Optical tweezers are a sensitive position and force transducer widely employed in physics and biology. In a focussed laser, forces due to radiation pressure enable to trap and manipulate small dielectric particles used as probes for various experiments. For sensitive biophysical measurements, microspheres are often used as a handle for the molecule of interest. The force range of optical traps well covers the piconewton forces generated by individual biomolecules such as kinesin molecular motors. However, cellular processes are often driven by ensembles of molecular machines generating forces exceeding a nanonewton and thus the capabilities of optical tweezers. In this thesis I focused, fifirst, on extending the force range of optical tweezers by improving the trapping e fficiency of the probes and, second, on applying the optical tweezers technology to understand the mechanics of molecular motors. I designed and fabricated photonically-structured probes: Anti-reflection-coated, high-refractive-index, core-shell particles composed of titania. With these probes, I significantly increased the maximum optical force beyond a nanonewton. These particles open up new research possibilities in both biology and physics, for example, to measure hydrodynamic resonances associated with the colored nature of the noise of Brownian motion. With respect to biophysical applications, I used the optical tweezers to study the mechanics of single kinesin-8. Kinesin-8 has been shown to be a very processive, plus-end directed microtubule depolymerase. The underlying mechanism for the high processivity and how stepping is affected by force is unclear. Therefore, I tracked the motion of yeast (Kip3) and human (Kif18A) kinesin-8s with high precision under varying loads. We found that kinesin-8 is a low-force motor protein, which stalled at loads of only 1 pN. In addition, we discovered a force-induced stick-slip motion, which may be an adaptation for the high processivity. Further improvement in optical tweezers probes and the instrument will broaden the scope of feasible optical trapping experiments in the future.

Les travaux présentés dans cette thèse s'intéressent à la géométrie fractale de processus stochastiques à travers le prisme d'un outil appelé l'analyse 2-microlocale. Ce dernier est issu d'une autre branche des mathématiques, l'analyse fonctionnelle et l'étude des équations aux dérivées partielles, et s'est avéré être pertinent pour décrire la géométrie fine de fonctions déterministes ou de processus aléatoires, généralisant notamment les exposants de Hölder classiques. Nous envisageons ainsi dans ce manuscrit différentes classes de processus, traitant en premier lieu le cas des martingales continues et de l'intégrale stochastique d'Ito. La régularité 2-microlocale de ces derniers fait notamment apparaître un autre concept, la pseudo frontière 2-microlocale, étroitement lié à son aîné. Nous appliquons également ce formalisme d'étude à une classe de processus gaussiens : le mouvement brownien multifractionnaire. Nous caractérisons ainsi sa régularité 2-microlocale et hölderienne, et déterminons dans un deuxième temps la forme générale de la dimension fractale de ses trajectoires. Dans notre étude portant sur les processus de Lévy, nous combinons le formalisme 2-microlocale à l'analyse multifractale, permettant alors de mettre en évidence des comportements géométriques n'étant pas captés par les outils usuels. Nous obtenons également en corollaire le spectre multifractal des processus fractionnaires de Lévy. Enfin, dans une dernière partie, nous nous intéressons à la définition et aux propriétés de certains processus de Markov multiparamètres, pouvant être plus généralement indicés par des ensembles. / The work presented in this thesis concerns the study of the fractal geometry of stochastic processes using the formalism of 2-microlocal analysis. The latter has been introduced in another branch of mathematics -functional analysis- but has also proved to be relevant to describe the geometry of deterministic functions or random processes, extending in particular the classic Hölder exponents. Several classes of processes are investigated in this manuscript, beginning with continuous martingales and Ito integrals. In particular, the characterisation of the 2-microlocal regularity of the latter leads to the introduction of a closely related concept: the pseudo 2-microlocal frontier. We also investigate using this formalism a class of Gaussian processes called multifractional Brownian motion and obtain a fine description of its Hölder and 2-microlocal behaviours. In addition, we characterize entirely the Hausdorff and Box dimensions of its graph. In our study of Lévy processes, we combine the 2-microlocal formalism and multifractal analysis to describe their regularity, exhibiting in particular some subtle geometrical behaviours which are not captured by classic tools. Furthermore, as a corollary of this result, we also determine the multifractal spectrum of another family of processes: the fractional Lévy processes. Lastly, we also define a class of multiparameter and set-indexed Markov processes and study its properties.

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

<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>

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This thesis compiles the publications I coauthored during my doctoral studies at University of Leipzig on the subject of optimizing thermodynamic systems, focusing on three optimization perspectives: maximum efficiency, maximum power, and maximum efficiency at given power. We considered two currently intensely studied models in finite-time thermodynamics, i.e., low-dissipation models and Brownian systems. The low-dissipation model is used to derive general bounds on the performance of real-world machines, while Brownian systems allow us to better understand the practical limits and features of small systems. First, we derived maximum efficiency at given power for various low-dissipation setups, with a particular focus on the behavior close to maximum power, which helps us to determine whether it is more beneficial to operate the system at maximum power, near maximum power or in a different regime. Then, we move to the design of maximum-efficiency and maximum-power protocols for Brownian systems under different boundary conditions. Particularly, when the constraints on control parameters are experimentally motivated, we presented a geometric method yielding maximum-efficiency and maximum-power protocols valid for systems with periodically scaled energy spectrum and otherwise arbitrary dynamics. Each chapter contains a short informal introduction to the matter as well as an outlook, pointing out the direction for our research in the future.