Limit theorems for coupled continuous time random walks

Becker-Kern, Peter; Meerschaert, Mark M.; Scheffler, Hans–Peter

doi:10.1214/aop/1079021462

Cited by 105 publications

(82 citation statements)

References 24 publications

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“…Recently, the theory of semi-Markov processes in continuous time has regained much interest. The literature concerning this topic is vast: see for example [3,28,29,37,38,41,42,64]. Consult also [18,49,56], where the theory has been extended to models of motions in heterogeneous media.…”

Section: Angelica Pachon Federico Polito and Costantino Ricciutimentioning

confidence: 99%

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On discrete-time semi-Markov processes

Pachón¹,

Polito²,

Ricciuti³

2021

Discrete &Amp; Continuous Dynamical Systems - B

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In the last years, several authors studied a class of continuous-time semi-Markov processes obtained by time-changing Markov processes by hitting times of independent subordinators. Such processes are governed by integrodifferential convolution equations of generalized fractional type. The aim of this paper is to develop a discrete-time counterpart of such a theory and to show relationships and differences with respect to the continuous time case. We present a class of discrete-time semi-Markov chains which can be constructed as time-changed Markov chains and we obtain the related governing convolution type equations. Such processes converge weakly to those in continuous time under suitable scaling limits.

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Section: Angelica Pachon Federico Polito and Costantino Ricciutimentioning

confidence: 99%

“…There is an extensive literature concerning the topics recalled in this section: see for example [3,26,28,29,40,37,41,38,42]. See also [20] for some applications and [25] for recent analytic results on fractional differential equations.…”

Section: Brief Review On Continuous-time Semi-markov Chainsmentioning

confidence: 99%

On discrete-time semi-Markov processes

Pachón¹,

Polito²,

Ricciuti³

2021

Discrete &Amp; Continuous Dynamical Systems - B

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“…1 a, one can assume that the CTRW particle arrives at at time , then it comes to at time due to the expansion of the medium and further followed by an instantaneous jump with length relative to because of the intrinsic motion. Since the particle does not move in the comoving coordinate x as long as the walker is in a state of rest, e.g., (‘+’ and ‘-’ respectively mean right and left limit), we have the expression of by combining with ( 2 ), The correlation and similarity between CTRW model and the traditional Lévy walk model are stated in [ 11 , 36 ]. Because of that, for Lévy walk under the action of constant force in non-static medium, as shown in Fig.…”

Section: Lévy Walk Under the Action Of An External Constant Force In ...mentioning

confidence: 99%

Lévy Walk Dynamics in an External Constant Force Field in Non-Static Media

Zhou

Deng

2022

J Stat Phys

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Based on the recognition of the huge change of the transport properties for diffusion particles in non-static media, we consider a Lévy walk model subjected to an external constant force in non-static media. Since the physical and comoving coordinates of non-static media are related by scale factor, we equivalently transfer the process from physical coordinate into comoving coordinate and derive the master equation governing the probability density function of the position of the particles in comoving coordinate. Utilizing the Hermite orthogonal polynomial expansions, some statistical properties are obtained, including the asymptotic behaviors of the first two moments in both coordinates and kurtosis. For some representative types of non-static media and Lévy walks, the striking and interesting phenomena originating from the interplay between non-static media, external force, and intrinsic stochastic motion are observed. The stationary distribution are also analyzed for some cases through numerical simulations.

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“…Our interest in the operator H s stems not only from the interesting analysis of these operators and their favourable connections to (local) linear degenerate parabolic operators, but also from the relevance of these operators to the study of anomalous diffusions. Indeed, from a probabilistic point of view, fractional powers of parabolic operators like the heat operator is closely connected to Continuous Time Random Walks (CTRWs), see [21] for a more physically oriented introduction and [1,2,5,18,19,20] for mathematical treatments. CTRWs are a kind of stochastic process in which random jumps in space are coupled with random waiting times.…”

Section: Introductionmentioning

confidence: 99%

Asymptotic mean value formulas, nonlocal space-time parabolic operators and anomalous tug-of-war games

Fjellström¹,

Nyström²,

Wang³

2022

Preprint

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The fractional heat operator (∂t − ∆x) s and Continuous Time Random Walks (CTRWs) are interesting and sophisticated mathematical models that can describe complex anomalous systems. In this paper, we prove asymptotic mean value representation formulas for functions with respect to (∂t −∆x) s and we introduce new nonlocal, nonlinear parabolic operators related to a tug-of-war which accounts for waiting times and space-time couplings. These nonlocal, nonlinear parabolic operators and equations can be seen as nonlocal versions of the evolutionary infinity Laplace operator.

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Limit theorems for coupled continuous time random walks

Cited by 105 publications

References 24 publications

On discrete-time semi-Markov processes

On discrete-time semi-Markov processes

Lévy Walk Dynamics in an External Constant Force Field in Non-Static Media

Asymptotic mean value formulas, nonlocal space-time parabolic operators and anomalous tug-of-war games

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