Ergodicity for stochastic 2D Boussinesq equations with a highly degenerate pure jump Levy noise
Jianhua Huang, Xuhui Peng, Xue Wang, Jiangwei Zhang
TL;DR
This work analyzes ergodicity for the 2D stochastic Boussinesq equations driven by highly degenerate pure-jump Lévy noise acting only in the temperature equation. By recasting the system in an abstract evolution form and applying Malliavin calculus with anticipative techniques, the authors establish equi-continuity of the Markov semigroup and weak irreducibility, and they perform a spectral analysis of the Malliavin covariance to obtain time-asymptotic smoothing. A key contribution is proving the invertibility of the Malliavin matrix on the forced modes through a Lévy-bracket framework, which underpins the e-property and irreducibility needed for ergodicity. Consequently, the paper proves the existence and uniqueness of an invariant measure, demonstrating ergodicity for this degenerate-noise SPDE, and extends ergodicity theory to highly degenerate Lévy perturbations in fluid models.
Abstract
This study aims to analyze the ergodicity for stochastic 2D Boussinesq equations and explore the impact of a highly degenerate pure jump Levy noise acting only in the temperature equation, this noise could appear on a few Fourier modes. By leveraging the equi-continuity of the semigroup-established through Malliavin calculus and an analysis of stochastic calculus-together with the weak irreducibility of the solution process, we prove the existence and uniqueness of the invariant measure. Moreover, we overcome the main challenge of establishing time asymptotic smoothing properties of the Markovian dynamics corresponding to this system by conducting spectral analysis of the Malliavin covariance matrix.