Young integrals and SPDEs

Antoine Lejay

Description

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

Sujets

• Poincaré
• Cartan
• Hölder condition

Informations

• Publié par: pefav
• Nombre de lectures: 10
• Langue: English

Extrait

Young

integrals and SPDEs

Antoine Lejay Project OMEGA, INRIA Lorraine IECN, Campus Scientique BP239–54506Vanduvre-les-NancyCEDEX,France [e-mail]

Massimiliano Gubinelli LaboratoireAnalyse,Geometrie&Applications–UMR7539 InstituteGalilee,UniversiteParis13, 93430 – Villetaneuse, France [e-mail]

Samy Tindel Institut Elie Cartan UniversiteHenriPoincare(Nancy) BP239–54506Vanduvre-les-NancyCEDEX,France [e-mail]

July 2004

Abstract

In this note, we study the non-linear evolution problem

dY t =−AY t dt+B(Y t )dX t ,

whereXis aγuoinntcotincfuustehtfonomarapemidlreH-oeter,with values in a distribution space, and−Athe 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.

1