Publications

2026

• Understanding the worst-kept secret of high-frequency trading
  
  • Pulido Sergio
  • Rosenbaum Mathieu
  • Sfendourakis Emmanouil
  Finance and Stochastics, Springer Verlag (Germany), 2026, 30, pp.329–396. Volume imbalance in a limit order book is often considered as a reliable indicator for predicting future price moves. In this work, we seek to analyse the nuances of the relationship between prices and volume imbalance. To this end, we study a market-making problem which allows us to view the imbalance as an optimal response to price moves. In our model, there is an underlying efficient price driving the mid-price, which follows the model with uncertainty zones. A single market maker knows the underlying efficient price and consequently the probability of a mid-price jump in the future. She controls the volumes she quotes at the best bid and ask prices. Solving her optimization problem allows us to understand endogenously the price-imbalance connection and to confirm in particular that it is optimal to quote a predictive imbalance. Our model can also be used by a platform to select a suitable tick size, which is known to be a crucial topic in financial regulation. The value function of the market maker's control problem can be viewed as a family of functions, indexed by the level of the market maker's inventory, solving a coupled system of PDEs. We show existence and uniqueness of classical solutions to this coupled system of equations. In the case of a continuous inventory, we also prove uniqueness of the market maker's optimal control policy. (10.1007/s00780-026-00585-9)
  DOI : 10.1007/s00780-026-00585-9
• Subadditivity and optimal matching of unbounded samples
  
  • Caglioti Emanuele
  • Goldman Michael
  • Pieroni Francesca
  • Trevisan Dario
  , 2026. We obtain new bounds for the optimal matching cost for empirical measures with unbounded support. For a large class of radially symmetric and rapidly decaying probability laws, we prove for the first time the asymptotic rate of convergence for the whole range of power exponents $p$ and dimensions $d$. Moreover we identify the exact prefactor when $p\le d$. We cover in particular the Gaussian case, going far beyond the currently known bounds. Our proof technique is based on approximate sub- and super-additivity bounds along a geometric decomposition adapted to some features the density, such as its radial symmetry and its decay at infinity.
• Long-time behaviour of a multidimensional age-dependent branching process with a singular jump kernel modelling telomere shortening
  
  • Olayé Jules
  • Tomasevic Milica
  Electronic Journal of Probability, Institute of Mathematical Statistics (IMS), 2026, 31. In this article, we investigate the ergodic behaviour of a multidimensional age-dependent branching process with a singular jump kernel, motivated by studying the phenomenon of telomere shortening in cell populations. Our model tracks individuals evolving within a continuous-time framework indexed by a binary tree, characterised by age and a multidimensional trait. Branching events occur with rates dependent on age, where offspring inherit traits from their parent with random increase or decrease in some coordinates, while the most of them are left unchanged. Exponential ergodicity is obtained at the cost of an exponential normalisation, despite the fact that we have an unbounded age-dependent birth rate that may depend on the multidimensional trait, and a non-compact transition kernel. These two difficulties are respectively treated by stochastically comparing our model to Bellman-Harris processes, and by using a weak form of a Harnack inequality. We conclude this study by giving examples where the assumptions of our main result are verified. (10.1214/25-EJP1469)
  DOI : 10.1214/25-EJP1469
• Existence and orbital stability proofs of traveling wave solutions on an infinite strip for the suspension bridge equation
  
  • van der Aalst Lindsey
  • Cadiot Matthieu
  , 2026. <div><p>In this paper, we present a computer-assisted approach for constructively proving the existence of traveling wave solutions of the suspension bridge equation on the infinite strip Ω = R × (-d2, d2). Using a meticulous Fourier analysis, we derive a quantifiable approximate inverse A for the Jacobian DF(ū) of the PDE at an approximate traveling wave solution ū. Such approximate objects are obtained thanks to Fourier coefficient sequences and operators, arising from Fourier series expansions on a rectangle Ω0 = (-d1, d1) × (-d2, d2) for large d1. In particular, the challenging exponential nonlinearity of the equation is tackled using a rigorous control of the aliasing error when computing related Fourier coefficients. This allows to establish a Newton-Kantorovich approach, from which the existence of a true traveling wave solution of the PDE can be proven in a vicinity of ū. We successfully apply such a methodology in the case of the suspension bridge equation and prove the existence of multiple traveling wave solutions on Ω. Finally, given a proven solution ũ, a Fourier series approximation on Ω0 allows us to accurately enclose the spectrum of DF(ũ). Such a tight control provides the number of negative eigenvalues, which in turn, allows us to conclude about the orbital (in)stability of the traveling wave.</p></div>
• Proving symmetry of localized solutions and application to dihedral patterns in the planar Swift-Hohenberg PDE
  
