Research Themes
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.
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 started a PhD thesis in 2020 entitled "Multiscale Modeling and simulations of branching in mycelian networks - shaping and controlling hyphae" on this subject. I co supervise him 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.
Stochastic partial differential equations
In [CC2018] we uses the newly constructed "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.