Stochastic Averaging and Diffusion Creations for Stochastic Equations.

我们研究了一个系统的演化，在这个系统中，可观测量在不同的时间尺度上相互作用和演化：一些变量变化缓慢，而另一些变量移动得更快并振荡。其目的是捕捉的平均影响的快变量上的慢变化的，并获得一个近似的动力学描述的慢变量的演变在很长一段时间。近似动力学称为有效运动。本文研究了一个由依赖于快变量y的向量ELD驱动的变量x的微分方程：x_ = 1$ f（x; y$）其中$是一个小数字，表示尺度的分离和持续时间的阶数.当y是一个马尔可夫过程，满足强混合条件，混合率至少是二次的，并且某些定量遍历定理成立时，随机变量族x$t的统计性质，光滑地依赖于时间，近似地由一个扩散过程给出，该扩散过程通常以不可微的方式依赖于t。这称为扩散创建，这是一个经典的问题，在今天仍然是，甚至更加相关。该项目的目的是研究这些类型的现象时，快速随机过程是不是马尔可夫。当快变量来自于一个充分混沌的确定性动力系统时，也研究了混沌的产生[2]。新奇与方法。我们开始采取快速变量的分数布朗运动驱动的随机微分方程的解决方案。对于这样的随机微分方程的不变测度的存在性和收敛速度最近已被研究。例如，对于加性噪声的情形，文献[1]中得到了遍历性，收敛速度也是已知的，但并不能很好地应用到标准工作中，甚至忽略了快速系统具有记忆的问题。参见[1]。然后对乘法情形进行了进一步的研究。在这里，我们提出了一种方法，用于处理粗糙微分方程[2]，希望通过的问题，这是非常困难的，以获得良好的率。这项研究是在对齐EPSRC的战略主题数学科学和研究领域：数学分析，统计和应用概率。[1]M.海尔分数布朗运动驱动的随机微分方程的遍历性。安。可能。2005. [2]D.凯莉和我。墨尔本。随机微分方程的光滑逼近。2016. [3]I.谢弗列夫峰Friz，A.科列帕诺夫岛墨尔本和H.张某多尺度系统、均匀化和粗糙路径。2017年，Arxiv。

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.