Dynamical phase transition in drifted Brownian motion

TL;DR

This paper investigates a dynamical phase transition in drifted Brownian motion, revealing a switch between different occupation behaviors in the long-time limit, with connections to quantum phase transitions.

Contribution

It demonstrates a novel dynamical phase transition in occupation fluctuations of drifted Brownian motion without low-noise limits, linking stochastic dynamics to quantum phase phenomena.

Findings

Identification of a phase transition between confined and ergodic motion

Connection between dynamical phase transition and quantum phase transitions

Analysis of variations including geometric Brownian motion in finance

Abstract

We study the occupation fluctuations of drifted Brownian motion in a closed interval, and show that they undergo a dynamical phase transition in the long-time limit without an additional low-noise limit. This phase transition is similar to wetting and depinning transitions, and arises here as a switching between paths of the random motion leading to different occupations. For low occupations, the motion essentially stays in the interval for some fraction of time before escaping, while for high occupations the motion is confined in an ergodic way in the interval. This is confirmed by studying a confined version of the model, which points to a further link between the dynamical phase transition and quantum phase transitions. Other variations of the model, including the geometric Brownian motion used in finance, are considered to discuss the role of recurrent and transient motion in dynamical phase transitions.