Asymptotic properties of the derivative of self-intersection local time of fractional Brownian motion

TL;DR

This paper investigates the asymptotic behavior of the derivative of self-intersection local time for fractional Brownian motion with Hurst parameter between 2/3 and 1, establishing central limit theorems and convergence results for its approximations.

Contribution

It provides new limit theorems and convergence results for the derivative of self-intersection local time of fractional Brownian motion, extending understanding of its asymptotic properties.

Findings

Central limit theorem for the approximation of the derivative of self-intersection local time.

Convergence of the $q$-th chaotic component in $L^2$ for certain Hurst parameters.

Different renormalization factors depending on the Hurst parameter range.

Abstract

Let $\{B_{t}\}_{t\geq0}$ be a fractional Brownian motion with Hurst parameter $\frac{2}{3}<H<1$. We prove that the approximation of the derivative of self-intersection local time, defined as \begin{align*} \alpha_{\varepsilon} &= \int_{0}^{T}\int_{0}^{t}p'_{\varepsilon}(B_{t}-B_{s})\text{d}s\text{d}t, \end{align*} where $p_\varepsilon(x)$ is the heat kernel, satisfies a central limit theorem when renormalized by $\varepsilon^{\frac{3}{2}-\frac{1}{H}}$. We prove as well that for $q\geq2$, the $q$-th chaotic component of $\alpha_{\varepsilon}$ converges in $L^{2}$ when $\frac{2}{3}<H<\frac{3}{4}$, and satisfies a central limit theorem when renormalized by a multiplicative factor $\varepsilon^{1-\frac{3}{4H}}$ in the case $\frac{3}{4}<H<\frac{4q-3}{4q-2}$.