Diffusive clustering in an infinite system of hierarchically interacting diffusions

We study a system of linearly interacting diffusions on the interval [0,1], indexed by an infinite hierachical group with parameter N, modeling the evolution of gene frequencies accounting for migration and resampling. Fisher-Wright diffusions are the typical example for the diffusion term. A particular choice of the migration term guaranties that we are in the so-called diffusive clustering regime. The processes in the whole class considered and starting with a shift-ergodic initial law have qualitative properties just as in the Fisher-Wright case (universality). Also, the initial law plays here no role. Clusters of components with values either close to 0 or close to 1 grow on various different scales (diffusive clustering). More precisely, at time N"t#->##infinity# (spatial) cluster sizes measured in a hierarchical distance grow like #alpha#t (i.e. consist of N"#alpha#"t components) where 1-#alpha# element of (0,1) is the hitting time of the traps (0,1) for a time transformed Fisher-Wright diffusion. Nevertheless, the single components oscillate infinitely often between values close to 0 and close to 1, but in such a way that they spend fraction one of their time together close to the boundary. Diffusive clustering phenomena we believe are in fact common to many interacting systems and were first discussed by Cox and Griffeath (1986) for the planar simple voter model. On the way, we prove some scaling results on a specific coalescing random walk with delay on the hierarchical group. (orig.)Available from TIB Hannover: RR 5549(23)+a / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekSIGLEDEGerman

Stability of weak numerical schemes for stochastic differential equations

The paper considers numerical stability and convergence of weak schemes solving stochastic differential equations. A relatively strong notion of stability for a special type of test equations is proposed. These are stochastic differential equations with multiplicative noise. For different explicit and implicit schemes the regions of stability are also examined. (orig.)Available from TIB Hannover: RR 5549(57)+a / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekSIGLEDEGerman

A functional law of large numbers for Boltzmann type stochastic particle systems

A large system of particles is studied. Its time evolution is determined as the superposition of two components. The first component is the independent motion of each particle. The second component is the random interaction mechansim between pairs of particles. The intensity of the interaction depends on the state of the system and is assumed to be bounded. Convergence of the empirical measures is proved as the number of particles tends to infinity. The limiting deterministic measure-valued funtion is characterized as the unique solution of a nonlinear equation of the Boltzmann type. (orig.)Available from TIB Hannover: RR 5549(93)+a / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekSIGLEDEGerman

Zur mathematischen Modellierung eines Halbleiterinjektionslasers mit Hilfe der Maxwellschen Gleichungen bei gegebener Stromverteilung

A semiconductor injection laser is described by basic equations which consist of the Maxwell equations and charge carrier conservation law in the active zone. It is shown that the corresponding initial boundary value problem for this system is well posed. (orig.)SIGLEAvailable from TIB Hannover: RR 5549(63)+a / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekDEGerman

Estimates of Weyl sums over subsequences of natural numbers

SIGLEAvailable from TIB Hannover: RR 5549(101)+a / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekDEGerman

On the functional equations of dynamical theta functions I

The twisted geodesic flow of compact locally symmetric spaces of rank one gives rise to a series of meromorphic functions on the complex plane satisfying simple functional equations. These results are discussed as part of geometric quantization and index theory. (orig.)Available from TIB Hannover: RR 5549(64)+a / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekSIGLEDEGerman

Stability of numerical schemes for stochastic differential equations with multiplicative noise

A notion of stability for a special type of test equations is proposed. These are stochastic differential equations with multiplicative noise for which there is a connection between the parameters in the drift and diffusion coefficient. By means of the Euler scheme and two different implicit Euler schemes a method to find the regions of stability is also examined. (orig.)Available from TIB Hannover: RR 5549(39)+a / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekSIGLEDEGerman

On the regularization of the ill-posed logarithmic kernel integral equation of the first kind

The logarithmic kernel integral equation of the first kind is investigated as improperly posed problem considering its right-hand side as observed quantity in a suitable space with a weaker norm. The improperly posed problem is decomposed into a well-posed one, extensively studied in the literature (cf. e.g. [11], [13], [14]), and an ill-posed imbedding problem. For the ill-posed part a modified truncated singular value decomposition regularization method is proposed that allows an easily performable a-posteriori parameter choice. The whole problem is then solved by combining the regularization method with a numerical procedure from [13] for the well-posed part. Finally, an error estimate is given revealing the influence of the observation error on the approximation error of the numerical procedure. For a specification of the discretization parameter as a known function of the noise level only, the optimal convergence order is achieved. (orig.)Available from TIB Hannover: RR 5549(46)+a / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekSIGLEDEGerman

On the computation of hyperbolic sets and their invariant manifolds

We describe a method for finding periodic orbits contained in a hyperbolic invariant set and of constructing their local stable and unstable manifolds, suitable to implement in a computer. (orig.)Available from TIB Hannover: RR 5549(68)+a / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekSIGLEDEGerman

Free energy estimates and asymptotic behaviour of reaction-diffusion processes

We prove a class of inequalities closely related to Poincares Inequality. Roughly speaking, these inequalities state that 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. (orig.)Available from TIB Hannover: RR 5549(20)+a / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekSIGLEDEGerman