The Ornstein Uhlenbeck Bridge and Applications to Markov Semigroups

Abstract

For an arbitrary Hilbert space-valued Ornstein-Uhlenbeck process we construct the Ornstein-Uhlenbeck Bridge connecting a starting point $x$ and an endpoint $y$ that belongs to a certain linear subspace of full measure. We derive also a stochastic evolution equation satisfied by the OU Bridge and study its basic properties. The OU Bridge is then used to investigate the Markov transition semigroup associated to a nonlinear stochastic evolution equation with additive noise. We provide an explicit formula for the transition density and study its regularity. Given the Strong Feller property and the existence of an invariant measure we show that the transition semigroup maps $L^p$ functions into continuous functions. We also show that transition operators are $q$-summing for some $q>p>1$, in particular of Hilbert-Schmidt type.