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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@article{AFST_2021_6_30_3_463_0, author = {Bailleul, Ismael}, title = {On the definition of a solution to a rough differential equation}, journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques}, pages = {463--478}, publisher = {Universit\'e Paul Sabatier, Toulouse}, volume = {Ser. 6, 30}, number = {3}, year = {2021}, doi = {10.5802/afst.1681}, language = {en}, url = {http://archive.numdam.org/articles/10.5802/afst.1681/} }
TY - JOUR AU - Bailleul, Ismael TI - On the definition of a solution to a rough differential equation JO - Annales de la Faculté des sciences de Toulouse : Mathématiques PY - 2021 SP - 463 EP - 478 VL - 30 IS - 3 PB - Université Paul Sabatier, Toulouse UR - http://archive.numdam.org/articles/10.5802/afst.1681/ DO - 10.5802/afst.1681 LA - en ID - AFST_2021_6_30_3_463_0 ER -
%0 Journal Article %A Bailleul, Ismael %T On the definition of a solution to a rough differential equation %J Annales de la Faculté des sciences de Toulouse : Mathématiques %D 2021 %P 463-478 %V 30 %N 3 %I Université Paul Sabatier, Toulouse %U http://archive.numdam.org/articles/10.5802/afst.1681/ %R 10.5802/afst.1681 %G en %F AFST_2021_6_30_3_463_0
Bailleul, Ismael. On the definition of a solution to a rough differential equation. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 30 (2021) no. 3, pp. 463-478. doi : 10.5802/afst.1681. http://archive.numdam.org/articles/10.5802/afst.1681/
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