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Ismael Bailleul; Massimiliano Gubinelli
Unbounded rough drivers
Annales de la faculté des sciences de Toulouse Sér. 6, 26 no. 4 (2017), p. 795-830, doi: 10.5802/afst.1553
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Résumé - Abstract

We propose a theory of linear differential equations driven by unbounded operator-valued rough signals. As an application we consider rough linear transport equations and more general linear hyperbolic symmetric systems of equations driven by time-dependent vector fields which are only distributions in the time direction.

Bibliography

[1] Ismaël Bailleul, “Flows driven by rough paths”, Rev. Mat. Iberoam. 31 (2015) no. 3, p. 901-934 Article
[2] Ismaël Bailleul & Sebastian Riedel, “Rough flows”, https://arxiv.org/abs/1505.01692v1, 2015
[3] Hakima Bessaih, Massimiliano Gubinelli & Francesco Russo, “The evolution of a random vortex filament”, Ann. Probab. 33 (2005) no. 5, p. 1825-1855 Article
[4] Michael Caruana & Peter K. Friz, “Partial differential equations driven by rough paths”, J. Differ. Equations 247 (2009) no. 1, p. 140-173 Article
[5] Michael Caruana, Peter K. Friz & Harald Oberhauser, “A (rough) pathwise approach to a class of non-linear stochastic partial differential equations”, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 28 (2011) no. 1, p. 27-46 Article
[6] Rémi Catellier, “Rough linear transport equation with an irregular drift”, https://arxiv.org/abs/1501.03000, 2014
[7] Rémi Catellier & Massimiliano Gubinelli, “Averaging along irregular curves and regularisation of ODEs”, https://arxiv.org/abs/1205.1735v2, 2014
[8] Laure Coutin & Antoine Lejay, “Perturbed linear rough differential equations”, Ann. Math. Blaise Pascal 21 (2014) no. 1, p. 103-150 Article
[9] Aurélien Deya, Massimiliano Gubinelli, Martina Hofmanová & Samy Tindel, “A priori estimates for rough PDEs with application to rough conservation laws”, https://arxiv.org/abs/1604.00437, 2016
[10] Aurélien Deya, Massimiliano Gubinelli & Samy Tindel, “Non-linear rough heat equations”, Probab. Theory Relat. Fields 153 (2012) no. 1-2, p. 97-147 Article
[11] Joscha Diehl, Peter K. Friz & Wilhelm Stannat, “Stochastic partial differential equations: a rough paths view”, Ann. Fac. Sci. Toulouse 26 (2017) no. 4, p. 911-947 Article
[12] Ronald J. DiPerna & Pierre-Louis Lions, “Ordinary differential equations, transport theory and Sobolev spaces”, Invent. Math. 98 (1989) no. 3, p. 511-547 Article
[13] Denis Feyel, Arnaud De La Pradelle & Gabriel Mokobodzki, “A non-commutative sewing lemma”, Electron. Commun. Probab. 13 (2008), p. 24-34 Article
[14] Denis Feyel & Arnaud de La Pradelle, “Curvilinear integrals along enriched paths”, Electron. J. Probab. 11 (2006), p. 860-892 Article
[15] Franco Flandoli, “The interaction between noise and transport mechanisms in PDEs”, Milan J. Math. 79 (2011) no. 2, p. 543-560 Article
[16] Peter K. Friz & Benjamin Gess, “Stochastic scalar conservation laws driven by rough paths”, https://arxiv.org/abs/1403.6785v1, 2014
[17] Peter K. Friz & Harald Oberhauser, “Rough path limits of a Wong-Zakai type with a modified drift term”, J. Funct. Anal. 256 (2009) no. 10, p. 3236-3256 Article
[18] Peter K. Friz & Harald Oberhauser, “On the splitting-up method for rough (partial) differential equations”, J. Differ. Equations 251 (2011) no. 2, p. 316-338 Article
[19] Peter K. Friz & Harald Oberhauser, “Rough path stability of (semi-)linear SPDEs”, Probab. Theory Relat. Fields 158 (2014) no. 1-2, p. 401-434 Article
[20] Peter K. Friz & Nicolas B. Victoir, Multidimensional stochastic processes as rough paths. Theory and applications, Cambridge Studies in Advanced Mathematics 120, Cambridge University Press, 2010
[21] Benjamin Gess & Panagiotis E. Souganidis, “Scalar conservation laws with multiplie rough fluxes”, https://arxiv.org/abs/1406.2978v2, 2014
[22] Massimiliano Gubinelli, “Controlling rough paths”, J. Funct. Anal. 216 (2004) no. 1, p. 86-140 Article
[23] Massimiliano Gubinelli, “Rough solutions for the periodic Korteweg–de Vries equation”, Commun. Pure Appl. Anal. 11 (2012) no. 2, p. 709-733 Article
[24] Massimiliano Gubinelli, Peter Imkeller & Nicolas Perkowski, “Paracontrolled distributions and singular PDEs”, https://arxiv.org/abs/1210.2684v3, 2014
[25] Massimiliano Gubinelli & Samy Tindel, “Rough evolution equations”, Ann. Probab. 38 (2010) no. 1, p. 1-75 Article
[26] Massimiliano Gubinelli, Samy Tindel & Iván Torrecilla, “Controlled viscosity solutions of fully nonlinear rough PDEs”, https://arxiv.org/abs/1403.2832, 2014
[27] Martin Hairer, “Rough stochastic PDEs”, Commun. Pure Appl. Math. 64 (2011) no. 11, p. 1547-1585
[28] Martin Hairer, “Solving the KPZ equation”, Ann. Math. 178 (2013) no. 2, p. 559-664 Article
[29] Martin Hairer, “A theory of regularity structures”, Invent. Math. 198 (2014) no. 2, p. 269-504 Article
[30] Martin Hairer, Jan Maas & Hendrik Weber, “Approximating rough stochastic PDEs”, https://arxiv.org/abs/1202.3094v1, 2012
[31] Martin Hairer & Hendrik Weber, “Rough Burgers-like equations with multiplicative noise”, Probab. Theory Relat. Fields 155 (2013) no. 1-2, p. 71-126, erratum in ibid. 157 (2013), no. 3-4, p. 1011–1013 Article
[32] Yaozhongand Hu & Khoa N. Lê, “Nonlinear Young integrals and differential systems in Hölder media”, https://arxiv.org/abs/1404.7582v1, 2014
[33] Stanislav Nicolayevich Kruzhkov, “First order quasilinear equations in several independent variables”, Math. USSR, Sb. 10 (1970), p. 217-243 Article
[34] Pierre-Louis Lions, Benoît Perthame & Panagiotis E. Souganidis, “Scalar conservation laws with rough (stochastic) fluxes”, Stoch. Partial Differ. Equ., Anal. Comput. 1 (2013) no. 4, p. 664-686
[35] Pierre-Louis Lions, Benoît Perthame & Panagiotis E. Souganidis, “Scalar conservation laws with rough (stochastic) fluxes: the spatially dependent case”, https://arxiv.org/abs/1403.4424, 2014
[36] Terry J. Lyons, “Differential equations driven by rough signals”, Rev. Mat. Iberoam. 14 (1998) no. 2, p. 215-310 Article
[37] Terry J. Lyons, Michael Caruana & Thierry Lévy, Differential equations driven by rough paths, Lecture Notes in Mathematics 1908, Springer, 2007
[38] Terry J. Lyons & Zhongmin Qian, System control and rough paths, Oxford Mathematical Monographs, Clarendon Press, 2002
[39] Michael Reed & Barry Simon, Methods of modern mathematical physics, Academic Press, 1980
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