Multilevel large deviations
Authors
- Dawson, Donald A.
- Gärtner, Jürgen
2010 Mathematics Subject Classification
- 60F10 60K35 60J60
Keywords
- large deviations, hierarchical systems, interacting diffusions, empirical measures, randomly pertubated systems, McKean-Vlasov interaction
DOI
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WIAS Preprint No. 1698, (1992)About some coercive inequalities for elementary elliptic and parabolic operators.
Authors
- Koshelev, A.
DOI
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WIAS Preprint No. 1698, (1992)The singularity spectrum of self-affine fractals with a Bernoulli measure.
Authors
- Schmeling, Jörg
ORCID: 0000-0001-6956-9463 - Siegmund-Schultze, Rainer
DOI
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WIAS Preprint No. 1698, (1992)Flow and reactive transport in porous media induced by well injection: similarity solution.
Authors
- Knabner, Peter
- van Duijn, C.J.
2010 Mathematics Subject Classification
- 35K65 35K57 35R35
Keywords
- degenerate parabolic equation, flow in porous media, similarity solution, asymptotic analysis
DOI
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WIAS Preprint No. 1698, (1992)Proceedings of the 1. workshop on stochastic numerics.
Authors
- Platen, Eckhard
2010 Mathematics Subject Classification
- 60H10
Keywords
- stochastic differential equations, numerical solution, statistical methods, simulation, limit theorems, financial mathematics
DOI
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WIAS Preprint No. 1698, (1992)International Symposium "Operator Equations and Numerical Analysis" September 28 - October 2, 1992 Gosen (nearby Berlin).
Authors
- Prößdorf, Siegfried (ed.)
DOI
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WIAS Preprint No. 1698, (1992)The discrete spectrum of the Dirac operators on certain symmetric spaces.
Authors
- Seifarth, Sönke
DOI
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WIAS Preprint No. 1698, (1992)Hölder continuity of the holonomy maps for hyperbolic basic sets II
Authors
- Schmeling, Jörg
ORCID: 0000-0001-6956-9463
2010 Mathematics Subject Classification
- 37D99
Keywords
- generic properties, holonomy mappings
DOI
Appeared in
- Math. Nachr., 170 (1994), pp. 211--225
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WIAS Preprint No. 1698, (1992)A super-Brownian motion with a single point catalyst.
Authors
- Dawson, Donald A.
- Fleischmann, Klaus
2010 Mathematics Subject Classification
- 60J80 60J55 60G57
Keywords
- Point-catalytic medium, critical branching, super-Brownian motion, superprocess, measure-valued branching, Hausdorff dimension, occupation time, occupation density, local extinction
DOI
Appeared in
- Stochastic Process. Appl., 49 (1994), pp. 3--40
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WIAS Preprint No. 1698, (1992)Stochastic systems of particles with weights and approximation of the Boltzmann equation. The Markov process in the spatially homogeneous case.
Authors
- Wagner, Wolfgang
2010 Mathematics Subject Classification
- 60K35 76P05 82C40
Keywords
- stochastic particle systems, convergence of empirical measures, Boltzmann equation
DOI
Abstract
A class of stochastic systems of particles with variable weights is studied. The corresponding empirical measures are shown to converge to the solution of the spatially homogeneous Boltzmann equation. In a certain sense, this class of stochastic processes generalizes the "Kac master process" ([4]).
Appeared in
- Stochastic Anal. Appl., 12 (1994), pp. 639--659
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WIAS Preprint No. 1698, (1992)On the convergence of algebraically defined multigrid methods.
Authors
- Fuhrmann, Jürgen
ORCID: 0000-0003-4432-2434
Keywords
- Multigrid methods, Algebraic multigrid, Schar complement iterative methods, Iterative methods for convection-diffusion equations
DOI
Abstract
Based on the theory for multigrid methods with nonnested spaces and noninherited quadratic forms, a V-cycle convergence proof for an algebraically defined multigrid method using the approximation and smoothing property from the theory of algebraic multigrid is given. The estimation of the approximation property is carried out by means of strengthened Cauchy inequalities. Further, a method is suggested which allows to construct multigrid algorithms for special nonsymmetric problems. The ideas of the paper are illustrated by some examples of multigrid methods for problems with strongly varying coefficients in two- and three-dimensional rectangular domains.
