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

Let \Bₜ, t0\ be a d-dimensional Brownian motion. We prove that the approximation of the higher derivative of renormalized self-intersection local time ₀^1₀^s (p^ (|k|) ₃, (Bₒ-Bₑ) -Ep^ (|k|) ₃, (Bₒ-Bₑ) ) drds, where the multiindex k= (k₁, , k₃), p₃, ^ (|k|) (x₁, x₂, , xd): =^k₁ₗ䃑^k₂ₗ䃒 ^kdₗ₃p₃, (x₁, x₂, , xd) and p₃, (x) =1 (2) ^{d/2}e^-|x|^{22}, xᵈ, satisfies the central limit theorems when renormalized by (1) ^-1 in the case d=2, |k|=1 and by ^d+|k|-3{2} in the case d 3, |k| 1, which gives a complete answer to the conjecture of Markowsky In S\'eminaire de Probabiliti\'es 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.