In this thesis, we discuss some topics in random matrix theory which have applications to probability, statistics and quantum information theory. In Chapter 2, by relying on the spectral properties of an associated adjacency matrix, we find the distribution of the maximum of a Dyck path and show that it has the same distribution function as the unsigned Brownian excursion which was first derived in 1976 by Kennedy. We obtain a large and moderate deviation principle for the law of the maximum of a random Dyck path. Our result extends the results of Chung, Kennedy and Khorunzhiy and Marckert. In Chapter 3, we discuss a method of sampling called the Gibbs-slice sampler. This method is based on Neal's slice sampling combined with Gibbs sampling. In Chapter 4, we discuss several examples which have applications in physics and quantum information theory.

Orientador: Gerson Francisco / Banca: Victo dos Santos Filho / Banca: Fernando Manoel Ramos / Resumo: Realizamos neste trabalho uma exposição geral da Teoria do Movimento Browniano, desde suas primeiras observações, feitas no âmbito da Biologia, até sua completa descrição seundo as leis da Mecânica estatística, formulação esta efetuada por Einstein em 1905. Com base nestes princípios físicos analisamos a Teoria do Movimento Browniano de Einstein como sendo um processo estocástico, o que permite sua generalização para um processo de Lévy. Fazemos uma exposição da Teoria de Lévy, e aplicamo-la em seguida na análise de dados provenientes do índice IBOVESPA. Camparamos os resultados com as distribuições empíricas e a modelada via distribuição gaussiana, demonstrando efetivamente que a série financeira analisada apresenta um comportamento não-gaussiano. / Abstracts: We review in this work the foundations of the Theory of Brownian Motion, from the first observations made in Biology to its complete description according to the laws of Statistical Mechanics performed by einstein in 1905. Afterwards we discuss the Einstein's Theory of Brownian Motion as a stochastic process, since this connection allows its generalization to a Lévy process. After a brief review of Lévy Theory we analyse IBOVESPA data within this framework. We compare the outcomes with the empirical and gaussian distributions, showing effectively that the analyzed financial series behaves exactly as a non-gaussian stochastic process. / Mestre

L'objet de cette thèse est l'étude quantitative du mouvement des particules intracellulaires, comme les protéines ou les molécules. L'estimation du mouvement des particules au sein de la cellule est en effet d'un intérêt majeur en biologie cellulaire puisqu'il permet de comprendre les interactions entre les différents composants de la cellule. Dans cette thèse, nous modélisons les trajectoires des particules avec des processus stochastiques puisque le milieu intra-cellulaire est soumis à de nombreux aléas. Les diffusions, des processus à trajectoires continues, permettent de modéliser un large panel de mouvements intra-cellulaires. Les biophysiciens distinguent trois principaux types de diffusion: le mouvement brownien, la super-diffusion et la sous-diffusion. Ces différents types de mouvement correspondent à des scénarios biologiques distincts. Le déplacement d'une particule évoluant sans contrainte dans le cytosol ou dans le plasma membranaire est modélisée par un mouvement brownien; la particule ne se déplace pas dans une direction précise et atteint sa destination en un temps long en moyenne. Les particules peuvent aussi être propulsées par des moteurs moléculaires le long des microtubules et filaments d'actine du cytosquelette de la cellule. Leur mouvement est alors modélisé par des super-diffusions. Enfin, la sous-diffusion peut être observée dans deux situations: i/ lorsque la particule est confinée dans un micro domaine, ii/ lorsqu’elle est ralentie par l'encombrement moléculaire et doit se frayer un chemin parmi des obstacles mobiles ou immobiles. Nous présentons un test statistique pour effectuer la classification des trajectoires en trois groupes: brownien, super-diffusif et sous-diffusif. Nous développons également un algorithme pour détecter les ruptures de mouvement le long d’une trajectoire. Nous définissons les temps de rupture comme les instants où la particule change de régime de diffusion (brownien, sous-diffusif ou super-diffusif). Enfin, nous associons une méthode de regroupement avec notre procédure de test pour identifier les micro domaines dans lesquels des particules sont confinées. De telles zones correspondent à des lieux d’interactions moléculaires dans la cellule. / In this thesis, we are interested in quantifying the dynamics of intracellular particles, as proteins or molecules, inside living cells. In fact, inference on the modes of mobility of molecules is central in cell biology since it reflects the interactions between the structures of the cell. We model the particle trajectories with stochastic processes as the interior of a living cell is a fluctuating environment. Diffusions are stochastic processes with continuous paths and can model a large range of intracellular movements. Biophysicists distinguish three main types of diffusions, namely Brownian motion, superdiffusion and subdiffusion. These different diffusion processes correspond to distinct biological scenarios. A particle evolving freely inside the cytosol or along the plasma membrane is modelled by Brownian motion; the particle does not travel along any particular direction and can take a very long time to go to a precise area in the cell. Active intracellular transport can overcome this difficulty so that motion is faster and direct specific. In this case, particles are carried by molecular motors along microtubular filament networks and their motion is modelled with superdiffusions. Subdiffusion can be observed in two cases i/ when the particle is confined in a microdomain, ii/ when the particle is hindered by molecular crowding and encounters dynamic or fixed obstacles. We develop a statistical test for classifying the observed trajectories into the three groups of diffusion of interest namely Brownian motion, super-diffusion and subdiffusion. We also design an algorithm to detect the changes of dynamics along a single trajectory. We define the change points as the times at which the particle switches from one diffusion type (Brownian motion, superdiffusion or subdiffusion) to another. Finally, we combine a clustering algorithm with our test procedure to identify micro domains that is zones where the particles are confined. Molecular interactions of great importance for the functioning of the cell take place in such areas.

