Paracontrolled calculus and regularity structures II

Bailleul, Ismael; Hoshino, Masato

Journal de l’École polytechnique — Mathématiques, Volume 8 (2021), pp. 1275-1328.

We prove a general equivalence statement between the notions of models and modeled distributions over a regularity structure, and paracontrolled systems indexed by the regularity structure. This takes in particular the form of a parameterization of the set of models over a regularity structure by the set of reference functions used in the paracontrolled representation of these objects. A number of consequences are emphasized. The construction of a modeled distribution from a paracontrolled system is explicit, and takes a particularly simple form in the case of the regularity structures introduced by Bruned, Hairer and Zambotti for the study of singular stochastic partial differential equations.

Nous démontrons un énoncé général d’équivalence entre les notions de modèles et de distributions modelées définis sur une structure de régularité et la notion de système paracontrôlé indexé par cette structure de régularité. Cet énoncé donne en particulier une paramétrisation de l’ensemble des modèles sur une structure donnée par l’ensemble des fonctions de référence utilisées dans la représentation paracontrôlée de ces objets. Un certain nombre de conséquences sont données. La construction d’une distribution modelée à partir d’un système paracontrôlé est explicite et prend une forme particulièrement simple dans le cadre des structures de régularités introduites par Bruned, Hairer et Zambotti pour l’étude des équations aux dérivées partielles stochastiques singulières.

Classification: 60L30, 60L40
Keywords: Regularity structures, models, modeled distributions/functions, paracontrolled calculus, parameterization
Mot clés : Structures de régularités, modèles, distributions/fonctions modelées, calcul paracontrôlé, paramétrisation

[1] Bahouri, Hajer; Chemin, Jean-Yves; Danchin, Raphaël Fourier analysis and nonlinear partial differential equations, Grundlehren Math. Wiss., 343, Springer, Heidelberg, 2011 | DOI | MR | Zbl

[2] Bailleul, Ismael Regularity of the Itô-Lyons map, Confluentes Math., Volume 7 (2015) no. 1, pp. 3-11 | DOI | Numdam | MR | Zbl

[3] Bailleul, Ismael; Bernicot, Frédéric High order paracontrolled calculus, Forum Math. Sigma, Volume 7 (2019), e44, 94 pages | DOI | MR | Zbl

[4] Bailleul, Ismael; Bernicot, Frédéric; Frey, Dorothee Space-time paraproducts for paracontrolled calculus, 3D-PAM and multiplicative Burgers equations, Ann. Sci. École Norm. Sup. (4), Volume 51 (2018) no. 6, pp. 1399-1456 | DOI | MR | Zbl

[5] Bailleul, Ismael; Hoshino, Masato Paracontrolled calculus and regularity structures I, J. Math. Soc. Japan, Volume 73 (2021) no. 2, pp. 553-595 | DOI | MR | Zbl

[6] Bruned, Yvain; Chandra, Ajay; Chevyrev, Ilya; Hairer, Martin Renormalising SPDEs in regularity structures, J. Eur. Math. Soc. (JEMS), Volume 23 (2021) no. 3, pp. 869-947 | DOI | MR | Zbl

[7] Bruned, Yvain; Hairer, Martin; Zambotti, Lorenzo Algebraic renormalisation of regularity structures, Invent. Math., Volume 215 (2019) no. 3, pp. 1039-1156 | DOI | MR

[8] Caravenna, Francesco; Zambotti, Lorenzo Hairer’s reconstruction theorem without regularity structures, EMS Surv. Math. Sci., Volume 7 (2020) no. 2, pp. 207-251 | DOI | MR | Zbl

[9] Chandra, Ajay; Hairer, Martin An analytic BPHZ theorem for regularity structures, 2018 | arXiv

[10] Chartier, Philippe; Hairer, Ernst; Vilmart, Gilles Algebraic structures of B-series, Found. Comput. Math., Volume 10 (2010) no. 4, pp. 407-427 | DOI | MR | Zbl

[11] Connes, Alain; Kreimer, Dirk Hopf algebras, renormalization and noncommutative geometry, Comm. Math. Phys., Volume 199 (1998) no. 1, pp. 203-242 | DOI | MR | Zbl

[12] Driver, Bruce K. A Cameron-Martin type quasi-invariance theorem for Brownian motion on a compact Riemannian manifold, J. Funct. Anal., Volume 110 (1992) no. 2, pp. 272-376 | DOI | MR | Zbl

[13] Gubinelli, Massimiliano; Imkeller, Peter; Perkowski, Nicolas Paracontrolled distributions and singular PDEs, Forum Math. Pi, Volume 3 (2015), e6, 75 pages | DOI | MR | Zbl

[14] Hairer, Martin A theory of regularity structures, Invent. Math., Volume 198 (2014) no. 2, pp. 269-504 | DOI | MR | Zbl

[15] Hairer, Martin; Pardoux, Étienne Fluctuations around a homogenised semilinear random PDE, Arch. Rational Mech. Anal., Volume 239 (2021) no. 1, pp. 151-217 | DOI | MR | Zbl

[16] Hoshino, Masato Commutator estimates from a viewpoint of regularity structures, RIMS Kôkyûroku Bessatsu, Volume B79 (2020), pp. 179-197 | Zbl

[17] Hoshino, Masato Iterated paraproducts and iterated commutator estimates in Besov spaces (Adv. Stud. Pure Math.), Volume 87, Mathematical Society of Japan, Tokyo, 2021 (to appear) | arXiv

[18] Liu, Chong; Prömel, David J.; Teichmann, Josef Stochastic analysis with modelled distributions, Stochastic Partial Differ. Equ. Anal. Comput., Volume 9 (2021) no. 2, pp. 343-379 | DOI | MR

[19] Lyons, Terry; Qian, Zhongmin Flow equations on spaces of rough paths, J. Funct. Anal., Volume 149 (1997) no. 1, pp. 135-159 | DOI | MR | Zbl

[20] Lyons, Terry; Victoir, Nicolas An extension theorem to rough paths, Ann. Inst. H. Poincaré Anal. Non Linéaire, Volume 24 (2007) no. 5, pp. 835-847 | DOI | Numdam | MR | Zbl

[21] Martin, Jörg Refinements of the solution theory for singular SPDEs, Ph. D. Thesis, H.U. Berlin (2018)

[22] Martin, Jörg; Perkowski, Nicolas A Littlewood-Paley description of modelled distributions, J. Funct. Anal., Volume 279 (2020) no. 6, p. 108634, 22 | DOI | MR | Zbl

[23] Singh, Harprit; Teichmann, Josef An elementary proof of the reconstruction theorem, 2018 | arXiv

[24] Tapia, Nikolas; Zambotti, Lorenzo The geometry of the space of branched rough paths, Proc. London Math. Soc. (3), Volume 121 (2020) no. 2, pp. 220-251 | DOI | MR | Zbl

[25] Unterberger, Jérémie Hölder-continuous rough paths by Fourier normal ordering, Comm. Math. Phys., Volume 298 (2010) no. 1, pp. 1-36 | DOI | Zbl