Abstract: Let $B^{H}$ be a fractional Brownian motion with Hurst index $\frac{1}{2}\leq H< 1$. In this paper, we consider the equation (called the Ornstein-Uhlenbeck process with a linear self-repelling drift) $\begin{equation*} {\rm d}X_{t}^{H}={\rm d}B_{t}^{H}+\sigma X_t^{H}{\rm d}t+\nu {\rm d}t-\theta \left(\int_{0}^{t}(X_t^{H}-X_{s}^{H}){\rm d}s\right){\rm d}t, \end{equation*}$ where $\theta<0$, $\sigma,\nu \in \mathbb{R}$. The process is an analogue of {self-attracting} diffusion (Cranston, Le Jan. Math Ann, 1995, 303: 87-93). Our main aim is to study the large time behaviors of the process. We show that the solution $X^H$ diverges to infinity as $t$ tends to infinity, and obtain the speed at which the process $X^H$ diverges to infinity as $t$ tends to infinity.

Key words: fractional Brownian motion, stochastic difference equations, rate of convergence, asymptotic