Conditioned random walks from Kac-Moody root systems
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.
Cite
@article{arxiv.1306.3082,
title = {Conditioned random walks from Kac-Moody root systems},
author = {Cédric Lecouvey and Emmanuel Lesigne and Marc Peigné},
journal= {arXiv preprint arXiv:1306.3082},
year = {2013}
}
Comments
30 pages, minor corrections