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

Abstract

Let BH={BtH,t≥0} be a fractional Brownian motion with Hurst index 12≤H<1. In this paper, we consider the linear self-attracting diffusion: dXtH=dBtH+σXtHdt−θ∫0tXsH−XuHdsdt+νdt with X0H=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−σθHXtH−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 XH.