Refine
Has Fulltext
- yes (30) (remove)
Year of publication
- 2012 (30) (remove)
Document Type
- Preprint (26)
- Doctoral Thesis (4)
Language
- English (30)
Is part of the Bibliography
- yes (30)
Keywords
- Gibbs point processes (2)
- Heat equation (2)
- Riemannian manifold (2)
- counting process (2)
- infinite divisibility (2)
- infinitely divisible point processes (2)
- reciprocal class (2)
- Boundary value methods (1)
- Boundary value problems for first order systems (1)
- Brownian bridge (1)
- Brownian motion (1)
- CCR-algebra (1)
- Cluster Entwicklung (1)
- Cluster expansion (1)
- Determinantal point processes (1)
- Dirac-harmonic maps (1)
- Dirac-harmonische Abbildungen (1)
- Dirac-type operator (1)
- Dirichlet form (1)
- Dirichlet problem (1)
- Duality formula (1)
- Dualitätsformeln (1)
- Eigenvalues (1)
- Elliptic operators (1)
- Eyring-Kramers Formel (1)
- Feynman-Kac formula (1)
- Finite difference method (1)
- Fischer-Riesz equations (1)
- Fourth order Sturm-Liouville problem (1)
- Gibbssche Punktprozesse (1)
- Gradient flow (1)
- Gradientenfluss (1)
- Green formula (1)
- Green's operator (1)
- Heat Flow (1)
- Hypoelliptic operators (1)
- Lefschetz number (1)
- Levy Maß (1)
- Levy measure (1)
- Levy process (1)
- Liouville theorem (1)
- Lipschitz domains (1)
- Lévy measure (1)
- Malliavin calculus (1)
- Metastabilität (1)
- Montel theorem (1)
- Numerov's method (1)
- Perturbed complexes (1)
- Renormalized integral (1)
- Semiklassische Spektralasymptotik (1)
- Spin Geometrie (1)
- Spin Geometry (1)
- Sturm-Liouville problems (1)
- Tunneleffekt (1)
- Vitali theorem (1)
- Wave operator (1)
- Wiener measure (1)
- Wärmefluss (1)
- Zählprozesse (1)
- absorbing boundary (1)
- boundary layer (1)
- boundary regularity (1)
- boundary value problems (1)
- characteristic boundary point (1)
- characteristic points (1)
- cluster expansion (1)
- coercivity (1)
- completeness (1)
- conditional Wiener measure (1)
- continuous time Markov chain (1)
- curvature (1)
- cusp (1)
- dbar-Neumann problem (1)
- decay of eigenfunctions (1)
- determinantal point processes (1)
- determinantische Punktprozesse (1)
- discontinuous Robin condition (1)
- discrete Witten complex (1)
- diskreter Witten-Laplace-Operator (1)
- duality formulae (1)
- elliptic boundary conditions (1)
- finsler distance (1)
- generalized Laplace operator (1)
- globally hyperbolic spacetime (1)
- hitting times (1)
- hypoelliptic estimate (1)
- interaction matrix (1)
- jump process (1)
- low-lying eignvalues (1)
- metastability (1)
- nichtlineare partielle Differentialgleichung (1)
- nonlinear partial differential equations (1)
- nonsmooth curves (1)
- path integral (1)
- permanental- (1)
- regularisation (1)
- rescaled lattice (1)
- reziproke Klassen (1)
- root functions (1)
- semi-classical difference operator (1)
- semiclassical spectral asymptotics (1)
- singular integral equations (1)
- spectral kernel function (1)
- stochastic mechanics (1)
- stochastische Mechanik (1)
- strongly pseudoconvex domains (1)
- the first boundary value problem (1)
- time duality (1)
- tunneling (1)
- unendlich teilbare Punktprozesse (1)
- unendliche Teilbarkeit (1)
Institute
- Institut für Mathematik (30) (remove)
We develop the method of Fischer-Riesz equations for general boundary value problems elliptic in the sense of Douglis-Nirenberg. To this end we reduce them to a boundary problem for a (possibly overdetermined) first order system whose classical symbol has a left inverse. For such a problem there is a uniquely determined boundary value problem which is adjoint to the given one with respect to the Green formula. On using a well elaborated theory of approximation by solutions of the adjoint problem, we find the Cauchy data of solutions of our problem.
Asymptotic solutions of the Dirichlet problem for the heat equation at a characteristic point
(2012)
The Dirichlet problem for the heat equation in a bounded domain is characteristic, for there are boundary points at which the boundary touches a characteristic hyperplane t = c, c being a constant. It was I.G. Petrovskii (1934) who first found necessary and sufficient conditions on the boundary which guarantee that the solution is continuous up to the characteristic point, provided that the Dirichlet data are continuous. This paper initiated standing interest in studying general boundary value problems for parabolic equations in bounded domains. We contribute to the study by constructing a formal solution of the Dirichlet problem for the heat equation in a neighbourhood of a characteristic boundary point and showing its asymptotic character.
