Large Time Behavior on the Linear Self-Interacting Diffusion Driven by Sub-Fractional Brownian Motion II: Self-Attracting Case - Guo, Rui; Gao, Han; Jin, Yang; Yan, Litan (Frontiers in Physics, 2022)
In this study, as a continuation to the studies of the self-interaction diffusion driven by subfractional Brownian motion SH, we analyze the asymptotic behavior of the linear self-attracting diffusion: dXtH=dStH−θ∫0t(XtH−XsH)dsdt+νdt,X0H=0, where θ > 0 and ν∈R are two parameters. When θ < 0, the solution of this equation is called self-repelling. Our main aim is to show the solution XH converges to a normal random variable X∞H with mean zero as t tends to infinity and obtain the speed at which the process XH converges to X∞H as t tends to infinity.