Renormalized self-intersection local time for fractional Brownian motion

Abstract

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 time l= ∫0T ∫0t δ(BtH - B sH) ds dt - E(∫0T ∫0t δ(BtH - B sH) 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 ε tends to zero, where lε is an approximation of l, defined through (2), and r(ε) = |logeε|-1 if H = 3/(2d), and r(ε) = ε d-3/(2H) if 3/(2d) < H. © Institute of Mathematical Statistics, 2005.