On sharp large deviations for the bridge of a general diffusion

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 probability are independent of the drift b, providedit satisfies a simple condition that is always satisfied in dimension 1. On the other hand we produce an example where this assumption is not satisfied and the drift is actually influential.