On discrete-time arrival processes and related random motions

TL;DR

This paper classifies discrete-time arrival processes, analyzes their properties, and applies findings to random walks on lattices, revealing behaviors like superdiffusion and universal steady states.

Contribution

It introduces a comprehensive framework for classifying and analyzing stopped renewal processes, including their asymptotic behaviors and applications to lattice random walks.

Findings

Superdiffusive (ballistic) behavior in biased walks

Convergence to a universal non-equilibrium steady state (NESS)

Integral representation of NESS involving alpha-stable distributions

Abstract

We consider three kinds of discrete-time arrival processes: transient, intermediate and recurrent, characterized by a finite, possibly finite and infinite number of events, respectively. In this context, we study renewal processes which are stopped at the first event of a further independent renewal process whose inter-arrival time distribution can be defective. If this is the case, the resulting arrival process is of an intermediate nature. For non-defective absorbing times, the resulting arrival process is transient, i.e.\ stopped almost surely. For these processes we derive finite time and asymptotic properties. We apply these results to biased and unbiased random walks on the d-dimensional infinite lattice and as a special case on the two-dimensional triangular lattice. We study the spatial propagator of the walker and its large time asymptotics. In particular, we observe the emergence of a superdiffusive (ballistic) behavior in the case of biased walks. For geometrically distributed stopping times, the propagator converges to a stationary non-equilibrium steady state (NESS), which is universal in the sense that it is independent of the stopped process. In dimension one, for both light- and heavy-tailed step distributions, the NESS has an integral representation involving alpha-stable distributions.