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Related Experiment Videos

Analytical results for random walk persistence

Sire1, Majumdar, Rudinger

  • 1Laboratoire de Physique Quantique (UMR C5626 du CNRS), Universite Paul Sabatier, 31062, Toulouse Cedex, France.

Physical Review. E, Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics
|October 25, 2000
PubMed
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This study calculates the persistence exponent (θ) for Gaussian and non-Gaussian processes, revealing connections to quantum mechanics and providing new expressions for persistence probability.

Area of Science:

  • Statistical Physics
  • Probability Theory
  • Quantum Mechanics

Background:

  • The persistence exponent (θ) quantifies the probability of a random process remaining on one side of its initial value.
  • Previous work introduced this problem for nearly Markovian Gaussian processes.

Purpose of the Study:

  • To present a detailed calculation of the persistence exponent (θ) for nearly Markovian Gaussian processes.
  • To derive resummed perturbative and nonperturbative expressions for θ.
  • To extend the calculation of θ to non-Gaussian processes and explore its connection to quantum mechanics.

Main Methods:

  • Resummed perturbative and nonperturbative calculations.
  • Establishing a connection between persistence problems and quantum mechanical energy eigenfunctions.

Related Experiment Videos

  • Analysis of Gaussian and non-Gaussian processes.

    • Main Results:

    • Derived new perturbative and nonperturbative expressions for the persistence exponent (θ).
    • Suggested a link between the persistence exponent and the independent interval approximation.
    • Extended persistence exponent calculations to non-Gaussian processes via quantum mechanics analogy.
    • Provided expressions for θ(X0), the probability of remaining above a scaled initial value.

    Conclusions:

    • The study provides a comprehensive analysis of the persistence exponent for various processes.
    • The findings highlight a deep connection between statistical persistence phenomena and quantum mechanics.
    • New theoretical tools are offered for studying persistence in complex systems.