Energy estimate for the wave equation driven by a fractional Gaussian noise

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2007, 2007(Special): 92-101. doi: 10.3934/proc.2007.2007.92

Energy estimate for the wave equation driven by a fractional Gaussian noise

Boris P. Belinskiy ^1, and Peter Caithamer ^2,

1.	Department of Mathematics, University of Tennessee at Chattanooga, 615 McCallie Avenue, Chattanooga, TN 37403-2598, United States
2.	Department of Mathematics & Actuarial Science, Indiana University Northwest, 3400 Broadway, Gary, IN 46408, United States

Received August 2006 Revised May 2007 Published September 2007

Abstract
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We consider a general linear stochastic wave equation driven by fractional-in-time noise and study its energy. We provide a mild solution for the wave equation in terms its Fourier expansion. We calculate the expected energy and give asymptotic results for the expected energy for large and small times and as the Hurst parameter, H, approaches 1/2. These results are phrased in terms of the norms of powers of the differential operator times powers of the spatial covariance operator.

Keywords: Fractional Browian Motion., Wave Equation, Energy.

Mathematics Subject Classification: Primary: 60H15; Secondary: 35R6.

Citation: Boris P. Belinskiy, Peter Caithamer. Energy estimate for the wave equation driven by a fractional Gaussian noise. Conference Publications, 2007, 2007 (Special) : 92-101. doi: 10.3934/proc.2007.2007.92

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