Optimization and Growth in First-Passage Resetting

Abstract

We combine the processes of resetting and first-passage to define \emph{first-passage resetting}, where the resetting of a random walk to a fixed position is triggered by a first-passage event of the walk itself. In an infinite domain, first-passage resetting of isotropic diffusion is non-stationary, with the number of resetting events growing with time as $\sqrt{t}$. We calculate the resulting spatial probability distribution of the particle analytically, and also obtain this distribution by a geometric path decomposition. In a finite interval, we define an optimization problem that is controlled by first-passage resetting; this scenario is motivated by reliability theory. The goal is to operate a system close to its maximum capacity without experiencing too many breakdowns. However, when a breakdown occurs the system is reset to its minimal operating point. We define and optimize an objective function that maximizes the reward (being close to maximum operation) minus a penalty for each breakdown. We also investigate extensions of this basic model to include delay after each reset and to two dimensions. Finally, we study the growth dynamics of a domain in which the domain boundary recedes by a specified amount whenever the diffusing particle reaches the boundary after which a resetting event occurs. We determine the growth rate of the domain for the semi-infinite line and the finite interval and find a wide range of behaviors that depend on how much the recession occurs when the particle hits the boundary.

31 pages in IOP format, 7 figures. Version 2: various additions and corrections; for publication in JSTAT

References in corpus (4)

• First Passage Under Restart
• Random walks with preferential relocations to places visited in the past and their application to biology
• Stochastic Search with Poisson and Deterministic Resetting
• Interacting Brownian Motion with Resetting

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• Random walks on complex networks with first-passage resetting
• Encounter-based approach to diffusion with resetting
• First passage in discrete-time absorbing Markov chains under stochastic resetting
• Target Search Optimization by Threshold Resetting
• Manipulating phases in many-body interacting systems with subsystem resetting
• Stochastic resetting of a population of random walks with resetting-rate-dependent diffusivity
• Optimal switching strategies for navigation in stochastic settings
• First-Passage-Driven Boundary Recession
• How target distributions shape optimal stochastic resetting