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Path properties of superprocesses

University of British Columbia

Superprocesses are measure valued diffusions that arise as high density limits of particle systems undergoing spatial motion and critical branching. The most closely studied superprocess is super Brownian motion where the underlying spatial motion is Brownian. In chapter 1 we describe the approximating particle systems, the nonstandard model for a superprocess and some known path properties of super Brownian motion. Super Brownian motion is effectively determined by its closed support. In chapter 2 we use the approximating particle systems to derive new path properties for the support process. We find the growth rate of the support for the process started at a point mass. We give a representation for the measure at a fixed time in terms of its support. We show that the support at a fixed time is nearly a totally disconnected set. Finally we calculate the Hausdorff dimension of the range of the process over random time sets. A superprocess can be characterised as the solution to a martingale problem and in chapter 3 we use this characterisation to study the properties of general superprocesses. We investigate when the real valued process given by the measure of a half space under a super symmetric stable process is a semimartingale. We give a description of the behaviour of a general superprocess and its support near extinction. Finally we consider the problem of recovering the spatial motion from a path of the superprocess.

Text

2010-10-18

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http://hdl.handle.net/2429/29308

Mathematics

University of British Columbia

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