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.
Highlights
Modelled distributions are the spine of Hairer’s theory of regularity structures [14]: they constitute a way to describe locally generalized functions of certain degrees of regularity by means of functions (“modelled distributions”) taking values in a graded vector space (“regularity structure”), which satisfy certain graded estimates
One of the key insight of the theory of regularity structures is that the solutions of some singular stochastic partial differential equations, like the KPZ equation or the 2D parabolic Anderson model, are more suitable described using an enlarged basis of monomials
Other motivations to work with specific Besov norms on the spaces of modelled distributions recently arose in the work [17] of Hairer and Labbé, where the solution to the multiplicative stochastic heat equation starting from a Dirac mass is constructed, and in the work [6] of Cannizzaro, Friz and Gassiat, where Malliavin calculus is implemented in the context of regularity structures
Summary
Modelled distributions are the spine of Hairer’s theory of regularity structures [14]: they constitute a way to describe locally generalized functions of certain degrees of (ir-) regularity by means of functions (“modelled distributions”) taking values in a graded vector space (“regularity structure”), which satisfy certain graded estimates. In the abstract setting of regularity structures, the so-called reconstruction operator provides a way to continuously map the modelled distributions to generalized Rd valued functions on space-time, which is the assertion of the celebrated reconstruction theorem, see Theorem 3.10 in [14]. We shall prove here that the Besov space Dγp,q of modelled distributions (for p, q ≥ 2) has the martingale type 2 and the UMD property, respectively, see Proposition 3.2 Since this suffices to set up a rich stochastic integration theory as needed for the treatment of stochastic partial differential equations with Brownian drivers like in the books of Da Prato–Peszat–Zabczyk [10,25], the results in the second part of the article pave the way to combine the powerful tools of stochastic integration and Hairer’s theory of regularity structure in a novel way. The space p is the Banach space of all sequences (xn)n∈N such that n∈N |xn|p < ∞ and the corresponding norm is denoted by · p
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