The distribution of particles in a rarefied gas in a vessel can be described by the Boltzmann equation. As an approximation of the solution to this equation, Caprino, Pulvirenti and Wagner [3] constructed a random N-particle system. In the equilibrium case, they prove in [3] that the L1-distance between the density function of k particles in the N-particle process and the k-fold product of the solution to the stationary Boltzmann equation is of order 1/N. They do this in order to show that the N-particle system converges to the system described by the stationary Boltzmann equation as the number of particles tends to infinity. This is different from the standard approach of describing convergence of an N-particle system. Usually, convergence in distribution of random measures or weak convergence of measures over the space of probability measures is used. The purpose of the present thesis is to compare different modes of convergence of the N-particle system as N tends to infinity assuming stationarity.

This thesis is split in two different topics. Firstly, we study anomalous transport from kinetic models. Secondly, we consider the equations coming from weak wave turbulence theory and we study them via mean-field limits of finite stochastic particle systems. $\textbf{Anomalous transport from kinetic models.}$ The goal is to understand how fractional diffusion arises from kinetic equations. We explain how fractional diffusion corresponds to anomalous transport and its relation to the classical diffusion equation. In previous works it has been seen that particles systems undergoing free transport and scattering with the media can give rise to fractional phenomena in two cases: firstly, if in the dynamics of the particles there is a heavy-tail equilibrium distribution; and secondly, if the scattering rate is degenerate for small velocities. We use these known results in the literature to study the emergence of fractional phenomena for some particular kinetic equations. Firstly, we study BGK-type equations conserving not only mass (as in previous results), but also momentum and energy. In the hydrodynamic limit we obtain a fractional diffusion equation for the temperature and density making use of the Boussinesq relation and we also demonstrate that with the same rescaling fractional diffusion cannot be derived additionally for the momentum. But considering the case of conservation of mass and momentum only, we do obtain the incompressible Stokes equation with fractional diffusion in the hydrodynamic limit for heavy-tailed equilibria. Secondly, we will study diffusion phenomena arising from transport of energy in an anharmonic chain. More precisely, we will consider the so-called FPU-$\beta$ chain, which is a very simple model for a one-dimensional crystal in which atoms are coupled to their nearest neighbours by a harmonic potential, weakly perturbed by a nonlinear quartic potential. The starting point of our mathematical analysis is a kinetic equation; lattice vibrations, responsible for heat transport, are modelled by an interacting gas of phonons whose evolution is described by the Boltzmann Phonon Equation. Our main result is the derivation of an anomalous diffusion equation for the temperature. $\textbf{Weak wave turbulence theory and mean-field limits for stochastic particle systems.}$ The isotropic 4-wave kinetic equation is considered in its weak formulation using model homogeneous kernels. Existence and uniqueness of solutions is proven in a particular setting. We also consider finite stochastic particle systems undergoing instantaneous coagulation-fragmentation phenomena and give conditions in which this system approximates the solution of the equation (mean-field limit).

Koagulation ist physikalisch bedeutsam für eine Vielzahl von technischen und naturwissenschaftlichen Anwendungen und bezeichnet die paarweise Verschmelzung von Clustern unterschiedlicher Masse. Der zeitliche Verlauf der Clusterkonzentration läßt sich durch Smoluchowskis Koagulationsgleichung beschreiben, einem unendliches System nichtlinearer Differentialgleichungen. Ausgangspunkt dieser Arbeit ist eine nichtlineare maßwertige Gleichung, die die Koagulations- und andere kinetische Gleichungen beinhaltet und verschiedene physikalische und chemische Mechanismen integriert. Sie ermöglicht einen allgemeinen Zugang zu Fragen bezüglich der Existenz von Lösungen und ihrer Approximation durch stochastische Partikelsysteme. Die Teilchensysteme werden dabei als reguläre Sprungprozesse modelliert, welche eine Menge diskreter Maße auf einem lokal-kompakten Raum als Zustandsraum besitzen. Die Arbeit untergliedert sich in drei Teile: Unter geeigneten Voraussetzungen an die Sprungraten werden zunächst für wachsende Teilchenzahlen Approximations- und Konvergenzaussagen unter Verwendung von Kompaktheitsargumenten, Martingaltheoremen und Lokalisierungstechniken bewiesen. Ihre Anwendung auf die Koagulationsgleichung mit Fragmentation, Quellen und Senken erlaubt anschließend die Herleitung neuer Existenzresultate und stochastischer Algorithmen. Der letzte Abschnitt illustriert die numerischen Eigenschaften und die Effizienz der neuen Algorithmen im Vergleich zu bisherigen Monte Carlo Methoden und ihre besondere Eignung zur Analyse des Gelationsphänomens, einem Phasenübergang, welcher zum Masseverlust im Clustersystem führt. / Coagulation is an important physical process for a wide range of technical and scientific applications and denotes the pairwise merging of clusters with different mass. The dynamic behaviour of the cluster concentration can be described by Smoluchowski's coagulation equation which is an infinite system of nonlinear differential equations. In this thesis we start with a nonlinear measure-valued equation generalizing the coagulation and other kinetic equations and integrating various physical and chemical processes. This equation allows a unified treatment of questions concerning existence of solutions and their approximation by means of stochastic particle systems. Here, the particle systems are defined as regular jump processes living on a set of point measures on a locally compact space. The thesis consists of three parts: First of all, approximation and convergence results for suitable jump rates and increasing particle numbers are proved by means of compactness theorems, martingale techniques and localizing procedures. Then, an application to the coagulation equation with fragmentation, source and efflux terms leads to new existence results and stochastic algorithms. Finally, their numerical features and efficiency are compared to known Monte Carlo methods and their specific convergence properties are presented with respect to a phase transition which is called gelation and leads to a loss of total cluster mass.

