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

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 $$ \int_{0}^{1}\int_{0}^{s}\left(p^{(|k|)}_{d,ε}(B_{s}-B_{r})-E[p^{(|k|)}_{d,ε}(B_{s}-B_{r})]\right)drds, $$ where the multiindex $k=(k_{1},\cdots,k_{d})$, $ p_{d,ε}^{(|k|)}(x_1,x_2,\cdots,x_d):=\partial^{k_1}_{x_1}\partial^{k_2}_{x_2}$ $\cdots\partial^{k_d}_{x_d}p_{d,ε}(x_1,x_2,\cdots,x_d)$ and $p_{d,ε}(x)=\frac{1}{(2πε)^{d/2}}e^{-\frac{|x|^{2}}{2ε}}, x\in\mathbb{R}^d$, satisfies the central limit theorems when renormalized by $(\log\frac{1}ε)^{-1}$ in the case $d=2$, $|k|=1$ and by $ε^{\frac{d+|k|-3}{2}}$ in the case $d\geq 3$, $|k|\geq 1$, which gives a complete answer to the conjecture of Markowsky [In Séminaire de Probabilitiés \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.

Theorems & Definitions (37)

• Theorem 1.1
• Remark 1.2
• Theorem 1.3
• Theorem 1.4
• Lemma 2.1
• proof
• Proposition 3.1
• proof
• Proposition 3.2
• proof