[pt] Neste trabalho aplicamos o método RPA renormalizada ao modelo de Hubbard estendido para a cadeia unidimensional com preenchimento meio. Inicialmente verificamos a inúncia do termo de interação entre primeiros vizinhos, V , nas susceptibilidades de carga e longitudinal de spin e comparamos com os resultados de Hubbard puro. Para pequenas interações entre os primeiros vizinhos, a influência de V é muito pequena. À medida que V cresce, seu efeito é de diminuir a frequência dos plasmons e dificultar o aparecimento de magnons. Em seguida determinamos as susceptibilidades de spin e de carga, numericamente, para U = 3,4 e 5 com V entre 0 e próximo de U. Partindo da fase de onda de densidade de spin (SDW) e quando V cresce, o pico de plasmon na susceptibilidade de cargas em 2kF se aproxima de ômega = 0, indicando uma instabilidade para formação de uma onda de densidade de carga (CDW). Isto permite determinar a fronteira de separação entre as duas fases. De acordo com a RPA convencional, esta transformação ocorre em U = 2V mas o processo de renormalização desloca esta fronteira ligeiramente para o interior da região da fase CDW, em conformidade com outros métodos que não fazem a aproximação de campo médio. / [en] In this work we apply the Random Phase Approximation (RPA) to the Extended Hubbard Model in one dimension. Initially we investigate the effect of the first neighbor interaction term, V , on the spin and charge susceptibilities. When V is small, its influence is also very small but as it grows, it tends to lower the plasmon frequency and to inhibit the condition for the existence of magnons. We also have calculated numerically both the spin and the charge susceptibilities for U = 3,4 and 5 with V in the region (0; U). Starting from the SDW region and increasing V we found that the plasmon peak in the charge susceptibility for 2kF tends to omega = 0, indicating an instability against the formation of a CDW ground state. This allows us to determine the phase separation line between the two phases. According to the conventional RPA, this transition occurs for U = 2V but due to the renormalization process, we nd that this frontier shifts slightly to the interior of the CDW region, in agreement with methods not using the mean eld approximation.

We introduce renormalized integrals which generalize conventional measure theoretic integrals. One approximates the integration domain by measure spaces and defines the integral as the limit of integrals over the approximating spaces. This concept is implicitly present in many mathematical contexts such as Cauchy's principal value, the determinant of operators on a Hilbert space and the Fourier transform of an L^p function. We use renormalized integrals to define a path integral on manifolds by approximation via geodesic polygons. The main part of the paper is dedicated to the proof of a path integral formula for the heat kernel of any self-adjoint generalized Laplace operator acting on sections of a vector bundle over a compact Riemannian manifold.

Numerical simulation is becoming increasingly used to support theoretical effort into understanding the turbulence problem. We develop theoretical ideas related to the transfer and dissipation of energy, which clarify long-standing issues with the energy balance in isotropic turbulence. These ideas are supported by results from large scale numerical simulations. Due to the large number of degrees of freedom required to capture all the interacting scales of motion, the increase in computational power available has only recently allowed flows of interest to be realised. A parallel pseudo-spectral code for the direct numerical simulation (DNS) of isotropic turbulence has been developed. Some discussion is given on the challenges and choices involved. The DNS code has been extensively benchmarked by reproducing well established results from literature. The DNS code has been used to conduct a series of runs for freely-decaying turbulence. Decay was performed from a Gaussian random field as well as an evolved velocity field obtained from forced simulation. Since the initial condition does not describe developed turbulence, we are required to determine when the field can be considered to be evolved and measurements are characteristic of decaying turbulence. We explore