Topics in Probability

9626590莱利调查的主要焦点将是状态空间的粗几何如何约束由状态空间支持的随机过程的行为，特别是对双曲几何中随机行走和随机增长过程(例如分支随机行走和接触过程)可能的行为进行分类。将特别注意在欧几里得状态空间中不能发生的行为，例如在每个紧致子集中最终灭绝的指数种群增长。关于各种有限状态马尔可夫链的混合速度的相关问题也将被研究，直接目的是有助于理解所谓的“截止现象”。随机增长过程是结构化环境中种群增长的原始模型，因此在种群生物学中具有天然的意义。在理论物理的各个领域，特别是统计力学中，它们也是重要的模型。由于“马尔可夫链蒙特卡罗”方法在模拟研究中的广泛应用，有限状态马尔可夫链的混合速率问题具有重要的实际意义。***

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. ***