Central limit theorems for the derivatives of self-intersection local time for $d$-dimensional Brownian motion

TL;DR

This paper establishes central limit theorems for derivatives of self-intersection local time of d-dimensional Brownian motion, providing a complete answer to a conjecture and analyzing Wiener chaos components.

Contribution

It proves CLTs for derivatives of self-intersection local time in various dimensions and derivatives, confirming a conjecture and extending understanding of these stochastic processes.

Findings

CLTs hold for derivatives when renormalized appropriately in 2D and higher dimensions.

Complete resolution of Markowsky's conjecture from 2012.

Wiener chaos components also satisfy CLTs under suitable normalization.

Abstract

Let $\{B_t,t\geq0\}$ be a d-dimensional Brownian motion. We prove that the approximation of the higher derivative of renormalized self-intersection local time $$ \int_{0}^{1}\int_{0}^{s}\left(p^{(|k|)}_{d,\epsilon}(B_{s}-B_{r})-E[p^{(|k|)}_{d,\epsilon}(B_{s}-B_{r})]\right)drds, $$ where the multiindex $k=(k_{1},\cdots,k_{d})$, $ p_{d,\epsilon}^{(|k|)}(x_1,x_2,\cdots,x_d):=\partial^{k_1}_{x_1}\partial^{k_2}_{x_2}$ $\cdots\partial^{k_d}_{x_d}p_{d,\epsilon}(x_1,x_2,\cdots,x_d)$ and $p_{d,\epsilon}(x)=\frac{1}{(2\pi\epsilon)^{d/2}}e^{-\frac{|x|^{2}}{2\epsilon}}, x\in\mathbb{R}^d$, satisfies the central limit theorems when renormalized by $(\log\frac{1}{\epsilon})^{-1}$ in the case $d=2$, $|k|=1$ and by $\epsilon^{\frac{d+|k|-3}{2}}$ in the case $d\geq 3$, $|k|\geq 1$, which gives a complete answer to the conjecture of Markowsky [In S\'{e}minaire de Probabiliti\'{e}s \uppercase\expandafter{\romannumeral10\romannumeral50\romannumeral4} (2012) 141-148 Springer]. We as well prove that its m-th Wiener chaotic component satisfies the central limit theorems when renormalized by a multiplicative factor in different cases.