Multi-Scale Stochastic Dynamics with Fractional Noise

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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.