First Passage under Restart

Abstract

First passage under restart has recently emerged as a conceptual framework suitable for the description of a wide range of phenomena, but the endless variety of ways in which restart mechanisms and first passage processes mix and match hindered the identification of unifying principles and general truths. Hope that these exist came from a recently discovered universality displayed by processes under optimal, constant rate, restart-but extensions and generalizations proved challenging as they marry arbitrarily complex processes and restart mechanisms. To address this challenge, we develop a generic approach to first passage under restart. Key features of diffusion under restart-the ultimate poster boy for this wide and diverse class of problems-are then shown to be completely universal.

Continuous-time random walks with reset events

“…More recently, in 2011, Evans and Majumdar [16,17] studied a diffusing model with a resetting term in a Fokker-Planck equation, derived from microscopical considerations. Afterwards, several analysis and generalizations of this formulation have been performed, including: The incorporation of an absorbing state [18]; the generalizations to d-spatial dimensions [19]; the presence of a general potential [20]; the inclusion of time dependency in the resetting rate [21] or a general distribution for the reset time [22]; a study of large deviations in Markovian processes [23]; a comparison with deterministic resetting [24]; the relocation to a previously position [25]; analyses on general properties of the first-passage time [26,27]; or the possibility that internal properties drive the reset mechanism of the system [28].…”

Reaction–diffusion with stochastic decay rates

“…For example, we will suggest below a stochastic Michaelis-Menten scheme defined by coupling ∆ and τ . Secondly, as we discuss below, results from the study of a broad class of stochastic processes can be applied directly to the general formulation of our model [29].…”

“…It is worth noting that the random decay time S takes the form of the generic first passage time (FPT) under reset [29,32]. Here the "passage" is completion of a reaction (decay) at time ∆, and the reset time τ begins the reaction anew.…”

Integral fluctuation theorems for stochastic resetting systems

“…Resetting can be quite helpful in search problems in which a particle can diffuse far away from the target in the wrong direction. Resetting prevents such realizations from occurring, thereby removing realizations which take an exceedingly large times to reach their target [1,2,15,18,19].…”

“…Indeed, dynamical systems with resetting have been employed as models for diverse situations such as searching for lost possessions, foraging for food in the wild, stochastic phenotype switching, optimal search algorithms, and random catastrophic events [10][11][12][13][14]. Part of the interest is due to the neat mathematical structure of resetting, but most of the interest is due to the usefulness of resetting in search problems [1][2][3][15][16][17][18][19]. It is now well understood that the inclusion of resetting can drastically affect the distribution of search times.…”