Criticality and Nonlinearity in Interacting Particle Systems and Stochastic Partial Differential Equations

该项目涉及大型随机系统的时间演化，重点关注它们的非线性和临界行为。这个项目中研究的系统代表了自然现象，例如晶体生长、磁区的演化、在嘈杂环境中演化的路径以及随机搅拌的流体。通常，这些系统表现出非线性和临界性。非线性是指宏观观测的随机性与潜在的微观随机性以一种非线性的方式相关的行为。临界性是指当底层微观模型中的某些参数达到某些临界值时，给定系统表现出截然不同的宏观行为的现象。时间演化的随机系统形成了一大堆概率理论，然而这里讨论的现象类型处于当前标准理论的前沿。该项目的目标是揭示与上述范围有关的数学结构，并完善现有的理论来研究这些系统。具体地说，该项目研究了三种类型的模型：具有移动边界的粒子相互作用模型、临界处的随机偏微分方程组(SPDEs)和随机六种顶点类型模型。具有移动边界的粒子系统在偏微分方程组中产生了Stefan-S问题，在一种特殊情况下，这些系统与反应扩散粒子系统的临界点有关。首席研究员试图开发更稳健的工具，不需要显式的平稳分布，并适用于上述临界点。对于某些参数，SPDEs表现出临界性，其中关于驾驶噪声的解变得不可测量。本研究旨在研究这类SPDEs的几个具体例子的相关函数、正则性和局部性质。随机六顶点模型是冰型模型的特化，可以表示为马尔可夫过程。它有许多退化，包括完全不对称的简单排斥过程。作为了解这些模型的极限形状的第一步，主要研究者计划利用这些模型的马尔可夫结构来研究大偏差。

The project concerns the time evolution of large, stochastic systems, with a focus on their nonlinear and critical behaviors. The systems studied in this project are representative of natural phenomena, such as crystal growths, evolution of magnetic domains, paths evolving in a noisy environment, and randomly stirred fluids. Often these systems exhibit nonlinearity and criticality. Nonlinearity refers to the behavior that the randomness of the macroscopic observables are related to the underlying microscopic randomness in a nonlinear fashion. Criticality refers to the phenomenon that a given system exhibits drastically distinct macroscopic behaviors when certain parameters in the underlying microscopic model reach some critical values. Time-evolutionary stochastic systems form a large body of probability theory, yet the type of phenomena considered here sits on the frontiers of current standard theories. The goal of this project is to unveil the mathematical structure pertaining to the aforementioned scopes, and to refine the existing theories to study these systems.In concrete terms, the project studies three types of models: interacting particles with moving boundaries, stochastic partial differential equaitons (SPDEs) at their criticality, and stochastic six vertex-types models. Particle systems with moving boundaries give rise to Stefan?s problem in PDE, and in one particular case those systems relate to the critical point of a reaction-diffusion particle system. The principal investigator seeks to develop more robust tools that do not require explicit stationary distributions and apply to the aforementioned critical point. SPDEs exhibit criticality for certain parameters, where the solutions become non-measurable with respect to the driving noise. This research aims at studying the correlation functions, regularity, and local properties of a few specific examples of such SPDEs. The stochastic six-vertex model is a specialization of the ice-type models that can be formulated as a Markov process. It hosts a number of degenerations, including the totally asymmetric simple exclusion process. As a first step toward understanding the limiting shape of these models, the principal investigator plans to study the large deviations utilizing the Markov structures of these models.