# Survival probability of stochastic processes beyond persistence exponents

## TL;DR

This paper derives explicit formulas for the prefactor in the algebraic decay of survival probability for stochastic processes, linking it to mean first-passage times, and confirms the results with simulations, including for non-Markovian processes.

## Contribution

It provides the first explicit expressions for the prefactor in survival probability decay, extending understanding to non-Markovian processes like Fractional Brownian Motion.

## Key findings

- Analytical expressions for the prefactor $S_0$ are derived.
- Results agree well with numerical simulations.
- The approach applies to strongly correlated processes like Fractional Brownian Motion.

## 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.