Grant: Canada Research Chairs

The objective of the proposal is to understand how the long­ term behaviour of random processes relates to the geometry of the underlying space. The emphasis is on developing robust and widely applicable methods to obtain quantitative estimates on the behaviour of diffusions, random walks, and jump processes, or more generally Markov processes. Markov processes are used to model a wide range of phenomena in physics, biology, statistics, and finance. Further, these processes are used as computational tools and for simulations in computer science. Therefore quantitative estimates describing the behaviour of a Markov processes are of fundamental importance. Here are a few key questions towards the understanding of dependence of the behaviour of Markov process in relation to its geometry: 1) Which properties of a Markov process are stable under perturbations? 2) How is the behaviour of a Markov process affected by natural transformations of the space such as change of metric or time change? 3) Is is possible to improve the behaviour of a Markov process by a reparametrization of space and time? If so, what does the `best' geometry look like? Here are a few novel projects of interest from the proposal that aim to address the above questions: (1) Constructing diffusions with prescribed behaviour of exit time and volume growth profiles. (2) Obtain heat kernel estimates for the trace jump process of reflected diffusions (3) Given a metric measure space, obtain conditions for existence or non-existence of diffusion processes satsifying nice (sub-Gaussian) heat kernel bounds (4) Develop a theory of `blow-up' of Dirichlet forms that would allow for instance to define diffusion processes on Gromov-Hausdorff limit spaces (5) Developing non-linear potential theory on fractals by constructing analogues of Sobolev spaces, energy measures, harmonic functions. (6) Use the non-linear potential theory developed in (5) to obtain results on quasisymmetric uniformization problems. These projects aim to improve our understanding of the interactions between behaviour of random processes and their underlying geometry. Many of the projects proposed above will be carried out with graduate students and postdoctoral researchers as part of their training. The training of bright young researchers is an important part of the proposal.