En chimiotaxie, le modèle parabolique-parabolique classique de Keller-Segel en dimension d décrit l’évolution en temps de la densité d'une population de cellules et de la concentration d'un attracteur chimique. Cette thèse porte sur l’étude des équations de Keller-Segel parabolique-parabolique par des méthodes probabilistes. Dans ce but, nous construisons une équation différentielle stochastique non linéaire au sens de McKean-Vlasov dont le coefficient dont le coefficient de dérive dépend, de manière singulière, de tout le passé des lois marginales en temps du processus. Ces lois marginales couplées avec une transformation judicieuse permettent d’interpréter les équations de Keller-Segel de manière probabiliste. En ce qui concerne l'approximation particulaire il faut surmonter une difficulté intéressante et, nous semble-t-il, originale et difficile chaque particule interagit avec le passé de toutes les autres par l’intermédiaire d'un noyau espace-temps fortement singulier. En dimension 1, quelles que soient les valeurs des paramètres de modèle, nous prouvons que les équations de Keller-Segel sont bien posées dans tout l'espace et qu'il en est de même pour l’équation différentielle stochastique de McKean-Vlasov correspondante. Ensuite, nous prouvons caractère bien posé du système associé des particules en interaction non markovien et singulière. Nous établissons aussi la propagation du chaos vers une unique limite champ moyen dont les lois marginales en temps résolvent le système Keller-Segel parabolique-parabolique. En dimension 2, des paramètres de modèle trop grands peuvent conduire à une explosion en temps fini de la solution aux équations du Keller-Segel. De fait, nous montrons le caractère bien posé du processus non-linéaire au sens de McKean-Vlasov en imposant des contraintes sur les paramètres et données initiales. Pour obtenir ce résultat, nous combinons des techniques d'analyse d’équations aux dérivées partielles et d'analyse stochastique. Finalement, nous proposons une méthode numérique totalement probabiliste pour approcher les solutions du système Keller-Segel bi-dimensionnel et nous présentons les principaux résultats de nos expérimentations numériques. / The standard d-dimensional parabolic--parabolic Keller--Segel model for chemotaxis describes the time evolution of the density of a cell population and of the concentration of a chemical attractant. This thesis is devoted to the study of the parabolic--parabolic Keller-Segel equations using probabilistic methods. To this aim, we give rise to a non linear stochastic differential equation of McKean-Vlasov type whose drift involves all the past of one dimensional time marginal distributions of the process in a singular way. These marginal distributions coupled with a suitable transformation of them are our probabilistic interpretation of a solution to the Keller Segel model. In terms of approximations by particle systems, an interesting and, to the best of our knowledge, new and challenging difficulty arises: each particle interacts with all the past of the other ones by means of a highly singular space-time kernel. In the one-dimensional case, we prove that the parabolic-parabolic Keller-Segel system in the whole Euclidean space and the corresponding McKean-Vlasov stochastic differential equation are well-posed in well chosen space of solutions for any values of the parameters of the model. Then, we prove the well-posedness of the corresponding singularly interacting and non-Markovian stochastic particle system. Furthermore, we establish its propagation of chaos towards a unique mean-field limit whose time marginal distributions solve the one-dimensional parabolic-parabolic Keller-Segel model. In the two-dimensional case there exists a possibility of a blow-up in finite time for the Keller-Segel system if some parameters of the model are large. Indeed, we prove the well-posedness of the mean field limit under some constraints on the parameters and initial datum. Under these constraints, we prove the well-posedness of the Keller-Segel model in the plane. To obtain this result, we combine PDE analysis and stochastic analysis techniques. Finally, we propose a fully probabilistic numerical method for approximating the two-dimensional Keller-Segel model and survey our main numerical results.