Stochastic Spatial Processes

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The project is a study of a class of stochastic processes used to model large systems with many interacting "agents" (cells, individuals, components, particles, plants). Typically the agents are located at the nodes of a network, either a homogeneous, heterogeneous or random graph, and transmission of ``information'' (infection, rumor, traits, etc.) between agents is random but obeys some simple local rule. Interest in processes of this type has been spurred in recent years by the introduction of ``small world'' random graphs used as models of the world wide web and the internet, and attempts to analyze large social networks. There is now a rapidly growing research literature on these new models, primarily outside of mathematics. An example of the type of question considered in this literature is: given a particular network and interaction mechanism, will a rumor or trait spread rapidly throughout the network or will it quickly die out? A second, related question is: will a given system relax into a quasi-equilibrium which maintains diversity for a very long time. There are rigorous results available for such questions for some ``classic'' homogeneous lattice systems, but analyzing the newer more heterogeneous models raises significant mathematical challenges. The research in this project will identify key features of these models, i.e. features of the network and of the interaction mechanism, which will determine their long-term behavior, and develop rigorous mathematical methods to validate corresponding predictions. The project will provide a rigorous foundation from which further research can be based.More generally, the project is concerned with how large stochastic systems based on local rules develop over time. Systems studied will vary from a spatial model for the evolution of genealogical traits in an infinite population located in homogeneous (geographic) space, both continuous and discrete, to a variety of stochastic models on heterogeneous random graphs. The proposed research will develop rigorous mathematical methodologies for determining whether or not predictions made based on heuristic or meanfield arguments are valid. The research will make use of a range of mathematical techniques from branching processes, interacting particle systems, percolation theory, finite Markov chains, random walks, and random graphs. In particular the project will make use of, and further extend, results on the behavior of rapidly mixing finite Markov chains.