Random processes and their underlying geometry

拟议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有关。重点是发展稳健和广泛适用的方法，以获得对马尔可夫过程行为的定量估计，特别是扩散、随机行走和跳跃过程。马尔可夫过程的行为是通过估计其转移概率密度来捕捉的。对于黎曼流形(例如光滑曲面)上的布朗运动，这种转移密度称为热核；换句话说，热方程的基本解。更一般地，我们使用术语‘热核’来表示马尔可夫过程的转移核。Harnack不等式是椭圆型和抛物型偏微分方程解的经典正则性估计，在这一研究领域中起着至关重要的作用。最近，Harnack不等式被用于研究分形上的扩散和跳跃过程，以及随机介质中的传输性质。热核及其估计在概率论、分析和偏微分方程组、几何和数学物理等数学领域中扮演着重要的角色。提出的研究的首要主题是发展稳健的方法来证明一大类马尔可夫过程的定量估计。定量估计包括热核估计、Harnack不等式和泛函估计。人们特别感兴趣的是开发对扰动具有健壮性的方法；例如，在图中添加或移除一些边，或者在黎曼流形的度量中适当地改变(拟等距)。另一个长期目标是开发稳健的方法来获得热核估计，即使在Harnack不等式失败的情况下也是如此。最终，我们打算使用这些稳健的方法来计算随机空间上的马尔可夫过程的行为，例如一致无限平面三角剖分(UIPT)。物理学家对UIPT上随机游走行为的预测仍有待证实，因此为本提案中考虑的一些研究提供了一个具体的试验场。更广泛地说，这项研究的目的之一是了解随机介质中的传输现象。提案中的随机过程在许多应用中起着关键作用。马尔可夫过程被用来对物理学、生物学、统计学和金融学中的各种现象进行建模。此外，这些过程被用作计算机科学中的计算工具和模拟(马尔科夫链蒙特卡罗和其他随机算法)。培训聪明、年轻的研究人员是这一提议的重要组成部分。通过参与这项研究，研究生和博士后研究员将能够为我们提供随机过程基本方面的知识。

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