Singular stochastic PDEs

Abstract

We present a series of recent results on the well-posedness of very singular parabolic stochastic partial differential equations. These equations are such that the question of what it even means to be a solution is highly non-trivial. This problem can be addressed within the framework of the recently developed theory of "regularity structures", which allows to describe candidate solutions locally by a "jet", but where the usual Taylor polynomials are replaced by a sequence of custom-built objects. In order to illustrate the theory, we focus on the particular example of the Kardar-Parisi-Zhang equation, a popular model for interface propagation.

Proceedings of the ICM

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• Spectrum of the totally asymmetric simple exclusion process on a periodic lattice -- first excited states
• Random tensors, propagation of randomness, and nonlinear dispersive equations
• Large scale behaviour of 3D continuous phase coexistence models
• Some recent progress in singular stochastic PDEs
• Paracontrolled quasilinear SPDEs
• Magnus expansions and pseudospectra of Master Equations
• Global well-posedness of complex Ginzburg-Landau equation with a space-time white noise