Comptes Rendus
Hölder–Sobolev regularity of solutions to a class of SPDE's driven by a spatially colored noise
[Régularité Hölder–Sobolev des solutions d'une classe d'E.D.P.S. dirigées par un bruit coloré]
Comptes Rendus. Mathématique, Volume 334 (2002) no. 10, pp. 869-874.

Dans cette Note nous présentons des résultats nouveaux concernant l'équivalence, l'existence et la régularité spatio–temporelle conjointe de diverses notions de solution relatives à une classe d'équations aux dérivées partielles stochastiques semilinéaires non autonomes définies dans un ouvert régulier borné convexe D d et dirigées par un bruit coloré en la variable spatiale défini à partir d'un processus de Wiener à valeurs dans L2(D).

In this Note we present new results regarding the equivalence, the existence and the joint space–time regularity properties of various notions of solution to a class of non-autonomous, semilinear, stochastic partial differential equations defined on a smooth, bounded, convex domain D d and driven by a spatially colored noise defined from an L2(D)-valued Wiener process.

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Accepté le :
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DOI : 10.1016/S1631-073X(02)02359-2

Marta Sanz-Solé 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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     title = {H\"older{\textendash}Sobolev regularity of solutions to a class of {SPDE's} driven by a spatially colored noise},
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Marta Sanz-Solé; Pierre-A. Vuillermot. Hölder–Sobolev regularity of solutions to a class of SPDE's driven by a spatially colored noise. Comptes Rendus. Mathématique, Volume 334 (2002) no. 10, pp. 869-874. doi : 10.1016/S1631-073X(02)02359-2. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(02)02359-2/

[1] A. Chojnowska-Michalik Stochastic differential equations in Hilbert spaces, Banach Center Publ., Volume 5 (1979), pp. 53-73

[2] R.C. Dalang Extending martingale measure stochastic integral with applications to spatially homogeneous S.P.D. E's, Electronic J. Probab., Volume 4 (1999), pp. 1-29

[3] G. Da Prato; J. Zabczyk Stochastic Equations in Infinite Dimensions, Encyclopedia of Mathematics and its Applications, 44, Cambridge University Press, Cambridge, 1992

[4] D.A. Dawson; L.G. Gorostiza Solutions of evolution equations in Hilbert space, J. Differential Equations, Volume 68 (1987), pp. 299-319

[5] S.D. Eidelman; S.D. Ivasis̆en Investigation of the Green matrix for a homogeneous parabolic boundary value problem, Trans. Moscow Math. Soc., Volume 23 (1970), pp. 179-242

[6] N.V. Krylov; B.L. Rozovskii Stochastic evolution equations, J. Soviet Math., Volume 16 (1981), pp. 1233-1277

[7] J.A. León Stochastic evolution equations with respect to semimartingales in Hilbert space, Stochastics, Volume 27 (1989), pp. 1-21

[8] O. Lévêque, Hyperbolic stochastic partial differential equations driven by boundary noises, Thèse EPFL 2452, Lausanne, 2001

[9] E. Pardoux, Équations aux dérivées partielles stochastiques nonlinéaires monotones : Étude de solutions fortes de type Itô, Thèse de l'Université Paris–Orsay 1556, Paris, 1975

[10] S. Peszat; J. Zabczyk Nonlinear stochastic wave and heat equations, Probab. Theory Related Fields, Volume 116 (2000), pp. 421-443

[11] M. Sanz-Solé, P.-A. Vuillermot, Equivalence and Hölder–Sobolev regularity of solutions for a class of non-autonomous stochastic partial differential equations, 2002, in preparation

[12] J.B. Walsh An introduction to stochastic partial differential equations, École d'Été de Probabilités de Saint-Flour XIV, Lecture Notes in Math., 1180, Springer, New York, 1986, pp. 265-439

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