the use of power-law decay of the total energy and compare with the use of dynamic quantities such as the peak dissipation rate, maximum transport power and velocity derivative skewness. We then show how this choice of evolved time affects the measurement of statistics. In doing so, it is found that the Taylor dissipation surrogate, u^3 / L, is a better surrogate for the maximum inertial flux than dissipation. Stationary turbulence has also been investigated, where we ensure that the energy input rate remains constant for all runs and variation is only introduced by modifying the fluid viscosity (and lattice size). We present results for Reynolds numbers up to Rλ = 335 on a 1024^3 lattice. Using different methods of vortex identification, the persistence of intermittent structure in an ensemble average is considered and shown to be reduced as the ensemble size increases. The longitudinal structure functions are computed for smaller lattices directly from an ensemble of realisations of the real-space velocity field. From these, we consider the generalised structure functions and investigate their scaling exponents using direct analysis and extended self-similarity (ESS), finding results consistent with the literature. An exploitation of the pseudo-spectral technique is used to calculate second- and third-order structure functions from the energy and transfer spectra, with a comparison presented to the real-space calculation. An alternative to ESS is discussed, with the second-order exponent found to approach 2/3. The dissipation anomaly is then considered for both forced and free-decay. Using different choices of the evolved time for a decaying simulation, we show how the behaviour of the dimensionless dissipation coefficient is affected. The Karman-Howarth equation (KHE) is studied and a derivation of a work term presented using a transformation of the Lin equation. The balance of energy represented by the KHE is then investigated using the pseudo-spectral method mentioned above. The consequences of this new input term for the structure functions are discussed. Based on the KHE, we develop a model for the behaviour of the dimensionless dissipation coefficient that predicts Cɛ= Cɛ(∞)+CL/RL. DNS data is used to fit the model. We find Cɛ(∞) = 0.47 and CL = 19.1 for forced turbulence, with excellent agreement to the data. Theoretical methods based on the renormalization group and statistical closures are still being developed to study turbulence. The dynamic RG procedure used by Forster, Nelson and Stephen (FNS) is considered in some detail and a disagreement in the literature over the method and results is resolved here. An additional constraint on the loop momentum is shown to cause a correction to the viscosity increment such that all methods of evaluation lead to the original result found by FNS. The application of statistical closure and renormalized perturbation theory is discussed and a new two-time model probability density functional presented. This has been shown to be self-consistent to second order and to reproduce the two-time covariance equation of the local energy transfer (LET) theory. Future direction of this work is discussed.

Diese Arbeit befasst sich mit dem Problem der nichtperturbativen Renormierung des Axialstroms eines leichten und eines Bottom-Quarks. Solche nichtperturbativen Berechnungen koennen nur in der Gitter-QCD durchgefuehrt werden, d. h. die kontinuierliche Raumzeit wird durch ein vierdimensionales hyperkubisches Gitter ersetzt. Da einerseits die Kantenlaenge des Gitters groesser sein muss als typische physikalische Laengenskalen des Problems, andererseits aber der durch die Gitterkonstante eingefuehrte Energie-Cutoff groesser sein muss als die Masse des b-Quarks, sind fuer dieses Problem Gittergroessen erforderlich, fuer die die heutige Computerleistung nicht ausreicht. Es ist daher sinnvoll, das B-Meson in der statischen Naeherung zu untersuchen, um dann zwischen dieser Naeherung und leichten Quarkmassen interpolieren zu koennen. Die Renormierung des Axialstroms in der statischen Naeherung ist skalenabhaengig. Um zu vermeiden, Rechnungen ueber einen grossen Energiebereich hinweg auf einem einzigen Gitter durchfuehren zu muessen, wird als Renormierungsverfahren das SF-Schema vorgeschlagen, in dem die Renormierungsskala