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Cost of excursions until first return for random walks and L\'evy flights: an exact general formula

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arXiv:2403.16152v1 Announce Type: new
Abstract: We consider a discrete-time random walk 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 returns to 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. The tail behavior of the distribution is universal and independent of the details of the model. Applications are given to the motion of an active 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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