Publications

2026

• A functional inequalities approach for the field-road diffusion model with (symmetric) nonlinear exchanges
  
  • Alfaro Matthieu
  • Chainais-Hillairet Claire
  • Nabet Flore
  Applied Mathematics Letters, Elsevier, 2026, 180, pp.109980. In this note, we consider the so-called field-road diffusion model in a bounded domain, consisting of two parabolic PDEs posed on sets of different dimensions and coupled through (symmetric) nonlinear exchange terms. We propose a new and rather direct functional inequalities approach to prove the exponential decay of a relative entropy, and thus the convergence of the solution towards the stationary state selected by the total mass of the initial datum. (10.1016/j.aml.2026.109980)
  DOI : 10.1016/j.aml.2026.109980
• Surrogate-Based Strategies for Accelerated Bayesian Calibration of Computer Codes With Complete Maximum a Posteriori Estimation of Model Error
  
  • Kahol Omar
  • Le Maître Olivier
  • Congedo Pietro M
  • Denimal Goy Enora
  Journal of Mechanical Design, American Society of Mechanical Engineers, 2026, 148 (9). The calibration of a computer code is a process that reduces the uncertainty of model parameters by matching the code’s predictions to experimental observations of a quantity of interest. A more faithful representation of the global uncertainty is achieved by including a model error term, a discrepancy between the physical system and the computer code. The recently proposed complete maximum a posteriori (CMP) method is able to infer both a posterior distribution of the model parameters and a model error term, improving upon traditional frameworks. On the other hand, the CMP method relies on an optimization step which increases the cost of complex calibration problems. This article proposes a surrogate-based strategy to reduce the computational cost of the CMP method. First, we build a surrogate model of the model error’s hyperparameters using Gaussian processes. Second, we propose an iterative algorithm that builds a training set in regions of the parameter space that are more likely, reducing the overall cost of the algorithm and improving the accuracy of the surrogate. The proposed strategy is applied to four different examples, including a design problem in solid mechanics and a complex test case in fluid dynamics. The results show that the proposed strategy is able to accelerate the CMP method without losing accuracy, making it suitable for real-world applications. In an industrial application, we demonstrate a speed-up of almost 100 compared to the original CMP method. (10.1115/1.4071071)
  DOI : 10.1115/1.4071071
• 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
  Computers and Fluids, Elsevier, 2026, 317, pp.107193. We present a robust computational framework for the numerical solution of a hyperbolic 6-equation single-velocity two-phase system. The system'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. (10.1016/j.compfluid.2026.107193)
  DOI : 10.1016/j.compfluid.2026.107193
• Optimal filtering in closed manifolds- a deterministic perspective
  
  • Le Ruz Gaël
  • Moireau Philippe
  , 2026. In this paper, we adopt an optimal control viewpoint to formulate a rigorous deterministic filtering theory when the dynamics and the observations are defined on manifolds. Therefore, our result extends the Mortensen observer to closed manifolds, namely a compact manifold without boundary, in both continuous and discrete time, where the second ultimately yields a convergent time discretization of the first. The resulting observer requires the computation of the viscosity solution of a Hamilton-Jacobi-Bellman equation on the state manifold, which we illustrate on the sphere.
• A simple rigorous integrator for semilinear parabolic PDEs
  
  • Berg Jan Bouwe van Den
  • Breden Maxime
  , 2026. Simulations of the dynamics generated by partial differential equations (PDEs) provide approximate, numerical solutions to initial value problems. Such simulations are ubiquitous in scientific computing, but the correctness of the results is usually not guaranteed. We propose a new method for the rigorous integration of parabolic PDEs, i.e., the derivation of rigorous and explicit error bounds between the numerically obtained approximate solution and the exact one, which is then proven to exist over the entire time interval considered. These guaranteed error bounds are obtained a posteriori, using a fixed point reformulation based on a piece-wise in time constant approximation of the linearization around the numerical solution. Our setup leads to relatively simple-to-understand estimates, which has several advantages. Most critically, it allows us to optimize various aspects of the proof, and in particular to provide an adaptive time-stepping strategy. In case the solution converges to a stable hyperbolic equilibrium, we are also able to prove this convergence, applying our rigorous integrator with a final, infinitely long timestep. We showcase the ability of our method to rigorously integrate over relatively long time intervals, and to capture non-trivial dynamics, via examples on the Swift--Hohenberg equation, the Ohta--Kawasaki equation and the Kuramoto--Sivashinsky equation. We expect that the simplicity and efficiency of the approach will enable generalization to a wide variety of other parabolic PDEs, as well as applications to boundary value problems.
• A 3D-shell model of left atrial electromechanics
  
