@book{Kulik2015, author = {Kulik, Alexei Michajlovič}, title = {Introduction to Ergodic rates for Markov chains and processes}, editor = {Roelly, Sylvie}, publisher = {Universit{\"a}tsverlag Potsdam}, address = {Potsdam}, isbn = {978-3-86956-338-1}, issn = {2199-4951}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-79360}, publisher = {Universit{\"a}t Potsdam}, pages = {ix, 122 S.}, year = {2015}, abstract = {The present lecture notes aim for an introduction to the ergodic behaviour of Markov Processes and addresses graduate students, post-graduate students and interested readers. Different tools and methods for the study of upper bounds on uniform and weak ergodic rates of Markov Processes are introduced. These techniques are then applied to study limit theorems for functionals of Markov processes. This lecture course originates in two mini courses held at University of Potsdam, Technical University of Berlin and Humboldt University in spring 2013 and Ritsumameikan University in summer 2013. Alexei Kulik, Doctor of Sciences, is a Leading researcher at the Institute of Mathematics of Ukrainian National Academy of Sciences.}, language = {en} } @unpublished{DereudreMazzonettoRoelly2015, author = {Dereudre, David and Mazzonetto, Sara and Roelly, Sylvie}, title = {An explicit representation of the transition densities of the skew Brownian motion with drift and two semipermeable barriers}, volume = {4}, number = {9}, publisher = {Universit{\"a}tsverlag Potsdam}, address = {Potsdam}, issn = {2193-6943}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-80613}, pages = {23}, year = {2015}, abstract = {In this paper we obtain an explicit representation of the transition density of the one-dimensional skew Brownian motion with (a constant drift and) two semipermeable barriers. Moreover we propose a rejection method to simulate this density in an exact way.}, language = {en} } @unpublished{Conforti2015, author = {Conforti, Giovanni}, title = {Reciprocal classes of continuous time Markov Chains}, volume = {4}, number = {8}, publisher = {Universit{\"a}tsverlag Potsdam}, address = {Potsdam}, issn = {2193-6943}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-78234}, pages = {198}, year = {2015}, abstract = {In this work we study reciprocal classes of Markov walks on graphs. Given a continuous time reference Markov chain on a graph, its reciprocal class is the set of all probability measures which can be represented as a mixture of the bridges of the reference walks. We characterize reciprocal classes with two different approaches. With the first approach we found it as the set of solutions to duality formulae on path space, where the differential operators have the interpretation of the addition of infinitesimal random loops to the paths of the canonical process. With the second approach we look at short time asymptotics of bridges. Both approaches allow an explicit computation of reciprocal characteristics, which are divided into two families, the loop characteristics and the arc characteristics. They are those specific functionals of the generator of the reference chain which determine its reciprocal class. We look at the specific examples such as Cayley graphs, the hypercube and planar graphs. Finally we establish the first concentration of measure results for the bridges of a continuous time Markov chain based on the reciprocal characteristics.}, language = {en} } @unpublished{LyTarkhanov2015, author = {Ly, Ibrahim and Tarkhanov, Nikolai Nikolaevich}, title = {Asymptotic expansions at nonsymmetric cuspidal points}, volume = {4}, number = {7}, publisher = {Universit{\"a}tsverlag Potsdam}, address = {Potsdam}, issn = {2193-6943}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-78199}, pages = {11}, year = {2015}, abstract = {We study asymptotics of solutions to the Dirichlet problem in a domain whose boundary contains a nonsymmetric conical point. We establish a complete asymptotic expansion of solutions near the singular point.}, language = {en} } @unpublished{ElinShoikhetTarkhanov2015, author = {Elin, Mark and Shoikhet, David and Tarkhanov, Nikolai Nikolaevich}, title = {Analytic semigroups of holomorphic mappings and composition operators}, volume = {4}, number = {6}, publisher = {Universit{\"a}tsverlag Potsdam}, address = {Potsdam}, issn = {2193-6943}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-77914}, pages = {30}, year = {2015}, abstract = {In this paper we study the problem of analytic extension in parameter for a semigroup of holomorphic self-mappings of the unit ball in a complex Banach space and its relation to the linear continuous semigroup of composition operators. We also provide a brief review around this topic.}, language = {en} } @unpublished{FedosovTarkhanov2015, author = {Fedosov, Boris and Tarkhanov, Nikolai Nikolaevich}, title = {Deformation quantisation and boundary value problems}, volume = {4}, number = {5}, publisher = {Universit{\"a}tsverlag Potsdam}, address = {Potsdam}, issn = {2193-6943}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-77150}, pages = {27}, year = {2015}, abstract = {We describe a natural construction of deformation quantisation on a compact symplectic manifold with boundary. On the algebra of quantum observables a trace functional is defined which as usual annihilates the commutators. This gives rise to an index as the trace of the unity element. We formulate the index theorem as a conjecture and examine it by the classical harmonic oscillator.}, language = {en} } @unpublished{Tarkhanov2015, author = {Tarkhanov, Nikolai Nikolaevich}, title = {A spectral theorem for deformation quantisation}, volume = {4}, number = {4}, publisher = {Universit{\"a}tsverlag Potsdam}, address = {Potsdam}, issn = {2193-6943}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-72425}, pages = {8}, year = {2015}, abstract = {We present a construction of the eigenstate at a noncritical level of the Hamiltonian function. Moreover, we evaluate the contributions of Morse critical points to the spectral decomposition.