Ces travaux de thèse ont pour but d'étudier l'état de dispersion de particules colloïdales dans des cristaux liquides nématiques lyotropes. Ces solvants organisés sont constitués de micelles nanométriques anisotropes. Dans un premier temps, nous montrons qu'il est possible de réaliser des suspensions cinétiquement stables en jouant notamment sur la forme des inclusions micrométriques. Un modèle, développé dans le cadre de cette étude, permet de rendre compte de nos observations. Dans un second temps, nous nous intéressons à l'influence de la diminution de taille de particules sur l'état de dispersion du système. A l'échelle nanométrique, le mouvement brownien, anisotrope dans ce type de milieu, semble gouverner les phénomènes observés. / The present PhD work aims at studying the dispersion state of colloidal particles in lyotropic nematic liquid crystals. These organized solvents are made of anisotropic nanometric micelles. Firstly, we show that kinetically stable suspensions may be achieved by playing on the shape of micrometric inclusions in particular. A model, which is developed for this study, can catch well our observations. Secondly, we are interested in the influence of a diminution of the particle size on the dispersion state of the system. At the nanometric scale, the Brownian motion, which is anisotropic in such media, seems to govern the observed phenomena.

Cette thèse est consacrée à l'étude asymptotique d'objets liés au mouvement brownien sur le groupe unitaire en grande dimension, ainsi qu'à l'étude, dans le cadre des probabilités libres, des versions non-commutatives de ces objets. Elle se subdivise essentiellement en trois parties.Dans le chapitre 2, nous résolvons le problème initial de cette thèse, à savoir la convergence de la transformation de Hall sur le groupe unitaire vers la transformation de Hall libre, lorsque la dimension tend vers l'infini. Pour résoudre ce problème, nous établissons des théorèmes d'existence de noyaux de transition pour la convolution libre. Enfin, nous utilisons ces résultats pour prouver que, pareillement au mouvement brownien sur le groupe unitaire, le mouvement brownien sur le groupe linéaire converge en distribution non-commutative vers sa version libre. Nous étudions les fluctuations autour de cette convergence dans le chapitre 3. Le chapitre 4 présente un morphisme entre les mesures infiniment divisibles pour la convolution libre additive d'une part et multiplicative de l'autre. Nous montrons que ce morphisme possède une version matricielle qui s'appuie sur un nouveau modèle de matrices aléatoires pour les processus de Lévy libres multiplicatifs. / This thesis focuses on the asymptotic of objects related to the Brownian motion on the unitary group in large dimension, and on the study, in free probability, of the non-commutative versions of those objects. It subdivides into essentially three parts.In Chapter 2, we solve the original problem of this thesis: the convergence of the Hall transform on the unitary group to the free Hall transform, as the dimension tends to infinity. To solve this problem, we establish theorems of existence of transition kernel for the free convolution. Finally, we use these results to prove that, exactly as the Brownian motion on the unitary group, the Brownian motion on the linear group converges in noncommutative distribution to its free version. Then we study the fluctuations around this convergence in Chapter 3. Chapter 4 presents a homomorphism between infinitely divisible measures for the free convolution, in respectively the additive case and the multiplicative case. We show that this homomorphism has a matricialversion which is based on a new model of random matrices for the free multiplicative Lévy processes.

In this diploma thesis a general purpose application was developed in order to analyse economic data emitted by the Prague Stock Exchange. The application was written in the Maple programming language. The purpose of this application is to simulate possible future development of the securities. The main part of the application is a user-friendly graphical user interface.

Diplomová práce se zabývá problematikou obyčejných stochastických diferenciálních rovnic. Po souhrnu teorie stochastických procesů, zejména tzv. Brownova pohybu je zaveden stochastický Itôův integrál, diferenciál a tzv. Itôova formule. Poté je definováno řešení počáteční úlohy stochastické diferenciální rovnice a uvedena věta o existenci a jednoznačnosti řešení. Pro případ lineární rovnice je odvozen tvar řešení a rovnice pro jeho střední hodnotu a rozptzyl. Závěr tvoří rozbor vybraných rovnic.

京都大学 / 0048 / 新制・課程博士 / 博士(理学) / 甲第19468号 / 理博第4128号 / 新制||理||1594(附属図書館) / 32504 / 京都大学大学院理学研究科数学・数理解析専攻 / (主査)准教授 矢野 孝次, 教授 上田 哲生, 教授 重川 一郎 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM

No description available.

No description available.

