We consider a system of infinitely many hard balls in Rd undergoing Brownian motions and submitted to a smooth pair potential. It is modelized by an infinite-dimensional Stochastic Differential Equation with a local time term. We prove that the set of all equilibrium measures, solution of a Detailed Balance Equation, coincides with the set of canonical Gibbs measures associated to the hard core potential added to the smooth interaction potential.

This thesis contains four papers and two introductory chapters. It is mainly devoted to problems concerning random growth models related to the Loewner differential equation. In Paper I we derive a rate of convergence of the Loewner driving function for loop-erased random walk to Brownian motion with speed 2 on the unit circle, the Loewner driving function for radial SLE(2). Thereby we provide the first instance of a formal derivation of a rate of convergence for any of the discrete models known to converge to SLE. In Paper II we use the known convergence of (radial) loop-erased random walk to radial SLE(2) to prove that the scaling limit of loop-erased random walk excursion in the upper half plane is chordal SLE(2). Our proof relies on a version of Wilson’s algorithm for weighted graphs together with a Beurling-type hitting estimate for random walk excursion. We also establish and use the convergence of the radial SLE path to the chordal SLE path as the bulk point tends to a boundary point. In the final section we sketch how to extend our results to more general domains. In Paper III we prove an upper bound on the optimal Hölder exponent for the chordal SLE path parameterized by capacity and thereby establish the optimal exponent as conjectured by J. Lind. We also give a new proof of the lower bound. Our proofs are based on sharp estimates of moments of the derivative of the inverse SLE map. In particular, we improve an estimate of G. F. Lawler. In Paper IV we consider radial Loewner evolutions driven by unimodular Lévy processes. We rescale the hulls of the evolution by capacity, and prove that the weak limit of the rescaled hulls exists. We then study a random growth model obtained by driving the Loewner equation with a compound Poisson process with two real parameters: the intensity of the underlying Poisson process and a localization parameter of the Poisson kernel which determines the jumps. A particular choice of parameters yields a growth process similar to the Hastings-Levitov HL(0) model. We describe the asymptotic behavior of the hulls with respect to the parameters, showing that growth tends to become localized as the jump parameter increases. We obtain deterministic evolutions in one limiting case, and Loewner evolution driven by a unimodular Cauchy process in another. We also show that the Hausdorff dimension of the limiting rescaled hulls is equal to 1.

This report presents an analysis of a partial differential equation, resulting from population model with age structure. The existence and uniqueness of a solution to the equation are proved. We look at stability of the solution. The asymptotic behaviour of the solution is treated. The report also contains a section about the connection between the solution to the age structure population model and a simple model without age structure.

The evolution of the price of two financial assets may be modeled by correlated geometric Brownian motion with additional, independent, finite activity jumps. Similarly, the evolution of the price of one financial asset may be modeled by a stochastic volatility process and finite activity jumps. The value of a contingent claim, written on assets where the underlying evolves by either of these two-factor processes, is given by the solution of a linear, two-dimensional, parabolic, partial integro-differential equation (PIDE). The focus of this thesis is the development of new, efficient numerical solution approaches for these PIDE's for both linear and non-linear cases. A localization scheme approximates the initial-value problem on an infinite spatial domain by an initial-boundary value problem on a finite spatial domain. Convergence of the localization method is proved using a Green's function approach. An implicit, finite difference method discretizes the PIDE. The theoretical conditions for the stability of the discrete approximation are examined under both maximum and von Neumann analysis. Three linearly convergent, monotone variants of the approach are reviewed for the constant coefficient, two-asset case and reformulated for the non-constant coefficient, stochastic volatility case. Each monotone scheme satisfies the conditions which imply convergence to the viscosity solution of the localized PIDE. A fixed point iteration solves the discrete, algebraic equations at each time step. This iteration avoids solving a dense linear system through the use of a lagged integral evaluation. Dense matrix-vector multiplication is avoided by using an FFT method. By using Green's function analysis, von Neumann analysis and maximum analysis, the fixed point iteration is shown to be rapidly convergent under typical market parameters. Combined with a penalty iteration, the value of options with an American early exercise feature may be computed. The rapid convergence of the iteration is verified in numerical tests using European and American options with vanilla payoffs, and digital, one-touch option payoffs. These tests indicate that the localization method for the PIDE's is effective. Adaptations are developed for degenerate or extreme parameter sets. The three monotone approaches are compared by computational cost and resulting error. For the stochastic volatility case, grid rotation is found to be the preferred approach. Finally, a new algorithm is developed for the solution of option values in the non-linear case of a two-factor option where the jump parameters are known only to within a deterministic range. This case results in a Hamilton-Jacobi-Bellman style PIDE. A monotone discretization is used and a new fixed point, policy iteration developed for time step solution. Analysis proves that the new iteration is globally convergent under a mild time step restriction. Numerical tests demonstrate the overall convergence of the method and investigate the financial implications of uncertain parameters on the option value.

