Operations on Modelled Distributions

Abstract

The original motivation for the development of the theory of regularity structures was to provide robust solution theories for singular stochastic PDEs like the KPZ equation or the dynamical $\varPhi_{3}^{4}$ model. The idea is to reformulate them as fixed point problems in some space ${\mathcal{D}}^{\gamma }$ (or rather a slightly modified version that takes into account possible singular behaviour near time 0) based on a suitable random model in a regularity structure purpose-built for the problem at hand. In order to achieve this this chapter provides a systematic way of formulating the standard operations arising in the construction of the corresponding fixed point problem (differentiation, multiplication, composition by a regular function, convolution with the heat kernel) as operations on the spaces ${\mathcal{D}}^{\gamma }$.