Title: Stochastic evolution equations with multiplicative fractional noise Author: Jana Šnupárková Departement: Department of Probability and Mathematical Statistics Supervisor: prof. RNDr. Bohdan Maslowski, DrSc. Supervisor's e-mail address: maslow@karlin.mff.cuni.cz Abstract: The fractional Gaussian noise is a formal derivative of a fractional Brownian motion with Hurst parameter H ∈ (0, 1). An explicit formula for a solution to stochastic differential equations with a multiplicative fractional Gaussian noise in a separable Hilbert space is given. The large time behaviour of the solution is studied. In addition, equations of this type with a nonlinear perturbation of a drift part are investigated in the case H > 1/2. Keywords: Fractional Brownian Motion, Stochastic Differential Equations in Hilbert Space, Explicit Formula for Solution

Stochastic Evolution Equations Petr Čoupek Doctoral Thesis Abstract Linear stochastic evolution equations with additive regular Volterra noise are studied in the thesis. Regular Volterra processes need not be Gaussian, Markov or semimartingales, but they admit a certain covariance structure instead. Particular examples cover the fractional Brownian motion of H > 1/2 and, in the non-Gaussian case, the Rosenblatt process. The solution is considered in the mild form, which is given by the variation of constants formula, and takes values either in a separable Hilbert space or the space Lp(D, µ) for large p. In the Hilbert-space setting, existence, space-time regularity and large-time behaviour of the solutions are studied. In the Lp setting, existence and regularity is studied, and in concrete cases of stochastic partial differential equations, the solution is shown to be a space-time continuous random field.

>Magister Scientiae - MSc / We present various methods of pricing Asian options. The methods include Monte Carlo simulations designed using control and antithetic variates, numerical solution of partial differential equation and using lower bounds.The price of the Asian option is known to be a certain risk-neutral expectation. Using the Feynman-Kac theorem, we deduce that the problem of determining the expectation implies solving a linear parabolic partial differential equation. This partial differential equation does not admit explicit solutions due to the fact that the distribution of a sum of lognormal variables is not explicit. We then solve the partial differential equation numerically using finite difference and Monte Carlo methods.Our Monte Carlo approach is based on the pseudo random numbers and not deterministic sequence of numbers on which Quasi-Monte Carlo methods are designed. To make the Monte Carlo method more effective, two variance reduction techniques are discussed.Under the finite difference method, we consider explicit and the Crank-Nicholson’s schemes. We demonstrate that the explicit method gives rise to extraneous solutions because the stability conditions are difficult to satisfy. On the other hand, the Crank-Nicholson method is unconditionally stable and provides correct solutions. Finally, we apply the pricing methods to a similar problem of determining the price of a European-style arithmetic basket option under the Black-Scholes framework. We find the optimal lower bound, calculate it numerically and compare this with those obtained by the Monte Carlo and Moment Matching methods.Our presentation here includes some of the most recent advances on Asian options, and we contribute in particular by adding detail to the proofs and explanations. We also contribute some novel numerical methods. Most significantly, we include an original contribution on the use of very sharp lower bounds towards pricing European basket options.

Cette thèse se concentre sur trois sujets liés aux SLE(k) processus. La première partie concerne le processus dipolar SLE(k) et la mesure de restriction conforme à la bande ; La deuxième partie porte sur la propriété de connectivité de la mesure de la boucle brownienne ; Et la troisième partie porte sur le spectre des moyens intégrés généralisés du processus entier intérieur des processus Loewner piloté par un processus Lévy. / This thesis focuses on three topics related to the SLE(k) processes. The first part is about the dipolar SLE(k)process and the conformal restriction measure on the strip ; the second part is about the connectivity propertyof the Brownian loop measure ; and the third part is about the generalized integral means spectrum of the innerwhole plane Loewner processes driven by a Lévy process.

