Brownian sheet and time inversion -- From $G$-orbit to $L(G)$-orbit

TL;DR

This paper provides a concise proof that a conditioned space-time Brownian motion in a Weyl chamber corresponds to the radial part of a Brownian sheet on a Lie algebra, using time inversion and stochastic calculus.

Contribution

It offers a simplified proof of a previous result linking Brownian motion conditioned in Weyl chambers to Brownian sheets via time inversion.

Findings

Established the distributional equivalence using a new proof technique

Connected the radial part process to the coadjoint action of loop groups

Simplified the understanding of Brownian motion in affine Kac-Moody settings

Abstract

We have proved in a previous paper that a space-time Brownian motion conditioned to remain in a Weyl chamber associated to an affine Kac-Moody Lie algebra is distributed as the radial part process of a Brownian sheet on the compact real form of the underlying finite dimensional Lie algebra, the radial part being defined considering the coadjoint action of a loop group on the dual of a centrally extended loop algebra. We present here a very brief proof of this result based on a time inversion argument and on elementary stochastic differential calculus.