BIROn - Birkbeck Institutional Research Online

    On the fluid limit of the Continuous-Time Random Walk with General Lévy Jump Distribution Functions

    Cartea, Alvaro and del-Castillo-Negrete, D. (2007) On the fluid limit of the Continuous-Time Random Walk with General Lévy Jump Distribution Functions. Working Paper. Birkbeck, University of London, London, UK.

    [img]
    Preview
    Text
    26910.pdf - Draft Version

    Download (346kB) | Preview

    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, 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´evy-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 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: Birkbeck Schools and Departments > School of Business, Economics & Informatics > Economics, Mathematics and Statistics
    Depositing User: Administrator
    Date Deposited: 26 Mar 2019 15:44
    Last Modified: 29 Jul 2019 00:27
    URI: http://eprints.bbk.ac.uk/id/eprint/26910

    Statistics

    Downloads
    Activity Overview
    80Downloads
    50Hits

    Additional statistics are available via IRStats2.

    Archive Staff Only (login required)

    Edit/View Item Edit/View Item