Conditioned random walks from Kac-Moody root systems

Abstract:

Random paths are time continuous interpolations of random walks. By using the Littelmann path model, we associate to each irreducible highest weight module of a Kac Moody algebra ${\mathfrak {g}}$ a random path ${\mathcal {W}}.$ Under suitable hypotheses, we make explicit the probability of the event $E$: “${\mathcal {W}}$ never exits the Weyl chamber of ${\mathfrak {g}}$.” We then give the law of the random walk defined by ${\mathcal {W}}$ conditioned by the event $E$ and prove this law can be recovered by applying to ${\mathcal {W}}$ a path transform of Pitman type. This generalizes the main results of Neil O’Connell (2003) and the authors (2012) to Kac Moody root systems and arbitrary highest weight modules. Our approach here is new and more algebraic than in the aforementioned works. We indeed fully exploit the symmetry of our construction under the action of the Weyl group of ${\mathfrak {g}}$ which permits us to avoid delicate generalizations of our previous results on renewal theory.