## Rough path metrics on a Besov–Nikolskii-type scale

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- by Peter K. Friz and David J. Prömel PDF
- Trans. Amer. Math. Soc.
**370**(2018), 8521-8550 Request permission

## Abstract:

It is known, since the seminal work [T. Lyons, Differential equations driven by rough signals, Rev. Mat. Iberoamericana, 14 (1998)], that the solution map associated to a controlled differential equation is locally Lipschitz continuous in $q$-variation, resp., $1/q$-Hölder-type metrics on the space of rough paths, for any regularity $1/q \in (0,1]$.

We extend this to a new class of Besov–Nikolskii-type metrics, with arbitrary regularity $1/q\in (0,1]$ and integrability $p\in [ q,\infty ]$, where the case $p\in \{ q,\infty \}$ corresponds to the known cases. Interestingly, the result is obtained as a consequence of known $q$-variation rough path estimates.

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## Additional Information

**Peter K. Friz**- Affiliation: Technische Universität Berlin and Weierstrass Institute Berlin, Germany
- MR Author ID: 656436
**David J. Prömel**- Affiliation: Eidgenössische Technische Hochschule Zürich, Switzerland
- Address at time of publication: Mathematical Institute, University of Oxford, Oxford OX2 6GG, United Kingdom
- Received by editor(s): October 21, 2016
- Received by editor(s) in revised form: March 30, 3017
- Published electronically: August 9, 2018
- Additional Notes: The first author was partially supported by the European Research Council through CoG-683164 and DFG research unit FOR2402.

The second author gratefully acknowledges financial support of the Swiss National Foundation under Grant No. 200021_163014. - © Copyright 2018 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**370**(2018), 8521-8550 - MSC (2010): Primary 34A34, 60H10; Secondary 26A45, 30H25, 46N20
- DOI: https://doi.org/10.1090/tran/7264
- MathSciNet review: 3864386