Steady-state moments under resetting to a distribution

TL;DR

This paper derives universal formulas for steady-state moments in stochastic resetting processes, showing they depend linearly on the moments of the resetting distribution, with explicit results for Brownian and run-and-tumble particles.

Contribution

It introduces a universal linear relation for steady-state moments under resetting, applicable to various processes, and provides explicit formulas for specific particle models.

Findings

Steady-state moments are linear combinations of resetting distribution moments.

Universal coefficients are independent of the resetting distribution.

Explicit formulas for moments of Brownian and run-and-tumble particles are derived.

Abstract

The non-equilibrium steady states emerging from stochastic resetting to a distribution is studied. We show that for a range of processes, the steady-state moments can be expressed as a linear combination of the moments of the distribution of resetting positions. The coefficients of this series are universal in the sense that they do not depend on the resetting distribution, only underlying dynamics. We consider the case of a Brownian particle and a run-and-tumble particle confined in a harmonic potential, where we derive explicit closed-form expressions for all moments for any resetting distribution. Numerical simulations are used to verify the results, showing excellent agreement.