Random Search with Resetting: A Unified Renewal Approach.

We provide a unified renewal approach to the problem of random search for several targets under resetting. This framework does not rely on specific properties of the search process and resetting procedure, allows for simpler derivation of known results, and leads to new ones. Concentrating on minimizing the mean hitting time, we show that resetting at a constant pace is the best possible option if resetting helps at all, and derive the equation for the optimal resetting pace. No resetting may be a better strategy if without resetting the probability of not finding a target decays with time to zero exponentially or faster. We also calculate splitting probabilities between the targets, and define the limits in which these can be manipulated by changing the resetting procedure. We moreover show that the number of moments of the hitting time distribution under resetting is not less than the sum of the numbers of moments of the resetting time distribution and the hitting time distribution without resetting.

[1]  S. Belan.  Restart Could Optimize the Probability of Success in a Bernoulli Trial. , 2017, Physical review letters.

[2]  Feller William,et al.  An Introduction To Probability Theory And Its Applications , 1950 .

[3]  J. G. Wendel.  Hitting Spheres with Brownian Motion , 1980 .

[4]  Daniel Campos,et al.  Stochastic Foundations in Movement Ecology , 2014 .

[5]  L. Stone.  Theory of Optimal Search , 1975 .

[6]  Satya N. Majumdar,et al.  Optimal diffusive search: nonequilibrium resetting versus equilibrium dynamics , 2012, 1212.4096.

[7]  Stephan Eule,et al.  Non-equilibrium steady states of stochastic processes with intermittent resetting , 2015, 1510.07876.

[8]  Shlomi Reuveni,et al.  Optimal Stochastic Restart Renders Fluctuations in First Passage Times Universal. , 2015, Physical review letters.

[9]  Javier Villarroel,et al.  Directed random walk with random restarts: The Sisyphus random walk. , 2016, Physical review. E.

[10]  Anupam Kundu,et al.  Diffusion under time-dependent resetting , 2015, 1512.08211.

[11]  Satya N. Majumdar,et al.  Diffusion with resetting in arbitrary spatial dimension , 2014, 1404.4574.

[12]  M. Srinivasan.  Search for Research , 2014, Journal of conservative dentistry : JCD.

[13]  Tsuyoshi Murata,et al.  {m , 1934, ACML.

[14]  Sidney Redner,et al.  A guide to first-passage processes , 2001 .

[15]  V P Shkilev.  Continuous-time random walk under time-dependent resetting. , 2017, Physical review. E.

[16]  Shamik Gupta,et al.  Diffusion with stochastic resetting at power-law times. , 2015, Physical review. E.

[17]  Satya N. Majumdar,et al.  Diffusion with optimal resetting , 2011, 1107.4225.

[18]  Satya N. Majumdar,et al.  Persistence and first-passage properties in nonequilibrium systems , 2013, 1304.1195.

[19]  Javier Villarroel,et al.  Continuous-time random walks with reset events , 2017, 1706.04812.

[20]  S. Redner.  A guide to first-passage processes , 2001 .

[21]  Sung Chul Bae,et al.  When Brownian diffusion is not Gaussian. , 2012, Nature Materials.

[22]  Lukasz Kusmierz,et al.  First Order Transition for the Optimal Search Time of Lévy Flights with Resetting. , 2014, Physical review letters.

[23]  Vicenç Méndez,et al.  Phase transitions in optimal search times: How random walkers should combine resetting and flight scales. , 2015, Physical review. E, Statistical, nonlinear, and soft matter physics.

[24]  A. Polyanin.  Handbook of Linear Partial Differential Equations for Engineers and Scientists , 2001 .

[25]  Satya N Majumdar,et al.  Diffusion with stochastic resetting. , 2011, Physical review letters.

[26]  Andreas Schadschneider,et al.  Diffusion with resetting in bounded domains , 2015 .

[27]  Ewa Gudowska-Nowak,et al.  Optimal first-arrival times in Lévy flights with resetting. , 2015, Physical review. E, Statistical, nonlinear, and soft matter physics.

[28]  Arnab Pal,et al.  First Passage under Restart. , 2016, Physical review letters.

[29]  Uttam Bhat,et al.  Stochastic search with Poisson and deterministic resetting , 2016, 1605.08812.

[30]  Andrea Falcón-Cortés,et al.  Localization Transition Induced by Learning in Random Searches. , 2017, Physical review letters.

[31]  M. Shlesinger.  Mathematical physics: Search research , 2006, Nature.

[32]  E. Revilla,et al.  A movement ecology paradigm for unifying organismal movement research , 2008, Proceedings of the National Academy of Sciences.

[33]  Gene-Wei Li,et al.  Central dogma at the single-molecule level in living cells , 2011, Nature.

[34]  of Partial Absorption on Diusion with Resetting , 2013 .