Brownian fluctuations in space-time with applications to vibrations of rods

In this paper Brownian fluctuations in space-time are considered. Time is assumed to run alternately forward and backward, the alternance being marked by a Poisson process with rate λ. It is shown that the law of this motion is a solution of a fourth-order partial differential equation. Furthermore the law of this movement in the presence of an absorbing barrier is derived. The equation ruling the movement analysed, when λ = 0 and is submitted to the change t' = −it, reduces to the equation of vibrations of rods. This fact is exploited to obtain the solution of boundary value problems concerning the equation of vibrating beams by means of Brownian motion techniques.