In this thesis we investigate large deviation and path properties of a few models within the Kardar-Parisi-Zhang (KPZ) universality class. The KPZ equation is the central object in the KPZ universality class. It is a stochastic PDE describing various objects in statistical mechanics such as random interface growth, directed polymers, interacting particle systems. In the first project we study one point upper tail large deviations of the KPZ equation 𝜢(t,x) started from narrow wedge initial data. We obtain precise expression of the upper tail LDP in the long time regime for the KPZ equation. We then extend our techniques and methods to obtain upper tail LDP for the asymmetric exclusion process model, which is a prelimit of the KPZ equation. In the next direction, we investigate temporal path properties of the KPZ equation. We show that the upper and lower law of iterated logarithms for the rescaled KPZ temporal process occurs at a scale (log log 𝑡)²/³ and (log log 𝑡)¹/³ respectively. We also compute the exact Hausdorff dimension of the upper level sets of the solution, i.e., the set of times when the rescaled solution exceeds 𝛼(log log 𝑡)²/³. This has relevance from the point of view of fractal geometry of the KPZ equation. We next study superdiffusivity and localization features of the (1+1)-dimensional continuum directed random polymer whose free energy is given by the KPZ equation. We show that for a point-to-point polymer of length 𝑡 and any 𝑝 ⋲ (0,1), the point on the path which is 𝑝𝑡 distance away from the origin stays within a 𝑂(1) stochastic window around a random point 𝙈_𝑝,𝑡 that depends on the environment. This provides an affirmative case of the folklore `favorite region' conjecture. Furthermore, the quenched density of the point when centered around 𝙈_𝑝,𝑡 converges in law to an explicit random density function as 𝑡 → ∞ without any scaling. The limiting random density is proportional to 𝑒^{-𝓡(x)} where 𝓡(x) is a two-sided 3D Bessel process with diffusion coefficient 2. Our proof techniques also allow us to prove properties of the KPZ equation such as ergodicity and limiting Bessel behaviors around the maximum. In a follow up project, we show that the annealed law of polymer of length 𝑡, upon 𝑡²/³ superdiffusive scaling, is tight (as 𝑡 → ∞) in the space of 𝐶([0,1]) valued random variables. On the other hand, as 𝑡 → 0, under diffusive scaling, we show that the annealed law of the polymer converges to Brownian bridge. In the final part of this thesis, we focus on an integrable discrete half-space variant of the CDRP, called half-space log-gamma polymer.We consider the point-to-point log-gamma polymer of length 2𝑁 in a half-space with i.i.d.Gamma⁻¹(2𝛳) distributed bulk weights and i.i.d. Gamma⁻¹(𝛼+𝛳) distributed boundary weights for 𝛳 > 0 and 𝛼 > -𝛳. We establish the KPZ exponents (1/3 fluctuation and 2/3 transversal) for this model when 𝛼 ≥ 0. In particular, in this regime, we show that after appropriate centering, the free energy process with spatial coordinate scaled by 𝑁²/³ and fluctuations scaled by 𝑁¹/³ is tight. The primary technical contribution of our work is to construct the half-space log-gamma Gibbsian line ensemble and develop a toolbox for extracting tightness and absolute continuity results from minimal information about the top curve of such half-space line ensembles. This is the first study of half-space line ensembles. The 𝛼 ≥ 0 regime correspond to a polymer measure which is not pinned at the boundary. In a companion work, we investigate the 𝛼 < 0 setting. We show that in this case, the endpoint of the point-to-line polymer stays within 𝑂(1) window of the diagonal. We also show that the limiting quenched endpoint distribution of the polymer around the diagonal is given by a random probability mass function proportional to the exponential of a random walk with log-gamma type increments.

<p>The goal of this thesis is to establish generalized isoperimetric inequalities on first exit time distributions as well as expectations of L\'evy processes.</p> <p>Firstly, we prove inequalities on first exit time distributions in the case that the L\'evy process is an $\alpha$-stable symmetric process $A_t$ on $\R^d$, $\alpha\in(0,2]$. Given $A_t$ and a bounded domain $D\subset\R^d$, we present a proof, based on the classical Brascamp-Lieb-Luttinger inequalities for multiple integrals, that the distribution of the first exit time of $A_t$ from $D$ increases under Steiner symmetrization. Further, it is shown that when a sequence of domains $\{D_m\}$ each contained in a ball $B\subset\R^d$ and satisfying the $\varepsilon$-cone property converges to a domain $D'$ with respect to the Hausdorff metric, the sequence of distributions of first exit times for Brownian motion from  $D_m$  converges to the distribution of the exit time of Brownian motion from $D'$. The second set of results in this thesis extends the theorems from \cite{BanMen} by proving generalized isoperimetric inequalities on expectations of L\'evy processes in the case of Steiner symmetrization.% using the Brascamp-Lieb-Luttinger inequalities used above. </p> <p>These results will then be used to establish inequalities involving distributions of first exit times of $\alpha$-stable symmetric processes $A_t$ from triangles and quadrilaterals. The primary application of these inequalities is verifying a conjecture from Ba\~nuelos for these planar domains. This extends a classical result of P\'olya and Szeg\"o to the fractional Laplacian with Dirichlet boundary conditions.</p>

No description available.