Publications

2026

• 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.
• 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
• On Reference Set Selection for Constrained Multiobjective Optimization Problems
  
  • Opravš Luka
  • Brockhoff Dimo
  • Tušar Tea
  , 2026, pp.493-501. The Pareto set and Pareto front of a continuous multiobjective optimization problem typically contain infinitely many solutions. Since dealing with infinite sets is impractical, they are commonly approximated by finite reference sets. In this way, the performance of multiobjective optimization algorithms can be assessed using quality indicators. However, the selection of a reference set depends on the intended goal, such as achieving a uniform distribution of solutions along the Pareto set or Pareto front, or optimizing a specific quality indicator. In this paper, we investigate and compare six strategies for reference set construction in constrained bi-objective problems from a recently proposed test problem generator. The approaches are evaluated with respect to multiple quality indicators and computational cost. The experiments are performed on test problems with irregular Pareto sets and fronts. From these experiments, we conclude that the choice of strategy should depend on the intended goal, with the approach that aims to maximize the hypervolume indicator value standing out as the best trade-off in terms of speed and indicator accuracies. (10.1145/3795095.3805160)
  DOI : 10.1145/3795095.3805160
• A Navier-Type Boundary Condition for Boltzmann Equation
  
  • Aoki Kazuo
  • Giovangigli Vincent
  • Golse François
  • Kosuge Shingo
  , 2026. We investigate a two term asymptotic expansion of the Boltzmann boundary condition at a solid wall using a physical kinetic equation of solid-gas interactions.
• Finding Simple Proofs for First-Order Optimization
  
  • Thomsen Daniel Berg
  • Upadhyaya Manu
  • Goujaud Baptiste
  • Dieuleveut Aymeric
  • Taylor Adrien
  , 2026. Progress in mathematics often requires more than a certificate of truth: it requires proof structures that are transparent, checkable, and reusable. Automated systems can increasingly certify that a result is true; what they typically return, however, is a dense certificate rather than an interpretable, reusable proof structure. Recent work on performance estimation problems has shown that performance bounds and complexity analyses of first-order optimization methods can be discovered by searching over a structured space of Lagrangian dual certificates. We cast the search for simpler proof structures as a second-stage optimization problem over these certificates. Starting from dual certificates, we develop post-processing procedures using tools from sparse optimization and statistical learning. We measure complexity through features such as active hypotheses and residual structure, and introduce methods based on exhaustive sparsification, weighted ℓ₁-type heuristics, and semidefinite programming (SDP) formulations for discovering simple proofs and intermediate lemmas. Examples on gradient descent, proximal methods, and fast-gradient methods show that these procedures can autonomously prune redundant inequalities, reveal structured proof patterns, and, in the proximal setting, recover Lyapunov functions as intermediate lemmas that lead to simple, streamlined proofs. By distilling dense machine-generated certificates into compact proof structures, this workflow acts as a pre-processing step for the final proof, reducing the complexity that must be managed during human interpretation, reuse, and formalization.
• Stochastic modeling of hematopoiesis and Myeloproliferative Neoplasms : hydrodynamic limit and statistical analysis
  
  • Fernández Baranda Ana
  , 2026. This thesis combines theoretical modeling and data-driven approaches to deepen the understanding of hematopoiesis (blood cells formation) and Myeloproliferative Neoplasms (MPN, a group of blood cancers).We first model hematopoiesis with continuum differentiation by studying the hydrodynamic limit of a stochastic slow-fast compartmental model. The multiscale model considers three types of cells (stem, immature and mature cells) and integrates cell differentiation and cell proliferation, both regulated through a feedback loop. As the number of immature compartments goes to infinity, the convergence in law of the stem and mature cells population size processes and of the empirical measures of the immature cells dynamics, conveniently rescaled, is proven. The partial differential equations limiting system consists of a unique triplet involving coupled functions and a measure, which are solutions of a deterministic measure valued equation with boundary dynamics.Then, through age of detection data, we characterize the age-structured emergence and the development of MPNs arising from the most common driver mutation. A model is defined for the distribution of the age of diagnostic by considering it as the sum of two independent components: the time for the mutation to occur and enter proliferation, and the time corresponding to the expansion of the clonal population until diagnosis. Through an EM algorithm to fit the data, this work proves an aging mechanism in the emergence of the mutation. The proposed model further allows to obtain quantitative results on these times and compare two types of MPN.Lastly, we use longitudinal sequencing data to study the effects of different additional mutations in MPN to support treatment selection. A mixed-effect model combining natural growth of the cancerous cells population, the reduction of this growth by different treatments, and the effects of each additional mutation is constructed. This model is fitted to data using the software Monolix to identify the effects of secondary mutations.
• 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%.
• Some Contributions to the Diophantine and Cohomological Study of Function Fields
  
