Browse issues
no. 3
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, Serie 6, Volume 29 (2020) no. 3, pp. 721-759.

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.

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.

Received:
Accepted:
Published online:
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)
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
@article{AFST_2020_6_29_3_721_0,
     author = {Ismael Bailleul and Remi Catellier},
     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},
     volume = {Ser. 6, 29},
     number = {3},
     year = {2020},
     doi = {10.5802/afst.1644},
     zbl = {07206358},
     language = {en},
     url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1644/}
}
TY  - JOUR
AU  - Ismael Bailleul
AU  - Remi Catellier
TI  - Non-explosion criteria for rough differential equations driven by unbounded vector fields
JO  - Annales de la Faculté des sciences de Toulouse : Mathématiques
PY  - 2020
SP  - 721
EP  - 759
VL  - 29
IS  - 3
PB  - Université Paul Sabatier, Toulouse
UR  - https://afst.centre-mersenne.org/articles/10.5802/afst.1644/
DO  - 10.5802/afst.1644
LA  - en
ID  - AFST_2020_6_29_3_721_0
ER  - 
%0 Journal Article
%A Ismael Bailleul
%A Remi Catellier
%T Non-explosion criteria for rough differential equations driven by unbounded vector fields
%J Annales de la Faculté des sciences de Toulouse : Mathématiques
%D 2020
%P 721-759
%V 29
%N 3
%I Université Paul Sabatier, Toulouse
%U https://afst.centre-mersenne.org/articles/10.5802/afst.1644/
%R 10.5802/afst.1644
%G en
%F AFST_2020_6_29_3_721_0
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, Serie 6, Volume 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

Cited by Sources: