Stochastic analysis with modelled distributions

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.Key words and phrases: UMD and M-type 2 Banach spaces, regularity structures, stochastic integration in Banach spaces, stochastic partial differential equations. MSC 2010 Classification: Primary: 35R60, 46E35, 60H15; Secondary: 46N30, 60H05. 1 collections of "Taylor" coefficients around each point w.r.t. a set of abstract monomials [You36] Laurence C. Young. An inequality of the Hölder type, connected with Stieltjes integration.

Paracontrolled calculus and regularity structures II

“…In the setting of pathspace analysis on manifolds, this kind of pathwise dynamics provided a clean understanding of Driver's flow equation on pathspace, in relation with quasi-invariance questions for Wiener measure on pathspace over a compact Riemannian manifold [12,19,2]. One may also make sense of classical stochastic PDEs on the space of models or modeled distributions, as in Liu, Prömel and Teichmann's work [18].…”

“…The first proof that this is possible for any choice of Hölder control h was found by Lyons and Victoir [20], for geometric rough paths, using the axiom of choice. This unexpected device stimulated further explorations of this questions, and different proofs not using the axiom of choice were given subsequently [25,14,24,18]. Unterberger constructs in [25] a rough path above h using paraproduct-like tools.…”

“…Unterberger constructs in [25] a rough path above h using paraproduct-like tools. Hairer uses in [14] the reconstruction theorem for that purpose, while Liu, Prömel and Teichmann use in [18] a version of the reconstruction theorem for Sobolev models and the notion of Sobolev rough path to extend Lyons-Victoir extension in their setting. Tapia and Zambotti used in [24] an explicit form of the Baker-Campbell-Hausdorff formula to bypass the use of the axiom of choice in the construction of a lift, and gave a parameterization of the set of all branched rough paths above h. Theorem 1 provides a direct access to such an extension result, in so far as the family of trees with only one node with decoration j, for 1 j , is a subset of the generator set G +…”

A Sobolev rough path extension theorem via regularity structures

“…This section is devoted to show that every path of suitable Sobolev regularity can be lifted to a weakly geometric rough path possessing exactly the same Sobolev regularity. To prove this statement, we proceed via an approach based on Hairer's reconstruction theorem appearing in the theory of regularity structures [Hai14], which requires not only to use a Sobolev topology on the space of modelled distributions, as introduced in [HL17] and [LPT21b] (see also [HR20]), but additionally to generalize the definition of models from the originally assumed Hölder bounds to more general Sobolev bounds. For a further discussion on this point we refer the end of Subsection 3.1.…”

“…While the rough path lift of a Hölder continuous path is a known and fairly simple application of Hairer's reconstruction theorem ([Hai14, Theorem 3.10]), see [FH14,Proposition 13.23] or [Bra19], lifting a Sobolev path lies outside the current framework of regularity structures and thus requires some serious additional effort. Indeed, we need to use a Sobolev topology on the space of modelled distributions, as introduced in [HL17] and [LPT21b] (see also [HR20]) and additionally to generalize the definition of models from the originally required Hölder bounds to some more general Sobolev bounds. In other words, we cannot apply Hairer's reconstruction theorem directly and instead need to generalize the essential features of Hairer's reconstruction operator to our setting allowing for Sobolev models and Sobolev modelled distributions, see Remark 3.11 for a more detailed discussion.…”