Optimal Resetting Brownian Bridges

Benjamin De Bruyne; Satya N. Majumdar; Gregory Schehr

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arXiv · 2201.01994

Optimal Resetting Brownian Bridges

Benjamin De Bruyne·Satya N. Majumdar·Gregory Schehr

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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.

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Benjamin De Bruyne, Satya N. Majumdar, Gregory Schehr. 2022-01-06. Optimal Resetting Brownian Bridges. https://arxiv.org/abs/2201.01994

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