Stochastic Processes on Rough Spaces and Geometric Properties of Random Sets

随着时间的推移，信息是如何在系统内传播的？这与系统的内在结构有何关系？开发数学模型是为了理解和预测高度复杂的过程和结构的行为。通过增加模型的“粗糙度”，可以得到更丰富、更真实的模型；在这里，我们可能会想到像互联网这样的大型网络的复杂分支。此外，纳入随机性有助于更好地描述组成模型的元素之间的交互，例如不同Web服务器之间的交互。与随机过程和结构分析有关的数学领域称为概率论。这个项目的目的是应用概率论来更好地理解随机过程、几何以及粗糙空间和随机几何模型分析之间的相互联系。该奖项支持的研究旨在研究粗糙空间中的随机过程和随机集的泛函。PI将分析类分形空间固有的扩散过程的行为，在这些空间中，导数或正则度量等经典概念不那么直接，甚至不可用。将特别注意泛函不平等和梯度估计的研究，以进一步深入了解这一过程与基本空间几何之间的联系。此外，PI还将分析由具有长范围相关性的点过程建模的大型空间数据集的属性。这里，焦点集中在随机覆盖集的几何和位势理论方面，以及用于罕见事件检测的统计量的渐近行为。PI期望开发的方法和技术将在数学物理、材料和数据科学的当前问题上取得进展。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

How does information spread within a system over time? How is this related to the intrinsic structure of the system? Mathematical models are developed to understand and make predictions about the behavior of highly complex processes and structures. Richer and more realistic models can be obtained by increasing their 'roughness'; here we may think of the intricate branching of large networks like the Internet. In addition, incorporating randomness can help to better describe the interactions between the elements that compose the model, for instance between different web servers. The area of mathematics concerned with the analysis of random processes and structures is called probability theory. This project aims to apply probability theory to better understand the interconnections between the random processes, the geometry, and the analysis of rough spaces and random geometric models. The research supported by this award aims to investigate stochastic processes in rough spaces and functionals of random sets. The PI will analyze the behavior of diffusion processes intrinsic to fractal-like spaces, where classical concepts such as derivatives or canonical measures are less straightforward or not even available. Special attention will be paid to the study of functional inequalities and gradient estimates to gain further insight into the connections between the process and the geometry of the underlying space. Besides, the PI will analyze properties of large spatial data sets modeled by point processes with long range dependencies. Here, the focus lies on geometric and potential-theoretic aspects of random covering sets and the asymptotic behavior of statistics employed in the detection of rare events. The PI expects that the methods and techniques developed will lead to progress on current questions in mathematical physics, material and data sciences.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.