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Related Experiment Videos

Persistence of a continuous stochastic process with discrete-time sampling.

S N Majumdar1, A J Bray, G C Ehrhardt

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

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|July 20, 2001
PubMed
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We introduce discrete-time persistence for stochastic processes. For Gaussian Markov processes, we found the probability of no zero-crossings decays exponentially with time, and calculated it precisely.

Area of Science:

  • Stochastic processes
  • Statistical physics
  • Time series analysis

Background:

  • Understanding the behavior of stochastic processes is crucial in various scientific fields.
  • Zero-crossing analysis is a key metric for characterizing the dynamics of continuous processes.
  • Previous studies often focused on continuous-time analysis, leaving gaps in discrete-time scenarios.

Purpose of the Study:

  • To introduce and define "discrete-time persistence" for stochastic processes.
  • To analyze the probability of no zero-crossings in discrete time for Gaussian Markov processes.
  • To investigate the phenomenon of "alternating persistence" and its implications.

Main Methods:

  • Mathematical formulation of discrete-time persistence.
  • Analysis of a Gaussian Markov process with relaxation rate mu.

Related Experiment Videos

  • Derivation of the decay rate for persistence probability using parameter a = exp(-mu Delta T).
  • Calculation of the persistence probability function rho(a) to high precision.

    • Main Results:

    • The persistence probability for large n decays as [rho(a)](n), where a = exp(-mu Delta T).
    • The function rho(a) was computed with high precision.
    • A non-zero probability of infinite-time no zero-crossings was identified for a > 1 (unstable potentials).

    Conclusions:

    • Discrete-time persistence provides a novel framework for analyzing stochastic process dynamics.
    • The derived decay rate offers precise predictions for the likelihood of extended periods without zero-crossings.
    • The study elucidates behavior in unstable potentials, expanding the applicability of persistence analysis.