Survival probability of random walks and L\'evy flights with stochastic resetting

TL;DR

This paper analyzes the survival probability of symmetric random walks and Lévy flights with stochastic resetting, revealing universal behaviors and providing exact and asymptotic results for various step length distributions.

Contribution

It derives universal formulas for survival probabilities with resetting, applicable to both finite and infinite variance distributions, including lattice-based walks.

Findings

Survival probability is universal, independent of step distribution.

Exact finite-time and asymptotic results are derived.

Lattice walk behaviors are characterized with algebraic methods.

Abstract

We perform a thorough analysis of the survival probability of symmetric random walks with stochastic resetting, defined as the probability for the walker not to cross the origin up to time $n$. For continuous symmetric distributions of step lengths with either finite (random walks) or infinite variance (L\'evy flights), this probability can be expressed in terms of the survival probability of the walk without resetting, given by Sparre Andersen theory. It is therefore universal, i.e., independent of the step length distribution. We analyze this survival probability at depth, deriving both exact results at finite times and asymptotic late-time results. We also investigate the case where the step length distribution is symmetric but not continuous, focussing our attention onto arithmetic distributions generating random walks on the lattice of integers. We investigate in detail the example of the simple Polya walk and propose an algebraic approach for lattice walks with a larger range.