A note on planar Brownian motion. (English) Zbl 0758.60083

The author calculates the joined density of the total winding \(\theta_ t\) and the radius \(R_ t\) of the planar Brownian motion \(Z_ t\) by a standard method, that is to say by solving explicitly the heat equation, and without using for example the calculation by Yor (1980) of the conditional density of \(\theta_ t\) knowing \(R_ t\). From this density she then deduces an expression for \(\mathbb{P}(T>t\), \(R_ t\in dr\), \(\theta_ t\in d\theta)\), where \(T\) is the exit time of \(Z\) from a cone \(\{0<\theta<\beta\}\), and an equivalent as \(t\to\infty\) of \(\mathbb{P}(T>t)\).
Reviewer: J.Franchi (Paris)


60J65 Brownian motion
60J35 Transition functions, generators and resolvents
60F05 Central limit and other weak theorems
58J65 Diffusion processes and stochastic analysis on manifolds
35K05 Heat equation
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