发表机构
Institute of Applied Physics and Computational Mathematics; School of Mathematics and Statistics, Wuhan University（应用物理与计算数学研究所; 武汉大学数学与统计学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对带乘性Lévy噪声的二维随机原始方程，分析其长期分布动力学与小噪声渐近，证明拉回测度吸引子的存在唯一性及上半连续性，并建立纯跳模型的中偏差原理。
AI 中文摘要
我们研究由高斯噪声和乘性Lévy噪声驱动的二维非自治随机原始方程的长期分布动力学及小噪声渐近行为，同时针对纯跳模型建立其中偏差。当噪声足够小时，一致矩界与指数加权终端估计给出拉回吸收族与紧性；而下半连续的垂直矩泛函在弱极限下保持可容许性。我们证明在概率测度的弱拓扑下存在唯一的拉回测度吸引子，并确立当两种噪声分量均消失时该吸引子的上半连续性。对于跳驱动方程，我们在$\boldsymbol{\textit{D}}([0,T];H)\bigcap L^2(0,T;V)$中建立速度为$a^2(\boldsymbol{\textit{\u03B5}})/\boldsymbol{\textit{\u03B5}}$的中偏差原理。证明结合骨架映射的连续性与基于熵界、截断、鞅估计及直接垂直估计的受控随机收敛，避免了跳系数垂直导数的Lipschitz连续性要求。
英文摘要
We study the long-term distributional dynamics and small-noise asymptotics of two-dimensional nonautonomous stochastic primitive equations driven by Gaussian and multiplicative Lévy noise, together with moderate deviations for the purely jump model. For sufficiently small noise, uniform moment bounds and an exponentially weighted terminal estimate yield a pullback absorbing family and tightness, while a lower semicontinuous vertical-moment functional preserves admissibility under weak limits. We prove the existence and uniqueness of a pullback measure attractor in the weak topology of probability measures and establish its upper semicontinuity as both noise components vanish. For the jump-driven equation, we establish a moderate deviation principle in $\mathcal D([0,T];H)\cap L^2(0,T;V)$ with speed $a^2(ε)/ε$. The proof combines continuity of the skeleton map with controlled stochastic convergence based on entropy bounds, truncation, martingale estimates, and direct vertical estimates, avoiding Lipschitz continuity of the vertical derivative of the jump coefficient.
