Trag2026

8 juin 2026, 08:00 → 10 juin 2026, 18:00 Europe/Paris

Institut Mathématiques de Toulouse

The aim of this conference is to bring together researchers in rough analysis and related fields from France and beyond. Rough path theory was introduced in 1994 by T. Lyons building on the concept of path signature. Over the last thirty years, the field has undergone major development. Some significant recent advances have been the study of singular PDEs (M. Hairer's Fields Medal in 2014), and of regularization by noise phenomena. More recently, path signatures have found numerous applications in the study of complex, high-dimensional data. 

Rough analysis forms one of the research axes of the RT MATRISK, which brings together the French community of researchers working on mathematics related to risk management.

The purpose of this conference is to review the progress in the field, to allow the French research community to exchange ideas, and to provide young doctoral and postdoctoral students with an opportunity to network.

 

The conference will take place in Toulouse.

 

Accomodation costs for participants will be covered. 

 

Invited speakers :

Horatio Boedihardjo (Warwick)

Joscha Diehl (Greifswald)

 Chengcheng Ling (Augsburg)

 

Organizing committee :

Laure Coutin (IMT Toulouse)

Romain Duboscq (IMT Toulouse)

Paul Gassiat (Univ. Gustave Eiffel)

 

Sponsors :

CIMI Toulouse

IMT Toulouse

Institut Universitaire de France

CNRS Mathématiques (RT MATRISK)

lun. 8 juin

14:00 → 15:00 Horatio Boedihardjo

Decay rate of logarithmic signature

It was conjectured by T. Lyons and N. Sidorova that, with the exceptions of straight lines, the logarithmic signatures of tree-reduced bounded variation paths have infinite radius of convergence. This conjecture was confirmed in the same work for certain types of paths and the general BV case remains unsolved.
In this talk, we develop a deeper understanding towards the Lyons-Sidorova conjecture. Joint work with Xi Geng and Sheng Wang.

15:00 → 15:30 Coffee break

15:30 → 16:30 Nathan Benichou

Unbounded nonconvex Young differential inclusions: existence of a measurable selection of solutions

We study the differential inclusion $\text{d} z_t\in F(z_t)\text{d} x_t$, with initial condition $z_0=\xi$,
where $F$ is a nonconvex-valued multifunction, and $x$ a path of bounded $q$-variation,
for some $1\le q<2$, extending the work of Bailleul, Brault and Coutin (2020).
We obtain existence of local and global solutions to this inclusion without assuming $F$ bounded.
If $z(\xi,x)$ denotes such a solution, we obtain measurability of $z$ with respect to $x$ and $\xi$.
To establish this, we introduce a Skorokhod-type distance and prove that Young integration is continuous with respect to it.
By the way, we prove that a compact-valued $\gamma$-Hölder map $F$ has, for any $p>1/\gamma$ and $\xi\in F(0)$,
a selection $f(\xi)$ of bounded $p$-variation, started at $\xi$, such that $f$ is measurable in $\xi$.

16:30 → 17:30 Łukasz Mądry

Ergodicity of fractional SDEs with singular drift

In this talk I will present a result on the construction of the unique invariant measure of the singular SDE with fractional Brownian noise (fBm), equipped with a linear damping. We build up on the theory of regularisation by noise, developed in recent years by Catellier, Gubinelli, Galeati and many others, and merge it with ergodic theory of fractional SDEs, studied by Hairer and his coauthors. We establish tightness in the usual regime of weak existence ($\alpha > 1/2-1/(2H)$, where $\alpha$ is Besov-Holder regularity of the drift and H is Hurst index of fBm), and uniqueness of invariant measure under usual condition on well-posedness for singular SDEs driven by fractional Brownian motion ($\alpha > 1 - 1/(2H)$). To this end we employ a modification of stochastic sewing, which also allows us to show Gaussian tails of the solution. Our approach does not require any assumption on the size of the drift with respect to the damping strength.
This is a joint work with Avi Mayorcas (Bath University, UK), https://arxiv.org/abs/2511.20556

19:00 → 20:30 Dinner cocktail

mar. 9 juin

09:50 → 10:50 Alexandre Richard

Directed polymers in a singular fractional environment: well-posedness and free-energy asymptotics

Directed polymers in random environments describe paths which are subject to competition between entropy, which favours diffusion, against the energy of a random, disordered medium, which favours localisation. After a brief word on polymer models in continuous space-time media, I will turn to environments that are irregular and long-range correlated, modelled by a space-time fractional Gaussian noise which is too rough for path-by-path integration. I will explain how to give meaning to the Hamiltonian and the polymer measure, and then describe the long-time behaviour of the free energy, leading to the conclusion that this polymer is subject to strong disorder.
Joint work with Julien Enguehard.

