CAREER: Research and training in stochastic dynamics

Duke University, Durham NC

Investigators

Abstract

The project involves analysis of two types of stochastic dynamics. First, the PI will study solutions to stochastic differential equations which undergo rare, random transitions between two or more regions of the state space. The goal of this work is to understand the statistics of these transitions, especially in the small-noise regime: what are the typical pathways by which the transitions occur? what is the typical time required for a transition? These issues are very relevant to problems in chemistry and molecular dynamics, as well as many other physical systems exhibiting metastable behavior. The second type of stochastic dynamics which the PI will study has to do with stochastic interacting particle systems. Specific systems to be studied involve random motion, growth, and selection, as in models of evolution, population genetics, adaptive dynamics. The PI will study continuum limits and large-time limits for such systems. In particular, this will illuminate new relations between interacting particle systems and continuum free boundary problems. When viewed at a certain scale, many physical and biological systems seem to behave randomly or are influenced by small random fluctuations. Mathematical models of such systems involve probability theory. Nevertheless, it is very difficult to use these mathematical models to efficiently predict the system behavior over a long period of time or over a large spatial region. Therefore, a fundamental scientific and mathematical problem is to understand how random dynamics or interactions at one spatial or temporal scale influence a system at another spatial or temporal scale. This research project develops mathematical tools for predicting and describing the macroscopic behavior of certain systems which behave randomly at a microscopic level. The specific systems to be studied are motivated by problems in chemistry and by models of biological evolution. One common feature in these systems is the appearance of random, perhaps rare, transitions: a chemical reaction occurs or a cell produces a mutation. The educational component of the project includes the training of graduate and undergraduate students at the intersection of probability, analysis, and applications, preparing them for careers in STEM-related disciplines.