  • Gambardella Felipe
  , 2026. This thesis is divided into two parts. The first part is devoted to Kato–Kuzumaki's conjecture. It includes a survey of cohomological dimension and of the current state of the conjecture, followed by a study of its behaviour under field extensions. Transfer principles are developed with respect to function fields and tame valued fields. In the case of function fields, these rely on explicit considerations of hypersurfaces and a refinement of ideas of Lang and Nagata, while in the tame valued setting they follow from the Ax–Kochen/Ershov principles of model theory. These results yield the conjecture in full generality for several arithmetically significant fields, such as mathbf{C}(x_1,dots, x_n)(! (t_1)! )dots (! (t_m)! ) and the perfection of mathbf{F}_p(! (t_1)! )dots(! (t_m)! ), as well as partially in some cases, including mathbf{F}_p(x_1,dots, x_n). We investigate further the conjecture for one-variable function fields over higher local fields, namely mathbf{Q}_p(! (t_1)! )dots (! (t_n)! )(x). A special case is established by extending arguments of Izquierdo and Lucchini Arteche for p-adic function fields, based on arithmetic duality for motivic cohomology and the vanishing of certain Tate-Shafarevich groups. This extension requires an explicit computation of certain Tate–Shafarevich groups over function fields over higher local~fields. parThe second part studies local–global properties over geometric global fields. Let K be a finite extension of either mathbf{C}(! (t)! )(x) or mathbf{C}(! (x,y)! ). It is shown that homogeneous spaces of operatorname{SL}_n whose geometric stabiliser is an iterated semidirect product of finite abelian K-groups satisfy approximation away from a finite explicit set of places. This is achieved via a weak form of Chebotarev's density theorem in the setting of geometric semi-global fields, introduced in this work.
• A Tight Theory of Error Feedback Algorithms in Distributed Optimization
  
  • Thomsen Daniel Berg
  • Taylor Adrien
  • Dieuleveut Aymeric
  , 2026. Communication costs are a major bottleneck in distributed learning and first-order optimization. A common approach to alleviate this issue is to compress the gradient information exchanged between agents. However, such compression typically degrades the convergence guarantees of gradient-based methods. Error feedback mechanisms provide a simple and computationally cheap remedy for this issue, but numerous variants have been proposed, and their relative performance remains poorly understood. This paper provides tight convergence analyses for two of the main error-feedback algorithms from the literature, the classic Error Feedback method (EF) and Error Feedback 21 (EF21), by identifying optimal step-size choices and constructing optimal Lyapunov functions tailored to each method. The results hold independently of the number of agents and recover the known best guarantees possible in the single-agent regime.
• 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.
• Spline interpolation on compact riemannian manifolds
  
  • Sire Charlie
  • Pereira Mike
  • Romary Thomas
  Numerical Algorithms, Springer Verlag, 2026. Spline interpolation is a widely used class of methods for solving interpolation problems by constructing smooth interpolants that minimize a regularized energy functional involving the Laplacian operator. While many existing approaches focus on Euclidean domains or the sphere, relying on the spectral properties of the Laplacian, this work introduces a method for spline interpolation on general manifolds by exploiting its equivalence with kriging. Specifically, the proposed approach uses finite element approximations of random fields defined over the manifold, based on Gaussian Markov Random Fields and a discretization of the Laplace-Beltrami operator on a triangulated mesh. This framework enables the modeling of spatial fields with smooth variations and local anisotropies via domain deformation. The method is first validated on the sphere using both analytical test cases and a pollution-related study, and is compared to the classical spherical harmonics-based method. Additional experiments on the surface of a cylinder further illustrate the generality of the approach. (10.1007/s11075-026-02415-7)
  DOI : 10.1007/s11075-026-02415-7
• 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 $d/\beta<1$ 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.
• Stochastic-structural modelling of particle-laden turbulent flows based on wavelet reconstruction
  