On introduit le concept d'une variété aléatoire stationnaire avec l'objectif de traiter de façon unifiée les résultats sur les variétés avec un group d'isométries transitif, les variétés avec quotient compact, et les feuilles génériques d'un feuilletage compact. On démontre des inégalités entre la vitesse de fuite, l'entropie du mouvement brownien et la croissance de volume de la variété aléatoire, en généralisant des résultats d'Avez, Kaimanovich, et Ledrappier. Dans la deuxième partie on démontre que la fonction feuille d'un feuilletage compact est semicontinue, en obtenant comme conséquences le théorème de stabilité local de Reeb, une partie du théorème de structure local pour les feuilletages à feuilles compactes d'Epstein, et un théorème de continuité d'Álvarez et Candel. / We introduce the concept of a stationary random manifold with the objective of treating in a unified way results about manifolds with transitive isometry group, manifolds with a compact quotient, and generic leaves of compact foliations. We prove inequalities relating linear drift and entropy of Brownian motion with the volume growth of such manifolds, generalizing previous work by Avez, Kaimanovich, and Ledrappier among others. In the second part we prove that the leaf function of a compact foliation is semicontinuous, obtaining as corollaries Reeb's local stability theorem, part of Epstein's the local structure theorem for foliations by compact leaves, and a continuity theorem of Álvarez and Candel.

