Diffusions on Combinatorially Structured State Spaces

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This project involves developing new methods to study diffusions with combinatorially structured state-spaces and is motivated by stochastic processes that arise in population genetics and phylogenetics. Combinatorial structure arises in these areas because one would like to model, for example, how the genetic makeup of a population changes over time, or to develop processes that move efficiently through the space of phylogenetic trees in search of the most likely one for a group of organisms. One of the challenges in understanding these processes is determining how changes to the combinatorial structure accumulate over time. This project seeks to develop new methods for understanding such processes when their behavior is driven by the rapid accumulation of small changes. Graduate and undergraduate students will be engaged in this project.Set partitions, integer partitions, and integer compositions play a central role in the study of combinatorial stochastic processes. Markov processes on set partitions and integer partitions are well studied, but the theory for integer composition-valued processes is not as well developed. This project will develop methods for studying diffusion analogues of integer composition-valued processes, which are interval partition-valued processes, with a focus on processes related to the two-parameter extension of the infinitely-many-neutral-alleles diffusion model. Using the connection between interval-partitions and continuum trees through bead-crushing constructions, this project will also develop methods for constructing continuum-tree valued diffusions. This project will develop both pathwise constructions and analytic characterizations of these processes as well as establish invariance principles.This project is jointly funded by Probability program and the Established Program to Stimulate Competitive Research (EPSCoR).This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.