Fractional Diffusion Equation for a Power-Law-Truncated Levy Process

Details Fractional Diffusion Equation for a Power-Law-Truncated Levy Process Fractional Diffusion Equation for a Power-Law-Truncated Levy Process

2003-09-19

arxiv.org/abs/cond-mat/0309464v1

Physics

Condensed Matter

Scientific paper

10.1016/j.physa.2003.12.044

Truncated Levy flights are stochastic processes which display a crossover from a heavy-tailed Levy behavior to a faster decaying probability distribution function (pdf). Putting less weight on long flights overcomes the divergence of the Levy distribution second moment. We introduce a fractional generalization of the diffusion equation, whose solution defines a process in which a Levy flight of exponent alpha is truncated by a power-law of exponent 5 - alpha. A closed form for the characteristic function of the process is derived. The pdf of the displacement slowly converges to a Gaussian in its central part showing however a power law far tail. Possible applications are discussed.

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