Super-Brownian motion conditioned on the total mass

We study the super-Brownian motion (Xt) conditioned on the total mass Z=∫0+∞Xt(1)dt as the continuous limit of a system of branching trajectories which genealogical structure is a Galton-Watson tree conditioned on the number of its vertices. We characterize this process by a martingale problem and give a “snake” representation. Then we apply these results to a process that we call super-Brownian excursion. Its integral is the so-called Integrated Super-Brownian Excursion that has appeared recently as limit of several systems of statistical mechanics.