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Survival probability of stochastic processes beyond persistence exponents
- 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.
- Subjects :
- Condensed Matter - Statistical Mechanics
Subjects
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