}, language = {en} } @unpublished{LyTarkhanov2015, author = {Ly, Ibrahim and Tarkhanov, Nikolai Nikolaevich}, title = {A Rad{\´o} theorem for p-harmonic functions}, volume = {4}, number = {3}, publisher = {Universit{\"a}tsverlag Potsdam}, address = {Potsdam}, issn = {2193-6943}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-71492}, pages = {10}, year = {2015}, abstract = {Let A be a nonlinear differential operator on an open set X in R^n and S a closed subset of X. Given a class F of functions in X, the set S is said to be removable for F relative to A if any weak solution of A (u) = 0 in the complement of S of class F satisfies this equation weakly in all of X. For the most extensively studied classes F we show conditions on S which guarantee that S is removable for F relative to A.}, language = {en} } @unpublished{AlsaedyTarkhanov2015, author = {Alsaedy, Ammar and Tarkhanov, Nikolai Nikolaevich}, title = {Weak boundary values of solutions of Lagrangian problems}, volume = {4}, number = {2}, publisher = {Universit{\"a}tsverlag Potsdam}, address = {Potsdam}, issn = {2193-6943}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-72617}, pages = {24}, year = {2015}, abstract = {We define weak boundary values of solutions to those nonlinear differential equations which appear as Euler-Lagrange equations of variational problems. As a result we initiate the theory of Lagrangian boundary value problems in spaces of appropriate smoothness. We also analyse if the concept of mapping degree of current importance applies to the study of Lagrangian problems.}, language = {en} } @unpublished{ConfortiRoelly2015, author = {Conforti, Giovanni and Roelly, Sylvie}, title = {Reciprocal class of random walks on an Abelian group}, volume = {4}, number = {1}, publisher = {Universit{\"a}tsverlag Potsdam}, address = {Potsdam}, issn = {2193-6943}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-72604}, pages = {22}, year = {2015}, abstract = {Processes having the same bridges as a given reference Markov process constitute its reciprocal class. In this paper we study the reciprocal class of a continuous time random walk with values in a countable Abelian group, we compute explicitly its reciprocal characteristics and we present an integral characterization of it. Our main tool is a new iterated version of the celebrated Mecke's formula from the point process theory, which allows us to study, as transformation on the path space, the addition of random loops. Thanks to the lattice structure of the set of loops, we even obtain a sharp characterization. At the end, we discuss several examples to illustrate the richness of reciprocal classes. We observe how their structure depends on the algebraic properties of the underlying group.}, language = {en} } @phdthesis{Ziese2014, author = {Ziese, Ramona}, title = {Geometric electroelasticity}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-72504}, school = {Universit{\"a}t Potsdam}, pages = {vi, 113}, year = {2014}, abstract = {In this work a diffential geometric formulation of the theory of electroelasticity is developed which also includes thermal and magnetic influences. We study the motion of bodies consisting of an elastic material that are deformed by the influence of mechanical forces, heat and an external electromagnetic field. To this end physical balance laws (conservation of mass, balance of momentum, angular momentum and energy) are established. These provide an equation that describes the motion of the body during the deformation. Here the body and the surrounding space are modeled as Riemannian manifolds, and we allow that the body has a lower dimension than the surrounding space. In this way one is not (as usual) restricted to the description of the deformation of three-dimensional bodies in a three-dimensional space, but one can also describe the deformation of membranes and the deformation in a curved space. Moreover, we formulate so-called constitutive relations that encode the properties of the used material. Balance of energy as a scalar law can easily be formulated on a Riemannian manifold. The remaining balance laws are then obtained by demanding that balance of energy is invariant under the action of arbitrary diffeomorphisms on the surrounding space. This generalizes a result by Marsden and Hughes that pertains to bodies that have the same dimension as the surrounding space and does not allow the presence of electromagnetic fields. Usually, in works on electroelasticity the entropy inequality is used to decide which otherwise allowed deformations are physically admissible and which are not. It is alsoemployed to derive restrictions to the possible forms of constitutive relations describing the material. Unfortunately, the opinions on the physically correct statement of the entropy inequality diverge when electromagnetic fields are present. Moreover, it is unclear how to formulate the entropy inequality in the case of a membrane that is subjected to an electromagnetic field. Thus, we show that one can replace the use of the entropy inequality by the demand that for a given process balance of energy is invariant under the action of arbitrary diffeomorphisms on the surrounding space and under linear rescalings of the temperature. On the one hand, this demand also yields the desired restrictions to the form of the constitutive relations. On the other hand, it needs much weaker assumptions than the arguments in physics literature that are employing the entropy inequality. Again, our result generalizes a theorem of Marsden and Hughes. This time, our result is, like theirs, only valid for bodies that have the same dimension as the surrounding space.