Diagram-free approach for convergence of tree-based models in Regularity Structures

TL;DR

This paper introduces a diagram-free method for proving the convergence of tree-based models in Regularity Structures, expanding the applicability to more singular SPDEs and providing new insights into the algebraic foundations of the theory.

Contribution

It develops a diagram-free approach at the decorated tree level, broadening the scope of convergence results for Regularity Structures and clarifying algebraic aspects.

Findings

Broadened class of singular SPDEs covered by the approach

Simplified convergence proofs without diagrams

Enhanced understanding of algebraic structures in Regularity Structures

Abstract

In this work, we translate at the level of decorated trees some of the crucial arguments which have been used in arXiv:2112.10739 for proposing a diagram-free approach for the convergence of the model in Regularity Structures. This allows us to broaden the perspective and enlarge the scope of singular SPDEs covered by this approach. It also sheds new light on algebraic structures introduced in the foundational paper of Martin Hairer on Regularity structures which was used later for recursively described renormalised models.