  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.
