×

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
PDF BibTeX XML Cite
Full Text: DOI