  • Blanco Dominic
  • Cadiot Matthieu
  , 2026. <div><p>In this article, we extend the framework developed in [14] to allow for rigorous proofs of existence of smooth, localized solutions in semi-linear partial differential equations possessing both space and non-space group symmetries. We demonstrate our approach on the Swift-Hohenberg model. In particular, for a given symmetry group G, we construct a natural Hilbert space H l G containing only functions with G-symmetry. In this space, products and differential operators are well-defined allowing for the study of autonomous semi-linear PDEs. Depending on the properties of G, we derive a Newton-Kantorovich approach based on the construction of an approximate inverse around an approximate solution, u0. More specifically, combining a meticulous analysis and computer-assisted techniques, the Newton-Kantorovich approach is validated thanks to the computation of some explicit bounds. The strategy for constructing u0, the approximate inverse, and the computation of these bounds will depend on the properties of G and its maximal square lattice space subgroup, H. More specifically, we consider three cases: G is a space group which can be represented on the square lattice, G is not a space group which can be represented on the square lattice and the symmetry of H isolates the solution, and where G is not a space group which can be represented on the square lattice and the symmetry of H does not isolate the solution. We demonstrate the methodology on the 2D Swift-Hohenberg PDE by proving the existence of various dihedral localized patterns. The algorithmic details to perform the computer-assisted proofs can be found on Github [4].</p></div>
• Proving the existence of localized patterns and saddle node bifurcations in 1D activator-inhibitor type models
  
  • Blanco Dominic
  • Cadiot Matthieu
  • Fassler Daniel
  , 2026. <div><p>In this paper, we present a general framework for constructively proving the existence and stability of stationary localized 1D solutions and saddle-node bifurcations in activatorinhibitor systems using computer-assisted proofs. Specifically, we develop the necessary analysis to compute explicit upper bounds required in a Newton-Kantorovich approach. Given an approximate solution ū, this approach relies on establishing that a well-chosen fixed point map is contracting on a neighborhood ū. For this matter, we construct an approximate inverse of the linearization around ū, and establish sufficient conditions under which the contraction is achieved. This provides a framework for which computer-assisted analysis can be applied to verify the existence and local uniqueness of solutions in a vicinity of ū, and control the linearization around ū. Furthermore, we extend the method to rigorously establish saddle-node bifurcations of localized solutions for the same type of models, by considering a well-chosen zero-finding problem. This depends on the rigorous control of the spectrum of the linearization around the bifurcation point. Finally, we demonstrate the effectiveness of the framework by proving the existence and stability of multiple steady-state patterns in various activatorinhibitor systems, as well as a saddle-node bifurcation in the Glycolysis model.</p></div>
• Hybrid FEM/IPDG semi-implicit schemes for time domain electromagnetic wave propagation in non cylindrical coaxial cables
  
  • Beni Hamad Akram
  • Imperiale Sébastien
  • Joly Patrick
  ESAIM: Mathematical Modelling and Numerical Analysis, Société de Mathématiques Appliquées et Industrielles (SMAI) / EDP, 2026. In this work, we develop an efficient numerical method for solving 3D Maxwell's equations in non-cylindrical coaxial cables. The main challenge arises from the elongated geometry of the computational domain, which induces strong anisotropy between the longitudinal direction (along the cable) and the transverse directions (within the cross-sections). This leads to the use of highly anisotropic meshes, where the longitudinal mesh size is much larger than the transverse one.<p>Our objective is to design a numerical scheme that is explicit in the longitudinal direction, with a CFL stability condition depending only on the longitudinal mesh size. In a previous work, we achieved this for cylindrical cables by employing prismatic edge elements, 1D quadrature for longitudinal mass lumping, and a hybrid explicit/implicit time discretization. The present paper extends this approach to non-cylindrical cables, addressing several new difficulties with the following key ingredients: (1) representing the cable as a deformation of a reference cylindrical cable and employing mapping techniques between the physical and reference domains; (2) using an anisotropic space discretization that combines an interior penalty discontinuous Galerkin (IPDG) method in the transverse directions with a conforming finite element method in the longitudinal direction; (3) utilizing prismatic edge elements on a prismatic mesh of the reference cable; and (4) adapting the construction of the hybrid explicit-implicit time discretization to the new structure of the semidiscrete problem. From a theoretical perspective, the main difficulty lies in the stability analysis, which requires extending and adapting standard techniques for DG methods in space and energy methods in time.</p>
• Inexact subgradient methods for semialgebraic functions
  