  • Morhain Clément
  • Letournel Roxane
  • Massot Marc
  • Vié Aymeric
  Computers and Fluids, Elsevier, 2026. Reduced-order modelling and simulation of turbulent particle-laden flows is required in numerous configurations, where the resolution of the whole spectrum of turbulent scales through DNS is out of reach. Whereas structural or stochastic models have been derived in order to provide a synthetic turbulent model for the non-resolved scales of the fluid flow field, reproducing particle dynamics is challenging because it requires capturing both spatial and temporal correlations. We present a reduced-order framework that combines wavelet-based structural modelling with stochastic evolution. Using compactly supported divergence-free wavelets within a multiresolution analysis, the method provides direct control over spatial structures and correlations of synthetic multiscale incompressible velocity fields. In contrast to Fourier modes, the wavelet basis functions are localized in space and spectrally non-sharp in Fourier space, and spread over a range of wavenumbers, which requires a dedicated procedure to enforce a prescribed turbulent energy spectrum. The stochastic evolution of wavelet coefficients further ensures consistent temporal correlations. The proposed framework is evaluated in homogeneous isotropic turbulence under a fully reduced setting, where all turbulent scales must be provided by the model. When coupled to a disperse phase in the one-way coupled framework, results show that it reproduces particle preferential concentration across a wide range of Stokes numbers as well as the pair-dispersion regimes, achieving similar agreement with DNS data as for classical Fourier-based Kinematic Simulation. This establishes a physically consistent turbulence model, which combines structural fidelity with stochastic dynamics, providing an alternative framework for synthetic turbulence modelling and investigating particle–turbulence interactions resolution.
• Phase Transition in Convex Relaxations for Graph Alignment
  
  • Massoulié Laurent
  • Varma Sushil Mahavir
  • Vassaux Louis
  • Waldspurger Irène
  , 2026, 336. We study the graph alignment problem for correlated Gaussian Orthogonal Ensemble (GOE) matrices, where the goal is to recover a hidden vertex permutation given two correlated symmetric Gaussian matrices $(A, B)$ with correlation $1/\sqrt{1+\sigma^2}$. While the maximum likelihood estimator is information-theoretically optimal, its computation, which reduces to a quadratic assignment problem, is intractable. Motivated by this, we analyze convex relaxations based on minimizing $\|AX - XB\|_F$ over the set of doubly stochastic matrices and the unit hypercube. We show that when the correlation parameter satisfies $\sigma = o(n^{-1/2}/\log^4 n)$, the solution of either relaxation $(X^\star)$ concentrates around the ground-truth permutation matrix $(\Pi^\star)$, i.e., $\|X^\star-\Pi^\star\|_F^2 = o(n)$, implying recovery of all but a vanishing fraction of vertices after simple post-processing. Combined with existing lower bounds, our results precisely characterize that $\|X^\star-\Pi^\star\|_F^2$ transitions from $o(n)$ for $\sigma = \tilde{o}(n^{-1/2})$ to $\Omega(n)$ for $\sigma = \tilde{\Omega}(n^{-1/2})$. In doing so, our analysis significantly tightens prior results and extends them beyond doubly stochastic relaxations.
• Fourier-Laplace Transforms of the Brownian Signature via Riccati Equations on the Tensor Algebra
  
  • Abi Jaber Eduardo
  • 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.
• A two-scale two-phase flow model for the separate-to-disperse phase transition in atomizing flows
  
