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Properties of Semigroups of Measures on Lie Groups

Summary: Some L2
Properties of Semigroups of Measures
on Lie Groups
David Applebaum,
Probability and Statistics Department,
University of Sheffield,
Hicks Building, Hounsfield Road,
Sheffield, England, S3 7RH
e-mail: D.Applebaum@sheffield.ac.uk
We investigate the induced action of convolution semigroups of
probability measures on Lie groups on the L2-space of Haar measure.
Necessary and sufficient conditions are given for the infinitesimal gen-
erator to be self-adjoint and the associated symmetric Dirichlet form is
constructed. We show that the generated Markov semigroup is trace-
class if and only if the measures have a square-integrable density. Two
examples are studied in some depth where the spectrum can be ex-
plicitly computed, these being the n-torus and Riemannian symmetric
pairs of compact type.
Key words and phrases Lie group, Lie algebra, convolution semigroup,


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


Collections: Mathematics