Publications

2026

• Stability analysis of a new curl-based full field reconstruction method in 2D isotropic nearly-incompressible elasticity
  
  • Chibli Nagham
  • Genet Martin
  • Imperiale Sébastien
  Inverse Problems, IOP Publishing, 2026. In time-harmonic elastography, the shear modulus is typically inferred from full field displacement data by solving an inverse problem based on the time-harmonic elastodynamic equation. In this paper, we focus on nearly incompressible media, which pose robustness challenges, especially in the presence of noisy data. Restricting ourselves to 2D and considering an isotropic, linearly deforming medium, we reformulate the problem as a non-autonomous hyperbolic system and, through theoretical analysis, establish existence, uniqueness, and stability of the inverse problem. To ensure robustness with noisy data, we propose a least-squares approach with regularization. The convergence properties of the method are verified numerically using in silico data.
• Improved well-posedness for the limit flow of differentiation of roots of polynomials
  
  • Bertucci Charles
  • Pesce Valentin
  , 2026. In this paper, we study the partial differential equation on the circle that was heuristically obtained by Steinerberg [32] on the real line and which represents the evolution of the density of the roots of polynomials under differentiation. After integrating the partial differential equation in question, we observe that it can be treated with the theory of viscosity solutions. This equation at hand is a non linear parabolic integro-differential equation which involves the elliptic operator called the half-Laplacian. Due to the singularity of the equation, we restrict our study to strictly positive initial condition. We obtain a comparison principle for solutions of the primitive equation which yields uniqueness, existence, continuity with respect to initial condition. We also present heuristics to justify that the system of particles indeed approximates the solution of the equation.
• A stochastic use of the Kurdyka-Lojasiewicz property: Investigation of optimization algorithms behaviours in a non-convex differentiable framework
  
  • Fest Jean-Baptiste
  • Repetti Audrey
  • Chouzenoux Emilie
  Foundations of Data Science, American Institute of Mathematical Sciences, 2026, 9, pp.164-191. Asymptotic analysis of generic stochastic algorithms often relies on descent conditions. In a convex setting, some technical shortcuts can be considered to establish asymptotic convergence guarantees of the associated scheme. However, in a non-convex setting, obtaining similar guarantees is usually more complicated, and relies on the use of the Kurdyka-Łojasiewicz (KŁ) property. While this tool has become popular in the field of deterministic optimization, it is much less widespread in the stochastic context and the few works making use of it are essentially based on trajectory-by-trajectory approaches. In this paper, we propose a new framework for using the KŁ property in a non-convex stochastic setting based on conditioning theory. We show that this framework allows for deeper asymptotic investigations on stochastic schemes verifying some generic descent conditions. We further show that our methodology can be used to prove convergence of generic stochastic gradient descent (SGD) schemes, and unifies conditions investigated in multiple articles of the literature. (10.3934/fods.2025016)
  DOI : 10.3934/fods.2025016
• Spatio-temporal thermalization and adiabatic cooling of guided light waves
  
  • Zanaglia Lucas
  • Garnier Josselin
  • Carusotto Iacopo
  • Doya Valérie
  • Michel Claire
  • Picozzi Antonio
  Physical Review Letters, American Physical Society, 2026, 136, pp.053802. We propose and theoretically characterize three-dimensional spatio-temporal thermalization of a continuous-wave classical light beam propagating along a multi-mode optical waveguide. By combining a non-equilibrium kinetic approach based on the wave turbulence theory and numerical simulations of the field equations, we anticipate that thermalizing scattering events are dramatically accelerated by the combination of strong transverse confinement with the continuous nature of the temporal degrees of freedom. In connection with the blackbody catastrophe, the thermalization of the classical field in the continuous temporal direction provides a novel intrinsic mechanism for adiabatic cooling and spatial beam condensation. This process of adiabatic cooling is distinct from other mechanisms of thermalization and provides new insights into the dynamics of far-from-equilibrium closed systems and their route to thermalization. (10.1103/mqzh-w2gh)
  DOI : 10.1103/mqzh-w2gh
• Nonparametric regression on Riemannian manifolds under an alpha -mixing process
  
