Stochastic PDEs, interacting particle systems and large deviations

随机偏微分方程、相互作用的粒子系统和大偏差

• 批准号: 2592873
• 金额: --
• 依托单位: University of Oxford
• 依托单位国家: 英国
• 项目类别: Studentship
• 财政年份: 2020
• 资助国家: 英国
• 起止时间: 2020 至 无数据
• 项目状态: 未结题
• 来源: https://gtr.ukri.org/projects?ref=studentship-2592873
• 关键词: Stochastic; PDEs; interacting; particle; systems

This project falls within the EPSRC research areas Mathematical Analysis, Statistics and Applied Probability, Mathematical Physics.Partial differential equations are one of the key tools to describe our reality. However, any model of the real world should take into account uncertainty or random fluctuations, so that most often the correct description of a physical phenomenon is in fact given by a stochastic PDE. Indeed, stochastic PDEs have crucial relevance for mathematical physics (quantum field theory, statistical mechanics, fluid dynamics), mathematical biology, mathematical finance, statistics and data science.Diffusion equations are a particular type of deterministic or stochastic PDEs used to describe the macroscopic behavior of many micro-particles interacting with each other as time passes. The study of these interacting particle systems is on its own of great interest for all of the above-mentioned subjects and many others such as computational neuroscience or population dynamics. These systems consider one or more equations for each single particle, and one should regard them as the discrete analogous of SPDEs. Indeed, a general feature of particle systems is that, in the limit as the number of particles increases up to infinity, the common behavior of the particles can be described by a single deterministic equation, which is often a nonlinear diffusion PDE. The associated stochastic PDE arises then to describe the fluctuations of the particle process about this deterministic limit.The convergence of the particle system towards the limiting model prescribed by a single equation is of interest both for a better understanding of the PDE, its stochastic version and the system itself, and for computational reasons, allowing for a quick, compact description of the phenomenon under consideration. In this setting, understanding and quantifying the occurrence of rare events that differ from the limiting behavior suggested by this convergence result is of course relevant for theoretical reasons, and even more crucial for the concrete application of these models in the real world. This is the object of study of large deviation theory.From a technical point of view, the main difficulty in the study of stochastic PDEs is their lack of regularity. Generally speaking, solutions to linear SPDEs are not function valued and need to be made sense of in the space of distributions. In the nonlinear setting, most often it is not even clear how to make sense of these equations and what to call a 'solution'.In this framework, the aim of the project is the understanding of these stochastic diffusion equations, both from the point of view of abstract PDE theory, answering questions such as well-posedness, regularity and continuity with respect to the initial data; and in terms of their connection with particle systems, addressing problems about the modelling of natural phenomena such as the convergence towards a limiting behavior or the occurrence of rare events not predicted by the limiting model.The mathematics used to tackle these problems lies at the interface of probability and analysis. At the continuum level, the combination of deep PDE theory and recent tools from probability is needed to handle the roughness of these SPDEs; the need for sharp probabilistic arguments is even more evident at the discrete level. Finally, both aspects crucially rely on a good understanding of the physical phenomena one is trying to describe.

该项目属于EPSRC的研究领域：数学分析、统计与应用概率、数学物理。偏微分方程是描述现实的关键工具之一。然而，现实世界的任何模型都应该考虑到不确定性或随机波动，因此，对物理现象的最正确描述实际上是由随机偏微分方程给出的。事实上，随机偏微分方程与数学物理（量子场论、统计力学、流体动力学）、数学生物学、数学金融、统计学和数据科学有着至关重要的相关性。扩散方程是一种特定类型的确定性或随机偏微分方程，用于描述许多微粒随着时间推移相互作用的宏观行为。这些相互作用的粒子系统的研究本身对所有上述学科和许多其他学科如计算神经科学或种群动力学都有很大的兴趣。这些系统为每个单个粒子考虑一个或多个方程，人们应该将它们视为spde的离散类比。事实上，粒子系统的一个普遍特征是，当粒子数量增加到无穷大时，粒子的共同行为可以用一个确定性方程来描述，这通常是一个非线性扩散偏微分方程。随后产生了相关的随机偏微分方程来描述粒子过程在这个确定性极限附近的波动。粒子系统向由单个方程规定的极限模型收敛，对于更好地理解PDE，其随机版本和系统本身，以及计算原因都很有意义，允许对所考虑的现象进行快速，紧凑的描述。在这种情况下，理解和量化与该收敛结果所建议的极限行为不同的罕见事件的发生当然具有理论意义，而且对于这些模型在现实世界中的具体应用更为重要。这就是大偏差理论的研究对象。从技术角度看，研究随机偏微分方程的主要困难是其缺乏规律性。一般来说，线性spde的解不是函数值，需要在分布空间中理解。在非线性环境中，大多数情况下，我们甚至不清楚如何理解这些方程，以及如何将其称为“解”。在这个框架下，该项目的目的是理解这些随机扩散方程，从抽象PDE理论的角度出发，回答诸如初始数据的适定性、规律性和连续性等问题；就它们与粒子系统的联系而言，解决关于自然现象建模的问题，例如向极限行为的收敛或极限模型无法预测的罕见事件的发生。用于解决这些问题的数学是概率和分析的结合。在连续体水平上，需要结合深度偏微分方程理论和最新的概率工具来处理这些偏微分方程的粗糙度；在离散的层面上，对明确的概率论证的需求甚至更为明显。最后，这两个方面都至关重要地依赖于对所要描述的物理现象的充分理解。