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Limit theorems are proved for the winding numbers of a three-dimensional Brownian motion around certain curves in space. In particular, the joint asymptotic distribution of the winding numbers around two curves is obtained. This joint distribution generalizes the asymptotic law of the winding numbers of a planar Brownian motion around two points, which has recently been given by Pitman and Yor. The limiting distributions are closely related to the time spent by a linear Brownian motion above and below a multiple of its maximum process. Proofs rely on stochastic calculus for continuous semi-martingales.

Enlacements du mouvement brownien autour des courbes de l’espace