mit der inversen Kantenlaenge des Raumzeitvolumens identifiziert wird. Das zentrale Objekt dieses Schemas ist die Step-Scaling-Funktion, die die Renormierungskonstanten bei verschiedenen Skalen miteinander in Beziehung setzt. Ein wesentlicher Punkt dieser Arbeit ist die O(a)-Verbesserung, die die Diskretisierungsfehler reduziert. Nach einer Erklaerung dieses Verfahrens fuer Eichfelder und leichte Quarks wird die statische Approximation im Kontinuum und auf dem Gitter eingefuehrt, und die in der Gittertheorie erforderlichen O(a)-Verbesserungsterme werden diskutiert. Fuer die eigentliche Renormierung werden Schroedinger-Funktional- Randbedingungen analog zum Fall leichter Quarks auch fuer die statische Approximation eingefuehrt, und die durch diese Randbedingungen notwendige zusaetzliche O(a)-Verbesserung diskutiert. Anschliessend wird durch eine Renormierungsbedingung das SF-Schema fuer den statischen Axialstrom definiert. Im weiteren Verlauf der Arbeit steht die Entwicklung geigneter Korrelationsfunktionen in der Einschleifennaeherung im Mittelpunkt. In dieser Naeherung wird zunaechst der renormierte statische Axialstrom im Gitter-MS-Schema untersucht, und seine Beziehung zum Axialstrom zweier leichter Quarks in der Stromalgebra-Normierung berechnet. Hierbei wird Uebereinstimmung mit einem Ergebnis anderer Autoren aus einer anderen Methode festgestellt. Aus diesem Ergebnis wird die endliche Renormierung zwischen dem statischen Strom im Gitter-MS-Schema und dem statischen Strom im MS-bar-Schema bestimmt. Diese Renormierungskonstante wird dann benutzt, um den Umrechnungsfaktor vom SF-Schema in das MS-bar-Schema in der Einschleifennaeherung zu berechnen. Die Zweischleifennaeherung der anomalen Dimension des statischen Axialstroms im SF-Schema wird dann durch Umrechnung aus dem MS-bar-Schema bestimmt. Diese Groesse ist vor allem deshalb wichtig, weil sie benoetigt wird, um bei Energieskalen von 10-100 GeV aus nichtperturbativ gewonnenen Ergebnissen den renormierungsgruppeninvarianten statischen Axialstrom zu berechnen. Es zeigt sich, dass der Zweischleifenwert dieser anomalen Dimension klein ist. Ein weiterer Untersuchungsgegenstand sind die Diskretisierungsfehler in der Step-Scaling-Funktion, die in der Einschleifennaeherung berechnet wurden. Sie stellen sich ebenfalls als klein heraus. Abschliessend wird der Einschleifen-Koeffizient des O(a)-Verbesserungsterms fuer den statischen Axialstrom berechnet. Hierbei ergibt sich Uebereinstimmung mit einem frueheren Ergebnis anderer Autoren. / The axial current of a light and a heavy quark is studied in the static approximation, with the aim of defining a non-perturbative renormalization scheme. To keep lattice artifacts small, O(a) improvement in the static approximation is discussed in detail. It is explained how a finite size scheme can be used to avoid the necessity of accommodating a large energy range on a single lattice in the determination of the scale dependence of the renormalized static-light axial current. To that end, Schroedinger functional boundary conditions are imposed on the static quark field, and a renormalization condition is formulated. As a central object of the SF scheme, the "step scaling function", connecting the renormalization constants at different scales, is introduced. A large part of this thesis is dedicated to the expansion of suitable correlation functions to one loop order of perturbation theory. Using these expansions, the finite renormalization constants connecting the static-light axial current in the lattice MS scheme and the light-light axial current normalized by current algebra relations is calculated at one loop order. From this result, the relation of the renormalized static-light axial current in the SF scheme to the MS-bar-renormalized static-light axial current is derived. Using that relation, the static-light axial current's two loop anomalous dimension in the SF scheme, which is needed for the calculation of the renormalization group invariant current, is calculated by conversion from the MS-bar scheme. Further studies made in this thesis are the determination of discretization errors in the step scaling function at one loop order, and the calculation of an improvement coefficient for the static-light axial current at one loop order of perturbation theory.

