Coordination Funds

粗糙路径与随机偏微分方程（SPDEs）的相互作用，自大约3年前撰写研究单元的初步建议以来，一直在继续，成为现代概率论和分析交叉中最活跃的领域之一。这在很大程度上源于这样一个事实，即对一个具有许多变量的进化系统进行建模几乎不可避免地导致（偏）微分方程，然而，我们现在面临许多情况（从统计物理学和量子场理论到神经科学和金融市场），在这些情况下，所有的平滑假设——经典微分方程理论以一种或另一种形式依赖的假设——都被严重违反了。简单的例子包括燃烧纸张的相变或固定收益市场收益率曲线的发展。另一个方面是这些例子的核心：内在随机性和由此产生的对统计描述的需求。在过去的几十年里，随机偏微分方程领域得到了极大的重视，这一点也不奇怪。在其经典形式中，SPDE理论的基础在30多年前就已经确立，它完全依赖于伊藤在希尔伯特空间环境下的（基于鞅的）随机分析。整整20年前，Lyons的粗糙路径理论出现了一场革命（一开始是缓慢的），该理论用于常微分微分方程。他意识到受噪声影响的方程的分析不适定性可以通过确定噪声的普遍提升来解决，而噪声的精确结构是由方程决定的。在由布朗噪声驱动的ode的情况下，这相当于增加了Levy的面积，从而导致决定性的“路径”SDE理论。不到10年前，Gubinelli-Tindel（2010）和Caruana-Friz（2009）成功地提出了粗糙路径意义下噪声驱动的偏微分微分方程的第一个公式。2012年，海勒用粗糙路径来求解KPZ（燃烧片）方程，这一方程由于时空粗糙噪声而严重不适定。不久之后，他提出将粗糙路径推广到“规则结构理论”，在某种意义上，手头的每个SPDE问题都引出了自己定制的代数/解析粗糙路径类型框架。许多其他spde，特别是来自统计物理和量子场论的spde，可以第一次被分析。（因为这些作品，海勒获得了2014年菲尔兹奖。）在一个平行的发展中，Gubinelli, Imkeller， Perkowski发起了一种辅助控制方法，方便地基于谐波分析的现有工具，来解决许多具有无限维噪声的类似问题。本研究单元的总体目标是继续专注于推进我们对粗糙路径、规则结构和随机偏微分方程之间重要相互作用的理解。

The interplay of rough paths with stochastic partial differential equations (SPDEs) has continued, since writing the initial proposal of the research unit some 3 years ago, to rise to one of the most active areas in the intersection of modern probability theory and analysis. Much stems from the fact that the modelling of an evolving system with many variables leads almost inevitably to (partial) differential equations, and yet, we now face many situations (ranging from statistical physics and quantum field theory to neuroscience and financial markets) in which all smoothness assumptions - on which classical theories of differential equations rely in one form or another - are a fortiori violated. Easy (to state) examples include the phase transition of a burning paper sheet or the development of yield curves in fixed income markets. Another aspect is central to these examples: the intrinsic randomness and the resulting need for a statistical description. It comes as no surprise that the field of stochastic partial differential equations has massively gained importance over the last decades. In its classical form, the foundations of which were settled 30+ years ago, SPDE theory fully relied on Ito’s (martingale based) stochastic analysis in a Hilbert space setting.A (slow in the beginning) revolution came in the form of Lyons’ rough path theory, exactly 20 years ago, formulated for ordinary differential differential equations. He realized that analytical ill-posedness of equations subjected to noise can be tamed by identifying a universal lift of that noise, the precise structure of which is dictated by the equation. In the case of ODEs driven by Brownian noise, this amounts to add Levy’s area, leading to a decisive “pathwise” SDE theory. Less than 10 years ago, Gubinelli–Tindel (2010) and Caruana–Friz (2009) succeeded with first formulations of partial differential differential equations driven by noise in the rough path sense. In 2012, Hairer famously used rough paths to solve the KPZ (burning sheet) equation, severely ill-posed due to space-time rough noise. Soon afterwards, he proposed a generalization of rough paths to “a theory of regularity structures”, where - in a sense - each SPDE problem at hand induces its own tailormade algebraic/analytic rough path type framework. Many other SPDEs, especially from statistical physics and quantum field theory could then be - for the first time - analyzed. (For these works Hairer was awarded the 2014 Fields medal.) In a parallel development, Gubinelli, Imkeller, Perkowski initiated the paracontrolled approach, conveniently based on existing tools from harmonic analysis, to a number of similar problems with infinite dimensional noise.The overall aim of this research unit is a continued focus to advance our understanding of the important interplay of rough paths, regularity structures and stochastic partial differential equations.