Exact solutions of the problem of a one-dimensional random walk of a particle with finite free-motion speed

Abstract

We consider a one-dimensional walk of a particle with finite speed of free motion. The universal description of this walk is proposed in terms of multiple convolutions of distributions of free paths. Some cases are discussed where the convolutions are expressed in terms of elementary or special functions, called exact solutions. As a particular case, we obtain Monin’s solution for the symmetric walk with exponential distribution of the free path, and extend it to the asymmetric case.