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

Let ₜ,t≥0\ be a d-dimensional Brownian motion. We prove that the approximation of the higher derivative of renormalized self-intersection local time ∫₀¹∫₀ˢ(p(|k|)d,ε(Bₛ-Bᵣ)-E[p(|k|)d,ε(Bₛ-Bᵣ)])drds, where the multiindex k=(k₁,⋯,kd), pd,ε(|k|)(x₁,x₂,⋯,xd):=∂k₁x₁∂k₂x₂ ⋯∂kdxdpd,ε(x₁,x₂,⋯,xd) and pd,ε(x)=1(2πε)d/2e^-|x|²2ε, xᵈ, satisfies the central limit theorems when renormalized by (log1/ε)⁻¹ in the case $d=2$, $|k|=1$ and by εd+|k|-3/2 in the case d≥ 3, |k|≥ 1, which gives a complete answer to the conjecture of Markowsky [In S\'{e}minaire de Probabiliti\'{e}s {10504} (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.