Asymptotic analysis and diffusion limit of the Persistent Turning Walker model

Patrick Cattiaux, Djalil Chafaï, Sébastien Motsch

Research output: Contribution to journalArticlepeer-review

4 Scopus citations


The Persistent Turning Walker Model (PTWM) was introduced by Gautrais et al. in Mathematical Biology for the modelling of fish motion. It involves a nonlinear pathwise functional of a non-elliptic hypo-elliptic diffusion. This diffusion solves a kinetic Fokker-Planck equation based on an Ornstein-Uhlenbeck Gaussian process. The long time "diffusive" behavior of this model was recently studied by Degond and Motsch using partial differential equations techniques. This model is however intrinsically probabilistic. In the present paper, we show how the long time diffusive behavior of this model can be essentially recovered and extended by using appropriate tools from stochastic analysis. The approach can be adapted to many other kinetic " probabilistic" models.

Original languageEnglish (US)
Pages (from-to)17-31
Number of pages15
JournalAsymptotic Analysis
Issue number1-2
StatePublished - 2010
Externally publishedYes


  • Animal behavior
  • Central limit theorems
  • Gaussian and Markov processes
  • Hypo-elliptic diffusions
  • Invariance principles
  • Kinetic Fokker-Planck equations
  • Mathematical Biology
  • Poisson equation

ASJC Scopus subject areas

  • Mathematics(all)


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