  • Ruz Oscar
  • Brito-Pacheco Carlos
  • Vidrascu Marina
  • Chapelle Dominique
  • Fernández Miguel Angel
  Biomechanics and Modeling in Mechanobiology, Springer Verlag, 2026. The thin-walled nature of the atrial myocardium can lead to artificial stiffening when full 3D electromechanical models are discretized using standard finite elements. In this work, we propose an electromechanical model of the left atrium based on a 3D-shell formulation that overcomes these limitations. The model incorporates both passive and active components of atrial tissue mechanics, while atrioventricular interaction is described by the coupling with a 0D electromechanical model of the left ventricle. The proposed approach is assessed under physiological and pathological conditions and systematically compared with the standard full 3D formulation. The results demonstrate the superior robustness and computational efficiency of the proposed 3D-shell electromechanical model.
• Strong law of large numbers and condition for supercritical branching processes
  
  • Bansaye Vincent
  • Berah Tresnia
  • Cloez Bertrand
  Probability Theory and Related Fields, Springer Verlag, 2026, pp.38 p.. We consider branching processes for structured populations: each individual is characterised by a type or trait which belongs to a general measurable state space. We focus on the supercritical recurrent case, where the population may survive and grow and the trait distribution converges to a probability measure. The branching process is then expected to be driven by the positive triplet of first eigenvalue problem of the first moment semigroup. Under the assumption of convergence of the renormalized semigroup in weighted total variation norm, we prove strong convergence of the renormalized empirical measure and non-degeneracy of the limit of the martingale. Convergence is obtained under an LlogL condition which provides a Kesten-Stigum result in infinite dimension and relaxes the uniform convergence of the first moment semigroup in the work of Asmussen and Hering in 1976. The techniques of proof combine families of martingales and contraction properties and the truncation procedure of Asmussen and Hering. These results unify part of the literature and capture new situations, as illustrated by absorbed branching diffusion, the house of cards model and some growth-fragmentation processes. (10.1007/s00440-026-01524-7)
  DOI : 10.1007/s00440-026-01524-7
• Diffusion-based Annealed Boltzmann Generators : benefits, pitfalls and hopes
  
  • Grenioux Louis
  • Noble Maxence
  Transactions on Machine Learning Research Journal, [Amherst Massachusetts]: OpenReview.net, 2022, 2026. Sampling configurations at thermodynamic equilibrium is a central challenge in statistical physics. Boltzmann Generators (BGs) tackle it by combining a generative model with a Monte Carlo (MC) correction step to obtain asymptotically unbiased samples from an unnormalized target. Most current BGs use classic MC mechanisms such as importance sampling, which both require tractable likelihoods from the backbone model and scale poorly in high-dimensional, multi-modal targets. We study BGs built on annealed Monte Carlo (aMC), which is designed to overcome these limitations by bridging a simple reference to the target through a sequence of intermediate densities. Diffusion models (DMs) are powerful generative models and have already been incorporated into aMC-based recalibration schemes via the diffusion-induced density path, making them appealing backbones for aMC-BGs. We provide an empirical meta-analysis of DM-based aMC-BGs on controlled multi-modal Gaussian mixtures (varying mode separation, number of modes, and dimension), explicitly disentangling inference effects from learning effects by comparing (i) a perfectly learned DM and (ii) a DM trained from data. Even with a perfect DM, standard integrations using only first-order stochastic denoising kernels fail systematically, whereas second-order denoising kernels can substantially improve performance when covariance information is available. We further propose a deterministic aMC integration based on first-order transport maps derived from DMs, which outperforms the stochastic first-order variant at higher computational cost. Finally, in the learned-DM setting, all DM-aMC variants struggle to produce accurate BGs; we trace the main bottleneck to inaccurate DM log-density estimation.
• Deciphering the Replication-Division Coordination in E. coli: A Unified Mathematical framework for Systematic Model Comparison
  