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WIAS Preprint No. 1698, (1992)Quantization and measurability in gauge theory and gravity.
Authors
- Schmelzer, Ilja
DOI
Abstract
Considering a mental experiment with a superposition of quasiclassical basic states for gauge theory and gravitation we obtain nonclassical quantum observables. They can be interpreted as the difference of the gauge potentials of the basic states in gauge theory and a homeomorphism of the metrcis of the basic states in general relativity. It is possible to consider gauge- and coordinate conditions (f.e. Lorentz gauge and harmonic coordinates) as new physical equations for these obervables. For gravity we obtain in this way an interesting quasi-classical generalization of general relativity with two times - the time of general relativity and the harmonic time.
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WIAS Preprint No. 1698, (1992)Zur direkten Lösung linearer Gleichungssysteme auf Shared und Distributed Memory Systemen.
Authors
- Hebermehl, Georg
2010 Mathematics Subject Classification
- 65F05 65Y05 65Y10
Keywords
- Gauß-Elimination, Rank-r LU Update Verfahren, blockzyklische Datenaufteilung, Topologie, Kommunikation, Shared Memory System, Distributed Memory System, Message Passing
DOI
Abstract
Der Gaußsche Algorithtmus zur Lösung linearer Gleichungssysteme Ax = b wird für Shared Memory Systeme als Rank-r LU Update Verfahren und bei blockzyklischer Aufteilung von A für Distributed Memory Systeme als Message Passing Implementation vorgestellt.
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WIAS Preprint No. 1698, (1992)Iterative Verfahren für lineare Gleichungssysteme mit schwach besetzten Koeffizientenmatrizen.
Authors
- Schlundt, Rainer
ORCID: 0000-0002-4424-4301
2010 Mathematics Subject Classification
- 65F10
Keywords
- große linare Systeme, iterative Verfahren, Krylov-Unterraum-Methoden, GMRES-Algorithmus, QMR-Algorithmus
DOI
Abstract
Für die Lösung großer linearer Gleichungssysteme mit schwach besetzten Koeffizientenmatrizen werden das GMRES-Verfahren und die QMR-Methode vorgestellt. Beide iterativen Verfahren basieren auf Krylov-Unterraum-Methoden. Es werden sowohl die Gram-Schmidt als auch die Householder-Orthogonalisierung für GMRS betrachtet. Das QMR-Verfahren wird mit dem look-ahead Lanczos-Algorithmus kombiniert. Ein einfacher Vergleich zwischen GMRES und QMR wird angegeben.
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WIAS Preprint No. 1698, (2012)Representation of hysteresis operators for vector-valued continuous monotaffine input functions by functions on strings
Authors
- Klein, Olaf
ORCID: 0000-0002-4142-3603
2010 Mathematics Subject Classification
- 47J40
Keywords
- hysteresis Operators, vectorial hysteresis, string representation
DOI
Abstract
In Brokate-Sprekels-1996, it was shown that scalar-valued hysteresis operators for scalar-valued continuous piecewise monotone input functions can be uniquely represented by functionals defined on the set of all finite alternating strings of real numbers. Using this representation, various properties of these hysteresis operators were investigated. In this work, it is shown that a similar representation result can be derived for hysteresis operators dealing with inputs in a general topological linear vector space. Introducing a new class of functions, the so-called emphmonotaffine functions, which can be considered as a vector generalization of monotone scalar functions, and the convexity triple free strings on a vector space as a generalization of the alternating strings allows to formulate the corresponding representation result. As an example for the application of the representation result, a vectorial formulation of the second and third Madelung rule are discussed.
