Multi-indices coproducts from ODEs to singular SPDEs

TL;DR

This paper introduces explicit formulae for coproducts associated with multi-indices in ODEs and singular SPDEs, connecting algebraic structures with practical computations.

Contribution

It provides new explicit formulae for coproducts in the context of multi-indices, applicable to both ODEs and singular SPDEs, based on dual product adjoints.

Findings

Derived explicit formulae for coproducts using symmetry factors

Connected coproducts to dual products with simple formulae

Applicable to algebraic structures in ODEs and SPDEs

Abstract

In this work, we introduce explicit formulae for the coproducts at play for multi-indices in ODEs and in singular SPDEs. The two coproducts described correspond to versions of the Butcher-Connes-Kreimer and extraction/contraction coproducts with multi-indices. The main idea is to use the fact that these coproducts are the adjoints of dual products for which one has explicit simple formulae. We are able to derive the explicit formulae via an inner product defined from a symmetry factor easily computable for multi-indices.