}, language = {en} } @unpublished{FedchenkoTarkhanov2014, author = {Fedchenko, Dmitry and Tarkhanov, Nikolai Nikolaevich}, title = {An index formula for Toeplitz operators}, volume = {3}, number = {12}, publisher = {Universit{\"a}tsverlag Potsdam}, address = {Potsdam}, issn = {2193-6943}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-72499}, pages = {24}, year = {2014}, abstract = {We prove a Fedosov index formula for the index of Toeplitz operators connected with the Hardy space of solutions to an elliptic system of first order partial differential equations in a bounded domain of Euclidean space with infinitely differentiable boundary.}, language = {en} } @phdthesis{Vu2014, author = {Vu, Dinh Phuong}, title = {Using video study to investigate eighth-grade mathematics classrooms in Vietnam}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-72464}, school = {Universit{\"a}t Potsdam}, pages = {273}, year = {2014}, abstract = {The International Project for the Evaluation of Educational Achievement (IEA) was formed in the 1950s (Postlethwaite, 1967). Since that time, the IEA has conducted many studies in the area of mathematics, such as the First International Mathematics Study (FIMS) in 1964, the Second International Mathematics Study (SIMS) in 1980-1982, and a series of studies beginning with the Third International Mathematics and Science Study (TIMSS) which has been conducted every 4 years since 1995. According to Stigler et al. (1999), in the FIMS and the SIMS, U.S. students achieved low scores in comparison with students in other countries (p. 1). The TIMSS 1995 "Videotape Classroom Study" was therefore a complement to the earlier studies conducted to learn "more about the instructional and cultural processes that are associated with achievement" (Stigler et al., 1999, p. 1). The TIMSS Videotape Classroom Study is known today as the TIMSS Video Study. From the findings of the TIMSS 1995 Video Study, Stigler and Hiebert (1999) likened teaching to "mountain ranges poking above the surface of the water," whereby they implied that we might see the mountaintops, but we do not see the hidden parts underneath these mountain ranges (pp. 73-78). By watching the videotaped lessons from Germany, Japan, and the United States again and again, they discovered that "the systems of teaching within each country look similar from lesson to lesson. At least, there are certain recurring features [or patterns] that typify many of the lessons within a country and distinguish the lessons among countries" (pp. 77-78). They also discovered that "teaching is a cultural activity," so the systems of teaching "must be understood in relation to the cultural beliefs and assumptions that surround them" (pp. 85, 88). From this viewpoint, one of the purposes of this dissertation was to study some cultural aspects of mathematics teaching and relate the results to mathematics teaching and learning in Vietnam. Another research purpose was to carry out a video study in Vietnam to find out the characteristics of Vietnamese mathematics teaching and compare these characteristics with those of other countries. In particular, this dissertation carried out the following research tasks: - Studying the characteristics of teaching and learning in different cultures and relating the results to mathematics teaching and learning in Vietnam - Introducing the TIMSS, the TIMSS Video Study and the advantages of using video study in investigating mathematics teaching and learning - Carrying out the video study in Vietnam to identify the image, scripts and patterns, and the lesson signature of eighth-grade mathematics teaching in Vietnam - Comparing some aspects of mathematics teaching in Vietnam and other countries and identifying the similarities and differences across countries - Studying the demands and challenges of innovating mathematics teaching methods in Vietnam - lessons from the video studies Hopefully, this dissertation will be a useful reference material for pre-service teachers at education universities to understand the nature of teaching and develop their teaching career.}, language = {en} } @unpublished{DereudreRoelly2014, author = {Dereudre, David and Roelly, Sylvie}, title = {Path-dependent infinite-dimensional SDE with non-regular drift : an existence result}, volume = {3}, number = {11}, publisher = {Universit{\"a}tsverlag Potsdam}, address = {Potsdam}, issn = {2193-6943}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-72084}, pages = {27}, year = {2014}, abstract = {We establish in this paper the existence of weak solutions of infinite-dimensional shift invariant stochastic differential equations driven by a Brownian term. The drift function is very general, in the sense that it is supposed to be neither small or continuous, nor Markov. On the initial law we only assume that it admits a finite specific entropy. Our result strongly improves the previous ones obtained for free dynamics with a small perturbative drift. The originality of our method leads in the use of the specific entropy as a tightness tool and on a description of such stochastic differential equation as solution of a variational problem on the path space.