Ergodicity for 2D Navier-Stokes equations with a degenerate pure jump noise

TL;DR

This paper proves the ergodicity of 2D Navier-Stokes equations driven by highly degenerate pure jump Lévy noise, solving a longstanding problem by establishing uniqueness of the invariant measure using advanced stochastic calculus techniques.

Contribution

It demonstrates ergodicity for 2D Navier-Stokes equations with minimal noise directions, extending results to non-Gaussian jump noise cases and employing Malliavin calculus.

Findings

Proved ergodicity for equations with noise in as few as four directions.

Established the uniqueness of the invariant measure.

Applied Malliavin calculus and anticipating stochastic calculus techniques.

Abstract

In this paper, we establish the ergodicity for stochastic 2D Navier-Stokes equations driven by a highly degenerate pure jump L\'evy noise. The noise could appear in as few as four directions. This gives an affirmative anwser to a longstanding problem. The case of Gaussian noise was treated in Hairer and Mattingly [\emph{Ann. of Math.}, 164(3):993--1032, 2006]. To obtain the uniqueness of invariant measure, we use Malliavin calculus and anticipating stochastic calculus to establish the equi-continuity of the semigroup, the so-called {\em e-property}, and prove some weak irreducibility of the solution process.