Asymptotic analysis and diffusion limit of the Persistent Turning Walker Model

Patrick Cattiaux (1) , Djalil Chafai (1, 2, 3) , Sébastien Motsch (1)

1 IMT - Institut de Mathématiques de Toulouse UMR5219
2 UPTE - Physiopathologie et Toxicologie Expérimentales
3 LAMA - Laboratoire d'Analyse et de Mathématiques Appliquées

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 & 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.

Mots clés: Mathematical Biology; animal behavior; hypo-elliptic diffusions; kinetic Fokker-Planck equations; Poisson equation; invariance principles; central limit theorems; Gaussian and Markov processes

Domaines: Equations aux dérivées partielles [math.AP]; Probabilités [math.PR]; Physique mathématique [math-ph]; Physique mathématique [math-ph]

Patrick Cattiaux, Djalil Chafai, Sébastien Motsch. Asymptotic analysis and diffusion limit of the Persistent Turning Walker Model. Asymptotic Analysis, 2010, 67 (1-2), pp.17-31. ⟨10.3233/ASY-2009-0969⟩. ⟨hal-00336405v2⟩