Some asymptotics related to generalized self-intersection local times of the Brownian motion

Let $\Big( B(t)\Big) _{0\leqslant t \leqslant 1}$ be the Brownian motion in $\mathbb{R}^d$. The double self-intersection local time at a point $u\in\mathbb{R}^d\setminus \{0\}$ is the local time at $u$ of the random field $X(s,t)=B(t)-B(s)$. We denote it by $\rho (u)$. Formally, it can be written as
$$ \rho(u)= \int _{\Delta_2} \delta_u \big(B(t)-B(s) \big) dsdt, $$
where $\Delta_2= \{ (s,t)\in [0,1]^2\, ;\, s<t\}. $ If $d\geqslant 4$, then $\rho (u)$ is a positive generalized Wiener function (an element from some Sobolev space constructed over the Wiener space) which can be represented as a measure $\theta _u$ on the Wiener space:
$$ W_0^d=\left\lbrace \omega :[0,1]\to \mathbb{R}^d\,\big| \omega \text{ is continuous and } \omega(0)=0\right\rbrace. $$
We provide some asymptotics for the generalized self-intersection local time $\rho (u)$, when $u$ tends to $0$, in terms of the measures $\theta _u$. The main results are related to the measure of the whole space $\theta_u(W_0^d)$ and the capacity of the support of $\theta _u$.