Random processes and their underlying geometry

提出的研究计划的目标是了解随机过程的长期行为如何与底层空间的几何形状相关。重点是开发鲁棒和广泛适用的方法，以获得对马尔可夫过程行为的定量估计，特别是扩散，随机漫步和跳跃过程。

The objective of the proposed research program 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 Markov processes, and in particular diffusions, random walks, and jump processes. The behaviour of a Markov process is captured through estimates of its transition probability density. For Brownian motion on a Riemannian manifold (e.g., a smooth surface), this transition density is called the heat kernel; in other words, the fundamental solution of the heat equation. More generally, we use the term heat kernel' to denote the transition kernel of a Markov process. Harnack inequalities are classical regularity estimates of solutions to elliptic and parabolic PDEs, and play an essential role in this area of research. More recently, Harnack inequalities have been used to study diffusions and jump processes on fractals, and transport properties in random media. The heat kernel and its estimates play an important role in diverse areas of Mathematics including Probability Theory, Analysis and PDEs, Geometry, and Mathematical Physics The overarching theme of the proposed research is to develop robust methods to prove quantitative estimates for a large class of Markov processes. The quantitative estimates include heat kernel estimates, Harnack inequalities, and functional inequalities. It is of particular interest to develop methods that are robust to perturbations; for instance, the addition or removal of a few edges in a graph, or suitable changes (quasiisometry) in the metric of a Riemannian manifold. A further long-term goal is to develop robust methods to obtain heat kernel estimates even when the Harnack inequality fails. Ultimately, we intend to use these robust methods to compute the behaviour of Markov process on random spaces, such as the uniform infinite planar triangulation (UIPT). The predictions of physicists on the behaviour of random walk on UIPT remain to be proved and thus provide a concrete testing ground for some of the research considered in this proposal. More generally, one of the aims of this research is to understand transport phenomena in random media. The random processes in the proposal play a key role in many applications. 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 (Markov Chain Monte Carlo and other randomized algorithms). The training of bright, young researchers is an important component of this proposal. By participating in this research, graduate students and postdoctoral fellows will be equipped to contribute to our knowledge of the fundamental aspects of random processes