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

Yor, M.

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.

Reviewer: I.C.Vladimirescu

MSC:

60J65	Brownian motion
60J55	Local time and additive functionals

Keywords:

convergence in distribution

Citations:

Zbl 0549.00007