Ergodicity of Stochastic Shell Models Driven by Pure Jump Noise

Abstract

In the present paper we study a stochastic evolution equation for shell (sabra and GOY) models with pure jump Lévy noise $L=\sum_{k=1}^\infty l_k(t)e_k$ on a Hilbert space $\mathbb{H}$. Here $\{l_k; k\in \mathbb{N}\}$ is a family of independent and identically distributed (i.i.d.) real-valued pure jump Lévy processes and $\{e_k; k\in \mathbb{N}\}$ is an orthonormal basis of $\mathbb{H}$. We mainly prove that the stochastic system has a unique invariant measure. For this aim we show that if the Lévy measure of each component $l_k(t)$ of $L$ satisfies a certain order and a nondegeneracy condition and is absolutely continuous with respect to the Lebesgue measure, then the Markov semigroup associated with the unique solution of the system has the strong Feller property. If, furthermore, each $l_k(t)$ satisfies a small deviation property, then 0 is accessible for the dynamics independently of the initial condition. Examples of noises satisfying our conditions are a family of i.i.d. tempered Lévy noises $\{l_k; k\in \mathbb{N}\}$ and $\{l_k=W_k\circ G_k + G_k; k\in \mathbb{N} \}$, where $\{G_k; k \in \mathbb{N}\}$ (resp., $\{W_k; k\in \mathbb{N}\}$) is a sequence of i.i.d. subordinator gamma (resp., real-valued Wiener) processes with Lévy density $f_G(z)=(\vartheta z)^{-1} e^{-\frac z\vartheta} \mathds{1}_{z>0}$. The proof of the strong Feller property relies on the truncation of the nonlinearity and the use of a gradient estimate for the Galerkin system of the truncated equation. The gradient estimate is a consequence of a Bismut--Elworthy--Li type formula that we prove in Appendix A of the paper.