## Renormalisation et convergence en loi pour les temps locaux d’intersection du mouvement brownien dans $${\mathbb{R}}^ 3$$.(French)Zbl 0569.60075

Sémin. de probabilités XIX, Univ. Strasbourg 1983/84, Proc., Lect. Notes Math. 1123, 350-365 (1985).
[For the entire collection see Zbl 0549.00007.]
In this paper a generalization, in a modified form, of a result obtained by Varadhan for Brownian motions $$(B_ t$$, $$t\geq 0)$$ having values in $${\mathbb{R}}^ 2$$ to the case of Brownian motions having values in $${\mathbb{R}}^ 3$$ is given. More exactly, it is proved that: $(B_ t;\quad (\log | y|^{-1})^{-1/2}\{2\pi \alpha (y;T_ t)- t| y|^{-1}\};\quad t\geq 0)\to^{(d)}_{y\to 0}(B_ t;\quad 2\beta_ t,\quad t\geq 0),$ where $$(\beta_ t$$, $$t\geq 0)$$ is a real Brownian motion starting from 0, independent of B, and (d) indicates the convergence in distribution associated with the topology of compact convergence on the canonical space.