Regularity structure
Martin Hairer's theory of regularity structures provides a framework for studying a large class of subcritical parabolic stochastic partial differential equations arising from quantum field theory. The framework covers the Kardar–Parisi–Zhang equation, the equation and the parabolic Anderson model, all of which require renormalization in order to have a well-defined notion of solution.
A key advantage of regularity structures over previous methods is its ability to pose the solution of singular non-linear stochastic equations in terms of fixed-point arguments in a space of “controlled distributions” over a fixed regularity structure. The space of controlled distributions lives in an analytical/algebraic space that is constructed to encode key properties of the equations at hand. As in many similar approaches, the existence of this fixed point is first poised as a similar problem where the noise term is regularised. Subsequently, the regularisation is removed as a limit process. A key difficulty in these problems is to show that stochastic objects associated to these equations converge as this regularisation is removed.
Hairer won the 2021 Breakthrough Prize in mathematics for introducing regularity structures.