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

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