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A Stability Property of Stochastic Vibration

Part of: Stochastic analysis

Published online by Cambridge University Press:  14 July 2016

M. Elshamy
M. Elshamy*
Affiliation:
Alabama A&M University
*
∗ Postal address: Department of Mathematics, Alabama A&M University, Normal, AL 35762, USA.
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Abstract

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Let u(t,x) be the displacement at time t of a point x on a string; the time variable t varies in the interval I≔[0,T] and the space variable x varies in the interval J≔[0,L], where T and L are fixed positive constants. The displacement u(t,x) is the solution to a stochastic wave equation. Two forms of random excitations are considered, a white noise in the initial condition and a nonlinear random forcing which involves the formal derivative of a Brownian sheet. In this article, we consider the continuity properties of solutions to this equation. Smoothness characteristics of these random fields, in terms of Hölder continuity, are also investigated.

Keywords

Stochastic wave equationBrownian sheetconvergence

MSC classification

Primary: 60H15: Stochastic partial differential equations 60H30: Applications of stochastic analysis (to PDE, etc.)
Type
Research Article
Information
Journal of Applied Probability , Volume 44 , Issue 2 , June 2007 , pp. 444 - 457
Copyright
Copyright © Applied Probability Trust 2007 

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