  We point out a connection between fusion coefficients and random walks in a
fixed level alcove associated to the root system of an affine Lie algebra and
use this connection to solve completely the Dirichlet problem on such an alcove
for a large class of simple random walks. We establish a correspondence between
the hypergroup of conjugacy classes of a compact Lie group and the fusion
hypergroup. We prove that a random walk in an alcove, obtained with the help of
fusion coefficients, converges, after a proper normalization, towards the
radial part of a Brownian motion on a compact Lie group.
