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

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.

References in corpus (21)

• Fractional calculus and continuous-time finance II: the waiting-time distribution
• Uncoupled continuous-time random walks: Solution and limiting behavior of the master equation
• A practical guide to Prabhakar fractional calculus
• Monte Carlo simulation of uncoupled continuous-time random walks yielding a stochastic solution of the space-time fractional diffusion equation
• Why fractional derivatives with nonsingular kernels should not be used
• Operational Calculus for the general fractional derivatives with the Sonine kernels
• The Space-Fractional Poisson Process
• The Continuous Time Random Walk, still trendy: Fifty-year history, state of art, and outlook
• General fractional calculus and Prabhakar's theory
• New identities for the partial Bell polynomials
• Fractional advection-diffusion-asymmetry equation
• Should I stay or should I go? Zero-size jumps in random walks for Lévy flights
• Compositions, Random Sums and Continued Random Fractions of Poisson and Fractional Poisson Processes
• Mittag-Leffler Waiting Time, Power Laws,Rarefaction, Continuous Time Random Walk, Diffusion Limit
• Nonlocal biased random walks and fractional transport on directed networks
• Generalized fractional Poisson process and related stochastic dynamics
• On recurrence of random walks with long-range steps generated by fractional Laplacian matrices on regular networks and simple cubic lattices
• Continuous time random walk and diffusion with generalized fractional Poisson process
• On Discrete Time Prabhakar-Generalized Fractional Poisson Processes and Related Stochastic Dynamics
• Biased continuous-time random walks with Mittag-Leffler jumps
• Derivation of Bell Polynomials of the Second Kind