Ergodicity and Averaging of Fractional Dynamics

本文研究分数阶驱动随机微分方程解的长时间行为。对该领域的兴趣是由Hairer的一篇开创性文章引发的，他建立了一个复杂的框架，允许转移经典马尔可夫过程理论的大多数概念。给定分数阶随机微分方程不变测度的存在唯一性，时间t律的解向不变测度的收敛速度是很有意义的。Hairer的原始工作获得了加性噪声情况下的代数速率，随后Fontbona和Panloup分别为大赫斯特参数和Deya，Panloup和Tindel分别为小赫斯特参数建立了乘性噪声。已知的最快收敛到不变测度（不包括平凡的情况下，无处不在的一致压缩漂移）是由于最近的工作Panloup和理查德的加性噪声。然而，所有这些速率都明显慢于通常表现出适当度量的指数衰减的Ito扩散。因此，我们看到相当大的潜力，这些结果的改善，无论是加性和乘性noises.In许多其他的，一个重要的应用程序，利用随机过程的遍历性是研究的平均原则的快慢系统。考虑到伯克霍夫定理，很自然地可以期望在宏观尺度上运动的遍历分数环境中的系统可以很好地近似于有效的自治动力学。据我们所知，以前没有对这种分数多尺度系统的研究，我们的研究结果在气候科学中得到了应用，以前的马尔可夫模型产生的预测与观测数据严重不匹配。

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.