  We obtain a formula for the distribution of the first exit time of Brownian
motion from the alcove of an affine Weyl group. In most cases the formula is
expressed compactly, in terms of Pfaffians. Expected exit times are derived in
the type \~A case. The results extend to other Markov processes. We also give
formulas for the real eigenfunctions of the Dirichlet and Neumann Laplacians on
alcoves, observing that the `Hot Spots' conjecture of J. Rauch is true for
alcoves.
