Conditioned random walks from Kac-Moody root systems

TL;DR

This paper extends the study of conditioned random walks associated with Kac-Moody algebras using the Littelmann path model, providing explicit probabilities and laws through a novel symmetry-based approach.

Contribution

It introduces a new method leveraging Weyl group symmetry to analyze conditioned random paths in Kac-Moody root systems, generalizing previous results.

Findings

Explicit probability of paths never exiting the Weyl chamber

Law of conditioned random walk derived via generalized Pitmann transform

New symmetry-based approach replaces traditional renewal theory methods

Abstract

Random paths are time continuous interpolations of random walks. By using Littelmann path model, we associate to each irreducible highest weight module of a Kac Moody algebra g a random path W. Under suitable hypotheses, we make explicit the probability of the event E: W never exits the Weyl chamber of g. We then give the law of the random walk defined by W conditioned by the event E and proves this law can be recovered by applying to W the generalized Pitmann transform introduced by Biane, Bougerol and O'Connell. This generalizes the main results of [10] and [16] to Kac Moody root systems and arbitrary highest weight modules. Moreover, we use here a completely new approach by exploiting the symmetry of our construction under the action of the Weyl group of g rather than renewal theory and Doob's theorem on Martin kernels.