On s’intéresse dans cette thèse à montrer que la solution approchée, par la méthode des volumes finis, converge vers la solution renormalisée de problèmes elliptiques ou paraboliques à donnée L1. Dans la première partie nous étudions une équation de convection-diffusion ellliptique à donnée L1. En adaptant la stratégie développée pour les solutions renormaliséesà la méthode des volumes finis, nous montrons que la solution approchée converge vers l’unique solution renormalisée.Dans la deuxième partie nous nous intéressons à un problème parabolique nonlinéaire à donnée L1. En utilisant une version discrète de résultats de compacité classiques, nous montrons que les résultats obtenues dans le cas elliptique restentvrais dans le cas parabolique. Dans la troisième partie nous montrons des résultats similaires pour une équationparabolique doublement non-linéaire à donnée L1. Le caractère doublement nonlinéaire de l’équation crée des difficultés supplémentaires par rapport à la partie précédente, notamment car la règle de dérivation en chaîne ne s’applique pas dansle cas discret. Enfin, dans la quatrième partie, nous utilisons les résultats établis précédemment pour étudier un système de type thermoviscoélasticité. Nous montrons que la solution approchée, obtenue par un schéma éléments finis-volumes finis, converge vers une solution faible-renormalisée du système. / In this thesis we are interested in proving that the approximate solution, obtained by the finite volume method, converges to the unique renormalized solution of elliptic and parabolic equations with L1 data. In the first part we study an elliptic convection-diffusion equation with L1 data. Mixing the strategy developed for renormalized solution and the finite volume method,we prove that the approximate solution converges to the unique renormalized solution. In the second part we investigate a nonlinear parabolic equation with L1 data. Using a discrete version of classical compactness results, we show that the results obtaines previously in the elliptic case hold true in the parabolic case. In the third part we prove similar results for a doubly nonlinear parabolic equation with L1 data. The doubly nonlinear character of the equation makes new difficulties with respect to the previous part, especially since the chain rule formula does not apply in the discrete case. Finaly, in the fourth part we use the results established previously to investigate a system of thermoviscoelasticity kind. We show that the approximate solution,obtaines by finite element-finite volume scheme, converges to a weak-renormalized solution of the system.

In this work we calculate the binding energies and root-mean-square radii for A−body nuclear bound state systems, where A ≥ 3. To study three−body systems, we employ the three−dimensional differential Faddeev equations with nucleon-nucleon semi-realistic potentials. The equations are solved numerically. For this purpose, the equations are transformed into an eigenvalue equation via the orthogonal collocation procedure using triquintic Hermite splines. The resulting eigenvalue equation is solved using the Restarted Arnoldi Algorithm. Ground state binding energies of the 3H nucleus are determined. For A > 3, the Potential Harmonic Expansion Method is employed. Using this method, the Schr¨odinger equation is transformed into coupled Faddeev-like equations. The Faddeevlike amplitudes are expanded on the potential harmonic basis. To transform the resulting coupled differential equations into an eigenvalue equation, we employ again the orthogonal collocation procedure followed by the Gauss-Jacobi quadrature. The corresponding eigenvalue equation is solved using the Renormalized Numerov Method to obtain ground state binding energies and root-mean-square radii of closed shell nuclei 4He, 8Be, 12C, 16O and 40Ca. / Physics / M. Sc. (Physics)

Esta tese compõe-se de contribuições à física das propriedades eletrônicas das heteroestruturas semicondutoras. São investigadas propriedades eletrônicas das duas hetero­estruturas básicas: o poço quântico e a super-rede. Considera-se o poço quântico dopado com impurezas rasas e estudam-se as suas propriedades eletrônicas nos regimes de poço fraca e altamente dopado. No caso de baixa densidade de impurezas é feita uma simulação Monte Carlo. É utilizado um modelo semi-clássico de band de impureza. A interação elétron-elétron é incluída de forma exata e são calculadas as seguintes propriedades do estado fundamental à temperatura zero: densidade de estados de uma partícula, distribuição de carga, energia de Fermi e distribuição do campo elétrico sobre os doadores neutros, todas em função do grau de compensação, da densidade de impurezas e da largura do poço. É observada uma. grande dependência com a compensação. Os resultados são explicados à luz da competição entre os efeitos de desordem e confinamento. É observada a ocorrência de Coulomb Gap característico de sistemas bidimensionais. Mostra-se que a. distribuição de carga possui largura e constante de decaimento determinados independentemente pela compensação e pela concentração de impurezas, respectivamente. Tais resultados são importantes para a caracterização de poços quânticos puros. No limite altamente dopado parte-se de um modelo light-binding desordenado e calcula­se a densidade de estados de uma partícula formada devido ao overlapping entre os estados localizados; utiliza-se o método de Matsubara e Toyosawa. para a obtenção da média sobre configurações. Discutem-se os efeitos da desordem diagonal introduzida pelo potencial de confinamento os quais são comparados com os da. desordem não-diagonal. São apresentados resultados para a densidade de estados em