A scaling proof for Walsh's Brownian motion extended arc-sine law

TL;DR

This paper provides a new scaling-based proof of the extended arc-sine law for Walsh's Brownian motion, generalizing previous methods from one-dimensional Brownian motion to multivariate cases and discussing related skew Bessel processes.

Contribution

It introduces a novel proof technique for the extended arc-sine law applicable to Walsh's Brownian motion, extending prior results to multivariate settings.

Findings

Proof of the extended arc-sine law for Walsh's Brownian motion

Generalization of scaling arguments to multivariate processes

Discussion on time spent positive by skew Bessel processes

Abstract

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 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.