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Path properties and convergence of interacting superprocesses

University of British Columbia

Dawson-Watanabe superprocesses are stochastic models for populations undergoing spatial migration and random reproduction. Recently E. Perkins (1993, 1995) introduced an infinite dimensional stochastic calculus in order to characterize superprocesses in which both the reproduction mechanism and the spatial motion of each individual are allowed to depend on the state of the entire population, i.e. superprocesses with interactions. This work consists of three independent chapters. In the first chapter we show that interactive superprocesses arise as diffusion approximations of interacting particle systems. We construct an approximating system of interacting particles and show that it converges (weakly) to a limit which is exactly the superprocess with interactions. This result depends very intimately upon the structure of the particle systems. In the second chapter we study some path properties of a class of one-dimensional interactive superprocesses. These are random measures in the real line that evolve in time. We employ the aforementioned stochastic calculus to show that they have a density with respect to the Lebesgue measure. We also show that this density, function is jointly continuous in space and time and compute its modulus of continuity. Along with the proof we develop a technique that can be used to solve some related problems. As an application we investigate path properties of a one-dimensional super-Brownian motion in a random environment. In the third chapter we investigate the local time of a very general class of one-dimensional interactive superprocesses. We apply Perkins' stochastic calculus to show that the local time exists and possesses a jointly continuous version.

Thesis/Dissertation

application/pdf

2009-03-17

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http://hdl.handle.net/2429/6160

Mathematics

University of British Columbia

UBCV

DSpace