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

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 Doobs space-time harmonic transformation of a space-time Brownian motion.