Cette thèse porte sur le problème de transport optimal incompressible, un problème introduit par Brenier à la fin des années 80 dans le but de décrire l’évolution d’un fluide incompressible et non-visqueux de façon lagrangienne, ou autrement dit en fixant l’état initial et l’état final de ce fluide, et en minimisant une certaine fonctionnelle sur un ensemble de dynamiques admissibles. Ce manuscrit contient deux parties.Dans la première, on étudie la dépendance du problème de transport optimal incompressible par rapport aux données. Plus précisément, on étudie la dépendance du champ de pression (le multiplicateur de Lagrange associé à la contrainte d’incompressibilité) par rapport à la mesure γ prescrivant l’état initial et l’état final du fluide. On montre au Chapitre 2 par des méthodes variationnelles que le gradient de la pression, en tant qu’élément d’un espace proche du dual des fonctions C^1, dépend de γ de façon hölderienne pour la distance de Monge-Kantorovic. En contrepartie, on montre au Chapitre 4 que pour tout r > 1, le champ de pression dans l'espace de Lebesgue L^r_t L^1_x ne peut pas être une fonction lipschitzienne de γ. Ce résultat est lié au caractère mal-posé de l’équation d’Euler cinétique, une équation cinétique s’interprétant comme l’équation d’optimalité dans le problème de transport optimal incompressible, à laquelle le Chapitre 3 de cette thèse est dédié.Dans une seconde partie, on s’intéresse à la régularisation entropique du problème de transport optimal incompressible, introduit par Arnaudon, Cruzeiro, Léonard et Zambrini en 2017 sous le nom de problème de Brödinger. On prouve au Chapitre 5 que comme dans le cas du transport optimal incompressible, on peut associer à toute solution un champ scalaire de pression agissant comme multiplicateur de Lagrange pour la contrainte d’incompressibilité. On montre ensuite au Chapitre 6 que lorsque le paramètre de régularisation tend vers zéro, le problème de Brödinger converge vers le problème de transport optimal incompressible au sens de la Γ-convergence, et avec convergence des champs de pression. Ce dernier chapitre est issu d'un travail effectué en collaboration avec L. Monsaingeon. / This thesis focuses on Incompressible Optimal Transport, a minimization problem introduced by Brenier in the late 80's, aiming at describing the evolution of an incompressible and inviscid fluid in a Lagrangian way , i.e. by prescribing the state of the fluid at the initial and final times and by minimizing some functional among the set of admissible dynamics. This text is divided into two parts.In the first part, we study the dependence of this optimization problem with respect to the data. More precisely, we analyse the dependence of the pressure field, the Lagrange multiplier corresponding to the incompressibility constraint, with respect to the endpoint conditions, described by a probability measure γ determining the state of the fluid at the initial and final times. We show in Chapter 2 by purely variational methods that the gradient of the pressure field, as an element of a space that is close to the dual of C^1, is a Hölder continuous function of γ for the Monge-Kantorovic distance. On the other hand, we prove in Chapter 4 that for all r>1 the pressure field, as an element of L^r_t L^1_x, cannot be a Lipschitz continuous function of γ for the Monge-Kantorovic distance. This last statement is linked to an ill-posedness result proved in Chapter 3 for the so-called kinetic Euler equation, a kinetic PDE interpreted as the optimality equation of the Incompressible Optimal Transport problem.In the second part, we are interested in the entropic regularization of the Incompressible Optimal Transport problem: the so-called Brödinger problem, introduced by Arnaudon, Cruzeiro, Léonard and Zambrini in 2017. On the one hand, we prove in Chapter 5 that similarly to what happens in the Incompressible Optimal Transport case, to a solution always corresponds a scalar pressure field acting as the Lagrange multiplier for the incompressibility constraint. On the other hand, we prove in Chapter 6 that when the diffusivity coefficient tends to zero, the Brödinger problem converges towards the Incompressible Optimal Transport problem in the sense of Gamma-convergence, and with convergence of the pressure fields. The results of Chapter 6 come from a joint work with L. Monsaingeon.

