Renormalized self-intersection local time for fractional Brownian motion

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Let Bₜʰ be a d-dimensional fractional Brownian motion with Hurst parameter H∈(0,1). Assume d≥2. We prove that the renormalized self-intersection local timeℓ=∫₀ᵗ∫₀ᵗδ(Bₜʰ-Bₛʰ) ds dt -E(∫₀ᵗ∫₀ᵗδ(Bₜʰ-Bₛʰ) ds dt) 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≥3/2d, r(ε)ℓε converges in distribution to a normal law N(0,Tσ²), as εtends to zero, where ℓε is an approximation of ℓ, defined through (2), and r(ε)=|ε|⁻¹ if H=3/(2d), and r(ε)=εᵈ⁻³/⁽²ʰ⁾ if 3/(2d)<H.