The stochastic Cauchy problem driven by a cylindrical Lévy process

Abstract

In this work, we derive sufficient and necessary conditions for the existence of a weak and mild solution of an abstract stochastic Cauchy problem driven by an arbitrary cylindrical Lévy process. Our approach requires to establish a stochastic Fubini result for stochastic integrals with respect to cylindrical Lévy processes. This approach enables us to conclude that the solution process has almost surely scalarly square integrable paths. Further properties of the solution such as the Markov property and stochastic continuity are derived.

Variational solutions of stochastic partial differential equations with cylindrical Lévy noise

“…Consequently, Y c k (t) and thus M c k (t) are well defined, continuous and weakly square-integrable. By (17), the (deterministic) process P c k is well defined. Since R c k = L − M c k − P c k it follows that the series in the definition of R c k converges and that R c k (t) : U → L 0 (Ω; R) is continuous for all t 0.…”

“…Brzeźniak and Zabczyk [6], Peszat and Zabczyk [22] and Priola and Zabczyk [25]. A general 2880 TOMASZ KOSMALA AND MARKUS RIEDLE framework of linear equations with additive noise modelled by arbitrary cylindrical Lévy processes is developed in Riedle [28] and Kumar and Riedle [16] and [17]. The case of an SPDE with a multiplicative perturbation is only considered in Riedle [27], however under the restrictive assumption of weak square-integrability of the driving cylindrical Lévy process.…”

Stable cylindrical Lévy processes and the stochastic Cauchy problem

“…exists, the stochastic process (T (t)x 0 + Y (t) : t 0) can be considered as a mild solution. By a result in [8] it also follows that the mild solution is a weak solution; however, this is of less concern in this work. More precisely, if L is a cylindrical Lévy processes with characteristics (0, 0, ν), for an orthonormal basis (e k ) k∈AE of U ; see [14,Th.5.10].…”

Modelling Lévy space‐time white noises

“…Both generalisations, cylindrical Lévy processes and Lévy space-time white noises, serve as a model for random perturbations of complex dynamical systems. These applications can be found for cylindrical Lévy processes, for example, in the monograph in Peszat and Zabczyk [34] or in Kumar and Riedle [30], and for Lévy space-time white noise in Applebaum and Wu [3], Chong [11], Chong and Kevei [12] and Dalang and Humeau [15], among many others. Another approach to model such perturbed dynamical systems, for example, parabolic stochastic partical differential equations, is provided by the recently introduced ambit fields, presented in the monograph [6] by Barndorff-Nielsen, Benth and Veraart, and their relations to SPDE investigated in [7] by the same authors.…”