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

Brassesco, Stella

Ann. Probab. 20, No. 3, 1498-1503 (1992).

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

Keywords:

exit time from a cone; planar Brownian motion; heat equation