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Related Experiment Videos

Persistence of a continuous stochastic process with discrete-time sampling: non-Markov processes.

George C M A Ehrhardt1, Alan J Bray, Satya N Majumdar

  • 1Department of Physics and Astronomy, University of Manchester, Manchester, M13 9PL, United Kingdom.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|May 15, 2002
PubMed
Summary

This study analyzes discrete-time persistence in stochastic processes. Discrete sampling affects smooth processes like diffusion less than random walkers, with new methods for calculating persistence probabilities.

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Area of Science:

  • Statistical Physics
  • Stochastic Processes
  • Time Series Analysis

Background:

  • Persistence, or the probability of no zero crossings, is crucial for understanding stochastic processes.
  • Continuous processes sampled at discrete times introduce complexities in analyzing persistence.
  • Previous work established a framework for continuous-time persistence, but discrete-time effects require further investigation.

Purpose of the Study:

  • To investigate the impact of discrete time sampling on the persistence probability of stochastic processes.
  • To develop and extend methods for calculating discrete persistence exponents for various random processes.
  • To compare the sensitivity of different processes to discrete sampling effects.

Main Methods:

  • Utilized the independent interval approximation to analyze the variation of the discrete persistence exponent with sampling time.

Related Experiment Videos

  • Extended a previously developed matrix method to determine persistence probabilities for 2D random walks and 1D random-acceleration problems.
  • Investigated the case of alternating persistence (a<0).

    • Main Results:

    • Derived the relationship between continuous and discrete persistence exponents, showing dependence on sampling interval (DeltaT).
    • Demonstrated that smooth processes like diffusion are less affected by discrete sampling than random walkers or randomly accelerated particles.
    • Calculated persistence probabilities (rho(a)) for two-dimensional random walks and one-dimensional random-acceleration problems, including the alternating persistence case.

    Conclusions:

    • Discrete time sampling has a quantifiable impact on persistence probabilities, varying in significance depending on the underlying process.
    • The developed methods provide a robust framework for analyzing discrete-time persistence in complex stochastic systems.
    • Experimental measurements of persistence should account for discrete sampling, especially for non-smooth processes.