Topics in Probability

小行星9626590 调查的主要重点将是如何粗糙的几何形状的状态空间约束的状态空间所支持的随机过程的行为，特别是在分类的行为可能为随机游动和随机增长过程（如分支随机游动和接触过程）在双曲几何。 特别注意的行为，不能发生在欧几里德状态空间，如指数人口增长，最终灭绝在每个紧凑的子集。 关于各种有限状态马尔可夫链的混合率的相关问题也将进行调查，直接的目的是促进对所谓的“截止现象”的理解。 随机增长过程作为结构化环境中种群增长的粗略模型，因此在种群生物学中具有天然的兴趣。 它们也是重要的模型在理论物理的各个领域，特别是统计力学。 关于混合速率的问题 有限状态马尔可夫链具有实际的重要性，因为在模拟研究中越来越多地使用“马尔可夫链蒙特卡罗”方法。 ***

9626590 Lalley The main focus of the investigation will be on how the coarse geometry of a state space constrains the behavior of stochastic processes supported by the state space, in particular, on categorizing the behaviors possible for random walks and random growth processes (such as branching random walks and contact processes) in hyperbolic geometries. Particular attention will be paid to behaviors that cannot occur in Euclidean state spaces, such as exponential population growth with eventual extinction in every compact subset. Related problems regarding the mixing rate for various finite state Markov chains will also be investigated, with the immediate aim of contributing to the understanding of the so-called "cutoff phenomenon". Random growth processes serve as crude models of population growth in structured environments, and thus are of natural interest in population biology. They are also important as models in various areas of theoretical physics, in particular, statistical mechanics. Problems concerning mixing rates of finite-state Markov chains are of practical importance because of the increasing use of "Markov chain Monte Carlo" methods in simulation studies. ***