We show in this note that the Itô-Lyons solution map associated to a rough differential equation is Fréchet differentiable when understood as a map between some Banach spaces of controlled paths. This regularity result provides an elementary approach to Taylor-like expansions of Inahama-Kawabi type for solutions of rough differential equations depending on a small parameter, and makes the construction of some natural dynamics on the path space over any compact Riemannian manifold straightforward, giving back Driver’s flow as a particular case.
@article{CML_2015__7_1_3_0,
author = {Bailleul, Isma\"el},
title = {Regularity of the It\^o-Lyons map},
journal = {Confluentes Mathematici},
volume = {7},
year = {2015},
pages = {3-11},
doi = {10.5802/cml.15},
language = {en},
url = {http://dml.mathdoc.fr/item/CML_2015__7_1_3_0}
}
Bailleul, Ismaël. Regularity of the Itô-Lyons map. Confluentes Mathematici, Tome 7 (2015) pp. 3-11. doi : 10.5802/cml.15. http://gdmltest.u-ga.fr/item/CML_2015__7_1_3_0/
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