Stochastic processes with spatial constraints

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The proposal focuses on infinite stochastic systems whose evolution is spatially constrained. It involves research on space-time inhomogeneous contact processes and branching random walks. It also involves a study of quasi-stationary distribution which is an essential concept related to equilibrium properties of a process conditioned to stay inside a sub-state-space. In an earlier work, we established a relation between quasi-stationary distributions and a Fleming-Viot particle system, which promises a novel, constructive and useful approach to quasi-stationarity. The proposed research on the subject will build on and significantly extend this work. Many natural and social phenomena arise as a result of interaction of large number of components (particles, humans, viruses, plants,...). Due to immense complexity of these systems, in order to make them more accessible, one assigns random components to these interactions and studies corresponding stochastic models. As suitable models for various physical systems, interacting stochastic systems have been studied extensively over the last decades. They were found also to be very useful models in epidemiology. The proposed project is related to infinite stochastic systems whose evolution is spatially constrained. Some mathematical questions to be addressed can be translated as:What happens with an infection if it spreads using only a certain number of individuals and contacts among them? How does reduction of a habitat affect distributions of a plant species? Beside its mathematical significance, the proposed research has very important applications in biogeography and evolution theory and hopefully it will generate fruitful collaborative interdisciplinary research.