How target distributions shape optimal stochastic resetting

TL;DR

This paper studies how the spatial distribution of a target influences the optimal stochastic resetting strategy for a searcher performing Brownian motion in a finite domain, revealing phase transitions and criteria for optimality.

Contribution

It introduces a framework to determine the optimal bulk resetting rate based on target distribution, highlighting the impact of target shape on search efficiency.

Findings

Optimal resetting strategy exhibits a second-order transition.

Derived mathematical criteria for various target distributions.

Target distribution critically influences search optimization.

Abstract

We investigate the search of a target with a given spatial distribution in a finite one-dimensional domain. The searcher follows Brownian dynamics and is always reset to its initial position when reaching the boundaries of the domain (boundary resetting). In addition, the searcher may be reset to its initial position from any internal point of the domain (bulk resetting). Specifically, we look for the optimal strategy for bulk resetting, i.e., the spatially dependent bulk resetting rate that minimizes the average search time. The best search strategy exhibits a second-order transition from vanishing to nonvanishing bulk resetting when varying the target distribution. The obtained mathematical criteria are further analyzed for different monoparametric families of distributions, which sheds light on the properties that control the optimal strategy for bulk resetting. Our work paves new research lines in the study of search processes, emphasizing the relevance of the target distribution for optimal search strategies, and identifies a successful framework to address these questions.