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# A Stochastic Calculus Approach for the Brownian Snake

Published:2000-02-01
Printed: Feb 2000
• Jean-Stéphane Dhersin
• Laurent Serlet
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## Abstract

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 - 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]