Survival probability of stochastic processes beyond persistence exponents

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∕ 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, which is quantitatively essential, remains poorly characterized, especially for non-Markovian processes. Here we derive explicit expressions for S 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 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.