La première partie de cette thèse s’inscrit dans le domaine des cartes aléatoires, qui est un sujet à la frontière des probabilités, de la combinatoire et de la physique statistique. Nos travaux complètent une série de résultats de convergence de différents modèles de cartes aléatoires vers la carte brownienne, qui est un espace métrique compact aléatoire. Plus précisément, on montre que la limite d’échelle d’une carte de loi uniforme sur l’ensemble des cartes biparties enracinées à n arêtes, munie de la distance de graphe renormalisée par (2n)^(−1/4), est, au sens de Gromov–Hausdorff, la carte brownienne. Pour prouver ce résultat, les arguments importants sont d’une part l’utilisation d’une bijection combinatoire entre cartes biparties et arbres multitypes, et d’autre part des théorèmes de convergence pour les arbres de Galton–Watson multitypes étiquetés. Dans un deuxième temps, le but est de présenter une théorie des excursions pour le mouvement brownien indexé par l’arbre brownien. De manière analogue à la théorie d’Itô des excursions pour le mouvement brownien, chaque excursion correspond à une composante connexe du complémentaire des zéros du mouvement brownien indexé par l’arbre, et l’excursion est définie comme un processus indexé par un arbre continu. On explique comment mesurer la longueur de la frontière de ces excursions, de sorte que la famille de ces longueurs coïncide avec les sauts d’un processus de branchement à temps continu de mécanisme de branchement stable d’indice 3/2. De plus, conditionnellement aux longueurs des frontières, les excursions sont indépendantes et leur loi conditionnelle est déterminée à l’aide d’une mesure d’excursion explicite que l’on introduit et décrit. Dans ce travail, le serpent brownien apparaît comme un outil particulièrement important. / The first part of this thesis concerns the area of random maps, which is a topic in between probability theory, combinatorics and statistical physics. Our work complements several results of convergence of various classes of random maps to the Brownian map, which is a random compact metric space. More precisely, we prove that the scaling limit of a map which is uniformly distributed over the class of rooted planar maps with n edges, equipped with the graph distance rescaled by (2n)^(−1/4), is, in the Gromov-Hausdorff sense, the Brownian map. To establish this result, the main arguments are the use of a combinatorial bijection between bipartite maps and multitype trees, together with convergence theorems for Galton-Watson multitype trees. We then aim to develop an excursion theory for Brownian motion indexed by the Brownian tree. Analogous to the Itô excursion theory for Brownian motion, each excursion corresponds to a connected component of the complement of the zero set of the tree-indexed Brownian motion, and the excursion is defined as a process indexed by a continuous tree. We explain how to measure the length of the boundary of these excursions, in a way that the collection of these lengths coincides with the collection of jumps of a continuous-state branching process with a 3/2-stable branching mechanism. Moreover, conditionally on the boundary lengths, the excursions are independent and their conditional distribution is determined in terms of an excursion measure that we introduce and study. In this work, the Brownian snake appears as a particularly important tool.

The change point problem is a problem where a process changes regimes because a parameter changes at a point in time called the change point. The objective of this problem is to estimate the change point and each of the parameters of the stochastic process. In this thesis, we examine the change point problem for two classes of stochastic processes. First, we consider the volatility change point problem for stochastic diffusion processes driven by Brownian motions. Then, we consider the drift change point problem for Ornstein-Uhlenbeck processes driven by _-stable Levy motions. In each problem, we establish the consistency of the estimators, determine asymptotic behavior for the changing parameters, and finally, we perform simulation studies to computationally assess the convergence of parameters. / Includes bibliography. / Dissertation (Ph.D.)--Florida Atlantic University, 2020. / FAU Electronic Theses and Dissertations Collection

京都大学 / 新制・課程博士 / 博士(理学) / 甲第22977号 / 理博第4654号 / 新制||理||1669(附属図書館) / 京都大学大学院理学研究科数学・数理解析専攻 / (主査)教授 日野 正訓, 教授 泉 正己, 准教授 楠岡 誠一郎 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM

