Asymptotic behaviour of random walks with correlated temporal structure

Magdziarz, Marcin; Szczotka, Władysław; Żebrowski, Piotr

doi:10.1098/rspa.2013.0419

Cited by 19 publications

(15 citation statements)

References 33 publications

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“…In this paper we generally assume that waiting times and jump lengths are uncorrelated. For a discussion of the correlated case, we refer to [64][65][66][67][68][69][70]. A natural parametrisation of such a random walk is obtained in terms of the number n of jumps performed.…”

Section: Anomalous Processes With General Waiting Timesmentioning

confidence: 99%

Feynman–Kac equation for anomalous processes with space- and time-dependent forces

Cairoli¹,

Baule²

2017

J. Phys. A: Math. Theor.

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Functionals of a stochastic process Y (t) model many physical time-extensive observables, for instance particle positions, local and occupation times or accumulated mechanical work. When Y (t) is a normal diffusive process, their statistics are obtained as the solution of the celebrated Feynman-Kac equation. This equation provides the crucial link between the expected values of diffusion processes and the solutions of deterministic second-order partial differential equations. When Y (t) is non-Brownian, e.g., an anomalous diffusive process, generalizations of the Feynman-Kac equation that incorporate power-law or more general waiting time distributions of the underlying random walk have recently been derived. A general representation of such waiting times is provided in terms of a Lévy process whose Laplace exponent is directly related to the memory kernel appearing in the generalized Feynman-Kac equation. The corresponding anomalous processes have been shown to capture nonlinear mean square displacements exhibiting crossovers between different scaling regimes, which have been observed in numerous experiments on biological systems like migrating cells or diffusing macromolecules in intracellular environments. However, the case where both spaceand time-dependent forces drive the dynamics of the generalized anomalous process has not been solved yet. Here, we present the missing derivation of the Feynman-Kac equation in such general case by using the subordination technique. Furthermore, we discuss its extension to functionals explicitly depending on time, which are of particular relevance for the stochastic thermodynamics of anomalous diffusive systems. Exact results on the work fluctuations of a simple non-equilibrium model are obtained. An additional aim of this paper is to provide a pedagogical introduction to Lévy processes, semimartingales and their associated stochastic calculus, which underlie the mathematical formulation of anomalous diffusion as a subordinated process.

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Section: Anomalous Processes With General Waiting Timesmentioning

confidence: 99%

Feynman–Kac equation for anomalous processes with space- and time-dependent forces

Cairoli¹,

Baule²

2017

J. Phys. A: Math. Theor.

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“…Governing equations for the densities of the CTRW limits and the related fractional Cauchy problems were analyzed in [2,4,20,23,33,37,45]. Some recent results for particular classes of correlated and coupled CTRWs can be found in [17,24,30,31,47] The trajectories of CTRW are step functions, thus they are discontinuous. However, the usual physical requirement for a mathematical model is to have continuous realizations.…”

Section: Introductionmentioning

confidence: 99%

Limit theorems and governing equations for Lévy walks

Magdziarz

Scheffler

Straka

et al. 2015

Stochastic Processes and their Applications

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Abstract. The Lévy Walk is the process with continuous sample paths which arises from consecutive linear motions of i.i.d. lengths with i.i.d. directions. Assuming speed 1 and motions in the domain of β-stable attraction, we prove functional limit theorems and derive governing pseudo-differential equations for the law of the walker's position. Both Lévy Walk and its limit process are continuous and ballistic in the case β ∈ (0, 1). In the case β ∈ (1, 2), the scaling limit of the process is β-stable and hence discontinuous. This case exhibits an interesting situation in which scaling exponent 1/β on the process level is seemingly unrelated to the scaling exponent 3 − β of the second moment. For β = 2, the scaling limit is Brownian motion.

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“…Examples are found in financial market dynamics [5], human motion patterns [6], and so on. Recently, these facts impel one to introduce the correlated CTRWs [7][8][9][10][11][12][13][14][15][16][17][18].…”

Section: Introductionmentioning

confidence: 99%

“…Magdziarz et al generalized these two models by combining the underlying correlation mechanisms for waiting times { } in the following manner [12]:…”

Section: Introductionmentioning

confidence: 99%

Generalized Diffusion Equation Associated with a Power-Law Correlated Continuous Time Random Walk

Shi

2019

Advances in Mathematical Physics

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In this work, a generalization of continuous time random walk is considered, where the waiting times among the subsequent jumps are power-law correlated with kernel function M(t)=tρ(ρ>-1). In a continuum limit, the correlated continuous time random walk converges in distribution a subordinated process. The mean square displacement of the proposed process is computed, which is of the form 〈x2(t)〉∝tH=t1/(1+ρ+1/α). The anomy exponent H varies from α to α/(1+α) when -1<ρ<0 and from α/(1+α) to 0 when ρ>0. The generalized diffusion equation of the process is also derived, which has a unified form for the above two cases.

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Asymptotic behaviour of random walks with correlated temporal structure

Cited by 19 publications

References 33 publications

Feynman–Kac equation for anomalous processes with space- and time-dependent forces

Feynman–Kac equation for anomalous processes with space- and time-dependent forces

Limit theorems and governing equations for Lévy walks

Generalized Diffusion Equation Associated with a Power-Law Correlated Continuous Time Random Walk

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