This work is a study of the relationship between Brownian motion and elementary, linear partial differential equations. In the text, I have shown that Brownian motion is a Markov process, and that Brownian motion itself, and certain Stochastic processes involving Brownian motion are also martingales. In particular, Dynkin's formula for Brownian motion was shown. Using Dynkin's formula and Brownian motion, I then constructed solutions for the classical Dirichlet problem and the heat equation, given by Δu=0 and ut= 1/2Δu+g, respectively. I have shown that the bounded solution is unique if Brownian motion will always exit the domain of the function once it has started at a point in the domain. The heat equation also has a unique bounded solution.

The characterization of fine particles is an area of immense significance to many industrial endeavors. It has been estimated that 70% of all industrial processes deal with fine particles at some point in the process. A natural phenomenon occurring in these processes is colloidal aggregation. This study examines aggregation in colloidal systems in order to characterize, examine, and control this occurrence in industrial processes. The study of particle aggregation has been broken into many different areas, such as collision mechanisms, interaction energy etc, but a complete model that integrates these different aspects has never been fully realized. A new model is required to accurately predict the aggregation behavior of colloidal particles. In this work, a new model is developed that integrates Smoluchowski kinetics, total interaction energy between particles, and stability ratios for perikinetic and orthokinetic collision mechanisms. The total particle interaction energy necessary for the calculation of stability ratios is represented by the summation of electrostatic and van der Waals interactions. The electrostatic interactions are modeled using DLVO theory, the linear Poisson-Boltzmann equation, and a numerical solution for the non-linear Poisson-Boltzmann Equation, while the van der Waals interactions are represented by Hamaker theory. The mathematical model is solved using an adjustable discretion technique, which is tested against a specific analytic solution, and yields an assessment of the error intrinsic in the discretization method. The basis of the mathematical model is a population balance framework. The model developed in this study is general in many respects, but could be readily applied to many different aggregation systems with minor modification. A comparison of the mathematical model with previous experiments conducted by Scott Fisher (1998) is carried out for the perikinetic and orthokinetic transport-limited aggregation regimes. The fractal nature of solid-sphere aggregates is considered when comparing the mathematical model predictions with experimental measurements. The previous experiments that are used for comparison utilized polystyrene particles ranging from 100 nm to 500 nm in initial diameter, several initial particle concentrations, and various stirring rates. Zeta potential measurements are presented in order to set the range of transport-limited aggregation. An assessment of the results of the mathematical model with the experimental results show good agreement for transport-limited aggregation within the perikinetic and orthokinetic transport-limited aggregation, with average particle sizes ranging from 100 nm to well over 2 microns.

Plusieurs mécanismes biologiques utilisent la polymérisation des filaments d'actine comme moteur mécanique. L'énergie chimique libérée à l'addition d'un monomère dans le filament est convertie en travail mécanique et une force est générée. Les filaments ainsi formés s'organisent grâce à des protéines liant l'actine et forment des structures qui diffèrent par leurs propriétés mécaniques et élastiques mais aussi de leurs fonctions dans les différents processus biologiques. Notre système expérimental permet d'étudier le lien entre les propriétés mécaniques et les mécanismes à l'origine de la production de la force. La polymérisation des filaments est directement initiée sur la surface de particules magnétiques. En présence d'un champ magnétique, ces dernières s'organisent en chaîne par des interactions dipôle-dipôle, et une force magnétique compressive est induite sur les filaments qui polymérisent. La polymérisation écartent les particules au cours du temps et en fonction de la force appliquée, la vitesse d'écartement des particules est ralentie. En suivant l'évolution de la distance entre particules, nous détaillons la relation force-vitesse et les propriétés mécaniques des filaments.

We prove in this paper an existence result for infinite-dimensional stationary interactive Brownian diffusions. The interaction is supposed to be small in the norm ||.||∞ but otherwise is very general, being possibly non-regular and non-Markovian. Our method consists in using the characterization of such diffusions as space-time Gibbs fields so that we construct them by space-time cluster expansions in the small coupling parameter.

