A Lie-Trotter product formula for Ornstein-Uhlenbeck semigroups in infinite dimensions

Abstract

Abstract. We prove a Lie-Trotter product formula for the Ornstein-Uhlenbeck semigroup associated with the stochastic linear Cauchy problem dX(t) = AX(t) dt + dW (t), t 0,Here A is the generator of a C 0 −semigroup on a separable real Banach space E and {W (t)} t 0 is an E-valued Brownian motion.

A Pettis-type integral and applications to transition semigroups

“…In order to treat transition semigroups on C b (E), several approaches have been proposed in the literature. We mention the theory of weakly continuous semigroups of Cerrai [6], the theory of bi-continuous semigroups of Kühnemund [18], see also [9,19] for applications in the context of transition semigroups, and the theory of π-semigroups by Priola [25]. It should be noted that in these approaches additional assumptions, in particular continuity and equicontinuity assumptions, are made which ensure that a Riemann integral can be used to compute the Laplace transform.…”

Kolmogorov equations in infinite dimensions: Well-posedness and regularity of solutions, with applications to stochastic generalized Burgers equations

“…In comparison to other constructions of semigroups on weighted function spaces using locally convex topologies and the concept of bicontinuous semigroups (cf. [36] and the references therein), we emphasize that our spaces are (separable) Banach spaces so, as spaces with one single norm, are easier to handle.…”

Perturbation for a class of transition semigroups on the Hölder space Cb,locθ(H)

“…This semigroup is extensively studied by many authors, we refer to Da Prato [3][4][5], Da Prato and Zabczyk [6][7][8], Goldys and Kocan [12], Kühnemund and van Neerven [16], Lunardi [18], Metafune et al [19], Priola [20], Rhandi [22], van Neerven [23] and van Neerven and Zabczyk [24]. The proposition below will be useful in the following and can be found, e.g., in Cerrai [2].…”