  Let $n$ particles move in standard Brownian motion in one dimension, with the
process terminating if two particles collide. This is a specific case of
Brownian motion constrained to stay inside a Weyl chamber; the Weyl group for
this chamber is $A_{n-1}$, the symmetric group. For any starting positions, we
compute a determinant formula for the density function for the particles to be
at specified positions at time $t$ without having collided by time $t$. We show
that the probability that there will be no collision up to time $t$ is
asymptotic to a constant multiple of $t^{-n(n-1)/4}$ as $t$ goes to infinity,
and compute the constant as a polynomial of the starting positions. We have
analogous results for the other classical Weyl groups; for example, the
hyperoctahedral group $B_n$ gives a model of $n$ independent particles with a
wall at $x=0$.
  We can define Brownian motion on a Lie algebra, viewing it as a vector space;
the eigenvalues of a point in the Lie algebra correspond to a point in the Weyl
chamber, giving a Brownian motion conditioned never to exit the chamber. If
there are $m$ roots in $n$ dimensions, this shows that the radial part of the
conditioned process is the same as the $n+2m$-dimensional Bessel process. The
conditioned process also gives physical models, generalizing Dyson's model for
$A_{n-1}$ corresponding to ${\mathfrak s}{\mathfrak u}_n$ of $n$ particles
moving in a diffusion with a repelling force between two particles proportional
to the inverse of the distance between them.
