ON REGULARITY OF SUPERPROCESSES

Three theorems on regularity of measure-valued processes X with branching property are established which improve earlier results of Fitzsimmons [F1] and the author [D5]. The main difference is that we treat X as a family of random measures associated with finely open sets Q in time-space. Heuristically, X describes an evolution of a cloud of infinitesimal particles. To every Q there corresponds a random measure X(tau) which arises if each particle is observed at its first exit time from Q. (The state X(t) at a fixed time t is a particular case.) We consider a monotone increasing family Q(t) of finely open sets and we establish regularity properties of X(t)BAR = X(taut) as a function of t. The results are used in [D6], [D7] and [DIO] for investigating the relations between superprocesses and non-linear partial differential equations. Basic definitions on Markov processes and superprocesses are introduced in Sect. 1. The next three sections are devoted to proving the regularity theorems. They are applied in Sect. 5 to study parts of a superprocess. The relation to the previous work is discussed in more detail in the concluding section. It may be helpful to look briefly through this section before reading Sects. 2-5.