Publications

2025

• Maximal Entropy Random Walks in Z: Random and non-random environments
  
  • Duboux Thibaut
  • Gerin Lucas
  • Offret Yoann
  Journal of Statistical Physics, Springer Verlag, 2025, 192 (10), pp.140. The Maximal Entropy Random Walk (MERW) is a natural process on a finite graph, introduced a few years ago with motivations from theoretical physics. The construction of this process relies on Perron-Frobenius theory for adjacency matrices. Generalizing to infinite graphs is rather delicate, and in this article, we study in detail specific models of the MERW on Z with loops, for both random and non-random loops. Thanks to an explicit combinatorial representation of the corresponding Perron-Frobenius eigenvectors, we are able to precisely determine the asymptotic behavior of these walks. We show, in particular, that essentially all MERWs on Z with loops have positive speed. (10.1007/s10955-025-03516-8)
  DOI : 10.1007/s10955-025-03516-8
• Improved Polynomial Bounds and Acceleration of GMRES by Solving a min-max Problem on Rectangles, and by Deflating
  
  • Spillane Nicole
  • Szyld Daniel B
  , 2025. Polynomial convergence bounds are considered for left, right, and split preconditioned GMRES. They include the cases of Weighted and Deflated GMRES for a linear system Ax = b. In particular, the case of positive definite A is considered. The well-known polynomial bounds are generalized to the cases considered, and then reduced to solving a min-max problem on rectangles on the complex plane. Several approaches are considered and compared. The new bounds can be improved by using specific deflation spaces and preconditioners. This in turn accelerates the convergence of GMRES. Numerical examples illustrate the results obtained.
• Extended reference prior theory for objective and practical inference, application to robust and auditable seismic fragility curve estimation
  
  • van Biesbroeck Antoine
  , 2025. Reference prior theory provides a principled framework for objective Bayesian inference, aiming to minimize subjective input and allow data-based information to drive the estimates distribution. For this reason, the application of this theory to the estimation of seismic fragility curves is particularly relevant. Indeed, these curves are essential elements of seismic probabilistic risk assessment studies; they express the probability of failure of a mechanical structure as a function of indicators that define seismic scenarios. Since they inform critical decisions in infrastructure safety, a complete auditability of the pipeline that leads to the estimates of these curves is required.This thesis investigates the interplay between reference prior theory and seismic fragility curves estimation, yielding original contributions in these two domains. First, we complement the theoretical foundations of reference priors by developing novel constructions of them. Our goal is to support their objectivity while improving their practical applicability. Our results take the form of theoretical contributions in this domain that are based on a generalized definition of the mutual information. Our approaches tackle the principal issues of reference priors, namely their improper characteristic or that of their posterior, and their complex formulation for practical use.Second, we revisit the estimation of seismic fragility curves based on the prominent probit-lognormal model in a context where the data are particularly sparse. Our goal is to conduct a Bayesian estimation of seismic fragility curves that leverages the optimization of every sort of information, including the a priori one, in order to provide estimates that are robust and auditable. Our results highlight the limitations and irregularities of the model and propose methods that provide accurate and efficient estimates of the curves. The evaluations of our approaches are carried out on different case studies taken from the nuclear industry.This thesis builds a strong link between these two domains. The application to seismic fragility curves not only motivated theoretical developments but also directly benefited them, ultimately producing a more robust, interpretable, and verifiable estimation framework.
• Improving the scalability of a high-order atmospheric dynamics solver based on the deal. II library
  
  • Orlando Giuseppe
  • Benacchio Tommaso
  • Bonaventura Luca
  , 2025, 267, pp.227-236. We present recent advances on the massively parallel performance of a numerical scheme for atmosphere dynamics applications based on the deal.II library. The implicit-explicit discontinuous finite element scheme is based on a matrix-free approach, meaning that no global sparse matrix is built and only the action of the linear operators on a vector is actually implemented. Following a profiling analysis, we focus on the performance optimization of the numerical method and describe the impact of different preconditioning and solving techniques in this framework. Moreover, we show how the use of the latest version of the deal.II library and of suitable execution flags can improve the parallel performance. (10.1016/j.procs.2025.08.249)
  DOI : 10.1016/j.procs.2025.08.249
• A reproducible comparative study of categorical kernels for Gaussian process regression, with new clustering-based nested kernels
  
