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A theory of regularity structures. (English) Zbl 1332.60093
The author presents a theory in which stochastic partial differential equations (typically with polynomial non-linearities), which are ill-posed due to low regularity of the initial data or roughness of the driving noise, can be reformulated, solved and analysed. The theory consists of an algebraic framework and a calculus for standard operations such as multiplication, composition with smooth functions, integration etc. The theory is consistent with the classical concept of SPDEs as solutions obtained under the theory are limits of classical solutions to suitably regularised problems and also existing results on singular SPDEs (KPZ, stochastic quantisation, Burgers) can be recovered within the new theory. The results can be applied for instance to the continuous parabolic Anderson model in higher dimensions, the stochastic quantisation of \(\Phi^4\) quantum field theory in dimension \(3\), KPZ-type equations or the Navier-Stokes equation with singular forcing. The author links the theory with the theories of rough paths, white noise analysis, Bony’s paraproduct and Colombeau’s generalised functions.

MSC:
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
81S20 Stochastic quantization
82C28 Dynamic renormalization group methods applied to problems in time-dependent statistical mechanics
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