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

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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.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Ottawa, 585 King Edward Avenue, Ottawa, ON, K1N 6N5, Canada

    Raluca M. Balan

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  1. Raluca M. Balan
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Correspondence to Raluca M. Balan.

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Research supported by a grant from the Natural Sciences and Engineering Research Council of Canada.

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Balan, R.M. The Stochastic Wave Equation with Multiplicative Fractional Noise: A Malliavin Calculus Approach. Potential Anal 36, 1–34 (2012). https://doi.org/10.1007/s11118-011-9219-z

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  • DOI: https://doi.org/10.1007/s11118-011-9219-z

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