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On the definition of a solution to a rough differential equation
Ismael Bailleul1
1 Institut de Recherche Mathematiques de Rennes, 263 Avenue du General Leclerc, 35042 Rennes (France)
Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 30 (2021) no. 3, pp. 463-478.

There are several ways of defining what it means for a path to solve a rough differential equation. The most commonly used notion is due to Davie; it involves a Taylor expansion property that only makes sense a priori in a given coordinate system. Bailleul’s definition [4] is coordinate independent. Cass and Weidner [9] recently proved that the two definitions are actually equivalent, using deep algebraic insights on rough paths. We provide in this note an algebraic-free elementary short proof of this fact.

Différentes notions de solution d’une équations différentielle rugueuses sont disponibles dans la littérature. La notion la plus utilisée est due à Davie ; elle met en jeu un développement de Taylor, dont le sens est lié à un choix de coordonnées. La définission donnée par Bailleul dans [4] ne dépend pas d’un choix de coordonnées. Cass et Weidner [9] ont récemment démontré que les deux définitions sont en fait équivalentes, en s’appuyant sur des résultats algébriques élaborés. On donne dans cette note une démonstration courte et élémentaire de ce fait.

Received:
Accepted:
Published online:
DOI: 10.5802/afst.1681

Ismael Bailleul 1

1 Institut de Recherche Mathematiques de Rennes, 263 Avenue du General Leclerc, 35042 Rennes (France)
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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Ismael Bailleul. On the definition of a solution to a rough differential equation. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 30 (2021) no. 3, pp. 463-478. doi : 10.5802/afst.1681. https://afst.centre-mersenne.org/articles/10.5802/afst.1681/

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