A new regularity result for Ornstein-Uhlenbeck generators and applications

Perturbations of bi-continuous semigroups

“…for all x ∈ X and for some constant C > 0, which shows that the third term in (9) converges to zero in the topology τ as h → 0. Putting (8) and (9) together we obtain…”

“…To deal with such semigroups the notion of bi-continuous semigroups has been introduced recently by F. ). Among the semigroups that fit into this setting are adjoint semigroups ( [20,24]), evolution semigroups on C b (R), semigroups induced by flows ( [13][14][15]), implemented semigroups ( [2,3]) and the Ornstein-Uhlenbeck semigroup on C b (H) ( [9,10,20,25]). In [21,22], F. Kühnemund obtained generation and approximation theorems for such semigroups (see also [1,6]).…”

Optimal regularity results in spaces of Hölder continuous functions for some infinite dimensional Ornstein–Uhlenbeck semigroup

“…We start by recalling some known results. The Schauder estimate (i) was proved in [1] whereas (ii) was proved in [3]. Clearly (ii) is a counterpart of (2.2) in the Hö lder setting.…”

“…Finally, Section 3 is devoted to study (1.7) in spaces of Hö lder continuous functions. We first recall previous optimal regularity result proved in [1] and [3] and then we present a new optimal regularity result. This last result will allow us to take into account a new kind of perturbations of the OrnsteinUhlenbeck di¤usion process for which it is will possible to prove existence and uniqueness of an associated martingale problems, arguing as in [13].…”

Invariant measures and regularity properties of perturbed Ornstein–Uhlenbeck semigroups

“…The Ornstein-Uhlenbeck semigroups corresponding to problem (1) for F ≡ 0 is studied in, e.g., Da Prato [5,6], Da Prato, Zabczyk [9][10][11], van Neerven [25] and van Neerven, Zabczyk [26]. For γ = 0 the general case, using perturbation arguments, is then treated by Da Prato in [5,6] and also by Chojnowska-Michalik [2], Chojnowska-Michalik, Goldys [3,4], Goldys, Kocan [16], Rhandi [23]. Our investigation is inspired by Da Prato's paper [6] where he considers the critical case γ = 1/2 and works in the space of bounded uniformly continuous functions on H .…”