Cette thèse est consacrée à l'étude mathématique de quelques modèles d'équations aux dérivées partielles issues de la physique des plasmas. On s'intéresse principalement à l'analyse théorique de différents régimes asymptotiques de systèmes d'équations cinétiques de type Vlasov-Poisson-Fokker-Planck. Dans un premier temps, en présence d'un champ magnétique extérieur on se concentre sur l'approximation des électrons sans masse fournissant des modèles réduits lorsque le rapport me{mi entre la masse me d'un électron et la masse mi d'un ion tend vers 0 dans les modèles. Suivant le régime considéré, on montre qu'à la limite les solutions vérifient des modèles hydrodynamiques de type convection-diffusion ou sont données par des densités de type Maxwell-Boltzmann-Gibbs, suivant l'intensité des collisions dans la mise à l'échelle. En utilisant les propriétés hypocoercives et hypoelliptiques des équations, on est capable d'obtenir des taux de convergence en fonction du rapport de masse. Dans un second temps, par des méthodes similaires, on montre la convergence exponentielle en temps long vers l'équilibre des solutions du système de Vlasov-Poisson-Fokker-Planck sans champ magnétique avec des taux explicites en les paramètres du modèles. Enfin, on conçoit un nouveau type de schéma volumes finis pour des équations de convection-diffusion non-linéaires assurant le bon comportement en temps long des solutions discrètes. Ces propriétés sont vérifiées numériquement sur plusieurs modèles dont l'équation de Fokker-Planck avec champ magnétique / This thesis is devoted to the mathematical study of some models of partial differential equations from plasma physics. We are mainly interested in the theoretical study of various asymptotic regimes of Vlasov-Poisson-Fokker-Planck systems. First, in the presence of an external magnetic field, we focus on the approximation of massless electrons providing reduced models when the ratio me{mi between the mass me of an electron and the mass mi of an ion tends to 0 in the equations. Depending on the scaling, it is shown that, at the limit, solutions satisfy hydrodynamic models of convection-diffusion type or are given by Maxwell-Boltzmann-Gibbs densities depending on the intensity of collisions. Using hypocoercive and hypoelliptic properties of the equations, we are able to obtain convergence rates as a function of the mass ratio. In a second step, by similar methods, we show exponential convergence of solutions of the Vlasov-Poisson-Fokker-Planck system without magnetic field towards the steady state, with explicit rates depending on the parameters of the model. Finally, we design a new type of finite volume scheme for a class of nonlinear convection-diffusion equations ensuring the satisfying long-time behavior of discrete solutions. These properties are verified numerically on several models including the Fokker-Planck equation with magnetic field

Cette thèse est consacrée à l'analyse asymptotique d'équations aux dérivées partielles issues de modèles de déplacement sous-diffusif en biologie cellulaire. Notre motivation biologique est fondée sur les nombreuses observation récentes de protéinescytoplasmiques dont le déplacement aléatoire dévié de la diffusion Fickienne normale. Dans la première partie, nous étudions la décroissance auto-similaire de la solution d'une équation de renouvellement à queue lourde vers un état stationnaire. Les idéesmises en jeu sont inspirées de méthodes d'entropie relative. Nos principaux apports sont la preuve d'un taux de décroissance en norme L1 vers la loi de l'arc-sinus et l'introduction d'une fonction pivot spécifique dans une méthode d'entropie relative.La seconde partie porte sur la limite hyperbolique d'une équation de renouvellement structurée en âge et à sauts en espace. Nous y prouvons un résultat de « stabilité » : les solutions des problèmes rééchelonnés à ε > 0 convergent lorsque ε --> 0 vers la solution de viscosité de l'équation de Hamilton-Jacobi limite des problèmes à ε > 0. Les outilsmis en jeu proviennent de la théorie des équations de Hamilton-Jacobi.Ce travail présente trois idées intéressantes. La première est celle de prouver le résultat de convergence sur la condition de bord du problème plutôt que d'utiliser des fonctions test perturbées. La deuxième consiste en l'introduction de termes correcteurslogarithmiques en temps dans des estimations a priori ne découlant pas directementdu principe du maximum. Cela est dû à la non-existence d'un équilibre du problèmehomogène en espace. La troisième est une estimation précise de la décroissance de l'influence de la condition initiale sur le terme de renouvellement. Elle correspond à une estimation fine d'une version non-locale de la dérivée temporelle de la solution. Au cours de cette thèse, des simulations numériques de type Monte Carlo, schémas volumes finis, Lax-Friedrichs et Weighted Essentially Non Oscillating ont été réalisées. / This thesis is devoted to the asymptotic analysis of partial differential equations modelling subdiffusive random motion in cell biology. The biological motivation for this work is the numerous recent observations of