses/by-nc-nd/4.0/) |
Keywords: | Feller semigroup; Pseudo-differential operator; Symbol; Fourier transform; Peter–Weyl theorem; Lie group; Lie algebra; Convolution semigroup; Courrège–Hunt operator; Sobolev space; Dirichlet form; Beurling–Deny representation |
Dates: |
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Institution: | The University of Sheffield |
Academic Units: | The University of Sheffield > Faculty of Science (Sheffield) > School of Mathematics and Statistics (Sheffield) |
Depositing User: | Symplectic Sheffield |
Date Deposited: | 14 Nov 2016 14:14 |
Last Modified: | 21 Mar 2018 01:32 |
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 |
Related URLs: | |
Open Archives Initiative ID (OAI ID): | oai:eprints.whiterose.ac.uk:106771 |