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.
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 into these interactions and studies corresponding stochastic models. As suitable models for various physical systems, interacting stochastic systems have been studied extensively in the last decades. This project was focused on infinite stochastic systems whose dynamics is spatially constrained. Such processes (in particular, branching random walk with barriers) have an application in computational models of evolution that we studied in collaboration with a group of bio-physicists. In these models formation of clusters is an analog of speciation. Long-time behavior of the system is studied with respect to several parameters, with focus on mutability (maximum mutation size) and its role in modulating speciation. Our most recent work in this area includes the demonstration of clustering of organisms (analogous to speciation) even on a selectively neutral landscape, where all organisms have identical fitness, which was a surprising result. The poject has also brought a new light in the area of stochastic dependence. Dependence of two random quantities is most commonly measured through the correlation coefficient (and the correlation matrix in the case of more than two variables). Working on attainable correlations we have obtained results that beside thier theoretical merit allow new simulation methods as well. This research has the potential of a huge impact in all areas where stochastic components are considered, such as physics, engineering, ecology, finance, and in particular, the generation of synthetic optimization problems. Theoretical outcomes of the project are published or submitted for publication. They have also been presented on many national and international conferences. Topics of this project have been incorporated as a part of advanced undegraduate and graduate reading courses. Two Ph.D. dissertations are partly related to the project. In addition, it has been created a circle of female mathematicians has that has been contributing in improving our academic environment.