脉冲型强迫下经典Lorenz流的随机稳定性
Stochastic stability of the classical Lorenz flow under impulsive type forcing
摘要：
我们引入了一种新的类型的随机扰动的经典洛伦兹流，以更好地模拟缓慢变化的现象，如人为强迫在气候学和证明随机稳定的未扰动流。扰动以脉冲方式作用于系统，因此不具有在\cite{Ki}、\cite{Ke}、\cite{Me}中已经讨论的扩散类型。也就是说，给定未扰动流的横截面$\mathcal{M}$，每当系统的轨迹穿过$\mathcal{M}$时，相速度场就被从未扰动流的适当邻域随机采样的新相速度场改变。由此产生的随机演变，因此描述了一个分段确定性的马尔可夫过程。然后，在随机动力系统和半马尔可夫过程的框架下，对无扰动流的随机稳定性进行了证明。
英文摘要：
We introduce a novel type of random perturbation for the classical Lorenz flow in order to better model phenomena slowly varying in time such as anthropogenic forcing in climatology and prove stochastic stability for the unperturbed flow. The perturbation acts on the system in an impulsive way, hence is not of diffusive type as those already discussed in \cite{Ki}, \cite{Ke}, \cite{Me}. Namely, given a cross-section $\mathcal{M}$ for the unperturbed flow, each time the trajectory of the system crosses $\mathcal{M}$ the phase velocity field is changed with a new one sampled at random from a suitable neighborhood of the unperturbed one. The resulting random evolution is therefore described by a piecewise deterministic Markov process. The proof of the stochastic stability for the umperturbed flow is then carryed on working either in the framework of the Random Dynamical Systems or in that of semi-Markov processes.
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