Publications

2025

• Forward stochastic integration for adapted processes w.r.t. Riemann-Liouville fractional Brownian motion (Full version)
  
  • da Costa Paulo Henrique
  • Ohashi Alberto
  • Russo Francesco
  , 2025. This paper provides the time-dependent $L^2$-martingale representation of the forward stochastic integral where the driving noise is the Riemann-Liouville fractional Brownian motion with parameter $\frac{1}{2} < H < 1$ and the integrand is a square-integrable adapted process. As a by-product, we obtain the exact $L^2$-isometry of the forward stochastic integrals based on suitable conditions on time-dependent martingale representations of adapted integrands combined with the Nelson's stochastic derivative of the underlying Gaussian driving noise.
• Trapped modes in electromagnetic waveguides
  
  • Bonnet-Ben Dhia Anne-Sophie
  • Chesnel Lucas
  • Fliss Sonia
  , 2025. We consider the Maxwell's equations with perfect electric conductor boundary conditions in three-dimensional unbounded domains which are the union of a bounded resonator and one or several semi-infinite waveguides. We are interested in the existence of electromagnetic trapped modes, i.e. $L^2$ solutions of the problem without source term. These trapped modes are associated to eigenvalues of the Maxwell's operator, that can be either below the essential spectrum or embedded in it. First for homogeneous waveguides, we present different families of geometries for which we can prove the existence of eigenvalues. Then we exhibit certain non homogeneous waveguides with local perturbations of the dielectric constants that support trapped modes. Let us mention that some of the mechanisms we propose are very specific to Maxwell's equations and have no equivalent for the scalar Dirichlet or Neumann Laplacians.
• Spurious resonances for substructured FEM-BEM coupling
  
  • Boisneault Antonin
  • Bonazzoli Marcella
  • Claeys Xavier
  • Marchand Pierre
  , 2025. We are interested in time-harmonic acoustic scattering by an impenetrable obstacle in a medium where the wavenumber is constant in an exterior unbounded subdomain and is possibly heterogeneous in a bounded subdomain. The associated Helmholtz boundary value problem can be solved by coupling the Finite Element Method (FEM) in the heterogeneous subdomain with the Boundary Element Method (BEM) in the homogeneous subdomain. Recently, we designed and analyzed a new substructured FEM-BEM formulation, called Generalized Optimized Schwarz Method (GOSM). Unfortunately, it is well known that, even when the initial boundary value problem is well-posed, the variational formulation of classical FEM-BEM couplings can be ill-posed for certain wavenumbers, called spurious resonances. In this paper, we focus on the Johnson-Nédélec and Costabel couplings and show that the GOSM derived from both is not immune to that issue. In particular, we give an explicit expression of the kernel of the local operator associated with the interface between the FEM and BEM subdomains. That kernel and the one of classical FEM-BEM couplings are simultaneously non-trivial.
• No-Regret Gaussian Process Optimization of Time-Varying Functions
  
  • Mauduit Eliabelle
  • Berthier Eloïse
  • Simonetto Andrea
  , 2025. Sequential optimization of black-box functions from noisy evaluations has been widely studied, with Gaussian Process bandit algorithms such as GP-UCB guaranteeing no-regret in stationary settings. However, for time-varying objectives, it is known that no-regret is unattainable under pure bandit feedback unless strong and often unrealistic assumptions are imposed. In this article, we propose a novel method to optimize time-varying rewards in the frequentist setting, where the objective has bounded RKHS norm. Time variations are captured through uncertainty injection (UI), which enables heteroscedastic GP regression that adapts past observations to the current time step. As no-regret is unattainable in general in the strict bandit setting, we relax the latter allowing additional queries on previously observed points. Building on sparse inference and the effect of UI on regret, we propose W-SparQ-GP-UCB, an online algorithm that achieves no-regret with only a vanishing number of additional queries per iteration. To assess the theoretical limits of this approach, we establish a lower bound on the number of additional queries required for no-regret, proving the efficiency of our method. Finally, we provide a comprehensive analysis linking the degree of time-variation of the function to achievable regret rates, together with upper and lower bounds on the number of additional queries needed in each regime. (10.48550/arXiv.2512.00517)
  DOI : 10.48550/arXiv.2512.00517
• Scattering from a random thin coating of nanoparticles: the Dirichlet case
  
