Rough flows

Ismaël Bailleul (1) , Sebastian Riedel (2)

1 IRMAR - Institut de Recherche Mathématique de Rennes
2 TU - Technical University of Berlin / Technische Universität Berlin

We introduce in this work a concept of rough driver that somehow provides a rough path-like analogue of an enriched object associated with time-dependent vector fields. We use the machinery of approximate flows to build the integration theory of rough drivers and prove well-posedness results for rough differential equations on flows and continuity of the solution flow as a function of the generating rough driver. We show that the theory of semi-martingale stochastic flows developed in the 80's and early 90's fits nicely in this framework, and obtain as a consequence some strong approximation results for general semimartingale flows and provide a fresh look at large deviation theorems for 'Gaussian' stochastic flows.

Domaines: Probabilités [math.PR]

Ismaël Bailleul, Sebastian Riedel. Rough flows. Journal of the Mathematical Society of Japan, 2019, 71 (3), pp.915-978. ⟨10.2969/jmsj/80108010⟩. ⟨hal-01278749⟩