Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

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

Jiahui Zhu, Zdzisław Brzeźniak, and Erika Hausenblas

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Abstract

We consider a Banach space $(E,\|\cdot\|)$ such that, for some $q\geq2$, the function $x\mapsto\|x\|^{q}$ 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 $C_{0}$-semigroup $S$ of contraction type on $(E,\|\cdot\|)$. Finally we consider a compensated Poisson random measure $\tilde{N}$ on a measurable space $(Z,\mathcal{Z})$.

We study the following stochastic convolution process

\[u(t)=\int_{0}^{t}\!\int_{Z}S(t-s)\xi(s,z)\tilde{N}(\mathrm{d}s,\mathrm{d} z),\quad t\geq0,\] where $\xi:[0,\infty)\times\Omega\times Z\rightarrow E$ is an $\mathbb{F}\otimes\mathcal{Z}$-predictable function.

We prove that there exists a càdlàg modification $\tilde{u}$ of the process $u$ which satisfies the following maximal type inequality

\[\mathbb{E}\sup_{0\leq s\leq t}\|\tilde{u}(s)\|^{q^{\prime}}\leq C\mathbb{E}(\int_{0}^{t}\!\int_{Z}\|\xi(s,z)\|^{p}N(\mathrm{d}s,\mathrm{d}z))^{\frac{q^{\prime}}{p}},\] for all $q^{\prime}\geq q$ and $1<p\leq2$ with $C=C(q,p)$.

Résumé

On considère un espace de Banach $(E,\|\cdot\|)$ tel que, pour $q\geq2$, la fonction $x\mapsto\|x\|^{q}$ 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 considère également un $C_{0}$-semigroupe de contraction $S$ sur $(E,\|\cdot\|)$. Finalement, on considère une mesure de Poisson compensée $\tilde{N}$ sur un espace mesurable $(Z,\mathcal{Z})$.

On étudie le processus stochastique suivant :

\[u(t)=\int_{0}^{t}\!\int_{Z}S(t-s)\xi(s,z)\tilde{N}(\mathrm{d}s,\mathrm{d} z),\quad t\geq0,\] où $\xi:[0,\infty[\times\Omega\times Z\rightarrow E$ est une fonction $\mathbb{F}\otimes\mathcal{Z}$-prévisible.

On prouve qu’il existe une modification càdlàg $\tilde{u}$ du processus $u$ qui vérifie l’inégalité de type maximale suivante :

\[\mathbb{E}\sup_{0\leq s\leq t}\|\tilde{u}(s)\|^{q^{\prime}}\leq C\mathbb{E}(\int_{0}^{t}\!\int_{Z}\|\xi(s,z)\|^{p}N(\mathrm{d}s,\mathrm{d}z))^{\frac{q^{\prime}}{p}},\] pour tout $q^{\prime}\geq q$ et $1<p\leq2$ avec $C=C(q,p)$.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 53, Number 2 (2017), 937-956.

Dates
Received: 9 September 2014
Revised: 30 December 2015
Accepted: 30 January 2016
First available in Project Euclid: 11 April 2017

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1491897752

Digital Object Identifier
doi:10.1214/16-AIHP743

Mathematical Reviews number (MathSciNet)
MR3634281

Zentralblatt MATH identifier
1372.60075

Subjects
Primary: 60H15: Stochastic partial differential equations [See also 35R60] 60F10: Large deviations 60H05: Stochastic integrals 60G57: Random measures 60J75: Jump processes

Keywords
Stochastic convolution Martingale type $p$ Banach space Poisson random measure

Citation

Zhu, Jiahui; Brzeźniak, Zdzisław; Hausenblas, Erika. Maximal inequalities for stochastic convolutions driven by compensated Poisson random measures in Banach spaces. Ann. Inst. H. Poincaré Probab. Statist. 53 (2017), no. 2, 937--956. doi:10.1214/16-AIHP743. https://projecteuclid.org/euclid.aihp/1491897752


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