Appeared in
- Adv. Math. Sci. Appl., 22 (2012) pp. 471-500, with new title: Representation of hysteresis operators acting on vector-valued monotaffine functions
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WIAS Preprint No. 1698, (1992)Piecewise polynomial collocation for the double layer potential equation over polyhedral boundaries. Part I: The wedge, Part II: The cube.
Authors
- Rathsfeld, Andreas
ORCID: 0000-0002-2029-5761
2010 Mathematics Subject Classification
- 45L10 65R20
Keywords
- potential equation, collocation
DOI
Abstract
In this paper we consider a piecewise polynomial method for the solution of the double layer potential equation corresponding to Lapalce's equation in a three-dimensional wedge. We prove the stability for our method in case of special triangulations over the boundaty.
Appeared in
- Boundary Value Problems and Integral Equations on Nonsmooth Domains, M. Costabel, M. Dauge , S. Nicaise, eds., vol. 167 of Lecture Notes in Pure and Applied Mathematics, Marcel Dekker, Inc., New York, 1994, pp. 218--253
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WIAS Preprint No. 1698, (1992)Asymptotic equivalence of density estimation and white noise.
Authors
- Nussbaum, Michael
2010 Mathematics Subject Classification
- 62G07 62B15 62G20
Keywords
- Nonparametric experiments, deficiency distance, curve estimation, likelihood ratio process, Hungarian construction, asymptotic minimax risk, exact constants, Hellinger loss, linear wavelets estimators
DOI
Abstract
Signal recovery in Gaussian white noise with variance tending to zero has served for some time as a representative model for nonparametric curve estimation, having all the essential traits in a purified form. The equivalence has mostly been stated informally, but an approximation in the sense of Le Cam's deficiency distance Δ would make it precise. Then two models are asymptotically equivalent for all purposes of statistical decision with bounded loss. In nonparametrics, a first result of this kind has recently been established for Gaussian regression (Brown and Low, 1992). We consider the analogous problem for the experiment given by n i. i. d. observations having density ƒ on the unit interval. Our basic result concerns the parameter space of densities which are in a Sobolev class of order 4 and uniformly bounded away from zero. We show that an i. i. d. sample of size n with density ƒ is globally asymptotically equivalent to a white noise experiment with trend ƒ1/2 and variance 1⁄4n-1. This represents a nonparametric analog of Le Cam's heteroskedastic Gaussian approximation in the finite dimensional case. The proof utilizes empirical process techniques, especially the Hungarian construction. White noise models on ƒ and log ƒ are also considered, allowing for various "automatic" asymptotic risk bounds in the i. i. d. model from white noise. As first applications we discuss linear wavelet estimators of a density and exact constants for Hellinger loss.
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WIAS Preprint No. 1698, (1992)A minimum-distance estimator for diffusion processes with ergodic properties.
Authors
- Dietz, H. M.
- Kutoyants, Y.
2010 Mathematics Subject Classification
- 62M05 60J60
Keywords
- minimum distance estimator, diffusion process, ergodic property, consistency, asymptotic normality
DOI
Abstract
Suppose one observes one path of a stochastic process X = (Xt)t ≥ 0 which is known to solve an equation of the form
dXθt = S(θ, Xθt)dt + dWt, t ≥ 0, θ ∈ Θ ⊂ ℝd (0.1)
with a given coefficient functional S and given initial condition X0, where Θ is a non-void bounded open subset of ℝd. In order to estimate the true but unknown parameter θ0 the paper proposes the minimum distance estimator (MDE) ̂θT given by
̂θT ∈ arg inf θ ∈ Θ ∫T0 (Xt-X(θ)t)2dt, T > 0, (0.2)
where
X (θ)t ≔ X0 + ∫t0 S(θ,Xu) du, t ≥ 0 (0.3)
and studies its asymptotic behaviour as T → ∞. Under the main assumption that the observed process has an ergodic property and some further (less restrictive) conditions it is shown that ̂θT is strongly consistent and - in case d = 1 - asymptotically normal. In particular, the results apply to models where S(θ,x) = S(θ-x). Several examples and a comparison with likelihood estimation are added.
