Stochastic PDEs and Regularity Structures

Many physically relevant stochastic PDEs — including the KPZ equation $\partial_t h = \Delta h + |\nabla h|^2 + \xi$ for interface growth and the $\Phi^4_3$ model from quantum field theory — are classically ill-posed because the noise $\xi$ is too irregular for the nonlinear terms to make sense. The theory of regularity structures provides a systematic way to assign meaning to these equations. We build a graded algebra of generalized Taylor expansions, define a notion of modelled distributions, and show how a renormalization procedure yields convergent solutions. The abstract framework is illustrated concretely with the KPZ equation \in one spatial dimension.