We study the ``Brownian snake'' introduced by Le Gall, and also studied by Dynkin, Kuznetsov, Watanabe. We prove that It\^o'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.

MSC Classifications:

60J25, 60G44, 60J80, 60J60 show english descriptions Continuous-time Markov processes on general state spaces
Martingales with continuous parameter
Branching processes (Galton-Watson, birth-and-death, etc.)
Diffusion processes [See also 58J65] 60J25 - Continuous-time Markov processes on general state spaces
60G44 - Martingales with continuous parameter
60J80 - Branching processes (Galton-Watson, birth-and-death, etc.)
60J60 - Diffusion processes [See also 58J65]

top of page | contact us | social media | privacy | site map