Publication: On the fluid limit of the continuous-time random walk with general Lévy jump distribution functions
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2007-04
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Birkbeck, University of London, School of Economics, Mathematics and Statistics
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Abstract
The continuous time random walk (CTRW) is a natural generalization of the Brownian random
walk that allows the incorporation of waiting time distributions ψ(t) and general jump distribution functions η(x). There are two well-known fluid limits of this model in the uncoupled case. For exponential
decaying waiting times and Gaussian jump distribution functions the fluid limit leads to
the diffusion equation. On the other hand, for algebraic decaying waiting times, and
algebraic decaying jump distributions, corresponding to Lévy stable processes, the
fluid limit leads to the fractional diffusion equation of order α in space and order β in time. However,
these are two special cases of a wider class of models. Here we consider the CTRW for the most
general Lévy stochastic processes in the Lévy-Khintchine representation for the jump distribution
function and obtain an integro-differential equation describing the dynamics in the fluid limit. The
resulting equation contains as special cases the regular and the fractional diffusion equations. As an
application we consider the case of CTRWs with exponentially truncated Lévy jump distribution
functions. In this case the fluid limit leads to a transport equation with exponentially truncated fractional
derivatives which describes the interplay between memory, long jumps, and truncation effects
in the intermediate asymptotic regime. The dynamics exhibits a transition from superdiffusion to
subdiffusion with the crossover time scaling as τc ∼ λ −α/β where 1/λ is the truncation length scale.
The asymptotic behavior of the propagator (Green’s function) of the truncated fractional equation
exhibits a transition from algebraic decay for t << τc to stretched Gaussian decay for t >> τc