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• W2022671909 abstract "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,ε1+ε map (in the sense of Fréchet) between Ωp(V) and Ωp(W), where y is the solution to the differential equationdyt=f(yt)dxt,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. Soient p∈[1,2), et α tel que α∈(p−1,1−ε). Soient V et W deux espaces euclidiens. On désigne par Ωp(V) l'espace des chemins continus à valeurs dans V et de p-variation finie. Soit f:W→Hom(V,W) une application de classe Lip(k+α+ε) au sens de E.M. Stein, avec k∈N et k⩾1. Dans cet article, nous montrons que l'application de Itô I:Ωp(V)→Ωp(W), définie par I(x)=y, où y est la solution de l'équation différentielle suivante :dyt=f(yt)dxt,y0=a,est localement de classe Ck,ε1+ε au sens de Fréchet. Cela nous permet de construire des processus de type mouvement brownien fractionnaire ainsi que des mouvements browniens de dimension infinie sur l'espace des chemins de p-variation finie. Comme corollaire, nous obtenons, dans le cas particulier où p=1, que l'application de développement de l'espace des chemins de 1-variation finie sur Rd dans l'espace des chemins de 1-variation finie sur une variété riemannienne compacte d-dimensionnelle est une bijection régulière." @default.
• W2022671909 created "2016-06-24" @default.
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• W2022671909 date "2006-07-01" @default.
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• W2022671909 title "Smoothness of Itô maps and diffusion processes on path spaces (I)" @default.
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• W2022671909 doi "https://doi.org/10.1016/j.ansens.2006.07.001" @default.
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