Summary
The existence of a joint asymptotic distribution for the windings of a three-dimensional Brownian motion around a finite number of straight lines is obtained. This complements the recent studies, by Pitman- Yor, and the authors, of the joint asymptotic distribution for the windings of planar Brownian motion around a finite number of points.
The following principle governs the passage from results in the plane to results in space:
Let B be a three-dimensional Brownian motion, and P 1, ..., P k, k planes which intersect two by two. Then, the convergences in distribution concerning the planar Brownian motions B i (1≦i≦k), defined respectively as the orthogonal projections of B on P i (1≦i≦k), take place jointly, and the corresponding limit variables are independent.
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Le Gall, J.F., Yor, M. Etude asymptotique des enlacements du mouvement brownien autour des droites de l'espace. Probab. Th. Rel. Fields 74, 617–635 (1987). https://doi.org/10.1007/BF00363519
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DOI: https://doi.org/10.1007/BF00363519