Geometric Brownian Motion under Stochastic Resetting: A Stationary yet Non-ergodic Process

TL;DR

This paper investigates how stochastic resetting transforms geometric Brownian motion into a stationary yet non-ergodic process, revealing distinct long-term regimes and the conditions for self-averaging behavior.

Contribution

It demonstrates that resetting induces stationarity in GBM while preserving non-ergodicity, and identifies regimes and optimal resetting rates affecting long-term dynamics.

Findings

Resetting makes GBM stationary but non-ergodic.

Three long-time regimes depend on resetting strength.

Self-averaging occurs in a specific regime with optimal resetting.

Abstract

We study the effects of stochastic resetting on geometric Brownian motion (GBM), a canonical stochastic multiplicative process for non-stationary and non-ergodic dynamics. Resetting is a sudden interruption of a process, which consecutively renews its dynamics. We show that, although resetting renders GBM stationary, the resulting process remains non-ergodic. Quite surprisingly, the effect of resetting is pivotal in manifesting the non-ergodic behavior. In particular, we observe three different long-time regimes: a quenched state, an unstable and a stable annealed state depending on the resetting strength. Notably, in the last regime, the system is self-averaging and thus the sample average will always mimic ergodic behavior establishing a stand alone feature for GBM under resetting. Crucially, the above-mentioned regimes are well separated by a self-averaging time period which can be minimized by an optimal resetting rate. Our results can be useful to interpret data emanating from stock market collapse or reconstitution of investment portfolios.