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WIAS Preprint No. 1698, (1992)Higher order approximate Markov chain filters.
Authors
- Kloeden, Peter E.
- Platen, Eckhard
- Schurz, Henri
2010 Mathematics Subject Classification
- 60G35 93E11
Keywords
- approximate discrete time filters for continuous time finite-state Markov chains, Wiener process
DOI
Abstract
The aim of this paper is to construct higher order approximate discrete time filters for continuous time finite-state Markov chains with observations that are perturbed by the noise of a Wiener process.
Appeared in
- Cambanis, Stamatis (ed.) et al., Stochastic processes: a festschrift in honour of Gopinath Kallianpur. New York: Springer-Verlag. 181-190 (1993).
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WIAS Preprint No. 1698, (1992)Hedging of options under discrete observation on assets with stochastic volatility.
Authors
- Di Masi, G. B.
- Platen, Eckhard
- Runggaldier, W. J.
2010 Mathematics Subject Classification
- 60H10
Keywords
- Stochastic differential equations, option pricing, stochastic volatility, discrete observation
DOI
Abstract
The paper considers the hedging of contingent claims on assets with stoachstic volatilities when the asset price is only observable at discrete time instants. Explicit fomulae are given for risk-minimizing hedging strategies.
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WIAS Preprint No. 1698, (1992)Balanced implicit methods for stiff stochastic systems: An introduction and numerical experiments.
Authors
- Milstein, Grigori N.
- Platen, Eckhard
- Schurz, Henri
2010 Mathematics Subject Classification
- 60H10
Keywords
- Stochastic differential equations, implicit numerical methods, stiff equations, simulation experiments
DOI
Abstract
The paper introduces implicitness in stochastic terms of numerical methods for solving of stiff stochastic differential equations and especially a class of fully implicit methods, the balanced methods. Their order of strong convergence is proved. Systematic numerical experiments compare the numerical behaviour of these schemes with that of different other schemes. A wide class of model equations are also provided as one by-product in order to test numerical methods in the case of stochastic stiffness in the given system.
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WIAS Preprint No. 1698, (1992)On optimal random nets.
Authors
- Mathé, Peter
ORCID: 0000-0002-1208-1421
2010 Mathematics Subject Classification
- 65C05 41A65 65J05
Keywords
- ε-entropy, Sobolev embeddings, random nets
DOI
Abstract
The possibility to approximate bounded linear mappings between Banach spaces depends on the degree of compactness. One way to measure this degree of compactness is the scale of entropy numbers, cf. [CS90]. In the usual (worst-case) setting of numerical analysis this scale has been studied for a long time. Recent reserach is concerned with the study of the so-called average-case and randomized (Monte Carlo) settings. We propose the respective counterparts of the entropy numbers in these settings and obtain their behavior for Sobolev embeddings. It turns out that, at least in this situation, randomly chosen nets may not improve the approximability of operators in the Monte Carlo setting. However, we can use the results to improve previous estimates for average Kolmogorov numbers, as obtained in [Mat91].
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WIAS Preprint No. 1698, (1992)Wavelet approximation methods for pseudodifferential equations I: stability and convergence.
Authors
- Dahmen, Wolfgang
- Prößdorf, Sigfried
- Schneider, R.
2010 Mathematics Subject Classification
- 65R20 65N35 65N30 45E05 45E10 41A25 41A63 47G30
Keywords
- Refinable functions, wavelets, periodic pseudodifferential operators, generalized Galerkin-Petrov schemes, discrete commutator property, stability analysis, convergence estimates
DOI
Abstract
This is the first part of two papers which are concerned with generalized Petrov-Galerkin schemes for elliptic periodic pseudodifferential equations in ℝn covering classical Galerkin methods, collocation, and quasiinterpolation. These methods are based on a general setting of multiresolution analysis, i.e., of sequences of nested spaces which are generated by refinable functions. In this part we develop a general stability and convergence theory for such a framework which recovers and extends many previously studied special cases. The key to the analysis is a local principle due to the second author. Its applicability relies here on a sufficiently general version of a so called discrete commutator property. These results establish important prerequisites for developing and analysing in the second part mehods for the fast solution of the resulting linear systems. These methods are based on compressing the stiffness matrices relative to wavelet bases for the given multiresolution analysis.
