Many stochastic PDEs - the dynamical Phi^4 equation, the KPZ equation, and the parabolic Anderson model - are too singular for their nonlinear terms to make classical sense, because the solution itself is only a distribution and distributions cannot be multiplied. A solution is instead built as the limit of mollified equations corrected by diverging counter-terms, and theories such as regularity structures and paracontrolled distributions make that limit well-posed by expanding the unknown around explicit Gaussian objects whose local behaviour it is required to mimic. The renormalization constants are not bookkeeping: they encode the rate at which temperature approaches criticality or the reference frame that must be tracked in the microscopic particle and interface models whose scaling limits these equations universally are.
A comprehensive 41-page survey by Ivan Corwin and Hao Shen covering recent developments in singular stochastic partial differential equations from a stochastic analysis perspective. The paper reviews…