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

Let $\{B_t,t\ge 0\}$ be a d-dimensional Brownian motion. We prove that the approximation of the higher derivative of renormalized self-intersection local time $${\int }_0^1{\int }_0^s(p_{d,\epsilon }^{(\mathrm{\mid }k\mathrm{\mid })}(B_s-B_r)-E[p_{d,\epsilon }^{(\mathrm{\mid }k\mathrm{\mid })}(B_s-B_r)])drds,$$ where the multiindex $k=(k_1,\cdots ,k_d)$, $p_{d,\epsilon }^{(\mathrm{\mid }k\mathrm{\mid })}(x_1,x_2,\cdots ,x_d):={\mathrm{\partial }}_{x_1}^{k_1}{\mathrm{\partial }}_{x_2}^{k_2}$ $\cdots {\mathrm{\partial }}_{x_d}^{k_d}{p}_{d,\epsilon }(x_1,x_2,\cdots ,x_d)$ and $p_{d,\epsilon }(x)=\frac{1}{(2\pi \epsilon {)}^{d\mathrm{/}2}}{e}^{-\frac{\mathrm{\mid }x{\mathrm{\mid }}^2}{2\epsilon }},x\in {\mathbb{R}}^d$, satisfies the central limit theorems when renormalized by $(\log \frac{1}{\epsilon }{)}^{-1}$ in the case $d=2$, $\mathrm{\mid }k\mathrm{\mid }=1$ and by ${\epsilon }^{\frac{d+\mathrm{\mid }k\mathrm{\mid }-3}{2}}$ in the case $d\ge 3$, $\mathrm{\mid }k\mathrm{\mid }\ge 1$, 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.