Lebesgue Approximation of Superprocesses

Abstract

Superprocesses are certain measure-valued Markov processes, whose distributions can be characterized by two components: the branching mechanism and the spatial motion. It is well known that some basic superprocesses are scaling limits of various random spatially distributed systems near criticality.

We consider the Lebesgue approximation of superprocesses. The Lebesgue approximation means that the processes at a fixed time can be approximated by suitably normalized restrictions of Lebesgue measure to the small neighborhoods of their support. From this, we see that the processes distribute their mass over their support in a deterministic and "uniform" manner. It is known that the Lebesgue approximation holds for the most basic Dawson-Watanabe superprocesses but fails for certain superprocesses with discontinuous spatial motion.

In this dissertation we first prove that the Lebesgue approximation holds for superprocesses with Brownian spatial motion and a stable branching mechanism. Then we generalize the Lebesgue approximation even further to superprocesses with Brownian spatial motion and a regularly varying branching mechanism. We believe that the Lebesgue approximation holds for superprocesses with Brownian spatial motion and any "reasonable" branching mechanism. Our present results may be regarded as some progress towards a complete proof of this very general conjecture.