  • Perrin Alexandre
  • Doumic Marie
  • El Karoui Meriem
  • Méléard Sylvie
  , 2025. Despite extensive research, the quantitative principles that govern the coordination between DNA replication and cell division in bacteria remain debated. Multiple theoretical models have been proposed, some postulating that a single regulatory process is sufficient to ensure replication–division coordination, while others argue that two concurrent processes are required for robust control. In this work, we develop a unifying mathematical framework within which models can be consistently formulated, qualitatively analysed and quantitatively compared. This framework also allows us to propose a new double-process model. Through theoretical analysis, we establish the necessary and sufficient conditions under which single-process models can reproduce physiological cell behaviours. Beyond the correlation-based analyses extensively used to date, we further demonstrate within a comprehensive statistical framework that double-process models more accurately recapitulate experimental data across all growth conditions. Specifically, the new model we propose robustly captures the replication-division coordination in every growth regime, thereby providing a foundation for future mechanistic studies. (10.1101/2025.07.25.666816)
  DOI : 10.1101/2025.07.25.666816
• PhysioBlocks: A Python Library for Graph-Based Nodal Simulation of Dynamical Physiological Systems
  
  • Chapelle Dominique
  • Drieu Colin
  • Kimmig François
  , 2026. The PhysioBlocks Python library is designed to simulate the dynamics of physiological systems (in particular cardiovascular systems) represented by graphs of connected components - as e.g.for electrical circuits - in order to provide built-in modularity. Accordingly, a system is represented by a network of modules (blocks) connected by nodes in which they share quantities (degrees of freedom) and exchange fluxes. The global time-discrete system of equations is automatically assembled by using a Modified Nodal Analysis type of approach. The user can easily create a new network by combining existing blocks. At a more advanced level, new blocks can be defined. The library is distributed under the LGPL-3.0-only license, and the initial distribution focuses on providing building blocks associated with lumped-parameter models of the cardiovascular system.
• Nonparametric hazard rate estimation with associated kernels and minimax bandwidth choice
  
  • Breuil Luce
  • Kaakai Sarah
  , 2026. <div><p>In this paper, we introduce a general theoretical framework for nonparametric hazard rate estimation using associated kernels, whose shapes depend on the point of estimation. Within this framework, we establish rigorous asymptotic results, including a second-order expansion of the MISE, and a central limit theorem for the proposed estimator. We also prove a new oracle-type inequality for both local and global adaptive bandwidth selection, extending the Goldenshluger–Lepski method to the context of associated kernels. Our results propose a systematic way to construct and analyze new associated kernels. Finally, we show that the general framework applies to the Gamma kernel, and we provide several examples of applications on simulated data and experimental data for the study of aging. </p></div> (10.48550/arXiv.2509.24535)
  DOI : 10.48550/arXiv.2509.24535
• On the simulation of extreme events with neural networks
  
  • Allouche Michaël
  • Girard Stéphane
  • Gobet Emmanuel
  , 2026. This article aims at investigating the use of generative methods based on neural networks to simulate extreme events. Although very popular, these methods are mainly invoked in empirical works. Therefore, providing theoretical guidelines for using such models in extreme values context is of primal importance. To this end, we propose an overview of most recent generative methods dedicated to extremes, giving some theoretical and practical tips on their tail behaviour thanks to both extreme-value and copula tools.
• Asymptotics of a two-species particle system associated to the doubly parabolic Keller-Segel equation in the plane
  
