Survival probability of stochastic processes beyond persistence exponents

• Published in: Nature communications 2019-07, Vol.10 (1), p.2990-7, Article 2990
• Main Authors: Levernier, N., Dolgushev, M., Bénichou, O., Voituriez, R., Guérin, T.
• Format: Article
• Language: English
• Subjects: 639/766/530/2804; 639/766/94; Brownian motion; Computer simulation; Condensed Matter; Decay rate; Humanities and Social Sciences; Markov analysis; Markov processes; multidisciplinary; Physics; Probability theory; Random walk; Random walk theory; Science; Science (multidisciplinary); Stochastic models; Stochastic processes; Stochasticity
• ISSN: 2041-1723; 2041-1723
• DOI: 10.1038/s41467-019-10841-6

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 ) ~ S 0 ∕ t θ . 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 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. The survival probability of a random walker is the probability that a particular target has not been reached by time t . Here the authors produce a formula for the prefactor involved in the expression of the survival probability which is shown to hold for both Markovian and non-Markovian processes.