The Conditional Gaussian Nonlinear System (CGNS) is a broad class of nonlinear stochastic dynamical systems. Given the trajectories for a subset of state variables, the remaining follow a Gaussian distribution. Despite the conditionally linear structure, the CGNS exhibits strong nonlinearity, thus capturing many non-Gaussian characteristics observed in nature through its joint and marginal distributions. Desirably, it enjoys closed analytic formulae for the time evolution of its conditional Gaussian statistics, which facilitate the study of data assimilation and other related topics. In this paper, we develop a martingale-free approach to improve the understanding of CGNSs. This methodology provides a tractable approach to proving the time evolution of the conditional statistics by deriving results through time discretization schemes, with the continuous-time regime obtained via a formal limiting process as the discretization time-step vanishes. This discretized approach further allows for developing analytic formulae for optimal posterior sampling of unobserved state variables with correlated noise. These tools are particularly valuable for studying extreme events and intermittency and apply to high-dimensional systems. Moreover, the approach improves the understanding of different sampling methods in characterizing uncertainty. The effectiveness of the framework is demonstrated through a physics-constrained, triad-interaction climate model with cubic nonlinearity and state-dependent cross-interacting noise.

翻译：条件高斯非线性系统（CGNS）是一类广泛的非线性随机动力系统。当给定部分状态变量的轨迹时，其余变量服从高斯分布。尽管具有条件线性结构，CGNS通过其联合分布与边缘分布展现出强非线性特征，从而能够捕捉自然界中观测到的许多非高斯特性。值得关注的是，该系统条件高斯统计量的时间演化具有封闭的解析表达式，这为研究数据同化及其他相关课题提供了便利。本文提出了一种非鞅论方法以深化对CGNS的理解。该方法通过时间离散化方案推导结果，并在离散时间步长趋于零时通过形式极限过程获得连续时间情形，从而为证明条件统计量的时间演化提供了一条可处理的路径。这种离散化方法还能进一步推导具有相关噪声的未观测状态变量最优后验采样的解析公式。这些工具对于研究极端事件和间歇性现象尤为宝贵，并适用于高维系统。此外，该方法深化了对不同采样方法在表征不确定性方面的理解。通过一个具有三次非线性和状态依赖交叉相互作用噪声的物理约束三元相互作用气候模型，验证了该框架的有效性。