# Extreme value statistics of ergodic Markov processes from first passage times in the large deviation limit

## TL;DR

This paper develops a framework linking extreme value statistics of ergodic Markov processes to their relaxation eigenspectra, providing bounds and approximations for long-time extreme value distributions in confining potentials.

## Contribution

It introduces a novel approach connecting extreme value functionals to relaxation eigenspectra and derives bounds for first passage densities in ergodic reversible Markov processes.

## Key findings

- Bounds on long-time first passage densities derived
- Large deviation limit accurately approximates extreme value statistics
- Applicable to Ornstein-Uhlenbeck and Bessel processes

## Abstract

Extreme value functionals of stochastic processes are inverse functionals of the first passage time -- a connection that renders their probability distribution functions equivalent. Here, we deepen this link and establish a framework for analyzing extreme value statistics of ergodic reversible Markov processes in confining potentials on the hand of the underlying relaxation eigenspectra. We derive a chain of inequalities, which bounds the long-time asymptotics of first passage densities, and thereby extrema, from above and from below. The bounds involve a time integral of the transition probability density describing the relaxation towards equilibrium. We apply our general results to the analysis of extreme value statistics at long times in the case of Ornstein-Uhlenbeck process and a 3-dimensional Brownian motion confined to a sphere, also known as Bessel process. We find that even on time-scales that are shorter than the equilibration time, the large deviation limit characterizing long-time asymptotics can approximate the statistics of extreme values remarkably well. Our findings provide a novel perspective on the study of extrema beyond the established limit theorems for sequences of independent random variables and for asymmetric diffusion processes beyond a constant drift.