Maximal inequalities for stochastic convolutions driven by compensated Poisson random measures in Banach spaces

Abstract

Résumé. We consider a Banach space (E, · ) such that, for some q ≥ 2, the function x → x q is of C 2 class and its k-th, k = 1, 2, Fréchet derivatives are bounded by some constant multiples of the (q − k)-th power of the norm. We also consider a C0-semigroup S of contraction type on (E, · ).Finally we consider a compensated Poisson random measureÑ on a measurable space (Z, Z).We study the following stochastic convolution processwhereWe prove that there exists a càdlàg modificationũ of the process u which satisfies the following maximal type inequalityfor all q ′ ≥ q and 1 < p ≤ 2 with C = C(q, p). Résumé.On considère un espace de Banach (E, · ) tel que, pour q ≥ 2, la fonction x → x q est de classe C 2 avec des dérivées k ième , k = 1, 2, au sens de Fréchet bornées par des constantes multiples de la puissance (q − k) de la norme. On considèreégalement un C0-semigroupe de contraction S sur (E, · ). Finalement, on considère une mesure de Poisson compenséeÑ sur un espace mesurable (Z, Z).Onétudie le processus stochastique suivant :On prouve qu'il existe une modification càdlàgũ du processus u qui vérifie l'inégalité de type maximale suivante :pour tout q ′ ≥ q et 1 < p ≤ 2 avec C = C(q, p).

Nonlinear stochastic partial differential equations of hyperbolic type driven by Lévy-type noises

“…Thus the stochastic convolution I τ (t) is well defined as well. Moreover, one can always assume that the stochastic convolution process I(t) and I τ (t), t ≥ 0 are H-valued càdlàg, see [35]. The following lemma verifies (11) in Definition 2.3 of a local mild solution.…”

“…maximal inequalities for stopped stochastic convolution processes. In our case, this would require to use Theorems 4.4, 4.5 and 5.1 from [35].…”

Maximal Inequalities and Exponential Estimates for Stochastic Convolutions Driven by Lévy-type Processes in Banach Spaces with Application to Stochastic Quasi-Geostrophic Equations

“…e tA L(E) ≤ 1. It is known (see [53]) that process X t , t ∈ [0, T ] is a unique strong solution to the following problem Hence, by applying the Itô formula (B.3) to ψ(·) = | · | p E and then using the fact that ψ ′ (x)(Ax) ≤ 0 for all x ∈ D(A) (see e.g. Lemma 4.7 in [53]), we obtain for t ∈ [0, T ], P-a.s.…”

“…The key tool employed in the proof of existence and uniqueness of a local mild solution theory in [18] is a type of maximal inequality that was recently developed by the first two authors of this paper and Hausenblas in [53]. It's worth mentioning that [53] deals with maximal inequalities with respect to stronger norms. More precisely, the p-th power of the norm is assumed to be of C 2 .…”

Stochastic evolution equations in Banach spaces and applications to the Heath–Jarrow–Morton–Musiela equations

“…The norm in the space γ(H, X) will be denoted by · γ(H,X) . We say that a Banach space X with the norm · X satisfies H p condition, see [32] for details, if for some p ≥ 2, the function ψ : x X → ψ(x) = x p X ∈ R is of C 2 class on X (in the Fréchet derivative sense) and there exist constants K 1 (p), K 2 (p) > 0 depending on p such that for every x ∈ X, ψ (x) ≤ K 1 (p) x p−1 X and ψ (x) ≤ K 2 (p) x p−2 X .…”