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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.
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https://hal.archives-ouvertes.fr/hal-00833657
Contributor : Cédric Lecouvey Connect in order to contact the contributor
Submitted on : Sunday, December 22, 2013 - 9:20:30 AM
Last modification on : Tuesday, January 11, 2022 - 5:56:07 PM
Long-term archiving on: : Saturday, March 22, 2014 - 10:35:28 PM

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  • HAL Id : hal-00833657, version 3
  • ARXIV : 1306.3082

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Cédric Lecouvey, Emmanuel Lesigne, Marc Peigné. Conditioned random walks from Kac-Moody root systems. Transactions of the American Mathematical Society, American Mathematical Society, 2016. ⟨hal-00833657v3⟩

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