  • Carpintero Perez Raphaël
  • Da Veiga Sébastien
  • Garnier Josselin
  , 2025. Designing categorical kernels is a major challenge for Gaussian process regression with continuous and categorical inputs. Despite previous studies, it is difficult to identify a preferred method, either because the evaluation metrics, the optimization procedure, or the datasets change depending on the study. In particular, reproducible code is rarely available. The aim of this paper is to provide a reproducible comparative study of all existing categorical kernels on many of the test cases investigated so far. We also propose new evaluation metrics inspired by the optimization community, which provide quantitative rankings of the methods across several tasks. From our results on datasets which exhibit a group structure on the levels of categorical inputs, it appears that nested kernels methods clearly outperform all competitors. When the group structure is unknown or when there is no prior knowledge of such a structure, we propose a new clustering-based strategy using target encodings of categorical variables. We show that on a large panel of datasets, which do not necessarily have a known group structure, this estimation strategy still outperforms other approaches while maintaining low computational cost.
• Differentiable Expectation-Maximisation and Applications to Gaussian Mixture Model Optimal Transport
  
  • Boïté Samuel
  • Tanguy Eloi
  • Delon Julie
  • Desolneux Agnès
  • Flamary Rémi
  , 2025. The Expectation-Maximisation (EM) algorithm is a central tool in statistics and machine learning, widely used for latent-variable models such as Gaussian Mixture Models (GMMs). Despite its ubiquity, EM is typically treated as a non-differentiable black box, preventing its integration into modern learning pipelines where end-to-end gradient propagation is essential. In this work, we present and compare several differentiation strategies for EM, from full automatic differentiation to approximate methods, assessing their accuracy and computational efficiency. As a key application, we leverage this differentiable EM in the computation of the Mixture Wasserstein distance $\mathrm{MW}_2$ between GMMs, allowing $\mathrm{MW}_2$ to be used as a differentiable loss in imaging and machine learning tasks. To complement our practical use of $\mathrm{MW}_2$, we contribute a novel stability result which provides theoretical justification for the use of $\mathrm{MW}_2$ with EM, and also introduce a novel unbalanced variant of $\mathrm{MW}_2$. Numerical experiments on barycentre computation, colour and style transfer, image generation, and texture synthesis illustrate the versatility of the proposed approach in different settings.
• A robust computational framework for the mixture-energy-consistent six-equation two-phase model with instantaneous mechanical relaxation terms
  
  • Orlando Giuseppe
  • Haegeman Ward
  • Pelanti Marica
  • Massot Marc
  , 2025. We present a robust computational framework for the numerical solution of a hyperbolic 6-equation single-velocity two-phase model. The model's main interest is that, when combined with instantaneous mechanical relaxation, it recovers the solution of the 5-equation model of Kapila. Several numerical methods based on this strategy have been developed over the years. However, neither the 5- nor 6-equation model admits a complete set of jump conditions because they involve non-conservative products. Different discretizations of these terms in the 6-equation model exist. The precise impact of these discretizations on the numerical solutions of the 5-equation model, in particular for shocks, is still an open question to which this work provides new insights. We consider the phasic total energies as prognostic variables to naturally enforce discrete conservation of total energy and compare the accuracy and robustness of different discretizations for the hyperbolic operator. Namely, we discuss the construction of an HLLC approximate Riemann solver in relation to jump conditions. We then compare an HLLC wave-propagation scheme which includes the non-conservative terms, with Rusanov and HLLC solvers for the conservative part in combination with suitable approaches for the non-conservative terms. We show that some approaches for the discretization of non-conservative terms fit within the framework of path-conservative schemes for hyperbolic problems. We then analyze the use of various numerical strategies on several relevant test cases, showing both the impact of the theoretical shortcomings of the models as well as the importance of the choice of a robust framework for the global numerical strategy.
• An all-topology two-fluid model for two-phase flows derived through Hamilton's Stationary Action Principle
  
