Symmetries for the gKPZ equation via multi-indices

TL;DR

This paper analyzes the symmetries of the one-dimensional gKPZ equation using multi-indices, simplifying previous approaches and completing the study of the chain rule in this context.

Contribution

It introduces multi-indices to study symmetries of the gKPZ equation, providing a simpler alternative to decorated trees and completing prior research on the chain rule.

Findings

Computed the dimension of symmetry-related spaces for gKPZ

Showed multi-indices simplify symmetry analysis

Extended previous work on the chain rule for gKPZ

Abstract

In this work, we study the two main symmetries for the one-dimensional generalised KPZ equation (gKPZ): the chain rule and the It\^o Isometry. We consider the equation in the full-subcritical regimes and use multi-indices that avoid an over-parametrization of the renormalised equation to compute the dimension of the two spaces associated with these two symmetries. Our proof is quite elementary and shows that multi-indices provide in this case a simplification in comparison to the results obtained via decorated trees. It also completes the program on the study of the chain rule initiated in arxiv:1902.02884 and continued in arxiv:2403.17066.