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 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 $$ dX^H_t=dS^H_t-θ\left(\int_0^t(X^H_t-X^H_s)ds\right)dt+ν dt,\quad X^H_0=0, $$ where $θ>0$ and $ν\in {\mathbb R}$ are two parameters. When $θ<0$ 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 $X^H_∞$ with mean zero, as $t$ tends to infinity, and obtain the speed at which the process $X^H$ converges to $X^H_∞$, as $t$ tends to infinity.