PATH PROCESSES AND HISTORICAL SUPERPROCESSES

A superprocess X over a Markov process zeta can be obtained by a passage to the limit from a branching particle system for which zeta-describes the motion of individual particles. The historical process zeta triple-overdot for zeta is the process whose state at time t is the path of zeta over time interval [0, t]. The superprocess X triple-overdot over zeta triple-overdot-called the historical superprocess over zeta-reflects not only the particle distribution at any fixed time but also the structure of family trees. The principal property of a historical process zeta triple-overdot is that zeta triple-overdot s is a function of zeta triple-overdot t for all s < t. Every process with this property is called a path process. We develop a theory of superprocesses over path processes whose core is the integration with respect to measure-functionals. By applying this theory to historical superprocesses we construct the first hitting distributions and prove a special Markov property for superprocesses.