This thesis investigates several phenomena that are associated with (self-)thermophoretic Janus particles with hemispheres made from different materials serving as a paradigm for active propul- sion on the microscale. (i) The dynamics of a single Janus sphere in the external temperature field created by an immobilized heat source is studied. I show that the particle’s angular velocity is solely determined by the temperature profile on the equator between the Janus particle’s hemispheres and their phoretic mobility contrast. (ii) The distinct polarization-density patterns observed for active-particle suspensions in activity landscapes are addressed. The results of my approximate theoretical model agree well with exact numerical and measurement data for a thermophoretic microswimmer, and can serve as a template for more complex applications. The essential physics behind the formal results is robustly captured and elucidated by a schematic two-species “run- and-tumble” model. (iii) I investigate coarse-grained models of suspended self-thermo- phoretic microswimmers. Starting from atomistic molecular dynamics simulations, the coarse-grained de- scription of the fluid in terms of a local molecular temperature field is verified, and effective nonequilibrium temperatures characterizing the particle’s so called hot Brownian motion are mea- sured from simulations. They are theoretically shown to remain relevant for any further spatial coarse-graining towards a hydrodynamic description of the entire suspension as a homogeneous complex fluid. / In dieser Arbeit untersuche ich mehrere Phänomene, die im Zusammenhang mit (selbst-)thermo- phoretischen Janusteilchen auftreten. Diese Teilchen bestehen aus zwei Halbkugeln mit unter- schiedlichen Materialeigenschaften und dienen in dieser Arbeit als Musterbeispiel für aktive Fort- bewegung auf der Mikroskala. (i) Die Dynamik eines einzelnen Janusteilchens im externen Temper- aturfeld einer ortsfesten Heizquelle wird untersucht. Es wird gezeigt, dass die Winkelgeschwindigkeit des Teilchens ausschließlich durch das Temperaturprofil am Äquator zwischen den Hemisphären des Janusteilchens und dem Unterschied ihrer phoretischen Mobilitäten bestimmt wird. (ii) Ich befasse mich mit den charakteristischen Polarisations- und Dichteprofilen, die für aktive Teilchen in Aktivitätslandschaften beobachtet werden. Die Ergebnisse meines approximativen theoretis- chen Modells stimmen gut mit exakten numerischen Lösungen und Messdaten für einen ther- mophoretischen Mikroschwimmer überein und können als Vorlage für komplexere Anwendungen dienen. Die wesentliche Physik hinter den formalen Ergebnissen wird durch ein schematisches Zwei-Spezies-“Run-and-Tumble”-Modell erfasst und erklärt. (iii) Ich untersuche Coarse-Graining- Modelle von suspendierten selbst-thermophoretischen Mikroschwimmern. Ausgehend von atom- istischen molekulardynamischen Simulationen wird die grobkörnige (coarse-grained) Beschreibung des Fluids in Form eines lokalen molekularen Temperaturfeldes verifiziert. Anschließend berechne ich effektive Nichtgleichgewichtstemperaturen, die die sogenannte heiße Brownsche Bewegung der Teilchen charakterisieren, und vergleiche diese mit Simulationsdaten. Es wird gezeigt, dass diese effektiven Temperaturen für jede weitere räumliche Vergröberung hin zu einer hydrodynamischen Beschreibung der gesamten Suspension als homogenes komplexes Fluid relevant bleiben.

This dissertation presents and explores a technique to confine and manipulate single and multiple nano-objects in solution by exploiting the thermophoretic interactions with local temperature gradients. The method named thermophoretic trap uses an all-optically controlled heating via plasmonic absorption by a gold nano-structure designed for this purpose. The dissipation of absorbed laser light to thermal energy generates a localized temperature field. The spatial localization of the heat source thereby leads to strong temperature gradients that are used to drive a particle or molecule into a desired direction. The behavior of nano-objects confined by thermal inhomogeneities is explored experimentally as well as theoretically. The monograph treats three major experimental stages of development, which essentially differ in the way the heating laser beam is shaped and controlled. In a first generation, a static heating of an appropriate gold structure is used to induce a steady temperature profile that exhibits a local minimum in which particles can be confined. This simple realization illustrates the working principle best. In a second step, the static heating is replaced. A focused laser beam is used to heat a smaller spatial region. In order to confine a particle, the beam is steered in circles along a circular gold structure. The trapping dynamics are studied in detail and reveal similarities to the well-established Paul trap. The largest part of the thesis is dedicated to the third generation of the trap. While the hardware is identical to the second generation, using the real-time information on the position of the trapped object to heat only particular sites of the gold structure strongly increases the efficiency of the trap compared to the earlier versions. Beyond that, the optical feedback control allows for an active shaping of the effective virtual trapping potential by applying modified feedback rules, including e.g. a double-well or a box-like potential. This transforms the formerly pure trapping device to a versatile technique for micro and nano-fluidic manipulation. The physical and technical contributions to the limits of the method are explored. Finally, the feasibility of trapping single macro-molecules is demonstrated by the confinement of lambda-DNA for extended time periods over which the molecules center-of-mass motion as well as its conformational dynamics can be studied.

In the thesis, multivariate fractional Brownian motions with possibly different Hurst indices in different coordinates are considered and a Girsanov-type theo- rem for these processes is shown. Two applications of this theorem to stochastic differential equations driven by multivariate fractional Brownian motions (SDEs) are given. Firstly, the existence of a weak solution to an SDE with a drift coeffi- cient that can be written as a sum of a regular and a singular part and a diffusion coefficient that is dependent on time and satisfies suitable conditions is shown. The results are applied for the proof of existence of a weak solution of an equation describing stochastic harmonic oscillator. Secondly, the Girsanov-type theorem is used to find the maximum likelihood scalar estimator that appears in the drift of an SDE with additive noise. 1