Affiliations: UMR5219 CNRS & Institut de Mathématiques de Toulouse, Université de Toulouse, 118 route de Narbonne, F-31062 Toulouse Cedex 09, France. E-mails: {cattiaux, sebastien.motsch}@math.univ-toulouse.fr | UMR181 INRA ENVT & Institut de Mathématiques de Toulouse, École Nationale Vétérinaire de Toulouse, Université de Toulouse, 23 chemin des Capelles, F-31076 Toulouse, Cedex 3, France. E-mail: chafai@math.univ-toulouse.fr
Abstract: 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.
