AUTHOR=Guo Rui , Gao Han , Jin Yang , Yan Litan 

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

JOURNAL=Frontiers in Physics

VOLUME=Volume 9 - 2021

YEAR=2022

URL=https://www.frontiersin.org/journals/physics/articles/10.3389/fphy.2021.791858

DOI=10.3389/fphy.2021.791858

ISSN=2296-424X

ABSTRACT=In this paper, as a continuation to the studies of the self-interaction diffusion driven by a sub-fractional Brownian motion $S^H$, we analyze the asymptotic behaviour of the linear self-attracting diffusion
$$
dX^H_t=dS^H_t-\theta\left(\int_0^t(X^H_t-X^H_s)ds\right)dt+\nu dt,\quad X^H_0=0,
$$
where $\theta>0$ and $\nu\in {\mathbb R}$ 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 variable $X^H_\infty$ with mean zero, as $t$ tends to infinity, and obtain the speed at which the process $X^H$ converges to $X^H_\infty$, as $t$ tends to infinity.