Ergodicity for the fractional Magneto-Hydrodynamic equations driven by a degenerate pure jump noise
Xue Wang, Jiangwei Zhang, Jianhua Huang
TL;DR
The paper addresses ergodicity of the 2D stochastic fractional MHD system on a torus under highly degenerate pure-jump Lévy noise. By integrating Malliavin calculus with anticipative stochastic techniques and a Hörmander-type Lie-bracket framework tailored to degenerate noise, the authors obtain non-degeneracy of the Malliavin covariance on the finite unstable subspace and derive gradient estimates for the Markov semigroup. They establish moment bounds for the solution and its Malliavin derivatives, prove invertibility of the Malliavin matrix on the pertinent subspace, and show the $e$-property and weak irreducibility, which together yield existence and uniqueness of an invariant measure (ergodicity). The results extend ergodic theory for degenerate SPDEs to coupled velocity-magnetic field systems driven by highly degenerate pure-jump noise, providing a rigorous statistical description of long-time behavior in fractional MHD models with jump perturbations.
Abstract
This paper is concerned with the ergodicity for stochastic 2D fractional magneto-hydrodynamic equations on the two-dimensional torus driven by a highly degenerate pure jump Lévy noise. We focus on the challenging case where the noise acts in as few as four directions, establishing new results for such high degeneracy. We first employ Malliavin calculus and anticipative stochastic calculus to demonstrate the equi-continuity of the semigroup (or so-called the e-property), and then verify the weak irreducibility of the solution process. Therefore, the uniqueness of invariant measure is proven.