Maximal regularity for stochastic convolutions driven by Lévy processes

Abstract

We generalize the maximal regularity result from Da Prato and Lunardi (Atti Accad Naz Lincei Cl Sci Fis Mat Natur Rend Lincei (9) Mat Appl 9(1):25-29, 1998) to stochastic convolutions driven by time homogenous Poisson random measures and cylindrical infinite dimensional Wiener processes.

Uniqueness in Law of the Itô Integral with Respect to Lévy Noise

“…Throughout this section we assume that E is a separable Banach space of martingale type p, where 1 < p ≤ 2, see for instance the Appendix of [3]. We also assume that (S, S) is a measurable space, ν ∈ M + (S) and (Ω, F, F, P), where F = (F t ) t≥0 , is a complete filtered probability space.…”

“…We assume that (S, S) is a Polish space endowed with the Borel σ algebra andη is a time homogeneous compensated Poisson random measure defined on a complete filtered probability space (Ω, F, F, P), where F = (F t ) t≥0 with intensity measure ν on S, to be specified later. Let us assume that 1 < p ≤ 2 and E is a separable Banach space of martingale type p, see for instance the Appendix of [3]. We consider the following (infinite-dimensional) Itô SDE…”

Strong solutions to stochastic hydrodynamical systems with multiplicative noise of jump type

“…Following the notation of [7], let M 2 (R + , L 2 (Z, ν, H)) be the class of all progressively measurable processes ξ :…”

Stochastic hydrodynamic-type evolution equations driven by Lévy noise in 3D unbounded domains—Abstract framework and applications

“…We follow the approach due to Brzeźniak and Hausenblas [11], [10], see also [28], [3] and [40]. Let us denote N := {0, 1, 2, ...}, N := N ∪ {∞}, R + := [0, ∞).…”

“…Definition 8.1. (see [3] and Appendix C in [11] Let us also recall basic properties of the stochastic integral with respect toη, see [11], [28] and [40] for details. Let E be a separable Hilbert space and let P be a predictable σ-field on [0, T ] × Ω.…”