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 semimartingale 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.
The first author was partly supported by the ANR project “Retour Post-doctorant”, no. 11–PDOC-0025. He also thanks the U.B.O. for their hospitality, part of this work was written there. The second author was partly supported by the DFG Research Unit FOR 2402.
"Rough flows." J. Math. Soc. Japan 71 (3) 915 - 978, July, 2019. https://doi.org/10.2969/jmsj/80108010