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Non-explosion criteria for rough differential equations driven by unbounded vector fields
Ismael Bailleul1 ; Remi Catellier2
1 Institut de Recherche Mathematiques de Rennes, 263 Avenue du General Leclerc, 35042 Rennes (France)
2 Université Côtes d’Azur, LJAD, Nice (France)
Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 29 (2020) no. 3, pp. 721-759.

On traite de façon simple dans cette note du problème de la non explosion des solutions d’équations différentielles rugueuses conduites par des champs de vecteurs non bornés et un p-rough path faiblement géométrique, pour p quelconque.

We give in this note a simple treatment of the non-explosion problem for rough differential equations driven by unbounded vector fields and weak geometric rough paths of arbitrary roughness.

Reçu le :
Accepté le :
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DOI : 10.5802/afst.1644
Ismael Bailleul 1 ; Remi Catellier 2

1 Institut de Recherche Mathematiques de Rennes, 263 Avenue du General Leclerc, 35042 Rennes (France)
2 Université Côtes d’Azur, LJAD, Nice (France)
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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     title = {Non-explosion criteria for rough differential equations driven by unbounded vector fields},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {721--759},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
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Ismael Bailleul; Remi Catellier. Non-explosion criteria for rough differential equations driven by unbounded vector fields. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 29 (2020) no. 3, pp. 721-759. doi : 10.5802/afst.1644. https://afst.centre-mersenne.org/articles/10.5802/afst.1644/

[1] Ismaël Bailleul A flows-based approach to rough differential equations (https://perso.univ-rennes1.fr/ismael.bailleul/files/M2Course.pdf)

[2] Ismaël Bailleul Flows driven by Banach space-valued rough paths, Séminaire de Probabilités XLVI (Lecture Notes in Mathematics), Volume 2123, Springer, 2014, pp. 195-205 | DOI | MR | Zbl

[3] Ismaël Bailleul Flows driven by rough paths, Rev. Mat. Iberoam., Volume 31 (2015) no. 3, pp. 901-934 | DOI | MR | Zbl

[4] Ismaël Bailleul; Rémi Catellier; François Delarue Solving mean field rough differential equations, Electron. J. Probab., Volume 25 (2020), 21, 51 pages | MR | Zbl

[5] Fabrice Baudoin Diffusion processes and stochastic calculus, EMS Textbooks in Mathematics, European Mathematical Society, 2014, xii+276 pages | Zbl

[6] Youness Boutaib; Lajos Gergely Gyurkó; Terry J. Lyons; Danyu Yang Dimension-free Euler estimates of rough differential equations, Rev. Roum. Math. Pures Appl., Volume 59 (2014) no. 1, pp. 25-53 | MR | Zbl

[7] Thomas Cass; Christian Litterer; Terry J. Lyons Integrability and tail estimates for Gaussian rough differential equations, Ann. Probab., Volume 41 (2013) no. 4, pp. 3026-3050 | DOI | MR | Zbl

[8] Thomas Cass; Marcel Ogrodnik Tail estimates for Markovian rough paths, Ann. Probab., Volume 45 (2017) no. 4, pp. 2477-2504 | DOI | MR | Zbl

[9] Thomas Cass; Martin P. Weidner Tree algebras over topological vector spaces in rough path theory (2017) (https://arxiv.org/abs/1604.07352v2)

[10] Kuo-Tsai Chen Integration of paths, geometric invariants and a generalized Baker–Hausdorff formula, Ann. Math., Volume 65 (1957), pp. 163-178 | DOI | MR | Zbl

[11] Laure Coutin; Antoine Lejay Perturbed linear rough differential equations, Ann. Math. Blaise Pascal, Volume 21 (2014) no. 1, pp. 103-150 | DOI | Numdam | MR | Zbl

[12] Laure Coutin; Antoine Lejay Sensitivity of rough differential equations: an approach through the Omega lemma, J. Differ. Equations, Volume 264 (2018) no. 6, pp. 3899-3917 | DOI | MR | Zbl

[13] Laure Coutin; Zhongmin Qian Stochastic analysis, rough path analysis and fractional Brownian motions, Probab. Theory Relat. Fields, Volume 122 (2002) no. 1, pp. 108-140 | DOI | MR | Zbl

[14] Alexander M. Davie Differential equations driven by rough paths: an approach via discrete approximation, AMRX, Appl. Math. Res. Express, Volume 2 (2007), pp. 9-40

[15] Aurélien Deya; Massimiliano Gubinelli; Martina Hofmanová; Samy Tindel A priori estimates for rough PDEs with application to rough conservation laws (2016) (https://arxiv.org/abs/1604.00437) | Zbl

[16] Bruce Driver Global existence of geometric rough flows (2018) (https://arxiv.org/abs/1810.03708)

[17] Bruce Driver On truncated logarithms of flows on a Riemannian manifold (2018) (https://arxiv.org/abs/1810.02414)

[18] Peter K. Friz; Martin Hairer A course on rough paths. With an introduction to regularity structures, Universitext, Springer, 2014, xiv+251 pages | Zbl

[19] Peter K. Friz; Sebastian Riedel Integrability of (non-)linear rough differential equations and integrals, Stochastic Anal. Appl., Volume 31 (2013) no. 2, pp. 336-358 | DOI | MR | Zbl

[20] Peter K. Friz; Nicolas B. Victoir Multidimensional stochastic processes as rough paths, Cambridge Studies in Advanced Mathematics, 120, Cambridge University Press, 2010 | MR | Zbl

[21] Antoine Lejay On rough differential equations, Electron. J. Probab., Volume 14 (2009) no. 12, pp. 341-364 | DOI | MR | Zbl

[22] Antoine Lejay Global solutions to rough differential equations with unbounded vector fields, Séminaire de probabilités XLIV (Lecture Notes in Mathematics), Volume 2046, Springer, 2012, pp. 215-246 | DOI | MR | Zbl

[23] Terry J. Lyons Differential equations driven by rough signals, Rev. Mat. Iberoam., Volume 14 (1998) no. 2, pp. 215-310 | DOI | MR | Zbl

[24] Terry J. Lyons; Michael Caruana; Thierry Lévy Differential equations driven by rough paths, Lecture Notes in Mathematics, 1908, Springer, 2007, xviii+109 pages (Lectures from the 34th Summer School on Probability Theory held in Saint-Flour, July 6–24, 2004, With an introduction concerning the Summer School by Jean Picard) | MR | Zbl

[25] Terry J. Lyons; Zhongmin Qian System control and rough paths, Oxford Mathematical Monographs, Clarendon Press, 2002 | Zbl

[26] Laurence C. Young An inequality of the Hölder type, connected with Stieltjes integration, Acta Math., Volume 67 (1936) no. 1, pp. 251-282 | DOI | Zbl

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