Affine Lie algebras and a conditioned space-time Brownian motion in an affine Weyl chamber

TL;DR

This paper constructs a Markov process based on affine Lie algebra representations and shows its convergence to a conditioned space-time Brownian motion within an affine Weyl chamber.

Contribution

It introduces a novel Markov process derived from affine Lie algebra modules and proves its convergence to a conditioned Brownian motion.

Findings

Convergence of the Markov process to a conditioned Brownian motion.

Explicit construction of the process using affine Lie algebra characters.

Insight into the interplay between algebraic structures and stochastic processes.

Abstract

We construct a sequence of Markov processes on the set of dominant weights of an affine Lie algebra $\mathfrak{g}$ considering tensor product of irreducible highest weight modules of $\mathfrak{g}$ and specializations of the characters involving the Weyl vector $\rho$. We show that it converges towards a space-time Brownian motion with a drift, conditioned to remain in a Weyl chamber associated to the root system of $\mathfrak{g}$.