Heterophase polymerization is a technique widely used for the synthesis of high performance polymeric materials with applications including paints, inks, adhesives, synthetic rubber, biomedical applications and many others. Due to the heterogeneous nature of the process, many different relevant length and time scales can be identified. Each of these scales has a direct influence on the kinetics of polymerization and on the physicochemical and performance properties of the final product. Therefore, from the point of view of product and process design and optimization, the understanding of each of these relevant scales and their integration into one single model is a very promising route for reducing the time-to-market in the development of new products, for increasing the productivity and profitability of existing processes, and for designing products with improved performance or cost/performance ratio. The process considered is the synthesis of structured or composite polymer particles by multi-stage seeded emulsion polymerization. This type of process is used for the preparation of high performance materials where a synergistic behavior of two or more different types of polymers is obtained. Some examples include the synthesis of core-shell or multilayered particles for improved impact strength materials and for high resistance coatings and adhesives. The kinetics of the most relevant events taking place in an emulsion polymerization process has been investigated using suitable numerical simulation techniques at their corresponding time and length scales. These methods, which include Molecular Dynamics (MD) simulation, Brownian Dynamics (BD) simulation and kinetic Monte Carlo (kMC) simulation, have been found to be very powerful and highly useful for gaining a deeper insight and achieving a better understanding and a more accurate description of all phenomena involved in emulsion polymerization processes, and can be potentially extended to investigate any type of heterogeneous process. The novel approach of using these kinetic-based numerical simulation methods can be regarded as a complement to the traditional thermodynamic-based macroscopic description of emulsion polymerization. The particular events investigated include molecular diffusion, diffusion-controlled polymerization reactions, particle formation, absorption/desorption of radicals and monomer, and the colloidal aggregation of polymer particles. Using BD simulation it was possible to precisely determine the kinetics of absorption/desorption of molecular species by polymer particles, and to simulate the colloidal aggregation of polymer particles. For diluted systems, a very good agreement between BD simulation and the classical theory developed by Smoluchowski was obtained. However, for concentrated systems, significant deviations from the ideal behavior predicted by Smoluchowski were evidenced. BD simulation was found to be a very valuable tool for the investigation of emulsion polymerization processes especially when the spatial and geometrical complexity of the system cannot be neglected, as is the case of concentrated dispersions, non-spherical particles, structured polymer particles, particles with non-uniform monomer concentration, and so on. In addition, BD simulation was used to describe non-equilibrium monomer swelling kinetics, which is not possible using the traditional thermodynamic approach because it is only valid for systems at equilibrium. The description of diffusion-controlled polymerization reactions was successfully achieved using a new stochastic algorithm for the kMC simulation of imperfectly mixed systems (SSA-IM). In contrast to the traditional stochastic simulation algorithm (SSA) and the deterministic rate of reaction equations, instead of assuming perfect mixing in the whole reactor, the new SSA-IM determines the volume perfectly mixed between two consecutive reactions as a function of the diffusion coefficient of the reacting species. Using this approach it was possible to describe, using a single set of kinetic parameters, typical mass transfer limitations effects during a free radical batch polymerization such as the cage effect, the gel effect and the glass effect. Using multiscale integration it was possible to investigate the formation of secondary particles during the seeded emulsion polymerization of vinyl acetate over a polystyrene seed. Three different cases of radical generation were considered: generation of radicals by thermal decomposition of water-soluble initiating compounds, generation of radicals by a redox reaction at the surface of the particles, and generation of radicals by thermal decomposition of surface-active initiators "inisurfs" attached to the surface of the particles. The simulation results demonstrated the satisfactory reduction in secondary particles formation achieved when the locus of radical generation is controlled close to the particles surface. / Eine der industriell am meisten verwendeten Methoden zur Herstellung von Hochleistungspolymeren ist die Heterophasenpolymerisation. Industriell von besonderer Bedeutung ist die sogenannte Saatemulsionspolymerisation bei der kleine Saatteilchen durch die sequentielle Zugabe von weiteren Monomeren gezielt modifiziert werden, um Kompositpolymerteilchen mit den gewünschten mechanischen und chemischen Gebrauchseigenschaften herzustellen. Ein häufig auftretendes Problem während dieser Art der Heterophasenpolymerisation ist die Bildung von neuen, kleinen Teilchen im Polymerisationsverlauf. Diese sogenannte sekundäre Teilchenbildung muss vermieden werden, da sie die Herstellung der gewünschten Teilchen mit den angestrebten Eigenschaften verhindert. Ein spezieller Fall der Saatemulsionspolymerisation ist die Kombination von Vinylacetat als Monomer, das auf Saatteilchen aus Polystyrol polymerisieren soll. Die Unterdrückung der Teilchenneubildung ist in diesem Beispiel besonders schwierig, da Vinylacetat eine sehr hohe Wasserlöslichkeit besitzt. In der vorliegenden Arbeit wurden zur Lösung der Aufgabenstellung verschiedene numerische Simulierungsalgorithmen verwendet, die entsprechend den charakteristischen Längen- und Zeitskalen der im Verlauf der Polymerisation ablaufenden Prozesse ausgewählt wurden, um die passenden Bedingungen für die Unterdrückung der sekundären Teilchenbildung zu finden. Die verwendeten numerischen Methoden umfassen Molekulare Dynamik Simulationen, die benutzt werden, um molekulare Bewegungen zu berechnen; Brownsche Dynamik Simulationen, die benutzt werden, um die zufälligen Bewegungen der kolloidalen Teilchen und der molekularen Spezies zu beschreiben, und kinetische Monte Carlo Simulationen, die das zufällige Auftreten von individuellen physikalischen oder chemischen Ereignissen modellieren. Durch die Kombination dieser Methoden ist es möglich, alle für die Beschreibung der Polymerisation relevanten Phänomene zu berücksichtigen. Damit können nicht nur die Reaktionsgeschwindigkeit und die Produktivität des Prozesses simuliert werden sondern auch Aussagen bezüglich der physikalischen und chemischen Eigenschaften des Produktes sowie den Applikationseigenschaften getroffen werden. In dieser Arbeit wurden zum ersten Mal Modelle für die unterschiedlichen Längen- und Zeitskalen bei Heterophasenpolymerisationen entwickelt und erfolgreich zur Modellierung des Prozesses angewendet. Die Ergebnisse führten zu bedeutenden Verbesserungen der Theorie von Emulsionspolymerisationen insbesondere für die Beschreibung des Massenaustausches zwischen den Phasen (bspw. Radikaleintritt in und Radikalaustritt aus die Polymerteilchen), der Bildung von neuen Teilchen, und der Polymerisationskinetik unter den heterogenen Reaktionsbedingungen mit uneinheitlicher Durchmischung.

