Hostname: page-component-586b7cd67f-t7czq Total loading time: 0 Render date: 2024-11-24T04:27:26.375Z Has data issue: false hasContentIssue false
  • English
  • Français

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.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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 

References

Belinskiy, B. P. and Caithamer, P. (2001). Energy of a string driven by a two-parameter Gaussian noise white in time. J. Appl. Prob. 38, 960–974.CrossRefGoogle Scholar
Cabaña, E. M. (1972). On barrier problems for the vibrating string. Z. Wahrscheinlichkeitsth. 22, 13–24.CrossRefGoogle Scholar
Cabaña, E. M. (1991). A Gaussian process with parabolic variance. J. Appl. Prob. 28, 898–902.CrossRefGoogle Scholar
Elshamy, M. (1995). Randomly perturbed vibrations. J. Appl. Prob. 32, 417–428.CrossRefGoogle Scholar
Elshamy, M. (1996). Stochastic models of damped vibrations. J. Appl. Prob. 33, 1159–1168.CrossRefGoogle Scholar
Orsingher, E. (1984). Damped vibrations excited by white noise. Adv. Appl. Prob. 16, 562–584.CrossRefGoogle Scholar
Orsingher, E. (1989). On the maximum of Gaussian Fourier series emerging in the analysis of random vibrations. J. Appl. Prob. 26, 182–188.CrossRefGoogle Scholar
Walsh, J. B. (1986). An introduction to stochastic partial differential equations. In École d'Été de Probabilitiés de Saint-Flour, XIV—1984 (Lecture Notes Math. 1180), Springer, New York, pp. 265–439.CrossRefGoogle Scholar