Generalized diffusion and random search processes

Zhou, Tian; Trajanovski, Pece; Xu, Pengbo; Deng, Weihua; Sandev, Trifce; Kocarev, Ljupčo

doi:10.1088/1742-5468/ac841e

Cited by 5 publications

(4 citation statements)

References 73 publications

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“…Equation ( 8) is known as the subordination of the spatial process x by the temporal process for N [3,34,56,72]. The technique was used in computer simulation in the context of fractional Fokker-Planck dynamics [73], random diffusivity [58], population heterogeneity [74] and one-dimensional Brownian search problem [57]. As mentioned before, time intervals and displacements of walkers under investigation are IID with common PDFs ϕ(τ ) and f(χ), respectively.…”

Section: Calculation Of the Positional Distributionmentioning

confidence: 99%

“…Subordination methods, based on an integral transformation, present a way of solving fractional kinetic equations [3,[51][52][53][54][55][56][57][58][59]. To be more exact, when the mean waiting time diverges [34], subordination methods map a classical Fokker-Planck equation onto a fractional diffusion one with the fractional time derivative.…”

Section: Introductionmentioning

confidence: 99%

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Fractional advection diffusion asymmetry equation, derivation, solution and application

Wang,

Barkai

2024

J. Phys. A: Math. Theor.

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The non-Markovian continuous-time random walk model, featuring fat-tailed waiting times and narrow distributed displacements with a non-zero mean, is a well studied model for anomalous diffusion. Using an analytical approach, we recently demonstrated how a fractional space advection diffusion asymmetry equation, usually associated with Markovian L{'e}vy flights, describes the spreading of a packet of particles. Since we use Gaussian statistics for jump lengths though fat-tailed distribution of waiting times, the appearance of fractional space derivatives in the kinetic equation demands explanations provided in this manuscript. As applications we analyse the spreading of tracers in two dimensions, breakthrough curves investigated in the field of contamination spreading in hydrology and first passage time statistics. We present a subordination scheme valid for the case when the mean waiting time is finite and the variance diverges, which is related to L'evy statistics for the number of renewals in the process.

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Section: Calculation Of the Positional Distributionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Fractional advection diffusion asymmetry equation, derivation, solution and application

Wang,

Barkai

2024

J. Phys. A: Math. Theor.

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“…(2023) 073202 random search strategy, the stochastic resetting as a mechanism can enhance the search efficiency [35][36][37]. This work gave rise to extensive interests and studies in optimizing the search efficiency in the presence of stochastic resetting [35][36][37][38][39][40][41][42][43][44][45][46][47][48][49]. Meanwhile, transport and diffusion of classical stochastic processes and models under stochastic resetting have been intensively studied, e.g.…”

Section: Introductionmentioning

confidence: 99%

The Lévy walk with rests under stochastic resetting

Liu¹,

Hu²,

Bao³

2023

J. Stat. Mech.

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The Lévy walk with rests (LWR) model is a typical two-state stochastic process that has been widely and successfully adopted in the study of intermittent stochastic phenomena in physical and biological systems. Stochastic processes under resetting provide treatable and interesting schemes to study foraging and search strategies. In this manuscript, we focus on the anomalous diffusive behavior of the LWR under stochastic resetting. We consider both the case of instantaneous resetting, in which the particle stochastically returns to a given position immediately, and the case of noninstantaneous resetting, in which the particle returns to a given position with a finite velocity. The anomalous diffusive behaviors are analyzed and discussed by calculating the mean squared displacement analytically and numerically. Results reveal that the stochastic resetting can not only hinder the diffusion, where the diffusion evolves toward a saturation state, but also enhances it, where as compared with the LWR without resetting, the diffusion exponent surprisingly increases. As far as we know, the enhancement effect caused by stochastic resetting has not yet been reported. In addition, the resetting time probability density function (PDF) of the instantaneous resetting and the return time PDF of the noninstantaneous resetting are studied. Results reveal that the resetting time PDF could follow a power law provided that the sojourn time PDF is power-law distributed and the sojourn time with a heavier tail plays a decisive role in determining the resetting time PDF, whereas the shape of the return time PDF is determined by not only by the sojourn time PDF, but also by the return manner.

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“…We refer the reader to [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18] for a detailed overview of the processes mentioned above. These are just a few examples of the stochastic processes commonly employed in insurance modeling.…”

Section: Introductionmentioning

confidence: 99%

Simultaneous Exact Controllability of Mean and Variance of an Insurance Policy

Rajaram

Ritchey

2023

Mathematics

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We explore the simultaneous exact controllability of mean and variance of an insurance policy by utilizing the benefit St and premium Pt as control inputs to manage the policy value tV and the variance 2σt of future losses. The goal is to determine whether there exist control inputs that can steer the mean and variance from a prescribed initial state at t=0 to a prescribed final state at t=T, where the initial–terminal pair of states (0V,TV) and (2σ0,2σT) represent the mean and variance of future losses at times t=0 and t=T, respectively. The mean tV and variance 2σt are governed by Thiele’s and Hattendorff’s differential equations in continuous time and recursive equations in discrete time. Our study focuses on solving the problem of exact controllability in both continuous and discrete time. We show that our result can be used to devise control inputs St,Pt in the interval [0,T] so that the mean and variance partially track a specified curve tV=a(t) and 2σt=b(t), respectively, i.e., at a fine sampling of points in the time interval [0,T].

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Generalized diffusion and random search processes

Cited by 5 publications

References 73 publications

Fractional advection diffusion asymmetry equation, derivation, solution and application

Fractional advection diffusion asymmetry equation, derivation, solution and application

The Lévy walk with rests under stochastic resetting

Simultaneous Exact Controllability of Mean and Variance of an Insurance Policy

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