Stochastic processes on hypergraphs and dynamic graphs

Stochastic processes on hypergraphs and dynamic graphs Stochastic processes on hypergraphs and dynamic graphs Interacting particle systems are ideally suited to understand the dynamics of populations in spatial environments through the inclusion of local interactions. The PI will investigate a wide variety of stochastic spatial models of interest in ecology, sociology and neuroscience, based on interacting particle systems, but also including a mesoscopic scale and/or describing evolutions on dynamic environments. These models take the form of interacting particle systems on hypergraphs and interacting particle systems on dynamic graphs, also known as adaptive networks, and are more realistic in a number of contexts than traditional particle systems. Intellectual Merit Motivated by a variety of problems that arise from ecology, sociology and neuroscience, the primary objective of this project is to introduce and analyze mathematically stochastic spatial models that extend the traditional framework of interacting particle systems as the components under consideration evolve on hypergraphs or dynamic graphs rather than regular lattices or general connected graphs. Hypergraph structures are employed in this project to model systems including mesoscopic scales, e.g., populations that undergo catastrophic events consisting in the removal of large blocks, and opinion dynamics including the emergence of large discussion groups. Dynamic graph structures are employed to model systems of components that have the ability to shape their environment, e.g., populations of pathogens that have a harmful effect and populations of mutualists that have a beneficial effect on their habitat of hosts, and electrochemical signals that can strengthen or weaken the connections between neurons. The resulting models are usually more challenging to analyze mathematically than traditional particle systems but they are also more realistic in many subfields of applied sciences. Broader Impacts Since their introduction in the early seventies, interacting particle systems have a major impact in a number of fields, including physics, biology and sociology. They are particularly important to understand the long-term behavior of spatial systems through the inclusion of space in the form of local interactions, especially because it is known from past research that spatial models can result in predictions that differ from nonspatial models. This research project motivates and extends the study of interacting particle systems on general graphs, which is a rapidly growing subfield of probability theory since the past couple of years. Beyond the mathematical analysis of some specific models of interest in ecology, sociology and neuroscience, the aim of this research program is also to initiate the development of a theoretical framework inspired from interacting particle systems that would allow to understand general spatial processes involving the presence of multiple spatial scales and/or having the property to shape their spatial environment. Examples of such systems, that cannot be captured by traditional interacting particle systems, are ubiquitous in nature.0