Passive tracer in a flow corresponding to a two dimensional stochastic Navier Stokes equations

TL;DR

This paper establishes the law of large numbers and central limit theorem for passive tracers in a 2D stochastic Navier-Stokes flow, demonstrating statistical regularity of particle trajectories under stochastic fluid dynamics.

Contribution

It proves the law of large numbers and CLT for tracers in stochastic Navier-Stokes flows, extending spectral gap results to Lagrangian observations.

Findings

Law of large numbers for tracer trajectories

Central limit theorem for tracer trajectories

Spectral gap property for environment process

Abstract

In this paper we prove the law of large numbers and central limit theorem for trajectories of a 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 in [9]. In the present paper we show that a similar property holds for the environment process corresponding to the Lagrangian observations of the velocity. In consequence we conclude the law of large numbers and the central limit theorem for the tracer. The proof of the central limit theorem relies on the martingale approximation of the trajectory process.