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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Rough path metrics on a Besov-Nikolskii-type scale

Authors: Peter K. Friz and David J. Prömel
Journal: Trans. Amer. Math. Soc. 370 (2018), 8521-8550
MSC (2010): Primary 34A34, 60H10; Secondary 26A45, 30H25, 46N20
Published electronically: August 9, 2018
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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

David J. Prömel
Affiliation: Eidgenössische Technische Hochschule Zürich, Switzerland
Address at time of publication: Mathematical Institute, University of Oxford, Oxford OX2 6GG, United Kingdom

Keywords: Controlled differential equation, Besov embedding, Besov space, It\^o--Lyons map, $p$-variation, Riesz-type variation, rough path.
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.
Article copyright: © Copyright 2018 American Mathematical Society

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