S4F2 - Graduate Seminar on Stochastic Analysis – SS22

Rough Paths

Massimiliano Gubinelli, Francesco de Vecchi

The goal of this seminar is to overview the theory of rough paths by covering large portions of the book

Friz, Peter K., and Martin Hairer. A Course on Rough Paths: With an Introduction to Regularity Structures. Springer, 2014.

and also some complementary material from other sources. 

Rough path theory analyses the behaviour of non-linear dynamical systems controlled by an external "input" signal, in particular by introducing suitable topologies in the input signal in order to obtain good continuity properties on the effects this signal can have on the dynamical system behaviour. It is an exploration of a domain at the crossroads between analysis, probablity and algebra, but still quite elementary in its basic ingredients. As results one obtains a theory of integration of differential equations which can be applied to non-differentiable random functions and which cover cases out of reach to standard stochastic analysis based on Ito's calculus and martingales. This seminar is an ideal complement to the course of Stochastic Analysis.

The preliminary meeting has been in April 13th, 10-12 (in the Stochastics common room R4.050 in End60).

The seminar started May 20th on Fridays 10-12 and 14-16 in SR0.003. 

NOTE: From June 17th the seminar will take place on Fridays 10-12 in SR0.003 and 12-14 in SR0.008.

Calendar

1. 20/5 – Shi* – Chap. 2 - Space of Rough Paths (handout)

2. 27/5 - Rinaldi – Chap. 3 – BM as rough path (slides)

3. 27/5 - Meyer – Chap. 4 – Integ. against RP (handout)

4. 3/6 - De Vecchi – Chap 5. – Stoch int & Ito formula: Chap 6. – Doob-Meyer for RP

5. 3/6 - Fresta – Chap 7. – Controlled rough path (handout)

6. 17/6 - Ravot Licheri* – Sewing Lemma & Young DE (from CGZ, Chap 1 and Chap 2) (cfr. with Chap 8 of FH)

7. 17/6 - Cai* – Rough DE (Chap 3 from CGZ) + Chap 8 of FH. (various notions of solutions, stability, flow property, etc..)

8. 24/6 - Song – Chap 9. Link with Stochastic analysis (Wong-Zakai, Support theorem, large deviations) (handout)

9. 24/6 - Liu – Chap 10+ Chap 11 : Gaussian rough paths + integrability of norms (handout) (notes)

10. 1/7 - CANCELED / NO SEMINAR THIS WEEK - Martini – Chap 12: SPDEs : Feynman-Kac + flow transformation + Linear stochastic heat equation

11. 8/7 - Wang – paper: Branched rough paths (paper)

12. 15/7 - Brennecke – unbounded rough drivers (paper)

13. 15/7 - Galeati – reflected rough DE (paper) (handout)

Reference Material

• Friz, Peter K., and Martin Hairer. A Course on Rough Paths: With an Introduction to Regularity Structures. Springer, 2014.

• Deya, Aurélien, Massimiliano Gubinelli, Martina Hofmanová, and Samy Tindel. “One-Dimensional Reflected Rough Differential Equations.” Stochastic Processes and Their Applications 129, no. 9 (September 1, 2019): 3261–81. https://doi.org/10.1016/j.spa.2018.09.007.

• Bailleul, Ismael, and Massimiliano Gubinelli. “Unbounded Rough Drivers.” Annales de La Faculté Des Sciences de Toulouse Mathématiques 26, no. 4 (2017): 795–830. https://doi.org/10.5802/afst.1553.

• Gubinelli, Massimiliano. “Ramification of Rough Paths.” Journal of Differential Equations 248, no. 4 (2010): 693–721. https://doi.org/10.1016/j.jde.2009.11.015.

• Hairer, Martin, and David Kelly. “Geometric versus Non-Geometric Rough Paths.” Annales de l’Institut Henri Poincaré, Probabilités et Statistiques 51, no. 1 (February 2015): 207–51. https://doi.org/10.1214/13-AIHP564.

• Some unpublished material from lectures on RP by Caravenna, Gubinelli and Zambotti (CGZ)

Additional reference Material

• An advanced topics course on rough paths (link) 
• The wikipedia page on rough paths (link)
• Recent directions in rough path research. The proceedings of an Oberwolfach workshop held in 2020 (link)