  We compute analytically the probability $S(t)$ that a set of $N$ Brownian
paths do not cross each other and stay below a moving boundary $g(\tau)= W
\sqrt{\tau}$ up to time $t$. We show that for large $t$ it decays as a power
law $S(t) \sim t^{- \beta(N,W)}$. The decay exponent $\beta(N,W)$ is obtained
as the ground state energy of a quantum system of $N$ non-interacting fermions
in a harmonic well in the presence of an infinite hard wall at position $W$.
Explicit expressions for $\beta(N,W)$ are obtained in various limits of $N$ and
$W$, in particular for large $N$ and large $W$. We obtain the joint
distribution of the positions of the walkers in the presence of the moving
barrier $g(\tau) =W \sqrt{\tau}$ at large time. We extend our results to the
case of $N$ Dyson Brownian motions (corresponding to the Gaussian Unitary
Ensemble) in the presence of the same moving boundary $g(\tau)=W\sqrt{\tau}$.
For $W=0$ we show that the system provides a realization of a Laguerre
biorthogonal ensemble in random matrix theory. We obtain explicitly the average
density near the barrier, as well as in the bulk far away from the barrier.
Finally we apply our results to $N$ non-crossing Brownian bridges on the
interval $[0,T]$ under a time-dependent barrier $g_B(\tau)= W \sqrt{\tau(1-
\frac{\tau}{T})}$.
