Generalized Continuous-Time Random Walks, Subordination by Hitting Times, and Fractional Dynamics

Kolokoltsov, Vassili N.

doi:10.1137/s0040585x97983857

Cited by 69 publications

(78 citation statements)

References 21 publications

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“…There is an extensive literature in this field: general results for the scaling limits of CTRWs on the stochastic process level can be found in [6,7,29,35,37,44,45]. 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].…”

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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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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“…Note that some properties the Fokker-Planck equation with coordinate derivatives of noninteger order 1 < α < 2 d are considered in Refs. [2,8,[14][15][16][17][18].…”

Section: Resultsmentioning

confidence: 99%

“…The Fokker-Planck equation with fractional coordinate derivatives was also considered in Refs. [14][15][16][17][18].…”

Section: Introductionmentioning

confidence: 99%

Fokker–Planck equation with fractional coordinate derivatives

Zaslavsky¹

2008

Physica A: Statistical Mechanics and its Applications

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a b s t r a c tUsing the generalized Kolmogorov-Feller equation with long-range interaction, we obtain kinetic equations with fractional derivatives with respect to coordinates. The method of successive approximations, with averaging with respect to a fast variable, is used. The main assumption is that the correlation function of probability densities of particles to make a step has a power-law dependence. As a result, we obtain a Fokker-Planck equation with fractional coordinate derivative of order 1 < α < 2.

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“…For this reason, we use generalized functions, and then we only require that D α−j f (x) is of polynomial growth for every non-negative integer j. The interpretation of fractional derivatives and Fourier transforms as generalized functions is common in applications, see for example [5,7,18]. Example 2.2.…”

Section: The One Variable Casementioning

confidence: 99%