A Sobolev rough path extension theorem via regularity structures

Liu, Chong; Prömel, David J.; Teichmann, Josef

doi:10.48550/arxiv.2104.06158

Cited by 2 publications

(3 citation statements)

References 17 publications

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“…This motivates the need for a reconstruction theorem in the more general context of Besov spaces, which, as it turns out, has already been established in the formalism of regularity structures in [6, Theorem 3.1], using once again wavelets in its proof, and later in [15]. See also [11], where a similar reconstruction result is proposed and applied to the problem of lifting Sobolev paths to Sobolev rough paths.…”

Section: Motivation and Backgroundmentioning

confidence: 89%

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Besov Reconstruction

Broux,

Lee

2021

Preprint

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Reconstruction theorems tackle the problem of building a global distribution on R d or on a manifold, given a sufficiently coherent family of local approximations, see [5,6,1,14] for examples of such results. In this paper, we establish a reconstruction theorem in the Besov setting, generalising results both of [1] and [6]. While [6] is written in the context of regularity structures and exploits nontrivial results from wavelet analysis, our calculations follow the more elementary and more general approach of [1]. In particular, as in [1], our results are both stated and proved with tools from the theory of distributions. As an application, we present an alternative proof of a (Besov) Young multiplication theorem which does not require the use of paraproducts. Contents 1. Motivation and Background Outline 2. Notations 3. Main Results 3.1. The problem of reconstruction 3.2. Comparison of Theorem 3.4 with the literature 3.3. The problem of Young multiplication in Besov spaces 4. A general Besov reconstruction theorem 4.1. Statement of the result 4.2. Uniqueness of reconstruction 4.3. Existence for γ > 0 4.4. Reconstruction bound for γ > 0 4.5. The reconstruction for γ ≤ 0 4.6. The reconstruction is Besov 5. Proof of Theorem 3.4 from Theorem 4.5 6. Proof of Equation 3.12 from Theorem 4.5 Appendix A. Besov spaces Appendix B. A technical lemma on series 2020 Mathematics Subject Classification. 46F10, 60L30.

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Section: Motivation and Backgroundmentioning

confidence: 89%

“…13. Compare (3.7) with[11, Equation (3.1)].4. A general Besov reconstruction theoremNow let us turn to the statement and proof of our most general reconstruction result, Theorem 4.5 below.…”

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confidence: 99%

Besov Reconstruction

Broux,

Lee

2021

Preprint

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“…While the approach of Lyons and Victoir is non-constructive, an explicit approach based on so-called Fourier normal ordering was developed by J. Unterberger [23]. The latter approach is in a related spirit to the one relying on Hairer's regularity structures [6, Section 13], see also [1,15]. Recently, Tapia and Zambotti [22] generalized Lyons-Victoir extension theorem to the case of anisotropic Hölder continuous paths, i.e., allowing each component to have a different regularity.…”

Section: Introductionmentioning

confidence: 99%

Optimal Extension to Sobolev Rough Paths

2022

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We show that every $\mathbb {R}^{d}$ ℝ d -valued Sobolev path with regularity α and integrability p can be lifted to a weakly geometric rough path in the sense of T. Lyons with exactly the same regularity and integrability, provided α > 1/p > 0. Moreover, we prove the existence of unique rough path lifts which are optimal w.r.t. strictly convex functionals among all possible rough path lifts given a Sobolev path. This paves a way towards classifying rough path lifts as solutions of optimization problems. As examples, we consider the rough path lift with minimal Sobolev norm and characterize the Stratonovich rough path lift of a Brownian motion as optimal lift w.r.t. a suitable convex functional. Generalizations of the results to Besov spaces are briefly discussed.

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A Sobolev rough path extension theorem via regularity structures

Cited by 2 publications

References 17 publications

Besov Reconstruction

Besov Reconstruction

Optimal Extension to Sobolev Rough Paths

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