  • Haegeman Ward
  • Orlando Giuseppe
  • Kokh Samuel
  • Massot Marc
  , 2025. We present a novel multi-fluid model for compressible two-phase flows. The model is derived through a newly developed Stationary Action Principle framework. It is fully closed and introduces a new interfacial quantity, the interfacial work. The closures for the interfacial quantities are provided by the variational principle. They are physically sound and well-defined for all type of flow topologies. The model is shown to be hyperbolic, symmetrizable, and admits an entropy conservation law. Its non-conservative products yield uniquely defined jump conditions which are provided. As such, it allows for the proper treatment of weak solutions. In the multi-dimensional setting, the model presents lift forces which are discussed. The model constitutes a sound basis for future numerical simulations.
• A two-point phase recovering from holographic data on a single plane
  
  • Novikov Roman
  • Sivkin Vladimir
  , 2025. We consider a plane wave, a radiation solution, and the sum of these solutions (total solution) for the Helmholtz equation in an exterior region in $\mathbb{R}^d$ , $d \geq 2$. In this region, we consider a hyperplane $X$ with sufficiently large distance $s$ from the origin in $\mathbb{R}^d$. We give two-point local formulas for approximate recovering the radiation solution restricted to the plane $X$ from the intensity of the total solution at $X$, that is, from holographic data. The recovering is given in terms of the far-field pattern of the radiation solution with a decaying error term as $s \to +\infty$. A numerical implementation is also presented.
• On The Cutoff Phenomenon For Dyson-Laguerre Processes
  
  • Chan-Ashing Samuel
  , 2025. <div><p>We study the convergence to equilibrium in high dimensions, focusing on explicit bounds on mixing times and the emergence of the cutoff phenomenon for Dyson-Laguerre processes. These are interacting particle systems with non-constant diffusion coefficients, arising naturally in the context of sample covariance matrices. The infinitesimal generator of the process admits generalized Laguerre orthogonal polynomials as eigenfunctions.</p><p>Our analysis relies on several distances and divergences, including an intrinsic Wasserstein distance adapted to the non-Euclidean geometry of the process. Within this framework, we employ tools from Riemannian geometry and functional inequalities. In particular, we establish exponential decay and derive a regularization inequality for the intrinsic Wasserstein distance via comparison with relative entropy.</p></div>
• Kuramoto Mean Field Game with Intrinsic Frequencies
  
  • Carmona Rene
  • Cormier Quentin
  • Soner Mete
  , 2025. This paper studies a mean field game formulation of the classical Kuramoto model for synchronization. Our model captures the diversity within the population by considering random intrinsic frequencies, which allows us to study the impact of this heterogeneity on synchronization patterns and stability. Our findings contribute insights into the interplay between intrinsic frequency diversity and synchronization dynamics, offering a more realistic understanding of complex systems. The proposed framework has broad applications ranging from coupled oscillators in physics to social dynamics, and serves as a valuable tool for studying networks with distributed intrinsic frequencies.
• Positive Univariate Polynomials: SOS certificates, algorithms, bit complexity, and T-systems
  
  • Bender Matías
  • Di Dio Philipp
  • Tsigaridas Elias
  , 2025. We consider certificates of positivity for univariate polynomials with rational coefficients that are positive over (an interval of)~$\mathbb{R}$. Such certificates take the form of weighted sums of squares (SOS) of polynomials with rational coefficients. We build on the algorithm of Chevillard, Harrison, Jolde{\c{s}}, and Lauter~\cite{chml-usos-alg-11}, and we introduce a variant that we refer to as \usos. Given a polynomial of degree~$d$ with maximum coefficient bitsize~$\tau$, we show that \usos computes a rational weighted SOS representation in $\widetilde{\mathcal{O}}_B(d^3 + d^2 \tau)$ bit operations; the resulting certificate of posivitity involves rationals of bitsize $\widetilde{\mathcal{O}}(d^2 \tau)$. This improves the best-known complexity bounds by a factor of~$d$ and completes previous analyses. We also extend these results to certificates of positivity over arbitrary rational intervals, via a simple transformation. In this case as well, our techniques yield a factor-$d$ improvement in the complexity bounds. Along the same line, for univariate polynomials with rational coefficients, we introduce a new class of certificates, which we call \emph{perturbed SOS certificates}. They consist of a sum of two rational squares that approximates the input polynomial closely enough so that nonnegativity of the approximation implies the nonnegativity of the original polynomial. This computation has the same bit complexity and yields certificates of the same bitsize as in the weighted SOS case. We further investigate structural properties of these SOS decompositions. Relying on the classical result that any nonnegative univariate real polynomial is the sum of two squares of real polynomials, we show that the summands form an interlacing pair. Consequently, their real roots correspond to the Karlin points of the original polynomial on~$\mathbb{R}$, establishing a new connection with the T-systems studied by Karlin~\cite{Karlin-repr-pos-63}. This connection enables us to compute such decompositions explicitly. Previously, only existential results were known for T-systems. We obtain analogous results for positivity over $(0, \infty)$, and hence over arbitrary real intervals. Finally, we present our open-source Maple implementation of the \usos algorithm, together with experiments on various data sets demonstrating the efficiency of our approach.
• Limitation strategies for high-order discontinuous Galerkin schemes applied to an Eulerian model of polydisperse sprays
  
