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 given starting point x and an endpoint y provided y 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 defined by a stochastic evolution equation with additive noise. We provide an explicit formula for the transition density and study its regularity. These results are applied to show some basic properties of the transition semigroup. Given the strong Feller property and the existence of invariant measure we show that all L p functions are transformed into continuous functions, thus generalising the strong Feller property. We also show that transition operators are q -summing for some q > p > 1 , in particular of Hilbert–Schmidt type.