Ergodicity and Averaging of Fractional Dynamics

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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.