AI 中文总结
研究马尔可夫余圈长时间统计和小噪声渐近性，通过结合广义耦合与遍历理论方法给出指数混合准则，建立小噪声极限下的大偏差原理，该理论适用于非自治随机偏微分方程并举例说明。
AI 中文摘要
本文研究与标准博雷尔概率空间上保测动力系统建模的随机环境中的马尔可夫过程相关的马尔可夫余圈的长时间统计和小噪声渐近性。第一个结果为这类余圈的平稳测度的指数混合提供了一个抽象准则，用于具有可直接从先验估计验证的假设的随机偏微分方程应用。通过将广义耦合论证与遍历理论方法相结合克服环境的非均匀性。第二个结果在小噪声极限下为唯一平稳测度建立了具有良好速率函数的弗赖德林 - 温策尔大偏差原理。该抽象理论适用于非自治随机偏微分方程，并通过有界域上的二维纳维 - 斯托克斯方程和阻尼正弦 - 戈登方程两个例子进行说明。
英文摘要
This paper studies the long time statistics and small noise asymptotics of Markov cocycles associated with Markov processes in random environments modeled by measure preserving dynamical systems on a standard Borel probability space. Our first result provides an abstract criterion for exponential mixing of stationary measures for such cocycles, formulated toward SPDE applications with assumptions that can be verified directly from a priori estimates. To overcome the nonuniformity from the environment, we combine generalized coupling arguments with ergodic theoretic methods. This allows us to convert nonuniform estimates along the environment into contraction on a positive density set of times, and then upgrade this to all time contraction by introducing a block gap-counting argument. Our second result establishes a Freidlin--Wentzell large deviation principle(LDP) for the unique stationary measure in the small noise limit with a good rate function. For the upper bound, the noise is allowed to be degenerate, while the deterministic pullback attractor may have nontrivial dynamics. The abstract theory applies to nonautonomous SPDEs. We illustrate it with two examples: the two-dimensional Navier--Stokes equations on bounded domains and damped Sine--Gordon equations, where both the deterministic forcing and the degenerate additive noise depend on the random environment.
Comments57 pages
