Brownian motion, reflection groups and Tanaka formula

TL;DR

This paper extends Tanaka's formula to multidimensional Brownian motions within finite reflection groups, showing that projections onto Weyl chambers behave as reflected Brownian motions with a probabilistic proof.

Contribution

It provides a probabilistic proof that projections of Brownian motion onto Weyl chambers are reflected Brownian motions, extending Tanaka's formula to multiple dimensions.

Findings

Projection of Brownian motion onto Weyl chambers is a reflected Brownian motion.

The decomposition generalizes Tanaka's formula to higher dimensions.

Boundary processes are described via local times at zero of distances to facets.

Abstract

In the setting of finite reflection groups, we prove that the projection of a Brownian motion onto a closed Weyl chamber is another Brownian motion normally reflected on the walls of the chamber. Our proof is probabilistic and the decomposition we obtain may be seen as a multidimensional extension of Tanaka's formula for linear Brownian motion. The paper is closed with a description of the boundary process through the local times at zero of the distances from the initial process to the facets.