A Stochastic Calculus Approach for the Brownian Snake

We study the “Brownian snake” introduced by Le Gall, and also studied by Dynkin, Kuznetsov, Watanabe. We prove that Itô’s formula holds for a wide class of functionals. As a consequence, we give a new proof of the connections between the Brownian snake and super-Brownian motion. We also give a new definition of the Brownian snake as the solution of a well-posed martingale problem. Finally, we construct a modified Brownian snake whose lifetime is driven by a path-dependent stochastic equation. This process gives a representation of some super-processes.