  • Bolte Jérôme
  • Le Tam
  • Moulines Éric
  • Pauwels Edouard
  Mathematical Programming, Series A, Springer, 2026, 219, pp.1-27. Motivated by the extensive application of approximate gradients in machine learning and optimization, we investigate inexact subgradient methods subject to persistent additive errors. Within a nonconvex semialgebraic framework, assuming boundedness or coercivity, we establish that the method yields iterates that eventually fluctuate near the critical set at a proximity characterized by an $O(\epsilon^\rho)$ distance, where $\epsilon$ denotes the magnitude of subgradient evaluation errors, and $\rho$ encapsulates geometric characteristics of the underlying problem. Our analysis comprehensively addresses both vanishing and constant step-size regimes. Notably, the latter regime inherently enlarges the fluctuation region, yet this enlargement remains on the order of $\epsilon^\rho$. In the convex scenario, employing a universal error bound applicable to coercive semialgebraic functions, we derive novel complexity results concerning averaged iterates. Additionally, our study produces auxiliary results of independent interest, including descent-type lemmas for nonsmooth nonconvex functions and an invariance principle governing the behavior of algorithmic sequences under small-step limits. (10.1007/s10107-025-02245-w)
  DOI : 10.1007/s10107-025-02245-w
• Selective excitatory effects of γ‐glutamyl‐glycine, a glutathione metabolite, on GluN1 NMDA and mGlu5 receptors impacting synaptic plasticity
  
  • Andreu Sandro
  • Gizzi Victoria Sonia
  • Pasco Ugo
  • Sebih Fatiha
  • Zitouni Yattou
  • Jimenez Pablo
  • Rolland Valérie
  • Chavanieu Alain
  • Estaran Sébastien
  • Rousset Matthieu
  • Cens Thierry
  • Cazals Guillaume
  • Charnet Pierre
  • Guiramand Janique
  • Cohen‐solal Catherine
  • Ménard Claudine
  • de Jésus Ferreira Marie‐céleste
  • Chevallier Nathalie
  • Cantel Sonia
  • Vignes Michel
  • Roussel Julien
  British Journal of Pharmacology, Wiley, 2026. Background and Purpose: Gamma‐glutamyl dipeptides are essentially produced after glutathione (GSH) degradation. While several γ‐glutamyl dipeptides have been detected in different organs and body fluids, their biological activity remains elusive. This is the case for γ‐Glutamyl‐Glycine (γ‐Glu‐Gly) which has been found in different brain areas. γ‐Glutamyl‐glycine may bind to ionotropic glutamate receptors. Here, we have investigated whether γ‐Glu‐Gly could exhibit modulatory actions on excitatory synaptic transmission and plasticity by performing its functional characterization on glutamate, GABA and glycine receptors. Experimental Approach: Electrophysiological and calcium imaging experiments were performed on cells heterologously expressing either Glutamate, GABA or Glycine receptors and on cultured hippocampal neurons and acute hippocampal slices. The γ‐Glu‐Gly production by cultured cells and hippocampal slices was measured by LC‐MS analysis, following incubation with modulators of the glutathione metabolic cycle. Key Results: γ‐Glu‐Gly exerted partial agonist effects on both NMDA GluN1 and mGlu5 receptors, but without any binding to the glutamate binding domain on both AMPA and NMDA GluN2A receptors. γ‐Glu‐Gly was devoid of any effect on GABA, glycine and GluN3A receptors. γ‐Glu‐Gly was able to trigger a long‐term increase in synaptic transmission in the CA1 area of mouse hippocampus. Conclusion and implications γ‐Glu‐Gly partially retains the excitatory actions of glutamate on specific receptor sub‐types and may trigger plastic events in the hippocampus. Its accumulation under pathological conditions associated with oxidative stress and a high GSH consumption could thus interfere with endogenous synaptic transmission and disrupt natural plasticity, which is a hallmark of neurodegenerative diseases. (10.1111/bph.70650)
  DOI : 10.1111/bph.70650
• Évaluation de l'équilibre électrique du scénario négawatt 2022 à l'aide du modèle open source eoles
  
