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

TL;DR

This paper investigates the asymptotic behavior of the derivative of self-intersection local time for multidimensional fractional Brownian motion, establishing conditions for L^2 convergence and deriving three different central limit theorems based on Hurst parameter and dimension.

Contribution

It provides the first detailed analysis of the asymptotic properties and CLTs for the derivative of self-intersection local time in multidimensional fractional Brownian motion.

Findings

L^2 exit condition: H<3/(2(1+d))

Three CLTs for different H ranges and dimensions

Normalization factors depend on H and d

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.