Cover times with stochastic resetting

TL;DR

This work analyzes cover times for stochastic searches under resetting, deriving exact results for a one-dimensional Brownian search and providing accurate moment approximations for a wide class of processes in $d$ dimensions and on networks in the frequent resetting limit. The central finding is that the moments of the cover time $T_r$ are governed by the farthest parts of the target, with asymptotics $\mathbb{E}[(T_r)^m] \sim m!/(r p)^m$ where $p$ encodes the success probability to the farthest target, and general network analogues depend on geodesic distances and shortest-path products. The paper provides explicit formulas and asymptotics for diffusion, run-and-tumble, subdiffusion, diffusion on torus, and lattice networks, and furnishes practical guidance for selecting resetting rates to minimize mean cover times. These results connect to and extend the broader literature on resetting-enhanced search, offering a tractable framework to estimate optimal resetting and to understand when resetting reduces search time in complex domains. The findings have potential implications for algorithmic search, animal foraging, and other stochastic exploration tasks where resetting can be leveraged to accelerate exhaustive search operations.

Abstract

Cover times quantify the speed of exhaustive search. In this work, we compute exactly the mean cover time associated with a one-dimensional Brownian search under exponentially distributed resetting. We also approximate the moments of cover times of a wide range of stochastic search processes in $d$-dimensional continuous space and on an arbitrary discrete network under frequent stochastic resetting. These results hold for a large class of resetting time distributions and search processes including diffusion and Markov jump processes.

Figures (4)

• Figure 1: Representative illustration of a spatial domain $M$ (pictured, a 2D torus) with a stochastic searcher that starts at $U_0$. Dark purple indicates the searcher path and light purple indicates the corresponding region detected. Orange denotes the target region ($U_T$) and yellow denotes the shortest path (length $L$) that the searcher must travel to cover the farthest part of the target.
• Figure 2: MCTs of the interval $[-1,2]$ for a resetting Brownian motion with diffusivity $D=1$ and $L=2$ denoting the distance of the furthest target point from the searcher initial and resetting position. The left plot illustrates \ref{['etrrp']} (solid), \ref{['unc_ETr']} (dashed), and \ref{['conc_ETr']} (dash-dotted). The right plot illustrates the relative error of the frequent resetting result with respect to the exact unconstrained and constrained expressions. Resetting is exponentially distributed with rate $r>0$. See section \ref{['exact']} for details.
• Figure 3: MCTs of the interval $[-1,1]$ for a resetting RTP with velocity $v=1$, switching rate $\gamma=1$, and $L=1$ denoting the distance of the furthest target points from the searcher initial and resetting position. The left plot illustrates \ref{['32']} (solid), \ref{['unc_ETr_rtp']} (dashed), and \ref{['con_ETr_rtp']} (dash-dotted). The right plot illustrates the relative error of the frequent resetting result with respect to the exact unconstrained and constrained expressions. Resetting is exponentially distributed with rate $r>0$. See section \ref{['rtp']} for details.
• Figure 4: Left: The initial node of the searcher in pink, which can take its first step to any of the four yellow nodes; the simulated nested target sets $U_{T,j}$ in \ref{['UTj']} are for $j=0$ (purple), $j=1$ (orange), and $j=2$ (green). Right: The theoretical MCT in Corollary \ref{['cor5']} to simulations of that for the depicted nested target sets. Each point is the average of 20 simulations with jump rates $k=1$. See section \ref{['markov']} for details.

Theorems & Definitions (7)

• Theorem 1
• Lemma 2
• proof : Proof of Theorem \ref{['thm1']}
• Theorem 3
• Corollary 4
• Lemma 5
• proof : Proof of Theorem \ref{['thm_net']}