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

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.