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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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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


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:
  • MathSciNet review: 3864386