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WIAS Preprint No. 1698, (1992)Random approximation of finite sums.
Authors
- Mathé, Peter
ORCID: 0000-0002-1208-1421
2010 Mathematics Subject Classification
- 41A44 65C05 65D30
Keywords
- Monte Carlo integration, optimal error, least favorable distribution
DOI
Abstract
This paper is devoted to a detailed study of the randomized approximation of finite sums, i.e., sums ∑mj=1 xj, x ∈ ℝm, where m is supposed to be large, shall be approximated with information on n coordinates, only. The error is measured on balls in lmp, 1 ≤ p ≤ ∞. Main emphasis is laid on the exact solution of the problems stated below. In most cases we obtain both, an optimal method for the Monte Carlo setting and the description of least favorable distributions for the average case setting, exhibiting results obtained in a previous paper by the author, [Mat92]. Moreover, the solution of the finite-dimensional problem is applied to the Monte Carlo integration of continuous functions. Finally, this knowledge is used to study some of the properties, the optimal methods possess.
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WIAS Preprint No. 1698, (1992)Existence and uniqueness results for equations modelling transport of dopants in semiconductors.
Authors
- Glitzky, Annegret
ORCID: 0000-0003-1995-5491 - Gröger, Konrad
- Hünlich, Rolf
Keywords
- Transport of dopants in semiconductors, reaction-diffusion equations, Lypunov-function, a-priori estimates, global existence, uniqueness, asymptotic behaviour
DOI
Abstract
This paper is devoted to the analytical investigation of some non-linear reaction-diffusion system modelling the transport of dopants in semiconductors. Estimates by the energy functional and L∞-estimates obtained by a modified De Giorgi method imply global existence and uniqueness as well as results concerning the asymptotic behaviour.
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WIAS Preprint No. 1698, (1992)Boundary element discretization of Poincaré-Steklov operators.
Authors
- Schmidt, Gunther
2010 Mathematics Subject Classification
- 65N38 65N30 65N55 35J25
Keywords
- Galerkin method, Poincaré-Steklov operators, boundary element method, domain decomposition methods, finite element
DOI
Abstract
This paper is devoted to the construction of a discretization of Poincaré-Steklov (PS) operators for elliptic boundary value problems with the boundary element method (BEM). PS operators are natural mathematical tools for the investigation of boundary value problems and their numerical solution with domain decomposition (DD) methods based on the finite element (FE) solution of the subproblems (cf. [1], [9]). We will show that the discretizations of PS operators with a direct Galerkin BEM possess the same properties as the FE discretizations if the boundary elements satisfy some natural conditions. Hence the given construction provides a base for the analysis of different DD methods using the BE solution of subproblems, of the coupling of FE and BE methods and related problems.
Appeared in
- Numer. Math., 69 (1994), pp. 83--101
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WIAS Preprint No. 1698, (1992)The modeling of reactive solute transport with sorption to mobile and immobile sorbents.
Authors
- Knabner, Peter
- Kögel-Knabner, I.
- Totsche, K.U.
2010 Mathematics Subject Classification
- 76S05 76R50
Keywords
- flow in porous media, adsorption, contaminant transport, mathematical modeling
DOI
Abstract
This paper presents a mathematical model to describe the transport of reactive solutes with sorption to mobile and immobile sorbents. The mobile sorbent is considered to be reactive, too. The sorption processes mentioned are equilibrium and nonequilibrium processes. A transformation of the model in terms of total concentrations of solute and mobile sorbents is presented which simplifies the mathematical formulation. Effective isotherms, which describe the sorption to the immobile sorbent in the presence of a mobile sorbent and rate functions are introduced and their properties are discussed. The differences to existing approaches to model reactive solute transport are shown. Possible extensions are pointed out and the numerical approximation is sketched. The restrictions of the model as a consequence of the assumptions made on reactive solute transport are not due to mathematical reasons, but due to limitations of experimental information available.