  • Boucart Amandine
  • Fliss Sonia
  • Giovangigli Laure
  , 2025. We study the time-harmonic scattering by a heterogeneous object covered with a thin layer of randomly distributed sound-soft nanoparticles. The size of the particles, their distance between each other and the layer's thickness are all of the same order but small compared to the wavelength of the incident wave. Solving the Helmholtz equation in this context can be very costly and the simulation depends on the given distribution of particles. To circumvent this, we propose, via a multi-scale asymptotic expansion of the solution, an effective model where the layer of particles is replaced by an equivalent boundary condition. The coefficients that appear in this equivalent boundary condition depend on the solutions to corrector problems of Laplace type defined on unbounded random domains. Under the assumption that the particles are distributed given a stationary and mixing random point process, we prove that those problems admit a unique solution in the proper space. We then establish quantitative error estimates for the effec tive model and present numerical simulations that illustrate our theoretical results.
• Construction and performance of kinetic schemes for linear systems of conservation laws
  
  • Audusse Emmanuel
  • Boyaval Sébastien
  • Dubos Virgile
  • Le Minh-Hoang
  , 2025. We describe a methodology to build vectorial kinetic schemes, targetting the numerical solution of linear symmetric-hyperbolic systems of conservation laws -a minimal application case for those schemes. Precisely, we fully detail the construction of kinetic schemes that satisfy a discrete equivalent to a convex extension (an additional non-trivial conservation law) of the target system -the (linear) acoustic and elastodynamics systems, specifically -. Then, we evaluate numerically the convergence of various possible kinetic schemes toward smooth solutions, in comparison with standard finite-difference and finite-volume discretizations on Cartesian meshes. Our numerical results confirm the interest of ensuring a discrete equivalent to a convex extension, and show the influence of remaining parameter variations in terms of error magnitude, both for "first-order" and "second-order" kinetic schemes : the parameter choice with largest CFL number (equiv., smallest spurious diffusion in the equivalent equation analysis) has the smallest discretization error.
• Temporal and Spatial Decomposition for Prospective Studies in Energy Systems under Uncertainty
  
  • Martinez Parra Camila
  • de Lara Michel
  • Chancelier Jean-Philippe
  • Carpentier Pierre
  • Janin Jean-Marc
  , 2025. <div><p>The increasing penetration of renewable energy requires greater use of storage resources to manage system intermittency. As a result, there is growing interest in evaluating the opportunity cost of stored energy, or usage values, which can be derived by solving a multistage stochastic optimization problem. Stochasticity arises from net demand (the aggregation of demand and non-dispatchable generation), the availability of dispatchable generation, and inflows when the storage facilities considered are hydroelectric dams. We aim to compute these usage values for each market zone of the interconnected European electricity system, in the context of prospective studies currently conducted by RTE, the French TSO. The energy system is mathematically modelled as a directed graph, where nodes represent market zones and arcs represent interconnection links. In large energy systems, spatial complexity (thirty nodes in the system, each with at most one aggregated storage unit) compounds temporal complexity (a one-year horizon modelled with two timescales: weekly subproblems with hourly time steps). This work addresses three main sources of complexity: temporal, spatial, and stochastic. We tackle the multinode multistage stochastic optimisation problem by incorporating a spatio-temporal decomposition scheme. To efficiently compute usage values, we apply Dual Approximate Dynamic Programming (DADP), which enables tractable decomposition across both time and space. This approach yields nodal usage values that depend solely on the local state of each node, independently of the others. We conduct numerical studies on a realistic system composed of thirty nodes (modelling part of Europe) and show that DADP obtains competitive results when comparing with traditional methods like Stochastic Dual Dynamic Programming (SDDP).</p></div>
• Identification of bottom deformations of the ocean from surface measurements
  
  • Bourgeois Laurent
  • Mercier Jean-François
  • Terrine Raphaël
  Inverse Problems and Imaging, AIMS American Institute of Mathematical Sciences, 2025, 19 (6), pp.1075-1113. In this paper we consider a general scheme to solve two different inverse problems related to oceanography, that is retrieving either a tsunami or the shape of the seabed from the measurement of the free surface perturbation. We consider two dimensional geometries and linear potential models in the frequency regime. Such general scheme consists in firstly recovering the potential in the whole domain and secondly compute the seeked parameter at the bottom of the ocean, which in the two inverse problems is a function involved in a more or less complicated boundary condition. The first step amounts to solve an ill-posed Cauchy problem for the Laplace or Helmholtz equation, which we regularize by using a mixed formulation of the Tikhonov regularization and the Morozov principle to compute the regularization parameter. The computation of such Tikhonov-Morozov solution is based on an iterative method consisting in solving a sequence of weak formulations which are discretized with the help of a simple Lagrange type Finite Element Method. In the particular case of the acoustic model, we need to solve a Laplace-type equation associated with the noisy Neumann boundary data and compute the noise amplitude of its solution. A probabilistic method is proposed to obtain such amplitude of noise. Some numerical experiments show the feasibility of our strategy. (10.3934/ipi.2025008)
  DOI : 10.3934/ipi.2025008
• $T$-coercivity: a practical tool for the study of variational formulations in Hilbert spaces
  
