Asymptotic Behavior on a Linear Self-Attracting Diffusion Driven by Fractional Brownian Motion.

Let B H = { B t H , t ≥ 0 } be a fractional Brownian motion with Hurst index 1 2 ≤ H < 1 . In this paper, we consider the linear self-attracting diffusion: d X t H = d B t H + σ X t H d t − θ ∫ 0 t X s H − X u H d s d t + ν d t with X 0 H = 0 , where θ > 0 and σ , ν ∈ R are three parameters. The process is an analogue of the self-attracting diffusion (Cranston and Le Jan, Math. Ann.303 (1995), 87–93). Our main aim is to study the large time behaviors. We show that the solution t − σ θ H X t H − X ∞ H converges in distribution to a normal random variable, as t tends to infinity, and obtain two strong laws of large numbers associated with the solution X H . [ABSTRACT FROM AUTHOR]