Large Time Behavior on the Linear Self-Interacting Diffusion Driven by Sub-Fractional Brownian Motion II: Self-Attracting Case

In this study, as a continuation to the studies of the self-interaction diffusion driven by subfractional Brownian motion S ( H ), we analyze the asymptotic behavior of the linear self-attracting diffusion: d X t H = d S t H - theta integral 0 t ( X t H - X s H ) d s d t + nu d t , X 0 H = 0 ,where theta > 0 and nu & ISIN; R are two parameters. When theta < 0, the solution of this equation is called self-repelling. Our main aim is to show the solution X ( H ) converges to a normal random variable X & INFIN; H with mean zero as t tends to infinity and obtain the speed at which the process X ( H ) converges to X & INFIN; H as t tends to infinity.