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Probability theory is the study of random phenomena. Many dynamical systems with random influence, in nature or artificial complex systems, are better modeled by equations incorporating the intrinsic stochasticity involved. In probability theory, stochastic partial differential equations (SPDEs) generalize partial differential equations through random force terms and coefficients, while stochastic differential equations (SDEs) generalize ordinary differential equations. They are both abound in models involving Brownian motion throughout science, engineering and economics. However, Brownian motion is just one example of a random noisy input. The goal of this thesis is to make contributions in the study and applications of stochastic dynamical systems involving a wider variety of stochastic processes and noises. This is achieved by considering different models arising out of applications in thermal engineering, population dynamics and mathematical finance.<br><div><br></div><div>1. Power-type non-linearities in SDEs with rough noise: We consider a noisy differential equation driven by a rough noise that could be a fractional Brownian motion, a generalization of Brownian motion, while the equation's coefficient behaves like a power function. These coefficients are interesting because of their relation to classical population dynamics models, while their analysis is particularly challenging because of the intrinsic singularities. Two different methods are used to construct solutions: (i) In the one-dimensional case, a well-known transformation is used; (ii) For multidimensional situations, we find and quantify an improved regularity structure of the solution as it approaches the origin. Our research is the first successful analysis of the system described under a truly rough noise context. We find that the system is well-defined and yields non-unique solutions. In addition, the solutions possess the same roughness as that of the noise.<br></div><div><br></div><div>2. Parabolic Anderson model in rough environment: The parabolic Anderson model is one of the most interesting and challenging SPDEs used to model varied physical phenomena. Its original motivation involved bound states for electrons in crystals with impurities. It also provides a model for the growth of magnetic field in young stars and has an interpretation as a population growth model. The model can be expressed as a stochastic heat equation with additional multiplicative noise. This noise is traditionally a generalized derivative of Brownian motion. Here we consider a one dimensional parabolic Anderson model which is continuous in space and includes a more general rough noise. We first show that the equation admits a solution and that it is unique under some regularity assumptions on the initial condition. In addition, we show that it can be represented using the Feynman-Kac formula, thus providing a connection with the SPDE and a stochastic process, in this case a Brownian motion. The bulk of our study is devoted to explore the large time behavior of the solution, and we provide an explicit formula for the asymptotic behavior of the logarithm of the solution.<br></div><div><br></div><div>3. Heat conduction in semiconductors: Standard heat flow, at a macroscopic level, is modeled by the random erratic movements of Brownian motions starting at the source of heat. However, this diffusive nature of heat flow predicted by Brownian motion is not observed in certain materials (semiconductors, dielectric solids) over short length and time scales. The thermal transport in these materials is more akin to a super-diffusive heat flow, and necessitates the need for processes beyond Brownian motion to capture this heavy tailed behavior. In this context, we propose the use of a well-defined Lévy process, the so-called relativistic stable process to better model the observed phenomenon. This process captures the observed heat dynamics at short length-time scales and is also closely related to the relativistic Schrödinger operator. In addition, it serves as a good candidate for explaining the usual diffusive nature of heat flow under large length-time regimes. The goal is to verify our model against experimental data, retrieve the best parameters of the process and discuss their connections to material thermal properties.<br></div><div><br></div><div>4. Bond-pricing under partial information: We study an information asymmetry problem in a bond market. Especially we derive bond price dynamics of traders with different levels of information. We allow all information processes as well as the short rate to have jumps in their sample paths, thus representing more dramatic movements. In addition we allow the short rate to be modulated by all information processes in addition to having instantaneous feedbacks from the current levels of itself. A fully informed trader observes all information which affects the bond price while a partially informed trader observes only a part of it. We first obtain the bond price dynamic under the full information, and also derive the bond price of the partially informed trader using Bayesian filtering method. The key step is to perform a change of measure so that the dynamic under the new measure becomes computationally efficient.</div>

