Canad. J. Math. 52(2000), 92-118
Printed: Feb 2000
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.
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]