Publications

2019

• Reflected stochastic differential equations driven by $G$-Brownian motion in non-convex domains
  
  • Lin Yiqing
  • Soumana-Hima Abdoulaye
  Stochastics and Dynamics, World Scientific Publishing, 2019, 19 (3), pp.n°1950025. In this paper, we first review the penalization method for solving deterministic Skorokhod problems in non-convex domains and establish estimates for problems with $\alpha$-H\"older continuous functions. With the help of these results obtained previously for deterministic problems, we pathwisely define the reflected $G$-Brownian motion and prove its existence and uniqueness in a Banach space. Finally, multi-dimensional reflected stochastic differential equations driven by $G$-Brownian motion are investigated via a fixed-point argument. (10.1142/S0219493719500254)
  DOI : 10.1142/S0219493719500254
• A policy iteration algorithm for non-zero sum stochastic impulse games
  
  • Aïd René
  • Bernal Francisco
  • Mnif Mohamed
  • Zabaljauregui Diego
  • Zubelli Jorge P.
  ESAIM: Proceedings and Surveys, EDP Sciences, 2019, 65 (CEMRACS 2017). This work presents a novel policy iteration algorithm to tackle nonzero-sum stochastic impulse games arising naturally in many applications. Despite the obvious impact of solving such problems, there are no suitable numerical methods available, to the best of our knowledge. Our method relies on the recently introduced characterisation of the value functions and Nash equilibrium via a system of quasi-variational inequalities. While our algorithm is heuristic and we do not provide a convergence analysis, numerical tests show that it performs convincingly in a wide range of situations, including the only analytically solvable example available in the literature at the time of writing. (10.1051/proc/201965027)
  DOI : 10.1051/proc/201965027
• Copula-like Variational Inference
  
  • Hirt Marcel
  • Dellaportas Petros
  • Durmus Alain
  , 2019. This paper considers a new family of variational distributions motivated by Sklar's theorem. This family is based on new copula-like densities on the hypercube with non-uniform marginals which can be sampled efficiently, i.e. with a complexity linear in the dimension of state space. Then, the proposed variational densities that we suggest can be seen as arising from these copula-like densities used as base distributions on the hypercube with Gaussian quantile functions and sparse rotation matrices as normalizing flows. The latter correspond to a rotation of the marginals with complexity $\mathcal{O}(d \log d)$. We provide some empirical evidence that such a variational family can also approximate non-Gaussian posteriors and can be beneficial compared to Gaussian approximations. Our method performs largely comparably to state-of-the-art variational approximations on standard regression and classification benchmarks for Bayesian Neural Networks.
• Relationship between biodiversity and agricultural production
  
  • Brunetti Ilaria
  • Tidball Mabel
  • Couvet Denis
  Natural Resource Modeling, Rocky Mountain Mathematics Consortium, 2019, 32 (2), pp.e12204. The intensification of agriculture is one of the main causes of biodiversity loss. We model the interdependent relationship between agriculture and wild biodiversity providing regulating services to agriculture on farmed land. We suppose that while agriculture has a negative impact on wild biodiversity, the latter can increase agricultural production. Farmers act as myopic agents, who maximize their instantaneous profit without considering the negative effects of their practice on the evolution of biodiversity. Two unexpected results arise (a) a tax on inputs can have a positive effect on yield since it can be considered as a social signal helping farmers to avoid myopic behavior concerning the positive effect of biodiversity on yield; (b) increasing biodiversity productivity, a proposal of ecological intensification, affects negatively the level of biodiversity, a counter‐intuitive result; due to the fact that when biodiversity is more productive, farmers can maintain lower biodiversity to get the same yield. (10.1111/nrm.12204)
  DOI : 10.1111/nrm.12204
• A variational approach to nonlinear and interacting diffusions
  
