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Cost of excursions until first crossing of the origin for random walks and Lévy flights: an exact general formula (2403.16152v2)

Published 24 Mar 2024 in cond-mat.stat-mech and math.PR

Abstract: We consider a discrete-time random walk on a line starting at $x_0\geq 0$ where a cost is incurred at each jump. We obtain an exact analytical formula for the distribution of the total cost of a trajectory until the process crosses the origin for the first time. The formula is valid for arbitrary jump distribution and cost function (heavy- and light-tailed alike), provided they are symmetric and continuous. We analyze the formula in different scaling regimes, and find a high degree of universality with respect to the details of the jump distribution and the cost function. Applications are given to the motion of an active run-and-tumble particle in one dimension and extensions to multiple cost variables are considered. The analytical results are in perfect agreement with numerical simulations.

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  1. Closed-form resummation of the Lévy-flight series expression for Q(0, C)
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