Long time asymptotics of a Brownian particle coupled with a random environment with non-diffusive feedback force

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Abstract

We study the long time behavior of a Brownian particle moving in an anomalously diffusing field, the evolution of which depends on the particle position. We prove that the process describing the asymptotic behavior of the Brownian particle has bounded (in time) variance when the particle interacts with a subdiffusive field; when the interaction is with a superdiffusive field the variance of the limiting process grows in time as t(2 gamma-1), 1/2 <gamma <1. Two different kinds of superdiffusing (random) environments are considered: one is described through the use of the fractional Laplacian; the other via the Riemann-Liouville fractional integral. The subdiffusive field is modeled through the Riemann-Liouville fractional derivative. (C) 2011 Elsevier B.V. All rights reserved.

Original languageEnglish
Pages (from-to)844-884
Number of pages41
JournalStochastic Processes and their Applications
Volume122
Issue number3
DOIs
Publication statusPublished - Mar 2012

Keywords

  • Anomalous diffusion
  • Riemann-Liouville fractional derivative (integral)
  • Fractional Laplacian
  • Continuous time random walk
  • Levy flight
  • Scaling limit
  • Interface fluctuations
  • ANOMALOUS DIFFUSION
  • FRACTIONAL DYNAMICS
  • RANDOM-WALKS

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