Mouvement brownien, cônes et processus stables. (Brownian motion, cones and stable processes)
Let \(W=(W_ t,t\geq 0)\) denote a two-dimensional Brownian motion starting at 0 and, for \(0<\alpha <\pi\), let \(C_{\alpha}\) be a wedge in \({\mathbb{R}}^ 2\) with vertex 0 and angle \(2\alpha\). We consider the set of times t such that the path of W, up to time t, stays inside the translated wedge \(W_ t-C_{\alpha}\). It follows from recent results of \textit{K. Burdzy} [Ann. Probab. 13, 1006-1010 (1985; Zbl 0574.60053)] and \textit{M. Shimura} [J. Math. Kyoto Univ. 25, 433-443 (1985; Zbl 0582.60048)] that this set, which we denote by \(H_{\alpha}\), contains nonzero times if, and only if, \(\alpha >\pi /4\). Here we construct a measure, a local time, supported on \(H_{\alpha}\). For \(\pi /4<\alpha \leq \pi /2\), the Brownian motion W, time-changed by the inverse of this local time, is shown to be a two-dimensional stable process with index 2- \(\pi\) /2\(\alpha\). This result extends Spitzer's construction of the Cauchy process, which is recovered by taking \(\alpha =\pi /2.\) A formula which describes the behaviour of W before a time \(t\in H_{\alpha}\) is established and applied to the proof of a conjecture of Burdzy stated in the quoted paper. We also obtain a two-dimensional version of the famous theorem of Lévy concerning the maximum process of linear Brownian motion. Precisely, for \(0<\alpha <\pi /2\), let \(S_ t\) denote the vertex of the smallest wedge of the type \(z-C_{\alpha}\) which contains the path of W up to time t. The process \(S_ t-W_ t\) is shown to be a reflected Brownian motion in the wedge \(C_{\alpha}\), with oblique reflection on the sides. Finally, we investigate various extensions of the previous results to Brownian motion in \({\mathbb{R}}^ d\), \(d\geq 3\). Let \(C_{\Omega}\) be the cone associated with an open subset \(\Omega\) of the sphere \(S^{d-1}\), and let \(H_{\Omega}\) be defined as \(H_{\alpha}\) above. Sufficient conditions are given for \(H_{\alpha}\) to contain nonzero times, in terms of the first eigenvalue of the Dirichlet Laplacian on \(\Omega\).
Spitzer's construction of the Cauchy process
maximum process of linear Brownian motion
scientific article; zbMATH DE number 3988414
scientific article