Optimal potentials for diffusive search strategies

Kuśmierz, Łukasz; Bier, Martin; Gudowska–Nowak, Ewa

doi:10.1088/1751-8121/aa6769

Cited by 14 publications

(17 citation statements)

References 72 publications

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“…In [81] target distributions were considered and the optimal potential for a Langevin process (without resetting) for a given target distribution was computed exactly. The mean time to absorption was then compared to that of resetting with a constant rate r * optimised for the target distribution.…”

Section: Optimal Resettingmentioning

confidence: 99%

Stochastic resetting and applications

Evans

Majumdar

Schehr

2020

J. Phys. A: Math. Theor.

531

600

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In this Topical Review we consider stochastic processes under resetting, which have attracted a lot of attention in recent years. We begin with the simple example of a diffusive particle whose position is reset randomly in time with a constant rate r, which corresponds to Poissonian resetting, to some fixed point (e.g. its initial position). This simple system already exhibits the main features of interest induced by resetting: (i) the system reaches a nontrivial nonequilibrium stationary state (ii) the mean time for the particle to reach a target is finite and has a minimum, optimal, value as a function of the resetting rate r. We then generalise to an arbitrary stochastic process (e.g. Lévy flights or fractional Brownian motion) and non-Poissonian resetting (e.g. power-law waiting time distribution for intervals between resetting events). We go on to discuss multiparticle systems as well as extended systems, such as fluctuating interfaces, under resetting. We also consider resetting with memory which implies resetting the process to some randomly selected previous time. Finally we give an overview of recent developments and applications in the field.

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Section: Optimal Resettingmentioning

confidence: 99%

Stochastic resetting and applications

Evans

Majumdar

Schehr

2020

J. Phys. A: Math. Theor.

531

600

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“…Aside from the optimal rate of resetting, there exists a certain exponent present in the power-law distribution of waiting times between resets which in the case of an ordinarily diffusing particle minimizes its mean first-passage time to a given target position [50]. The first-passage problem under stochastic resetting has also been studied in a presence of various external potentials [72,21,73], in bounded domains [74,75] and under restart with branching [76].…”

Section: Introductionmentioning

confidence: 99%

Non-linear diffusion with stochastic resetting

Chelminiak

2021

Preprint

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Resetting or restart, when applied to a stochastic process, usually brings its dynamics to a timeindependent stationary state. In turn, the optimal resetting rate makes the mean time to reach a target to be the shortest one. These and other problems have been intensively studied in the case of ordinary diffusive processes during the last decade. In this paper we consider the influence of stochastic resetting on a diffusive motion modeled in terms of the non-linear differential equation. The reason for its non-linearity is the power-law dependence of the diffusion coefficient on the probability density function or, in another context, the concentration of particles. We first derive an exact formula for the mean squared displacement and show how it attains the steady-state value under the exponential resetting. Then, we analyse the steady-state properties of the probability density function. We also explore the first-passage properties for the non-linear diffusion affected by the exponential resetting and find the exact expressions for the survival probability, the mean firstpassage time and the optimal resetting rate, which minimizes the mean time needed for a particle to reach a pre-determined target. Finally, we find the universal property that the relative fluctuation in the mean first-passage time of optimally restarted non-linear diffusion is always equal to unity.

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“…Thus, the searcher is a Markov diffusion process generated by the operator L = D Our aim is to minimize this expected search time over all admissible drifts b ∈ D µ . We note that this same problem was recently considered in the physics literature [5]; for more on this, see Remark 1 after Theorem 2.…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 75%

Optimizing the drift in a diffusive search for a random stationary target

Pinsky¹

2019

Electron. J. Probab.

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Let a ∈ R denote an unknown stationary target with a known distribution µ ∈ P(R), the space of probability measures on R.A diffusive searcher X(·) sets out from the origin to locate the target.The time to locate the target is Ta = inf{t ≥ 0 : X(t) = a}. The searcher has a given constant diffusion rate D > 0, but its drift b can be set by the search designer from a natural admissible class Dµ of drifts. Thus, the diffusive searcher is a Markov process generated by theOur aim is to minimize this expected search time over all admissible drifts b ∈ Dµ. For measures µ that satisfy a certain balance condition between their restriction to the positive axis and their restriction to the negative axis, a condition satisfied, in particular, by all symmetric measures, we can give a complete answer to the problem. We calculate the above infimum explicitly, we classify the measures for which the infimum is attained, and in the case that it is attained, we calculate the minimizing drift explicitly. For measures that do not satisfy the balance condition, we obtain partial results.2010 Mathematics Subject Classification. 60J60.

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Optimal potentials for diffusive search strategies

Cited by 14 publications

References 72 publications

Stochastic resetting and applications

Stochastic resetting and applications

Non-linear diffusion with stochastic resetting

Optimizing the drift in a diffusive search for a random stationary target

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