Young integrals and SPDEs

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Young integrals and SPDEs Antoine Lejay Project OMEGA, INRIA Lorraine IECN, Campus Scientifique BP 239 – 54506 Vandœuvre-les-Nancy CEDEX, France Massimiliano Gubinelli Laboratoire Analyse, Geometrie & Applications – UMR 7539 Institute Galilee, Universite Paris 13, 93430 – Villetaneuse, France Samy Tindel Institut Elie Cartan Universite Henri Poincare (Nancy) BP 239 – 54506 Vandœuvre-les-Nancy CEDEX, France July 2004 Abstract In this note, we study the non-linear evolution problem dYt = ?AYtdt+B(Yt)dXt, where X is a ?-Holder continuous function of the time parameter, with values in a distribution space, and ?A the generator of an analytical semigroup. Then, we will give some sharp conditions on X in order to solve the above equation in a function space, first in the linear case (for any value of ? in (0, 1)), and then when B satisfies some Lipschitz type conditions (for ? > 1/2). The solution of the evolution problem will be understood in the mild sense, and the integrals involved in that definition will be of Young type. 1

  • evolution equation

  • all ? ?

  • dimensional fractional

  • linear additive

  • holder continuity

  • all ?

  • noise

  • solution can


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Young
integrals and SPDEs
Antoine Lejay Project OMEGA, INRIA Lorraine IECN, Campus Scientique BP23954506Vanduvre-les-NancyCEDEX,France Antoine.Lejay@iecn.u-nancy.fr
Massimiliano Gubinelli LaboratoireAnalyse,Geometrie&ApplicationsUMR7539 InstituteGalilee,UniversiteParis13, 93430 – Villetaneuse, France gubinell@math.univ-paris13.fr
Samy Tindel Institut Elie Cartan UniversiteHenriPoincare(Nancy) BP23954506Vanduvre-les-NancyCEDEX,France tindel@iecn.u-nancy.fr
July 2004
Abstract
In this note, we study the non-linear evolution problem
dYt=AYtdt+B(Yt)dXt,
whereXis aγuoinntcotincfuustehtfonomarapemidlreH-oeter,with values in a distribution space, andAthe generator of an analytical semigroup. Then, we will give some sharp conditions onXin order to solve the above equation in a function space, rst in the linear case (for any value ofγin (0,1)), and then whenBsatises some Lipschitz type conditions (forγ >1/ solution of the evolution problem2). The will be understood in themildsense, and the integrals involved in that denition will be of Young type.
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