A new definition of rough paths on manifolds

Boutaib, Youness; Lyons, Terry

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 31 (2022) no. 4, pp. 1223-1258.

Les variétés régulières ne sont pas bien adaptées à la généralisation du concept des chemins rugueux aux variétés. En effet, quand on travaille avec des applications régulières plutôt que des applications Lipschitz pour résoudre une équation différentielle rugueuse, on perd les estimations quantitatives qui contrôlent la convergence des itérations de Picard. De plus, étant donne une définition de chemins rugueux sur variétés, on ne peut en général résoudre des équations différentielles ordinaires ou rugueuses que de manière locale. Dans cet article, on rappelle d’abord les fondations de la géométrie différentielle Lipschitz, introduite dans [8], ainsi que les principaux résultats qui généralisent ceux de la théorie classique des chemins rugueux dans les espaces de Banach. Ensuite on donne un cadre minimal pour la définition des chemins rugueux sur une variété qui est moins rigide que la précédente et qui met l’accent sur le comportement local des chemins rugueux. Finalement, on explique comment ces idées peuvent être appliquées pour généraliser la définition de tout chemin coloré à une variété.

Smooth manifolds are not the suitable context for trying to generalize the concept of rough paths on a manifold. Indeed, when one is working with smooth maps instead of Lipschitz maps and trying to solve a rough differential equation, one loses the quantitative estimates controlling the convergence of the Picard sequence. Moreover, even with a definition of rough paths in smooth manifolds, ordinary and rough differential equations can only be solved locally in such case. In this paper, we first recall the foundations of the Lipschitz geometry, introduced in [8], along with the main findings that encompass the classical theory of rough paths in Banach spaces. Then we give what we believe to be a minimal framework for defining rough paths on a manifold that is both less rigid than the classical one and emphasized on the local behaviour of rough paths. We end by explaining how this same idea can be used to define any notion of coloured paths on a manifold.

Classification : 60L20, 58J65
Mots clés : Rough paths, rough analysis on manifolds

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