 
Summary: On The Infinitesimal Generators of
OrnsteinUhlenbeck Processes with Jumps in
Hilbert Space
David Applebaum
Probability and Statistics Department,
University of Sheffield,
Hicks Building, Hounsfield Road,
Sheffield, England, S3 7RH
email: D.Applebaum@sheffield.ac.uk
Abstract
We study Hilbert space valued OrnsteinUhlenbeck processes (Y (t), t
0) which arise as weak solutions of stochastic differential equations of
the type dY = JY + CdX(t) where J generates a C0 semigroup in
the Hilbert space H, C is a bounded operator and (X(t), t 0) is an
Hvalued L´evy process. The associated Markov semigroup is of gen
eralised Mehler type. We discuss an analogue of the Feller property
for this semigroup and explicitly compute the action of its generator
on a suitable space of twicedifferentiable functions. We also compare
the properties of the semigroup and its generator with respect to the
mixed topology and the topology of uniform convergence on compacta.
