Workshop "Rough Paths in Toulouse"

19 oct. 2017, 08:00 → 20 oct. 2017, 18:00 Europe/Paris

Salle 13 Bâtiment Mathématiques Appliquées (INSA Toulouse)

In recent years the theory of "rough paths" analysis has been more and more successful for solving problems at the interface of analysis and probability theory. The objective of this two-days workshop is to bring together scientific experts in this domain at the interface of Analysis, Probability theory and Geometry.

 

                              

 

 

Invited Speakers :

- Yvain Bruned (Imperial College)

- Thomas Cass (Imperial College)

- Rémi Catellier (Université de Nice Sophia Antipolis)

- Reda Chhaibi (Institut de Mathématiques de Toulouse)

- Ilya Chevyrev (University of Oxford)

- Xi Geng (Carnegie Mellon University)

- Peter Imkeller (Humboldt-Universität zu Berlin)

- Antoine Lejay (Institut Elie Cartan)

- Terry Lyons (University of Oxford)

- Nicolas Marie (Nanterre)  

- Sina Nejad (University of Oxford)

- Harald Oberhauser (University of Oxford)

- Imanol Perez (University of Oxford)

- Sebastian Riedel (Technische Universität Berlin)

- Dario Trevisan (Università degli Studi di Pisa)

- Danyu Yang (University of Oxford)

 

Slides

• Geng.pdf
• Imkeller.pdf
• Marie.pdf
• Perez.pdf
• Riedel.pdf
• Trevisan.pdf

jeudi 19 octobre

1 On Constrained Pathwise Stochastic Differential Equations

Let $C$ be a convex subset of $\mathbb R^d$. An interesting question is how to constrain the solution $X$ to a stochastic differential equation, driven by a process $B$, to stay in $C$. When $B$ is a Brownian motion, in Itô’s calculus framework, this problem has been solved by several methods. One of them is to put an invariance condition on the vector field of the SDE. Another one is to define $X$ as the solution of a Skorokhod reflexion problem. In this talk, we will extend these two methods when $B$ is a fractional Brownian motion in the rough paths framework. Co-authors: Laure Coutin, Paul Raynaud de Fitte and Charles Castaing.

Orateur: Nicolas Marie

2 TBA

Orateur: Sina Nejad

10:30 Coffee Break

3 TBA

Orateur: Terry Lyons

12:00 Lunch Break

4 Recent developments in discontinuous rough paths

In this talk, we will present the main features of rough paths theory in the discontinuous setting. We will discuss several notions of solutions to discontinuous RDEs and stability results which render the solution map continuous. We will also present an (enhanced) BDG inequality for lifts of càdlàg local martingales. Time permitted, we will discuss several applications, including robust Wong-Zakai-type theorems in the spirit of Kurtz-Pardoux-Protter, and weak convergence of stochastic flows which extends classical results of Kunita. Joint work with Peter Friz.

Orateur: Ilya Chevyrev

5 TBA

Orateur: Imanol Perez

15:15 Coffee Break

6 TBA

Orateur: Rémi Catellier

7 On the parametrization of level sets in the Heisenberg group

We introduce novel equations, in the spirit of Young-Rough Path theory, that parametrize level sets of intrinsically regular maps on the Heisenberg group with values in the plane. These equations can be seen as a sub-Riemannian counterpart to classical ODEs arising from the implicit function theorem. We show that they enjoy all the natural well-posedness properties, thus allowing for a ``good calculus'' on nonsmooth level sets, e.g., measuring their length. Examples and recent progress towards the higher co-dimension case will be discussed. Joint work with V. Magnani and E. Stepanov.

Orateur: Dario Trevisan

20:00 Conference Dinner

vendredi 20 octobre

8 TBA

Orateur: Sebastian Riedel

9 Delayed stochastic systems and Hormander spanning conditions

Malliavin's probabilistic proof of Hormander's criterion can be considerably simplified using some rough path theory - at the end. In our case, we are interested in non-Markovian SDEs, where the non-Markovian aspect finds its source in the presence of delays. As such - I shall present a framework for RDEs with delays. Extensions if time allows. - I will show the application to finding a simple spanning condition of "Hormander-type" for delayed SDEs/RDEs which garantees smoothness of densities. Malliavin calculus is kept to the minimum. This is extracted from a joint body of work with I. Ekren.

Orateur: Reda Chhaibi

10:30 Coffee Break

10 TBA

Orateur: Harald Oberhauser

11 Applications of tail estimates in rough path theory

We survey the results on tail estimates for different classes of stochastic rough paths. We present some recent applications of these results.

Orateur: Thomas Cass

12:30 Lunch Break

12 The tail asymptotics of the Brownian signature

In the groundbreaking work of B. Hambly and T. Lyons (Uniqueness for the signature of a path of bounded variation and the reduced path group, Ann. of Math., 2010), it has been conjectured that the geometry of a tree-reduced bounded variation path can be recovered from the tail asymptotics of its associated sequence of iterated path integrals. While this conjecture is still remaining open in the general deterministic case, in this talk we investigate a similar problem in the probabilistic setting for Brownian motion. It turns out that a martingale approach applied to the hyperbolic development of Brownian motion allows us to extract useful information from the tail asymptotics of Brownian iterated integrals, which can be used to determined the Brownian rough path along with its natural parametrization uniquely. This in particular strengthens the existing uniqueness results in the literature.

Orateur: Xi Geng

13 A rough path perspective on renormalisation

In this talk, we present the translation operator on rough paths which is the counterpart of the negative renormalisation described in the work Bruned-Hairer-Zambotti 2016 for regularity structures. Using a pre-Lie structure on rooted trees, one is able to derive the renormalised equation in the context of Rough SDEs. This approach also provides a method to derive the renormalised equation for SPDEs. This is a joint work with Ilya Chevyrev, Peter Friz and Rosa Preiss.

Orateur: Yvain Bruned

15:30 Coffee Break

14 Sensitivity of Rough Differential Equations

We present some result on the sensitivity of rough differential equations with respect to all the parameters. For this, we use the implicit function theorem together with an adaptation of the so-called omega lemma, which consists in studying the regularity of a function between two Banach spaces U and V which is transformed as a function mapping paths with values in U as a path with values in V. In particular, we show how the regularity of the vector field is transferred, up to some loss, to the one of the solution of the RDE. This talk is based on a joint work with Laure Coutin.

Orateur: Antoine Lejay

15 On the Hausdorff dimension of a very rough Weierstrass curve whose components are not controlled

We investigate geometric properties of Weierstrass curves with two components, representing series based on trigonometric functions. They are seen to be $\frac12$-Hölder continuous, and are not (para-)controlled with respect to each other in the sense of the recently established Fourier analytic approach of rough path analysis. Their graph is represented as an attractor of a smooth random dynamical system. Our argument that its graph has Hausdorff dimension 2 is in the spirit of Ledrappier-Young’s approach of the Hausdorff dimension of attractors. This is joint work with G. dos Reis (U Edinburgh) and O. Pamen (U Liverpool and AIMS Ghana).

Orateur: Peter Imkeller