Fluctuations in SPDEs and Interacting Particle Systems

顾名思义，相互作用粒子系统用于模拟相互作用的粒子系统的集体行为。粒子系统有广泛的应用，例如在经济学中描述选民对特定主题的意见;在生物学中模拟流行病的传播或在金融市场中模拟竞争公司的价值。我们通常假设粒子系统中存在噪声，这意味着粒子不会以确定性的方式移动，而是随机运动。这种随机性通常是通过使用随机微分方程（SDEs）描述单个粒子的演化来实现的。为了跟踪粒子的位置，考虑经验测量通常是有用的。经验测量表征了粒子在给定时间处于某个区域的经验概率-它既是时间的函数，也是粒子数量的函数。事实证明，在上述两者的适当联合标度下，经验测度收敛到极限测度。有趣的是，极限测度满足偏微分方程（PDE），也就是说，粒子密度在极限中以确定性的方式演化。为了激发随机偏微分方程（SPDE）的思想以及为什么在这种情况下需要它们，我们需要引入波动和大偏差原理的概念。如上所述，我们期望当我们增加系统中的粒子数量并允许系统运行更长的时间时，经验测度应该收敛到极限测度。我们将对以下问题感兴趣：给定一个非常大的时间和大量的粒子，粒子系统看起来与我们期望的极限行为非常不同的概率是多少？这些涨落概率可以用SPDE来表征，为了回答上述问题，需要考虑粒子系统需要多少"能量"才能从平衡态偏离到偏离态。接下来，我们简单介绍一下第一个项目。假设我们正在观察一个粒子系统，其中粒子根据环面上的独立布朗运动进行扩散。这意味着粒子是不可区分的，并且彼此不相互作用。该系统的经验密度（未标度）满足称为Dean川崎方程的SPDE。Ferhman和Gess（https：//doi.org/10.1007/s00205-019-01357-w）证明了具有截断（低空间频率）噪声和正则化非线性的更一般类别的SPDE的适定性。随后在arXiv：1910.11860中，他们也证明了SPDE系统的大偏差原理。我们的第一个目标是通过改变粒子系统的边界条件从环面到有界域来扩展这些论文的结果。我们将看看在不同的边界条件下，例如Dirichlet（粒子在边界处被杀死）或Neumann（粒子在边界处被反射）条件下，过程的极限行为可以说些什么。改变这个领域的一个动机是，我们可能能够对与真实的生活更相关的粒子系统进行建模。例如，在一个金融应用中，粒子代表公司的价值，人们可以在空间点0处观察狄利克雷边界条件来代表公司的破产。我们也可以考虑粒子沿整条真实的直线演化的情况，或者考虑初始数据更一般的情况。虽然我们无法预测后续的项目会是什么样子，但它们将与上面讨论的主题类似。我们的项目福尔斯属于EPSRC的"数学分析"领域。

As the name suggests, interacting particle systems are used to model the collective behaviour of a system of particles which interact with one another. Particle systems have a broad applications, for example in economics to describe voters' opinion on a specific topic; in biology to model the spread of an epidemic or in financial markets to model the values of competing firms. We typically assume that there is noise in the particle system, meaning that the particles don't move around in a deterministic way but are subject to random motion. This randomness is typically realised through describing the evolution of individual particles using stochastic differential equations (SDEs). To keep track of the position of the particles it is often useful to consider the empirical measure. The empirical measure characterises the empirical probability that the particles are in a certain region at a given time - it is both a function of time and number of particles. It turns out that under an appropriate joint scaling of both of the above, the empirical measure converges to a limiting measure. Interestingly, the limiting measure satisfies a partial differential equation (PDE), and that is to say that the density of particles evolves in a deterministic way in the limit. To motivate the idea of stochastic partial differential equations (SPDEs) and why they are needed in this context, we need to introduce the notion of fluctuations and large deviation principles. As noted above, we expect that as we increase the number of particles in our system and allow the system to run for a longer time, the empirical measure should converge to a limiting measure. We will be interested in the following question: Given a very large time and large number of particles, what is the probability that the system of particles looks very different to the limiting behaviour we would expect? These fluctuation probabilities can be characterised by SPDEs, and to answer the above question one needs to consider how much "energy" the system of particles needs to deviate from the equilibrium state to the deviated state. Next we briefly outline the first project. Suppose we are looking at a particle system where particles diffuse according to independent Brownian motions on a torus. This means that the particles are indistinguishable and don't interact with one another. The empirical density (not scaled) of this system satisfies a SPDE called the Dean Kawasaki equation. Ferhman and Gess (https://doi.org/10.1007/s00205-019-01357-w) proved the well posedness of a more general class of SPDEs with truncated (low spatial frequency) noise and regularised nonlinearity. Subsequently in arXiv:1910.11860 they also proved a large deviation principle for the SPDE system. Our first goal is to extend the results of these papers by changing the boundary conditions of the particle system from the torus to a bounded domain. We will look at what can be said about the limiting behaviour of the process for different boundary conditions, for example Dirichlet (particles being killed at boundary) or Neumann (particles reflected at boundary) conditions. One motivation for changing the domain is that we may be able to model particle systems that relate more to real life. For example, in a finance application where particles represent value of firms, one may view a Dirichlet boundary condition at spacial point 0 to represent bankruptcy of a firm. We might also consider what happens in the case that the particles evolve on the whole real line, or in the case of more general initial data. Whilst we can't predict what the subsequent projects will look like, they will be of a similar flavour to the topics discussed above. Our project falls within the EPSRC area of 'Mathematical analysis'.