Singular SPDEs: Approximation and Statistical Properties

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The powerful and novel theories of regularity structures and paracontrolled distributions have so far been used mostly for deriving existence, uniqueness and regularity results for singular stochastic partial differential equations (SPDEs). We feel that the time is now mature for a further exploration of the full power of these techniques: to extend them for deriving qualitative properties of the solutions, in particular physical effects such as aging and intermittency and alike. We will do this for two of the most prominent and promising equations, the Kardar-Parisi-Zhang equation and the parabolic Anderson model. By combining our expertise in aging and intermittency (J.-D.D. and W.K.) and paracontrolled distributions / regularity structures (N.P.) respectively, we dispose of a wide range of techniques which will allow us to gain a much better understanding of these equations.