Self-intersection local time of fractional Brownian motions-via chaos expansion

Abstract

Let $B_{1,t}^{H},\ldots ,B_{d,t}^{H}$ be $d$ independent fractional Brownian motions with Hurst parameter $H\in (0, 1)$. Denote $X_{t}=(B_{1,t}^{H},\ldots ,B_{d,t}^{H})$ and let $\delta$ be the Dirac delta function. It is shown that when $H< \mathrm{min}(3/(2d), 2/(d+2))$, the (renormalized) self-intersection local time of fractional Brownian motion, $\int _{0}^{T}\int _{0}^{t}\delta (X_{t}-X_{s})dsdt-\mathbb{E}\int _{0}^{T}\int _{0}^{t}\delta (X_{t}-X_{s})dsdt$, is in $D_{1,2}$, where $D_{1,2}$ is the Meyer-Watanabe test functional space, i.e. the $L^{2}$ space of “differentiable” functionals, whose precise meaning is given in Section 2.