Survival probability of stochastic processes beyond persistence exponents

N. Levernier, M. Dolgushev, O. Bénichou, R. Voituriez () and T. Guérin
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N. Levernier: University of Geneva
M. Dolgushev: CNRS/Sorbonne Université, 4 Place Jussieu
O. Bénichou: CNRS/Sorbonne Université, 4 Place Jussieu
R. Voituriez: CNRS/Sorbonne Université, 4 Place Jussieu
T. Guérin: University of Bordeaux, Unité Mixte de Recherche 5798, CNRS

Nature Communications, 2019, vol. 10, issue 1, 1-7

Abstract: Abstract For many stochastic processes, the probability $$S(t)$$ S ( t ) of not-having reached a target in unbounded space up to time $$t$$ t follows a slow algebraic decay at long times, $$S(t) \sim {S}_{0}/{t}^{\theta }$$ S ( t ) ~ S 0 ∕ t θ . This is typically the case of symmetric compact (i.e. recurrent) random walks. While the persistence exponent $$\theta$$ θ has been studied at length, the prefactor $${S}_{0}$$ S 0 , which is quantitatively essential, remains poorly characterized, especially for non-Markovian processes. Here we derive explicit expressions for $${S}_{0}$$ S 0 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 $${S}_{0}$$ S 0 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.

Date: 2019
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DOI: 10.1038/s41467-019-10841-6

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