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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 \(\alpha\). The solution is interpreted using the Skorohod integral. We show that the sufficient condition for the existence of the solution is \(\alpha>d-2\), which coincides with the condition obtained in Dalang (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\"older continuity, and we show that the solution is Malliavin differentiable of any order. When \(d \leq 2\), we prove that the first-order Malliavin derivative of the solution satisfies a certain integral equation.

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          Heat Equations with Fractional White Noise Potentials

          Y. Hu (2001)
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            The Non-Linear Stochastic Wave Equation in High Dimensions

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              Author and article information

              Journal
              28 May 2010
              Article
              1005.5275
              a5a0cb56-2f50-43ab-a1df-99d7071159d0

              http://arxiv.org/licenses/nonexclusive-distrib/1.0/

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              Custom metadata
              60H15, secondary 60H15, 60H07
              math.PR

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