$Ito s formula for linear fractional PDEs Jorge A Leon$

Ito's formula for linear fractional PDEs Jorge A Leon

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Ito's formula for linear fractional PDEs Jorge A. Leon ? Depto. de Control Automatico, CINVESTAV-IPN Apartado Postal 14-740, 07000 Mexico, D.F., Mexico Samy Tindel Institut Elie Cartan, Universite de Nancy 1 BP 239 – 54506 Vandœuvre-les-Nancy, France August 30, 2007 Abstract In this paper we introduce a stochastic integral with respect to the solution X of the fractional heat equation on [0, 1], interpreted as a divergence operator. This allows to use the techniques of the Malliavin calculus in order to establish an Ito-type formula for the process X. Keywords: heat equation, fractional Brownian motion, Ito's formula. MSC: 60H15, 60H07, 60G15 1 Introduction In the last few years, a great amount of effort has been devoted to a proper definition of stochastic PDEs driven by a general noise. For instance, the case of stochastic heat and wave equations in Rn driven by a Brownian motion in time, with some mild conditions on its spatial covariance, has been considered e.g. in [8, 18, 15], leading to some optimal results. More recently, the case of SPDEs driven by a fractional Brownian motion has been analyzed in [5, 10, 22] in the linear case, or in [12, 14, 19] in the non-linear case.

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Itˆo’sformulaforlinearfractionalPDEs JorgeA.Leon Depto.deControlAutomatico,CINVESTAV-IPN ApartadoPostal14-740,07000Mexico,D.F.,Mexico jleon@ctrl.cinvestav.mx

Samy Tindel InstitutElieCartan,UniversitedeNancy1 BP239–54506Vanduvre-les-Nancy,France tindel@iecn.u-nancy.fr

August 30, 2007

Abstract

In this paper we introduce a stochastic integral with respect to the solution Xof the fractional heat equation on [0,1], interpreted as a divergence operator. This allows to use the techniques of the Malliavin calculus in order to establish an Itˆo-typeformulafortheprocessX.

Keywords:ontiacfrniowBralnoitomnafs’oˆtI,aeethoi,nuqtaormula.

MSC:60H15, 60H07, 60G15

1 Introduction

In the last few years, a great amount of eort has been devoted to a proper denition of stochasticPDEs driven by a general noise. instance, Forthe case of stochastic heat and wave equations inRndriven by a Brownian motion in time, with some mild conditions on its spatial covariance, has been considered e.g. in [8, 18, 15], leading to some optimal results. More recently, the case ofSPDEs driven by a fractional Brownian motion has been analyzed in [5, 10, 22] in the linear case, or in [12, 14, 19] in the non-linear case. Notice that this kind of development can be related to the study of turbulent plasmas [6], where some non-diusiveSPDEs may appear. Partially supported by the CONACyT grant 45684-F