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

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