归功于分数布朗运动的简单结构与精美性质以及在各种科学领域的广泛应用，近些年来备受关注，相关的随机（偏）微分方程的研究也取得了长足的进步。然而，对于某些具有特定意义的随机（偏）微分方程渐近行为的研究特别少，主要原因是方程越具体其研究往往越精细，而分数布朗运动驱动的随机方程的研究与布朗运动驱动的随机方程相比，还缺乏一些工具与技巧，因此这些方程渐近行为的研究具有一定挑战性。..本项目旨在研究分数布朗运动驱动的几类随机（偏）微分方程的遍历性、大数定律、稳定性等一些渐近行为及其相关问题。主要内容阐述如下：一是分数布朗运动驱动的自交互扩散的收敛性、遍历性、有界性、大数定律等；二是分数布朗运动局部时的逼近及其离散刻划，并且建立这些逼近的收敛速度；三是分数噪声驱动的一类特殊随机方程的适定性与离散化以及用Yosida逼近研究抽象空间中分数布朗运动驱动的随机微分方程的渐近性。

In recent years fractional Brownian motion has become an object of intense study, due to its interesting properties, simple structures and its applications in various scientific areas, and moreover, relevant stochastic (partial) differential equations have also made considerable progress. However, there has been little systematic investigation on stochastic (partial) differential equations with some specific significances. The main reason for this, in our opinion, is that the more specific an equation is, the more detailed study of it will be conducted and the study of fractional Brownian motion still lack tools and techniques in comparison with Brownian motion. So, the studies for some asymptotic behaviors of these equations admit challenging...In this research, our objectives are to study some asymptotic behaviors of several kinds of stochastic (partial) differential equations driven by fractional Brownian motion and some related questions. Main topics are expounded as follows: 1. Some asymptotic behaviors of self-interacting diffusions driven by fractional Brownian motion such as ergodicity, the laws of large numbers, and stability. 2. As some related questions, we study also some approximations and discrete characterizations to fractional Brownian local times. Our aims are to introduce some rates of convergence about these approximations. 3. The well-posedness and discretization of a class of special stochastic equations driven by the fractional noise, and using Yosida's approximation to study asymptotic behaviors of stochastic differential equations in infinite dimensions driven by fractional Brownian motion.