Limit theorem for continuous-time random walks with two time scales

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

doi:10.1239/jap/1082999078

Cited by 55 publications

(42 citation statements)

References 11 publications

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“…The main equation (1) can be rewritten in the different forms involving material fractional derivatives [12,[29][30][31][32][33]. We write it in the form…”

Section: Gamma Pdf G(τ2λ)mentioning

confidence: 99%

“…In the strong ballistic case (0 < μ < 1), the integral term is in the balance with classical wave equation, while for the subballistic superdiffusive case (1 < μ < 2), the memory term is in the balance with the Cattaneo (telegraph) equation. One can perform various asymptotic analysis of (30) and (33), obtain the pseudodifferential equations for the walker's PDF position [30][31][32][33] and determine the shape of PDF profiles [34].…”

Section: Gamma Pdf G(τ2λ)mentioning

confidence: 99%

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Single integrodifferential wave equation for a Lévy walk

Fedotov

2016

Phys. Rev. E

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We derive the single integrodifferential wave equation for the probability density function of the position of a classical one-dimensional Lévy walk with continuous sample paths. This equation involves a classical wave operator together with memory integrals describing the spatiotemporal coupling of the Lévy walk. It is valid at all times, not only in the long time limit, and it does not involve any large-scale approximations. It generalizes the well-known telegraph or Cattaneo equation for the persistent random walk with the exponential switching time distribution. Several non-Markovian cases are considered when the particle's velocity alternates at the gamma and power-law distributed random times. In the strong anomalous case we obtain the asymptotic solution to the integrodifferential wave equation. We implement the nonlinear reaction term of Kolmogorov-PetrovskyPiskounov type into our equation and develop the theory of wave propagation in reaction-transport systems involving Lévy diffusion.

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“…The main equation (1) can be rewritten in the different forms involving material fractional derivatives [12,[29][30][31][32][33]. We write it in the form…”

Section: Gamma Pdf G(τ2λ)mentioning

confidence: 99%

Section: Gamma Pdf G(τ2λ)mentioning

confidence: 99%

Single integrodifferential wave equation for a Lévy walk

Fedotov

2016

Phys. Rev. E

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“…For heavy tail waiting times, convergence in the renewal theorem is slower than in the finite variance case, and thus it is advantageous to make a second-order correction. This is accomplished using the two-scale limit procedure introduced in Becker- Kern et al (2003) and Meerschaert and Scheffler (2004) for random walk models.…”

Section: Introductionmentioning

confidence: 99%

“…Time-fractional derivatives are connected with physical models of particle sticking and trapping . For events separated by waiting times with whose exceedence probability belongs to some sum-stable domain of attraction, a connection between limit theorems and fractional time derivatives was laid out in Becker-Kern et al (2003) and Meerschaert and Scheffler (2004). Specifically, we assume that the exceedence probability is regularly varying; this means, essentially, that the probability of waiting longer than time t > 0 falls off like a power law t − .…”

Section: Introductionmentioning

confidence: 99%

“…Our result stands in contrast to established results in the literature, because we consider a novel two-scale limit scenario. Limit theorems for continuous time random walks, where the waiting time between jumps is separated by random waiting times with heavy tails, were developed in Becker- Kern et al (2003) and Meerschaert and Scheffler (2004). In the present study, we adapt the method used there for the waiting time process, replacing the sum in those papers with the maximum.…”

Section: Introductionmentioning

confidence: 99%

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Extremal limit theorems for observations separated by random power law waiting times

Meerschaert

Stoev

2009

Journal of Statistical Planning and Inference

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This paper develops extreme value theory for random observations separated by random waiting times whose exceedence probability falls off like a power law. In the case where the waiting times between observations have an infinite mean, a limit theorem is established, where the limit is comprised of an extremal process whose time index is randomized according to the non-Markovian hitting time process for a stable subordinator. The resulting limit distributions are shown to be solutions of fractional differential equations, where the order of the fractional time derivative coincides with the power law index of the waiting time. The probability that the limit process remains below a threshold is also computed. For waiting times with finite mean but infinite variance, a two-scale argument yields a fundamentally different limit process. The resulting limit is an extremal process whose time index is randomized according to the first passage time of a positively skewed stable Lévy motion with positive drift. This two-scale limit provides a second-order correction to the usual limit behavior.

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Numerical analysis for Klein‐Gordon equation with time‐space fractional derivatives

Zhang

Wang

Zhou

2019

Math Methods in App Sciences

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We present and analyze two numerical schemes for solving a nonlinear Klein‐Gordon equation with time‐space fractional derivatives. Numerical methods are base on finite difference scheme in fractional derivative and Fourier‐spectral method in spatial variable. It is proved that the linearized method is conditionally stable while the nonlinearized one is unconditionally stable. In addition, the error estimate shows that the linearized method is in the order of scriptOfalse(normalΔt+Nβ−rfalse), and the nonlinearized method converge with the order scriptOfalse(normalΔt3−α+Nβ−rfalse), where normalΔt, N, β, and r are, respectively, step of time, polynomial degree, the fractional derivative in space, and regularity of u. Some numerical experiments are performed to demonstrate the theoretical results.

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Limit theorem for continuous-time random walks with two time scales

Cited by 55 publications

References 11 publications

Single integrodifferential wave equation for a Lévy walk

Single integrodifferential wave equation for a Lévy walk

Extremal limit theorems for observations separated by random power law waiting times

Numerical analysis for Klein‐Gordon equation with time‐space fractional derivatives

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