This thesis describes a new approach to biomolecular analysis, called the volume-amplified magnetic nanobead detection assay (VAM-DNA). It is a sensitive, specific magnetic bioassay that offers a potential platform for the development of low-cost, easy-to-use diagnostic devices. The VAM-NDA consists of three basic steps: biomolecular target recognition, enzymatic amplification of the probe-target complex using the rolling circle amplification (RCA) technique, and addition of target complementary probe-tagged magnetic nanobeads which exhibit Brownian relaxation behavior. Target detection is demonstrated by measuring the frequency-dependent complex magnetization of the magnetic beads. The binding of the RCA products (target DNA-sequence coils) to the bead surface causes a dramatic increase in the bead size, corresponding essentially to the size of the DNA coil (typically around one micrometer). This causes a decrease in the Brownian relaxation frequency, since it is inversely proportional to the hydrodynamic size of the beads. The concentration of the DNA coils is monitored by measuring the decrease in amplitude of the Brownian relaxation peaks of free beads. The parameters oligonucleotide surface coverage, bead concentration, bead size and RCA times were investigated in this thesis to characterize features of the assay. It was found that all of these parameters affect the outcome and efficiency of the assay. The possibility of implementing the assay on a portable, highly sensitive AC susceptometer platform was also investigated. The performance of the assay under these circumstances was compared with that using a superconducting quantum interference device (SQUID); the sensitivity of the assay was similar for both platforms. It is concluded that, the VAM-NDA opens up the possibility to perform biomolecular detection in point-of-care and outpatient settings on portable platforms similar to the one tested in this thesis. Finally, the VAM-NDA was used to detect Escherichia coli bacteria and the spores of Bacillus globigii, the non-pathogenic simulant of Bacillus anthracis. A limit of detection of at least 50 bacteria or spores was achieved. This shows that the assay has great potential for sensitive detection of biomolecules in both environmental and biomedical applications.

