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Damped vibrations excited by white noise

Published online by Cambridge University Press:  01 July 2016

Enzo Orsingher
Enzo Orsingher*
Affiliation:
University of Rome
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Abstract

Let v(x, t) denote the displacement of an infinitely long, idealized string performing damped vibrations caused by white noise.

Upper and lower bounds for the distribution of maxsv(x, s) and maxxv(x, t) are presented. The results are obtained by adapting Lévy-type inequalities and exploiting a connection of v(x, t) with the Ornstein-Uhlenbeck process through Slepian's theorem.

The case of forced-damped vibrations is also analysed. Finally, a section is devoted to the case of a semi-infinite string performing damped vibrations.

Keywords

ORNSTEIN-UHLENBECK PROCESSLÉVY-TYPE INEQUALITIESBROWNIAN MOTIONGREEN'S FUNCTION
Type
Research Article
Information
Advances in Applied Probability , Volume 16 , Issue 3 , September 1984 , pp. 562 - 584
Copyright
Copyright © Applied Probability Trust 1984 

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References

Cabaña, E. ?. (1970) On the vibrating string forced by white noise. Z. Wahrscheinlichkeitsth. 15, 111–130.CrossRefGoogle Scholar
Cabaña, E. M. (1972) On the barrier problem for the vibrating string. Z. Wahrscheinlichkeitsth. 22, 13–24.CrossRefGoogle Scholar
Magnus, W. and Oberhettinger, F. (1948) Formeln und Sätze für die Speziellen Funktionen der Mathematischer Physik. Springer-Verlag, Berlin.CrossRefGoogle Scholar
Marcus, M. B. and Shepp, L. A. (1971) Sample behavior of gaussian processes. Proc. 6th Berkeley Symp. Math. Statist. Prob. 2, 423–441.Google Scholar
Orsingher, E. (1982) Randomly forced vibrations of a string. Ann. Inst. H. Poincaré B 18 (4), 367–394.Google Scholar
Orsingher, E. (1983) Vibrations with random initial conditions.Google Scholar
Smirnov, V. I. (1970) Cours de mathématiques supérieures. Editions de Moscou, Moscow.Google Scholar