Survival probability of stochastic processes beyond persistence exponents.

Area of Science:

• Stochastic Processes
• Statistical Physics
• Random Walks

Background:

• The probability of not reaching a target in unbounded space often decays algebraically at long times for stochastic processes.
• While the persistence exponent is well-studied, the prefactor remains poorly characterized, especially for non-Markovian processes.

Purpose of the Study:

• To derive explicit expressions for the prefactor of the survival probability for compact random walks in unbounded space.
• To establish an analytic relation between this prefactor and the mean first-passage time in a confined volume.

Main Methods:

• Analytical derivation of explicit expressions for the prefactor.
• Establishing an analytic relation with mean first-passage time in a large confining volume.
• Comparison with numerical simulations for various processes.

Main Results:

• Explicit expressions for the prefactor [Formula: see text] were derived for compact random walks.
• The results show good agreement with numerical simulations, including for strongly correlated processes like Fractional Brownian Motion.
• A refined understanding of the statistics of longest first-passage events in unbounded space is provided.

Conclusions:

• The study provides a quantitative characterization of the prefactor in random walk survival probability.
• The established relation offers a new method for analyzing long-time behavior in stochastic processes.
• The findings are relevant for understanding phenomena involving persistence and first-passage times.