We introduce a theoretical framework for performing statistical hypothesis testing simultaneously over a fairly general, possibly uncountably infinite, set of null hypotheses. This extends the standard statistical setting for multiple hypotheses testing, which is restricted to a finite set. This work is motivated by numerous modern applications where the observed signal is modeled by a stochastic process over a continuum. As a measure of type I error, we extend the concept of false discovery rate (FDR) to this setting. The FDR is defined as the average ratio of the measure of two random sets, so that its study presents some challenge and is of some intrinsic mathematical interest. Our main result shows how to use the p-value process to control the FDR at a nominal level, either under arbitrary dependence of p-values, or under the assumption that the finite dimensional distributions of the p-value process have positive correlations of a specific type (weak PRDS). Both cases generalize existing results established in the finite setting, the latter one leading to a less conservative procedure. The interest of this approach is demonstrated in several non-parametric examples: testing the mean/signal in a Gaussian white noise model, testing the intensity of a Poisson process and testing the c.d.f. of i.i.d. random variables. Conceptually, an interesting feature of the setting advocated here is that it focuses directly on the intrinsic hypothesis space associated with a testing model on a random process, without referring to an arbitrary discretization.
The authors discuss the use of the discrepancy principle for statistical inverse problems, when the underlying operator is of trace class. Under this assumption the discrepancy principle is well defined, however a plain use of it may occasionally fail and it will yield sub-optimal rates. Therefore, a modification of the discrepancy is introduced, which takes into account both of the above deficiencies. For a variety of linear regularization schemes as well as for conjugate gradient iteration this modification is shown to yield order optimal a priori error bounds under general smoothness assumptions. A posteriori error control is also possible, however at a sub-optimal rate, in general. This study uses and complements previous results for bounded deterministic noise.
This thesis investigates the gradient flow of Dirac-harmonic maps. Dirac-harmonic maps are critical points of an energy functional that is motivated from supersymmetric field theories. The critical points of this energy functional couple the equation for harmonic maps with spinor fields. At present, many analytical properties of Dirac-harmonic maps are known, but a general existence result is still missing. In this thesis the existence question is studied using the evolution equations for a regularized version of Dirac-harmonic maps. Since the energy functional for Dirac-harmonic maps is unbounded from below the method of the gradient flow cannot be applied directly. Thus, we first of all consider a regularization prescription for Dirac-harmonic maps and then study the gradient flow. Chapter 1 gives some background material on harmonic maps/harmonic spinors and summarizes the current known results about Dirac-harmonic maps. Chapter 2 introduces the notion of Dirac-harmonic maps in detail and presents a regularization prescription for Dirac-harmonic maps. In Chapter 3 the evolution equations for regularized Dirac-harmonic maps are introduced. In addition, the evolution of certain energies is discussed. Moreover, the existence of a short-time solution to the evolution equations is established. Chapter 4 analyzes the evolution equations in the case that the domain manifold is a closed curve. Here, the existence of a smooth long-time solution is proven. Moreover, for the regularization being large enough, it is shown that the evolution equations converge to a regularized Dirac-harmonic map. Finally, it is discussed in which sense the regularization can be removed. In Chapter 5 the evolution equations are studied when the domain manifold is a closed Riemmannian spin surface. For the regularization being large enough, the existence of a global weak solution, which is smooth away from finitely many singularities is proven. It is shown that the evolution equations converge weakly to a regularized Dirac-harmonic map. In addition, it is discussed if the regularization can be removed in this case.
We introduce renormalized integrals which generalize conventional measure theoretic integrals. One approximates the integration domain by measure spaces and defines the integral as the limit of integrals over the approximating spaces. This concept is implicitly present in many mathematical contexts such as Cauchy's principal value, the determinant of operators on a Hilbert space and the Fourier transform of an L^p function. We use renormalized integrals to define a path integral on manifolds by approximation via geodesic polygons. The main part of the paper is dedicated to the proof of a path integral formula for the heat kernel of any self-adjoint generalized Laplace operator acting on sections of a vector bundle over a compact Riemannian manifold.
A linear differential operator L is called weakly hypoelliptic if any local solution u of Lu = 0 is smooth. We allow for systems, i.e. the coefficients may be matrices, not necessarily of square size. This is a huge class of important operators which covers all elliptic, overdetermined elliptic, subelliptic and parabolic equations. We extend several classical theorems from complex analysis to solutions of any weakly hypoelliptic equation: the Montel theorem providing convergent subsequences, the Vitali theorem ensuring convergence of a given sequence, and Riemann's first removable singularity theorem. In the case of constant coefficients we show that Liouville's theorem holds, any bounded solution must be constant and any L^p solution must vanish.
We study boundary value problems for linear elliptic differential operators of order one. The underlying manifold may be noncompact, but the boundary is assumed to be compact. We require a symmetry property of the principal symbol of the operator along the boundary. This is satisfied by Dirac type operators, for instance. We provide a selfcontained introduction to (nonlocal) elliptic boundary conditions, boundary regularity of solutions, and index theory. In particular, we simplify and generalize the traditional theory of elliptic boundary value problems for Dirac type operators. We also prove a related decomposition theorem, a general version of Gromov and Lawson's relative index theorem and a generalization of the cobordism theorem.