On the Rough Gronwall Lemma and Its Applications
We present a rough path analog of the classical Gronwall Lemma introduced recently by Deya, Gubinelli, Hofmanová, Tindel in arXiv:1604.00437,  and discuss two of its applications. First, it is applied in the framework of rough path driven PDEs in order to establish energy estimates for weak solutions. Second, it is used in order to prove uniqueness for reflected rough differential equations.
KeywordsRough paths Rough partial differential equations Reflected rough differential equations Rough gronwall lemma
2010 Mathematics Subject Classification60H15 35R60 35L65
Financial support by the DFG via Research Unit FOR 2402 is gratefully acknowledged.
- 1.Aida, S.: Rough differential equations containing path-dependent bounded variation terms (2016) arXiv:1608.03083
- 3.Bailleul, I., Gubinelli, M.: Unbounded rough drivers. arXiv:1501.02074 [math], January 2015
- 6.Deya, A., Gubinelli, M., Hofmanová, M., Tindel, S.: A priori estimates for rough PDEs with application to rough conservation laws. arXiv:1604.00437
- 7.Deya, A., Gubinelli, M., Hofmanová, M., Tindel, S.: One-dimensional reflected rough differential equations. arXiv:1610.07481
- 9.Friz, P.K., Hairer, M.: A Course on Rough Paths: With an Introduction to Regularity Structures. Springer, New York (2014)Google Scholar
- 10.Friz, P.K., Victoir, N.B.: Multidimensional Stochastic Processes As Rough Paths: Theory and Applications. Cambridge University Press, Cambridge (2010)Google Scholar
- 11.Gubinelli, M., Imkeller, P., Perkowski, N.: Paracontrolled distributions and singular PDEs. Forum Math. Pi 3:e6, 75 (2015)Google Scholar
- 15.Lyons, T.J., Caruana, M., Lévy, T.: Differential Equations Driven by Rough Paths. Lecture Notes in Mathematics, vol. 1908. 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. Springer, Berlin (2007)Google Scholar
- 16.Lyons, T.: On the nonexistence of path integrals. Proc. R. Soc. London Ser. A 432(1885), 281–290 (1991)Google Scholar
- 18.Young, L.C.: An inequality of the Hölder type, connected with Stieltjes integration. Acta Math. 67(1), 251–282 (1936)Google Scholar