基于中间随机正则化的完全耦合快-慢混沌系统的均匀化
Homogenization of Fully-Coupled Chaotic Fast-Slow Systems via Intermediate Stochastic Regularization
摘要：
本文研究了具有小时间尺度分离参数$ε$的耦合快慢常微分方程，使得对于慢变量的每个固定值，快动力学在遍历不变测度下是充分混沌的.收敛的慢过程的解决方案的均匀化随机微分方程（ESTA）在限制$ε$为零，明确的公式漂移和扩散系数，迄今为止，只获得的情况下，快速的动态发展独立。本文给出了耦合情形下慢变量一阶矩收敛的充分条件。我们的证明是基于一种新的方法，随机正则化和功能分析技术相结合，通过一个双重限制程序，涉及零噪声限制，以及考虑$ε$为零。我们还给出了漂移系数和扩散系数的精确公式。作为我们的理论的一个主要应用，我们研究\emph{weakly-coupled}系统，其中的耦合只发生在较低的时间尺度和我们的条件是更容易验证，只需要温和的，即求和，衰减的相关性。
英文摘要：
In this paper we study coupled fast-slow ordinary differential equations (ODEs) with small time scale separation parameter $ε$ such that, for every fixed value of the slow variable, the fast dynamics are sufficiently chaotic with ergodic invariant measure. Convergence of the slow process to the solution of a homogenized stochastic differential equation (SDE) in the limit $ε$ to zero, with explicit formulas for drift and diffusion coefficients, has so far only been obtained for the case that the fast dynamics evolve independently. In this paper we give sufficient conditions for the convergence of the first moments of the slow variable in the coupled case. Our proof is based upon a new method of stochastic regularization and functional-analytical techniques combined via a double limit procedure involving a zero-noise limit as well as considering $ε$ to zero. We also give exact formulas for the drift and diffusion coefficients for the limiting SDE. As a main application of our theory, we study \emph{weakly-coupled} systems, where the coupling only occurs in lower time scales and our conditions are more easily verifiable requiring only mild, namely summable, decay of correlations.
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