Stochastic PDEs, interacting particle systems and large deviations

该项目福尔斯EPSRC的研究领域数学分析，统计和应用概率，数学物理。偏微分方程是描述我们现实的关键工具之一。然而，任何真实的世界的模型都应该考虑不确定性或随机波动，因此大多数情况下，物理现象的正确描述实际上是由随机PDE给出的。随机偏微分方程在数学物理（量子场论、统计力学、流体动力学）、数学生物学、数学金融学、统计学和数据科学中有着重要的意义。扩散方程是一种特殊类型的确定性或随机偏微分方程，用于描述许多微观粒子随着时间的推移相互作用的宏观行为。对这些相互作用的粒子系统的研究本身就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）具有极大的兴趣。这些系统考虑每个单个粒子的一个或多个方程，并且应该将它们视为SPDE的离散类似物。事实上，粒子系统的一般特征是，在粒子数量增加到无穷大的极限下，粒子的共同行为可以由单个确定性方程描述，该方程通常是非线性扩散偏微分方程。相关联的随机偏微分方程的出现，然后描述波动的粒子过程，这个确定性的limit.The收敛的粒子系统的限制模型规定的一个单一的方程是感兴趣的更好地理解偏微分方程，其随机版本和系统本身，并为计算的原因，允许一个快速，紧凑的描述所考虑的现象。在这种情况下，理解和量化与这种收敛结果所暗示的极限行为不同的罕见事件的发生当然与理论原因有关，并且对于这些模型在真实的世界中的具体应用甚至更为关键。这就是大偏差理论的研究对象。从技术的角度看，随机偏微分方程研究的主要困难是它们缺乏规律性。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，通常甚至不清楚如何理解这些方程以及如何称之为“解”。在这个框架内，该项目的目的是从抽象PDE理论的角度理解这些随机扩散方程，回答诸如关于初始数据的适定性，正则性和连续性等问题;在与粒子系统的联系方面，解决自然现象的建模问题，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位于概率和分析的界面。在连续体水平上，需要将深度偏微分方程理论和最近的概率工具相结合，以处理这些SPDE的粗糙度;在离散水平上，对尖锐概率参数的需求更加明显。最后，这两个方面都依赖于对试图描述的物理现象的良好理解。

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.