  We provide sharp Large Deviation estimates for the probability of exit from a
domain for the bridge of a $d$-dimensional general diffusion process $X$, as
the conditioning time tends to $0$. This kind of results is motivated by
applications to numerical simulation. In particular we investigate the
influence of the drift $b$ of $X$. It turns out that the sharp asymptotics for
the exit time probability are independent of the drift, provided $b$ enjoyes a
simple condition that is always satisfied in dimension $1$. On the other hand,
we show that the drift can be influential if this assumption is not satisfied.
}
