Multi-Scale Stochastic Dynamics with Fractional Noise

分数布朗运动是具有相关增量的最简单的随机过程之一，因此被用来模拟在经济周期和数据网络等时间序列数据中观察到的普遍的长、短程依赖现象。它是一个具有平稳和相依增量的高斯过程，其协方差函数的衰减服从幂定律。它也是具有自相似指数的自相似的，我们用H.分数布朗运动表示，以及非高斯自相似过程，也被用于数学物理文献中研究临界现象(例如在统计物理中)。多尺度在数学模型中是普遍存在的。一个特别有趣的模型是双尺度慢/快随机系统，其中慢变量和快变量相互作用，并在不同的时间尺度上演化。快速变星是高度振荡的，以极小的速度移动。在这些系统中，慢变量模拟了在其自然时间尺度上演化的兴趣量。其目的是得到一个封闭的方程，称为有效方程，用于逼近慢变量。到目前为止，对慢/快系统的研究主要集中在布朗运动驱动的随机微分方程上。布朗运动是一个具有独立增量的过程。因此，使用它进行建模依赖于独立性假设，这在某些情况下是很自然的。在许多其他重要和具有挑战性的情况下，我们必须考虑噪音的相互依存关系。当参数H大于1/2时，分数布朗运动驱动的随机方程可以用Young积分理论来理解。当H等于1/2时，我们得到了经典的布朗运动驱动的随机微分方程组，它的解是马尔可夫甚至是扩散过程。在过去的20年左右的时间里，人们在粗糙路径理论中建立了对由分数布朗运动驱动的H&gt；1/4随机方程的理解。然而，由于还没有工具，对具有分数噪声的慢/快系统的研究还不够充分。随着随机分析的新发展，发展具有长、短程相关分数噪声的随机动力学的多尺度理论成为可能。我们将同时研究随机平均化和均质化制度。在前一种情况下，这种有效动力学是通过绝热传输考虑较大/快速移动的变量的持续效应而通过平均化过程获得的。有效方程通常与慢方程是同一类型的。然而，对于这一点，还没有一个足够好的极限理论。借助于最近在粗糙路理论中得到的一个结果，我们证明了第一个随机平均定理。同质化问题是关于平均值的波动。最近，我们建立了长程相关分数阶环境中随机常数组的非扩散有效动力学，从而偏离了经典的随机常数组扩散齐化理论。随着随机分析的最新发展，这些都成为可能。我们的项目是实施一个完整的方案，致力于具有分数噪声的慢/快系统。

A fractional Brownian motion is used to model the prevalent long and short-range dependence phenomena, observed in time-series data such as economic cycles and data networks, because it is one of the simplest stochastic processes with correlated increments. It is a Gaussian process with stationary and dependent increments; the decay of its covariance function follows the power law. It is also self-similar with self-similarity exponent, which we denote by H. Fractional Brownian motions, as well as the non-Gaussian self-similar processes, are also used in mathematical physics literature for studying the critical phenomena (e.g. in statistical physics). Multi-scales are ubiquitous in mathematical models. One especially interesting model is the two-scale Slow/fast stochastic system, in which the slow and fast variables interact with each other and evolve in different time scales. The fast variables are highly oscillatory, moving at the microscopic speed. In these systems, the slow variables model quantities of interest evolving in its natural time scale. The aim is to obtain a closed equation, called the effective equation, for approximating the slow variables. So far, the study of slow/fast systems has been predominantly focused on stochastic differential equations driven by Brownian Motions. A Brownian motion is a process with independent increments. Hence modelling with it relies on the independence assumption, which is natural in some cases. In many other important and challenging cases, we must consider the inter-dependence of the noise. Stochastic equations driven by fractional Brownian motions can be understood within the Young integration theory if the parameter H is greater than 1/2. If H equals 1/2, we have the classical stochastic differential equations driven by Brownian Motions, whose solutions are Markov or even diffusion processes. In the last 20 years or so, an understanding of stochastic equations driven by fractional Brownian motions with H>1/4 has been established within the rough path theory. However, the slow/fast systems with fractional noise have not been sufficiently studied, for there had not been the tools. With the new developments in Stochastic Analysis, it is possible to take on the challenge to develop a multi-scale theory of stochastic dynamics with both long, and short, range dependent fractional noise. We will study both the stochastic averaging and the homogenisation regimes. In the former case, this effective dynamics is obtained with an averaging procedure by taking care of the persistent effect coming from the larger/fast-moving variables through adiabatic transmission. The effective equation is usually the same type as the slow equation. However, there had not been a good enough limit theory for this. With the help of a very recently obtained results in the rough path theory, we proved the first stochastic averaging theorem. The homogenisation problem is about the fluctuations from the average. We recently established a non-diffusive effective dynamics for random ODEs in a long-range dependent fractional environment, thus departing from the classical diffusive homogenisation theory of random ODEs. These are made possible with recent developments in Stochastic Analysis. Our project is to implement a full programme devoted to slow/fast systems with fractional noise.