# 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.
Document type :
Journal articles
Domain :

https://hal.archives-ouvertes.fr/hal-03117919
Contributor : Claudine Le Vaou <>
Submitted on : Thursday, January 21, 2021 - 4:40:10 PM
Last modification on : Friday, January 22, 2021 - 3:30:16 AM

### Identifiers

• HAL Id : hal-03117919, version 1
• ARXIV : 2009.03419

### Citation

B. de Bruyne, J. Randon-Furling, S. Redner. Optimization and Growth in First-Passage Resetting. Journal of Statistical Mechanics: Theory and Experiment, IOP Publishing, 2021. ⟨hal-03117919⟩

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