McKean Vlasov Stochastic Partial Differential Equations

这是日常经验的一部分，物理系统倾向于走向平衡（或稳定状态）：物体落在地板上，钟摆停下来，牛奶和咖啡混合在一起制作卡布奇诺。平衡的概念对我们来说有点直观：一个给定的状态是一个系统的平衡，如果i）当系统从平衡开始时，系统不改变它的状态（如果笔在地板上，那么它就留在那里）; ii）如果我们让系统进化，那么它将逐渐“停止”，或者换句话说，它将朝着这样的平衡进化。（如果我们把它掉在地上，笔就会掉在地板上）。平衡的概念可以比上面描述的更具动态性：你的卡布奇诺不会自发地分裂成咖啡和牛奶，也就是说，它仍然是卡布奇诺，尽管其中的每一个分子仍然在移动。这就是概率发挥作用的地方，事实证明，平衡，即稳定状态（SS），往往更好地描述了概率测度-所谓的平衡测度（EM）-而不是点。研究一个EM的动态行为是遍历理论的中心问题，这是致力于建立准则下的动力学承认一个唯一的EM和分析收敛到这样的EM。这个研究项目的目的是为了加深我们对（无限维）遍历过程，（随机）偏微分方程及其与相互作用粒子系统的关系;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的宏伟目标迈出一些（进一步的）步骤，即具有多个SS的过程。特别是，我们将利用遍历理论的结果，我们将获得，以产生在非遍历过程的理论的理解。我们将考虑的一类过程是所谓的McKean-Vlasov（S）偏微分方程，它支持的研究interactingmulti-agent系统（IMAS）。这种过程在该提议的框架内是令人感兴趣的，因为它们可以由（S）PDE或IPS建模，并且它们可以表现出多个SS。更广泛地说，商业界对这些系统越来越感兴趣，这是因为它们能够灵活地描述各种各样的场景，事实上，IMAS理论在各种应用领域都产生了巨大的影响：在经济学中，代理人是交易者或公司，他们中的每一个人都有一个初始财富，在相互作用（交易）后更新;在生物学上的应用范围从鱼群和鸟群到肿瘤生长;在社会科学上，IMAS已经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技术，例如，更好地控制大群个体（机器人，粒子或细胞）的运动，以创建或避免形成期望/不期望的模式。这可能会影响公共空间的设计方式（例如，在大型集会或活动的情况下），基础设施的组织以及公共利益的许多相关方面。多SS的IMAS和过程也出现在机器人技术中，当目标是组织一组相对简单的机器人的协调运动，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.