The Stochastic Wave Equation with Multiplicative Fractional Noise: A Malliavin Calculus Approach |
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Authors: | Raluca M Balan |
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Institution: | 1. Department of Mathematics and Statistics, University of Ottawa, 585 King Edward Avenue, Ottawa, ON, K1N 6N5, Canada
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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 H?lder 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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