Problems in Stochastic Processes: Hyperbolic structures, Bayesian nonparametric estimation, and spatial epidemic and interspecies competition models

该项目的主要重点将是研究双曲群和图上的随机过程、随机漫步和分支随机漫步、接触过程、Ising模型和竞争过程。后四个过程表现出多个阶段，包括一个中间的弱生存阶段，这在整数格上是看不到的。研究的一个主要目标将是了解双曲几何如何限制弱生存和强生存之间的上相转变，以及这反过来如何与随机行走的格林函数的渐近行为相关。这将涉及以安科纳不等式、符号动力学和无限维Perron-Frobenius理论为中心的技术发展，以及无限代数泛函数方程系统的Lyapunov-Schmidt约简。该项目的第二个重点将是(a)贝叶斯非参数函数估计中出现的数学问题，以及(b)流行病和物种间竞争的随机模型。随机相互作用粒子系统作为模型被广泛应用于各种科学领域，特别是统计物理学和种群生物学，同时也应用于社会科学。希望在无限双曲几何中研究这些过程将最终有助于理解它们在有限“扩展者”和“小世界”网络中的行为，从而更好地模拟各种社会和生态系统中的相互作用。

The primary focus of the project will be the study of random processes random walks and branching random walks, contact processes, Ising model, and competition processes on hyperbolic groups and graphs. The last four of these processes exhibit multiple phases, including an intermediate phase of weak survival that is not seen on integer lattices. A major objective of the research will be to understand how hyperbolic geometry constrains the upper phase transition between weak and strong survival, and how this in turn is related to asymptotic behavior of the Green's functions of random walk. This will involve development of techniques centered around Ancona inequalities, symbolic dynamics and infinite-dimensional Perron-Frobenius theory, and the Lyapunov-Schmidt reduction of infinite systems of algebraic functional equations. Secondary foci of the project will be (a) mathematical problems arising in Bayesian nonparametric function estimation, and (b) stochastic models of epidemics and interspecies competition.Stochastic interacting particle systems are widely used as models in various areas of science, especially in statistical physics and in population biology, but also in the social sciences. It is hoped that studying such processes in infinite hyperbolic geometries will ultimately contribute to understanding their behavior in finite "expander" and "small-worlds" networks, which may better model interactions in various social and ecological systems.