#### Advanced Topics in Stochastic Analysis

V5F3 - WiSe 2015/16

### Rough paths and controlled paths

Wed 8-10 Endenicher Allee 60 - SemR 0.003

Thu 8-10 Endenicher Allee 60 - SemR 0.006

Notes:

The first lecture takes place on Tuesday 20 October, 8-10, SemR 0.003.

Change of room: since 26/11 Thursday lecture will take place in R0.006 (instead of R0.007)

This course is a introduction to the theory of rough paths, controlled paths and to recent developments in the analysis of singular SPDEs.

Rough paths were introduced by Lyons in the '90s as the right topology in which the map from a stochastic process to the solution of a stochastic differential equation driven by this process become continuous. This continuity, which is not present in Ito's theory of integration, allows to obtain quite directly a lot of results on the stability and fine properties of SDEs driven by Brownian motion (flows, large deviation, support theorems). But more importantly allows to consider more general driving signals, not necessarily semi-martingales, for example fractional Brownian motion. Later the notion of controlled paths has been introduced in order to simplify and extend the theory to a larger class of applications. Then, in the last few years, ideas stemming from these analysis were extended from stochastic differential equations to the analysis of stochastic partial differential equations leading to various approaches, including Hairer’s the theory of regularity structures.

#### Indicative list of the topics covered in the course

- Young integration and solution to differential equation.
- Beyond Young integration: the sewing map and algebraic integration.
- Controlled paths and solutions to rough differential equations. Stability and non-stability. Relation with the Ito theory. Convergence of certain random differential equations to SDEs.
- Iterated integrals: algebraic and analytic properties. Signature of a path. Rough paths.
- Application of controlled paths to singular SPDEs. The Stochastic Burgers equation.
- Applications of controlled paths to the low regularity theory of dispersive PDEs (Korteweg-de-Vries, Nonlinear Schrödinger equation).

### Prerequisites

Basic knowledge of stochastic analysis: Brownian motion, Ito integral, basic stochastic differential equations.

### Literature

- Lyons, Terry J., Michael J. Caruana, and Thierry Lévy. Differential Equations Driven by Rough Paths: Ecole d’Eté de Probabilités de Saint-Flour XXXIV-2004. 1st ed. Springer, 2007.
- Friz, Peter K., and Martin Hairer. A Course on Rough Paths. Universitext. Cham: Springer International Publishing, 2014.
- Gubinelli, M. “Controlling Rough Paths.” Journal of Functional Analysis 216, no. 1 (2004): 86–140.
- Gubinelli, Massimiliano. “Ramification of Rough Paths.” Journal of Differential Equations 248, no. 4 (2010): 693–721.
- Gubinelli, M. “Abstract Integration, Combinatorics of Trees and Differential Equations.” Proceedings of the Workshop “Combinatorics and Physics”, 2007. MPI Bonn.
- Gubinelli, M. “Rough Solutions for the Periodic Korteweg–de Vries Equation.” Comm. Pure Appl. Anal. 11, no. 2 (2012): 709–33.

# Lecture Sheets

- sheet-1-r2.pdfSheet 1. Introduction. Sewing map. Weighted norms. (rev 2 20151112)222 K
- sheet-2-r1 02.pdfSheet 2. Young differential equation. Asymptotics for the Euler scheme. (rev 1 20151126)225 K
- sheet-3-r1.pdfSheet 3. Rough paths. Controlled paths. (rev 1 20160121)246 K
- sheet-4-r1.pdfSheet 4. Rough differential equations. (rev 1 20160131)106 K