  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 $\frac12<H<\frac3{4}$.
