Asymptotic Properties of the Derivative of Self-Intersection Local Time of Multidimensional Fractional Brownian Motion
Jiazhen Gu, Jinchi Jiang, Qian Yu
TL;DR
The paper analyzes the first-order derivative of the self-intersection local time (DSLT) for a $d$-dimensional fractional Brownian motion with Hurst parameter $H\in(0,1)$. It introduces the regularized DSLT approximation $\alpha_{\varepsilon,t}^{(1)}(0)$ via the derivative of the heat kernel and derives sharp $L^2$ existence conditions: the limit exists in $L^2$ iff $H<\tfrac{3}{2(1+d)}$ for $d\ge3$ (with $d=2$ matching prior results). It then proves three central-limit theorems under distinct regimes: (i) for $H>\tfrac12$, $\\varepsilon^{\frac{d}{2}+1-1/H}\alpha_{\varepsilon,t}^{(1)}(0) \Rightarrow N(0,\sigma^2)$; (ii) for $\tfrac{3}{2(1+d)}<H<\tfrac12$, $\varepsilon^{\frac{d}{2}+\frac12-\frac{3}{4H}}\alpha_{\varepsilon,t}^{(1)}(0) \Rightarrow N(0,\hat{\sigma}^2)$; and (iii) at the critical $H=\tfrac{3}{2(1+d)}$, $(\log(1/\varepsilon))^{-1/2}\alpha_{\varepsilon,t}^{(1)}(0) \Rightarrow N(0,\bar{\sigma}^2)$. The proofs hinge on a detailed Wiener chaos expansion, contraction estimates, and a central-limit-theorem framework due to Hu and Nualart, supplemented by novel bounds for the covariance-structure of fBm increments. The results elucidate the delicate balance between dimension, roughness, and regularization in the asymptotics of DSLT, with explicit limit variances given by Beta-function– and integral-based expressions.
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 α_{\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)=(2π\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.