Motivated by the results concerning the regularity of solutions to the fractional Navier-Stokes system and questions about the influence of noise on the formation of singularities in hydrodynamic models, we have explored these two problems in the context of the fractional 1D Burgers equation. First, we performed highly accurate numerical computations to characterize the dependence of the blow-up time on the the fractional dissipation exponent in the supercritical regime. The problem was solved numerically using a pseudospectral method where integration in time was performed using a hybrid method combining the Crank-Nicolson and a three-step Runge-Kutta techniques. A highlight of this approach is automated resolution refinement. The blow-up time was estimated based on the time evolution of the enstrophy (H1 seminorm) and the width of the analyticity strip. The consistency of the obtained blow-up times was verified in the limiting cases. In the second part of the thesis we considered the fractional Burgers equation in the presence of suitably colored additive noise. This problem was solved using a stochastic Runge-Kutta method where the stochastic effects were approximated using a Monte-Carlo method. Statistic analysis of ensembles of stochastic solutions obtained for different noise magnitudes indicates that as the noise amplitude increases the distribution of blow-up times becomes non-Gaussian. In particular, while for increasing noise levels the mean blow-up time is reduced as compared to the deterministic case, solutions with increased existence time also become more likely. / Thesis / Master of Science (MSc)

Dans cette thèse, on s'intéresse à l'étude qualitative des solutions d'équations aux dérivées partielles hamiltoniennes par le biais de la théorie des mesures invariantes. L'existence d'une telle mesure pour une EDP fournit, en effet, des informations sur sa dynamique en temps long. Nous étudierons deux situations quelque peu "extrémales". Dans une première partie, nous nous intéressons aux équations ayant une infinité de lois de conservation et dans une seconde, aux équations dont on ne connaît qu'une seule loi de conservation non triviale.Nous étudions les premières équations par le biais de l'équation de Benjamin-Ono. Il s'agit d'un modèle de description des ondes internes dans un fluide de grande profondeur.Nous nous intéressons à la dynamique de cette équation sur l'espace C^infty(T) en lui construisant une mesure invariante sur cet espace. Par conséquent, une propriété de récurrence presque sûre (par rapport à cette mesure) est établie pour les solutions infiniment lisses de cette équation. Nous prouvons, ensuite, des propriétés de non-dégénérescence pour cette mesure. En effet, nous montrons que, via cette mesure, une infinité de fonctionnelles indépendantes ont des distributions absolument continues par rapport à la mesure de Lebesgue sur R. Enfin, nous montrons que cette mesure est de nature au moins $2$-dimensionnelle. Dans ce travail, nous avons utilisé l'approche Fluctuation-Dissipation-Limite (FDL) introduite par Kuksin-Shirikyan. Notons qu'une propriété de récurrence presque sûre a été établie pour les solutions de régularité Sobolev de l'équation de Benjamin-Ono, dans les travaux de Deng, Tzvetkov et Visciglia.Dans l'autre partie de la thèse, nous abordons l'équation de Klein-Gordon à non-linéarité cubique, c'est un exemple d'EDPs hamiltoniennes pour lesquelles il n'est connu qu'une seule loi de conservation non triviale. Cette équation modélise l'évolution d'une particule massive relativiste. Ici, nous considérons les cas où l'équation est posée sur le tore tri-dimensionnel ou sur un domaine borné de R^3 à bord assez régulier. Nous lui construisons une mesure invariante concentrée sur l'espace de Sobolev H^2, en utilisant toujours l'approche FDL. Un autre aspect de ce travail est d'étendre le cadre de cette approche au contexte des EDPs à une seule loi de conservation, en effet, dans les travaux antérieurs, l'approche FDL avait nécessité deux lois de conservation pour fonctionner. Puis nous établissons une propriété de non-dégénérescence pour la mesure construite. Par conséquent, une propriété de récurrence presque sûre, par rapport à la mesure construite, est prouvée. Notons que des travaux antérieurs dus à Burq-Tzvetkov, de Suzzoni, Bourgain-Bulut et Xu ont traité la question de mesure de Gibbs invariante pour des équations des ondes dans un contexte radial. / In this thesis, we are concerned with the qualitative study of solutions of Hamiltonian partial differential equations by the way of the invariant measures theory. Indeed, existence of such a measure provides some informations concerning the large time dynamics of the PDE in question. In this thesis we treat two "extremal" situations. In the first part, we consider equations with infinitely many conservation laws, and in the second, we study equations for which we know only one non-trivial conservation law.We study the first equations by considering the Benjamin-Ono equation. The latter is a model describing internal waves in a fluide of great depth.We are concerned with the dynamics of that equation on the space C^infty(T) by constructing for it an invariant measure on that space. Accordingly, an almost sure (w.r.t. this measure) recurrence property is established for infinitely smooth solutions of that equation. Then, we prove qualitative properties for the constructed measure by showing that there are infinitely many independent observables whose distributions via this measure are absolutely continuous w.r.t. the Lebesgue measure on R. Moreover, we establish that the measure is of at least 2-dimensional nature. In this work, we used the Fluctuation-Dissipation-Limit (FDL) approach introduced by Kuksin and Shirikyan. Notice that an almost sure recurrence property for the Benjamin-Ono equation was established on Sobolev spaces by Deng, Tzvetkov and Visciglia.In the second part of the thesis, we consider the cubic Klein-Gordon equation, which is an example of Hamiltonian PDEs for which we know only one conservation law. This equation models the evolution of a massive relativistic particle. Here, we consider both the case of the tri-dimensional periodic solutions and those defined on a bounded domain of R^3. In both settings, we construct an invariant measure concentrated on the Sobolev space H^2xH^1, again with use of the FDL approach. Another aspect of this work is to extend the FDL approach to the context of PDEs having only one conservation law; indeed, in previous works, this approach required two conservation laws. Qualitative properties for the measure and almost sure (w.r.t. this measure) recurrence for H^2-solutions are proven. Notice that previous works by Burq-Tzvetkov, de Suzzoni, Bourgain-Bulut and Xu have treated the invariant Gibbs measure problem in the radial symmetry context for waves equations.

