  The paper deals with the asymptotic behavior of the bridge of a Gaussian
process conditioned to stay in $n$ fixed points at $n$ fixed past instants. In
particular, functional large deviation results are stated for small time.
Several examples are considered: integrated or not fractional Brownian motion,
$m$-fold integrated Brownian motion. As an application, the asymptotic behavior
of the exit probability is studied and used for the practical purpose of the
numerical computation, via Monte Carlo methods, of the hitting probability up
to a given time.
