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Large and small deviations of a string driven by a two-parameter Gaussian noise white in time

Part of: Stochastic analysis Limit theorems

Published online by Cambridge University Press:  14 July 2016

Peter Caithamer
Peter Caithamer*
Affiliation:
United States Military Academy, West Point
*
∗Postal address: 952 N. Loomis Street, Naperville, IL 60563, USA. Email address: peter.caithamer@comcast.net
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Abstract

Upper as well as lower bounds for both the large deviations and small deviations of several sup-norms associated with the displacements of a one-dimensional string driven by a Gaussian noise which is white in time and has general spatial covariance are developed.

Keywords

Stochastic wave equationwhite noiselarge deviationsmall deviation

MSC classification

Primary: 60H15: Stochastic partial differential equations
Secondary: 60F10: Large deviations
Type
Research Papers
Information
Journal of Applied Probability , Volume 40 , Issue 4 , September 2003 , pp. 946 - 960
Copyright
Copyright © Applied Probability Trust 2003 

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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.Google Scholar
Biswas, S. K., and Ahmed, N. U. (1985). Stabilisation of systems governed by the wave equation in the presence of white noise. IEEE Trans. Automatic Control 30, 1043–1045.Google Scholar
Cabaña, E. (1970). The vibrating string forced by white noise. Z. Wahrscheinlichkeitsth. 15, 111–130.Google Scholar
Cabaña, E. (1972). On barrier problems for the vibrating string. Z. Wahrscheinlichkeitsth. 22, 13–24.Google Scholar
Conway, J. B. (1990). A Course in Functional Analysis. Springer, New York.Google Scholar
Dalang, R. C., and Frangos, N. E. (1998). The stochastic wave equation in two spatial dimensions. Ann. Prob. 26, 187–212.CrossRefGoogle Scholar
Dudley, R. (1967). The sizes of compact subsets of Hilbert space and continuity of Gaussian processes. J. Funct. Anal. 1, 290–330.Google Scholar
Durrett, R. (1995). Probability: Theory and Examples, 2nd edn. Duxbury, Belmont, CA.Google Scholar
Elshamy, M. (1995). Randomly perturbed vibrations. J. Appl. Prob. 32, 417–428.Google Scholar
Elshamy, M. (1996). Stochastic models of damped vibrations. J. Appl. Prob. 33, 1159–1168.Google Scholar
Fernique, X. (1975). Régularité des trajectoires des fonctions aléatoires gaussiennes. In école d'été de Probabilités de Saint-Flour IV (Lecture Notes Math. 480), ed. Hennequin, P.-L., Springer, Berlin, pp. 1–96.Google Scholar
Ledoux, M. (1996). Isoperimetry and Gaussian analysis. In Lectures on Probability Theory and Statistics (Lecture Notes Math. 1648), Springer, Berlin, pp. 165–294.CrossRefGoogle Scholar
Li, W., and Shao, Q. (2001). Gaussian processes: inequalities, small ball probabilities and applications. In Handbook of Statistics, Vol. 19, Stochastic Processes: Theory and Methods, eds Rao, C. R. and Shanbhag, D., North-Holland, Amsterdam, pp. 533–597.Google Scholar
Miller, R. N. (1990). Tropical data assimilation with simulated data: the impact of the tropical ocean and global atmosphere thermal array for the ocean. J. Geophys. Res. Oceans 95, 11461–11482.CrossRefGoogle Scholar
Orsingher, E. (1982). Randomly forced vibrations of a string. Ann. Inst. H. Poincaré B 18, 367–394.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.CrossRefGoogle Scholar