Diffusion Semigroups in Spaces of Continuous Functions with Mixed Topology

Abstract

We study transition semigroups and Kolmogorov equations corresponding to stochastic semilinear equations on a Hilbert space H. It is shown that the transition semigroup is strongly continuous and locally equicontinuous in the space of polynomially increasing continuous functions on H when endowed with the so-called mixed topology. As a result we characterize cores of certain second order differential operators in such spaces and show that they have unique extensions to generators of strongly continuous semigroups. Academic Press

On the Infinitesimal Generators of Ornstein–Uhlenbeck Processes with Jumps in Hilbert Space

“…I am grateful to Enrico Priola for valuable comments on an earlier draft of this article, N.H. Bingham for some useful remarks and Ben Goldys for helpful e-mail discussions and kindly allowing me an early preview of [14]. Many thanks are also due to the referees, who both made very helpful remarks.…”

“…The problem of lack of strong continuity can be overcome by working in a weaker topology than the usual one induced by the norm, and this has been carried out by a number of authors in different ways, for example the theory of "weakly continuous semigroups" is developed in [5], the notion of π-convergence is used in [29], bi-continuous semigroups are introduced in [20,21] and the mixed topology is utilised in [15], [16]. We use two topologies in this paper.…”

“…the finest locally convex topology that agrees on norm bounded sets with the topology of uniform convergence on compacts. This has previously been applied to study Mehler semigroups associated with Banach-space valued Ornstein-Uhlenbeck processes driven by Brownian motion in [16] (see also [15]). In this case all of the results established for the topology of uniform convergence on compacta continue to hold, however in addition the generator is closed and (at least when large jumps are well-behaved) has a convenient invariant core of cylinder functions.…”

“…We say that it is quasi-locally equicontinuous if for each T > 0 and for every p ∈ P I , there exists K p,T > 0 such that 14) for each u ∈ B, 0 ≤ t ≤ T . It is said to be quasi-equicontinuous if K p,T can be chosen to be independent of T .…”

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

“…For this reason many authors have studied strong continuity of {P(t)} t 0 in various locally convex topologies on C b (E), cf. [5], [6], [8], [11], [16]. They only consider the situation where E is a Hilbert space, in which case Itô calculus may be applied.…”