  This paper considers a new model of individual displacement, based on fish
motion, the so-called Persistent Turning Walker (PTW) model, which involves an
Ornstein-Uhlenbeck process on the curvature of the particle trajectory. The
goal is to show that its large time and space scale dynamics is of diffusive
type, and to provide an analytic expression of the diffusion coefficient. Two
methods are investigated. In the first one, we compute the large time
asymptotics of the variance of the individual stochastic trajectories. The
second method is based on a diffusion approximation of the kinetic formulation
of these stochastic trajectories. The kinetic model is a Fokker-Planck type
equation posed in an extended phase-space involving the curvature among the
kinetic variables. We show that both methods lead to the same value of the
diffusion constant. We present some numerical simulations to illustrate the
theoretical results.
