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

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