Stochastic Analysis with Modelled Distributions

TL;DR

This paper extends the theory of modelled distributions within Hairer's regularity structures by establishing a Besov topology, enabling advanced stochastic analysis and solutions for distribution-valued SPDEs.

Contribution

It introduces a Besov space framework for modelled distributions, proving the reconstruction theorem and developing stochastic integration theory in this setting.

Findings

Proves the reconstruction theorem on Besov spaces with negative regularity.

Shows that Besov spaces of modelled distributions are UMD Banach spaces and martingale type 2.

Provides a Fubini theorem for stochastic integration and reconstruction.

Abstract

Using a Besov topology on spaces of modelled distributions in the framework of Hairer's regularity structures, we prove the reconstruction theorem on these Besov spaces with negative regularity. The Besov spaces of modelled distributions are shown to be UMD Banach spaces and of martingale type $2$. As a consequence, this gives access to a rich stochastic integration theory and to existence and uniqueness results for mild solutions of semilinear stochastic partial differential equations in these spaces of modelled distributions and for distribution-valued SDEs. Furthermore, we provide a Fubini type theorem allowing to interchange the order of stochastic integration and reconstruction.