}, language = {en} } @phdthesis{Dyachenko2014, author = {Dyachenko, Evgeniya}, title = {Elliptic problems with small parameter}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-72056}, school = {Universit{\"a}t Potsdam}, year = {2014}, abstract = {In this thesis we consider diverse aspects of existence and correctness of asymptotic solutions to elliptic differential and pseudodifferential equations. We begin our studies with the case of a general elliptic boundary value problem in partial derivatives. A small parameter enters the coefficients of the main equation as well as into the boundary conditions. Such equations have already been investigated satisfactory, but there still exist certain theoretical deficiencies. Our aim is to present the general theory of elliptic problems with a small parameter. For this purpose we examine in detail the case of a bounded domain with a smooth boundary. First of all, we construct formal solutions as power series in the small parameter. Then we examine their asymptotic properties. It suffices to carry out sharp two-sided \emph{a priori} estimates for the operators of boundary value problems which are uniform in the small parameter. Such estimates failed to hold in functional spaces used in classical elliptic theory. To circumvent this limitation we exploit norms depending on the small parameter for the functions defined on a bounded domain. Similar norms are widely used in literature, but their properties have not been investigated extensively. Our theoretical investigation shows that the usual elliptic technique can be correctly carried out in these norms. The obtained results also allow one to extend the norms to compact manifolds with boundaries. We complete our investigation by formulating algebraic conditions on the operators and showing their equivalence to the existence of a priori estimates. In the second step, we extend the concept of ellipticity with a small parameter to more general classes of operators. Firstly, we want to compare the difference in asymptotic patterns between the obtained series and expansions for similar differential problems. Therefore we investigate the heat equation in a bounded domain with a small parameter near the time derivative. In this case the characteristics touch the boundary at a finite number of points. It is known that the solutions are not regular in a neighbourhood of such points in advance. We suppose moreover that the boundary at such points can be non-smooth but have cuspidal singularities. We find a formal asymptotic expansion and show that when a set of parameters comes through a threshold value, the expansions fail to be asymptotic. The last part of the work is devoted to general concept of ellipticity with a small parameter. Several theoretical extensions to pseudodifferential operators have already been suggested in previous studies. As a new contribution we involve the analysis on manifolds with edge singularities which allows us to consider wider classes of perturbed elliptic operators. We examine that introduced classes possess a priori estimates of elliptic type. As a further application we demonstrate how developed tools can be used to reduce singularly perturbed problems to regular ones.}, language = {en} } @unpublished{MakhmudovTarkhanov2014, author = {Makhmudov, Olimdjan and Tarkhanov, Nikolai Nikolaevich}, title = {The first mixed problem for the nonstationary Lam{\´e} system}, volume = {3}, number = {10}, publisher = {Universit{\"a}tsverlag Potsdam}, address = {Potsdam}, issn = {2193-6943}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-71923}, pages = {19}, year = {2014}, abstract = {We find an adequate interpretation of the Lam{\´e} operator within the framework of elliptic complexes and study the first mixed problem for the nonstationary Lam{\´e} system.}, language = {en} } @book{Pilipenko2014, author = {Pilipenko, Andrey}, title = {An introduction to stochastic differential equations with reflection}, series = {Lectures in pure and applied mathematics}, journal = {Lectures in pure and applied mathematics}, number = {1}, publisher = {Universit{\"a}tsverlag Potsdam}, address = {Potsdam}, isbn = {978-3-86956-297-1}, issn = {2199-4951}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-70782}, publisher = {Universit{\"a}t Potsdam}, pages = {ix, 75}, year = {2014}, abstract = {These lecture notes are intended as a short introduction to diffusion processes on a domain with a reflecting boundary for graduate students, researchers in stochastic analysis and interested readers. Specific results on stochastic differential equations with reflecting boundaries such as existence and uniqueness, continuity and Markov properties, relation to partial differential equations and submartingale problems are given. An extensive list of references to current literature is included. This book has its origins in a mini-course the author gave at the University of Potsdam and at the Technical University of Berlin in Winter 2013.}, language = {en} } @unpublished{ConfortiLeonardMurretal.2014, author = {Conforti, Giovanni and L{\´e}onard, Christian and Murr, R{\"u}diger and Roelly, Sylvie}, title = {Bridges of Markov counting processes : reciprocal classes and duality formulas}, volume = {3}, number = {9}, publisher = {Universit{\"a}tsverlag Potsdam}, address = {Potsdam}, issn = {2193-6943}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-71855}, pages = {12}, year = {2014}, abstract = {Processes having the same bridges are said to belong to the same reciprocal class. In this article we analyze reciprocal classes of Markov counting processes by identifying their reciprocal invariants and we characterize them as the set of counting processes satisfying some duality formula.}, language = {en} } @unpublished{FlandoliHoegele2014, author = {Flandoli, Franco and H{\"o}gele, Michael}, title = {A solution selection problem with small stable perturbations}, volume = {3}, number = {8}, publisher = {Universit{\"a}tsverlag Potsdam}, address = {Potsdam}, issn = {2193-6943}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-71205}, pages = {43}, year = {2014}, abstract = {The zero-noise limit of differential equations with singular coefficients is investigated for the first time in the case when the noise is a general alpha-stable process. It is proved that extremal solutions are selected and the probability of selection is computed. Detailed analysis of the characteristic function of an exit time form on the half-line is performed, with a suitable decomposition in small and large jumps adapted to the singular drift.