cytoplasmic proteins whose random motion deviates from normal Fickian diffusion. In the first part, we study the self-similar decay towards a steady state of the solution of a heavy-tailed renewal equation. The ideas therein are inspired from relative entropy methods. Our main contributions are the proof of an L1 decay rate towards the arc-sine distribution and the introduction of a specific pivot function in a relative entropy method.The second part treats the hyperbolic limit of an age-structured space-jump renewal equation. We prove a "stability" result: the solutions of the rescaled problems at ε > 0 converge as ε --> 0 towards the viscosity solution of the limiting Hamilton-Jacobi equation of the ε > 0 problems. The main mathematical tools used come from the theory of Hamilton-Jacobi equations. This work presents three interesting ideas. The first is that of proving the convergence result on the boundary condition of the studied problem rather than using perturbed test functions. The second consists in the introduction of time-logarithmic correction termsin a priori estimates that do not follow directly from the maximum principle. That is due to the non-existence of a suitable equilibrium for the space-homogenous problem. The third is a precise estimate of the decay of the inuence of the initial condition on the renewal term. This is tantamount to a refined estimate of a non-local version of the time derivative of the solution. Throughout this thesis, we have performed numerical simulations of different types: Monte Carlo, finite volume schemes, Lax-Friedrichs schemes and Weighted Essentially Non Oscillating schemes.

Our vision is as yet unsurpassed by machines because of the sophisticated representations of objects in our brains. This representation is vastly different from a pixel-based representation used in machine storages. It is this sophisticated representation that enables us to perceive two faces as very different, i.e, they are far apart in the “perceptual space”, even though they are close to each other in their pixel-based representations. Neuroscientists have proposed distances between responses of neurons to the images (as measured in macaque monkeys) as a quantification of the “perceptual distance” between the images. Let us call these neuronal dissimilarity indices of perceptual distances. They have also proposed behavioural experiments to quantify these perceptual distances. Human subjects are asked to identify, as quickly as possible, an oddball image embedded among multiple distractor images. The reciprocal of the search times for identifying the oddball is taken as a measure of perceptual distance between the oddball and the distractor. Let us call such estimates as behavioural dissimilarity indices. In this thesis, we describe a decision-theoretic model for visual search that suggests a connection between these two notions of perceptual distances. In the first part of the thesis, we model visual search as an active sequential hypothesis testing problem. Our analysis suggests an appropriate neuronal dissimilarity index which correlates strongly with the reciprocal of search times. We also consider a number of alternative possibilities such as relative entropy (Kullback-Leibler divergence), the Chernoff entropy and the L1-distance associated with the neuronal firing rate profiles. We then come up with a means to rank the various neuronal dissimilarity indices based on how well they explain the behavioural observations. Our proposed dissimilarity index does better than the other three, followed by relative entropy, then Chernoff entropy and then L1 distance. In the second part of the thesis, we consider a scenario where the subject has to find an oddball image, but without any prior knowledge of the oddball and distractor images. Equivalently, in the neuronal space, the task for the decision maker is to find the image that elicits firing rates different from the others. Here, the decision maker has to “learn” the underlying statistics and then make a decision on the oddball. We model this scenario as one of detecting an odd Poisson point process having a rate different from the common rate of the others. The revised model suggests a new neuronal dissimilarity index. The new dissimilarity index is also strongly correlated with the behavioural data. However, the new dissimilarity index performs worse than the dissimilarity index proposed in the first part on existing behavioural data. The degradation in performance may be attributed to the experimental setup used for the current behavioural tasks, where search tasks associated with a given image pair were sequenced one after another, thereby possibly cueing the subject about the upcoming image pair, and thus violating the assumption of this part on the lack of prior knowledge of the image pairs to the decision maker. In conclusion, the thesis provides a framework for connecting the perceptual distances in the neuronal and the behavioural spaces. Our framework can possibly be used to analyze the connection between the neuronal space and the behavioural space for various other behavioural tasks.

