Summary
Three theorems on regularity of measure-valued processesX with branching property are established which improve earlier results of Fitzsimmons [F1] and the author [D5]. The main difference is that we treatX as a family of random measures associated with finely open setsQ in time-space. Heuristically,X describes an evolution of a cloud of infinitesimal particles. To everyQ there corresponds a random measureX τ which arises if each particle is observed at its first exit time fromQ. (The stateX t at a fixed timet is a particular case.) We consider a monotone increasing familyQ t of finely open sets and we establish regularity properties of\(\bar X_t = X_{\tau _t } \) as a function oft. The results are used in [D6], [D7] and [D10] 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 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.
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References
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Partially supported by the National Science Foundation Grant DMS-8802667 and by The US Army Research Office through the Mathematical Sciences Institute at Cornell University