Physical systems far from thermo dynamic equilibrium present excitability and irreversibility. The excitability is responsible for the great sensitivity of these systems to external stimuli while the irreversibility is associated with energy dissipation. The thermal fluctuations, inevitable in any real system, arise due to the interaction between many particles of the system. For such systems one of the best approaches is given by the non-equilibrium Statistical Mechanics, since it is virtually impossible an individualized approach of the motion equations. Many works in the current literature use a Gaussian stochastic modeling (without correlations) to represent the fluctuations. However, there is a growing number of studies reporting the occurrence of correlated fluctuations, mainly related to biological systems. In this thesis we investigate the influence of non-Gaussian stochastic distribution on the properties for two representative excitable models. In the first model we study the influence of distribution on the neural dynamics through the stochastic resonance (SR) mechanism. In the second model we approach the ratchet effect (RE) on directed transport of particles. In both systems we use a non-Gaussian power-law distributed noise obtained through a random multiplicative process (RMP). This process allows a fine tuning of the asymptotic power-law decay exponent. The optimization conditions are reported. In particular, we show that the optimization conditions for resonance and directed transport in Brownian ratchets are reached for a finit decay exponent of the stochastic distribution that represents a Strong non-Gaussian character. As non-Gaussian fluctuations occur with great frequency in natural systems, we believe that the non-Gaussian character can optimize the efficiency on the stochastic transport mechanisms in micro and nanoscale. / Fundação de Amparo a Pesquisa do Estado de Alagoas / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / Sistemas físicos fora do equilíbrio termo dinâmico apresentam excitabilidade e irreversibilidade. A excitabilidade é responsável pela grande sensibilidade desses sistemas a estímulos externos enquanto a irreversibilidade está asso ciada à dissipação de energia. As flutuações térmicas, inevitáveis em qualquer sistema real, surgem devido à interação entre as inúmeras partículas do meio. Para tais sistemas uma das melhores abordagens é dada pela Mecânica Estatística de não-equilíbrio, uma vez que é praticamente impossível uma abordagem individualizada das equações de movimento. Muitos trabalhos na literatura atual utilizam uma modelagem estocástica gaussiana (sem correlação) para representar as flutuações. No entanto, há um número crescente de trabalhos que relatam a ocorrência de flutuações correlacionadas, principalmente em sistemas biológicos. Nesta tese nós investigamos a influência da distribuição estocástica não-gaussiana sobre as propriedades de dois modelos excitáveis representativos. No primeiro, estudamos a influência da distribuição sobre a dinâmica neural através do mecanismo de ressonância estocástica (RE). No segundo, abordamos o mecanismo do efeito catraca (EC) sobre o transporte direcionado de partículas. Nos dois sistemas utilizamos um ruído colorido não-gaussiano com distribuição tipo lei de potência obtido através de um processo multiplicativo aleatório (PMA). Esse processo permite o ajuste no do expoente de decaimento assintótico da lei de potência. As condições de otimização são relatadas. Em particular, obtivemos que as condições de otimização para a ressonância e para o transporte direcionado em catracas brownianas são atingidas para um valor finito do expoente da distribuição estocástica que representa um caráter fortemente não-gaussiano. Como flutuações não-gaussianas o correm com muita frequência nos sistemas naturais, acreditamos que o caráter não-gaussiano pode otimizar a eficiência dos mecanismos estocásticos de transporte em micro e nanoescala.

