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Survival probability of stochastic processes beyond persistence exponents

Abstract : For many stochastic processes, the probability S(t) of not-having reached a target in unbounded space up to time t follows a slow algebraic decay at long times, S(t)∼S0/tθ. This is typically the case of symmetric compact (i.e. recurrent) random walks. While the persistence exponent θ has been studied at length, the prefactor S0, which is quantitatively essential, remains poorly characterized, especially for non-Markovian processes. Here we derive explicit expressions for S0 for a compact random walk in unbounded space by establishing an analytic relation with the mean first-passage time of the same random walk in a large confining volume. Our analytical results for S0 are in good agreement with numerical simulations, even for strongly correlated processes such as Fractional Brownian Motion, and thus provide a refined understanding of the statistics of longest first-passage events in unbounded space.
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Submitted on : Friday, July 19, 2019 - 11:04:32 AM
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N. Levernier, M. Dolgushev, O. Benichou, R. Voituriez, T. Guérin. Survival probability of stochastic processes beyond persistence exponents. Nature Communications, Nature Publishing Group, 2019, 10 (1), pp.2990 (2019). ⟨10.1038/s41467-019-10841-6⟩. ⟨hal-02189196⟩

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