  • Fournier Nicolas
  • Tomasevic Milica
  , 2026. We consider the two-species particle system introduced by Stevens [22] related to the doubly parabolic Keller-Segel equation. It consists of N cells and of a varying number of chemoattractant particles. Cells diffuse in the plane and follow the (mollified) empirical gradient of concentration of chemoattractant. Chemoattractant particles are produced by cells at some constant rate, diffuse and disappear at some constant rate. We show that when the sensitivity of cells to the chemoattractant is small enough, under some rather weak condition on the family of mollifiers, this system approximates the parabolic-parabolic Keller-Segel equation as N → ∞. We also prove that when N is fixed and when the production rate of chemoattractant particles tends to infinity, this system approximates the (non-Markovian) one-species system introduced in [25] and further studied in [12].
• A sequential Bayesian approach to sparse Gaussian process quantile optimization: application to hydrofoil design
  
  • Nicolas Hugo
  • Le Maître Olivier
  , 2026. In previous work, we developed a sequential Bayesian approach to estimate conditional quantiles of a response variable--a problem framed as quantile regression [1]. The latent conditional quantile function was modeled via a sparse Gaussian process with inducing points. The asymmetric Laplace distribution was considered for the likelihood of the data. The inference of the posterior distribution over the inducing variables was recast as its Laplace approximation. The importance of inducing input locations to predictive accuracy was identified. As a result, we proposed a novel approach to optimally select their locations by leveraging the Gaussian process predictive variance. Moreover, we exploited uncertainties in the inducing variables to guide the acquisition of new observations of the response variable. Adaptive inducing point allocation and active learning were combined into a sequential algorithm, which further enabled automatic selection of the required number of inducing points. The present work adapts the above methodology, which focused on global predictive accuracy, to the problem of finding the optimum of the conditional quantile function--a problem framed as quantile optimization. New observations are selected using Thompson sampling by drawing samples from the posterior and choosing the corresponding optimal inputs. We leverage the decoupled strategy by [2] to efficiently sample from the sparse Gaussian quantile posterior within the Thompson sampling. This framework for quantile optimization is applied to the design of a hydrofoil under uncertainties. We focus on the hydrodynamic performance of the hydrofoil, with XFOIL as the flow solver, supplemented by analytical corrections to the hydrodynamic coefficients to account for free-surface proximity effects. Multiple sources of uncertainty are considered, including operational and manufacturing uncertainties. This application demonstrates the effectiveness of our algorithm and highlights its key advantages in high-dimensional uncertainty settings, as commonly encountered in engineering design [3].
• Twisted Schrödinger Bridge Matching
  
  • Noble Maxence
  • Scheid Marie
  • Janati Yazid
  • Moulines Eric
  • Durmus Alain
  , 2026. Over the past few years, diffusion-based Schrödinger bridge models have been proposed to approximate optimal transport dynamics between two prescribed boundary distributions, with successful applications to generative modeling. More precisely, these methods aim to estimate a path measure whose initial and terminal marginals match the two boundary distributions, while minimizing the Kullback-Leibler divergence with respect to a reference Markov process. In this work, we consider the generalized Schrödinger bridge problem, in which the reference process is a twisted Brownian motion, that is, a Feynman-Kac transform of a Brownian motion induced by a time-dependent differentiable potential. Building on the Iterative Markovian Fitting (IMF) paradigm, and in particular on its special case Diffusion Schrödinger Bridge Matching (DSBM), which corresponds to the zero potential case, we introduce Twisted Schrödinger Bridge Matching (TSBM), a diffusion-based method designed to handle both continuous-and discrete-time potentials. Unlike previous approaches, TSBM provides a rigorous extension of the IMF scheme to the generalized Schrödinger bridge problem. This derivation leads to a new bridge-matching loss that depends explicitly on the gradient of the potential and recovers the DSBM objective when the potential vanishes, yielding improved performance. We further introduce trajectory-based variance-reduction techniques that substantially stabilize optimization and may be useful beyond the present setting. Finally, we empirically demonstrate the benefits of TSBM for trajectory inference across increasingly high-dimensional settings, including crowd navigation and single-cell data.
• Quantifying the impact of different forms of stress on fungal growth: an inference method based on high-resolution pictures of the mycelial network
  
