$The d stochastic wave equation driven by a fractional Brownian sheet$

The d stochastic wave equation driven by a fractional Brownian sheet

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The 1-d stochastic wave equation driven by a fractional Brownian sheet Lluıs Quer-Sardanyons ? Project OMEGA, INRIA Lorraine Institut Elie Cartan, Campus Scientifique BP 239 – 54506 Vandœuvre-les-Nancy, France Samy Tindel Institut Elie Cartan, Universite de Nancy 1 BP 239 – 54506 Vandœuvre-les-Nancy, France 29th January 2007 Abstract In this paper, we develop a Young integration theory in dimension 2 which will allow us to solve a non-linear one dimensional wave equation driven by an arbitrary signal whose rectangular increments satisfy some Holder regularity conditions, for some Holder exponent greater than 1/2. This result will be applied to the fractional Brownian sheet. Keywords: wave equation, fractional Brownian sheet, Young integration. MSC: 60H15, 60G15, 35L05 ?Supported by an INRIA's Postdoc grant and grants BFM2003-01345 and HF2003-006, Direccion General de Investigacion, Ministerio de Educacion y Ciencia, Spain. 1

young integral

dimensional wave

noise has

been considered

holder exponent

positive real constant

equation driven

stochastic differential equations

brownian noise

brownian sheet

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The 1-d stochastic wave equation driven by a fractional Brownian sheet

LlusQuer-Sardanyons Project OMEGA, INRIA Lorraine Institut Elie Cartan, Campus Scientique BP239–54506Vanduvre-les-Nancy,France quer@iecn.u-nancy.fr

Samy Tindel InstitutElieCartan,UniversitedeNancy1 BP 239 – 54506 Vanduvre-l -Nancy, France es tindel@iecn.u-nancy.fr

29th January 2007

Abstract

In this paper, we develop a Young integration theory in dimension 2 which will allow us to solve a non-linear one dimensional wave equation driven by an arbitrary signalwhoserectangularincrementssatisfysomeHolderregularityconditions,for someHolderexponentgreaterthan1/2. This result will be applied to the fractional Brownian sheet.

Keywords:wave equation, fractional Brownian sheet, Young integration.

MSC:60H15, 60G15, 35L05

Supported by an INRIA’s Postdoc grant and grants BFM2003-01345 and HF2003-006, Direcci on GeneraldeInvestigacion,MinisteriodeEducacionyCiencia,Spain.

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