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Sewing, Reconstruction and Schauder in rough analysis and regularity structures

Abstract : In this thesis, we derive analytic results related to the theories of Rough Paths and Regularity Structures, with the point of view of germs, that is, families of local approximations of functions or distributions. We first establish a Sewing Lemma in the regime 0 < γ ≤ 1, giving a construction which is non unique nor canonical but still continuous. As a corollary, we exhibit a bicontinuous parametrisation of the set of Rough Paths by a product of Hölder spaces, generalising both the Lyons--Victoir extension theorem and a recent result by Tapia--Zambotti. Secondly, we propose a Reconstruction Theorem in the context of Besov spaces, generalising results of Hairer--Labbé and Caravenna--Zambotti. As a corollary, we provide a new proof of the multiplication theorem in Besov spaces without relying on paraproducts. Finally, we study the regularising properties of singular kernels against germs. A first result is the construction of a convolution map which acts on general coherent and homogeneous germs. We also revisit Hairer's multilevel Schauder estimates, providing a presentation and a proof which make only minimal references to the formalism of regularity structures.
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Submitted on : Tuesday, November 22, 2022 - 11:53:09 AM
Last modification on : Wednesday, November 23, 2022 - 1:02:07 PM


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  • HAL Id : tel-03865422, version 1


Lucas Broux. Sewing, Reconstruction and Schauder in rough analysis and regularity structures. Probability [math.PR]. Sorbonne Université, 2022. English. ⟨NNT : 2022SORUS278⟩. ⟨tel-03865422⟩



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