  • Wiem Nefzi
  • Salah Khardani
  • Françoise Yao Anne
  Communications in Statistics - Theory and Methods, Taylor & Francis, 2026, pp.1-16. This paper investigates the asymptotic properties of a kernel estimator of the regression function linking a real-valued response Y to a covariate X taking values in a Riemannian submanifold M. The estimator is an adaptation of the nonparametric regression estimator introduced by Pelletier (2006) for i.i.d. data on manifolds to the case where the observations (X t , Y t ) t∈Z form a stationary α-mixing process. Under suitable geometric regularity and mixing assumptions, we derive explicit expressions for the bias and the variance of the estimator, obtain the optimal bandwidth that minimizes the mean squared error, and establish the optimal rate of convergence in mean squared error. As a consequence, we prove both pointwise and uniform convergence in probability of the estimator on compact Riemannian manifolds under dependence. (10.1080/03610926.2026.2618625)
  DOI : 10.1080/03610926.2026.2618625
• Quantitative sensitivity analysis for Fokker-Planck equation with respect to the Wasserstein distance
  
  • Morange Martin
  , 2026. We analyze the sensitivity of solutions to the Fokker-Planck equation with respect to some unknown parameter. Our main result is to provide quantitative upper bounds for the $p$-Wasserstein distance $\mathcal{W}_p$ between two solutions with different parameters, for every $p \geq 2$. We are able to give two proofs of this result, the first relying on synchronous coupling between two solutions of an SDE, and another one that relies on the differentiation of Kantorovitch dual formulation of optimal transport. We also provide more specific bounds in the case of the overdamped Langevin process, for which we are able to compare convergence to the invariant measure and sensitivity to the parameter.
• The Mortensen observer on the space of probability measures
  
  • Morange Martin
  , 2026. We study a deterministic filtering problem formulated directly on the Wasserstein space of probability measures with finite second moment. Motivated by the Mortensen minimum-energy observer, we consider the reconstruction of an evolving probability density from partial observations by minimizing an action functional combining a kinetic transport cost and a time-dependent observation mismatch. The resulting value function is defined on the infinite-dimensional manifold (P_2(R^d), W_2) and satisfies a Hamilton-Jacobi-Bellman equation involving the Wasserstein gradient. Under suitable regularity and growth assumptions on the observation functional, we establish dynamic programming principles, continuity of the value function, existence of minimizing trajectories, and viscosity solution properties of the associated Hamilton-Jacobi equation. We provide two complementary notions of viscosity solutions: a geometric formulation based on subdifferentials in Wasserstein space, and a Hilbertian formulation inspired by Lions' lifting approach. This allows us to prove a comparison principle and uniqueness of solutions. Extensions to transport equations with drift are also discussed. Finally, we introduce a semi-Lagrangian scheme in order to approximate the value function, and show Γ-convergence of the scheme.
• Separation rates for the detection of synchronization of interacting point processes in a mean field frame. Application to neuroscience.
  
  • Tchouanti Josué
  • Löcherbach Éva
  • Reynaud-Bouret Patricia
  • Tanré Etienne
  Electronic Journal of Statistics, Shaker Heights, OH : Institute of Mathematical Statistics, 2026, 20 (1), pp.560--632. Permutation tests have been proposed by Albert et al. (2015) to detect dependence between point processes, modeling in particular spike trains, that is the time occurrences of action potentials emitted by neurons. Our present work focuses on exhibiting a criterion on the separation rate to ensure that the Type II errors of these tests are controlled non asymptotically. This criterion is then discussed in two major models in neuroscience: the jittering Poisson model and Hawkes processes having \(M\) components interacting in a mean field frame and evolving in stationary regime. For both models, we obtain a lower bound of the size \(n\) of the sample necessary to detect the dependency between two neurons. (10.1214/26-EJS2483)
  DOI : 10.1214/26-EJS2483