Cette thèse s’inscrit dans le domaine mathématique de l’analyse des équations aux dérivées partielles (EDP) non-linéaires stochastiques. Nous nous intéressons à des EDP paraboliques et hyperboliques que l’on perturbe stochastiquement au sens d’Itô. Il s’agit d’introduire l’aléatoire via l’ajout d’une intégrale stochastique (intégrale d’Itô) qui peut dépendre ou non de la solution, on parle alors de bruit multiplicatif ou additif. La présence de la variable de probabilité ne nous permet pas d’utiliser tous les outils classiques de l’analyse des EDP. Notre but est d’adapter les techniques connues dans le cadre déterministe aux EDP non linéaires stochastiques en proposant des méthodes alternatives. Les résultats obtenus sont décrits dans les cinq chapitres de cette thèse : Dans le Chapitre I, nous étudions une perturbation stochastique des équations de Barenblatt. En utilisant une semi- discrétisation implicite en temps, nous établissons l’existence et l’unicité d’une solution dans le cas additif, et grâce aux propriétés de la solution nous sommes en mesure d’étendre ce résultat au cas multiplicatif à l’aide d’un théorème de point fixe. Dans le Chapitre II, nous considérons une classe d’équations de type Barenblatt stochastiques dans un cadre abstrait. Il s’agit là d’une généralisation des résultats du Chapitre I. Dans le Chapitre III, nous travaillons sur l’étude du problème de Cauchy pour une loi de conservation stochastique. Nous montrons l’existence d’une solution par une méthode de viscosité artificielle en utilisant des arguments de compacité donnés par la théorie des mesures de Young. L’unicité repose sur une adaptation de la méthode de dédoublement des variables de Kruzhkov.. Dans le Chapitre IV, nous nous intéressons au problème de Dirichlet pour la loi de conservation stochastique étudiée au Chapitre III. Le point remarquable de l’étude repose sur l’utilisation des semi-entropies de Kruzhkov pour montrer l’unicité. Dans le Chapitre V, nous introduisons une méthode de splitting pour proposer une approche numérique du problème étudié au Chapitre IV, suivie de quelques simulations de l’équation de Burgers stochastique dans le cas unidimensionnel. / This thesis deals with the mathematical field of stochastic nonlinear partial differential equations’ analysis. We are interested in parabolic and hyperbolic PDE stochastically perturbed in the Itô sense. We introduce randomness by adding a stochastic integral (Itô integral), which can depend or not on the solution. We thus talk about a multiplicative noise or an additive one. The presence of the random variable does not allow us to apply systematically classical tools of PDE analysis. Our aim is to adapt known techniques of the deterministic setting to nonlinear stochastic PDE analysis by proposing alternative methods. Here are the obtained results : In Chapter I, we investigate on a stochastic perturbation of Barenblatt equations. By using an implicit time discretization, we establish the existence and uniqueness of the solution in the additive case. Thanks to the properties of such a solution, we are able to extend this result to the multiplicative noise using a fixed-point theorem. In Chapter II, we consider a class of stochastic equations of Barenblatt type but in an abstract frame. It is about a generalization of results from Chapter I. In Chapter III, we deal with the study of the Cauchy problem for a stochastic conservation law. We show existence of solution via an artificial viscosity method. The compactness arguments are based on Young measure theory. The uniqueness result is proved by an adaptation of the Kruzhkov doubling variables technique. In Chapter IV, we are interested in the Dirichlet problem for the stochastic conservation law studied in Chapter III. The remarkable point is the use of the Kruzhkov semi-entropies to show the uniqueness of the solution. In Chapter V, we introduce a splitting method to propose a numerical approach of the problem studied in Chapter IV. Then we finish by some simulations of the stochastic Burgers’ equation in the one dimensional case.