Stochastic Averaging and Diffusion Creations for Stochastic Equations.

We study the evolution of a system in which the observables interact and evolve at different time scales: some of the variables change slowly and the others move faster and oscillate. The aim is to capture the average influence of the fast variable on the slow varying ones and obtain an approximate dynamics for describing the evolution of the slow variable over a long time. The approximate dynamics is known as the effective motion. We study a differential equation in a variable x, driven by a vector eld depending on a fast variable y:
x_ = 1
$ f(x; y$) where $ is a small number indicating the separation of the scales and the order fo the duration of time. When y is a Markov process satisfying a strong mixing condition with mixing rate at least quadratic and for which certain quantitative ergodic theorems hold, the statistical properties of the family of random variables x$t, which depends smoothly on time, is approximately given by a diffusion process that typically depends on t in a non-differentiable way.

This is called diffusion creation, a classical problem which is still and even more relevant today. The aim of the project is to study these type of phenomenon when the fast stochastic process is not Markovian. Dffusion creation has also been studied when the fast variables coming from a deterministic dynamical system which is sufficiently chaotic [2].

Novelty and Methodology. We begin with taking the fast variables the solutions of a stochastic differential equation driven by fractional Brownian motions. For such SDEs the existence of the invariant measure and the rate of the convergence have been recently studied. For example, for the additive noise case, the ergodicity was obtained in [1], a rate of convergence is also known, but not suffciently good to t into the standard work, even we were to ignore the problem of that the fast system having memory. See [1]. Subsequent studied were then made on the multiplicative cases. Here we propose to use a method, used to dealing with rough differential equations [2], to hope to by pass the problem that it is extremely difficult to obtain a good rate.

This research is in alignment to EPSRC's strategic theme mathematical sciences and research area: Mathematical analysis, Statistics and applied probability.

References.
[1] M. Hairer. Ergodicity of Stochastic Differential Equations Driven by Fractional Brownian Motion. Ann. Probab. 2005.
[2] D. Kelly and I. Melbourne. Smooth approximation of stochastic differential equations Ann. Probab. 2016.
[3] I. Chevyrev, P. K. Friz, A. Korepanov, I. Melbourne, and H. Zhang. Multi-scale systems, homogenization, and rough paths. 2017, Arxiv.