McKean Vlasov Stochastic Partial Differential Equations

物理系统倾向于走向平衡(或稳定状态)，这是日常经验的一部分：物体掉落在地板上，钟摆停下来，牛奶和咖啡混合在一起制成卡布奇诺。平衡的概念对我们来说有些直观：给定的状态是系统的平衡，如果i)在平衡状态下开始时，系统不会改变其状态(如果笔在地板上，那么它就会留在那里)；ii)如果我们让系统进化，那么它将逐渐进入“停止”，或者换句话说，它将进化到这样的平衡(如果我们掉到地上，笔就会掉到地板上)。平衡的概念可能比上面描述的更有活力：你的卡布奇诺不会自发地分裂成咖啡和牛奶，也就是说，它将停留在卡布奇诺，尽管其中的每个分子仍将移动。这就是概率发挥作用的地方，事实证明，均衡，即稳态(SS)，通常更好地用概率度量--即所谓的平衡度量(EM)--来描述，而不是用点来描述。用一个EM来研究动力学行为是遍历理论的中心问题，该理论致力于建立动力学允许唯一EM的标准，并分析收敛到这样一个EM。这项研究项目旨在加深我们对由(随机)偏微分方程组(S)所模拟的(无限维)遍历过程理论及其与相互作用粒子系统(IPS)的关系的理解；此外，我们将研究一种可能的策略的可行性，以朝着建立非遍历过程理论，即具有多个SS的过程的宏伟目标迈进一些(进一步)步骤。特别是，我们将利用我们将获得的遍历理论结果，以产生对非遍历过程理论的理解。我们将考虑的过程类是所谓的McKean-Vlasov(S)偏微分方程组，它是交互多智能体系统(IMAS)研究的基础。这类过程在本提案的框架内很有意义，因为它们可以由(S)PDE或IPS建模，并且可以表现出多个SS。更广泛地说，数学界对这些系统的兴趣与日俱增，源于它们描述各种情景的灵活性，事实上，IMAS理论在各种应用领域产生了巨大影响：在经济学中，代理人是贸易商或公司，每个人的特征是初始财富，在互动(交易)后更新；在生物学方面，应用范围从养鱼和鸟类聚集到肿瘤生长；在社会科学中，IMAS模拟了意见形成和评级系统。值得注意的是，类似的原理也被用于控制工程和机器人学，例如，获得对大群个体(机器人、粒子或细胞)的运动的更好控制，以创建或避免形成合意/不合意的图案。这可能会对公共空间的设计方式(例如，在大型聚会或活动的情况下)、基础设施的组织以及公共利益的许多相关方面产生影响。具有多个SS的IMAS和过程也出现在机器人学中，当目标是组织一组相对简单的机器人的协调运动时，而不是使用更少的非交互但更复杂的机器。

It is a part of everyday experience that physical systems tend to move towards an equilibrium (or steady state): objects fallon the floor, the pendulum comes to a stop, milk and coffee mix together to make cappuccino. The concept of equilibrium issomewhat intuitive to us: a given state is an equilibrium for a system if i) when starting in equilibrium, the system doesn'tchange its state (if the pen is on the floor, then it stays there); ii) if we let the system evolve, then it will gradually come to a``stop" or, in other words, it will evolve towards such an equilibrium (if we drop it, the pen will fall on the floor). The conceptof equilibrium can be more dynamic than the one described above: your cappuccino is not going to spontaneously splitback into coffee and milk, i.e. it will stay cappuccino, although each single molecule in it will still be moving. This is whereprobability comes into play and it turns out that equilibria, i.e. steady states (SS), are often better described by probabilitymeasures - so called equilibrium measures (EMs) - rather than by points.Studying the behaviour of dynamics with one EM is the central concern of ergodic theory, which is devoted to establishingcriteria under which the dynamics admits a unique EM and to the analysis of convergence to such an EM. This researchproject is aimed at furthering our understanding of the theory of (infinite dimensional) ergodic processes modelled by(Stochastic) Partial Differential Equations (S)PDEs and of their relation to interacting particle systems (IPS); furthermore,we will investigate the feasibility of a possible strategy to move some (further) steps towards the ambitious goal ofestablishing a theory of non-ergodic processes, i.e. of processes with multiple SS. In particular we will leverage on theergodic theory results that we will obtain, in order to produce understanding in the theory of non-ergodic processes.The class of processes we will consider are so-called McKean-Vlasov (S)PDEs, which underpin the study of interactingmulti-agent system (IMAS). Such processes are of interest within the framework of this proposal because they can bemodelled by (S)PDEs or IPS and they can exhibit multiple SS. More broadly the growing interest of the mathematicalcommunity in these systems stems from their flexibility to describe a vast range of scenarios, and indeed the theory ofIMAS has had a huge impact in a variety of application fields: in economics agents are traders or companies, each ofthem characterized by an initial wealth, which is updated after interactions (trades); in biology applications range from fishschooling and bird flocking, to tumor growth; in the social sciences IMAS have modelled opinion formation and ratingsystems. Notably, analogous principles have been used also in control engineering and robotics, e.g. to gain bettercontrol on the motion of large groups of individuals (robots, particles or cells) to create or avoid the formation ofdesirable/undesirable patterns. This can have an impact on the way public spaces are designed (e.g. in case of largegatherings or events), on organization of infrastructures, and on many connected aspects of public interest. IMAS andprocesses with multiple SS also arise in robotics, when the aim is to organise the coordinated motion of a group ofrelatively simple robots rather than employing fewer non-interacting but more sophisticated machines.