Stochastic Spatial Processes

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We study a spatial version of the well-known normal distribution which applies to quite general populations distributed through space undergoing reproduction and migration. Examples include populations of a particular genetic type undergoing random replacement and mutation, diseases spreading through a population, a rare species competing for resources, or a liquid percolating through a porous medium. Although the local rules of evolution may be complex and quite different, it can be much easier to analyze the process at large space and time scales where these processes can exhibit universal behaviour. Much of my research is to find, justify and analyze a mathematically precise picture of this large scale evolution which describes all of the above phenomena in the large. By fitting a few parameters of this limit one can then predict large-scale behaviour of the underlying population in myriad settings. Of particular interest is the behaviour of the population near the edge of its range or the set of infected sites near the infection front of the disease. We can then analyze the fractal behaviour of this random front. The limiting random process itself is called a Dawson-Watanabe superprocess (its co-discover, Don Dawson, is a leading figure in Canadian mathematics), an important special case being super-Brownian motion (SBM). It exhibits some surprising regular behaviour, in spite of its inherent randomness. We study its qualitative and quantitative properties. We also develop methods to construct and characterize other random processes which behave locally like these fundamental objects.