Asymptotic analysis and diffusion limit of the Persistent Turning Walker Model

TL;DR

This paper analyzes the long-term diffusive behavior of the Persistent Turning Walker Model using stochastic analysis, extending previous PDE-based results and demonstrating the approach's applicability to other kinetic probabilistic models.

Contribution

It provides a probabilistic analysis of PTWM's diffusive limit, offering new insights beyond PDE methods and applicable to similar kinetic models.

Findings

Recovered diffusive behavior using stochastic analysis

Extended analysis to a broader class of kinetic probabilistic models

Provided a probabilistic framework complementing PDE approaches

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 & 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.