The idea of holography [1, 2] finds a concrete realization in form of the AdS/CFT correspondence [3, 4]. This duality relates a field theory with conformal symmetries to quantum gravity living in one higher dimension. In this thesis we study aspects of black hole quasinormal modes, higher spin theories and entanglement entropy in the context of this duality. In almost all cases we have been able to subject the duality to some precision tests. Quasinormal modes encode the spectrum of black holes and the time-scale of pertur- bations therein [5]. From the dual CFT viewpoint they are the poles of retarded Green's function (or peaks in the spectral function) [6]. Quasinormal modes were previously studied for scalar, gauge field and fermion fluctuations [7]. We solve for these quasinormal modes of higher spin (s _ 2) fields in the background of the BTZ black hole [8, 9]. We obtain an exact solution for a field of arbitrary spin s (integer or half-integer) in the BTZ background. This implies that the BTZ is perhaps the only known black hole background where such an analysis can be done analytically for all bosonic and fermionic fields. The quasinormal modes are shown to match precisely with the poles of the corresponding Green's function in the CFT living on the boundary. Furthermore, we show that one-loop determinants of higher spin fields can also be written as a product form [10] in terms of these quasinormal modes and this agrees with the same obtained by integrating the heat-kernel [11]. We then turn our attention to dualities relating higher-spin gravity to CFTs with W algebra symmetries. Since higher spin gravity does go beyond diffeomorphism invariance, one needs re_ned notions of the usual concepts in differential geometry. For example, in general relativity black holes are defined by the presence of the horizon. However, higher spin gravity has an enlarged group of symmetries of which the diffeomorphisms form a subgroup. The appropriate way of thinking of solutions in higher spin gravity is via characterizations which are gauge invariant [12, 13]. We study classical solutions embedded in N = 2 higher spin supergravity. We obtain a general gauge-invariant condition { in terms of the odd roots of the superalgebra and the eigenvalues of the holonomy matrix of the background { for the existence of a Killing spinor such that these solutions are supersymmetric [14]. We also study black holes in higher spin supergravity and show that the partition function of these black holes match exactly with that obtained from a CFT with the same asymptotic symmetry algebra [15]. This involved studying the asymptotic symmetries of the black hole and thereby developing the holographic dictionary for the bulk charges and chemical potentials with the corresponding quantities of the CFT. We finally investigate entanglement entropy in the AdS3/CFT2 context. Entanglement entropy is an useful non-local probe in QFT and many-body physics [16]. We analytically evaluate the entanglement entropy of the free boson CFT on a circle at finite temperature (i.e. on a torus) [17]. This is one of the simplest and well-studied CFTs. The entanglement entropy is calculated via the replica trick using correlation functions of bosonic twist operators on the torus [18]. We have then set up a systematic high temperature expansion of the Renyi entropies and determined their finite size corrections. These _nite size corrections both for the free boson CFT and the free fermion CFT were then compared with the one-loop corrections obtained from bulk three dimensional handlebody spacetimes which have higher genus Riemann surfaces (replica geometry) as its boundary [19]. One-loop corrections in these geometries are entirely determined by the spectrum of the excitations present in the bulk. It is shown that the leading _nite size corrections obtained by evaluating the one-loop determinants on these handlebody geometries exactly match with those from the free fermion/boson CFTs. This provides a test for holographic methods to calculate one-loop corrections to entanglement entropy. We also study conformal field theories in 1+1 dimensions with W-algebra symmetries at _nite temperature and deformed by a chemical potential (_) for a higher spin current. Using OPEs and uniformization techniques, we show that the order _2 correction to the Renyi and entanglement entropies (EE) of a single interval in the deformed theory is universal [20]. This universal feature is also supported by explicit computations for the free fermion and free boson CFTs { for which the EE was calculated by using the replica trick in conformal perturbation theory by evaluating correlators of twist fields with higher spin operators [21]. Furthermore, this serves as a verification of the holographic EE proposal constructed from Wilson lines in higher spin gravity [22, 23]. We also examine relative entropy [24] in the context of higher-spin holography [25]. Relative entropy is a measure of distinguishability between two quantum states. We confirm the expected short-distance behaviour of relative entropy from holography. This is done by showing that the difference in the modular Hamiltonian between a high-temperature state and the vacuum matches with the difference in the entanglement entropy in the short-subsystem regime.

