FOR 2402: Rough Paths, Stochastic Partial Differential Equations and Related Topics

粗糙路径与随机偏微分方程（SPDE）的相互作用一直在继续，因为写的初步建议的研究单位约3年前，上升到最活跃的领域之一，在现代概率论和分析的交叉。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演化系统进行建模几乎不可避免地会导致（偏）微分方程，然而，我们现在面临着许多情况（从统计物理学和量子场论到神经科学和金融市场），其中所有的光滑性假设--微分方程的经典理论以这种或那种形式依赖于这些假设--都被违反了。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。另一个方面是这些示例的核心：内在随机性以及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的重要性大大增加，这并不奇怪。在其经典形式中，SPDE理论的基础在30多年前就已经确定，它完全依赖于伊藤在希尔伯特空间环境中的（基于鞅的）随机分析。一场（开始时缓慢的）革命以里昂的粗糙路径理论的形式出现，正好是20年前，为常微分微分方程制定的。他意识到，受噪声影响的方程的分析不适定性可以通过识别该噪声的普遍提升来驯服，其精确结构由方程决定。在布朗噪声驱动的常微分方程的情况下，这相当于增加了利维面积，导致了决定性的“路径”理论。不到10年前，Gubinelli-Tindel（2010）和Caruana-Friz（2009）成功地首次提出了粗糙路径意义下噪声驱动的偏微分方程。在2012年，Hairer着名地使用粗糙路径来解决KPZ（燃烧片）方程，由于时空粗糙噪声而严重不适定。不久之后，他提出了一个概括的粗糙路径的“理论的规律性结构”，其中-在某种意义上-每个SPDE问题在手诱导自己量身定制的代数/分析粗糙路径类型的框架。许多其他SPDE，特别是统计物理学和量子场论的SPDE，可以第一次被分析。(For这些作品Hairer被授予2014年菲尔兹奖。在一个平行的发展，Gubinelli，Imkeller，Perkowski发起的paracontrolled的方法，方便地基于现有的工具，从谐波分析，一些类似的问题与无穷维noises.The研究单位的总体目标是一个持续的重点，以推进我们的理解粗糙路径，规律性结构和随机偏微分方程的重要相互作用。

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.