The Linear Self-Attracting Diffusion Driven by Weighted-Fractional Brownian Motion I: Large Time Behaviors

Abstract

Let $B^{a,b}$ be a weighted-fractional Brownian motion with Hurst indexes a and b such that $a>-1$ and $0<b<1\wedge (1+a)$. In this paper, we consider the linear self-attracting diffusion

$ dX^{a,b}_t=dB^{a,b}_t-\theta \left( \int _0^t(X^{a,b}_t-X^{a,b}_s)ds-\nu \right) dt $

with $X^{a,b}_0=0$, where $\theta >0$, $\nu \in {{\mathbb {R}}}$ are two real parameters. The model is an analogue of the linear self-attracting diffusion (Cranston and Le Jan in Math Ann 303:87–93, 1995). We study large time behavior of the model. For this seemingly trivial generalization, we find that the large time behavior is much more complex than that of fractional Brownian motion, which can not be observed in the model driven by fractional Brownian motion.

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Keywords

Weighted-fractional Brownian motion / Self-attracting diffusion / The law of large numbers / Asymptotic distribution / 60G22 / 60H07 / 60F05 / 62M09

Funding

National Natural Science Foundation of China(11971101)

RIGHTS & PERMISSIONS

School of Mathematical Sciences, University of Science and Technology of China and Springer-Verlag GmbH Germany, part of Springer Nature