L'objet de cette thèse est d'étendre le cadre théorique et les méthodes numériques du transport optimal à des objets plus généraux que des mesures de probabilité. En premier lieu, nous définissons des modèles de transport optimal entre mesures positives suivant deux approches, interpolation et couplage de mesures, dont nous montrons l'équivalence. De ces modèles découle une généralisation des métriques de Wasserstein. Dans une seconde partie, nous développons des méthodes numériques pour résoudre les deux formulations et étudions en particulier une nouvelle famille d'algorithmes de "scaling", s'appliquant à une grande variété de problèmes. La troisième partie contient des illustrations ainsi que l'étude théorique et numérique, d'un flot de gradient de type Hele-Shaw dans l'espace des mesures. Pour les mesures à valeurs matricielles, nous proposons aussi un modèle de transport optimal qui permet un bon arbitrage entre fidélité géométrique et efficacité algorithmique. / This thesis generalizes optimal transport beyond the classical "balanced" setting of probability distributions. We define unbalanced optimal transport models between nonnegative measures, based either on the notion of interpolation or the notion of coupling of measures. We show relationships between these approaches. One of the outcomes of this framework is a generalization of the p-Wasserstein metrics. Secondly, we build numerical methods to solve interpolation and coupling-based models. We study, in particular, a new family of scaling algorithms that generalize Sinkhorn's algorithm. The third part deals with applications. It contains a theoretical and numerical study of a Hele-Shaw type gradient flow in the space of nonnegative measures. It also adresses the case of measures taking values in the cone of positive semi-definite matrices, for which we introduce a model that achieves a balance between geometrical accuracy and algorithmic efficiency.

Dans cette thèse, nous étudions l'ensemble des solutions d'équations aux dérivées partielles résultant de modèles d'astrophysique et de biologie. Nous répondons aux questions de l'existence, mais aussi nous essayons de décrire le comportement de certaines familles de solutions lorsque les paramètres varient. Tout d'abord, nous étudions deux problèmes issus de l'astrophysique, pour lesquels nous montrons l'existence d'ensembles particuliers de solutions dépendant d'un paramètre à l'aide de la méthode de réduction de Lyapunov-Schmidt. Ensuite un argument de perturbation et le théorème du Point xe de Banach réduisent le problème original à un problème de dimension finie, et qui peut être résolu, habituellement, par des techniques variationnelles. Le reste de la thèse est consacré à l'étude du modèle Keller-Segel, qui décrit le mouvement d'amibes unicellulaires. Dans sa version plus simple, le modèle de Keller-Segel est un système parabolique-elliptique qui partage avec certains modèles gravitationnels la propriété que l'interaction est calculée au moyen d'une équation de Poisson / Newton attractive. Une différence majeure réside dans le fait que le modèle est défini dans un espace bidimensionnel, qui est expérimentalement consistant, tandis que les modèles de gravitationnels sont ordinairement posés en trois dimensions. Pour ce problème, les questions de l'existence sont bien connues, mais le comportement des solutions au cours de l'évolution dans le temps est encore un domaine actif de recherche. Ici nous étendre les propriétés déjà connues dans des régimes particuliers à un intervalle plus large du paramètre de masse, et nous donnons une estimation précise de la vitesse de convergence de la solution vers un profil donné quand le temps tend vers l'infini. Ce résultat est obtenu à l'aide de divers outils tels que des techniques de symétrisation et des inégalités fonctionnelles optimales. Les derniers chapitres traitent de résultats numériques et de calculs formels liés au modèle Keller-Segel / In this thesis we study the set of solutions of partial differential equations arising from models in astrophysics and biology. We answer the questions of existence but also we try to describe the behavior of some families of solutions when parameters vary. First we study two problems concerned with astrophysics, where we show the existence of particular sets of solutions depending on a parameter using the Lyapunov-Schmidt reduction method. Afterwards a perturbation argument and Banach's Fixed Point Theorem reduce the original problem to a finite-dimensional one, which can be solved, usually, by variational techniques. The rest of the thesis is de-voted to the study of the Keller-Segel model, which describes the motion of unicellular amoebae. In its simpler version, the Keller-Segel model is a parabolic-elliptic system which shares with some gravitational models the property that interaction is computed through an attractive Poisson / Newton equation. A major difference is the fact that it is set in a two-dimensional setting, which experimentally makes sense, while gravitational models are ordinarily three-dimensional. For this problem the existence issues are well known, but the behaviour of the solutions during the time evolution is still an active area of research. Here we extend properties already known in particular regimes to a broader range of the mass parameter, and we give a precise estimate of the convergence rate of the solution to a known profile as time goes to infinity. This result is achieved using various tools such as symmetrization techniques and optimal functional inequalities. The last chapters deal with numerical results and formal computations related to the Keller-Segel model