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

该项目关注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。本项目研究的系统具有代表性的自然现象，如晶体生长、磁畴演化、噪声环境下的路径演化以及随机搅拌流体。这些系统通常表现出非线性和临界性。非线性是指宏观可观测物的随机性与潜在的微观随机性以非线性的方式相互关联的行为。临界是指当系统微观模型中的某些参数达到某个临界值时，系统表现出截然不同的宏观行为的现象。时间进化的随机系统构成了概率论的一个大主体，然而这里所考虑的现象类型却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。具体而言，项目研究了三种类型的模型：具有移动边界的相互作用粒子、临界状态下的随机偏微分方程（SPDEs）和随机六顶点型模型。有移动边界的粒子系统会产生斯特凡吗？在一个特殊的情况下，这些系统与反应扩散粒子系统的临界点有关。首席研究员寻求开发更强大的工具，不需要明确的平稳分布，并适用于上述临界点。spde对某些参数表现出临界性，其解相对于驱动噪声变得不可测量。本研究旨在研究这类spde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一种专门化，可以表示为马尔可夫过程。它拥有许多退化，包括完全不对称的简单排斥过程。作为理解这些模型的极限形状的第一步，首席研究员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.