Ergodicity and Averaging of Fractional Dynamics

我们考虑分数驱动随机微分方程解的长时间行为。对这一领域的兴趣是由海尔的一篇开创性文章引发的，在这篇文章中，他建立了一个复杂的框架，允许传递经典马尔可夫过程理论的大多数概念。在分数阶随机微分方程的不变测度存在唯一性的情况下，解的时间-t律对不变测度的收敛速度是很有意义的。海尔最初的工作得到了加性噪声的代数率，后来Fontbona和Panoup分别为大的乘性噪声和小的Hurst参数建立了Deya、Panoup和Tindel的代数率。已知的收敛到不变度量的最快速度(不包括无处不在的一致压缩漂移的平凡情况)是由于PanLoup和Richard最近对加性噪声所做的工作。然而，所有这些速率都比它的扩散慢得多，后者通常表现出适当度规的指数衰减。因此，我们看到了这些结果在加性和乘性噪声方面有相当大的改进潜力。在许多其他应用中，利用随机过程的遍历性的一个重要应用是研究快慢系统的平均原理。考虑到Birkhoff定理，自然会期望在宏观尺度上运动的遍历分数环境中的系统能够很好地被有效的自治动力学所逼近。就我们所知，以前还没有对这种分数多尺度系统的研究，我们的结果在气候科学中得到了应用，在这种情况下，以前的马尔科夫模型产生的预测与观测数据不匹配是出了名的。

We are concerned with the long-time behavior of the solution to fractional-driven stochastic differential equations (SDEs). The interest in the field was sparked by a seminal article by Hairer where he built a sophisticated framework allowing to transfer most notions of classical Markov process theory. Given the existence and uniqueness of an invariant measure for fractional SDEs, the rate of convergence of the time-t law of the solution towards the invariant measure is of interest. The original work of Hairer obtained an algebraic rate for the additive noise case, which was later also established for multiplicative noise by Fontbona and Panloup for large and Deya, Panloup, and Tindel for small Hurst parameters, respectively. The fastest known convergence to the invariant measure (excluding the trivial case of an everywhere uniformly contractive drift) is due to recent work of Panloup and Richard for addtive noise. All of these rates are however significantly slower than for It\^o diffusions which often exhibit an exponential decay of appropriate metric. We thus see considerable potential for improvement of these results, both for additive and multiplicative noise.Among many others, an important application exploiting ergodicity of stochastic processes is the study of averaging principles for fast-slow systems. With Birkhoff's theorem in mind, it is natural to expect that a system in ergodic fractional environment, which moves on a macroscopic scale, is well approximated by an effective, autonomous dynamics. To the best of our knowledge, there has been no previous study of such fractional multi-scale systems before and our results find applications in climate science, in which previous Markovian models produce predictions notoriously mismatching observational data.