Stochastic dynamics for singularly perturbed PDEs with fractional Brownian motions

1、研究了混合分数阶布朗粗糙路径驱动下的快慢粗糙微分方程系统的平均原理。快分量由布朗运动驱动，慢分量由分数布朗运动驱动，Hurst指数为H （1/3 < H \leq 1/2）。结合粗糙路径理论的分数阶微积分方法和Khasminskii的经典时间离散化方法，证明了慢分量在L^1意义上强收敛于相应平均方程的解。粗糙路径理论框架下的快慢系统平均原理是一种新方法。本文的主要目的是研究一类双时间尺度泛函随机微分方程的平均原理，其中慢变过程包含一个Hurst参数为1/2<H<1的乘分数布朗噪声，快变过程是一个快速变化的扩散。我们要强调的是，本文提出的方法是基于这样一个事实，即关于分数阶布朗运动的随机积分，Hurst参数在（1/2,1）中可以用广义Stieltjes积分来定义。特别地，为了证明平均原理的一个极限定理，我们将引入停止时间来控制乘法分数布朗噪声的大小。然后，受Khasminskii方法的启发，在p阶矩均匀收敛的意义上发展了平均原理。

1, We devoted to studying the averaging principle for fast-slow system of rough differential equations driven by mixed fractional Brownian rough path. The fast component is driven by Brownian motion, while the slow component is driven by fractional Brownian motion with Hurst index H (1/3 < H \leq 1/2). Combining the fractional calculus approach to rough path theory and Khasminskii’s classical time discretization method, we prove that the slow component strongly converges to the solution of the corresponding averaged equation in the L^1 sense. The averaging principle for a fast-slow system in the framework of rough path theory seems new.2, The main goal of our work is to study an averaging principle for a class of two-time-scale functional stochastic differential equations in which the slow-varying process includes a multiplicative fractional Brownian noise with Hurst parameter 1/2<H<1 and the fast-varying process is a rapidly-changing diffusion. We would like to emphasize that the approach proposed in this paper is based on the fact that a stochastic integral with respect to fractional Brownian motion with Hurst parameter in (1/2 , 1) can be defined by a generalized Stieltjes integral. In particular, to prove a limit theorem for the averaging principle, we will introduce stopping times to control the size of the multiplicative fractional Brownian noise. Then, inspired by the Khasminskii’s approach, an averaging principle is developed in the sense of convergence in the p-th moment uniformly in time.