Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-28T07:43:58.907Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Part of: Stochastic analysis Hyperbolic equations and systems

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: [email protected]
Get access Rights & Permissions [Opens in a new window]

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

∗∗

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

References

Cabaña, E. (1972). On barrier problems for the vibrating string. Z. Wahrscheinlichkeitsth. 22, 13–24.CrossRefGoogle Scholar
Caithamer, P. (1999). On the distribution of self-similar stochastic processes defined by double Wiener–Itô integrals. Rad. Mat. 9, 125–134.Google Scholar
Dalang, R. C., and Frangos, N. E. (1998). The stochastic wave equation in two spatial dimensions. Ann. Prob. 26, 187–212.Google Scholar
Da Prato, G., and Zabczyk, J. (1992). Stochastic Equations in Infinite Dimensions. Cambridge University Press.Google Scholar
Elshamy, M. (1996). Stochastic models of damped vibrations. J. Appl. Prob. 33, 1159–1168.Google Scholar
Gyöngy, I., and Krylov, N. V. (1980). On stochastic equations with respect to semi-martingales I. Stochastics 4, 1–21.CrossRefGoogle Scholar
Houdré, C., Pérez-Abreu, V. and Üstünel, A. C. (1994). Multiple Wiener–Itô integrals: an introductory survey. In Chaos Expansions, Multiple Wiener–Itô Integrals and Their Applications, eds Houdré, C. and Pérez-Abreu, V., CRC Press, Boca Raton, FL, pp. 1–33.Google Scholar
Ledoux, M., and Talagrand, M. (1991). Probability in Banach Spaces (Ergeb. Math. Grenzgeb. 23). Springer, Berlin.Google Scholar
Oksendal, B. (1991). Stochastic Differential Equations: an Introduction with Applications, 3rd edn. Springer, Berlin.Google Scholar
Orsingher, E. (1984). Damped vibrations excited by white noise. Adv. Appl. Prob. 16, 562–584.Google Scholar
Orsingher, E. (1989). On the maximum of Gaussian Fourier series emerging in the analysis of random vibrations. J. Appl. Prob. 26, 182–188.Google Scholar
Plikusas, A. (1981). Some properties of the multiple Itô integral. Lithuanian Math. J. 21, 184–191.Google Scholar
Ross, S. (1995). Stochastic Processes, 2nd edn. John Wiley, New York.Google Scholar
Walsh, J. B. (1986). An introduction to stochastic partial differential equations. In Ecole d'Eté de Probabilités de Saint-Flour XIV (Lecture Notes Math. 1180), ed. Hennequin, P. L., Springer, Berlin, pp. 265–439.Google Scholar