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Energy of a string driven by a two-parameter Gaussian noise white in time

Part of: Hyperbolic equations and systems Stochastic analysis

Published online by Cambridge University Press:  14 July 2016

Boris P. Belinskiy  and
Peter Caithamer
Boris P. Belinskiy*
Affiliation:
University of Tennessee at Chattanooga
Peter Caithamer
Affiliation:
University of Tennessee at Chattanooga
*
∗ Postal address: Department of Mathematics, University of Tennessee, 615 McCallie Avenue, Chattanooga, TN 37403, USA. Email address: bbelinsk@cecasun.utc.edu
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Abstract

In this paper we consider the stochastic wave equation in one spatial dimension driven by a two-parameter Gaussian noise which is white in time and has general spatial covariance. We give conditions on the spatial covariance of the driving noise sufficient for the string to have finite expected energy and calculate this energy as a function of time. We show that these same conditions on the spatial covariance of the driving noise are also sufficient to guarantee that the energy of the string has a version which is continuous almost surely.

Keywords

Stochastic wave equationenergystochastic partial differential equations

MSC classification

Primary: 60H15: Stochastic partial differential equations
Secondary: 35L05: Wave equation
Type
Research Papers
Information
Journal of Applied Probability , Volume 38 , Issue 4 , December 2001 , pp. 960 - 974
Copyright
Copyright © by the Applied Probability Trust 2001 

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Footnotes

∗∗

Current address: Department of Mathematical Sciences, US Military Academy, West Point, NY 10996, USA.

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