On the linear fractional self-attracting diffusion

TL;DR

This paper introduces a new class of self-attracting diffusions driven by fractional Brownian motion with Hurst index between 1/2 and 1, analyzing their convergence, local times, and self-intersection properties.

Contribution

It extends the theory of self-attracting diffusions to fractional Brownian motion, providing new results on convergence and local time existence for different dimensions.

Findings

Convergence results for 1-dimensional fractional self-attracting diffusion.

Existence of weighted local time in 1D.

Existence of renormalized self-intersection local time in 2D for certain Hurst indices.

Abstract

In this paper, we introduce the linear fractional self-attracting diffusion driven by a fractional Brownian motion with Hurst index 1/2<H<1, which is analogous to the linear self-attracting diffusion. For 1-dimensional process we study its convergence and the corresponding weighted local time. For 2-dimensional process, as a related problem, we show that the renormalized self-intersection local time exists in L^2 if $\frac12<H<\frac3{4}$.