Nous avons tou(te)s été témoins des nuages d'étourneaux dans le ciel ou de la formation de bancs de poissons dans l'océan. Ce type d'organisation chez les êtres vivant se produit aussi à des échelles parfois invisibles pour l'oeil humain: celles des micro-organismes. Les suspensions de micro-nageurs présentent une dynamique riche. Elles peuvent former des structures cohérentes résultant d'un mouvement collectif, mélanger le fluides environnant et/ou modifier ses propriétés rhéologiques. Leurs comportements peuvent jouer un rôle important dans la survie, l'équilibre des espèces, leur stratégie trophique et même pour la fertilité animale. La diversité des phénomènes observés résulte de l'interaction complexe entre mécanismes de nage, processus physiologiques, processus chimiques et interactions hydrodynamiques. Comprendre et maîtriser les mécanismes impliqués fait nécessairement appel la Mécanique des Fluides. Les études expérimentales permettent de mettre en exergue certains phénomènes et parfois de les expliquer. Cependant la modélisation s'avère indispensable. Or, inclure une description fine des mécanismes de nages dans une suspension contenant des milliers (voire des millions) d'individus, implique de considérer une vaste gamme d'échelles couplées (typiquement du micron 10^-6m au millimètre 10^-3m). Décrire une physique multi-échelles pour ce type problème reste un défi majeur pour la modélisation numérique actuelle. Ainsi, dans le cadre de cette thèse nous nous proposons d'apporter une contribution dans cette direction. Nous montrerons dans une premiere partie qu'il est possible de reproduire les mécanismes de nage de façon satisfaisante à l'échelle du micro-organisme avec des modèles de différentes complexités. Nous présenterons ensuite nos développements pour étendre ces modèles a l'échelle de la suspension. Nous montrerons comment inclure simultanément les effets Browniens qui agissent sur les plus petite particules (10^-6m). Enfin, nous exploiterons l'outil mis en place pour simuler des suspensions actives. Sa capacité à reproduire certains résultats de la littérature à précision égale, à moindre coût et à plus grande échelle, permet de combler le fossé entre modèles individuels, travaux expérimentaux et modèles continus issus de la théorie cinétique. Forts de cet outil, nous tenterons de répondre à deux questions ouvertes dans la littérature expérimentale : l'origine des corrélations d'orientation dans les suspensions de microgouttes auto-propulsées et les mécanismes en jeu dans la diffusion des particules Browniennes dans les suspensions actives. / We have all witnessed the flocking of starlings in the sky and the schools of fish that form in the ocean. This kind of organization of living creatures is not limited to those that we see, but also occurs for those that we don’t : swimming microorganisms. Suspen- sions of micro-swimmers exhibit a rich dynamics. Their behaviors can play an important role in the survival of the group, its development, the balance between species, their trophic strategies and even animal fertility. They can form coherent structures due to collective motion, mix the surrounding fluid or modify its rheological properties. Such diversity results from the complex interplay between swimming strategies, physiological processes, chemical reactions and hydrodynamic interactions. Fluid Mechanics is there- fore essential to understand and master the mechanisms involved in these phenomena. While experimental studies bring out new findings and, sometimes, provide physical ex- planations, modeling remains essential. Yet, including an accurate description of the micro-swimmers in a suspension containing thousands (nay millions) individuals, requires considering a wide range of coupled scales (from one micron 10^−6m to several millimeters 10^−3m). What happens on large scales depends on sophisticated mechanisms occurring two or three orders of magnitude below. Therefore, the multiscale modeling of such phenomena is still a major challenge for the state-of-the-art numerical methods. This thesis aims at providing a contribution in that direction. In a first part, we will show that reproducing swimming mechanisms at the scale of the micro-swimmer can be achieved with various models spanning different levels of complexity. We will then present our developments to incorporate these models in an efficient framework for large scale simulations. We will show how to simultaneously account for the Brownian motion of the smallest particles (10^−6m). Our code reproduces known results from the literature with the same accuracy, but at lower cost and at larger scales, thus bridging a gap between particle-based models, experiments and continuum formulations from kinetic theory. Using the capabilities afforded by our method, we eventually address two open problems in the experimental literature : the origins of orientational correla- tions between interacting self-propelled micro-droplets and the mechanisms at play in the nonlinear enhancement of Brownian particle diffusion in active suspensions.

Cette thèse porte sur l'existence de champs browniens fractionnaires indexés par des variétés riemanniennes. Ces objets héritent des propriétés qui font le succès du mouvement brownien fractionnaire classique (H-autosimilarité des trajectoires ajustable, accroissements stationnaires), mais autorisent à considérer des applications où les données sont portées par un espace qui peut par exemple être courbé ou troué. L'existence de ces champs n'est assurée que lorsque la quantité 2H est inférieure à l'indice fractionnaire de la variété, qui n'est connu que dans un petit nombre d'exemples. Dans un premier temps nous donnons une condition nécessaire pour l'existence de champ brownien fractionnaire. Dans le cas du champ brownien (correspondant à H=1/2) indexé par des variétés qui ont des géodésiques fermées minimales, cette condition s'avère très contraignante : nous donnons des résultats de non-existence dans ce cadre, et montrons notamment qu'il n'existe pas de champ brownien indexé par une variété compacte non simplement connexe. La condition nécessaire donne également une preuve courte d'un fait attendu qui est la non-dégénérescence du champ brownien indexé par les espaces hyperboliques réels. Dans un second temps nous montrons que l'indice fractionnaire du cylindre est nul, ce qui constitue un exemple totalement dégénéré. Nous en déduisons que l'indice fractionnaire d'un espace métrique n'est pas continu par rapport à la convergence de Gromov-Hausdorff. Nous généralisons ce résultat sur le cylindre à un produit cartésien qui possède une géodésique fermée minimale, et donnons une majoration de l'indice fractionnaire de surfaces asymptotiquement proches du cylindre au voisinage d'une géodésique fermée minimale. / The aim of the thesis is the study of the existence of fractional Brownian fields indexed by Riemannian manifolds. Those fields inherit key properties of the classical fractional Brownian motion (sample paths with self-similarity of adjustable parameter H, stationary increments), while allowing to consider applications with data indexed by a space which can be for example curved or with a hole. The existence of those fields is only insured when the quantity 2H is inferior or equal to the fractional index of the manifold, which is known only in a few cases. In a first part we give a necessary condition for the fractional Brownian field to exist. In the case of the Brownian field (corresponding to H=1/2) indexed by a manifold with minimal closed geodesics this condition happens to be very restrictive. We give several nonexistence results in this situation. In particular we show that there exists no Brownian field indexed by a nonsimply connected compact manifold. Our necessary condition also gives a short proof of an expected result: we prove the nondegeneracy of fractional Brownian fields indexed by the real hyperbolic spaces. In a second part we show that the fractional index of the cylinder is null, which gives a totally degenerate case. We deduce from this result that the fractional index of a metric space is noncontinuous with respect to the Gromov-Hausdorff convergence. We generalise this result about the cylinder to a Cartesian product with a closed minimal geodesic. Furthermore we give a bound of the fractional index of surfaces asymptotically close to the cylinder in the neighbourhood of a closed minimal geodesic.

