Regularity of the Itô-Lyons map
Confluentes Mathematici, Volume 7 (2015) no. 1, p. 3-11

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.

DOI : https://doi.org/10.5802/cml.15
Classification:  34H99,  58J35,  60H99
@article{CML_2015__7_1_3_0,
     author = {Bailleul, Isma\"el},
     title = {Regularity of the It\^o-Lyons map},
     journal = {Confluentes Mathematici},
     publisher = {Institut Camille Jordan},
     volume = {7},
     number = {1},
     year = {2015},
     pages = {3-11},
     doi = {10.5802/cml.15},
     language = {en},
     url = {http://www.numdam.org/item/CML_2015__7_1_3_0}
}
Bailleul, Ismaël. Regularity of the Itô-Lyons map. Confluentes Mathematici, Volume 7 (2015) no. 1, pp. 3-11. doi : 10.5802/cml.15. http://www.numdam.org/item/CML_2015__7_1_3_0/

[1] Lyons, T., Differential equations driven by rough signals. Rev. Mat. Iberoamericana, 14 (2):215–310, 1998. | MR 1654527 | Zbl 0923.34056

[2] Lyons, T. and Qian, Z., System control and rough paths. Oxford University Press, 2002. | MR 2036784 | Zbl 1029.93001

[3] Lyons, T.J. and Caruana, M. and Lévy, Th., Differential equations driven by rough paths. Lecture Notes in Mathematics, 1908, Springer 2007. | MR 2314753 | Zbl 1176.60002

[4] Bailleul, I., Flows driven by Banach space-valued rough paths. Séminaire de Probabilités XLVI, LNM 2123, Springer, 2014. | MR 3330818

[5] Li, X.-D. and Lyons, T., Smoothness of Itô maps and diffusion process on path spaces. Ann. Scient. Ec. Norm. Sup., 39:649–677, 2006. | Numdam | MR 2290140 | Zbl 1127.60033

[6] Qian, Z. and Tudor, J., Differentiable structure and flow equations on rough path space. Bull. Sci. Math., 135(6-7): 695–732, 2011. | MR 2838098 | Zbl 1229.60072

[7] Friz, P. and Hairer, M., A short introduction to rough paths. Lect. Notes in Math., 2014.

[8] Feyel, D. and de La Pradelle, A., Curvilinear integrals along enriched paths. Electron. J. Probab., 11:860–892, 2006. | MR 2261056 | Zbl 1110.60031

[9] Gubinelli, M., Controlling rough paths. J. Funct. Anal., 216(1):86–140, 2004. | MR 2091358 | Zbl 1058.60037

[10] Gubinelli, M., Ramification of rough paths. J. Differential Equations, 248(4),693–721, 2010. | MR 2578445

[11] Inahama, Y., Malliavin differentiability of solutions of rough differential equations. arXiv:1312.7621, 2014. | MR 3229800 | Zbl 1296.60142

[12] Aida, S., Semi-classical limit of the bottom of spectrum of a Schrödinger operator on a path space over a compact Riemannian manifold. J. Funct. Anal., 251(1):59–121, 2007. | MR 2353701 | Zbl 1127.58014

[13] Inahama, Y. and Kawabi, H. Asymptotic expansion of the Laplace approximations for Itô functionals of Brownian rough paths. J. Funct. Anal., 243(1):270–322, 2007. | MR 2291439 | Zbl 1114.60062

[14] Inahama, Y., A stochastic Taylor-like expansion in the rough path theory. J. Theoret. Probab., 23(3):671–714, 2010. | MR 2679952 | Zbl 1203.60073

[15] Azencott, R., Formules the Taylor asymptotiques et développement asymptotique d’intégrales de Feynman. Sem. Probab., XVI:237–285, 1982. | Numdam | MR 658728 | Zbl 0484.60064

[16] Ben Arous, G., Méthode de Laplace et de la phse stationnaire sur l’espace de Wiener. Stochastics, 25(3):125–153, 1988. | MR 999365 | Zbl 0666.60026

[17] Lyons, T. and Qian, Z., Flow equations on spaces of rough paths. J. Funct. Anal., 149(1):135–159, 1997. | MR 1471102 | Zbl 0890.58090

[18] Driver, B., A Cameron-Martin type quasi-invariance theorem for Brownian motion on a compact manifold. J. Funct. Anal., 109:272–376, 1992. | MR 1194990 | Zbl 0765.60064

[19] Cass, T. and Driver, B. and Litterer, C., Constrained rough paths. arXiv:1402.4529, 2014.

[20] Hsu, E., Flows and quasi-invariance of the Wiener measure on path space. Proceedings of Symposia in Pure Mathematics, 57:265–279, 1995. | MR 1335476 | Zbl 0830.58035

[21] Lyons, T. and Qian, Z., Stochastic Jacobi fields and vector fields induced by varying area on path spaces. Prob. Th. Related Fields, 109:539–570, 1997. | MR 1483599 | Zbl 0903.60008

[22] Cass, T. and Friz, P., Densities for rough differential equations under Hörmander’s condition. Ann. Math., 171(3):2115–2141, 2010. | MR 2680405 | Zbl 1205.60105

[23] Cass, T. and Hairer, M. and Litterer, C. and Tindel, S. Smoothness of the density for solutions to Gaussian Rough Differential Equations. Arxiv preprint, 1209.3100, 2013. | MR 3298472

[24] Malliavin, P., Stochastic calculus of variation and hypoelliptic operators. Proceedings of the International Symposium on Stochastic Differential Equations (Res. Inst. Math. Sci., Kyoto Univ., Kyoto), 327(1):195–263, 1976. | MR 536013 | Zbl 0411.60060

[25] Norris, J., Two-parameter stochastic calculus and Malliavin’s integration-by-parts formula on Wiener space. Astérisque, 327(1):93–114, 2009. | MR 2642354 | Zbl 1201.60054