Branching processes in Levy processes: Laplace functionals of snakes and superprocesses

We use the exploration process introduced in a previous work to develop a n ew construction of superprocesses with a general branching mechanism. This construction depends on a path-valued process called the Levy snake, which is of independent interest. Our method of proof involves a calculation of t he Laplace functional of the occupation field of the Levy snake. This calcu lation relies on an evaluation of the corresponding moment functionals, whi ch requires precise information about the underlying genealogical structure .