  • Cédiey Hippolyte
  • Quirion Philippe
  • Baratgin Laure
  • Bustarret Quentin
  • de Oliveira-Gill Nilam
  • Perrier Quentin
  • Escribe Célia
  • Letz Thomas
  • Salomon Thierry
  • Shirizadeh Behrang
  Revue de l'OFCE, Presses de Sciences Po, 2026. Nous présentons la dernière version du modèle d'optimisation du système énergétique open source Eoles et nous l'utilisons pour évaluer dans quelle mesure le mix énergétique du scénario négaWatt 2022 peut satisfaire la demande d'énergie en France à l'horizon 2050, pour 19 années météorologiques. Nous obtenons que même sans recours aux interconnexions, la demande d'électricité n'excède la production que 3 à 4 heures par an en moyenne, ce qui ne dépasse que de très peu les critères de défaillance du Code de l'énergie. Pour éliminer toute heure de défaillance et assurer les besoins de réserves, une puissance supplémentaire de technologies pilotables de 13,8 GW est nécessaire, soit une augmentation de 39 % par rapport au scénario négaWatt. Nous étudions l'ajout de trois technologies pilotables : turbines à gaz (méthane ou hydrogène) et batteries, qui sont toutes proches en termes de coût total du système énergétique. Par ailleurs, l'équilibre électrique peut être atteint même en réduisant la capacité photovoltaïque sur toitures par rapport au scénario négaWatt. Le gain associé (3,4 Md€/an) est plus élevé que le surcoût entraîné par les capacités pilotables mentionnées ci-dessus (environ 1 Md€/an).
• Covariance-Informed Subspace: an Adaptive Gradient-Free Input Dimension Reduction Method for Bayesian Inference
  
  • Polette Nadège
  • Le Maître Olivier
  • Sochala Pierre
  • Gesret Alexandrine
  , 2026. This paper addresses the challenge of dimension reduction (DR) in Bayesian inference of high-resolution two-or three-dimensional fields, where a priori parametrizations require a large number of terms. The underlying idea is common to state-of-the-art methods in which the parameter space is decomposed into two subspaces, one informed by the likelihood and one constrained by the prior. DR techniques generally use gradient information from the log-likelihood to derive the corresponding subspaces. However, the gradient may be unavailable or expensive to compute accurately, for instance in the case of simulation-based inference. Inspired by approaches based on likelihood-informed subspaces, we develop a new DR method tailored for settings where gradient computation is not feasible. More specifically, we propose a gradient-free indicator for determining whether a direction is informed by the data. This indicator is derived from the posterior-to-prior covariance ratio introduced in Spantini et al. (2015). We show that, in the linear Gaussian case, this indicator combined with an approximate likelihood leads to a better posterior approximation. The method is then extended to nonlinear cases, and strategies to approximate the posterior covariance are detailed. We demonstrate the effectiveness of this DR through two high-dimensional inference problems arising from groundwater and atmospheric applications.
• Dense and nondense limits for uniform random intersection graphs
  
