We study stability of solutions for a randomly driven and degenerately damped version of the Lorenz ’63 model. Specifically, we prove that when damping is absent in one of the temperature components, the system possesses a unique invariant probability measure if and only if noise acts on the convection variable. On the other hand, if there is a positive growth term on the vertical temperature profile, we prove that there is no normalizable invariant state. Our approach relies on the derivation and analysis of nontrivial Lyapunov functions which ensure positive recurrence or null-recurrence/transience of the dynamics.

中文翻译：

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退化阻尼随机 Lorenz 系统中稳态的灵敏度

我们研究了 Lorenz '63 模型的随机驱动和退化阻尼版本的解决方案的稳定性。具体来说，我们证明了当其中一个温度分量没有阻尼时，当且仅当噪声作用于对流变量时，系统才具有唯一的不变概率测度。另一方面，如果在垂直温度分布上有一个正增长项，我们证明不存在可归一化的不变状态。我们的方法依赖于非平凡李雅普诺夫函数的推导和分析，这些函数确保动态的正递归或零递归/瞬态。