Harmonic functions on multiplicative graphs and inverse Pitman transform on infinite random paths

Cédric Lecouvey (1) , Emmanuel Lesigne (1) , Marc Peigné (1)

1 LMPT - Laboratoire de Mathématiques et Physique Théorique

We introduce and characterize central probability distributions on Littelmann paths. Next we establish a law of large numbers and a central limit theorem for the generalized Pitmann transform. We then study harmonic functions on multiplicative graphs defined from the tensor powers of finite-dimensional Lie algebras representations. Finally, we show there exists an inverse of the generalized Pitman transform defined almost surely on the set of infinite paths remaining in the Weyl chamber and explain how it can be computed.

Keywords: Random walks; Littelmann paths; Pitman transform; representations; law of large numbers

Domains: Combinatorics [math.CO]; Probability [math.PR]; Representation Theory [math.RT]

Cédric Lecouvey, Emmanuel Lesigne, Marc Peigné. Harmonic functions on multiplicative graphs and inverse Pitman transform on infinite random paths. Annales de la Faculté des Sciences de Toulouse. Mathématiques., 2018, 27 (3), pp.629-666. ⟨10.5802/afst.1580⟩. ⟨hal-01061664v3⟩