The evolution of the price of two financial assets may be modeled by correlated geometric Brownian motion with additional, independent, finite activity jumps. Similarly, the evolution of the price of one financial asset may be modeled by a stochastic volatility process and finite activity jumps. The value of a contingent claim, written on assets where the underlying evolves by either of these two-factor processes, is given by the solution of a linear, two-dimensional, parabolic, partial integro-differential equation (PIDE). The focus of this thesis is the development of new, efficient numerical solution approaches for these PIDE's for both linear and non-linear cases. A localization scheme approximates the initial-value problem on an infinite spatial domain by an initial-boundary value problem on a finite spatial domain. Convergence of the localization method is proved using a Green's function approach. An implicit, finite difference method discretizes the PIDE. The theoretical conditions for the stability of the discrete approximation are examined under both maximum and von Neumann analysis. Three linearly convergent, monotone variants of the approach are reviewed for the constant coefficient, two-asset case and reformulated for the non-constant coefficient, stochastic volatility case. Each monotone scheme satisfies the conditions which imply convergence to the viscosity solution of the localized PIDE. A fixed point iteration solves the discrete, algebraic equations at each time step. This iteration avoids solving a dense linear system through the use of a lagged integral evaluation. Dense matrix-vector multiplication is avoided by using an FFT method. By using Green's function analysis, von Neumann analysis and maximum analysis, the fixed point iteration is shown to be rapidly convergent under typical market parameters. Combined with a penalty iteration, the value of options with an American early exercise feature may be computed. The rapid convergence of the iteration is verified in numerical tests using European and American options with vanilla payoffs, and digital, one-touch option payoffs. These tests indicate that the localization method for the PIDE's is effective. Adaptations are developed for degenerate or extreme parameter sets. The three monotone approaches are compared by computational cost and resulting error. For the stochastic volatility case, grid rotation is found to be the preferred approach. Finally, a new algorithm is developed for the solution of option values in the non-linear case of a two-factor option where the jump parameters are known only to within a deterministic range. This case results in a Hamilton-Jacobi-Bellman style PIDE. A monotone discretization is used and a new fixed point, policy iteration developed for time step solution. Analysis proves that the new iteration is globally convergent under a mild time step restriction. Numerical tests demonstrate the overall convergence of the method and investigate the financial implications of uncertain parameters on the option value.

In this thesis we are going to study the energy shaping problem on controlled Lagrangian systems with degree of underactuation less than or equal to two. Energy shaping is a method of stabilization by designing a suitable feedback control force on the given controlled Lagrangian system so that the total energy of the feedback equivalent system has a non-degenerate minimum at the equilibrium. The feedback equivalent system can then be stabilized by a further dissipative force. Finding a feedback equivalent system requires solving a system of PDEs. The existence of solutions for this system of PDEs is guaranteed, under some conditions, in the case of one degree of underactuation. Higher degrees of underactuation, however, requires a more careful study on the system of PDEs, and we apply the formal theory of PDEs to achieve this purpose in the case of two degrees of underactuation. The thesis is divided into four chapters. First, we review the basic notion of energy shaping and state the results for the case of one degree of underactuation. We then devise a general scheme to solve the energy shaping problem with degree of underactuation equal to one, together with some examples to illustrate the general procedure. After that we review the tools from the formal theory of PDEs, as a preparation for solving the problem with two degrees of underactuation. We derive an equivalent involutive system of PDEs from which we can deduce the existence of solutions which suit the energy shaping requirement.

We propose an algebraic construction of $(d+1)$-point $D$-optimal designs for $d$th degree polynomial regression with weight function $omega(x)ge 0$ on the interval $[a,b]$. Suppose that $omega'(x)/omega(x)$ is a rational function and the information of whether the optimal support contains the boundary points $a$ and $b$ is available. Then the problem of constructing $(d+1)$-point $D$-optimal designs can be transformed into a differential equation problem leading us to a certain matrix including a finite number of auxiliary unknown constants, which can be solved from a system of polynomial equations in those constants. Moreover, the $(d+1)$-point $D$-optimal interior support points are the zeros of a certain polynomial which the coefficients can be computed from a linear system. In most cases the $(d+1)$-point $D$-optimal designs are also the approximate $D$-optimal designs.

Beam problem is very important for engineering theoretically and practically. In this thesis we study such kind of nonlinear 4-th order ordiniary differential equations with nonlinear boundary conditions. The well-posedness of these boundary value problems will be discussed. Moreover, we will design different schemes to solve them, through differential equation, integral equation or minimization. Each type can further be discretized by finite difference, finite element or spectral method, etc. In the end we will compare all methods and find the best one.

Im Folgenden wird die analytische Lösung der eindimensionalen, instationären Wärmeleitungsgleichung mit einer Randbedingung 3. Art gegeben. Die Außentemperatur wird dabei als harmonische Schwingung angenommen. Abhängig von den materialspezifischen Eigenschaften des Bauteils (Wärmeleitfähigkeit, Rohdichte, spezifische Wärmekapazität) kommt es zur Dämpfung und zeitlichen Verschiebung der Temperaturwelle im Bauteil. Die analytische Lösung liefert den raum- und zeitaufgelösten Temperaturverlauf innerhalb des Bauteils. Die analytische Lösung ist primär für die Kalibrierung und Validierung numerischer Approximationsverfahren relevant. Die zeitliche Verfügbarkeit von thermischer Speichermasse ist für die thermische Gebäude- und Raumsimulation von besonderer Wichtigkeit. Daher muss ein numerisches Berechnungsverfahren diese Prozesse gut abbilden können. Die hier gezeigte analytische Lösung kann daher zur Bewertung der Güte der gewählten numerischen Approximation verwendet werden. Zu diesem Zweck werden Ergebnisse beispielhaft für zwei getrennte Konstruktionen angegeben.

In this master thesis it is analyzed the system of four partial fluxions of the primary row of differential equations the row of which dwindles at the point of p+1, when p = 0. The system of partial fluxions of differential equations has been solved through the modified technique of summarized degree rows. Two cases were explored: general and when the system’s ratios initial matrix is a special structure matrix. The solutions of the system dependence on the ratios, which are near partial fluxions of the searched function, were explored. There were designed four families of the detached solutions and proved that each of them depends on one optional function of y and z variables. The structure of system of solutions at the malformation points depends not only on x degree, but also on y and z variables. There is no such type of malformation in the theory of dwindle simple differential equations, therefore this analyzed dwindle of the system of partial fluxion of differential equations is called quasi-regular malformation. Keywords: differential equation, partial derivation, quasi-regular malformation.