CAREER: Stochastic Spatial Systems

这个项目的重点是在三个环境中理解随机空间模型：化学反应，病毒噬菌体合作攻击细菌，以及多发性硬化症的病变形成。传统的建模方法往往忽略了随机性和异质性。这样的假设产生了易于处理的方程，但牺牲了正在建模的过程的显著特征。随机空间模型提供了一个更微妙的视角。探索理论前沿将使我们能够更好地利用它们尚未开发的潜力。这个项目协同结合了对本科生、初级研究人员和被监禁的学生的研究培训。对具有不对称扩散速率的湮灭和结合粒子系统的了解有限。目的一设计了描述这些系统相变的新方法，如耦合和多尺度渗流结构。目的二研究空间易感感染模型中的协同感染动力学问题。在稀疏随机图和整数格上的这些模型中大规模感染的出现将被研究。目的三为多发性硬化症轴突脱髓鞘提供一种新的模型。将利用随机增长模型的变体，如内部扩散限制聚集和伊甸园模型。其目标是通过研究这些模型中生长和抑制之间的相互作用来捕捉损害中枢神经系统的炎症过程的各个方面。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

This project focuses on understanding stochastic spatial models in three settings: chemical reactions, viral phages cooperating to attack bacteria, and lesion formation from multiple sclerosis. Traditional modeling approaches often omit randomness and heterogeneity. Such assumptions yield tractable equations but sacrifice salient features of the processes being modeled. Stochastic spatial models provide a more nuanced perspective. Exploring the theoretical frontier will allow us to better harness their untapped potential. This project synergistically incorporates research training for undergraduates, junior researchers, and students who are incarcerated.There is limited understanding of annihilating and coalescing particle systems with asymmetric diffusion rates. Objective One devises new methods, such as couplings and multi-scale percolation constructions, to describe phase-transitions in these systems. Objective Two involves cooperative infection dynamics in spatial Susceptible-Infected models. The emergence of large-scale infections in these models on sparse random graphs and integer lattices will be studied. Objective Three proposes a new modeling paradigm for axon demyelination due to multiple sclerosis. Variants of stochastic growth models such as internal diffusion-limited aggregation and the Eden model will be utilized. The goal is to capture aspects of inflammatory processes that damage the central nervous system by studying the interplay between growth and suppression in these models.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.