Renormalized self-intersection local time for fractional Brownian motion

Details Renormalized self-intersection local time for fractional Brownian motion Renormalized self-intersection local time for fractional Brownian motion

2005-06-29

arxiv.org/abs/math/0506592v1

Annals of Probability 2005, Vol. 33, No. 3, 948-983

Mathematics

Probability

Published at http://dx.doi.org/10.1214/009117905000000017 in the Annals of Probability (http://www.imstat.org/aop/) by the Ins

Scientific paper

10.1214/009117905000000017

Let B_t^H be a d-dimensional fractional Brownian motion with Hurst parameter H\in(0,1). Assume d\geq2. 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 -E\biggl(\int_0^T\int_0^t\delta (B_t^H-B_s^H) ds dt\biggr) 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\geq\frac{3}{2d}, r(\epsilon)\ell_{\epsilon} converges in distribution to a normal law N(0,T\sigma^2), as \epsilon tends to zero, where \ell_{\epsilon} is an approximation of \ell, defined through (2), and r(\epsilon)=|\log\epsilon|^{-1} if H=3/(2d), and r(\epsilon)=\epsilon^{d-3/(2H)} if 3/(2d)

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