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Parametric subordination in fractional diffusion processes

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      Abstract

      We consider simulation of spatially one-dimensional space-time fractional diffusion. Whereas in an earlier paper of ours we have developed the basic theory of what we call parametric subordination via three-fold splitting applied to continuous time random walk with subsequent passage to the diffusion limit, here we go the opposite way. Via Fourier-Laplace manipulations of the relevant fractional partial differential equation of evolution we obtain the subordination integral formula that teaches us how a particle path can be constructed by first generating the operational time from the physical time and then generating in operational time the spatial path. By inverting the generation of operational time from physical time we arrive at the method of parametric subordination. Due to the infinite divisibility of the stable subordinator, we can simulate particle paths by discretization where the generated points of a path are precise snapshots of a true path. By refining the discretization more and more fine details of a path become visible.

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      Fractional Calculus

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        Langevin equations for continuous time L\'{e}vy flights

        We consider the combined effects of a power law L\'{e}vy step distribution characterized by the step index \(f\) and a power law waiting time distribution characterized by the time index \(g\) on the long time behavior of a random walker. The main point of our analysis is a formulation in terms of coupled Langevin equations which allows in a natural way for the inclusion of external force fields. In the anomalous case for \(f<2\) and \(g<1\) the dynamic exponent \(z\) locks onto the ratio \(f/g\). Drawing on recent results on L\'{e}vy flights in the presence of a random force field we also find that this result is {\em independent} of the presence of weak quenched disorder. For \(d\) below the critical dimension \(d_c=2f-2\) the disorder is {\em relevant}, corresponding to a non trivial fixed point for the force correlation function.
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          Über den Fundamentalsatz in der Teorie der FunktionenE a (x)

           A Wiman (1905)
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            Author and article information

            Journal
            31 October 2012
            1210.8414

            http://arxiv.org/licenses/nonexclusive-distrib/1.0/

            Custom metadata
            26A33, 33E12, 33C60, 44A10, 45K05, 60G18, 60G50, 60G52, 60K05, 76R50
            Fractional Dynamics, Recent Advances (World Scientific, Singapore 2012) pp 227-261
            40 pages 14 Figures. Text overlap with our E-print arXiv:1104.4041
            math.PR cond-mat.stat-mech math-ph math.MP

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