RUI: Boundary and entropy of random walks on groups

Applications of random walks emerge in different branches of science and our daily life. In game theory, random walks can model shuffling a deck of cards to find optimal shuffle iterations. In finance, the unpredictability of price changes of stocks can be modeled by random walks. The project will apply rigorous mathematical tools for studying the long-term behavior of random walks. In particular, the following two problems will be investigated: first, how geometric or algebraic features can capture the long-term behavior of random walks. Second, how different quantities can describe the long-term behavior of a random walk. The research will engage undergraduate and graduate students. This engagement with students exposes them to contemporary areas of mathematics not part of the regular curriculum. The work will expand and refine existing results on random walk invariants such as boundaries and asymptotic entropies. These arise from the interaction between random walks on groups and their algebraic or geometric structures. The project will improve understanding of the Poisson boundary, which provides a representation of bounded harmonic functions on a group for a given probability measure. The current work will leverage recent developments, such as the theory of pivotal times, to identify the Poisson boundary of random walks on hyperbolic-like groups. Another goal is to investigate connections between quotients of the Poisson boundaries with subgroups of a given group. The area of research has strong connections to geometric group theory, ergodic theory, and information theory. This project is jointly funded by the DMS Probability program and the Established Program to Stimulate Competitive Research (EPSCoR). 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.