função do grau de confinamento e concentração de impurezas para poços e fios quânticos. Sâo estudadas as propriedades eletrônicas das super-redes sob campo magnético transversal à direção de crescimento. Mostra-se que esta configuração é ideal para o estudo das características básicas das super-redes: a estrutura de mini bandas e o tunelamento. Calculam-se as sub-bandas de condução utilizando a teoria de massa efetiva de muitas bandas. Introduz-se a idéia de massa efetiva renormalizada para barreiras semicondutoras. Comparam-se os resultados com dados experimentais de ressonância ciclotrônica. A ótima concordância obtida demonstra a grande importância e a utilidade do conceito de massa efetiva renormalizada para barreiras semicondutoras, que é uma maneira nova e simples de lidar com as soluções evanescentes. / This thesis is composed of contributions to the theory of electronic properties of semicon­ ductor heterostructures. Electronic properties of the basic two heterostructures (quantum well and superlattice) are investigated. A quantum well doped with shallow impurities is considered and its electronic properties are studied in both limits: lightly and heavily doped. In the first case a Monte Carlo simula­ tion technique is used. A semiclassical impurity band model is used . The electron-electron interaction is included exactly and properties of the ground state such as the density of single particle states, the charge distribution, the Fermi energy and the electric field di tribution on the neutra/ donors are calculated, all of them as a function of the degree of compensation, the impurity concentration and the width of the well. A great dependency with the compensation is observed. The results are explained by the competition between the effects of disorder and confinement. The existence of a Coulomb Gap is verified . The charge distribution is shown to have a width and decay rate given by the degree of compensation and impurity concentration, in this order. Such results are important to characterize pure quantum wells. On the heavily doped limit, a disordered tight-binding model is used and the density of states that is formed by the overlapping of localized states is calculated by using the method of Matsubara and Toyosawa for the configuration average. The diagonal disord er effect introduced by the confinement potential is considered and compared to that of the non­ diagonal disorder. Results of the density of states as a function of the degree of confinement and impurity concentration for quantum wells and wires are presented. The electronic propertie s of a superlattice under a magnetic field which is transversal to the growth direction are studied. Jt is shown that this configuration is id eal for the study of the basic characteristics of the superlattices: the subband structure and the tunneling. The conduction subbands are calculated by using the theory of many bands effective mass. The idea of renormalized effective mass for barriers is introduced. The obtained level spacings are compared with cyclotron resonance experimental data (infrared absorption). The good agreement obtained demonstrates the importance and usefulness of the renormalized effective mass, which is a new and simple way to handle evanescent waves.

Esta tese compõe-se de contribuições à física das propriedades eletrônicas das heteroestruturas semicondutoras. São investigadas propriedades eletrônicas das duas hetero­estruturas básicas: o poço quântico e a super-rede. Considera-se o poço quântico dopado com impurezas rasas e estudam-se as suas propriedades eletrônicas nos regimes de poço fraca e altamente dopado. No caso de baixa densidade de impurezas é feita uma simulação Monte Carlo. É utilizado um modelo semi-clássico de band de impureza. A interação elétron-elétron é incluída de forma exata e são calculadas as seguintes propriedades do estado fundamental à temperatura zero: densidade de estados de uma partícula, distribuição de carga, energia de Fermi e distribuição do campo elétrico sobre os doadores neutros, todas em função do grau de compensação, da densidade de impurezas e da largura do poço. É observada uma. grande dependência com a compensação. Os resultados são explicados à luz da competição entre os efeitos de desordem e confinamento. É observada a ocorrência de Coulomb Gap característico de sistemas bidimensionais. Mostra-se que a. distribuição de carga possui largura e constante de decaimento determinados independentemente pela compensação e pela concentração de impurezas, respectivamente. Tais resultados são importantes para a caracterização de poços quânticos puros. No limite altamente dopado parte-se de um modelo light-binding desordenado e calcula­se a densidade de estados de uma partícula formada devido ao overlapping entre os estados localizados; utiliza-se o método de Matsubara e Toyosawa. para a obtenção da média sobre configurações. Discutem-se os efeitos da desordem diagonal introduzida pelo potencial de confinamento os quais são comparados com os da. desordem não-diagonal. São apresentados resultados para a densidade de estados em função do grau de confinamento e concentração de impurezas para poços e fios quânticos. Sâo estudadas as propriedades eletrônicas das super-redes sob campo magnético transversal à direção de crescimento. Mostra-se que esta configuração é ideal para o estudo das características básicas das super-redes: a estrutura de mini bandas e o tunelamento. Calculam-se as sub-bandas de condução utilizando a teoria de massa efetiva de muitas bandas. Introduz-se a idéia de massa efetiva renormalizada para barreiras semicondutoras. Comparam-se os resultados com dados experimentais de ressonância ciclotrônica. A