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From shape to randomness: A classification of Langevin stochasticity


  • Eliazar, Iddo
  • Cohen, Morrel H.


The Langevin equation–perhaps the most elemental stochastic differential equation in the physical sciences–describes the dynamics of a random motion driven simultaneously by a deterministic potential field and by a stochastic white noise. The Langevin equation is, in effect, a mechanism that maps the stochastic white-noise input to a stochastic output: a stationary steady state distribution in the case of potential wells, and a transient extremum distribution in the case of potential gradients. In this paper we explore the degree of randomness of the Langevin equation’s stochastic output, and classify it à la Mandelbrot into five states of randomness ranging from “infra-mild” to “ultra-wild”. We establish closed-form and highly implementable analytic results that determine the randomness of the Langevin equation’s stochastic output–based on the shape of the Langevin equation’s potential field.

Suggested Citation

  • Eliazar, Iddo & Cohen, Morrel H., 2013. "From shape to randomness: A classification of Langevin stochasticity," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 392(1), pages 27-42.
  • Handle: RePEc:eee:phsmap:v:392:y:2013:i:1:p:27-42
    DOI: 10.1016/j.physa.2012.08.009

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