  In this article we establish some estimates related to the Gaussian densities
and to Hermite polynomials in order to obtain an almost sure estimate for each
term of the It\^{o}-Wiener expansion of the self-intersection local times of
the Brownian motion. In dimension $d\geqslant 4$ the self-intersection local
times of the Brownian motion can be considered as a family of measures on the
classical Wiener space. We provide some asymptotics relative to these measures.
Finally, we try to estimate the quadratic Wasserstein distance between these
measures and the Wiener measure.
