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Survival probability of stochastic processes beyond persistence exponents

Authors :
Levernier, N.
Dolgushev, M.
Bénichou, O.
Voituriez, R.
Guérin, T.
Source :
Nature Communications 2019
Publication Year :
2019

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, $S(t)\sim S_0/t^\theta$. This is typically the case of symmetric compact (i.e. recurrent) random walks. While the persistence exponent $\theta$ has been studied at length, the prefactor $S_0$, which is quantitatively essential, remains poorly characterized, especially for non-Markovian processes. Here we derive explicit expressions for $S_0$ 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 $S_0$ 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.

Details

Database :
arXiv
Journal :
Nature Communications 2019
Publication Type :
Report
Accession number :
edsarx.1907.03632
Document Type :
Working Paper
Full Text :
https://doi.org/10.1038/s41467-019-10841-6