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Related Experiment Videos

Average trajectory of returning walks.

Francesca Colaiori1, Andrea Baldassarri, Claudio Castellano

  • 1Dipartimento di Fisica, Università di Roma La Sapienza, and Istituto Nazionale per la Fisica della Materia, Unità di Roma 1, Piazzale Aldo Moro 2, I-00185 Roma, Italy. fran@pil.phys.uniroma1.it

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|June 1, 2004
PubMed
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The average shape of one-dimensional stochastic process trajectories during excursions follows a universal scaling form. This shape is semicircular for symmetric random walks and Levy flights, but damping or noise correlations alter this universal behavior.

Area of Science:

  • Statistical physics
  • Stochastic processes
  • Nonlinear dynamics

Background:

  • Understanding the behavior of stochastic processes is crucial in various scientific fields.
  • Characterizing trajectory shapes during excursions provides insights into underlying dynamics.
  • Previous studies have explored random walks, but a universal description of excursion shapes across different processes is lacking.

Purpose of the Study:

  • To compute and analyze the average shape of one-dimensional stochastic process trajectories during excursions.
  • To identify universal scaling forms governing these shapes.
  • To investigate the impact of factors like damping and noise correlations on trajectory shapes.

Main Methods:

  • Computational analysis of one-dimensional stochastic processes.

Related Experiment Videos

  • Calculation of average trajectory shapes in the (t,x) plane during excursions.
  • Numerical simulations to study the effects of damping and noise correlations.

    • Main Results:

    • The average shape of stochastic process trajectories during excursions obeys a universal scaling form.
    • For symmetric uncorrelated random walks, Levy flights, and biased random walks, the average shape is semicircular.
    • Linear damping leads to flat excursions, destroying the scaling behavior.
    • Noise correlations introduce non-trivial, asymmetric trajectory shapes.

    Conclusions:

    • A universal scaling law describes the average shape of stochastic process excursions.
    • The semicircular shape is a robust feature for many symmetric processes.
    • Deviations from universality arise due to damping and noise correlations, highlighting the sensitivity of excursion shapes to process parameters.