Generalized Ornstein–Uhlenbeck Processes

Abstract

Generalized Ornstein–Uhlenbeck processes constitute a large class of explicit examples of Markov processes in infinite-dimensional spaces with rich mathematical structures. Those processes may have non-trivial invariant measures, which make them become better candidates for infinite-dimensional reference processes than Levy processes. In this chapter, we first give a formulation of generalized Ornstein–Uhlenbeck processes in Hilbert spaces using the generalized Mehler semigroups introduced by Bogachev and Rockner (1995) and Bogachev et al. (1996). Then we give a systematic exploration of the structures of the generalized Mehler semigroups. Since such a semigroup can be defined by a linear semigroup and a skew convolution semigroup, we mainly discuss the latter. We shall see that a skew convolution semigroup is always formed with infinitely divisible probability measures. The key result is a characterization for the skew convolution semigroups in terms of infinitely divisible probability entrance laws. For centered skew convolution semigroups with finite second-moments, those entrance laws can be closed by probability measures on an enlarged Hilbert space. We also give some constructions for the generalized Ornstein–Uhlenbeck processes determined by closed entrance laws and study the corresponding Langevin type equations.