Variational solutions for a class of fractional stochastic partial differential equations

David Nualart; Pierre-A. Vuillermot

doi:10.1016/j.crma.2005.01.006

Partial Differential Equations/Probability Theory

Variational solutions for a class of fractional stochastic partial differential equations

Presented by: Pierre-Louis Lions

David Nualart ¹; Pierre-A. Vuillermot ²

¹ Facultat de Matemàtiques, Universitat de Barcelona, Gran Via, 585, 08007 Barcelona, Spain
² I.E.C.N., université Henri-Poincaré, Nancy 1, BP 239, 54506 Vandoeuvre-lès-Nancy cedex, France

Comptes Rendus. Mathématique, Volume 340 (2005) no. 4, pp. 281-286.

Abstract
Résumé

In this Note we present new results regarding the existence, the uniqueness and the equivalence of two notions of variational solution related to a class of non autonomous, semilinear, stochastic partial differential equations defined on an open bounded domain $D \subset R^{d}$ . The equations we consider are driven by an infinite-dimensional noise derived from an $L^{2} (D)$ -valued fractional Wiener process $W^{H}$ with Hurst parameter $H \in (\frac{1}{γ + 1}, 1)$ , where $γ \in (0, 1]$ denotes the Hölder exponent of the derivative of the nonlinearity that appears in the stochastic term.

Dans cette Note nous présentons des résultats nouveaux concernant l'existence, l'unicité et l'équivalence de deux notions de solution variationnelle relatives à une classe d'équations aux dérivées partielles stochastiques semilinéaires non autonomes définies dans un ouvert borné $D \subset R^{d}$ . Les équations que nous considérons sont dirigées par un bruit en dimension infinie déduit d'un processus de Wiener fractionnaire $W^{H}$ à valeurs dans $L^{2} (D)$ de paramètre de Hurst $H \in (\frac{1}{γ + 1}, 1)$ , où $γ \in (0, 1]$ est l'exposant de Hölder de la dérivée de la nonlinéarité apparaissant dans le terme stochastique.

Received: 2004-10-18
Accepted: 2005-01-04
Published online: 2005-02-05

DOI: 10.1016/j.crma.2005.01.006

Author's affiliations:

David Nualart ¹; Pierre-A. Vuillermot ²

¹ Facultat de Matemàtiques, Universitat de Barcelona, Gran Via, 585, 08007 Barcelona, Spain
² I.E.C.N., université Henri-Poincaré, Nancy 1, BP 239, 54506 Vandoeuvre-lès-Nancy cedex, France

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     author = {David Nualart and Pierre-A. Vuillermot},
     title = {Variational solutions for a class of fractional stochastic partial differential equations},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {281--286},
     publisher = {Elsevier},
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     number = {4},
     year = {2005},
     doi = {10.1016/j.crma.2005.01.006},
     language = {en},
}

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David Nualart; Pierre-A. Vuillermot. Variational solutions for a class of fractional stochastic partial differential equations. Comptes Rendus. Mathématique, Volume 340 (2005) no. 4, pp. 281-286. doi : 10.1016/j.crma.2005.01.006. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2005.01.006/

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