Stochastic dynamics for singularly perturbed PDEs with fractional Brownian motions

1、研究了混合分数布朗粗糙路径驱动的快慢粗糙微分方程组的平均原理，其中快分量由布朗运动驱动，慢分量由Hurst指数为H（1/3 < H \leq 1/2）的分数布朗运动驱动。结合粗糙路理论的分数阶微积分方法和Khasminskii的经典时间离散方法，我们证明了慢分量在L^1意义下强收敛于相应的平均方程的解.在粗糙路径理论的框架下研究了一类具有Hurst参数1/2<H<1的乘性分数布朗噪声的慢变过程和具有快变扩散的快变过程的双时间尺度泛函随机微分方程的平均原理.我们要强调的是，本文提出的方法是基于这样一个事实，即一个随机积分的分数布朗运动与赫斯特参数在（1/2，1）可以定义一个广义Stieltjes积分。特别地，为了证明平均原理的极限定理，我们将引入停止时间来控制乘性分数布朗噪声的大小。然后，Khasminskii的方法的启发，平均原则的意义上的收敛在第p时刻一致的时间。

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.