no. 6
Space-time paraproducts for paracontrolled calculus, 3d-PAM and multiplicative Burgers equations
[Paraproduits espace-temps pour le calcul paracontrôlé, 3d-PAM et équation de Burgers multiplicative]
Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 51 (2018) no. 6, pp. 1399-1456

We sharpen in this work the tools of paracontrolled calculus in order to provide a complete analysis of the parabolic Anderson model equation and Burgers system with multiplicative noise, in a 3-dimensional Riemannian setting, in either bounded or unbounded domains. With that aim in mind, we introduce a pair of intertwined space-time paraproducts on parabolic Hölder spaces, with good continuity, that happens to be pivotal and provides one of the building blocks of higher order paracontrolled calculus.

Nous enrichissons dans ce travail les outils du calcul paracontrôlé afin de fournir une analyse complète de l'équation du modèle parabolique d'Anderson et du système Burgers avec un bruit multiplicatif, dans un cadre riemannien de dimension 3, dans des domaines bornés ou non. Dans ce but, nous introduisons une paire de paraproduits espace-temps agissant sur les espaces de Hölder paraboliques, qui se révèle cruciale et fournit l'un des éléments constitutifs du calcul paracontrôlé d'ordre supérieur.

DOI : 10.24033/asens.2378
Classification : 60H15, 35R60, 35R01.
Keywords: Stochastic singular PDEs, semigroups, paraproducts, paracontrolled calculus, Parabolic Anderson Model equation, multiplicative stochastic Burgers equation
Mots-clés : EDPs singulières stochastiques, semi-groupes, paraproduits, calcul paracontrôlé, modèle parabolique d'Anderson, équation de Burgers avec bruit multiplicatif 
@article{ASENS_2018__51_6_1399_0,
     author = {Bailleul, Ismael and Bernicot, Fr\'ed\'eric and Frey, Dorothee},
     title = {Space-time paraproducts  for paracontrolled calculus, {3d-PAM}  and multiplicative {Burgers} equations},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     pages = {1399--1456},
     year = {2018},
     publisher = {Soci\'et\'e Math\'ematique de France. Tous droits r\'eserv\'es},
     volume = {Ser. 4, 51},
     number = {6},
     doi = {10.24033/asens.2378},
     mrnumber = {3940901},
     zbl = {1430.60053},
     language = {en},
     url = {https://www.numdam.org/articles/10.24033/asens.2378/}
}
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Bailleul, Ismael; Bernicot, Frédéric; Frey, Dorothee. Space-time paraproducts  for paracontrolled calculus, 3d-PAM  and multiplicative Burgers equations. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 51 (2018) no. 6, pp. 1399-1456. doi: 10.24033/asens.2378

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