  • Ciarlet Patrick
  , 2025.
• Surface Plasmon Polariton Excitation in Time-modulated Media
  
  • Raziman T. V.
  • Touboul Marie
  • Sapienza Riccardo
  • Craster Richard V.
  • Rodríguez-Fortuño Francisco J.
  , 2025. Surface plasmon polaritons (SPPs) are central to application areas such as sensing, energy harvesting, and nanoscale optics, and are typically excited via spatial structuring -- an approach lacking dynamic control. We demonstrate that step-like time modulation allows the excitation and out-coupling of SPPs through modulating a dispersive Drude-like metal thereby modelling realistic transparent conducting oxides; this establishes a pathway for the active control and extraction of plasmons in experimentally viable time-varying systems. Using finite-difference time-domain simulations we show that time modulation facilitates both the launching and radiation of surface plasmons with frequencies governed by the dispersion of the bounding media. Our results also reveal the generation of time-reflected waves and the emergence of a magnetostatic mode required for matching boundary conditions at the temporal interface.
• Low-thrust Interplanetary Trajectories with Missed Thrust Events: a Numerical Approach
  
  • Chancelier Jean-Philippe
  • Carpentier Pierre
  • Cohen Guy
  • Dargent Thierry
  • Epenoy Richard
  , 2025. The problem under consideration is to drive a spatial vehicle to a target at a given final time while minimizing fuel consumption. This is a classical optimal control problem in a deterministic setting. However temporary stochastic failures of the engine may prevent reaching the target after the engine usage is recovered. Therefore, a stochastic optimal control problem is formulated under the constraint of ensuring a minimal probability of hitting the target. This problem is modeled, improved and finally solved by dualizing the probability constraint and using an Arrow-Hurwicz stochastic algorithm. Numerical results concerning an interplanetary mission are presented.
• CONVERGENCE OF MONOTONE NUMERICAL SCHEMES FOR FIRST-ORDER HAMILTON-JACOBI EQUATIONS IN PROPER CAT(κ) SPACES
  
  • Bokanowski Olivier
  • Jerhaoui Othmane
  • Zidani Hasnaa
  , 2025. <div><p>In this article, we extend the viscosity-solution theory for first-order Hamilton-Jacobi equations to general proper CAT(κ) spaces, using test functions that are locally Lipschitz, directionally differentiable, and locally DC (difference of two semi-convex functions). We prove comparison and existence (via Perron's method) under hypotheses tailored to the metric structure of the state space. We also formulate monotone finite-difference schemes on proper CAT(κ) spaces and establish a convergence result à la Barles-Souganidis. Finally, several network-type examples (gluing of flat pieces) illustrate the construction and demonstrate the relevance and versatility of the approach.</p></div>
• Risk-Averse Control for Continuous-Time Stochastic System Under Signal Temporal Logic Constraints
  
  • Lai En
  • Bonalli Riccardo
  • Girard Antoine
  • Jean Frédéric
  , 2025. Signal Temporal Logic (STL) has become a powerful formalism for specifying complex temporal-spatial behaviors in autonomous systems. Handling STL constraints within stochastic setting has received increasing research interest but still poses challenges. This paper proposes a general framework to efficiently solve continuous-time nonlinear stochastic optimal control problems under chance STL constraints. The STL formulae are implemented through extended dynamics, yielding a more classical chance constraint on the terminal state uniquely that we reliably relax via Conditional Value-at-Risk. The resulting new optimal control problem is then solved using established algorithms from risk--averse control. The efficiency and feasibility of the proposed approach are demonstrated through numerical simulations.
• Asymptotic models for time-domain scattering by small particles of arbitrary shapes
  
