Super-Brownian motion conditioned on the total mass

We study the super-Brownian motion ( X t ) conditioned on the total mass Z = ∫ 0 + ∞ X t ( 1 ) d t 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.