  • Kuwata Lena
  • Chassereau Thibault
  • Chapeland-Leclerc Florence
  • David Pascal
  • Herbert Eric
  • Ruprich-Robert Gwenaël
  • Tomašević Milica
  • Véber Amandine
  , 2025. In a previous work, a complete methodology for monitoring the growth of a filamentous fungus was introduced, allowing the automated extraction of its graph structure and of its key statistics. In parallel, a stochastic growth-fragmentation model for the dynamics of such mycelial networks was introduced and studied. This simple model depends on three parameters: the elongation speed v of a single filament, the branching rate b_1 of a filament at its open end, and the per unit length rate b_2 at which a budding event happens and creates a new filament branching off from an existing one. Leaving aside the spatial structure of the mycelium and describing its structure essentially through the empirical measure of the lengths of its filament segments, the three parameters of the growth dynamics encoded in the model summarise the local balance between mass creation and regulation through non-linear mechanisms such as anastomosis or density-dependent growth modulation. In this work, we develop a generic statistical inference method based on the large-time behaviour of the stochastic model, and on the high-resolution pictures of the mycelial network obtained using the methodology for fungal growth monitoring, to reconstruct the effective growth parameters v, b_1 and b_2 from a single panorama of the filament network pictured after several hours of development and an empirical measurement of the exponential rate of increase of the number of branch points and apexes. We use this method to analyse the growth dynamics of Podospora anserina mycelia observed under standard conditions and when several forms of stress are applied, in order to quantify the effect of these stresses on the different mechanisms of fungal growth. By comparing the reconstructed effective parameters with the hyphal elongation speed and branching rates estimated from the much more complex dynamical tracking of individual filaments, we find that the effective branching rates we infer are in close agreement with the individual branching rates estimated from the dynamical tracking procedure. This suggests that, in this application at least, the effects of non-linear local regulation mechanisms on branching are well encoded by an effective Markovian rate during the exponential growth phase of the mycelium. By contrast, the effective elongation speed reconstructed with our model-based approach is approximately twice as low as the hyphal elongation speed estimated from the dynamical tracking procedure, a bias that may be explained by a mismatch between the definitions of ``elongation speed'' used in the two approaches. Nevertheless, our results show that the three parameters of our growth-fragmentation model, combined with experimental data that are now readily available, enable us to quantify different components of the exponential growth dynamics of an expanding mycelial network.
• Second-order optimally stable IMEX (pseudo-)staggered Galerkin discretization with application to depth-integrated lava flow simulations
  
  • 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
• Model Updating of Rotating Wind Turbines in Operation for Blade Condition Assessment
  
  • Delette Nina
  • Denimal Goy Enora
  • Pfister Jean-Lou
  • El Amri Mohamed Reda
  • Mevel Laurent
  , 2026. As wind turbines grow in size and complexity, Structural Health Monitoring (SHM) has become essential for efficient maintenance. Vibration-based Operational Modal Analysis (OMA) is particularly effective in this context; however, purely data-driven methods often lack information for detailed damage qualification. Model-based approaches, such as model updating, address this limitation by combining numerical models with experimental data to enhance fault localization and quantification. While most studies focus on support structures, this work aims to bridge the gap toward blade-mounted damage detection in operational conditions. Analyzing rotating blades is challenging as it violates the Linear Time-Invariant (LTI) assumptions required for standard OMA. To overcome this, recent advances are leveraged to treat Linear Time-Periodic (LTP) systems as equivalent LTI systems by considering the Fourier harmonics of Floquet modes. This work applies a model updating framework based on this equivalence to track changes in physical blade parameters as indicators of damage. The methodology is validated using the NREL 5MW turbine under operational conditions. The results demonstrate that the proposed two-step strategy successfully identifies the damaged blade, localizes the fault along its span, and quantifies its intensity with an accuracy of approximately 1%.
• Tightening the Score Matching Gap for Diffusion Models
  