  • Kachanovska Maryna
  • Savchuk Adrian
  , 2025. In this work, we investigate time-dependent wave scattering by multiple small particles of arbitrary shape. To approximate the solution of the associated boundary-value problem, we derive an asymptotic model that is valid in the limit as the particle size tends to zero. Our method relies on a boundary integral formulation, semi-discretized in space using a Galerkin approach with appropriately chosen basis functions, s.t. convergence is achieved as the particle size vanishes rather than by increasing the number of basis functions. Since the computation of the Galerkin matrix involves double integration over particles, the method can become computationally demanding when the number of obstacles is large. To address this, we also derive a simplified model and consider the Born approximation to improve computational efficiency. For each model, we provide an error analysis, supported and validated by numerical experiments.
• Local Multiple Traces Formulation for Transmission Problems in Linear Elasticity
  
  • Chaillat Stéphanie
  • Darbas Marion
  • Escapil-Inchauspé Paul
  • Jerez-Hanckes Carlos
  , 2025. We investigate the use of the local Multiple Trace Formulation (MTF) in solving time-harmonic elastic wave transmission problems. Originally devised for heterogeneous acoustic media, MTF recasts the boundary value problem as a well-posed system of first-kind boundary integral equations, naturally amenable to parallelization and preconditioning. The formulation takes independent displacement and traction unknowns per subdomain, enforces Calderón identities locally, and imposes transmission conditions weakly across interfaces. We restrict our analysis to single homogeneous scatterers, representing a foundational step toward the application of MTF techniques in heterogeneous elasticity transmission problems. For the sake of clarity, all derivations in the one-dimensional setting are carried out explicitly, with illustrative examples provided in two dimensions. We analyze the effects of frequency and material contrast on the convergence of the GMRES iterative solver. Finally, we present preliminary results for an elastic Calderón preconditioner and discuss its potential to further accelerate iterative solvers.
• Preconditioning of GMRES for Helmholtz problems with quasimodes
  
  • Dolean Victorita
  • Marchand Pierre
  • Modave Axel
  • Raynaud Timothée
  , 2025. Finite element methods are effective for Helmholtz problems involving complex geometries and heterogeneous media. However, the resulting linear systems are often large, indefinite, and challenging for iterative solvers, particularly at high wave numbers or near resonant conditions. We derive a GMRES convergence bound that incorporates the nonlinear behavior of the relative residual and relates convergence to harmonic Ritz values. This perspective reveals how small eigenvalues associated with quasimodes can hinder convergence, and when they cease to have an effect. These phenomena occur in domain decomposition, and we illustrate them through numerical experiments. We also combine domain decomposition methods with deflation techniques using (approximate) eigenvectors tailored to resonant regimes. Their impact on GMRES performance is evaluated.
• Hybrid FEM/IPDG semi-implicit schemes for time domain electromagnetic wave propagation in non cylindrical coaxial cables
  
  • Beni Hamad Akram
  • Imperiale Sébastien
  • Joly Patrick
  , 2025. In this work, we develop an efficient numerical method for solving 3D Maxwell's equations in non-cylindrical coaxial cables. The main challenge arises from the elongated geometry of the computational domain, which induces strong anisotropy between the longitudinal direction (along the cable) and the transverse directions (within the cross-sections). This leads to the use of highly anisotropic meshes, where the longitudinal mesh size is much larger than the transverse one.<p>Our objective is to design a numerical scheme that is explicit in the longitudinal direction, with a CFL stability condition depending only on the longitudinal mesh size. In a previous work, we achieved this for cylindrical cables by employing prismatic edge elements, 1D quadrature for longitudinal mass lumping, and a hybrid explicit/implicit time discretization. The present paper extends this approach to non-cylindrical cables, addressing several new difficulties with the following key ingredients: (1) representing the cable as a deformation of a reference cylindrical cable and employing mapping techniques between the physical and reference domains; (2) using an anisotropic space discretization that combines an interior penalty discontinuous Galerkin (IPDG) method in the transverse directions with a conforming finite element method in the longitudinal direction; (3) utilizing prismatic edge elements on a prismatic mesh of the reference cable; and (4) adapting the construction of the hybrid explicit-implicit time discretization to the new structure of the semidiscrete problem. From a theoretical perspective, the main difficulty lies in the stability analysis, which requires extending and adapting standard techniques for DG methods in space and energy methods in time.</p>
• Quadratization and convexification in polynomial binary optimization
  
