Asymptotic Properties of the Derivative of Self-Intersection Local Time of Multidimensional Fractional Brownian Motion

Abstract

Let \{B_t^H,t\geq0\} be a d-dimensional fractional Brownian motion. We prove that the approximation of the first-order derivative of self-intersection local time, defined as \alpha_{\varepsilon,t}^{(1)}(0)=-\int_0^t\int_0^sp_\varepsilon^{(1)}(B_s^H-B_r^H)\d r\d s, where p_\varepsilon^{(1)}(x_1,\cdots,x_d):=\partial _{x_1}p(x_1,\cdots,x_d) and p_\varepsilon(x)=(2\pi\varepsilon)^{-d/2}e^{|x|^2/2\varepsilon},x\in\mathbb{R}^d, d\geq2 is the heat kernel, exits in L^2 sense if and only if H<\frac{3}{2(1+d)} and satisfies three different central limit theorems when normalized by \varepsilon^{\frac d2+1-\frac1H} for H>\frac12 and d\geq2, normalized by \varepsilon^{\frac d2+\frac12-\frac 3{4H}} for \frac{3}{2(1+d)}<H<\frac12 and d\geq3, and normalized by \log(1/\varepsilon)^{-\frac12} for the critical case H=\frac{3}{2(1+d)} and d\geq3.