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(_0^t(X^H_t-X^H_s)ds)dt+ν dt,X^H_0=0, where θ>0 and νR 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.