On the definition of a solution to a rough differential equation

Bailleul, Ismael

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 30 (2021) no. 3, pp. 463-478.

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.

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.

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