  • Ait-Ameur Katia
  • Essadki Mohamed
  • Massot Marc
  • Pichard Teddy
  ESAIM: Mathematical Modelling and Numerical Analysis, Société de Mathématiques Appliquées et Industrielles (SMAI) / EDP, 2025, 59 (5), pp.2349-2383. In this paper, we tackle the modeling and numerical simulation of polydisperse sprays. Starting from a kinetic description for point particles, we focus on an Eulerian high-order geometric method of moment (GeoMOM) in size and consider a system of partial differential equations on a vector of successive fractional size moments of order 0 to N/2, N &gt; 2, over a compact size interval. These moments correspond to physical quantities, which can be interpreted in terms of the geometry of the interface at small scale. There exists a stumbling block for the usual approaches using high-order moment methods resolved with high-order numerical methods: the transport algorithm does not naturally preserve the moment space. Indeed, reconstruction of moments by polynomials inside computational cells can create N-dimensional vectors which can fail to be moment vectors. We thus propose a new approach, as well as an algorithm, which is arbitrarily high-order in space and time with limited numerical diffusion, including at the boundaries of the state space, where a specific study is proposed. It allows to accurately describe the advection process and naturally preserves the moment space, at a reasonable computational cost. We show that such an approach is competitive compared to second order finite volume schemes, where limiters generate numerical diffusion and clipping at extrema. An accuracy study assesses the order of the method as well as the low level of numerical diffusion on structured meshes. We focus in this paper on cartesian meshes and 2D test cases are presented where the accuracy and efficiency of the approach are assessed. (10.1051/m2an/2025057)
  DOI : 10.1051/m2an/2025057
• Sample-Efficient Reinforcement Learning : Exploration, Imitation, and Online Learning
  
  • Tiapkin Daniil
  , 2025. Reinforcement learning (RL) provides a formal framework for sequential decision-making in an unknown environment. An agent learns a policy to maximize a cumulative reward signal by interacting with a Markov Decision Process (MDP). This thesis investigates three extensions to the standard RL setting, providing new algorithms and associated theoretical analyses.First, we study the problem of exploration in the absence of a reward signal. We consider two distinct formulations of maximum entropy exploration and propose computationally efficient algorithms. For these algorithms, we derive sample complexity bounds that improve upon previously known results. Our analysis further establishes that learning the maximum visitation entropy policy is statistically less demanding than both a common variant of reward-free exploration and the task of learning an optimal policy in a standard rewarded MDP.Second, we address the problem of accelerating reinforcement learning using expert demonstrations, a setting motivated by modern applications such as Reinforcement Learning from Human Feedback (RLHF). We analyze the framework of demonstration-regularized RL, where the agent’s learning objective is augmented with a Kullback-Leibler penalty for deviating from a reference policy learned from the demonstrations. We prove that this approach leads to a linear reduction in sample complexity with respect to the number of demonstrations. This result establishes how expert data can improve the efficiency of the subsequent RL task.Third, we address the problem of learning in MDPs with rewards selected by an (oblivious) adversary. We propose a new algorithm that uses a black-box online linear optimization oracle, making it easy to implement. We provide a regret analysis showing that its performance bound matches the minimax lower bound for the stochastic (non-adversarial) case in its dependence on the number of states, actions, and the number of episodes.Collectively, the contributions of this thesis consist of novel algorithms for various problems in reinforcement learning, accompanied by proofs of their performance in terms of sample complexity and regret bounds.
• Second-order Optimally Stable IMEX (pseudo-)staggered Galerkin discretization: application to lava flow modeling
  