ótima concordância obtida demonstra a grande importância e a utilidade do conceito de massa efetiva renormalizada para barreiras semicondutoras, que é uma maneira nova e simples de lidar com as soluções evanescentes. / This thesis is composed of contributions to the theory of electronic properties of semicon­ ductor heterostructures. Electronic properties of the basic two heterostructures (quantum well and superlattice) are investigated. A quantum well doped with shallow impurities is considered and its electronic properties are studied in both limits: lightly and heavily doped. In the first case a Monte Carlo simula­ tion technique is used. A semiclassical impurity band model is used . The electron-electron interaction is included exactly and properties of the ground state such as the density of single particle states, the charge distribution, the Fermi energy and the electric field di tribution on the neutra/ donors are calculated, all of them as a function of the degree of compensation, the impurity concentration and the width of the well. A great dependency with the compensation is observed. The results are explained by the competition between the effects of disorder and confinement. The existence of a Coulomb Gap is verified . The charge distribution is shown to have a width and decay rate given by the degree of compensation and impurity concentration, in this order. Such results are important to characterize pure quantum wells. On the heavily doped limit, a disordered tight-binding model is used and the density of states that is formed by the overlapping of localized states is calculated by using the method of Matsubara and Toyosawa for the configuration average. The diagonal disord er effect introduced by the confinement potential is considered and compared to that of the non­ diagonal disorder. Results of the density of states as a function of the degree of confinement and impurity concentration for quantum wells and wires are presented. The electronic propertie s of a superlattice under a magnetic field which is transversal to the growth direction are studied. Jt is shown that this configuration is id eal for the study of the basic characteristics of the superlattices: the subband structure and the tunneling. The conduction subbands are calculated by using the theory of many bands effective mass. The idea of renormalized effective mass for barriers is introduced. The obtained level spacings are compared with cyclotron resonance experimental data (infrared absorption). The good agreement obtained demonstrates the importance and usefulness of the renormalized effective mass, which is a new and simple way to handle evanescent waves.

Les travaux de cette thèse s’inscrivent dans le contexte des systèmes de particules. Nous considérons différents systèmes physiques, décrits de manière continue, et dont la dynamique est modélisée par des équations aux dérivées partielles décrivant l’évolution temporelle de certaines quantités macroscopiques ou microscopiques, selon l’échelle de description envisagée. Dans une première partie, nous nous intéressons à une équation de type coagulation-fragmentation cinétique. Nous obtenons un résultat d’existence globale en temps, dans le cadre des solutions renormalisées de DiPerna-Lions, pour toute donnée initiale vérifiant les estimations naturelles et possédant une norme L1 et une norme Lp (p > 1) finies. La deuxième partie traite de méthodes de moments. L’objectif de ces méthodes est d’approcher un modèle cinétique par un nombre fini d’équations portant sur des quantités dépendant uniquement de la variable d’espace, et la question est de savoir comment fermer le système obtenu pour obtenir une bonne approximation de la solution du modèle cinétique. Dans un cadre linéaire, nous obtenons une méthode de fermeture explicite conduisant à un résultat de convergence rapide. Enfin, dans une troisième partie, nous travaillons sur la modélisation du trafic routier avec prise en compte de la congestion à l’aide d’un système hyperbolique avec contraintes, issu de la dynamique des gaz sans pression. En modifiant convenablement ce système, nous parvenons à modéliser des phénomènes de trafic routier "multi-voies", comme l’accélération, et la création de zones de vide. Un résultat d’existence et de stabilité des solutions de ce modèle modifié est démontré. / The context of this thesis is particle systems. We deal with different physical systems, described continuously, whose dynamics are modeled by partial differential equations. These equations follow the evolution in time of macroscopic or microscopic quantities, according to scale description. In the first part, we consider a kinetic model for coagulation-fragmentation. We obtain a global existence result, using the notion of DiPerna-Lions renormalized solutions, for initial data satisfying the natural physical bounds, and assumptions of finite L1 and Lp norm (for some p > 1). The second part deals with methods of moments. The aim of these methods is to approximate a kinetic model by a finite number of equations whose unknowns depend only on the space variable. The question is : how to close this system to get a good approximation of the solution of the kinetic model ? In a linear setting, we obtain an explicit method with linear closure relations, which leads to a fast convergence result. In the last part, we work on modeling of traffic jam taking into account the congestion, using a hyperbolic system with constraints, which occurs in the dynamics of a pressureless gas. By suitably modifying this system, we can model "multi-lane" phenomena, like acceleration, and creation of vacuum. An existence and stability result is proved on this new model.

