Coordination Funds

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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.