This paper studies typical dynamics and fluctuations for a slow–fast dynamical system perturbed by a small fractional Brownian noise. Based on an ergodic theorem with explicit rates of convergence, which may be of independent interest, we characterize the asymptotic dynamics of the slow component to two orders (i.e. the typical dynamics and the fluctuations). The limiting distribution of the fluctuations turns out to depend upon the manner in which the small-noise parameter is taken to zero relative to the scale-separation parameter. We study also an extension of the original model in which the relationship between the two small parameters leads to a qualitative difference in limiting behavior. The results of this paper provide an approximation, to two orders, to dynamical systems perturbed by small fractional Brownian noise and incorporating multiscale effects.

中文翻译：

---

分数布朗运动驱动的慢-快系统的典型动力学和波动分析

本文研究了受小分数布朗噪声扰动的慢-快动态系统的典型动力学和波动。基于具有显式收敛速度的遍历定理，这可能是独立的兴趣，我们将慢分量的渐近动力学描述为两个数量级（即典型动力学和波动）。波动的极限分布最终取决于小噪声参数相对于尺度分离参数取零的方式。我们还研究了原始模型的扩展，其中两个小参数之间的关系导致限制行为存在质的差异。本文的结果提供了两个数量级的近似值，