The long time behavior of the fractional Ornstein-Uhlenbeck process with linear self-repelling drift

Abstract

Let B^H be a fractional Brownian motion with Hurst index \({1 \over 2} \le H < 1\). In this paper, we consider the equation (called the Ornstein-Uhlenbeck process with a linear self-repelling drift)

$${\rm{d}}X_t^H = dB_t^H + \sigma X_t^H{\rm{d}}t + \nu {\rm{d}}t - \theta \left( {\int_0^t {(X_{^t}^H - X_s^H){\rm{d}}s} } \right){\rm{d}}t,$$

where θ < 0, σ, v ∈ 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.