Brownian paths in an alcove and the Littelmann path model

TL;DR

This paper explores the relationship between Littelmann paths and Brownian motion within affine Lie algebra frameworks, aiming to develop a Pitman-type theorem for Brownian motion in alcoves associated with affine Weyl groups.

Contribution

It introduces initial results linking Littelmann paths and Brownian paths in affine Lie algebra settings, proposing a foundation for future probabilistic theorems.

Findings

Establishes connections between Littelmann paths and Brownian motion in affine Lie algebra contexts

Proposes a framework for a Pitman-type theorem in this setting

Lays groundwork for further research in probabilistic models related to affine Weyl groups

Abstract

We present some results about connections between Littelmann paths and Brownian paths in the framework of affine Lie algebras. We expect that they will be the first steps on a way which could hopefully lead to a Pitman type theorem for a Brownian motion in an alcove associated to an affine Weyl group.