First passage time statistics of non-Markovian random walker: Onsager's regression hypothesis approach

TL;DR

This paper develops an analytical framework based on Onsager's regression hypothesis to accurately compute first passage time statistics for non-Markovian stochastic processes, extending understanding beyond traditional Markovian assumptions.

Contribution

It introduces a novel approach linking non-Markovian dynamics to relaxation principles, enabling precise calculations of first passage times and distributions for complex systems.

Findings

Confirmed non-trivial scaling for fractional Brownian motion

Derived a formula for first passage time distribution across all time scales

Provided a quantitative description of position probabilities with absorbing boundaries

Abstract

First passage time plays a fundamental role in dynamical characterization of stochastic processes. Crucially, our current understanding on the problem is almost entirely relies on the theoretical formulations, which assume the processes under consideration are Markovian, despite abundant non-Markovian dynamics found in complex systems. Here we introduce a simple and physically appealing analytical framework to calculate the first passage time statistics of non-Markovian walkers grounded in a fundamental principle of nonequilibrium statistical physics that connects the fluctuations in stochastic system to the macroscopic law of relaxation. Pinpointing a crucial role of the memory in the first passage time statistics, our approach not only allows us to confirm the non-trivial scaling conjectures for fractional Brownian motion, but also provides a formula of the first passage time distribution in the entire time scale, and establish the quantitative description of the position probability distribution of non-Markovian walkers in the presence of absorbing boundary.