Asymmetric random walks with bias generated by discrete-time counting\n processes Article Swipe
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· 2021
· Open Access
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· DOI: https://doi.org/10.1016/j.cnsns.2021.106121
· OA: W3215151200
We introduce a new class of asymmetric random walks on the one-dimensional\ninfinite lattice. In this walk the direction of the jumps (positive or\nnegative) is determined by a discrete-time renewal process which is independent\nof the jumps. We call this discrete-time counting process the `it generator\nprocess' of the walk. We refer the so defined walk to as `Asymmetric\nDiscrete-Time Random Walk' (ADTRW). We highlight connections of the\nwaiting-time density generating functions with Bell polynomials. We derive the\ndiscrete-time renewal equations governing the time-evolution of the ADTRW and\nanalyze recurrent/transient features of simple ADTRWs (walks with unit jumps in\nboth directions). We explore the connections of the recurrence/transience with\nthe bias: Transient simple ADTRWs are biased and vice verse. Recurrent simple\nADTRWs are either unbiased in the large time limit or `strictly unbiased' at\nall times with symmetric Bernoulli generator process. In this analysis we\nhighlight the connections of bias and light-tailed/fat-tailed features of the\nwaiting time density in the generator process. As a prototypical example with\nfat-tailed feature we consider the ADTRW with Sibuya distributed waiting times.\nWe also introduce time-changed versions: We subordinate the ADTRW to a\ncontinuous-time renewal process which is independent from the generator process\nand the jumps to define the new class of `Asymmetric Continuous Time Random\nWalk' (ACTRW). This new class - apart of some special cases - is not a\nMontroll--Weiss continuous-time random walk (CTRW). ADTRW and ACTRW models may\nopen large interdisciplinary fields in anomalous transport, birth-death models\nand others.\n
