## Journal of the Mathematical Society of Japan

### Rough flows

#### Abstract

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.

#### Note

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.

#### Article information

Source
J. Math. Soc. Japan, Volume 71, Number 3 (2019), 915-978.

Dates
First available in Project Euclid: 28 May 2019

https://projecteuclid.org/euclid.jmsj/1559030414

Digital Object Identifier
doi:10.2969/jmsj/80108010

Mathematical Reviews number (MathSciNet)
MR3984248

#### Citation

BAILLEUL, Ismaël; RIEDEL, Sebastian. Rough flows. J. Math. Soc. Japan 71 (2019), no. 3, 915--978. doi:10.2969/jmsj/80108010. https://projecteuclid.org/euclid.jmsj/1559030414

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