Research Themes

Mathematical Modelling for Water Resource Management

This recent research direction arises from a collaboration with Eau d’Azur, the public water utility of the Nice metropolitan area, and focuses on modelling questions related to the drinking water network.

In particular, we study the long-term forecasting of groundwater levels, using both classical time series methods and more advanced tools such as path signatures. Related projects are currently in progress, concerning the modelling and analysis of data from water distribution networks, including distribution forecasting using time series on graphs and learning-based methods.

This work is carried out in collaboration with Eau d’Azur and includes the supervision of several students: the M2 internship of Simone Singh, co-supervised with Yassine Laguel, on network-related modelling questions, and the PhD thesis of Biagio Tropiano, co-supervised with Elena Di Bernardino, on groundwater level forecasting.

This collaboration is also presented in a CNRS info.

Peano's example of non-uniqueness

Regularization by noise

A large part of my work is deveoted to the regularization by niose phenemenon. It is well known that wellposedness of (partial) differential equation may fails when the coefficients of the problem are not regular enough. The addition of a noise (Brownian motion, Lévy noise...) may restore uniqueness (and even existence).

Together with several coauthors, I studied how the addition of fractional noise in a pathwise manner to the usual ODE for quite irregular drift (see [CG2016]) allows to restore existence, and even uniqueness.

I extended this result to the corresponding transport PDE in [C2016], using a new definition of solutions thanks to rough paths theory. In [CH2022] we use the same techniques, together with Volterra-type rough integration to deal with the gPAM equation.

Finally, in [CD2022] using Malliavin calculus, we extend the previous result to rough differential equations with singular drift driven by Gaussian rough.

biological modelization

This part of my research take place in the DREAMS/NEMATIC consortium. The aim of this project is to study, model and control the growth of random network at several scales. A key example is the mycellium of P. Anserina. A first work has lead to the publication of [D2020].

The apporach of this project is strongly pluridisciplinary, as it involves biologists, mathematician, physisists, geographers...

With two coauthors [CDR2021] we have proposed and study a model for the growth of a self-interacting random network (via branching diffusions), and its limiting large scale behaviour.

Together with L. Béthencourt and E. Tanré [BCT2024] we have developped a setting to control in an optimal fashion the volume occupied by trajectories of Brownian particules. It is a first step toward a description of the dynamics of a self-interacting random network as minimizers of certain functionals.

Sébastian Baudelet has defended his PhD thesis in 2025 entitled "Multiscale Modeling and simulations of branching in mycelian networks" on this subject, co supervised with Claire Guerrier and Thierry Goudon.

Rough paths theory

In a series a works with different coauthors, we study various properties of rough paths and rough differential equations. In [BC2017] we propose a rough flow-based approach to treat homogeneization phenomenon in random ODEs. It allows to focus on the homogeneization of simple object to characterize the limit dynamic.

In [BC2019] we exhibit a general criterion for non-explosion of solutions of rough differential equations. Unlike in the determinstic setting, a linear-growth condition is not enough, and one has to require stronger conditions.

Finally in [BCD2020] and [BCD2021], using Lions differential calculus we developped the theory for distribution dependent rough differential equations of the form

where B is a random rough path. We are then able to deal we the propagation of chaos phenomenon for the corresponding the interacting particles system.

Anton Baeza has started a PhD thesis on these subject in 2024, co-supervised with François Delarue

Stochastic partial differential equations

In [CC2018] we used the "paracontrolled calculus" of Gubinelli, Imkeller and Perkowski to study the ϕ^4 stochastic quantisation equation in dimension 3 :

The difficulty was to defined the cube of the solution, which is only a distribution and not a function, and to construct all the needed stochastic terms which where necessary to perform the analysis of the previous equation.

Similar but less involved techniques where used in [CH2022] to deal with the regularization by noise phenomenon for the gPAM equation.

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