Brett Geiger
University of Houston
Math Department
Large Deviations and Rare Events
My main research interests lie in the area of large deviations with a focus on applying large deviation principles to stochastic models with small random fluctuations (noise). In such systems, small noise gives rise to positive probabilities of rare events of high interest, such as excursions from nominally stable states, transitions from metastable states, and fixations of genotypes in bacterial populations. Obtaining information on such events from direct simulations is difficult and computationally inefficient. Large deviations theory quantifies these probabilities as well as provides the most likely way in which the system achieves rare events, which uses a "large deviations principle." A key question that arises is whether these principles can be applied to a probabilistic model in an explicit and computationally efficient way. This is the key question that I aim to answer.
Nonstationary Dynamical Systems
Nonequilibrium dynamics refers to situations in which the dynamical model varies in time. The study of nonequilibrium dynamics is motivated by our desire to understand the dynamics of systems that evolve in time-varying environments. Examples of such systems include Sinai billiards with moving scatterers and open systems (systems with holes) in which mass is allowed to escape. The field of random dynamical systems treats the case in which the statistical properties of the variability of the model are assumed to be known. In contrast, no a priori knowledge of the statistical properties of the evolution of the dynamical model are assumed in the case of nonequilibrium dynamics. Therefore, the study of the statistical properties a time-dependent system becomes a natural approach to take in order to understand the system over time. One property of particular interest is the behavior of the distances between pairs of initial distributions as distributions evolve.
Submitted Papers and Papers in Preparations
**Preliminary drafts for papers in preparation are available upon request.**
Large Deviations for Gaussian Diffusions with Delay (arXiv preprint)
Application of Large Deviations to Genetic Evolution of Bacterial Populations (In preparation)
Nonstationary Open Dynamical Systems (In preparation)