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

2024/05/01 by Xuhui Peng, Peng, Xuhui, Jianliang Zhai +3 · 1 citation
Engineering · Economics, Econometrics and Finance · Mathematics · #Stability and Controllability of Differential Equations #Stochastic processes and financial applications #Navier-Stokes equation solutions

paper · pdf · doi:10.48550/arxiv.2405.00414

Abstract

In this paper, we establish the ergodicity for stochastic 2D Navier-Stokes equations driven by a highly degenerate pure jump Lévy 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 [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.

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