On the fluid limit of the ContinuousTime Random Walk with General Lévy Jump Distribution Functions
Cartea, Alvaro and delCastilloNegrete, D. (2007) On the fluid limit of the ContinuousTime Random Walk with General Lévy Jump Distribution Functions. Working Paper. Birkbeck, University of London, London, UK.

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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 wellknown 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, ψ ∼ t−(1+β), and algebraic decaying jump distributions, η ∼ x−(1+α), corresponding to L´evy 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 L´evy stochastic processes in the L´evyKhintchine representation for the jump distribution function and obtain an integrodifferential 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 L´evy 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.
Metadata
Item Type:  Monograph (Working Paper) 

Additional Information:  BWPEF 0708 
School:  School of Business, Economics & Informatics > Economics, Mathematics and Statistics 
Depositing User:  Administrator 
Date Deposited:  26 Mar 2019 15:44 
Last Modified:  12 Jan 2023 04:00 
URI:  https://eprints.bbk.ac.uk/id/eprint/26910 
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