  • Gatti Federico
  • Orlando Giuseppe
  , 2025. We present second-order optimally stable Implicit-Explicit (IMEX) Runge-Kutta (RK) schemes with application to a modified set of shallow water equations that can be used to model the dynamics of lava flows. The schemes are optimally stable in the sense that they satisfy, at the space-time discretization level, a condition analogous to the L-stability of Runge-Kutta methods for ordinary differential equations. A novel (pseudo-)staggered Galerkin scheme is introduced, which can be interpreted as an extension of the classical two-step Taylor-Galerkin (TG2) scheme. The method is derived by combining a von Neumann stability analysis with a Lax-Wendroff procedure. For the discretization of the non-conservative terms that characterize the lava flow model, we employ the Path-Conservative (PC) method. The proposed scheme is evaluated on a number of relevant test cases, demonstrating accuracy, robustness, and well-balancing properties for the lava flow model. (10.13140/RG.2.2.29911.53925)
  DOI : 10.13140/RG.2.2.29911.53925
• Asymptotic Analysis of a class of abstract stiff wave propagation problems
  
  • Imperiale Sébastien
  Multiscale Modeling and Simulation: A SIAM Interdisciplinary Journal, Society for Industrial and Applied Mathematics, 2025, 23 (3), pp.1389-1416. This work addresses the mathematical analysis, by means of asymptotic analysis, of a class of linear wave propagation problems with singular stiff terms represented by a single small parameter. This abstract setting is defined using linear operators in Hilbert spaces. In this setting, we show, under some assumptions on the structure of the wave propagation problems, weak and strong convergence of solutions with respect to the small parameter towards the solution of a well-defined limit problem. Applications in nearly incompressible elastodynamics, elastic waves in thin plates, piezoelectricity, and homogenization are presented. (10.1137/24M1679458)
  DOI : 10.1137/24M1679458
• Random modeling for the formation process of limit order books
  
  • Sfendourakis Emmanouil
  , 2025. This thesis aims to better understand the dynamics of limit order books and connect them to rational behavior of market participants. The first chapter examines how optimal market-making strategies explain the predictive power of imbalance on high-frequency price movements. We develop a model where a single market maker is aware of an efficient price driving mid-price dynamics. She controls the volumes posted at the best bid and ask prices to maximize daily profit. Her value function is the solution of a system of coupled PDEs. We prove the existence and uniqueness of smooth solutions of this system, and derive her control in Markovian form. In the case of continuous inventories, we also show uniqueness of her optimal control policy. This model is also used by the exchange to choose an optimal tick size to attract liquidity on its platform. In the second chapter, we derive the limit order book's shape within the Glosten-Milgrom framework by modeling the interactions between three kinds of agents: an informed trader, a noise trader and several market makers. The study focuses on the non-zero tick size case. We are able to recover stylized facts such as the dependence of the spread on the tick size, and the predictive power of imbalance on short-term price moves. The model also allows computing valuable quantities for the market maker: the average waiting time until the next trade, and the value of an order in the queue. The final chapter develops market models that replicate observed macroscopic correlations that are observed between asset prices. We postulate the existence of a Brownian unobserved efficient price having the macroscopic correlation structure. Our study focuses on two sub-models: a signal-driven price model where the mid-price jump rates depend on the efficient price and an observable signal, and the queue-reactive model dependent on the efficient price via the intensities of the order arrivals. We show that, at the long-time scale, the mid-price becomes indistinguishable from the efficient price. We also develop a maximum likelihood procedure to infer the parameters of the model. We use our model to backtest strategies in an optimal liquidation context.
• Adaptive mesh refinement quantum algorithm for Maxwell's equations
  
  • Fressart Elise
  • Nowak Michel
  • Spillane Nicole
  , 2025. Algorithms that promise to leverage resources of quantum computers efficiently to accelerate the finite element method have emerged. However, the finite element method is usually incorporated into a high-level numerical scheme which allows the adaptive refinement of the mesh on which the solution is approximated. In this work, we propose to extend adaptive mesh refinement to the quantum formalism, and apply our method to the solution of Maxwell’s equations. An important step in this procedure is the computation of error estimators, which guide the refinement. By using block-encoding, we propose a way to compute these estimators with quantum circuits. We present first numerical experiments on a 2D geometry.
• Curvature in chemotaxis: A model for ant trail pattern formation
  