}, language = {en} } @unpublished{PornsawadBoeckmann2014, author = {Pornsawad, Pornsarp and B{\"o}ckmann, Christine}, title = {Modified iterative Runge-Kutta-type methods for nonlinear ill-posed problems}, volume = {3}, number = {7}, publisher = {Universit{\"a}tsverlag Potsdam}, address = {Potsdam}, issn = {2193-6943}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-70834}, pages = {30}, year = {2014}, abstract = {This work is devoted to the convergence analysis of a modified Runge-Kutta-type iterative regularization method for solving nonlinear ill-posed problems under a priori and a posteriori stopping rules. The convergence rate results of the proposed method can be obtained under H{\"o}lder-type source-wise condition if the Fr{\´e}chet derivative is properly scaled and locally Lipschitz continuous. Numerical results are achieved by using the Levenberg-Marquardt and Radau methods.}, language = {en} } @phdthesis{Kollosche2014, author = {Kollosche, David}, title = {Gesellschaft, Mathematik und Unterricht : ein Beitrag zum soziologisch-kritischen Verst{\"a}ndnis der gesellschaftlichen Funktionen des Mathematikunterrichts}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-70726}, school = {Universit{\"a}t Potsdam}, year = {2014}, abstract = {Die vorliegende Studie untersucht die gesellschaftliche Rolle des gegenw{\"a}rtigen Mathematikunterrichts an deutschen allgemeinbildenden Schulen aus einer soziologisch-kritischen Perspektive. In Zentrum des Interesses steht die durch den Mathematikunterricht erfahrene Sozialisation. Die Studie umfasst unter anderem eine Literaturdiskussion, die Ausarbeitung eines soziologischen Rahmens auf der Grundlage des Werks von Michel Foucault und zwei Teilstudien zur Soziologie der Logik und des Rechnens. Abschließend werden Dispositive des Mathematischen beschrieben, die darlegen, in welcher Art und mit welcher pers{\"o}nlichen und gesellschaftlichen Folgen der gegenw{\"a}rtige Mathematikunterricht eine spezielle Geisteshaltung etabliert.}, language = {de} } @unpublished{ConfortiDaiPraRoelly2014, author = {Conforti, Giovanni and Dai Pra, Paolo and Roelly, Sylvie}, title = {Reciprocal class of jump processes}, volume = {3}, number = {6}, publisher = {Universit{\"a}tsverlag Potsdam}, address = {Potsdam}, issn = {2193-6943}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-70776}, pages = {30}, year = {2014}, abstract = {Processes having the same bridges as a given reference Markov process constitute its reciprocal class. In this paper we study the reciprocal class of compound Poisson processes whose jumps belong to a finite set A in R^d. We propose a characterization of the reciprocal class as the unique set of probability measures on which a family of time and space transformations induces the same density, expressed in terms of the reciprocal invariants. The geometry of A plays a crucial role in the design of the transformations, and we use tools from discrete geometry to obtain an optimal characterization. We deduce explicit conditions for two Markov jump processes to belong to the same class. Finally, we provide a natural interpretation of the invariants as short-time asymptotics for the probability that the reference process makes a cycle around its current state.}, language = {en} } @unpublished{HoegelePavlyukevich2014, author = {H{\"o}gele, Michael and Pavlyukevich, Ilya}, title = {Metastability of Morse-Smale dynamical systems perturbed by heavy-tailed L{\´e}vy type noise}, volume = {3}, number = {5}, publisher = {Universit{\"a}tsverlag Potsdam}, address = {Potsdam}, issn = {2193-6943}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-70639}, pages = {27}, year = {2014}, abstract = {We consider a general class of finite dimensional deterministic dynamical systems with finitely many local attractors each of which supports a unique ergodic probability measure, which includes in particular the class of Morse-Smale systems in any finite dimension. The dynamical system is perturbed by a multiplicative non-Gaussian heavytailed L{\´e}vy type noise of small intensity ε > 0. Specifically we consider perturbations leading to a It{\^o}, Stratonovich and canonical (Marcus) stochastic differential equation. The respective asymptotic first exit time and location problem from each of the domains of attractions in case of inward pointing vector fields in the limit of ε-> 0 has been investigated by the authors. We extend these results to domains with characteristic boundaries and show that the perturbed system exhibits a metastable behavior in the sense that there exits a unique ε-dependent time scale on which the random system converges to a continuous time Markov chain switching between the invariant measures. As examples we consider α-stable perturbations of the Duffing equation and a chemical system exhibiting a birhythmic behavior.}, language = {en} } @unpublished{SultanovKalyakinTarkhanov2014, author = {Sultanov, Oskar and Kalyakin, Leonid and Tarkhanov, Nikolai Nikolaevich}, title = {Elliptic perturbations of dynamical systems with a proper node}, volume = {3}, number = {4}, publisher = {Universit{\"a}tsverlag Potsdam}, address = {Potsdam}, issn = {2193-6943}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-70460}, pages = {12}, year = {2014}, abstract = {The paper is devoted to asymptotic analysis of the Dirichlet problem for a second order partial differential equation containing a small parameter multiplying the highest order derivatives. It corresponds to a small perturbation of a dynamical system having a stationary solution in the domain. We focus on the case where the trajectories of the system go into the domain and the stationary solution is a proper node.