  • Dupuis Benjamin
  • Farghly Tyler
  • Haddouche Maxime
  • Durmus Alain
  • Simsekli Umut
  , 2026. Diffusion models (DMs) are a state-of-the-art generative method to approximately sample from an unknown distribution. Their training and evaluation primarily rely on an Evidence Lower Bound (ELBO), which relates the Kullback-Leibler (KL) divergence of model samples to the score matching loss along the path, which serves as a tractable surrogate. The difference between sample quality and the score matching loss produced by this bound leads to the \emph{score matching gap}, which is known to be tight in the worst-case but not descriptive of sample quality in general. In this work, we provide a theoretical analysis of this gap, developing tighter bounds for three metrics: KL divergence, reverse KL divergence, and Wasserstein distance, effectively exploiting the regularity of the class of score estimators. Our results suggest that the quality of the score approximation has more impact on closing the score matching gap for low noise scales. To obtain these bounds, our key technical insight is to exploit the contraction properties of the backward processes. In particular, we rely on entropy flows, logarithmic Sobolev inequalities and reflection couplings, rigorously linking the ergodicity of the Langevin diffusion to the score matching gap problem.
• Affine Structure of the Brownian Signature
  
  • Jaber Eduardo Abi
  • Attal Elie
  • Sotnikov Dimitri
  , 2026. We establish an infinite-dimensional affine transform theory for the time-augmented Brownian signature. Our first main result shows that, for a suitable class of linear functions of the signature, the conditional Fourier-Laplace transform admits an entire signature expansion. We prove that the associated coefficients solve an infinite-dimensional linear differential equation on the extended tensor algebra. Our second main result shows that the logarithm admits a local signature expansion whose coefficients satisfy a Riccati equation on the extended tensor algebra, revealing a generalized affine structure of the Brownian signature in a genuinely path-dependent setting. In contrast to conventional affine processes, we show that this representation is intrinsically local: zeros of the Fourier-Laplace transform in the complex plane prevent any global expansion. To recover global representations, we introduce a new class of randomized Riccati equations with path-dependent terminal conditions through a recentering argument. Furthermore, we establish uniqueness of solutions to the linear and Riccati equations within a suitable class of solutions. Our results provide a theoretical framework for transform methods in non-Markovian settings, with applications to the computation of conditional distributions.
• Stochastic control with signatures via Riccati equations on the tensor algebra
  
  • Abi Jaber Eduardo
  • Attal Elie
  • Sotnikov Dimitri
  , 2026. We solve in semi-explicit form a class of non-Markovian stochastic optimal control problems with pathdependent rewards, using path signatures. We reformulate the control problem as the computation of Laplace transforms of signature functionals thanks to the Boué-Dupuis representation. Exploiting recent signature representations of such transforms on tensor algebras, we determine the value process and the optimal control through an infinite-dimensional system of Riccati equations on the extended tensor algebra. We establish an explicit feedback representation of the optimal control and the value process as an infinite linear combination of the time-extended signature of the controlled process, with time-dependent coefficients. The expansions being intrinsically local, we propose a dynamic recentering algorithm to ensure a global representation over the entire time horizon. We illustrate the approach on genuinely path-dependent, non-linear examples that go beyond the tractable linear-quadratic setting, including the tracking of linear functionals of the signature and signature lifts of Volterra control problems.
• Strong law of large numbers and L log L condition for supercritical branching processes
  
