Ergodicity for stochastic equations of Navier–Stokes type

Zdzisław Brzeźniak,Tomasz Komorowski,Szymon Peszat

doi:10.1214/21-ecp443

Abstract

In this note we consider a simple example of a finite dimensional system of stochastic differential equations driven by a one dimensional Wiener process with a drift, that displays some similarity with the stochastic Navier-Stokes Equations (NSEs), and investigate its ergodic properties depending on the strength of the drift. If the latter is sufficiently small and lies below a critical threshold, then the system admits a unique invariant probability measure which is Gaussian. If, on the other hand, the strength of the noise drift is larger than the threshold, then in addition to a Gaussian invariant probability measure, there exist another one. In particular, the generator of the system is not hypoelliptic.

Highlights

Study of ergodic properties of dynamical systems is of profound importance from both applied and theoretical standpoints
In this note we consider a simple example of a finite dimensional system of stochastic differential equations driven by a one dimensional Wiener process with a drift, that displays some similarity with the stochastic Navier-Stokes Equations (NSEs), and investigate its ergodic properties depending on the strength of the drift
Flandoli [14], who showed the existence of an invariant probability measure for the 2D Navier–Stokes equations (NSE) driven by an additive Gaussian noise

Summary

Introduction

Study of ergodic properties of dynamical (inclusive random) systems is of profound importance from both applied and theoretical standpoints. The culminating work on this topic is due to Hairer and Mattingly [21] who, using a new concept of an asymptotically strong Feller semigroup, proved that the Markov process generated by the stochastic NSEs on a 2D torus has a unique invariant probability measure provided the Gaussian perturbation is of mean 0 and acts on at least two modes that are of different length and whose integer linear combinations generate the two dimensional integer lattice Such a system can be called a hypoelliptic. The uniqueness of the invariant measure for the stochastic Navier-Stokes equations (on a 2D torus) when external force f = κe is equal to 0, the noise is onedimensional and viscosity is large with respect to the diffusion coefficient, is known, see e.g. the paper [12, Theorem 1] by E, Mattingly and Sinai.

By the law of large numbers we deduce that

Consequently we have the estimate