Asymmetric random walks with bias generated by discrete-time counting processes

TL;DR

This paper introduces a new class of asymmetric random walks driven by discrete-time renewal processes, analyzing their recurrence, transience, bias, and connections to fat-tailed waiting times, with applications in anomalous transport.

Contribution

The paper proposes the Asymmetric Discrete-Time Random Walk (ADTRW) model, linking bias and tail properties of waiting times, and extends it to continuous time as ACTRW, broadening the scope of anomalous transport models.

Findings

Transient ADTRWs are biased; recurrent ADTRWs are unbiased or strictly unbiased.

Bias correlates with fat-tailed waiting time distributions.

The introduced models differ from traditional CTRWs, offering new insights into anomalous transport.

Abstract

We introduce a new class of asymmetric random walks on the one-dimensional infinite lattice. In this walk the direction of the jumps (positive or negative) is determined by a discrete-time renewal process which is independent of the jumps. We call this discrete-time counting process the `it generator process' of the walk. We refer the so defined walk to as `Asymmetric Discrete-Time Random Walk' (ADTRW). We highlight connections of the waiting-time density generating functions with Bell polynomials. We derive the discrete-time renewal equations governing the time-evolution of the ADTRW and analyze recurrent/transient features of simple ADTRWs (walks with unit jumps in both directions). We explore the connections of the recurrence/transience with the bias: Transient simple ADTRWs are biased and vice verse. Recurrent simple ADTRWs are either unbiased in the large time limit or `strictly unbiased' at all times with symmetric Bernoulli generator process. In this analysis we highlight the connections of bias and light-tailed/fat-tailed features of the waiting time density in the generator process. As a prototypical example with fat-tailed feature we consider the ADTRW with Sibuya distributed waiting times. We also introduce time-changed versions: We subordinate the ADTRW to a continuous-time renewal process which is independent from the generator process and the jumps to define the new class of `Asymmetric Continuous Time Random Walk' (ACTRW). This new class - apart of some special cases - is not a Montroll--Weiss continuous-time random walk (CTRW). ADTRW and ACTRW models may open large interdisciplinary fields in anomalous transport, birth-death models and others.