Les suspensions colloïdales ont été étudiées par simulations numériques à partir de deux modèles : la dynamique Brownienne (BD) et la SRD-MD (Stochastic Rotation Dynamics - Molecular Dynamics). Ces études ont consisté à reprendre des travaux existants pour les porter sur GPU, tout en cherchant différentes optimisations possibles adaptées à ces simulations. Une amélioration de la recherche de voisinage de la littérature a pu être utilisée pour toutes ces simulations de type BD. Une simulation de SRD-MD avec couplage de force qui n'avait pas encore été parallélisée sur GPU dans la littérature, a été implémentée en utilisant un nouveau schéma de décomposition adapté à cette simulation, améliorant considérablement les performances. Ces simulations ont pu donner lieu par la suite à des études sur des suspensions colloïdales plus complexes : une hétéroagrégation entre deux suspensions avec des particules de même taille, une hétéroagrégation entre deux populations de colloïdes de tailles très différentes, et en dehors des suspensions colloïdales, une simulation de nanoalliages. Enfin, le modèle de SRD a été adapté afin d'être utilisé dans le cadre d'animation physique de fluide réaliste dans le contexte de l'informatique graphique. Des adaptations du modèle pour y incorporer des notions comme la gestion de la compressibilité, de la tension de surface ont dues être apportées. Des premiers résultats ont pu permettre de réaliser quelques simulations, dont une chute d'eau dans une verre. / Colloïdal suspensions have been studied by means of numerical simulation, using two physical models : Brownian dynamics and Stochastic Rotation Dynamics - Molecular Dynamics. These studies consist in parallizing colloïdal simulations from previous studies on GPU, and find some new optimisations for these specific simulations. An improvement of the neigborhood search has been implemented in all our BD type simulations. A SRD-MD with force coupling have been implemented for the first time in the literature, using a new decomposition scheme, which improves significantly its performances. Then, theses simulations have been adapted to study more complex colloidal suspensions : an interfacial heteroaggregation of colloidal suspensions, a heteroaggregation between two types of particles with a large size ratio, and outside this context, a nanoalloy simulation. Finally, the SRD model has been adapted to realistic fluid animtion from computer science context. Theses adaptations require to add to SRD model, the notion of compressibility and surface tension. First results have been released, like a pouring water into a glass simulation.

