The affine Lie algebra $\hat{\mathfrak{sl}_2}(\C)$ and a conditioned space-time Brownian motion

TL;DR

This paper constructs a sequence of Markov processes based on affine Lie algebra representations and demonstrates their convergence to a conditioned space-time Brownian motion, linking algebraic structures with stochastic processes.

Contribution

It introduces a novel Markov process on affine Lie algebra weights and proves its convergence to a space-time Brownian motion conditioned by harmonic transformation.

Findings

Convergence of Markov processes to conditioned Brownian motion

Connection between affine Lie algebra representations and stochastic processes

New method for constructing space-time Brownian motions

Abstract

We construct a sequence of Markov processes on the set of dominant weights of the Affine Lie algebra $\hat{\mathfrak{sl}_2}(\C)$ which involves tensor product of irreducible highest weight modules of $\hat{\mathfrak{sl}_2}(\C)$ and show that it converges towards a Doob's space-time harmonic transformation of a space-time Brownian motion.