}, language = {en} } @unpublished{AizenbergTarkhanov2014, author = {Aizenberg, Lev A. and Tarkhanov, Nikolai Nikolaevich}, title = {An integral formula for the number of lattice points in a domain}, volume = {3}, number = {3}, publisher = {Universit{\"a}tsverlag Potsdam}, address = {Potsdam}, issn = {2193-6943}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-70453}, pages = {7}, year = {2014}, abstract = {Using the multidimensional logarithmic residue we show a simple formula for the difference between the number of integer points in a bounded domain of R^n and the volume of this domain. The difference proves to be the integral of an explicit differential form over the boundary of the domain.}, language = {en} } @unpublished{GairingHoegeleKosenkovaetal.2014, author = {Gairing, Jan and H{\"o}gele, Michael and Kosenkova, Tetiana and Kulik, Alexei Michajlovič}, title = {On the calibration of L{\´e}vy driven time series with coupling distances : an application in paleoclimate}, volume = {3}, number = {2}, publisher = {Universit{\"a}tsverlag Potsdam}, address = {Potsdam}, issn = {2193-6943}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-69781}, pages = {18}, year = {2014}, abstract = {This article aims at the statistical assessment of time series with large fluctuations in short time, which are assumed to stem from a continuous process perturbed by a L{\´e}vy process exhibiting a heavy tail behavior. We propose an easily implementable procedure to estimate efficiently the statistical difference between the noisy behavior of the data and a given reference jump measure in terms of so-called coupling distances. After a short introduction to L{\´e}vy processes and coupling distances we recall basic statistical approximation results and derive rates of convergence. In the sequel the procedure is elaborated in detail in an abstract setting and eventually applied in a case study to simulated and paleoclimate data. It indicates the dominant presence of a non-stable heavy-tailed jump L{\´e}vy component for some tail index greater than 2.}, language = {en} } @phdthesis{Kroencke2013, author = {Kr{\"o}ncke, Klaus}, title = {Stability of Einstein Manifolds}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-69639}, school = {Universit{\"a}t Potsdam}, year = {2013}, abstract = {This thesis deals with Einstein metrics and the Ricci flow on compact mani- folds. We study the second variation of the Einstein-Hilbert functional on Ein- stein metrics. In the first part of the work, we find curvature conditions which ensure the stability of Einstein manifolds with respect to the Einstein-Hilbert functional, i.e. that the second variation of the Einstein-Hilbert functional at the metric is nonpositive in the direction of transverse-traceless tensors. The second part of the work is devoted to the study of the Ricci flow and how its behaviour close to Einstein metrics is influenced by the variational be- haviour of the Einstein-Hilbert functional. We find conditions which imply that Einstein metrics are dynamically stable or unstable with respect to the Ricci flow and we express these conditions in terms of stability properties of the metric with respect to the Einstein-Hilbert functional and properties of the Laplacian spectrum.}, language = {en} } @unpublished{DyachenkoTarkhanov2014, author = {Dyachenko, Evgueniya and Tarkhanov, Nikolai Nikolaevich}, title = {Singular perturbations of elliptic operators}, volume = {3}, number = {1}, publisher = {Universit{\"a}tsverlag Potsdam}, address = {Potsdam}, issn = {2193-6943}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-69502}, pages = {21}, year = {2014}, abstract = {We develop a new approach to the analysis of pseudodifferential operators with small parameter 'epsilon' in (0,1] on a compact smooth manifold X. The standard approach assumes action of operators in Sobolev spaces whose norms depend on 'epsilon'. Instead we consider the cylinder [0,1] x X over X and study pseudodifferential operators on the cylinder which act, by the very nature, on functions depending on 'epsilon' as well. The action in 'epsilon' reduces to multiplication by functions of this variable and does not include any differentiation. As but one result we mention asymptotic of solutions to singular perturbation problems for small values of 'epsilon'.}, language = {en} } @unpublished{FedchenkoTarkhanov2013, author = {Fedchenko, Dmitry and Tarkhanov, Nikolai Nikolaevich}, title = {A Class of Toeplitz Operators in Several Variables}, issn = {2193-6943}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-68932}, year = {2013}, abstract = {We introduce the concept of Toeplitz operator associated with the Laplace-Beltrami operator on a compact Riemannian manifold with boundary. We characterise those Toeplitz operators which are Fredholm, thus initiating the index theory.}, language = {en} } @unpublished{RoellyRuszel2013, author = {Roelly, Sylvie and Ruszel, Wioletta M.}, title = {Propagation of Gibbsianness for infinite-dimensional diffusions with space-time interaction}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-69014}, year = {2013}, abstract = {We consider infinite-dimensional diffusions where the interaction between the coordinates has a finite extent both in space and time. In particular, it is not supposed to be smooth or Markov. The initial state of the system is Gibbs, given by a strong summable interaction. If the strongness of this initial interaction is lower than a suitable level, and if the dynamical interaction is bounded from above in a right way, we prove that the law of the diffusion at any time t is a Gibbs measure with absolutely summable interaction. The main tool is a cluster expansion in space uniformly in time of the Girsanov factor coming from the dynamics and exponential ergodicity of the free dynamics to an equilibrium product measure.