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Pseudo Differential Operators and Markov Semigroups on Compact Lie Groups

Summary: Pseudo Differential Operators and Markov
Semigroups on Compact Lie Groups
David Applebaum,
School of Mathematics and Statistics,
University of Sheffield,
Hicks Building, Hounsfield Road,
Sheffield, England, S3 7RH
e-mail: D.Applebaum@sheffield.ac.uk
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 op-
erators 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´evy pro-
cesses. We study a family of L´evy-type linear operators on general Lie


Source: Applebaum, David - Department of Probability and Statistics, University of Sheffield


Collections: Mathematics