  We use Kashiwara crystal basis theory to associate a random walk W to each
irreducible representation V of a simple Lie algebra. This is achieved by
endowing the crystal attached to V with a (possibly non uniform) probability
distribution compatible with its weight graduation. We then prove that the
generalized Pitmann transform defined by Biane, Bougerol and O'Connell for
similar random walks with uniform distributions yields yet a Markov chain. When
the representation is minuscule, and the associated random walk has a drift in
the Weyl chamber, we establish that this Markov chain has the same law as W
conditionned to never exit the cone of dominant weights. At the heart of our
proof is a quotient version of a renewal theorem that we state in the context
of general random walks in a lattice.
