A Littlewood-Paley description of modelled distributions

TL;DR

This paper bridges Hairer's regularity structures and Gubinelli et al.'s paracontrolled calculus by providing a Littlewood-Paley framework for modelled distributions, akin to Besov spaces.

Contribution

It introduces a Littlewood-Paley description of modelled distributions, connecting two major theories in regularity analysis.

Findings

Establishes a fundamental link between regularity structures and paracontrolled calculus.

Provides a Besov-like description of modelled distributions.

Enhances understanding of the structure of distributions in stochastic analysis.

Abstract

We exhibit a fundamental link between Hairer's theory of regularity structures and the paracontrolled calculus of Gubinelli, Imkeller and Perkowski. By using paraproducts we provide a Littlewood-Paley description of the spaces of modelled distributions in regularity structures that is similar to the Besov description of classical H\"older spaces.