Stochastic Processes on Rough Spaces and Geometric Properties of Random Sets
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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.