  • Bansaye Vincent
  • Berah Tresnia
  • Cloez Bertrand
  , 2025. We consider branching processes for structured populations: each individual is characterized by a type or trait which belongs to a general measurable state space. We focus on the supercritical recurrent case, where the population may survive and grow and the trait distribution converges. The branching process is then expected to be driven by the positive triplet of first eigenvalue problem of the first moment semigroup. Under the assumption of convergence of the renormalized semigroup in weighted total variation norm, we prove strong convergence of the normalized empirical measure and non-degeneracy of the limiting martingale. Convergence is obtained under an Llog L condition which provides a Kesten-Stigum result in infinite dimension and relaxes the uniform convergence assumption of the renormalized first moment semigroup required in the work of Asmussen and Hering in 1976. The techniques of proofs combine families of martingales and contraction of semigroups and the truncation procedure of Asmussen and Hering. We also obtain L^1 convergence of the renormalized empirical measure and contribute to unifying different results in the literature. These results greatly extend the class of examples where a law of large numbers applies, as we illustrate it with absorbed branching diffusion, the house of cards model and some growth-fragmentation processes.
• On the range of competing random walks
  
  • Baccara Maxence
  , 2026. We consider $N$ independent random walks $X^1,\dots,X^N$ in the lattice $\mathbb{Z}^d$ and prove limit theorems for the competitive range $\mathcal{R}_n^k$ of the $k$-th random walk $X^k$, which corresponds to the number of distinct sites that it has discovered before any of the other $X^\ell$, $\ell\ne k$, up to time $n$. This is a natural object to study foraging mechanisms in population ecology, in which context it is also natural to ask how the effect of competition for the access to resources affects the number of resources consumed by each individual. We work with random walks in the domain of attraction of a $\beta$-stable law and focus on the regime $d/\beta\in[1,3/2)$, in which classical results for the range show that the fluctuations are described by the renormalized self-intersection local time of the limiting process. We establish a central limit theorem in which a competition term emerges, thus answering the two previous questions we asked. We end the paper with a brief discussion on the remaining regimes $d/\beta\ge3/2$, in which the fluctuations are Gaussian and are not affected by the competition, and $d/\beta&lt;1$ in which no strong law of large numbers holds and we expect the effect of the competition to strongly affect the first-order asymptotics.
• Sequential sparse Gaussian process quantile regression
  
  • Nicolas Hugo
  • Le Maître Olivier
  , 2026. Quantile regression aims to estimate the conditional quantiles of a response variable from observed data. In a Bayesian setting, Gaussian process quantile regression provides uncertainty quantification but faces significant computational challenges due to the nonconjugacy of the asymmetric Laplace likelihood and the cost of posterior inference. We develop a sparse Gaussian process framework in which the quantile function is represented through a reduced set of inducing variables and posterior inference is performed using a Laplace approximation. A decomposition of the predictive uncertainty into conditional-prior and posterior-induced variance components is then exploited to drive two complementary adaptive mechanisms: inducing-input infilling and data acquisition. These mechanisms are combined within a sequential algorithm that allocates computational effort toward the dominant source of predictive uncertainty and adaptively controls model complexity. Numerical experiments on benchmark problems demonstrate the accuracy of the Laplace approximation, the benefits of variance-based inducing-input placement, and the effectiveness of the proposed sequential enrichment strategy compared with predefined data-acquisition strategies.
• Threatening excursions in large population quasi-stationary birth and death systems. On a question of Antonio Galves.
  
  • Collet Pierre
  • Martínez Servet
  • Méléard Sylvie
  , 2026. We consider time continuous multispecies birth and death processes in a regime of large populations. The jump rates depend on a large scaling parameter K modeling the charge capacity. When K tends to infinity, the process is close (in finite time) to a dynamical system containing a non zero global attracting equilibrium and zero as unstable equilibrium. For each fixed K, extinction in finite time occurs almost surely and a quasi-stationary distribution occurs naturally in the study of the statistics over times scales which are large but smaller than the extinction time scale. Before this catastrophic event the process makes many unsuccessful large deviations attempts with time scales corresponding to how far it deviates from the quasi-equilibrium. The paper concerns the statistical description of these typical trajectories starting from the quasi-stationary distribution until extinction. An unusual mixing property yields large time scale behavior for the process starting from a fixed state. We give a precise statistical description of the successive exit times of the process rescaled by K from a neighborhood of the equilibrium of the dynamical system in a clumping time scale and prove their asymptotic Poisson distribution. We also give a precise description of the asymptotic distribution of the successive records until extinction.