Souvenir collector's walk: The distribution of the number of steps of a continuous-time random walk ending at a given position.

Abstract

We consider a random walker performing a continuous-time random walk (CTRW) with a symmetric step lengths' distribution possessing a finite second moment and with a power-law waiting time distribution with finite or diverging first moment. The problem we pose concerns the distribution of the number of steps of the corresponding CTRW conditioned on the final position of the walker at some long time t. For positions within the scaling domain of the probability density function (PDF) of final displacements, the distributions of the number of steps show a considerable amount of universality, and are different in the cases when the corresponding CTRW corresponds to subdiffusion and to normal diffusion. We moreover note that the mean value of the number of steps can be obtained independently and follows from the solution of the Poisson equation whose right-hand side depends on the PDF of displacements only. This approach works not only in the scaling domain but also in the large deviation domain of the corresponding PDF, where the behavior of the mean number of steps is very sensitive to the details of the waiting time distribution beyond its power-law asymptotics.