Decay rate of iterated integrals of branched rough paths
Annales de l'I.H.P. Analyse non linéaire, Tome 35 (2018) no. 4, pp. 945-969.

Iterated integrals of paths arise frequently in the study of the Taylor's expansion for controlled differential equations. We will prove a factorial decay estimate, conjectured by M. Gubinelli, for the iterated integrals of non-geometric rough paths. We will explain, with a counter example, why the conventional approach of using the neoclassical inequality fails. Our proof involves a concavity estimate for sums over rooted trees and a non-trivial extension of T. Lyons' proof in 1994 for the factorial decay of iterated Young's integrals.

DOI : 10.1016/j.anihpc.2017.09.002
Mots clés : Branched rough paths, Non-geometric rough paths, Iterated integrals
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     title = {Decay rate of iterated integrals of branched rough paths},
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Boedihardjo, Horatio. Decay rate of iterated integrals of branched rough paths. Annales de l'I.H.P. Analyse non linéaire, Tome 35 (2018) no. 4, pp. 945-969. doi : 10.1016/j.anihpc.2017.09.002. http://www.numdam.org/articles/10.1016/j.anihpc.2017.09.002/

[1] Connes, A.; Kreimer, D. Hopf algebras, renormalization and noncommutative geometry, Commun. Math. Phys., Volume 199 (1998) no. 1, pp. 203–242 | DOI | MR | Zbl

[2] Hara, K.; Hino, M. Fractional order Taylor's series and the neo-classical inequality, Bull. Lond. Math. Soc., Volume 42 (2010), pp. 467–477 | DOI | MR | Zbl

[3] Hairer, M.; Kelly, D. Geometric versus non-geometric rough paths, Ann. Inst. Henri Poincaré Probab. Stat., Volume 51 (2015) no. 1, pp. 207–251 | DOI | Numdam | MR | Zbl

[4] Gubinelli, M. Ramification of rough paths, J. Differ. Equ., Volume 248 (2010), pp. 693–721 | DOI | MR | Zbl

[5] Gubinelli, M. Rooted trees for 3D Navier–Stokes equation, Dyn. Partial Differ. Equ., Volume 3 (2006) no. 2, pp. 161–172 | DOI | MR | Zbl

[6] Lyons, T. Differential equations driven by rough signals (I): an extension of an inequality of L.C. Young, Math. Res. Lett., Volume 1 (1994), pp. 451–464 | DOI | MR | Zbl

[7] Lyons, T. Differential equations driven by rough signals, Rev. Mat. Iberoam., Volume 14 (1998) no. 2, pp. 215–310 | MR | Zbl

[8] Lyons, T.; Caruana, M.; Lévy, T. Differential Equations Driven by Rough Paths, Springer, 2007 | DOI | MR | Zbl

[9] Young, L.C. An inequality of Hölder type connected with Stieltjes integration, Acta Math., Volume 67 (1936), pp. 251–282 | DOI | MR | Zbl

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