Metastable states in Brownian energy landscape

TL;DR

This paper investigates the long-term behavior of a Brownian motion in a one-dimensional random environment, revealing a stationary renewal process structure in the depths of visited wells and analyzing their statistical properties.

Contribution

It provides a detailed description of the renewal process of well depths in Brownian energy landscapes and characterizes their limit behavior and fluctuations.

Findings

The depths of visited wells form a stationary renewal process in logarithmic scale.

The empirical density of well depths converges almost surely.

Fluctuations of the empirical density are characterized statistically.

Abstract

Random walks and diffusions in symmetric random environment are known to exhibit metastable behavior: they tend to stay for long times in wells of the environment. For the case that the environment is a one-dimensional two-sided standard Brownian motion, we study the process of depths of the consecutive wells of increasing depth that the motion visits. When these depths are looked in logarithmic scale, they form a stationary renewal cluster process. We give a description of the structure of this process and derive from it the almost sure limit behavior and the fluctuations of the empirical density of the process.