Quasi-generalised KPZ equation

TL;DR

This paper derives a renormalised form of the quasi-generalised KPZ equation driven by space-time white noise, advancing the mathematical framework for solving such stochastic PDEs with regularity structures.

Contribution

It introduces an algebraic machinery for systematically removing non-local counterterms and explicitly expressing the renormalised equation, extending the regularity structures approach.

Findings

Provides a systematic algebraic method for renormalisation

Ensures the solution satisfies the chain rule and Itô isometry

Achieves global in time solutions for the equation

Abstract

We derive the renormalised equation for the quasi-generalised KPZ equation with space-time white noise. We complement the program initiated by Gerencs\'er and Hairer for solving quasi-linear equations using regularity structures by an algebraic machinery that gives a systematic tool to remove non-local counterterms and provide a precise expression of the renormalised equation that is consistent with the semilinear case. The solution theory satisfies the chain rule and a natural notion of It\^o isometry, which can be combined to obtain global in time solution.