Renormalized self-intersection local time for fractional Brownian motion

Abstract

Let B_t^H be a d-dimensional fractional Brownian motion with Hurst parameter H\in(0,1). Assume d\geq2. We prove that the renormalized self-intersection local time\ell=\int_0^T\int_0^t\delta(B_t^H-B_s^H) ds dt -E\biggl(\int_0^T\int_0^t\delta (B_t^H-B_s^H) ds dt\biggr) exists in L^2 if and only if H<3/(2d), which generalizes the Varadhan renormalization theorem to any dimension and with any Hurst parameter. Motivated by a result of Yor, we show that in the case 3/4>H\geq\frac{3}{2d}, r(\epsilon)\ell_{\epsilon} converges in distribution to a normal law N(0,T\sigma^2), as \epsilon tends to zero, where \ell_{\epsilon} is an approximation of \ell, defined through (2), and r(\epsilon)=|\log\epsilon|^{-1} if H=3/(2d), and r(\epsilon)=\epsilon^{d-3/(2H)} if 3/(2d)<H.