  • Crama Yves
  • Elloumi Sourour
  • Lambert Amélie
  • Rodríguez-Heck Elisabeth
  Journal of Combinatorial Optimization, Springer Verlag, 2025, 50 (3), pp.28. In this paper, we discuss several reformulations and solution approaches for the problem of minimizing a polynomial in binary variables (P). We review and integrate different literature streams to describe a methodology consisting of three distinct phases, together with several possible variants for each phase. The first phase determines a recursive decomposition of each monomial of interest into pairs of submonomials, down to the initial variables. The decomposition gives rise to a so-called quadratization scheme. The second phase builds a quadratic reformulation of (P) from a given quadratization scheme, by associating a new auxiliary variable with each submonomial that appears in the scheme. A quadratic reformulation of (P) is obtained by enforcing relations between the auxiliary variables and the monomials that they represent, either through linear constraints or through penalty terms in the objective function. The resulting quadratic problem (QP) is non-convex in general and is still difficult to solve. At this stage we introduce the third phase of the resolution process, which consists in convexifying (QP). We consider different types of convexification methods, including complete linearization or quadratic convex reformulations. Theoretical properties of the different phases are recalled from the literature or are further clarified. Finally, we present some experimental results to illustrate the discussion. (10.1007/s10878-025-01334-y)
  DOI : 10.1007/s10878-025-01334-y
• Non-Destructive Ultrasonic Evaluation of Asphalt Concrete Waterproofing in Hydraulic Structures: Experimental and Numerical Study
  
  • Maifia Nabil
  • Belmokhtar Mohamed
  • Villain Géraldine
  • Vautrin Denis
  • Taillade Frédéric
  • Cothenet Alexis
  e-Journal of Nondestructive Testing, Nondestructive Testing (NDT), 2025, 30 (10). Some hydraulic structures use a multi-layer asphalt concrete coating to ensure waterproofing. Over time, defects may appear in this coating due to ageing of the materials or environmental conditions. In order to prevent further degradation phenomena that could eventually compromise the stability of the structure, we aim to develop a non-destructive monitoring system based on mechanical waves propagation for early detection of such defects. A first step of this study consists in characterizing the presence of defects in terms of measurable mechanical parameters. To this end, six coating samples taken from both healthy and degraded areas of a real embankment dam are considered. They are analysed using a ultrasound acquisition system. The initial results are promising, showing a difference in dynamic modulus between healthy and degraded areas. (10.58286/31682)
  DOI : 10.58286/31682
• Examples of non-scattering inhomogeneities
  
  • Chesnel Lucas
  • Haddar Houssem
  • Li Hongjie
  • Xiao Jingni
  Inverse Problems and Imaging, AIMS American Institute of Mathematical Sciences, 2025, 19 (5). We consider the scattering of waves by a penetrable inclusion embedded in some reference medium. We exhibit examples of materials and geometries for which non-scattering frequencies exist, i.e. for which at some frequencies there are incident fields which produce null scattered fields outside of the inhomogeneity. We show in particular that certain domains with corners or even cusps can support non-scattering frequencies. We relate the latter, for some inclusions, to resonance frequencies for Dirichlet or Neumann cavities. We also find situations where incident non-scattering fields solve the Helmholtz equation in a neighbourhood of the inhomogeneity and not in the whole space. In relation with invisibility, we give examples of inclusions of anisotropic materials which are non-scattering for all real frequencies. We prove that corresponding material indices must have a special structure on the boundary.
• Approaches and limitations of Non-Destructive Testing for evaluating bituminous concrete using wave propagation: A review from the laboratory test to in situ case study
  
  • Belmokhtar Mohamed
  • Maifia Nabil
  • Villain Géraldine
  • Vautrin Denis
  • Cothenet Alexis
  e-Journal of Nondestructive Testing, Nondestructive Testing (NDT), 2025, 30 (10). This work addresses the challenges of using non-destructive testing (NDT) methods based on mechanical wave propagation as alternatives to conventional destructive tests for evaluating the mechanical behavior of bituminous concrete materials and structures. Given the viscoelastic nature of these materials, the paper first outlines the fundamental principles of their mechanical behavior and the key parameters to be identified. The study then focuses on two categories of NDT techniques involving wave propagation: laboratory methods, such as impact resonance testing and ultrasonic excitation; and in-situ methods, referring to geophysical techniques. Finally, the paper presents a comprehensive literature review of these NDT applied to bituminous concrete and discusses their respective advantages and limitations. (10.58286/31648)
  DOI : 10.58286/31648
• Time-Varying Gaussian Process Bandit Optimization with Experts: No-Regret in Logarithmically-Many Side Queries
  
