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.
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DOI : 10.5802/cml.15
@article{CML_2015__7_1_3_0,
author = {Bailleul, Isma\"el},
title = {Regularity of the {It\^o-Lyons} map},
journal = {Confluentes Mathematici},
pages = {3--11},
publisher = {Institut Camille Jordan},
volume = {7},
number = {1},
year = {2015},
doi = {10.5802/cml.15},
language = {en},
url = {http://www.numdam.org/articles/10.5802/cml.15/}
}
Bailleul, Ismaël. Regularity of the Itô-Lyons map. Confluentes Mathematici, Tome 7 (2015) no. 1, pp. 3-11. doi : 10.5802/cml.15. http://www.numdam.org/articles/10.5802/cml.15/
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