Many fundamental Markov processes --- including random walk on group and diffusion on Riemannian manifold or Lie group --- can be viewed as defined by an underlying geometric structure. This project studies the intricate relationships between the properties of the process and the properties of the underlying geometry. Technically, this is done by obtaining precise estimates on the transition kernel of the process (the heat kernel). The aim is to gain a better understanding of the large scale properties of the process, based on the underlying geometry but also, in some cases, to explore the geometry with the help of the associated Markov process.
Markov processes play a fundamental role in modern scientific activities, from physics to biology to finance, where they model complex phenomena. They also play a basic role in computer simulation. This proposal studies basic properties of Markov processes by relating the properties of the process to the geometry of the space in which it evolves. How does heat diffuses in a large piece of alloy? and how does this depends on the shape of the piece and the perhaps varying nature of the alloy? What can one discover about the nature of the alloy by observing temperatures? These are, in spirit, some of the questions that are considered.
Randomness serves as an important tool to model the world we live in. In this project, specific random processes describing the evolution of certain objects over time are studied. An example of such a process is provided by the familiar action of shuffling a deck of cards. In this case, the object of interest is the deck of cards. The order of the cards evolves under repeated shuffles. Different shuffling methods leads to different processes and one of the questions of interest is to describe precisely how long it takes to mix the deck of cards. This example is of great importance because similar mixing procedure can be observed in nature and are relevant in a variety of scientific fields and in computational algorithms. In this project, techniques were developed to understand the behavior of basic examples such processes. In many instance, these processes evolve in a geometric environment and one is interested in discovering how the basic features of the environment explain and are reflected by the behavior of the random process. For instance, in a certain geometric environment, a particle moves around by taking certain random steps. It is of interest to understand (a) The typical displacement of the particle after n random steps (b) The probability that the particle returns back to its starting point after n steps (c) The extend of space that has been visited by the particle up to time. Results from this project provide answers to these questions in cases that were not understood before. In addition, general methods are proposed and developed to answer this type of questions in some large context. These techniques have potential applications to the study of models and problems connected to a variety of scientific areas such as biology, statistics, computer sciences and statistical physics. This project directly contributed to the training of graduate and postgraduate students and associates who worked under the PI supervision on specific problems and have now taken positions in academia. The outcomes of this project have been presented in international conferences and are expected to have an impact on the development of the field and on future work of the PI and others in this area of research.
