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Some properties of Brownian motion in a cone. (Quelques propriétés du mouvement Brownien dans un cone.) (English) Zbl 0812.60067

Consider an open convex cone \(C\) with vertex at 0 and Euclidean boundary \(\partial C\), suppose that \(\partial C\) is Lipschitz. The Martin boundary of \(C\) is then \(\partial_ M C = \partial C \cup \{\infty\}\); let \(h\) be the unique (up to a constant factor) minimal harmonic function corresponding to \(\infty\). The Doob’s \(h\)-transform leads to a process \(X\) that can be thought of as a Brownian motion in \(C\) conditioned to avoid \(\partial C\). The author studies the hitting probabilities of cones for \(X\). The most striking result concerns the special case where \(C = \{z \in \mathbb{C} : | \text{Arg} z | < \pi/6\}\). If \(X\) starts from 0 and if \(J\) stands for the future infimum process of \(X\) for the natural order in \(C\), then \({\mathfrak F} (3J-X)\) is a real-valued Brownian motion. This should be compared with the well-known theorem of J. W. Pitman [Adv. Appl. Probab. 7, 511-526 (1975; Zbl 0332.60055)]: if \(R\) is a three-dimensional Bessel process (that is, informally, a one-dimensional Brownian motion conditioned to stay positive) and if \(J\) now stands for the future infimum process of \(R\), then \(2J-R\) is a standard Brownian motion.
Reviewer: J.Bertoin (Paris)

MSC:

60J65 Brownian motion

Citations:

Zbl 0332.60055
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References:

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