  • Bassino Frédérique
  • Bouvel Mathilde
  • Féray Valentin
  • Gerin Lucas
  • Pierrot Adeline
  Electronic Journal of Probability, Institute of Mathematical Statistics (IMS), 2026, 31 (none). We obtain the scaling limits of random graphs drawn uniformly in three families of intersection graphs: permutation graphs, circle graphs, and unit interval graphs. The two first families typically generate dense graphs, in these cases we prove a.s. convergence to an explicit deterministic graphon. Uniform unit interval graphs are nondense and we prove convergence in the sense of Gromov-Prokhorov after normalization of the distances: the limiting object is the interval [0,1] endowed with a random metric defined through a Brownian excursion. Asymptotic results for the number of cliques of size k (k fixed) in a uniform random graph in each of these three families are also given. In all three cases, an important ingredient of the proof is that, for indecomposable graphs in each class (where the notion of indecomposability depends on the class), the combinatorial object defining the graph (permutation, matching, or intervals) is essentially unique. (10.1214/26-EJP1570)
  DOI : 10.1214/26-EJP1570
• Counterfactually Fair Regression via Optimal Transport
  
  • Lince Marie Generali
  • Gaucher Solenne
  • Vie Jill-Jênn
  • Loiseau Patrick
  , 2026. We consider the problem of learning a counterfactually fair regressor. We adopt a causal uncertainty view in which counterfactual fairness is defined with resampled noise. We focus on obtaining theoretical fairness guarantees for a new post-processing estimator. We begin by showing that counterfactual fairness is equivalent to satisfying demographic parity conditional on the latent variable. This allows us to provide a closed-form expression of the optimal fair regressor via a barycentric quantile map. In order to handle continuous latent variables, we propose a discretized post-processing method. Then, under mild regularity assumptions, we prove high-probability finite-sample fairness guarantees for our estimator, providing an unfairness decay at rate Õ(n -1/3 ), and establishing a matching risk bound of order Õ(n -1/3 ). We provide a matching lower bound on the excess risk of almost fair predictions. Finally, we extend our results to the setting of relaxed counterfactual fairness. We validate our approach on real-world and synthetic data.
• Validated Enclosure of Renormalization Fixed Points Via Chebyshev Series and the DFT
  
  • Breden Maxime
  • Gonzalez Jorge
  • Mireles James Jason D.
  Foundations of Computational Mathematics, Springer Verlag, 2026. This work develops a computational framework for proving existence, uniqueness, isolation, and stability results for degree d, real analytic, unimodal functions fixed by m-th order Feigenbaum-Coullet-Tresser-Cvitanović renormalization operators using Chebyshev series approximation. Here the order m of the operator refers to the number of function compositions involved in its definition, and the degree d of the fixed function is the number of derivatives vanishing at its (unique) critical point. The advantage of Chebyshev series is that they are naturally adapted to spaces of real analytic functions, in the sense that they converge on ellipses containing real intervals rather than on disks in C. This facilitates the development of a functional analytic approach independent of order and degree. The main disadvantage of working with Chebyshev series is that the operations of rescaling and composition, essential to the definition of the renormalization operators, are less straightforward for Chebyshev than for Taylor series. These difficulties are overcome via a combination of a-priori information about decay rates in Banach spaces of rapidly decaying coefficient sequences, with a-posteriori estimates on Chebyshev interpolation errors for analytic functions. We illustrate the utility of the method by proving existence, local uniquness, global non-uniqueness results, and interval enclosures of universal constants for fixed points of varying orders and degrees. (10.1007/s10208-026-09763-9)
  DOI : 10.1007/s10208-026-09763-9
• Both genome instability and replicative senescence stem from the shortest telomere in telomerase-negative cells
  
  • Berardi Prisca
  • Martinez-Fernandez Veronica
  • Rat Anaïs
  • Rosas Bringas Fernando R
  • Jolivet Pascale
  • Langston Rachel
  • Mattarocci Stefano
  • Maes Alexandre
  • Aspert Théo
  • Zeinoun Bechara
  • Casier Karine
  • Kazemier Hinke G
  • Charvin Gilles
  • Doumic Marie
  • Chang Michael
  • Teixeira Maria Teresa
  Nature Communications, Nature Publishing Group, 2026, 17 (1), pp.4271. In the absence of telomerase, telomere shortening triggers replicative senescence, a tumor suppressor mechanism that is also associated with oncogenic genomic instability. Yet, the precise mechanism that connects these seemingly opposing forces remains poorly understood. To directly study the complex interplay between senescence, telomere dynamics, and genomic instability, we develop a system in Saccharomyces cerevisiae to generate and track telomeres of precise length in the absence of telomerase. Using single-telomere and single-cell analyses combined with mathematical modeling, we identify a threshold length at which telomeres switch into dysfunction. A single shortest telomere below the threshold length is necessary and sufficient to trigger the onset of replicative senescence in a majority of cells. At population level, fluctuation assays establish that rare genomic instability arises predominantly in cis to the shortest telomere as Pol32-dependent non-reciprocal translocations that result in re-elongation of the shortest telomere and likely transient escape from senescence. The switch of the shortest telomere into dysfunction and subsequent processing in telomerase-negative cells thus serves as the mechanistic link between replicative senescence onset, genomic instability and the initiation of post-senescence survival. (10.1038/s41467-026-70352-z)
  DOI : 10.1038/s41467-026-70352-z
• Finite element modelling for the reproduction of dynamic OCE measurements in the cornea
  
