A stochastic reconstruction theorem

In a recent landmark paper, Khoa Le (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. Our formulation makes use of the distributional viewpoint of Caravenna--Zambotti (2021).