  • Haegeman Ward
  • Orlando Giuseppe
  • Kokh Samuel
  • Massot Marc
  , 2026. An original two-scale, isothermal compressible two-phase flow model with surface tension is presented. The model allows for a unified description of the separate interface and disperse phase regimes. The inter-scale mass transfer terms, activated when the local curvatures exceed a physical and grid-independent length threshold, allow for the transition from the former regime, to the latter, through atomization. This mass transfer process is obtained through a pressure relaxation towards a modified Laplace law such that local curvatures do not exceed the prescribed threshold. It leads to a local and dissipative regularization of the large-scale interface, while retaining a sub-scale representation of the small-scale flow features. The backbone of the model is derived through the use of Hamilton's Stationary Action Principle. The source terms are derived such that the inter-scale mass transfer is dissipative for the extended thermodynamics, which includes the surface energies at both scales. The methodology that is developed allows for the derivation of a thermodynamically consistent model which admits a supplementary conservation law for the entropy and real characteristics.
• Rémy's diffusion on Brownian trees
  
  • Curien Nicolas
  • Marzouk Cyril
  , 2026. Rémy's algorithm is a famous recursive construction of uniform random binary trees of growing size by a local grafting operation. In this work we construct a continuous version, a new local diffusion on the space of real trees of growing Brownian Continuum Random Trees (CRT's). It appears as the scaling limit of a variant of Rémy's algorithm due to Bacher, Bodini, and Jacquot. Once the trees are rescaled to have constant mass, this diffusion uncovers an ergodic dynamics on trees with the Brownian CRT as unique invariant law.
• Constrained non-linear estimation and links with stochastic filtering
  
  • Chaintron Louis-Pierre
  • Mertz Laurent
  • Moireau Philippe
  • Zidani Hasnaa
  SIAM Journal on Control and Optimization, Society for Industrial and Applied Mathematics, 2026, 64 (3), pp.1974-2000. This article studies the problem of estimating the state variable of non-smooth subdifferential dynamics constrained in a bounded convex domain given some real-time observation. On the one hand, we show that the value function of the estimation problem is a viscosity solution of a Hamilton Jacobi Bellman equation whose sub and super solutions have different Neumann type boundary conditions. This intricacy arises from the non-reversibility in time of the non-smooth dynamics, and hinders the derivation of a comparison principle and the uniqueness of the solution in general. Nonetheless, we identify conditions on the drift (including zero drift) coefficient in the non-smooth dynamics that make such a derivation possible. On the other hand, we show in a general situation that the value function appears in the small noise limit of the corresponding stochastic filtering problem by establishing a large deviation result. We also give quantitative approximation results when replacing the non-smooth dynamics with a smooth penalised one. (10.1137/25M1730934)
  DOI : 10.1137/25M1730934
• Model selection for imperfect computer codes under discrepancy-aware calibration: application to IATE closure modelling
  
  • Janati Idrissi Sanae
  • Rodio Maria Giovanna
  • Congedo Pietro Marco
  , 2026. Selecting among competing computer models is challenging when all candidates are imperfect representations of reality. In discrepancy-aware Bayesian calibration, classical evidence-based criteria (e.g. Bayes factors) can become difficult to interpret. In particular, marginal likelihood integrates over flexible discrepancy components and can exhibit pronounced sensitivity to prior assumptions on discrepancy hyperparameters. In this paper, we revisit model selection under additive discrepancy and propose a posterior-averaged error criterion that quantifies the magnitude of the residual systematic correction required by each model after calibration. We develop the criterion within a modular calibration framework, i.e. the Full Maximum a Posteriori (FMP) approximation, and analyze its empirical robustness to hyperparameter priors on a pedagogical example, and compare it to Bayes factors and information criteria. We then apply the methodology to the modeling of a PDE problem arising in multiphase flow simulations. The objective is to select among alternative closure models for source and sink terms in the two-group Interfacial Area Transport Equation (IATE), using experimental interfacial area measurements across multiple flow regimes in large-diameter vertical air-water pipe flows. Results show that Bayes-factor conclusions can change markedly depending on whether discrepancy is included and on the discrepancy prior, whereas the posterior-averaged error provides a stable and physically interpretable ranking by isolating irreducible model bias.