Orientador: Marcus Aloizio Martinez de Aguiar / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Fisica Gleb Wataghin / Made available in DSpace on 2018-08-06T21:05:02Z (GMT). No. of bitstreams: 1 Bonanca_MarcusViniciusSegantini_D.pdf: 10284922 bytes, checksum: 28ea976c05e0eadcda732211e40afb25 (MD5) Previous issue date: 2006 / Resumo: Neste trabalho, estudamos de que maneira e sob que condições um sistema caótico com apenas dois graus de liberdade produz efeitos irreversíveis como dissipação, termalização e, do ponto de vista quântico, perda de coerência em um sistema simples a ele acoplado. Na formulação clássica do problema, descrevemos analiticamente o comportamento do fluxo de energia em Resposta Linear e apontamos o ingrediente talvez principal que um sistema caótico possui para causar irreversibilidade: correlações que decaem exponencialmente. Mostramos que é possível descrever o equilíbrio assintótico inclusive com uma temperatura, o que é não-intuitivo em se tratando de sistemas pequenos. Esse último resultado completa o paralelo entre o movimento Browniano usual e o modelo proposto. Formulamos o problema do ponto de vista quântico via o formalismo de Funcionais de Influência. Mostramos que este formalismo é mesmo adequado pois a influência do sistema caótico é descrita pelas contrapartidas quânticas das mesmas funções que encontramos na Resposta Linear clássica. Calculamos semiclassicamente essas funções e mostramos que os termos em mais baixa ordem da aproximação semiclássica evoluem conforme a dinâmica clássica caótica. As escalas de tempo da análise clássica se mostram fundamentais para a resolução dos cálculos assim como a análise semiclássica das funções de correlação. Mostramos que efeitos de dissipação e perda de coerência, no contexto quântico, são possíveis devido ao caráter caótico do sistema / Abstract: We study here how and under which conditions a chaotic system with only two degrees of freedom can produce irreversible phenomena such as dissipation, thermalization and, from the quantum point of view, decoherence in a simple system coupled to it. In the classical formulation of the problem, we describe analytically the behavior of the energy ux in Linear Response regime and we point the main ingredient for a chaotic system to produce irreversible effects: correlations with exponential decay. We show that it is possible to describe the asymptotic equilibrium even with a temperature, which seems to be a counter intuitive result for systems with few degrees of freedom. We formulate the problem from the quantum point of view using In uence Functionals approach. We show the formalism is very adequate since the chaotic system in uence is described by quantum analogues of the same functions we obtain in the Linear Response approach to the classical problem. We calculate those functions semiclassically and we show the lowest order terms of the semiclassical approximation evolve as given by classical chaotic dynamics. The time scales of the classical analysis are shown to be very important for the resolution of the quantum problem as well as the semiclassical analysis of the correlation functions. We show that dissipative and decoherence effects, in the quantum regime, are possible due to the chaotic dynamics of the system / Doutorado / Física Estatistica e Termodinamica / Doutor em Ciências

L'objectif de cette thèse porte sur l'étude des phénomènes de blocage dans un flux particules à faible densité dans un canal. Le blocage est induit par la géométrie du canal. L'essentiel de mes travaux concerne la description des situations où le blocage est contrôlé par les limites en capacité d'un canal. Le paramètre pertinent pour ce phénomène est donné par le nombre de particules minimum, N, conduisant à l'interruption du flux de particules. Un modèle stochastique simple introduit par Gabrielli et al. (PRL. 110, 170601, 2013) illustre ce comportement: des particules arrivent aléatoirement selon une distribution de Poisson à l'entrée d'un canal unidimensionnel et le traversent avec un temps constant, noté t. Le blocage survient lorsque N particules sont simultanément sur le pont. Le travail de cette thèse à été d'étudier les extensions de ce modèle. Les observables du système sont la probabilité de survie, le flux sortant ainsi que la statistique sur les particules sorties avant le blocage. Les différentes études ont permis pour le cas N>2, pour une distribution homogène quelconque et inhomogène d'entrée, pour un système de multi-canaux ainsi que pour une durée finie de blocage d'obtenir des résultats analytiques exactes ainsi que des approximations à l'aide d'outils statistique. Le dernier projet de cette thèse porte sur l'étude microscopique des phénomènes de blocage. Le modèle simple que nous avons étudié est un système bidimensionnel de particules browniennes soumis à une force de traînée et se déplaçant dans un canal avec rétrécissement. La présence d'un obstacle au milieu du canal peut causer un colmatage selon les valeurs des différents paramètres du système. / This manuscript presents a study of blocking phenomenon in particulate streams flowing through anarrow channel. In particular, it examines situations in which blocking is controlled by the limitedcarrying capacity of the channel. It builds on a simple stochastic model, introduced by Gabrielli etal. (Phys. Rev. Lett. 110, 170601, 2013), in which particles arrive randomly according to a Poissondistribution at the entrance of a one-dimensional channel with an intensity λ and, unless interrupted,exit after a transit time, τ. Blocking occurs instantaneously when N=2 particles are simultaneouslypresent in the channel. The quantities of interest include the probability that the channel is still openat time t (survival probability) and the flux and total number of exiting particles. The thesisexamines a number of generalizations including when more than two particles must be present toinduce blockage, N>2, a time dependent intensity, a finite blocking time, and multi-channelsystems. We obtain exact and approximate analytical results using tools such as the masterequations describing the evolution of the n-particle partial probabilities, large deviation theory andqueuing theory. The theoretical results are validated by comparison with the results of numericalsimulations. The final chapter of the thesis uses a different approach, namely a brownian dynamics simulation of a two dimensional system of soft particles subjected to an external driving and dragforces. The presence of an obstacle in the middle of the channel can cause irreversible orintermittent clogging depending on the system geometry, temperature and particle stiffness.