Survival probability of stochastic processes beyond persistence exponents

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.