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Nonindependent continuous-time random walks.

Miquel Montero1, Jaume Masoliver

  • 1Departament de Física Fonamental, Universitat de Barcelona, Diagonal 647, Barcelona, Spain. miquel.montero@ub.edu

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|February 1, 2008
PubMed
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This study explores correlated continuous-time random walks (CTRWs), moving beyond independent variables. Introducing correlations, even sign-dependent ones, reveals novel behaviors like altered distribution shapes in random walk models.

Area of Science:

  • Statistical Physics
  • Stochastic Processes
  • Mathematical Modeling

Background:

  • Standard continuous-time random walk (CTRW) models assume independent jumps and time intervals.
  • Real-world systems often exhibit correlations between successive steps or time durations.
  • Existing CTRW frameworks are limited in addressing these non-independent processes.

Purpose of the Study:

  • To theoretically investigate non-independent continuous-time random walks (CTRWs).
  • To analyze the impact of correlations between consecutive jumps and/or time intervals.
  • To develop analytical methods for handling correlated CTRW scenarios.

Main Methods:

  • Developed a theoretical framework for non-independent CTRWs.
  • Derived an exact solution for CTRWs with sign-dependent jump correlations.

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  • Employed analytical techniques and approximations for general correlated cases.

    • Main Results:

    • An exact solution was obtained for CTRWs where correlations depend on jump signs.
    • Correlated CTRWs can exhibit transitions from unimodal to bimodal probability distributions.
    • Analytical tools were established for analyzing more complex correlated random walks.

    Conclusions:

    • Correlations in CTRWs, even simple ones, significantly alter system dynamics.
    • The developed methods enable the study of more realistic correlated stochastic processes.
    • This work provides a foundation for analyzing complex systems with memory effects.