SPD

Abstract

We prove the law of large numbers and central limit theorem for trajectories of particle carried by a two dimensional Eulerian velocity field. The field is given by a solution of a stochastic Navier--Stokes system with a non-degenerate noise. The spectral gap property, with respect to Wasserstein metric, for such a system has been shown by Hairer and Mattingly. We show that a similar property holds for the environment process corresponding to the Lagrangian observations of the velocity. The proof of the central limit theorem relies on the martingale approximation of the trajectory process.