La thèse est composée de deux parties.Dans la première partie, nous allons étudier le flot quasi-invariant défini par une équation différentielle stochastique de Stratanovich avec le dérive ayant seulement la BV-régularitésur un espace euclidien, en généralisant des résultats de L. Ambrosio sur l'existence,unicité et stabilité des flots lagrangiens associés aux équations différentielles ordinaires[Invent. Math. 158 (2004), 227{260]. Comme une application d'un résultat de stabilité,nous allons construire une solution explicite à l'equation de transport stochastique enterme de flot stochastique. La différentiabilité approximative du flot sera aussi investie,lorsque le dérive possµede une régularité de Sobolev.Dans la deuxième partie, nous allons généraliser la théorie de DiPerna-Lions aux cas desvariétés riemanniennes complètes. Nous allons utiliser le semi-groupe de la chaleur pourrégulariser des fonctions et des champs de vecteur. L'estimation sur le commutateur seraobtenue par la méthode probabiliste. Une application de cette estimation est de prouverl'unicité des solutions à l'équation de transport à l'aide du concept des solutions renormal-isables. L'équation différentielle ordinaire associée à un champ de vecteur de régularité deSobolev sera enfin résolue en adoptant une méhode due à L. Ambrosio. La fin de cett par-tie consacre à la construction des processus de diffusion, par la méthode de la variation deconstante, sur une variété riemannienne complète, ayant comme générateur, un opérateurelliptique contenant le dérive non-régulier. Pour cela, nous allons donner des conditionssur la courbure pour que le flot horizontal canonique soit un flot de difféomorphismes / The thesis mainly consists of two parts.In the first part, we study the quasi-invariant flow generated by the Stratonovich stochas-tic differential equation with BV drift coefficients in the Euclidean space. We generalizethe results of Ambrosio [Invent. Math. 158 (2004), 227{260] on the existence, uniquenessand stability of regular Lagrangian flows of ordinary differential equations to Stratonovichstochastic differential equations with BV drift coefficients. As an application of the sta-bility result, we construct an explicit solution to the corresponding stochastic transportequation in terms of the stochastic flow. The approximate differentiability of the flow isalso studied when the drift coefficient has some Sobolev regularity.In the second part, we generalize the DiPerna-Lions theory in the Euclidean space to thecomplete Riemannian manifold. We define the commutator on the complete Riemannianmanifold which is a probabilistic version of the one in the DiPerna-Lions theory, andestablish the commutator estimate by the probabilistic method. As a direct applicationof the commutator estimate, we investigate the uniqueness of solutions to the transportequation by the method of the renormalized solution. Following Ambrosio's method, weconstruct the DiPerna-Lions flow on the Riemannian manifold. In order to construct thediffusion process associated to an elliptic operator with irregular drift on the completeRiemannian manifold, we give some conditions which guarantee the strong completenessof the horizontal flow. Finally, we construct the diffusion process with the drift coefficienthaving only Sobolev regularity.Besides, we present a brief introduction of the classical theory on the ordinary differentialequation in the smooth case and the quasi-invariant flow of homeomorphisms under theOsgood condition before the first part; and we recall some basic tools and results whichare widely used throughout the whole thesis after the second part.