We use the exploration process introduced in a previous work to develop a new construction of superprocesses with a general branching mechanism. This construction depends on a path-valued process called the Lévy snake, which is of independent interest. Our method of proof involves a calculation of the Laplace functional of the occupation field of the Lévy snake. This calculation relies on an evaluation of the corresponding moment functionals, which requires precise information about the underlying genealogical structure.
"Branching processes in Lévy processes: Laplace functionals of snakes and superprocesses." Ann. Probab. 26 (4) 1407 - 1432, October 1998. https://doi.org/10.1214/aop/1022855868