ON REGULARITY OF SUPERPROCESSES

Three theorems on regularity of measure-valued processes X with branch ing property are established which improve earlier results of Fitzsimm ons [F1] and the author [D5]. The main difference is that we treat X a s a family of random measures associated with finely open sets Q in ti me-space. Heuristically, X describes an evolution of a cloud of infini tesimal particles. To every Q there corresponds a random measure X(tau ) which arises if each particle is observed at its first exit time fro m Q. (The state X(t) at a fixed time t is a particular case.) We consi der a monotone increasing family Q(t) of finely open sets and we estab lish regularity properties of X(t)BAR = X(taut) as a function of t. Th e results are used in [D6], [D7] and [DIO] for investigating the relat ions between superprocesses and non-linear partial differential equati ons. Basic definitions on Markov processes and superprocesses are intr oduced 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 s uperprocess. The relation to the previous work is discussed in more de tail in the concluding section. It may be helpful to look briefly thro ugh this section before reading Sects. 2-5.