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In this thesis, we investigate the Lp-solvability of a class of (possibly) degenerate stochastic integro-differential equations (SIDEs) of parabolic type, which includes the Zakai equation in nonlinear filtering for jump diffusions and the Kolmogorov equations for jump diffusions. We first study the solvability of integro-differential equations in the same type but without randomness. Then we present an Itˆo formula for the Lp-norm of jump processes having stochastic differentials in Lp-spaces, which can be used to study the solvability of SIDEs. In the last chapter, existence and uniqueness of the solutions to SIDEs are established in Bessel potential spaces.

In this dissertation we consider several applications of evolutionary game theory. Using the replicator equation as a starting point we explore broadly the question of how interactions between agents in heterogeneous populations influence how the demographics of the population change over time. We explore in depth how mutation mechanisms can lead to oscillatory population dynamics (via Hopf bifurcations) in systems with natural cyclicity. Building on previous work examining the evolutionary dynamics of the game Rock-Paper-Scissors with one mutation mechanism, we obtain similar results using a more biologically inspired model of mutation. The evolutionary dynamics of the class of social dilemmas known as public goods games are analyzed in detail. In particular, we study these games in cases where a continuum of possible strategies is available to players. A strategy in this context is simply the amount that a player contributes to the creation of a public good. Many questions arise here: What effect do mutations have on the dynamics? What happens when the value of the public good doesn't depend linearly on the sum of the contributions to its creation? What happens if an ecological component is included, so that growth rates are partly determined by available space and resources? We answer each of these questions as well as several others and see what implications can be drawn from such models regarding the persistence of pro-social behaviour. An analysis of evolutionary game assay experiments is presented also. In these experiments heterogeneous populations of cancer cells are monitored during their evolution. At regular intervals during each trial the numbers of cells of each type are estimated by photographic means. Important characteristics of the population's state are inferred from the resulting data. However, mutational effects may be obscuring the true counts of different types of cells during the experiment, leading to errors in the conclusions that are drawn. We present a speculative population model of the experiment and show, under the assumption of its correctness, that mutation need not be worried about very much when interpreting the experimental results.

Integro-Differential Equations (IDEs) are a novel way of dynamically modeling spatio-temporal data. IDEs are characterized by a kernel which controls the spatial and temporal associations. The ubiquitous choice for kernel has been Gaussian. We explore advantages of more flexible kernel choices. One-dimensional space is considered initially, replacing the Gaussian IDE kernel with more flexible parametric families of distributions. The kernels are chosen based on stochastic partial differential equation approximations which connect characteristics of the kernel with interpretable physical properties of the underlying process controlling the data. Next, Dirichlet process mixtures of normal distributions are used to model non-parametrically the IDE kernel. Computational issues arise using non-parametric kernels which are solved using Hermite polynomials and Hamiltonian Monte Carlo sampling. To develop flexible modeling in two-dimensional space, we propose bivariate stable distributions as IDE kernels. By using Bernstein polynomials as a prior for the measure defining the bivariate stable, a wide variety of shapes can be achieved. Bivariate stable kernels will be shown to outperform the Gaussian kernel by comparing K-step ahead predictions for Pacific sea surface temperature anomalies. Through study of properties for the proposed models, and empirical investigation with synthetic and real data, we demonstrate that the methodology has the potential to significantly improve the inference and forecasting capacity of IDE models based on Gaussian kernels.

On the first part, we consider nonlinear operators I depending on a family of nonlocal linear operators [mathematical equations]. We study the solutions of the Dirichlet initial and boundary value problems [mathematical equations]. We do not assume even symmetry for the kernels. The odd part bring some sort of nonlocal drift term, which in principle competes against the regularization of the solution. Existence and uniqueness is established for viscosity solutions. Several Hölder estimates are established for u and its derivatives under special assumptions. Moreover, the estimates remain uniform as the order of the equation approaches the second order case. This allows to consider our results as an extension of the classical theory of second order fully nonlinear equations. On the second part, we study two phase problems posed over a two dimensional cone generated by a smooth curve [mathematical symbol] on the unit sphere. We show that when [mathematical equation] the free boundary avoids the vertex of the cone. When [mathematical equation]we provide examples of minimizers such that the vertex belongs to the free boundary.

This thesis is divided into two independent parts. The first part concerns the 3D Navier-Stokes equations. The second part deals with regularity issues for a family of integro-differential equations. In the first part of this thesis, we consider weak solutions of the 3D Navier-Stokes equations with L² initial data. We prove that ([Nabla superscript alpha])u is locally integrable in space-time for any real [alpha] such that 1 < [alpha] < 3. Up to now, only the second derivative ([Nabla]²)u was known to be locally integrable by standard parabolic regularization. We also present sharp estimates of those quantities in local weak-L[superscript (4/([alpha]+1))]. These estimates depend only on the L² norm of the initial data and on the domain of integration. Moreover, they are valid even for [alpha] ≥ 3 as long as u is smooth. The proof uses a standard approximation of Navier-Stokes from Leray and blow-up techniques. The local study is based on De Giorgi techniques with a new pressure decomposition. To handle the non-locality of fractional Laplacians, Hardy space and Maximal functions are introduced. In the second part of this thesis, we consider non-local integro-differential equations under certain natural assumptions on the kernel, and obtain persistence of Hölder continuity for their solutions. In other words, we prove that a solution stays in C[superscript beta] for all time if its initial data lies in C[superscript beta]. Also, we prove a C[superscript beta]-regularization effect from [mathematical equation] initial data. It provides an alternative proof to the result of Caffarelli, Chan and Vasseur [10], which was based on De Giorgi techniques. This result has an application for a fully non-linear problem, which is used in the field of image processing. In addition, we show Hölder regularity for solutions of drift diffusion equations with supercritical fractional diffusion under the assumption [mathematical equation]on the divergent-free drift velocity. The proof is in the spirit of Kiselev and Nazarov [48] where they established Hölder continuity of the critical surface quasi-geostrophic (SQG) equation by observing the evolution of a dual class of test functions.

This thesis introduces mathematical models to understand the role of peer influenceon the spread of alcohol abuse. Our first model involves an integro-differential equation (IDE) model and is based on the Braun et al [BWPBG] network model. IDEproblems often arise in studying non-local interactions between populations [M]. Weprovide a mathematical formula to explain the relationship between a constant population resilience level and the initial average peer influence, a detailed analysis of a regularized partial differential equation approximation to the IDE problem traveling wave solutions, and a mathematical model for treatment that would be available to severe alcoholics.We also propose a new network model that shows similar dynamics with the one in Braun et al (2006). However, this new approach does not have a certain degeneracy we observed in the Braun model. This work is the first step toward the development of continuum models for the dynamics of social networks. There are many exciting new directions to pursue including applications in obesity and gang violence as well as evolution of network connection dynamics.