}, language = {en} } @unpublished{GairingHoegeleKosenkovaetal.2013, author = {Gairing, Jan and H{\"o}gele, Michael and Kosenkova, Tetiana and Kulik, Alexei Michajlovič}, title = {Coupling distances between L{\´e}vy measures and applications to noise sensitivity of SDE}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-68886}, year = {2013}, abstract = {We introduce the notion of coupling distances on the space of L{\´e}vy measures in order to quantify rates of convergence towards a limiting L{\´e}vy jump diffusion in terms of its characteristic triplet, in particular in terms of the tail of the L{\´e}vy measure. The main result yields an estimate of the Wasserstein-Kantorovich-Rubinstein distance on path space between two L{\´e}vy diffusions in terms of the couping distances. We want to apply this to obtain precise rates of convergence for Markov chain approximations and a statistical goodness-of-fit test for low-dimensional conceptual climate models with paleoclimatic data.}, language = {en} } @unpublished{CattiauxFradonKuliketal.2013, author = {Cattiaux, Patrick and Fradon, Myriam and Kulik, Alexei Michajlovič and Roelly, Sylvie}, title = {Long time behavior of stochastic hard ball systems}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-68388}, year = {2013}, abstract = {We study the long time behavior of a system of two or three Brownian hard balls living in the Euclidean space of dimension at least two, submitted to a mutual attraction and to elastic collisions.}, language = {en} } @unpublished{LyTarkhanov2013, author = {Ly, Ibrahim and Tarkhanov, Nikolai Nikolaevich}, title = {Generalised Beltrami equations}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-67416}, year = {2013}, abstract = {We enlarge the class of Beltrami equations by developping a stability theory for the sheaf of solutions of an overdetermined elliptic system of first order homogeneous partial differential equations with constant coefficients in the Euclidean space.}, language = {en} } @unpublished{ShlapunovTarkhanov2013, author = {Shlapunov, Alexander and Tarkhanov, Nikolai Nikolaevich}, title = {Sturm-Liouville problems in domains with non-smooth edges}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-67336}, year = {2013}, abstract = {We consider a (generally, non-coercive) mixed boundary value problem in a bounded domain for a second order elliptic differential operator A. The differential operator is assumed to be of divergent form and the boundary operator B is of Robin type. The boundary is assumed to be a Lipschitz surface. Besides, we distinguish a closed subset of the boundary and control the growth of solutions near this set. We prove that the pair (A,B) induces a Fredholm operator L in suitable weighted spaces of Sobolev type, the weight function being a power of the distance to the singular set. Moreover, we prove the completeness of root functions related to L.}, language = {en} } @unpublished{Wallenta2013, author = {Wallenta, Daniel}, title = {A Lefschetz fixed point formula for elliptic quasicomplexes}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-67016}, year = {2013}, abstract = {In a recent paper with N. Tarkhanov, the Lefschetz number for endomorphisms (modulo trace class operators) of sequences of trace class curvature was introduced. We show that this is a well defined, canonical extension of the classical Lefschetz number and establish the homotopy invariance of this number. Moreover, we apply the results to show that the Lefschetz fixed point formula holds for geometric quasiendomorphisms of elliptic quasicomplexes.}, language = {en} } @phdthesis{DiGesu2012, author = {Di Ges{\`u}, Giacomo}, title = {Semiclassical spectral analysis of discrete Witten Laplacians}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-65286}, school = {Universit{\"a}t Potsdam}, year = {2012}, abstract = {A discrete analogue of the Witten Laplacian on the n-dimensional integer lattice is considered. After rescaling of the operator and the lattice size we analyze the tunnel effect between different wells, providing sharp asymptotics of the low-lying spectrum. Our proof, inspired by work of B. Helffer, M. Klein and F. Nier in continuous setting, is based on the construction of a discrete Witten complex and a semiclassical analysis of the corresponding discrete Witten Laplacian on 1-forms. The result can be reformulated in terms of metastable Markov processes on the lattice.}, language = {en} } @unpublished{AlsaedyTarkhanov2013, author = {Alsaedy, Ammar and Tarkhanov, Nikolai Nikolaevich}, title = {Normally solvable nonlinear boundary value problems}, issn = {2193-6943}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-65077}, year = {2013}, abstract = {We study a boundary value problem for an overdetermined elliptic system of nonlinear first order differential equations with linear boundary operators. Such a problem is solvable for a small set of data, and so we pass to its variational formulation which consists in minimising the discrepancy. The Euler-Lagrange equations for the variational problem are far-reaching analogues of the classical Laplace equation. Within the framework of Euler-Lagrange equations we specify an operator on the boundary whose zero set consists precisely of those boundary data for which the initial problem is solvable. The construction of such operator has much in common with that of the familiar Dirichlet to Neumann operator. In the case of linear problems we establish complete results.