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WIAS Preprint No. 1698, (1992)Pointwise confidence intervals in nonparametric regression with heteroscedastic error structure.
Authors
- Neumann, Michael H.
2010 Mathematics Subject Classification
- 62G15 62G07 62E20
Keywords
- Nonparametric regression, asymptotic confidence interval, error in coverage probability, Edgeworth expansion, wild bootstrap
DOI
Abstract
We assume a nonparametric model with heteroscedastic error structure and consider pointwise confidence intervals for the mean. We construct confidence intervals by using quantiles from a Cornish-Fisher expansion and from the wild bootstrap distribution, with as well as without a subsequent bias correction. It turns out that pure undersmoothing, where the full smoothness is used by the initial estimator, outperforms the method with a subsequent bias correction.
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WIAS Preprint No. 1698, (1992)Large deviation probabilities for some rescaled superprocesses.
Authors
- Fleischmann, Klaus
- Kaj, I.
2010 Mathematics Subject Classification
- 60J80 60F10 60G57
Keywords
- Large Deviation, superprocess, cumulant equation, rate functional
DOI
Abstract
We consider a class of rescaled superprocesses and derive a full large deviation principle with a good convex rate functional defined on the measure state space. A relatively complete picture of the related non-linear reaction-diffusion equation is accomplished although the rate functional is only partly expressed in terms of solutions of the equation.
Appeared in
- Annales de l'I.H.P. Probabilités et statistiques, Volume 30 (1994) no. 4, p. 607-645
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WIAS Preprint No. 1698, (1992)The h-p-version of spline approximation methods for Mellin convolution equations.
Authors
- Elschner, Johannes
2010 Mathematics Subject Classification
- 65R20
Keywords
- Mellin convolution equations, spline approximation methods, h-p-version
DOI
Abstract
We consider the numerical solution of Mellin convolution equations on an interval by the h-p-version of spline approximation methods. Using a geometric mesh refinement towards the singularity of the integral equation, we prove stability and exponential convergence in the Lq norm, 1 ≤ q ≤ ∞, for Galerkin, collocation and Nyström methods based on piecewise polynomials.
Appeared in
- Integral Eq. Appl. 5, (1993), pp. 47-73
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WIAS Preprint No. 1698, (1992)Spectral properties of one-dimensional Schrödinger operators with potentials generated by substitutions.
Authors
- Bovier, Anton
- Ghez, Jean-Michel
2010 Mathematics Subject Classification
- 35J10
Keywords
- substitution, trace map, spectra of one-dimensional discrete Schrödinger operators
DOI
Abstract
We investigate one-dimensional discrete Schrödinger operators whose potentials are invariant under a substitution rule. The spectral properties of these operators can be obtained from the analysis of a dynamical system, called the trace map. We give a careful derivation of these maps in the general case and exhibit some specific properties. Under an additional, easily verifiable hypothesis concerning the structure of the trace map we present an analysis of their dynamical properties that allows us to prove that the spectrum of the underlying Schrödinger operator is singular and supported on a set of zero Lebesgue measure. A condition allowing to exclude point spectrum is also given. The application of our theorems is explained on a series of examples.
Appeared in
- Commun. Math. Phys. 158 (1993), pp. 45-66
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WIAS Preprint No. 1698, (1992)Free energy estimates and asymptotic behaviour of reaction-diffusion processes.
