论文标题
通过中间随机正规化对完全耦合混沌快速系统的均质化
Homogenization of Fully-Coupled Chaotic Fast-Slow Systems via Intermediate Stochastic Regularization
论文作者
论文摘要
在本文中，我们研究了具有较小的时间尺度分离参数$ε$的耦合的快速慢速微分方程（ODE），这样，对于慢速变量的每个固定值，快速动力学都足够混乱，而ergodic不变度。慢速过程与质量$ε$以零为零的均质随机微分方程（SDE）的解决方案的融合，具有显式和扩散系数的显式公式，仅对快速动力学独立地进化而获得了迄今为止获得的。在本文中，我们为耦合情况下慢变量的第一矩收敛提供了足够的条件。我们的证明是基于一种随机正则化和功能分析技术的新方法，该技术通过涉及零额限的双限制程序组合，并考虑$ε$至零。我们还为限制SDE的漂移和扩散系数提供了精确的公式。作为我们理论的主要应用，我们研究了\ emph {弱耦合}系统，其中耦合仅发生在较低的时间尺度上，并且我们的条件更容易验证，仅需要温和的相关性衰减。
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.
