Modelled distributions of Triebel–Lizorkin type

Abstract

A. In order to provide a local description of a regular function in a small neighbourhood of a point x, it is sufficient by Taylor's theorem to know the value of the function as well as all of its derivatives up to the required order at the point x itself. In other words, one could say that a regular function is locally modelled by the set of polynomials. The theory of regularity structures due to Hairer generalizes this observation and provides an abstract setup, which in the application of singular SPDE extends the set of polynomials by functionals constructed from, e.g., white noise. In this context, the notion of Taylor polynomials is lifted to the notion of so-called modelled distributions. The celebrated reconstruction theorem, which in turn was inspired by Gubinelli's sewing lemma, is of paramount importance for the theory. It enables to reconstruct a modelled distribution as a true distribution on R d which is locally approximated by this extended set of models or "monomials". In the original work of Hairer, the error is measured based on Hölder norms. This was then generalized to the whole scale of Besov spaces by Hairer and Labbé in subsequent papers. It is the aim of this work to adapt the analytic part of the theory of regularity structures to the scale of Triebel-Lizorkin spaces.

Paracontrolled calculus and regularity structures I

“…One advantage of the explicit construction of the reconstruction operator given by Theorem 14 is that this representation is flexible enough to work in other functional settings than the present B γ 88 -type space D γ pT , gq. The continuity properties of the paraproduct operator on Besov, Triebel-Lizorkin or Sobolev-Slobodeckij spaces are well-known, and allow for a direct approach to reconstruction in these spaces, in the line of the recent works [13,14,15,17].…”

Stochastic analysis with modelled distributions

“…As in the case of Hairer's original reconstruction theorem, these two regimes require different proofs but both can be proven along similar lines of arguments as used by Hairer's original proof on the existence of the reconstruction operator. Let us also mention that the reconstruction theorem was recently obtained by Hensel and Rosati [HR17] for Triebel-Lizorkin type spaces with positive regularity parameter. A natural application of the reconstruction operator applied to modelled distributions with negative regularity is Lyons-Victoir's extension theorem [LV07], cf.…”

A Sobolev rough path extension theorem via regularity structures

“…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], lifting a Sobolev path lies outside the current framework of regularity structures. Indeed, as we will see in Subsection 3.2, we need not only to use a Sobolev topology on the space of modelled distributions, as introduced in [HL17] and [LPT20] (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.…”

“…[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 not only to use a Sobolev topology on the space of modelled distributions, as introduced in [HL17] and [LPT20] (see also [HR20]), but 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.…”