Probability and Stochastic Processes

• Frank Gao
  Professor Gao's research area lies at the interface of probability theory and functional analysis. In particular, he studies small deviation probabilities for Gaussian random processes/fields, metric entropy of function spaces, and their applications to the geometric quantification of compact convex sets.

  The study of small deviations concerns the asymptotic behavior of the probability of the rare event that a random process stays in a small region over a certain period of time. From the Probability Theory point of view, the importance of small deviation probabilities lies in the fact that they determine rates of convergence in many limit laws. Their importance also lies in close connections with many other mathematical areas, such as approximation theory and geometric functional analysis.

  Metric entropy is a measurement of system complexity. More precisely, it measures the sizes of sets in metric spaces. However, standard metric entropy does not reflect how the size is spread out within a set. Currently, Professor Gao is working on the localization of metric entropy (in which the distribution of size is taken into consideration) and its applications in the study of local properties such as mean oscillation of random processes.

• Paul Joyce

• Steve Krone
  Professor Krone's research in probability focuses mostly on interacting particle systems. These are stochastic spatial models that live on discrete sets such as the d-dimensional integer lattice or other graphs. They originally arose in statistical physics and now constitute one of the main classes of models for spatially arranged biological populations.

  Krone's other work in probability deals with scaling limits that arise in population genetics. More information on these applications to biology can be found on the Bioinformatics and Mathematical Biology page.