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随机过程的研究随机游动和分支随机游动，接触过程，伊辛模型，双曲群和图上的竞争过程。这些过程中的最后四个过程表现出多个阶段，包括在整数格上看不到的弱生存的中间阶段。研究的一个主要目标将是了解双曲几何约束弱和强生存之间的上相变，以及这反过来又是如何与随机游走的绿色函数的渐近行为。这将涉及围绕安科纳不等式，符号动力学和无限维Perron Frobenius理论，以及代数函数方程的无穷系统的Lyapunov-Schmidt约化技术的发展。该项目的第二个焦点是（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被广泛用作各个科学领域的模型，特别是在统计物理学和人口生物学中，但也在社会科学中。希望在无限双曲几何中研究这些过程最终有助于理解它们在有限“扩展器”和“小世界”网络中的行为，这可能会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.