  • Bertucci Charles
  • Rakotomalala Matthias
  • Tomašević Milica
  Mathematical Models and Methods in Applied Sciences, World Scientific Publishing, 2025, 35 (12), pp.2695-2740. In this paper, we propose a new model of chemotaxis motivated by ant trail pattern formation, formulated as a coupled parabolic-parabolic local PDE system, for the population density and the chemical field. The main novelty lies in the transport term of the population density, which depends on the second-order derivatives of the chemical field. This term is derived as an anticipation-reaction steering mechanism of an infinitesimally small ant as its size approaches zero. We establish global-in-time existence and uniqueness for the model, and the propagation of regularity from the initial data. Then, we build a numerical scheme and present various examples that provide hints of trail formation. (10.1142/S0218202525500496)
  DOI : 10.1142/S0218202525500496
• Combining Molecular Beam and Plasmatron Data for Robust Calibration of Nitridation Mechanism
  
  • Capriati Michele
  • del Val Anabel
  • Schwartzentruber Thomas
  • Minton Timothy
  • Congedo Pietro Marco
  • Magin Thierry
  AIAA Journal, American Institute of Aeronautics and Astronautics, 2025, pp.1-15. The modeling of gas–surface interaction phenomena is crucial for accurately predicting the heat flux and the mass loss experienced by hypersonic vehicles. Gas–surface interactions refer to the phenomena occurring between the reacting gas and the thermal protection material. An important part of the modeling concerns the description of the surface chemical reactions. In this regard, we propose a novel methodology to infer the parameters underlying such surface chemistry models. It combines uncertainty quantification techniques with state-of-the-art modeling and different types of experiments. The methodology is used to calibrate, in a Bayesian sense, the rates of the elementary reactions between a nitrogen gas and a carbon surface. We rely on both molecular beam and plasma wind tunnel observations. The former provides detailed data on the chemical mechanisms but is characterized by pressures nonrepresentative of atmospheric entries. By contrast, plasma wind tunnel experiments are conducted at representative pressures but contain only macroscopic information. The parameters’ posterior distributions are then propagated through the models representing the two experiments. The calibrated model is able to satisfactorily explain both experiments, highlighting the robustness of the proposed methodology. (10.2514/1.J065221)
  DOI : 10.2514/1.J065221
• Monotone solutions to mean field games master equation in the L2-monotone setting
  
  • Meynard Charles
  , 2025. This paper is concerned with extending the notion of monotone solution to the mean field game (MFG) master equation to situations in which the coefficients are displacement monotone, instead of the previously introduced notion in the flat monotone regime. To account for this new setting, we work directly on the equation satisfied by the controls of the MFG. Following previous works, we define an appropriate notion of solution under which uniqueness and stability results hold for solutions without any differentiability assumption with respect to probability measures. Thanks to those properties, we show the existence of a monotone solution to displacement monotone mean field games under local regularity assumptions on the coefficients and sufficiently strong monotonicity. Albeit they are not the focus of this article, results presented are also of interest for mean field games of control and general mean field forward backward systems. In order to account for this last setting, we use the notion of L2−monotonicity instead of displacement monotonicity, those two notions being equivalent in the particular case of MFG.
• On some systems of partial differential equations of the mean field type
  