  • Merlini Giulia
  • Imperiale Sébastien
  • Allain Jean-Marc
  Journal of the Mechanics and Physics of Solids, Elsevier, 2026, 206, pp.106363. Recent advances in dynamic elastography, particularly through optical coherence tomography combined with transient excitations have enabled rapid, localized, and non-invasive mechanical data acquisition of the cornea. This dataopens the path to early-detection of pathologies and more accurate treatment. However, the analysis of the wave propagation is a complex mechanical problem: the cornea is a structure under pressure, with non-linear material behavior. Thus, computational analysis are needed to extract mechanical parameters from the data. In this study, we present a time-dependent finite element model for the reproduction of transient shear wave elastographic measurements in the cornea. The mechanical problem consists in a smallamplitude wave propagating in the cornea, largely deformed by intraocular pressure in physiological conditions. The model accounts for anisotropic, hyperelastic, and incompressible behavior of the cornea, as well as its accurate geometry, and the preloaded condition. We have implemented two different numerical approaches to solve first the static non-linear inflation of the cornea and then the linear wave propagation problem to reproduce the measurements. We investigate the impact of material anisotropy and prestress on wave propagation and demonstrate that intraocular pressure critically influences shear wave velocity. Additionally, by introducing a localized mechanical defect to simulate a pathological defect, we show that simulated shear wave can detect and quantify mechanical weaknesses, suggesting potential as a diagnostic tool to assess corneal health. (10.1016/j.jmps.2025.106363)
  DOI : 10.1016/j.jmps.2025.106363
• Decompositions of CSBPs via Poissonian Intertwining
  
  • Foucart Clément
  • Hénard Olivier
  ALEA : Latin American Journal of Probability and Mathematical Statistics, Instituto Nacional de Matemática Pura e Aplicada (Rio de Janeiro, Brasil) [2006-....], 2026, 23 (2), pp.1001. (10.30757/ALEA.v23-37)
  DOI : 10.30757/ALEA.v23-37
• Positive Markov processes in Laplace duality
  
  • Foucart Clément
  • Vidmar Matija
  Electronic Journal of Probability, Institute of Mathematical Statistics (IMS), 2026, 31 (none), pp.1-73. This article develops a general framework for Laplace duality between positive Markov processes in which the one-dimensional Laplace transform of one process can be represented through that of another. We show that a process admits a Laplace dual if and only if it satisfies a certain complete monotonicity condition. Moreover, we analyse how the conventions adopted for the values of 0⋅∞ and ∞⋅0 are reflected in the weak continuity/absorptivity properties of the processes in duality at the boundaries 0 and ∞. A broad class of generators admitting Laplace duals is identified, and we provide sufficient conditions under which the associated martingale problems are well-posed with the solutions being in duality at the level of their semigroups. Laplace duality is shown to furnish a unifying structure for several generalizations of continuous-state branching processes, e.g. those with immigration or evolving in random environments. Along the way, a theorem originally due to Ethier and Kurtz – connecting duality of generators to that of the associated semigroups – is refined, and we provide a concise proof of the Courrège form for the pointwise infinitesimal generator of a positive Markov process whose domain includes the exponential functions. The latter leads naturally to the notion of a Laplace symbol, which is a parsimonious encoding of the infinitesimal dynamics of the process. (10.1214/26-EJP1582)
  DOI : 10.1214/26-EJP1582
• Self-interacting approximation to McKean-Vlasov long-time limit: a Markov chain Monte Carlo method
  
