Diffusion with Resetting Inside a Circle

TL;DR

This paper analyzes a stochastic search process combining diffusion and boundary travel within a circle, incorporating resetting to optimize search time for a target on the boundary, with analytical and numerical results.

Contribution

It introduces a model of combined diffusion and boundary travel with resetting inside a circle, deriving analytical expressions for mean search time and optimal resetting rate.

Findings

Resetting can significantly reduce the mean search time.

Analytical formulas for optimal resetting rate are derived.

Numerical methods verify the theoretical results.

Abstract

We study the Brownian motion of a particle in a bounded circular 2-dimensional domain, in search for a stationary target on the boundary of the domain. The process switches between two modes: one where it performs a two-dimensional diffusion inside the circle and one where it travels along the one-dimensional boundary. During the diffusion, the Brownian particle resets to its initial position with a constant rate $r$. The Fokker-Planck formalism allows us to calculate the mean time to absorption (MTA) as well as the optimal resetting rate for which the MTA is minimized. From the derived analytical results the parameter regions where resetting reduces the search time can be specified. We also provide a numerical method for the verification of our results.