10:50 → 11:15 Coffee break

11:15 → 12:15 Antoine Lejay

Why trees in Runge-Kutta methods

While Runge-Kutta methods for solving ODE have been introduced around 1900, J. Butcher presented in the 1970's a set of algebraic techniques to check the order of a Runge-Kutta method.
This approach in now one of the pillars of the geometric integration theory. We give an intuitive account on why trees appear in such a problem and how transform is into an algebraic problem. From a joint work with Lorenzi Agabiti and Lorenzo Zambotti (LPSM, Paris).

12:15 → 14:00 Lunch break

14:00 → 15:00 Chengcheng Ling

Stroock--Varadhan martingale problem of Young stochastic differential equations

Under mild regularity assumptions, we prove that the martingale problem associated with the hybrid Young-Lyons-Itô differential equation admits a unique solution, thereby establishing probabilistic weak well-posedness. Our proof relies on analysis of the associated Kolmogorov equations, which are Young-type parabolic PDEs with irregular coefficients. The resulting theory for such Young-type parabolic PDEs is also of independent interest.
This talk is based on joint work with Fabio Bugini, Michele Coghi and Khoa Lê.

15:00 → 15:30 Coffee break

15:30 → 16:30 Dimitri Sotnikov

Chasing Stationarity: Exponentially Fading Memory Signature

We introduce the exponentially fading memory (EFM) signature, a time-invariant transformation of an infinite (possibly rough) path that serves as a mean-reverting analogue of the classical path signature. We construct the EFM-signature via rough path theory, carefully adapted to accommodate improper integration from minus infinity. The EFM-signature retains many of the key algebraic and analytical properties of classical signatures, including a suitably modified Chen identity, the linearization property, path-determinacy, and the universal approximation property. From the probabilistic perspective, the EFM-signature provides a "stationarized" representation, making it particularly well-suited for time-series analysis and signal processing overcoming the shortcomings of the standard signature. In particular, the EFM-signature of time-augmented Brownian motion evolves as a group-valued Ornstein-Uhlenbeck process. We establish its stationarity, Markov property, and exponential ergodicity in the Wasserstein distance, and we derive an explicit formula à la Fawcett for its expected value in terms of Magnus expansions. We also study linear combinations of EFM-signature elements and the computation of associated characteristic functions in terms of a mean-reverting infinite dimensional Riccati equation. We illustrate applications of the EFM signature in time series modeling through an SPX–VIX regression example and an electricity demand forecasting problem. This work is in collaboration with Eduardo Abi Jaber.

16:30 → 17:30 Anton Baeza

Gaussian driven Mckean-Vlasov equation

We study McKean--Vlasov equations driven by Gaussian rough noise. We exploit a regularization effect in the measure variable to transform the equation into one with a time-dependent vector field of complementary Young regularity.

The regularity gain is obtained through Gaussian integration by parts and the Duhamel formula from rough path theory, using the Cameron--Martin structure and the two-dimensional finite variation of the covariance. This yields deterministic stability estimates and an invariance property for the gained regularity. These results provide a pathwise framework for solving distribution-dependent rough equations under irregular Gaussian forcing.

19:00 → 21:00 Conference dinner

mer. 10 juin

09:00 → 10:00 Joscha Diehl

Tensor-to-Tensor Models with Fast Iterated Sum Features

We present a new class of tensor-to-tensor (in particular: image-to-image) models based on iterated sums.
Their efficient computation is inspired by recent breakthrough results in the field of permutation pattern counting.
Work in progress with R. Ibraheem, L. Schmitz and Y. Wue.

10:00 → 10:30 Coffee break

10:30 → 11:30 Aurélien Deya

An SPDE model: the $\Phi_3^4$ equation for the harmonic oscillator

We will first present the physical motivations behind this model, then turn to the comparison with its "standard" counterpart, in which the harmonic oscillator (on $\mathbb{R}^3$) is replaced by the Laplacian on the torus.

The results, from joint work with Reika Fukuizumi (Tokyo) and Laurent Thomann (Nancy), include the interpretation of the model in the renormalized sense, the global existence and uniqueness of a solution, the existence of an invariant measure, and finally its uniqueness in the weak nonlinearity regime.

11:30 → 12:30 Nicolas Moench

A general paracontrolled ansatz for singular SPDE

In this talk, we will present the paracontrolled approach to singular SPDEs and discuss its main ingredients. This framework enables us to give a definition for the ill-posed non-linearities from the equation at hand, and perform a fixed point argument in some suitable space of paracontrolled distributions. We will try to genaralize these ideas in order to construct a general paracontrolled ansatz for a broad class of subcritical SPDEs. A key ingredient in this construction is a family of stochastic fields indexed by decorated trees possessing good analytical and algebraic properties, we then estalish a paralinearisation formula for functions that are paracontrolled with respect to these fields. The singular products appearing in the equation are defined through a genarlised correcor lemma.
This talk is based on a joint work with Yvain Bruned.