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Randomly perturbed vibrations

Part of: Stochastic analysis Stochastic processes

Published online by Cambridge University Press:  14 July 2016

M. Elshamy
M. Elshamy*
Affiliation:
University of Alabama in Huntsville
*
∗Postal address: Department of Mathematical Sciences, The University of Alabama in Huntsville, Huntsville, AL 35899, USA.
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Abstract

Let uε(t, x) be the position at time t of a point x on a string, where the time variable t varies in an interval I: = [0, T], T is a fixed positive time, and the space variable x varies in an interval J. The string is performing forced vibrations and also under the influence of small stochastic perturbations of intensity ε. We consider two kinds of random perturbations, one in the form of initial white noise, and the other is a nonlinear random forcing which involves the formal derivative of a Brownian sheet. When J has finite endpoints, a Dirichlet boundary condition is imposed for the solutions of the resulting non-linear wave equation. Assuming that the initial conditions are of sufficient regularity, we analyze the deviations uε(t, x) from u0(t, x), the unperturbed position function, as the intensity of perturbation ε ↓ 0 in the uniform topology. We also discuss some continuity properties of the realization of the solutions uε(t, x).

Keywords

RANDOM PERTURBATIONSVIBRATIONSBROWNIAN SHEETRANDOM FIELDSLARGE DEVIATIONS

MSC classification

Primary: 60H15: Stochastic partial differential equations
Secondary: 60G60: Random fields
Type
Research Papers
Information
Journal of Applied Probability , Volume 32 , Issue 2 , June 1995 , pp. 417 - 428
Copyright
Copyright © Applied Probability Trust 1995 

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