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

TL;DR

This paper studies harmonic functions on multiplicative graphs, introduces central distributions on Littelmann paths, and establishes probabilistic limit theorems for the inverse Pitman transform on infinite paths within Lie algebra frameworks.

Contribution

It characterizes central distributions on Littelmann paths and constructs an almost sure inverse of the generalized Pitman transform for infinite paths in the Weyl chamber.

Findings

Law of large numbers for the generalized Pitman transform

Central limit theorem for the generalized Pitman transform

Existence and computability of the inverse transform on infinite paths

Abstract

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.