Workshop "Rough Paths in Toulouse"
from
Thursday, October 19, 2017 (8:00 AM)
to
Friday, October 20, 2017 (6:00 PM)
Monday, October 16, 2017
Tuesday, October 17, 2017
Wednesday, October 18, 2017
Thursday, October 19, 2017
9:00 AM
On Constrained Pathwise Stochastic Differential Equations

Nicolas Marie
On Constrained Pathwise Stochastic Differential Equations
Nicolas Marie
9:00 AM  9:45 AM
Room: Salle 13 Bâtiment Mathématiques Appliquées
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. Coauthors: Laure Coutin, Paul Raynaud de Fitte and Charles Castaing.
9:45 AM
TBA

Sina Nejad
TBA
Sina Nejad
9:45 AM  10:30 AM
Room: Salle 13 Bâtiment Mathématiques Appliquées
10:30 AM
Coffee Break
Coffee Break
10:30 AM  11:00 AM
Room: Salle 13 Bâtiment Mathématiques Appliquées
11:00 AM
TBA

Terry Lyons
TBA
Terry Lyons
11:00 AM  12:00 PM
Room: Salle 13 Bâtiment Mathématiques Appliquées
12:00 PM
Lunch Break
Lunch Break
12:00 PM  1:45 PM
Room: Salle 13 Bâtiment Mathématiques Appliquées
1:45 PM
Recent developments in discontinuous rough paths

Ilya Chevyrev
Recent developments in discontinuous rough paths
Ilya Chevyrev
1:45 PM  2:30 PM
Room: Salle 13 Bâtiment Mathématiques Appliquées
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 WongZakaitype theorems in the spirit of KurtzPardouxProtter, and weak convergence of stochastic flows which extends classical results of Kunita. Joint work with Peter Friz.
2:30 PM
TBA

Imanol Perez
TBA
Imanol Perez
2:30 PM  3:15 PM
Room: Salle 13 Bâtiment Mathématiques Appliquées
3:15 PM
Coffee Break
Coffee Break
3:15 PM  3:45 PM
Room: Salle 13 Bâtiment Mathématiques Appliquées
3:45 PM
TBA

Rémi Catellier
TBA
Rémi Catellier
3:45 PM  4:30 PM
Room: Salle 13 Bâtiment Mathématiques Appliquées
4:30 PM
On the parametrization of level sets in the Heisenberg group

Dario Trevisan
On the parametrization of level sets in the Heisenberg group
Dario Trevisan
4:30 PM  5:15 PM
Room: Salle 13 Bâtiment Mathématiques Appliquées
We introduce novel equations, in the spirit of YoungRough 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 subRiemannian counterpart to classical ODEs arising from the implicit function theorem. We show that they enjoy all the natural wellposedness properties, thus allowing for a ``good calculus'' on nonsmooth level sets, e.g., measuring their length. Examples and recent progress towards the higher codimension case will be discussed. Joint work with V. Magnani and E. Stepanov.
8:00 PM
Conference Dinner
Conference Dinner
8:00 PM  10:00 PM
Room: Restaurant "Chez Emile"
Friday, October 20, 2017
9:00 AM
TBA

Sebastian Riedel
TBA
Sebastian Riedel
9:00 AM  9:45 AM
Room: Salle 13 Bâtiment Mathématiques Appliquées
9:45 AM
Delayed stochastic systems and Hormander spanning conditions

Reda Chhaibi
Delayed stochastic systems and Hormander spanning conditions
Reda Chhaibi
9:45 AM  10:30 AM
Room: Salle 13 Bâtiment Mathématiques Appliquées
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 nonMarkovian SDEs, where the nonMarkovian 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 "Hormandertype" 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.
10:30 AM
Coffee Break
Coffee Break
10:30 AM  11:00 AM
Room: Salle 13 Bâtiment Mathématiques Appliquées
11:00 AM
TBA

Harald Oberhauser
TBA
Harald Oberhauser
11:00 AM  11:45 AM
Room: Salle 13 Bâtiment Mathématiques Appliquées
11:45 AM
Applications of tail estimates in rough path theory

Thomas Cass
Applications of tail estimates in rough path theory
Thomas Cass
11:45 AM  12:30 PM
Room: Salle 13 Bâtiment Mathématiques Appliquées
We survey the results on tail estimates for different classes of stochastic rough paths. We present some recent applications of these results.
12:30 PM
Lunch Break
Lunch Break
12:30 PM  2:00 PM
Room: Salle 13 Bâtiment Mathématiques Appliquées
2:00 PM
The tail asymptotics of the Brownian signature

Xi Geng
The tail asymptotics of the Brownian signature
Xi Geng
2:00 PM  2:45 PM
Room: Salle 13 Bâtiment Mathématiques Appliquées
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 treereduced 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.
2:45 PM
A rough path perspective on renormalisation

Yvain Bruned
A rough path perspective on renormalisation
Yvain Bruned
2:45 PM  3:30 PM
Room: Salle 13 Bâtiment Mathématiques Appliquées
In this talk, we present the translation operator on rough paths which is the counterpart of the negative renormalisation described in the work BrunedHairerZambotti 2016 for regularity structures. Using a preLie 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.
3:30 PM
Coffee Break
Coffee Break
3:30 PM  4:00 PM
Room: Salle 13 Bâtiment Mathématiques Appliquées
4:00 PM
Sensitivity of Rough Differential Equations

Antoine Lejay
Sensitivity of Rough Differential Equations
Antoine Lejay
4:00 PM  5:00 PM
Room: Salle 13 Bâtiment Mathématiques Appliquées
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 socalled 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.
5:00 PM
On the Hausdorff dimension of a very rough Weierstrass curve whose components are not controlled

Peter Imkeller
On the Hausdorff dimension of a very rough Weierstrass curve whose components are not controlled
Peter Imkeller
5:00 PM  6:00 PM
Room: Salle 13 Bâtiment Mathématiques Appliquées
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 LedrappierYoung’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).