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Covariant Mehler Semigroups in Hilbert Space David Applebaum

Summary: Covariant Mehler Semigroups in Hilbert Space
David Applebaum
Probability and Statistics Department,
University of Sheffield,
Hicks Building, Hounsfield Road,
Sheffield, England, S3 7RH
e-mail: D.Applebaum@sheffield.ac.uk
Dedicated to the memory of J.T.Lewis
We find necessary and sufficient conditions for a generalised Mehler
semigroup to be covariant under the action of a locally compact group.
These are then applied to implement "noise reduction" for Hilbert-
space valued Ornstein-Uhlenbeck processes driven by Levy processes.
1 Introduction
Generalised Mehler semigroups are beautiful objects which have attracted the
attention of both analysts and probabilists. They are semigroups of linear
operators (T(t), t 0) acting on the space of bounded continuous functions
on a real separable Hilbert space which are built from two components:
a C0-semigroup (S(t), t 0) acting in H with generator J,
a family (t, t 0) of probability measures on H satisfying the skew-


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


Collections: Mathematics