  • Du Kai
  • Ren Zhenjie
  • Suciu Florin
  • Wang Songbo
  Journal de Mathématiques Pures et Appliquées, Elsevier, 2026, 205, pp.103782. For a certain class of McKean-Vlasov processes, we introduce proxy processes that substitute the mean-field interaction with self-interaction, employing a weighted occupation measure. Our study encompasses two key achievements. First, we demonstrate the ergodicity of the self-interacting dynamics, under broad conditions, by applying the reflection coupling method. Second, in scenarios where the drifts are negative intrinsic gradients of convex mean-field potential functionals, we use entropy and functional inequalities to demonstrate that the stationary measures of the self-interacting processes approximate the invariant measures of the corresponding McKean-Vlasov processes. As an application, we show how to learn the optimal weights of a two-layer neural network by training a single neuron. (10.1016/j.matpur.2025.103782)
  DOI : 10.1016/j.matpur.2025.103782
• A REMARK ON SELF-ADJOINT PROBLEMS IN THE OPTIMIZATION OF NON-LINEAR MODELS
  
  • Égoire Allaire G R
  • Cherrière Théodore
  • Gauthey Thomas
  • Hage Hassan Maya
  • Mininger Xavier
  Journal of Optimization Theory and Applications, Springer Verlag, 2026, 208, pp.100. This article considers optimization problems under nonlinear partial differential equation (p.d.e.) constraints. It is assumed that the p.d.e. arises from minimizing a convex energy. We prove that the optimization problem is self-adjoint when the objective function is the dual energy. In other words, the differential of the objective function with respect to the optimization variable does not involve any adjoint state. This result generalizes the well-known fact that the so-called compliance is self-adjoint in the linear case. We also prove that in a large class of objective functions the dual energy is the only one which is self-adjoint.
• Spatio-temporal equilibrium thermodynamics of guided optical waves at positive and negative temperatures
  
  • Zanaglia Lucas
  • Garnier Josselin
  • Michel Claire
  • Doya Valérie
  • Ferraro Mario
  • Wabnitz Stefan
  • Carusotto Iacopo
  • Picozzi Antonio
  Physical Review A, American Physical Society, 2026, 113, pp.063533. Optical thermalization has been recently studied theoretically and experimentally in the 2D spatial evolution of (quasi-)monochromatic light waves propagating in multimode fibers. In this work, we investigate the spatio-temporal equilibrium properties of incoherent multimode optical waves through the analysis of the (2+1)D Bose-Einstein thermal distribution and the corresponding classical Rayleigh-Jeans approximation. In the classical regime, we perform numerical simulations of the nonlinear Schr\"odinger equation and demonstrate relaxation toward the spatio-temporal Rayleigh–Jeans equilibrium state, as described by the corresponding wave turbulence kinetic equation. Remarkable adiabatic cooling phenomena stemming from the high-frequency tails of the Rayleigh-Jeans distribution are discussed and the consequent limitations of the classical approximation are highlighted. To overcome these issues and properly include quantum effects, we make use of a quantum version of the nonlinear Schr\"odinger equation that is obtained from the general quantum theory of light propagating in nonlinear waveguides. The associated kinetic equation describes relaxation toward the spatio-temporal Bose-Einstein equilibrium distribution with a fully regular UV behaviour. The analysis of thermodynamic equilibrium properties reveals a strong dependence on the specific dispersion regime under consideration. In the anomalous dispersion regime, the system relaxes to positive-temperatures equilibrium states: as the number of modes of the waveguide increases, the fundamental spatial mode becomes macroscopically populated, while its temporal spectrum undergoes significant narrowing, ultimately leading to complete (2+1)D spatio-temporal condensation in the thermodynamic limit. By contrast, in the normal dispersion regime the system evolves toward negative-temperature equilibrium states characterized by a hybrid structure: the spatial equilibrium displays an inverted modal population, whereas the temporal spectrum remains peaked around the fundamental (carrier) optical frequency. In this regime, we predict that spatio-temporal light waves exhibit a phase transition to Bose-Einstein condensation at negative temperatures, which occurs by increasing the temperature above a negative critical value. Our work opens new avenues for future research, including the possibility for a dual spatio-temporal beam cleaning through full spatio-temporal light condensation, and lay the groundwork for the development of spatio-temporal optical thermodynamics. (10.1103/bf7m-chkd)
  DOI : 10.1103/bf7m-chkd
• Numerical Analysis of differential equations on weighted Sobolev spaces: beyond classical orthogonal polynomials
  
