Asymptotic behavior of a weighted self-repelling diffusion driven by fractional Brownian motion

Abstract

Let ${B}^{H}$ be a fractional Brownian motion with Hurst index $\frac{1}{2}\le H＜1$. In this paper, we consider the self-repelling diffusion ${X}_{t}^{H}={B}_{t}^{H}+\underset{0}{\overset{t}{{\displaystyle \int }}}\underset{0}{\overset{s}{{\displaystyle \int }}}g(u)\text{d}{X}_{u}^{H}\text{d}s+\nu t$,where $\nu \in ℝ$, and $g$ is a nonnegative Borel function. The process is an analogue of linear self-interacting diffusion (M. Cranston and Y. Le Jan Math. Ann. 303 (1995), 87-93.). Based on the asymptotic behavior of the weight function $g$ at infinity, we establish the large time behavior of the recursive convergence of the solution ${X}^{H}$. For example, when $g\in {C}^{\infty }({ℝ}_{+})$ and $0<g(t) \rightarrow+\infty(t \rightarrow+\infty)$, we demonstrate that there is a sequence $\left\{{\lambda }_{n}\right\}$ of positive real numbers such that $J_{t}^{H}(0 ; g):=g(t) e^{-G(t)} X_{t}^{H} \rightarrow \nu+\xi_{\infty}^{H}$ and $J_{t}^{H}(n ; g):=G(t)\left(J_{t}^{H}(n-1 ; g)-\lambda_{n-1}\left(\xi_{\infty}^{H}+\nu\right)\right) \longrightarrow \lambda_{n}\left(\xi_{\infty}^{H}+\nu\right) \quad(t \rightarrow+\infty)$ in ${L}^{2}$ and almost surely for every $n\in \{1,2,\dots \}$, where $G(t)=\underset{0}{\overset{t}{{\displaystyle \int }}}g(s)\text{d}s$ and $\xi_{\infty}^{H}:=\int_{0}^{\infty} g(r) e^{-G(r)} \mathrm{d} B_{r}^{H}$.

Keywords

Fractional brownian motion / Self-repelling diffusion / Malliavin calculus / Recursive convergence

Acknowledgements

The authors would like to thank the editor and the anonymous referee for their valuable suggestions and comments. The research is supported by the National Natural Science Foundation of China (Grant No. 11971101).

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