| Summary: | Let p ∈ [1, 2) and α, ε > 0 be such that α ∈ (p - 1, 1 - ε). Let V, W be two Euclidean spaces. Let Ωp (V) be the space of continuous paths taking values in V and with finite p-variation. Let k ∈ N and f : W → Hom (V, W) be a Lip (k + α + ε) map in the sense of E.M. Stein [Stein E.M., Singular integrals and differentiability properties of functions, Princeton Mathematical Series, vol. 30, Princeton University Press, Princeton, NJ, 1970]. In this paper we prove that the Itô map, defined by I (x) = y, is a local Ck, frac(ε, 1 + ε) map (in the sense of Fréchet) between Ωp (V) and Ωp (W), where y is the solution to the differential equationd yt = f (yt) d xt, y0 = a .This result strengthens the continuity results and Lipschitz continuity results in [Lyons T., Differential equations driven by rough signals. I. An extension of an inequality of L.C. Young, Math. Res. Lett. 1 (4) (1994) 451-464; Lyons T., Qian Z., System Control and Rough Paths, Oxford Mathematical Monographs, Clarendon Press, Oxford, 2002] particularly to the non-integer case. It allows us to construct the fractional like Brownian motion and infinite dimensional Brownian motions on the space of paths with finite p-variation. As a corollary in the particular case where p = 1, we obtain that the development from the space of finite 1-variation paths on Rd to the space of finite 1-variation paths on a d-dimensional compact Riemannian manifold is a smooth bijection. © 2006 Elsevier Masson SAS. All rights reserved.
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