  • Breden Maxime
  • Chu Hugo
  , 2026. We lay mathematical foundations for the Numerical Analysis of differential equations on Sobolev spaces weighted by a Gibbs probability measure $\nu$(dx) = e^-V(x)dx/$\mathcal{Z}$ on the real line. Over recent decades, the Functional Analysis of these spaces has been thoroughly developed to study Schrödinger-type equations and diffusion processes. While such equations should therefore be amenable to a numerical resolution with respect to orthogonal polynomials, this feat has only ever been achieved with respect to classical bases. We bridge this gap by showing that such equations can be solved with respect to suitable bases by factorising their leading linear component. In particular, we propose a new natural notion of Sobolev orthogonal polynomials, simpler and more tractable than those arising from the usual Sobolev inner product. In the case of $V$ being an even polynomial, we further establish quantitative estimates for the compactness of the embedding $H^1(\nu)\hookrightarrow L^2(\nu)$, uncovering a connection with the growth of the Jacobi recurrence coefficients, which are solutions of corresponding Painlevé-type discrete equations. As an application, we rigorously and tightly enclose solutions of the Gross--Pitaevskii equation with sextic potential and rigorously demonstrate the phenomenon of stochastic resonance via a computer-assisted proof.
• Entropic Mirror Monte Carlo
  
  • Cherradi Anas
  • Janati Yazid
  • Durmus Alain
  • Le Corff Sylvain
  • Petetin Yohan
  • Stoehr Julien
  Proceedings of the 43rd International Conference on Machine Learning, in Proceedings of Machine Learning Research, Zhang, Tong and Dudik, Miroslav and Jaggi, Martin and Agarwal, Alekh and Li, Sharon and Schuurmans, Dale and Zhu, Jerry and Berkenkamp, Felix and Dong, Hanze and Bietti, Alberto, 2026, 306, pp.19354--19375. Importance sampling is a Monte Carlo method which designs estimators of expectations under a target distribution using weighted samples from a proposal distribution. When the target distribution is complex, such as multimodal distributions in highdimensional spaces, the efficiency of importance sampling critically depends on the choice of the proposal distribution. In this paper, we propose a novel adaptive scheme for the construction of efficient proposal distributions. Our algorithm promotes efficient exploration of the target distribution by combining global sampling mechanisms with a delayed weighting procedure. The proposed weighting mechanism plays a key role by enabling rapid resampling in regions where the proposal distribution is poorly adapted to the target. Our sampling algorithm is shown to be geometrically convergent under mild assumptions and is illustrated through various numerical experiments.
• SAM-Guide : vers un premier Laser Run à destination de sportifs non-voyants
  
  • Romeo-Pakker Katerin
  • Huet Sylvain
  • Faugloire Elise
  • François Alouges
  • Abdellatif Ennaji
  • Christèle Lecomte
  • Wenqi Luo
  • Pellerin Denis
  • Fons Coline
  • Bruno Mantel
  • Mathieu Thomas
  • Sylvain Ferrand
  • Clément Francqueville
  • Graff Christian
  , 2026. Le projet ANR SAM-Guide vise à rendre le Laser Run praticable par des personnes aveugles et malvoyantes grâce à la combinaison multimodale de systèmes d'aide à la navigation et à l'atteinte de cible. Le dispositif fournit un retour auditif et/ou tactile en temps réel sur la position du corps et l'orientation du pistolet vers une cible virtuelle. Nous décrivons l'architecture du système et rapportons les premiers résultats obtenus auprès de participants avec ou sans capacité visuelle.
• torchsom: The Reference PyTorch Library for Self-Organizing Maps
  
  • Berthier Louis
  • Shokry Ahmed
  • Moreaud Maxime
  • Ramelet Guillaume
  • Moulines Eric
  Journal of Machine Learning Research, Microtome Publishing, 2026, 27 (190), pp.1–17.