  • Mauduit Eliabelle
  • Berthier Eloïse
  • Simonetto Andrea
  , 2025, 16017, pp.164-182. We study a time-varying Bayesian optimization problem with bandit feedback, where the reward function belongs to a Reproducing Kernel Hilbert Space (RKHS). We approach the problem via an upper-confidence bound Gaussian Process algorithm, which has been proven to yield no-regret in the stationary case. The time-varying case is more challenging and no-regret results are out of reach in general in the standard setting. As such, we instead tackle the question of how many additional observations asked to an expert are required to regain a no-regret property. To do so, we formulate the presence of past observation via an uncertainty injection procedure, and we reframe the problem as a heteroscedastic Gaussian Process regression. In addition, to achieve a no-regret result, we discard long outdated observations and replace them with updated (possibly very noisy) ones obtained by asking queries to an external expert. By leveraging and extending sparse inference to the heteroscedastic case, we are able to secure a no-regret result in a challenging time-varying setting with only logarithmically-many side queries per time step. Our method demonstrates that minimal additional information suffices to counteract temporal drift, ensuring efficient optimization despite time variation. (10.1007/978-3-032-06096-9_10)
  DOI : 10.1007/978-3-032-06096-9_10
• Non-unitary enhanced transfer efficiency in quantum walk search on complex networks
  
  • Nzongani Ugo
  • Simonetto Andrea
  • Molfetta Giuseppe Di
  , 2025. The task of finding an element in an unstructured database is known as spatial search and can be expressed as a quantum walk evolution on a graph. In this article, we modify the usual search problem by adding an extra trapping vertex to the graph, which is only connected to the target element. The walker evolution is a mix between classical and quantum walk search dynamics. The balance between unitary and non-unitary dynamics is tuned with a parameter, and we numerically show that depending on the graph topology and the connectivity of the target element, this hybrid approach can outperform a purely classical or quantum evolution for reaching the trapping site. We show that this behavior is only observed in the presence of an extra trapping site, and that depending on the topology, the increase of non-unitary operations can be compensated by increasing the strength of the quantum walk exploration. This compensation comes at the cost of reducing the searching feature of the evolution induced by the Hamiltonian. We also relate the optimal hybrid regime to the entropy's decay rate. As the introduction of non-unitary operations may be considered as noise, we interpret this phenomena as a noisy-assisted quantum evolution.
• Sampled-Based Guided Quantum Walk: Non-variational quantum algorithm for combinatorial optimization
  
  • Nzongani Ugo
  • Mermoud Dylan Laplace
  • Di Molfetta Giuseppe
  • Simonetto Andrea
  , 2025. We introduce SamBa-GQW, a novel quantum algorithm for solving binary combinatorial optimization problems of arbitrary degree with no use of any classical optimizer. The algorithm is based on a continuous-time quantum walk on the solution space represented as a graph. The walker explores the solution space to find its way to vertices that minimize the cost function of the optimization problem. The key novelty of our algorithm is an offline classical sampling protocol that gives information about the spectrum of the problem Hamiltonian. Then, the extracted information is used to guide the walker to high quality solutions via a quantum walk with a time-dependent hopping rate. We investigate the performance of SamBa-GQW on several quadratic problems, namely MaxCut, maximum independent set, portfolio optimization, and higher-order polynomial problems such as LABS, MAX-$k$-SAT and a quartic reformulation of the travelling salesperson problem. We empirically demonstrate that SamBa-GQW finds high quality approximate solutions on problems up to a size of $n=20$ qubits by only sampling $n^2$ states among $2^n$ possible decisions. SamBa-GQW compares very well also to other guided quantum walks and QAOA. (10.48550/arXiv.2509.15138)
  DOI : 10.48550/arXiv.2509.15138
• Verification theorem related to a zero sum stochastic differential game, based on a chain rule for non-smooth functions
  
  • Ciccarella Carlo
  • Russo Francesco
  , 2024. In the framework of stochastic zero-sum differential games, we establish a verification theorem, inspired by those existing in stochastic control, to provide sufficient conditions for a pair of feedback controls to form a Nash equilibrium. Suppose the validity of the classical Isaacs' condition and the existence of a (what is termed) quasi-strong solution to the Bellman-Isaacs (BI) equations. If the diffusion coefficient of the state equation is non-degenerate, we are able to show the existence of a saddle point constituted by a couple of feedback controls that achieve the value of the game: moreover, the latter is equal to the (necessarily unique) solution of the BI equations. A suitable generalization is available when the diffusion is possibly degenerate. Similarly we have also improved a well-known verification theorem in stochastic control theory. The techniques of stochastic calculus via regularization we use, in particular specific chain rules, are borrowed from a companion paper of the authors.