Paracontrolled calculus and regularity structures (II)

TL;DR

This paper establishes a fundamental equivalence between models and modelled distributions in regularity structures and paracontrolled systems, providing explicit constructions and parametrizations that enhance understanding of singular SPDEs.

Contribution

It introduces a general equivalence between models and paracontrolled systems over regularity structures, with explicit construction methods and parametrization results.

Findings

Equivalence between models and paracontrolled systems established

Explicit construction of modelled distributions from paracontrolled systems

Simplified form for regularity structures used in singular SPDEs

Abstract

We prove a general equivalence statement between the notions of models and modelled distributions over a regularity structure, and paracontrolled systems indexed by the regularity structure. This takes in particular the form of a parametrisation of the set of models over a regularity structure by the set of reference functions used in the paracontrolled representation of these objects. A number of consequences are emphasized. The construction of a modelled distribution from a paracontrolled system is explicit, and takes a particularly simple form in the case of the regularity structures introduced by Bruned, Hairer and Zambotti for the study of singular stochastic partial differential equations.