Comptes Rendus
Probability Theory/Partial Differential Equations
SPDEs in infinite dimension with Poisson noise
Comptes Rendus. Mathématique, Volume 339 (2004) no. 9, pp. 647-652.

In this Note we investigate stochastic partial differential equations in infinite dimension driven by a compensated Poisson random measure. Apart from the existence and uniqueness of mild solutions our main interest is directed towards their regularity w.r.t. the initial datum. Our main result is the first order Fréchet differentiability of the mild solution as a mapping from Lq to Hp, the space of predictable p-integrable processes, where q>p2. Higher order Fréchet differentiability can be proved similarly. As a consequence we obtain gradient estimates in infinite dimensions for the corresponding resolvents.

Dans cette Note nous analysons des équations stochastiques aux dérivées partielles en dimension infinie avec une diffusion décrite par une intégrale stochastique par rapport à une mesure aléatoire de Poisson compensée par la mesure d'intensité. Outre l'existence et l'unicité de la solution ‘mild’, notre principal intérêt concerne la régularité par rapport à la condition initiale. Le résultat principal est la différentiabilité au sens de Fréchet de la solution comme application de Lq vers l'espace des processus prévisibles X(t), t[0,T], tels que E[X(t)p]<q>p2. La différentiabilité d'ordre deux au sens de Fréchet peut être obtenue de la même façon.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2004.09.004
Claudia Knoche 1

1 Fakultät für Mathematik, Universität Bielefeld, Postfach 10 01 31, 33501 Bielefeld, Germany
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Claudia Knoche. SPDEs in infinite dimension with Poisson noise. Comptes Rendus. Mathématique, Volume 339 (2004) no. 9, pp. 647-652. doi : 10.1016/j.crma.2004.09.004. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2004.09.004/

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[5] C. Knoche, Stochastic integrals and stochastic differential equations with respect to compensated Poisson random measures in infinite dimensional Hilbert spaces, Preprint No. 03-06-119 des Forschungszentrums BiBoS (Bielefeld-Bonn-Stochastics), Universität Bielefeld, 2003

[6] C. Knoche, Mild solutions of SPDE's driven by Poisson noise in infinite dimensions and their dependence on initial datum, Doctor-degree thesis, Fakultät für Mathematik, Universität Bielefeld, 2004, in preparation

[7] V. Mandrekar, B. Rüdiger, Existence and uniqueness of path wise solutions for stochastic integral equations driven by non Gaussian noise on separable Banach spaces, Preprint No. 61 des SFB 611, Fakultät für Mathematik, Universität Bonn, 2003

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