Probability Theory and Related Fields

, Volume 76, Issue 4, pp 587–627 | Cite as

Mouvement brownien, cônes et processus stables

  • Jean-François Le Gall
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Summary

LetW=(Wt, t≧0) denote a two-dimensional Brownian motion starting at 0 and, for 0<α<π, letCα be a wedge in ℝ2 with vertex 0 and angle 2α. We consider the set of timest's such that the path ofW, up to timet, stays inside the translated wedgeWt-Cα. It follows from recent results of Burdzy and Shimura that this set, which we denote byHα, contains nonzero times if, and only if, α>π/4. Here we construct a measure, a local time, supported onHα. For π/4<α≦α/2, the Brownian motionW, time-changed by the inverse of this local time, is shown to be a two-dimensional stable process with index 2-π/2α. This results extends Spitzer's construction of the Cauchy process, which is recovered by taking α=π/2. A formula which describes the behaviour ofW before a timet∈Hα is established and applied to the proof of a conjecture of Burdzy. We also obtain a two-dimensional version of the famous theorem of Lévy concerning the maximum process of linear Brownian motion. Precisely, for 0<α<π/2, letSt denote the vertex of the smallest wedge of the typez-Cα which contains the path ofW up to timet. The processSt-Wt is shown to be a reflected Brownian motion in the wedgeCα, with oblique reflection on the sides. Finally, we investigate various extensions of the previous results to Brownian motion inRd, d≧3. LetCΩ be the cone associated with an open subset Ω of the sphereSd-1, and letHΩ be defined asHα above. Sufficient conditions are given forHΩ to contain nonzero times, in terms of the first eigenvalue of the Dirichlet Laplacian on Ω.

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Copyright information

© Springer-Verlag 1987

Authors and Affiliations

  • Jean-François Le Gall
    • 1
  1. 1.Laboratoire de ProbabilitésUniversité P. et M. CurieParis Cedex 05France

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