Insertion pre-Lie products and translation of rough paths based on multi-indices

TL;DR

This paper develops a diagram-free framework for rough paths using multi-indices, identifying a pre-Lie algebra structure and establishing algebraic morphisms to rooted tree Hopf algebras, advancing the mathematical understanding of rough path translation.

Contribution

It introduces a novel diagram-free approach to rough paths based on multi-indices, linking insertion pre-Lie algebra structures with Hopf algebra morphisms.

Findings

Identifies the insertion pre-Lie algebra of trees with multi-indices.

Establishes a Hopf algebra morphism to rooted tree Hopf algebra.

Provides a precise algebraic framework for rough path translation.

Abstract

We use the diagram-free approach to regularity structures introduced by Otto et. al. to build rough paths based on multi-indices. We identify the analogue of the insertion pre-Lie algebra of trees and use it to build the corresponding group of translations of rough paths. We make this identification precise by showing that the natural dictionary between trees and multi-indices is a pre-Lie morphism under insertion, which in turn yields a Hopf algebra morphism to the rooted tree Hopf algebra equipped with the extraction-contraction coproduct.