  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 $\mathfrak{g}$.
