Journal of the Mathematical Society of Japan

Rough flows

Ismaël BAILLEUL and Sebastian RIEDEL

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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
Received: 12 March 2018
First available in Project Euclid: 28 May 2019

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1559030414

Digital Object Identifier
doi:10.2969/jmsj/80108010

Mathematical Reviews number (MathSciNet)
MR3984248

Subjects
Primary: 60H10: Stochastic ordinary differential equations [See also 34F05] 60G44: Martingales with continuous parameter 34F05: Equations and systems with randomness [See also 34K50, 60H10, 93E03]

Keywords
stochastic flows rough flows semimartingale velocity fields

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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