Damped vibrations excited by white noise

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 max s v(x, s) and max x v(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.