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Potential Analysis

, Volume 36, Issue 1, pp 1–34 | Cite as

The Stochastic Wave Equation with Multiplicative Fractional Noise: A Malliavin Calculus Approach

  • Raluca M. Balan
Article

Abstract

We consider the stochastic wave equation with multiplicative noise, which is fractional in time with index H > 1/2, and has a homogeneous spatial covariance structure given by the Riesz kernel of order α. The solution is interpreted using the Skorohod integral. We show that the sufficient condition for the existence of the solution is α > d − 2, which coincides with the condition obtained in Dalang (Electr J Probab 4(6):29, 1999), when the noise is white in time. Under this condition, we obtain estimates for the p-th moments of the solution, we deduce its Hölder continuity, and we show that the solution is Malliavin differentiable of any order. When d ≤ 2, we prove that the first-order Malliavin derivative of the solution satisfies a certain integral equation.

Keywords

Stochastic wave equation Fractional Brownian motion Spatially homogeneous Gaussian noise Malliavin calculus 

Mathematics Subject Classifications (2010)

Primary 60H15; Secondary 60H07 

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Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of OttawaOttawaCanada

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