平稳遍历背景下的弱KAM理论
Weak KAM Theory topics in the stationary ergodic setting
摘要：
我们进行定性分析的临界方程与一个固定的遍历哈密顿通过随机版本的度量方法，其中封闭的随机平稳集的概念，从随机几何发出，起着重要的作用。我们的目的是给出一个适当的随机Aubry集的概念，挑选出存在精确或近似校正的特征条件，并写下它们的表示公式。对于最后一个任务，我们使用一个拉克斯型公式，适应于随机环境。这种材料可以被视为长期项目的第一步，以开发弱KAM理论的随机模拟，概括在周期性情况下所做的事情，或者更一般地说，当底层空间是紧致流形时。
英文摘要：
We perform a qualitative analysis of the critical equation associated with a stationary ergodic Hamiltonian through a stochastic version of the metric method, where the notion of closed random stationary set, issued from stochastic geometry, plays a major role. Our purpose is to give an appropriate notion of random Aubry set, to single out characterizing conditions for the existence of exact or approximate correctors, and write down representation formulae for them. For the last task, we make use of a Lax--type formula, adapted to the stochastic environment. This material can be regarded as a first step of a long--term project to develop a random analog of Weak KAM Theory, generalizing what done in the periodic case or, more generally, when the underlying space is a compact manifold.
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