A solution theory for quasilinear singular SPDEs

TL;DR

This paper extends the theory of regularity structures to construct local renormalised solutions for a broad class of quasilinear stochastic PDEs, including KPZ-type equations driven by space-time white noise.

Contribution

It generalizes previous results by enabling the construction of solutions for quasilinear SPDEs within the regularity structures framework, with simplified renormalization counterterms.

Findings

Constructed local renormalised solutions for quasilinear SPDEs

Renormalisation counterterms are local functionals of the solution

Framework applies to equations with KPZ-type nonlinearities

Abstract

We give a construction allowing to construct local renormalised solutions to general quasilinear stochastic PDEs within the theory of regularity structures, thus greatly generalising the recent results of [BDH16,FG16,OW16]. Loosely speaking, our construction covers quasilinear variants of all classes of equations for which the general construction of [Hai14,BHZ16,CH16] applies, including in particular one-dimensional systems with KPZ-type nonlinearities driven by space-time white noise. In a less singular and more specific case, we furthermore show that the counterterms introduced by the renormalisation procedure are given by local functionals of the solution. The main feature of our construction is that it allows to exploit a number of existing results developed for the semilinear case, so that the number of additional arguments it requires is relatively small.