Renormalized self-intersection local time for fractional Brownian motion

Let B-t(H) H be a d-dimensional fractional Brownian motion with Hurst parameter H E (0, 1). Assume d ≥ 2. We prove that the renormalized self-intersection local time L = ∫(T)(0) ∫(t)(0) δ(B-t(H) - B-s(H)) ds dt - E(∫(T)(0) ∫(t)(0) δ(B-t(H) -B-s(H)) ds dt) 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 ≥ 3/2d, r(ε)Lε converges in distribution to a normal law N(0, Tσ(2)), as E tends to zero, where Lε, is an approximation of L, defined through (2), and r(ε) = | logε|(-1) if H = 3/(2d), and r(ε) = ε(d-3/(2H)) if 3/(2d) < H.