  • Arnaudon Marc
  • del Moral Pierre
  Stochastic Analysis and Applications, Taylor & Francis: STM, Behavioural Science and Public Health Titles, 2019, 37 (5), pp.717-748. The article presents a novel variational calculus to analyze the stability and the propagation of chaos properties of nonlinear and interacting diffusions. This differential methodology combines gradient flow estimates with backward stochastic interpolations, Lyapunov linearization techniques as well as spectral theory. This framework applies to a large class of stochastic models including nonhomogeneous diffusions, as well as stochastic processes evolving on differentiable manifolds, such as constraint-type embedded manifolds on Euclidian spaces and manifolds equipped with some Riemannian metric. We derive uniform as well as almost sure exponential contraction inequalities at the level of the nonlinear diffusion flow, yielding what seems to be the first result of this type for this class of models. Uniform propagation of chaos properties w.r.t. the time parameter is also provided. Illustrations are provided in the context of a class of gradient flow diffusions arising in fluid mechanics and granular media literature. The extended versions of these nonlinear Langevin-type diffusions on Riemannian manifolds are also discussed. (10.1080/07362994.2019.1609985)
  DOI : 10.1080/07362994.2019.1609985
• High Pressure Flames with Multicomponent Transport
  
  • Giovangigli Vincent
  • Matuszewski Lionel
  • Gaillard Pierre
  , 2019. The thermodynamic formulation and the traditional formulation of multicomponent transport fluxes in high pressure fluids are discussed. The impact of high pressure transport models on mixing layers, premixed plane flames and strained diffusion flames is then investigated. Multicomponent fluxes in diffuse-interface transcritical diffusion flames are further addressed.
• Hamiltonian models of interacting fermion fields in Quantum Field Theory
  
  • Alvarez Benjamin
  • Faupin Jérémy
  • Guillot Jean-Claude
  Letters in Mathematical Physics, Springer Verlag, 2019, 109 (11), pp.2403-2437. We consider Hamiltonian models representing an arbitrary number of spin 1 / 2 fermion quantum fields interacting through arbitrary processes of creation or annihilation of particles. The fields may be massive or massless. The interaction form factors are supposed to satisfy some regularity conditions in both position and momentum space. Without any restriction on the strength of the interaction, we prove that the Hamiltonian identifies to a self-adjoint operator on a tensor product of antisymmetric Fock spaces and we establish the existence of a ground state. Our results rely on new interpolated $N_\tau $ estimates. They apply to models arising from the Fermi theory of weak interactions, with ultraviolet and spatial cutoffs. (10.1007/s11005-019-01193-9)
  DOI : 10.1007/s11005-019-01193-9
• An explicit Floquet-type representation of Riccati aperiodic exponential semigroups
  
  • Bishop Adrian N
  • del Moral Pierre
  International Journal of Control, Taylor & Francis, 2019, pp.1-9. The article presents a rather surprising Floquet-type representation of time-varying transition matri-ces associated with a class of nonlinear matrix differentialRiccati equations. The main difference withconventional Floquet theory comes from the fact that the underlying flow of the solution matrix is aperi-odic. The monodromy matrix associated with this Floquet representation coincides with the exponential(fundamental) matrix associated with the stabilizing fixedpoint of the Riccati equation. The second partof this article is dedicated to the application of this representation to the stability of matrix differentialRiccati equations. We provide refined global and local contraction inequalities for the Riccati exponentialsemigroup that depend linearly on the spectral norm of the initial condition. These refinements improveupon existing results and are a direct consequence of the Floquet-type representation, yielding whatseems to be the first results of this type for this class of models. (10.1080/00207179.2019.1590647)
  DOI : 10.1080/00207179.2019.1590647
• Moutard transforms for the conductivity equation
  
  • Grinevich Piotr G
  • Novikov Roman G
  Letters in Mathematical Physics, Springer Verlag, 2019, 109 (10), pp.2209-2222. We construct Darboux-Moutard type transforms for the two-dimensional conductivity equation. This result continues our recent studies of Darboux-Moutard type transforms for generalized analytic functions. In addition, at least, some of the Darboux-Moutard type transforms of the present work admit direct extension to the conductivity equation in multidimensions. Relations to the Schrödinger equation at zero energy are also shown. (10.1007/s11005-019-01183-x)
  DOI : 10.1007/s11005-019-01183-x