From an exact solution of dynamics in the vicinity of hard walls to extreme value statistics of non-Markovian processes

TL;DR

This paper provides an exact solution linking the dynamics near hard walls to extreme value statistics in stochastic processes, applicable to both Markovian and non-Markovian systems, with implications for understanding confinement and crossing probabilities.

Contribution

It introduces a universal mapping between confined steady-state distributions and extreme value statistics, extending to non-Markovian processes like run-and-tumble motion.

Findings

Derived noncrossing probabilities for Brownian motion.

Extended the analysis to non-Markovian processes.

Unified various known results within a single framework.

Abstract

We present an exact solution for one-dimensional overdamped dynamics near a hard wall, allowing us to connect steady-state distributions under confinement with the extreme value statistics of unconfined stochastic processes. This mapping holds regardless of the statistics of the noise driving the dynamics. We first apply this result within Brownian motion theory, deriving the noncrossing probability of a Brownian path with a specific family of curves, from which several well-known results in the field can be recovered in a unified way. We then extend the analysis to non-Markovian processes, using the mapping to a steady-state to compute the long-time noncrossing probability of a pair of run-and-tumble and Brownian particles.