Summary: Covariant Mehler Semigroups in Hilbert Space
Probability and Statistics Department,
University of Sheffield,
Hicks Building, Hounsfield Road,
Sheffield, England, S3 7RH
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 L´evy processes.
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 (µt, t 0) of probability measures on H satisfying the skew-