Ergodic properties of Brownian motion under stochastic resetting

TL;DR

This paper investigates the ergodic properties of one-dimensional Brownian motion with stochastic resetting, revealing transitions between different ergodic regimes and deriving a fractional integral equation for the particle density.

Contribution

It introduces a comprehensive analysis of ergodic transitions in Brownian motion with resetting, including the derivation of a fractional integral equation for the density.

Findings

Identification of two ergodic transitions based on waiting time moments.

Derivation of a fractional integral equation for particle density.

Discovery of rich ergodic behaviors beyond Brownian motion.

Abstract

We study ergodic properties of one-dimensional Brownian motion with resetting. Using generic classes of statistics of times between resets, we find respectively for thin/fat tailed distributions, the normalized/non-normalised invariant density of this process. The former case corresponds to known results in the resetting literature and the latter to infinite ergodic theory. Two types of ergodic transitions are found in this system. The first is when the mean waiting time between resets diverges, when standard ergodic theory switches to infinite ergodic theory. The second is when the mean of the square root of time between resets diverges and the properties of the invariant density are drastically modified. We then find a fractional integral equation describing the density of particles. This finite time tool is particularly useful close to the ergodic transition where convergence to asymptotic limits is logarithmically slow. Our study implies rich ergodic behaviors for this non-equilibrium process which should hold far beyond the case of Brownian motion analyzed here.