A scaling proof for Walsh’s Brownian motion extended arc-sine law. (English) Zbl 1323.60115

Vakeroudis, Stavros; Yor, Marc

Electron. Commun. Probab. 17, Paper No. 63, 9 p. (2012).

Summary: We present a new proof of the extended arc-sine law related to Walsh’s Brownian motion, known also as Brownian spider. The main argument mimics the scaling property used previously, in particular by D. Williams [Bull. Am. Math. Soc. 75, 1035–1036 (1969; Zbl 0266.60060)], in the 1-dimensional Brownian case, which can be generalized to the multivariate case. A discussion concerning the time spent positive by a skew Bessel process is also presented.

MSC:

60J65	Brownian motion
60J60	Diffusion processes
60G52	Stable stochastic processes
60J70	Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.)

Keywords:

arc-sine law; Walsh Brownian motion; skew Bessel process; stable variables; subordinators

Citations:

Zbl 0266.60060