The first part of this thesis studies tail probabilities forelliptical distributions and probabilities of extreme eventsfor multivariate stochastic processes. It is assumed that thetails of the probability distributions satisfy a regularvariation condition. This means, roughly speaking, that thereis a non-negligible probability for very large or extremeoutcomes to occur. Such models are useful in applicationsincluding insurance, finance and telecommunications networks.It is shown how regular variation of the marginals, or theincrements, of a stochastic process implies regular variationof functionals of the process. Moreover, the associated tailbehavior in terms of a limit measure is derived. The second part of the thesis studies problems related toparameter estimation in stochastic models with long memory.Emphasis is on the estimation of the drift parameter in somestochastic differential equations driven by the fractionalBrownian motion or more generally Volterra-type processes.Observing the process continuously, the maximum likelihoodestimator is derived using a Girsanov transformation. In thecase of discrete observations the study is carried out for theparticular case of the fractional Ornstein-Uhlenbeck process.For this model Whittle’s approach is applied to derive anestimator for all unknown parameters.

In this thesis we study the standard probabilistic model for hashing with linear probing. The main purpose is to determine the asymptotic distribution for the maximum displacement. Depending on the ratio between the number of items and the number of cells, there are several cases to consider. Paper I solves the problem for the special case of almost full hash tables. That is, hash tables where every cell but one is occupied. Paper II completes the analysis by solving the problem for all remaining cases. That is, for every case where the number of items divided by the number of cells lies in the interval [0,1]. The last two papers treat quite different topics. Paper III studies the area covered by the supremum process of Brownian motion. One of the main theorems in Paper I is expressed in terms of the Laplace transform of this area. Paper IV provides a new sufficient condition for a collection of independent random variables to be negatively associated when conditioned on their total sum. The condition applies to a collection of independent Borel-distributed random variables, which made it possible to prove a Poisson approximation that where essential for the completion of Paper II.

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.

The first passage time (FPT) problem for Brownian motion has been extensively studied in the literature. In particular, many incarnations of integral equations which link the density of the hitting time to the equation for the boundary itself have appeared. Most interestingly, Peskir (2002b) demonstrates that a master integral equation can be used to generate a countable number of new integrals via its differentiation or integration. In this thesis, we generalize Peskir's results and provide a more powerful unifying framework for generating integral equations through a new class of martingales. We obtain a continuum of new Volterra type equations and prove uniqueness for a subclass. The uniqueness result is then employed to demonstrate how certain functional transforms of the boundary affect the density function. Furthermore, we generalize a class of Fredholm integral equations and show its fundamental connection to the new class of Volterra equations. The Fredholm equations are then shown to provide a unified approach for computing the FPT distribution for linear, square root and quadratic boundaries. In addition, through the Fredholm equations, we analyze a polynomial expansion of the FPT density and employ a regularization method to solve for the coefficients. Moreover, the Volterra and Fredholm equations help us to examine a modification of the classical FPT under which we randomize, independently, the starting point of the Brownian motion. This randomized problem seeks the distribution of the starting point and takes the boundary and the (unconditional) FPT distribution as inputs. We show the existence and uniqueness of this random variable and solve the problem analytically for the linear boundary. The randomization technique is then drawn on to provide a structural framework for modeling mortality. We motivate the model and its natural inducement of 'risk-neutral' measures to price mortality linked financial products. Finally, we address the inverse FPT problem and show that in the case of the scale family of distributions, it is reducible to nding a single, base boundary. This result was applied to the exponential and uniform distributions to obtain analytical approximations of their corresponding base boundaries and, through the scaling property, for a general boundary.