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)

MSC:

 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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