Comptes Rendus
no. 9
Probability Theory/Partial Differential Equations
SPDEs in infinite dimension with Poisson noise
[ESDP en dimension infinie avec un bruit de Poisson.]

Présenté par : Paul Malliavin

Claudia Knoche1
1 Fakultät für Mathematik, Universität Bielefeld, Postfach 10 01 31, 33501 Bielefeld, Germany
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>p⩾2. 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]<∞ où q>p⩾2. La différentiabilité d'ordre deux au sens de Fréchet peut être obtenue de la même façon.

Reçu le :
Accepté le :
Publié le :
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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