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∈(0,1). Assume d≥2. We prove that the renormalized self-intersection local time $$\ell=\int_{0}^{T}\int_{0}^{t}\delta(B_{t}^{H}-B_{s}^{H})\,ds\,dt-\mathbb{E}\biggl(\int_{0}^{T}\int_{0}^{t}\delta (B_{t}^{H}-B_{s}^{H})\,ds\,dt\biggr)$$ exists in L² 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(ɛ)ℓ_ɛ converges in distribution to a normal law N(0,Tσ²), as ɛ tends to zero, where ℓ_ɛ is an approximation of ℓ, defined through (2), and r(ɛ)=|logɛ|⁻¹ if H=3/(2d), and r(ɛ)=ɛ^d−3/(2H) if 3/(2d)<H.