Lehrstuhl I: Analysis
Permanent URI for this collection
Browse
Recent Submissions
Item Non-uniformly parabolic equations and applications to the random conductance model(2021-07-30) Bella, Peter; Schäffner, MathiasWe study local regularity properties of linear, non-uniformly parabolic finite-difference operators in divergence form related to the random conductance model on Zd. In particular, we provide an oscillation decay assuming only certain summability properties of the conductances and their inverse, thus improving recent results in that direction. As an application, we provide a local limit theorem for the random walk in a random degenerate and unbounded environment.Item Analyse von Wellenpaketen in der periodischen nichtlinearen Schrödingergleichung durch Approximation mittels Coupled Mode Gleichungen(2018) Wahlers, Lisa; Dohnal, Tomas; Schweizer, BenDer Kern dieser Arbeit ist die Untersuchung von Wellenpaketen in periodischen Strukturen für unterschiedliche Dimensionen. Als asymptotische Skalierung betrachten wir Wellenpakete aus mehreren tragenden Blochwellen mit unterschiedlichen Gruppengeschwindigkeiten. Dadurch leiten wir ein Amplitudensystem erster Ordnung her, die Coupled Mode Gleichungen. Im eindimensionalen Fall haben die Coupled Mode Gleichungen eine Familie von lokalisierten Solitärwellen, welche durch die Geschwindigkeit parametrisiert sind. Weil diese Solitärwellen in der spektralen Lücke der Coupled Mode Gleichungen existieren, werden sie Gap Solitone genannt. Wir beschäftigen uns mit der Frage, ob für die Coupled Mode Gleichungen in höheren Dimensionen ebenfalls eine Familie von beweglichen Gap Solitonen existiert. Für den zweidimensionalen Fall und einen Ansatz aus vier Moden finden wir numerisch stehende Solitärwellen und beweisen anschließend die Existenz von lokalisierten Wellen für die Coupled Mode Gleichungen. Schließlich führen wir eine rigorose Rechtfertigung der Coupled Mode Gleichungen als asymptotisches Modell für die periodische nichtlineare Schrödingergleichung für den allgemeinen Fall von d Dimensionen durch.Item Traveling wave solutions of reaction-diffusion equations with x-dependent combustion type nonlinearities(2016-03) Badke, Sven; Schweizer, Ben; Röger, MatthiasWe investigate the existence and uniqueness of traveling wave solutions of the reaction-diffusion equation in periodic heterogeneous media. The reaction-diffusion equation is considered in nondivergence form with no first order term. Our traveling wave problem is considered in similar form in [1] by Xin in the special case that the reaction-term is given by a combustion nonlinearity ƒ = ƒ(u). We prove the existence of traveling wave solutions in case of a class of nonlinearities ƒ = ƒ(x, u), which are a generalization of a combustion nonlinearity. In particular, ƒ is allowed to depend explicitly on x. In case of an additional assumption on ƒ, we also prove a monotonicity result and a uniqueness result. References [1] X. Xin. Existence and uniqueness of travelling waves in a reaction-diffusion equation with combustion nonlinearity. Indiana Univ. Math. J., 40(3):985–1008, 1991.Item Uniqueness and regularity for porous media equations with x-dependent coefficients(2014) Koch, Jan-Christopher; Schweizer, Ben; Röger, MatthiasItem A worst-case optimization approach to impulse perturbed stochastic control with application to financial risk management(2012-08-07) Mönnig, Laurenz; Guiaş, Flavius; Schweizer, BenThis work presents the main ideas, methods and results of the theory of impulse perturbed stochastic control as an extension of the classic stochastic control theory. Apart from the introduction and the motivation of the basic concept, two stochastic optimization problems are the focus of the investigations. On the one hand we consider a differential game as analogue of the expected utility maximization problem in the situation with impulse perturbation, and on the other hand we study an appropriate version of a target problem. By dynamic optimization principles we characterize the associated value functions by systems of partial differential equations (PDEs). More precisely, we deal with variational inequalities whose single inequalities comprise constrained optimization problems, where the corresponding admissibility sets again are given by the seeked value functions. Using the concept of viscosity solutions as weak solutions of PDEs, we avoid strong regularity assumptions on the value functions. To use this concept as sufficient verification method, we additionally have to prove the uniqueness of the solutions of the PDEs. As a second major part of this work we apply the presented theory of impulse perturbed stochastic control in the field of financial risk management where extreme events have to be taken into account in order to control risks in a reasonable way. Such extreme scenarios are modelled by impulse controls and the financial decisions are made with respect to the worstcase scenario. In a first example we discuss portfolio problems as well as pricing problems on a capital market with crash risk. In particular, we consider the possibility of trading options and study their in uence on the investor's performance measured by the expected utility of terminal wealth. This brings up the question of crash-adjusted option prices and leads to the introduction of crash insurance. The second application concerns an insurance company which faces potentially large losses from extreme damages. We propose a dynamic model where the insurance company controls its risk process by reinsurance in form of proportional reinsurance and catastrophe reinsurance. Optimal reinsurance strategies are obtained by maximizing expected utility of the terminal surplus value and by minimizing the required capital reserves associated to the risk process.Item Waves in heterogeneous media: long time behavior and dispersive models(2011-09-12) Lamacz, Agnes; Schweizer, Ben; Röger, MatthiasItem On the local well-posedness of the Kadomtsev-Petviashvili II equation(2007-08-27T09:48:25Z) Hadac, Martin; Koch, Herbert; Saut, Jean-ClaudeItem Well-posedness results for dispersive equations with derivative nonlinearities(2006-08-28T12:14:46Z) Herr, Sebastian; Koch, Herbert; Kenig, Carlos E.Item Eine analytische Methode zur Punktereduktion und Flächenrekonstruktion(2005-09-27T09:31:45Z) Guias, Adina Aurelia; Koch, Herbert; Stöckler, Joachim