Authors
- Gröger, Konrad
2010 Mathematics Subject Classification
- 35K57 35K55 35K45
Keywords
- Reaction-diffusion systems, asymptotic behaviour, Lyapunov function
DOI
Abstract
We prove a class of inequalities closely related to Poincaré's Inequality. Roughly speaking, these inequalites state for many reaction-diffusion systems the free energy can be estimated by the corresponding dissipation rate. This allows to describe the asymptotic behaviour of such reaction-diffusion systems without using global uniform bounds for the concentrations.
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WIAS Preprint No. 1698, (1992)On uniqueness of solutions to the drift-diffusion-model of semiconductor devices.
Authors
- Gajewski, Herbert
2010 Mathematics Subject Classification
- 35K55 35K57 35Q99
Keywords
- drift-diffusion-model, device equations, uniqueness of transient solutions, energy functional, Lyapunov function
DOI
Abstract
We prove a uniqueness result for the drift-diffusion-model of semiconductor devices under weak regularity assumptions. Our proof rests on the convexity of the free energy functional and uses a new concavity argument.
Appeared in
- Math. Models Methods Appl. Sci., 4 (1994), pp.121-139.
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WIAS Preprint No. 1698, (1992)Diffusive clustering in an infinite system of hierarchically interacting diffusions.
Authors
- Fleischmann, Klaus
- Greven, Andreas
2010 Mathematics Subject Classification
- 60K35 60J60 60J15
Keywords
- Interacting diffusions, coalescing random walk, hierachial system, cluster formation, universality
DOI
Abstract
We study a countable system of interacting diffusions on the interval [0,1], indexed by a hierarchical group. A particular choice of the interaction guarantees, we are in the diffusive clustering regime. This means clusters of components with values either close to 0 or close to 1 grow on various different scales. However, single components oscillate infinitely often between values close to 0 and close to 1 in such a way that they spend fraction one of their time together and close to the boundary. The processes in the whole class considered and starting with a shift-ergodic initial law have the same qualitative properties (universality).
Appeared in
- Probab. Theor. Relat. Fields, 98 (1994), pp. 517--566
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WIAS Preprint No. 1698, (1992)The thermodynamics of the Curie-Weiss model with random couplings.
Authors
- Bovier, Anton
- Gayrard, Veronique
Keywords
- Curie-Weiss model, random graphs, disordered magnets, mean-field theory
DOI
Abstract
We study the Curie-Weiss version of an Ising spin system with random, positively biased, couplings. In particular the case where the couplings ∈ij take the values one with probability p and zero with probability 1 - p which describes the Ising model on a random graph is considered. We prove that if p is allowed to decrease with the system size N in such a way that Np(N) ↑ ∞ as N ↑ ∞, then the free energy converges (after trivial rescaling) to that of the standard Curie Weiss model, almost surely. Equally, the induced measures on the mean magnetizations converge to those of the Curie-Weiss model. Generalizations of this result to a wide class of distributions are detailed.
Appeared in
- J. Stat. Phys. 76 (1993), pp. 643-664
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WIAS Preprint No. 1698, (1992)Rigorous results on the thermodynamics of the dilute Hopfield model.
Authors
- Bovier, Anton
- Gayrard, Veronique
2010 Mathematics Subject Classification
- 92B20
Keywords
- Neural networks, Hopfield model, random graphs, mean-field theory
DOI
Abstract
We study the Hopfield model of an autoassociative memory on a random graph on N vertices where the probability of two vertices being joined by a link is p(N). Assuming that p(N) goes to zero more slowly than O(1/N), we prove the following results: 1) If the number of stored patterns, m(N), is small enough such that m(N)/(Np(N)) ↓ 0, as N ↑ ∞, then the free energy of this model converges, upon proper rescaling, to that of the standard Curie-Weiss model, for almost all choices of the random graph and the random patterns. 2) If in addition m(N) > ln N / ln 2, we prove that there exists, for T > 1, a Gibbs measure associated to each original pattern, whereas for higher temperatures the Gibbs measure is unique. The basic technical result in the proofs is a uniform bound on the difference between the Hamiltonian on a random graph and its mean value.
Appeared in
- J. Stat. Phys. 72 (1993), pp. 79-112
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