论文标题
Markovian慢速系统的大偏差原理
A large deviation principle for Markovian slow-fast systems
论文作者
论文摘要
我们证明，慢速变量在慢速系统中的慢速变量的大小偏差原理在趋向于无穷大的时间尺度分离的极限下。在我们考虑的限制方案中，慢变量对其确定性极限的收敛性以及快速变量与平衡的收敛性在相同的尺度上竞争。大偏差原理是通过将大偏差问题与汉密尔顿 - 雅各比 - 贝尔曼方程的解决方案联系起来的，该方程在同伴论文[Arxiv：1912.06579]中为其建立了良好的。 我们以行动综合形式施放了速率功能，并通过两种方式来解释拉格朗日。首先，就缓慢变量的速度和快速变量的分布的双重优化问题而言，精神类似于从收缩原理中获得的。其次，就与慢速系统相关的主元素价值问题而言。第一个表示特别证明，对于从大偏差原则的平均原理推导而有用。 作为我们一般结果的主要示例，我们认为经验度量 - 升华对耦合到紧凑型歧管上的快速扩散。我们证明了巨大的偏差，并以双重优化形式使用拉格朗日人来证明该系统中平均原理的有效性。
We prove pathwise large deviation principles of slow variables in slow-fast systems in the limit of time-scale separation tending to infinity. In the limit regime we consider, the convergence of the slow variable to its deterministic limit and the convergence of the fast variable to equilibrium are competing at the same scale. The large deviation principle is proven by relating the large deviation problem to solutions of Hamilton-Jacobi-Bellman equations, for which well-posedness was established in the companion paper [arXiv:1912.06579]. We cast the rate functions in action-integral form and interpret the Lagrangians in two ways. First, in terms of a double-optimization problem of the slow variable's velocity and the fast variable's distribution, similar in spirit to what one obtains from the contraction principle. Second, in terms of a principal-eigenvalue problem associated to the slow-fast system. The first representation proves in particular useful in the derivation of averaging principles from the large deviations principles. As main example of our general results, we consider empirical measure-flux pairs coupled to a fast diffusion on a compact manifold. We prove large deviations and use the Lagrangian in double-optimization form to demonstrate the validity of the averaging principle in this system.
