In the present paper we study a stochastic evolution equation for shell (SABRA \& GOY) models with pure jump \levy noise $L=\sum_{ k=1}^\infty l_k(t)e_k$ on a Hilbert space $\h$. Here $\{ l_k, k\in \mathbb{ N}\}$ is a family of independent and identically distributed (i.i.d.) real-valued pure jump \levy processes and $\{ e_k, k\in \mathbb{ N}\}$ is an orthonormal basis of $\h$. We mainly prove that the stochastic system has a unique invariant measure. For this aim we show that if the \levy measure of each component $l_k(t)$ of $L$ satisfies a certain order and a non-{ degeneracy} 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 \levy 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 \levy 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 (BEL) type formula that we prove in the Appendix A of the paper.
Categories: math.PR