Directed random walk with random restarts: The Sisyphus random walk

Montero, Miquel; Villarroel, Javier

doi:10.1103/physreve.94.032132

Cited by 78 publications

(78 citation statements)

References 20 publications

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“…As a model of an animal returning to its nest between multiple excursions, the inclusion of resets to a diffusive motion [5] is a first step in the formulation of a model cappable of capturing the global dynamics from the multiple internal states of the animals. After that, multiple papers have been devoted to study many types of processes with different resetting mechanisms , some of them focusing on the completion time of the processes [27][28][29][30], or using the reset to concatenate different processes [31,32].…”

Section: Introductionmentioning

confidence: 99%

Stochastic movement subject to a reset-and-residence mechanism: transport properties and first arrival statistics

Masó-Puigdellosas¹,

Campos²,

Méndez³

2019

J. Stat. Mech.

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In this work we consider a stochastic movement process with random resets to the origin followed by a random residence time there before the walker restarts its motion. First, we study the transport properties of the walker, we derive an expression for the mean square displacement of the overall process and study its dependence with the statistical properties of the resets, the residence and the movement. From this general formula, we see that the inclusion of the residence after the resets is able to induce super-diffusive to sub-diffusive (or diffusive) regimes and it can also make a sub-diffusive walker reach a constant MSD or even collapse. Second, we study how the reset-andresidence mechanism affects the first arrival time of different search processes to a given position, showing that the long time behavior of the reset and residence time distributions determine the existence of the mean first arrival time.

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Section: Introductionmentioning

confidence: 99%

Stochastic movement subject to a reset-and-residence mechanism: transport properties and first arrival statistics

Masó-Puigdellosas¹,

Campos²,

Méndez³

2019

J. Stat. Mech.

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“…There, a diffusive particle is studied when it may occasionally reset its position with a constant probability and the authors find that a non-equilibrium steady state (NESS) is reached and the mean first passage time of the overall process is finite and attains a minimum in terms of the resetting rate. The existence of a NESS has been further studied for different types of motion and resetting mechanisms [5][6][7][8][9][10][11][12][13][14][15][16][17][18], showing that they are not exclusive of diffusion with Markovian resets. Aside from these, other works have shown that the resetting does not always generate a NESS but transport is also possible when the resetting probability density function (PDF) is long-tailed [19][20][21][22] or when the resetting process is subordinated to the motion [10,12].…”

Section: Introductionmentioning

confidence: 99%

Transport properties of random walks under stochastic noninstantaneous resetting

2019

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Random walks with stochastic resetting provides a treatable framework to study interesting features about central-place motion. In this work, we introduce non-instantaneous resetting as a twostate model being a combination of an exploring state where the walker moves randomly according to a propagator and a returning state where the walker performs a ballistic motion with constant velocity towards the origin. We study the emerging transport properties for two types of reset time probability density functions (PDFs): exponential and Pareto. In the first case, we find the stationary distribution and a general expression for the stationary mean square displacement (MSD) in terms of the propagator. We find that the stationary MSD may increase, decrease or remain constant with the returning velocity. This depends on the moments of the propagator. Regarding the Pareto resetting PDF we also study the stationary distribution and the asymptotic scaling of the MSD for diffusive motion. In this case, we see that the resetting modifies the transport regime, making the overall transport sub-diffusive and even reaching a stationary MSD., i.e., a stochastic localization. This phenomena is also observed in diffusion under instantaneous Pareto resetting. We check the main results with stochastic simulations of the process.PACS numbers:

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“…This type of process may lead to an optimal resetting rate for the first-passage time as shown for the discrete model in [32], since now a reset does not always represent a penalty for the process: it can help the walker to regain its direction when it has reached a region which is far from its target.…”

Section: Alternate Process With Resetsmentioning

confidence: 99%

“…Furthermore, the possibility of having walkers relocated to a known position [30] or, more specifically, to a previous maximum [31], have been considered. Here we can also find the Sisyphus random walk [32], where a walker with oriented and deterministic step lengths is subject to random resets to the initial point.…”

Section: Introductionmentioning

confidence: 99%

Continuous-time random walks with reset events

2017

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In this paper, we consider a stochastic process that may experience random reset events which relocate the system to its starting position. We focus our attention on a one-dimensional, monotonic continuous-time random walk with a constant drift: the process moves in a fixed direction between the reset events, either by the effect of the random jumps, or by the action of a deterministic bias. However, the orientation of its motion is randomly determined after each restart. As a result of these alternating dynamics, interesting properties do emerge. General formulas for the propagator as well as for two extreme statistics, the survival probability and the mean first-passage time, are also derived. The rigor of these analytical results is verified by numerical estimations, for particular but illuminating examples.

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Directed random walk with random restarts: The Sisyphus random walk

Cited by 78 publications

References 20 publications

Stochastic movement subject to a reset-and-residence mechanism: transport properties and first arrival statistics

Stochastic movement subject to a reset-and-residence mechanism: transport properties and first arrival statistics

Transport properties of random walks under stochastic noninstantaneous resetting

Continuous-time random walks with reset events

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