Thesis (Ph.D.)--Australian National University, 2002.

Thesis (Ph. D.)--Missouri University of Science and Technology, 2008. / Vita. The entire thesis text is included in file. Title from title screen of thesis/dissertation PDF file (viewed August 21, 2008) Includes bibliographical references (p. 124-131).

Thesis (Ph. D.)--University of Hong Kong, 1997. / Includes bibliographical references (leaf 142-147).

Thesis (Ph.D)--Chemical Engineering, Georgia Institute of Technology, 2010. / Committee Chair: Victor Breedveld; Committee Member: Eric W. Weeks; Committee Member: Hang Lu; Committee Member: J. Carson Meredith; Committee Member: L. Andrew Lyon. Part of the SMARTech Electronic Thesis and Dissertation Collection.

Thesis (M.S.)--Worcester Polytechnic Institute. / Keywords: drawdowns, maximum drawdowns. Includes bibliographical references (p.23).

This dissertation discusses the time and place that Brownian motion on a Riemannian manifold first exit a normal ball of small radius. A general procedure is given for computing asymptotic expansions of joint moments of the first exit time and place random variables as the radius of the geodesic ball decreases to zero. The asymptotic expansion of the joint Laplace transform of exit time and spherical harmonics of exit position is derived for a ball of small radius. A generalized Pizetti’s formula is used to expand the solution of the related partial differential equations. These expansions are represented in terms of curvature in the manifold. Asymptotic Independence Conditions (AIC) and Asymptotic Uncorrelated Conditions (AUC) are defined for the joint distributions of exit time and place. Computations using the methods developed in this work demonstrate that AIC and AUC produce the same curvature conditions up to a certain level of asymptotics. It is conjectured that AUC implies AIC. Further, a generalized method is given for computing the Laplace transform, and therefore the moments of the exit time. This work is related to and also extends the work of M. Liao and H. R. Hughes in stochastic geometric analysis.

After more than a century, there are no analytical solutions for the Navier-Stokes equations to describe complex fluid behavior, and we often resort to different computational methods to find solutions under specific conditions. In particular, to address many biological questions, we need to use techniques which are accurate at the mesoscale regime and computationally efficient, since atomistic simulations are still incredibly computationally costly, and continuum methods based on Navier-Stokes present challenges with complicated moving boundaries, in the presence of fluctuations. Here, we use a novel particle-based coarse-grained method, known as MPCD, to study ciliated swimmers. Using experimentally measured beating patterns, we show how we recapitulate the emergence of metachronal waves (MCW) on planar surfaces, and present new results on curved surfaces. To quantitatively study these waves, we also analyzed their effect on beating intervals, energy fluctuations, and fluid motion. We then extended our model to realistic cellular geometries, using experimentally obtained Basal Bodies locations.\par In the second part of our study, we focused on the intracellular fluid motion, neglecting hydrodynamic interactions. We developed the Digital Confocal Microscopy Suite (DCMS) that can run on multiple platforms using GPUs and can input realistic cell shapes and optical properties of the confocal microscope. It has this ability to simulate both (Fluorescence Recovery After Photobleaching) FRAP and Fluorescence Correlation Spectroscopy (FCS) experiments, as well as the capability to model photo-switching of fluorophores, acquisition photo-bleaching, and reaction-diffusion systems. With this platform, in collaboration with the Vidali Lab, we were able to elucidate the role of boundaries in interpreting FRAP experiments in \textit{moss} and estimate the binding rates of myosin XI.

The goal of this thesis is to present a comprehensive study of the parabolic and hyperbolic Anderson models with constant initial condition, driven by a Gaussian noise which is fractional in space with index H > 1/2 or H < 1/2, and is either white in time, or fractional in time with index H_0 > 1/2. As a preliminary step, we study the linear stochastic heat and wave equations with the same type of noise. In the case H_0 > 1/2 and H < 1/2, we present a new result, regarding the solution of the parabolic Anderson model with general initial condition given by a measure.

After more than a century, there are no analytical solutions for the Navier-Stokes equations to describe complex fluid behavior, and we often resort to different computational methods to find solutions under specific conditions. In particular, to address many biological questions, we need to use techniques which are accurate at the mesoscale regime and computationally efficient, since atomistic simulations are still incredibly computationally costly, and continuum methods based on Navier-Stokes present challenges with complicated moving boundaries, in the presence of fluctuations. Here, we use a novel particle-based coarse-grained method, known as MPCD, to study ciliated swimmers. Using experimentally measured beating patterns, we show how we recapitulate the emergence of metachronal waves (MCW) on planar surfaces, and present new results on curved surfaces. To quantitatively study these waves, we also analyzed their effect on beating intervals, energy fluctuations, and fluid motion. We then extended our model to realistic cellular geometries, using experimentally obtained Basal Bodies locations.\par In the second part of our study, we focused on the intracellular fluid motion, neglecting hydrodynamic interactions. We developed the Digital Confocal Microscopy Suite (DCMS) that can run on multiple platforms using GPUs and can input realistic cell shapes and optical properties of the confocal microscope. It has this ability to simulate both (Fluorescence Recovery After Photobleaching) FRAP and Fluorescence Correlation Spectroscopy (FCS) experiments, as well as the capability to model photo-switching of fluorophores, acquisition photo-bleaching, and reaction-diffusion systems. With this platform, in collaboration with the Vidali Lab, we were able to elucidate the role of boundaries in interpreting FRAP experiments in \textit{moss} and estimate the binding rates of myosin XI.

We consider a stochastic functional differential equation with infinite memory driven by a fractional Brownian motion with Hurst parameter $H>1/2$. We prove an existence and uniqueness result of the solution to the stochastic differential equation. We investigate the dependence of the solution on the initial condition and the existence of finite moments of the solution. Furthermore we generalize these results to wider classes of stochastic differential equations. The stochastic integral with respect to fractional Brownian motion is defined as a pathwise Riemann-Stieltjes integral.