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

自从大约3年前撰写研究单位的初步建议以来，粗糙路径与随机偏微分方程(SPDEs)的相互作用一直在继续，成为现代概率理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，即对一个多变量演化系统的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(从统计物理和量子场论到神经科学和金融市场)，在这些情况下，所有的光滑假设--经典的微分方程式理论以这样或那样的形式依赖于这些假设--都遭到了强烈的违反。容易描述的例子包括燃烧的纸张的相变或固定收益市场的收益率曲线的发展。另一个方面是这些例子的核心：内在的随机性和由此产生的对统计描述的需要。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，这并不令人惊讶。在其经典形式中，SPDE理论的基础在30年前就已经奠定了，它完全依赖于希尔伯特空间中的Ito(基于鞅的)随机分析。一场(开始缓慢的)革命以Lyons的粗糙路径理论的形式到来，恰好是20年前，为常微分方程组建立的。他认识到，受噪声影响的方程的解析不适性可以通过识别噪声的普遍升力来驯服，其精确结构由方程决定。在布朗噪声驱动的颂歌的情况下，这相当于增加了利维的面积，导致了决定性的“路径”SDE理论。不到10年前，Gubinelli-Tindel(2010)和Caruana-Friz(2009)成功地首次建立了粗糙路径意义下的噪声驱动的偏微分方程组。2012年，海尔著名地使用粗糙路径求解KPZ(燃烧薄板)方程，由于时空粗糙噪声，KPZ方程严重不适定。不久之后，他提出将粗糙路径推广为“正则性结构理论”，在某种意义上，手头的每个SPDE问题都归纳出它自己的量身定制的代数/解析粗糙路径类型框架。然后，许多其他的SPDEs，特别是来自统计物理和量子场论的SPDEs，可以第一次被分析。(由于这些作品，海尔被授予2014年度菲尔兹奖。)在并行发展中，Gubinelli，Imkeller，Perkowski开创了基于调和分析的现有工具的准控制方法，用于许多类似的具有无限维噪声的问题。这个研究单元的总体目标是继续关注促进我们对粗糙路径、正则性结构和随机偏微分方程之间的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.