Ergodicity of stochastic shell models driven by pure jump noise

TL;DR

This paper proves ergodicity and the existence of a unique invariant measure for stochastic shell models driven by pure jump Lévy noise, under certain conditions on the noise's Lévy measure and small deviation properties.

Contribution

It establishes strong Feller property and ergodicity for shell models with pure jump Lévy noise, extending previous results to non-Gaussian noise settings.

Findings

Unique invariant measure exists for the system.

Strong Feller property holds under specified Lévy measure conditions.

0 is accessible for the dynamics regardless of initial state.

Abstract

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.