Regularity structures

Summary

A regularity structure is an algebraic framework, introduced by Martin Hairer, for describing functions and distributions locally by generalized Taylor expansions. The building blocks are abstract symbols with assigned degrees of regularity, together with a group that says how to re-expand them around another point. The theory gives a solution theory for singular stochastic partial differential equations, whose nonlinear terms are not defined classically. Examples include the Kardar–Parisi–Zhang equation and the dynamical $\Phi^4_3$ model.