Optimal Resetting Brownian Bridges

TL;DR

This paper introduces a resetting Brownian bridge model to analyze search processes with finite search time and returns to the start, deriving algorithms and identifying an optimal resetting rate that enhances search efficiency.

Contribution

It presents a novel model of resetting Brownian bridges, provides a rejection-free algorithm for simulation, and reveals a unique optimal resetting rate for search efficiency.

Findings

Existence of an optimal resetting rate $r^*$ for search efficiency.

Derived an explicit Langevin equation for generating resetting bridges.

Found that the mechanism for optimal resetting differs from unconstrained Brownian motion.

Abstract

We introduce a resetting Brownian bridge as a simple model to study search processes where the total search time $t_f$ is finite and the searcher returns to its starting point at $t_f$. This is simply a Brownian motion with a Poissonian resetting rate $r$ to the origin which is constrained to start and end at the origin at time $t_f$. We first provide a rejection-free algorithm to generate such resetting bridges in all dimensions by deriving an effective Langevin equation with an explicit space-time dependent drift $\tilde \mu({\bf x},t)$ and resetting rate $\tilde r({\bf x}, t)$. We also study the efficiency of the search process in one-dimension by computing exactly various observables such as the mean-square displacement, the hitting probability of a fixed target and the expected maximum. Surprisingly, we find that there exists an optimal resetting rate $r^*$ that maximizes the search efficiency, even in the presence of a bridge constraint. We show however that the physical mechanism responsible for this optimal resetting rate for bridges is entirely different from resetting Brownian motions without the bridge constraint.