  • Rakotomalala Matthias
  , 2025. He main objective of this thesis is thestudy of two new models of collective behavior invol-ving an infinite number of interacting particles. Thefirst model focuses on a phenomenon of chemo-taxis, namely the formation of trails in ant colonies.The second model addresses the structure of stra-tegic interactions within the framework of mean fieldgame theory. This thesis adopts a mean field-type ap-proach, which consists of replacing the complexity ofmicroscopic interactions within finite groups of par-ticles by an averaged interaction described throughfields. In accordance with this approach, the distribu-tion of an infinite cloud of particles is represented bya density. The description of certain phenomena re-quires the introduction of auxiliary fields, which, toge-ther with this density, obey evolutionary laws expres-sed through partial differential equations (PDEs).Chemotaxis models aim to describe the collective mo-tion of a large population of particles, insects, or bac-teria that use chemical signals to coordinate their mo-vement in space. The density of this population is mo-deled by a Fokker–Planck-type equation, coupled witha second equation that governs the evolution of thechemoattractant concentration.Mean field game theory seeks to model Nash equili-bria in systems with infinitely many players. A typicalmean field game system is described by a coupled setof equations: a Fokker–Planck equation for the evolu-tion of the players’ density and a Hamilton–Jacobi–Bellman equation representing the value function ofa representative player. This theory has proven fruit-ful for modeling a wide range of phenomena in fieldssuch as finance, economics, and crowd dynamics.For each of the phenomena under consideration, wederive a new model and establish the well-posednessof the associated partial differential equations. In thecase of the chemotaxis model, we further analyzequalitative properties of the long-time behavior of thePDE system. Finally, the last chapter of the thesis pre-sents an existence and uniqueness result for stochas-tic differential equations on Riemannian manifolds ofbounded geometry
• Functional Limit Theorems for the range of stable random walks
  
  • Baccara Maxence
  , 2025. <div><p>In this paper we establish Functional Limit Theorems for the range of random walks in Z d that are in the domain of attraction of a non-degenerate β-stable process in the weakly transient and recurrent regimes. These results complement the fluctuations obtained at fixed time and the functional limit Theorems obtained in the strongly transient regime.</p><p>The techniques involve original ideas of Le Gall and Rosen for fluctuations and allow to show tightness in some Hölder space, thus also providing sharp regularity results about the limiting processes.</p><p>The original motivation of this work is the description of functionals appearing in spatial ecology for consumption of resources induced by random motion. We apply our result to estimate the large fluctuations of energy and mortality for a simple prey predator model.</p></div>
• Factorization of polynomials over the symmetrized tropical semiring and Descartes' rule of sign over ordered valued fields
  
  • Akian Marianne
  • Gaubert Stephane
  • Tavakolipour Hanieh
  Journal of Pure and Applied Algebra, Elsevier, 2025, 229 (9), pp.51. The symmetrized tropical semiring is an extension of the tropical semifield, initially introduced to solve tropical linear systems using Cramer's rule. It is equivalent to the signed tropical hyperfield, which has been used in the study of tropicalizations of semialgebraic sets. Polynomials over the symmetrized tropical semiring, and their factorizations, were considered by Quadrat. Recently, Baker and Lorscheid introduced a notion of multiplicity for the roots of univariate polynomials over hyperfields. In the special case of the hyperfield of signs, they related multiplicities with Descartes' rule of signs for real polynomials. More recently, Gunn extended these multiplicity definitions and characterization to the setting of “whole idylls”. We investigate here the factorizations of univariate polynomial functions over symmetrized tropical semirings, and relate them to the multiplicities of roots over these semirings. We deduce Descartes' rule for “signs and valuations”, which applies to polynomials over a real closed field with a convex valuation and an arbitrary (divisible) value group. We show in particular that the inequality of Descartes' rule is tight when the value group is non-trivial. This extends a characterization of Gunn from the rank one case to arbitrary value groups, also answering the tightness question. Our results are obtained using the framework of semiring systems introduced by Rowen, together with model theory of valued fields. (10.1016/j.jpaa.2025.108055)
  DOI : 10.1016/j.jpaa.2025.108055
• Mathematical analysis of an observer for solving inverse source wave problem
  
  • Delaunay Tiphaine
  • Imperiale Sébastien
  • Moireau Philippe
  Inverse Problems and Imaging, AIMS American Institute of Mathematical Sciences, 2025. The objective of this work is to propose a method using observers to estimate a source term of a wave equation from internal measurements in a subdomain. The first part of the work consists in proving an identifiability result from classical observability conditions for wave equations. We show that the source reconstruction is an ill-posed inverse problem of degree 1 or 2 depending on the measurements type. This inverse problem is solved using observers -- a sequential strategy -- that is proven to be equivalent to a minimization of a cost functional with Tikhonov regularization. (10.3934/ipi.2025043)
  DOI : 10.3934/ipi.2025043