A stochastic reconstruction theorem

TL;DR

This paper extends the stochastic sewing lemma to a stochastic reconstruction theorem within Hairer's regularity structures, providing new tools for stochastic analysis and integration against white noise.

Contribution

It introduces a stochastic reconstruction theorem inspired by the stochastic sewing lemma, incorporating the distributional approach of Caravenna and Zambotti.

Findings

Establishes a stochastic reconstruction theorem applicable in regularity structures.

Proposes two variations motivated by different stochastic integration methods.

Connects the stochastic sewing lemma with Hairer's reconstruction framework.

Abstract

In a recent landmark paper, Khoa L\^e (2020) established a stochastic sewing lemma which since has found many applications in stochastic analysis. He further conjectured that a similar result may hold in the context of the reconstruction theorem within Hairer's regularity structures. The purpose of this article is to provide such a stochastic reconstruction theorem. We also discuss two variations of this theorem, motivated by different constructions of stochastic integration against white noise. Our formulation makes use of the distributional viewpoint of Caravenna--Zambotti (2021).