Moderate deviation principles for stochastic 2D hydrodynamics type systems with multiplicative Lévy noise
Yue Li, Shijie Shang
TL;DR
This work establishes a moderate deviation principle for abstract SPDEs driven by small Lévy noise, encompassing a broad class of 2D hydrodynamical-type models such as the stochastic Navier–Stokes and MHD equations. The authors adopt a weak convergence (Budhiraja–Dupuis–Gao–Maroulas) approach and construct skeleton equations to characterize the rate function, notably avoiding the compactness requirements on Gelfand triples by using finite-dimensional projections. The main result shows that $M^\varepsilon=\frac{u^\varepsilon-u^0}{a(\varepsilon)}$ satisfies a large deviation principle with speed $\varepsilon/a^2(\varepsilon)$ and rate function $I(\eta)=\inf_{\varphi:\eta=Y^\varphi}\{\tfrac{1}{2}\|\varphi\|_2^2\}$, where $Y^\varphi$ solves the skeleton equation $\frac{d}{dt}Y^\varphi+\mathcal{A}Y^\varphi+B(Y^\varphi,u^0)+B(u^0,Y^\varphi)=\int_Z G(t,u^0,z)\varphi(t,z)\nu(dz)$. The work broadens applicability to unbounded domains and contributes a robust MDP framework for SPDEs with Lévy noise, with potential use in statistical inference and uncertainty quantification for turbulent fluid systems.
Abstract
In this paper, we establish a moderate deviation principle for an abstract nonlinear equation forced by random noise of Lévy type. This type of equation covers many hydrodynamical models, including stochastic 2D Navier-Stokes equations, stochastic 2D MHD equations, the stochastic 2D magnetic Bérnard problem, and also several stochastic shell models of turbulence. This paper gets rid of the compact embedding assumption on the associated Gelfand triple. The weak convergence method plays an important role.