Winding number of a Brownian particle on a ring under stochastic resetting

TL;DR

This paper analyzes how stochastic resetting affects the winding number of a Brownian particle on a ring, deriving explicit formulas for mean first-completion times and showing how resetting can optimize the process.

Contribution

It provides a closed-form expression for the mean first-completion time under resetting and explores the non-steady-state distribution of winding numbers on a ring.

Findings

Mean first-completion time can be minimized by tuning resetting rate.

Distribution of winding numbers does not reach a steady state.

Total number of turns grows linearly with time.

Abstract

We consider a random walker on a ring, subjected to resetting at Poisson-distributed times to the initial position (the walker takes the shortest path along the ring to the initial position at resetting times). In the case of a Brownian random walker the mean first-completion time of a turn is expressed in closed form as a function of the resetting rate. The value is shorter than in the ordinary process if the resetting rate is low enough. Moreover, the mean first-completion time of a turn can be minimised in the resetting rate. At large time the distribution of winding numbers does not reach a steady state, which is in contrast with the non-compact case of a Brownian particle under resetting on the real line. The mean total number of turns (and the variance of the net number of turns) grow linearly with time, with a proportionality constant equal to the inverse of the mean first-completion time of a turn.