}, language = {en} } @unpublished{HoegeleRuffino2013, author = {H{\"o}gele, Michael and Ruffino, Paulo}, title = {Averaging along L{\´e}vy diffusions in foliated spaces}, issn = {2193-6943}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-64926}, year = {2013}, abstract = {We consider an SDE driven by a L{\´e}vy noise on a foliated manifold, whose trajectories stay on compact leaves. We determine the effective behavior of the system subject to a small smooth transversal perturbation of positive order epsilon. More precisely, we show that the average of the transversal component of the SDE converges to the solution of a deterministic ODE, according to the average of the perturbing vector field with respect to the invariant measures on the leaves (of the unpertubed system) as epsilon goes to 0. In particular we give upper bounds for the rates of convergence. The main results which are proved for pure jump L{\´e}vy processes complement the result by Gargate and Ruffino for Stratonovich SDEs to L{\´e}vy driven SDEs of Marcus type.}, language = {en} } @unpublished{KiselevTarkhanov2013, author = {Kiselev, Oleg and Tarkhanov, Nikolai Nikolaevich}, title = {The capture of a particle into resonance at potential hole with dissipative perturbation}, issn = {2193-6943}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-64725}, year = {2013}, abstract = {We study the capture of a particle into resonance at a potential hole with dissipative perturbation and periodic outside force. The measure of resonance solutions is evaluated. We also derive an asymptotic formula for the parameter range of those solutions which are captured into resonance.}, language = {en} } @phdthesis{Branding2012, author = {Branding, Volker}, title = {The evolution equations for Dirac-harmonic Maps}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-64204}, school = {Universit{\"a}t Potsdam}, year = {2012}, abstract = {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.}, language = {en} } @unpublished{MeleardRoelly2013, author = {M{\´e}l{\´e}ard, Sylvie and Roelly, Sylvie}, title = {Evolutive two-level population process and large population approximations}, issn = {2193-6943}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-64604}, year = {2013}, abstract = {We are interested in modeling the Darwinian evolution of a population described by two levels of biological parameters: individuals characterized by an heritable phenotypic trait submitted to mutation and natural selection and cells in these individuals influencing their ability to consume resources and to reproduce. Our models are rooted in the microscopic description of a random (discrete) population of individuals characterized by one or several adaptive traits and cells characterized by their type. The population is modeled as a stochastic point process whose generator captures the probabilistic dynamics over continuous time of birth, mutation and death for individuals and birth and death for cells. The interaction between individuals (resp. between cells) is described by a competition between individual traits (resp. between cell types). We are looking for tractable large population approximations. By combining various scalings on population size, birth and death rates and mutation step, the single microscopic model is shown to lead to contrasting nonlinear macroscopic limits of different nature: deterministic approximations, in the form of ordinary, integro- or partial differential equations, or probabilistic ones, like stochastic partial differential equations or superprocesses.}, language = {en} } @unpublished{LeonardRoellyZambrini2013, author = {L{\´e}onard, Christian and Roelly, Sylvie and Zambrini, Jean-Claude}, title = {Temporal symmetry of some classes of stochastic processes}, issn = {2193-6943}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-64599}, year = {2013}, abstract = {In this article we analyse the structure of Markov processes and reciprocal processes to underline their time symmetrical properties, and to compare them. Our originality consists in adopting a unifying approach of reciprocal processes, independently of special frameworks in which the theory was developped till now (diffusions, or pure jump processes). This leads to some new results, too.}, language = {en} } @unpublished{Roelly2013, author = {Roelly, Sylvie}, title = {Reciprocal processes : a stochastic analysis approach}, issn = {2193-6943}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-64588}, year = {2013}, abstract = {Reciprocal processes, whose concept can be traced back to E. Schr{\"o}dinger, form a class of stochastic processes constructed as mixture of bridges, that satisfy a time Markov field property. We discuss here a new unifying approach to characterize several types of reciprocal processes via duality formulae on path spaces: The case of reciprocal processes with continuous paths associated to Brownian diffusions and the case of pure jump reciprocal processes associated to counting processes are treated. This presentation is based on joint works with M. Thieullen, R. Murr and C. L{\´e}onard.}, language = {en} } @unpublished{NehringPoghosyanZessin2013, author = {Nehring, Benjamin and Poghosyan, Suren and Zessin, Hans}, title = {On the construction of point processes in statistical mechanics}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-64080}, year = {2013}, abstract = {By means of the cluster expansion method we show that a recent result of Poghosyan and Ueltschi (2009) combined with a result of Nehring (2012) yields a construction of point processes of classical statistical mechanics as well as processes related to the Ginibre Bose gas of Brownian loops and to the dissolution in R^d of Ginibre's Fermi-Dirac gas of such loops. The latter will be identified as a Gibbs perturbation of the ideal Fermi gas. On generalizing these considerations we will obtain the existence of a large class of Gibbs perturbations of the so-called KMM-processes as they were introduced by Nehring (2012). Moreover, it is shown that certain "limiting Gibbs processes" are Gibbs in the sense of Dobrushin, Lanford and Ruelle if the underlying potential is positive. And finally, Gibbs modifications of infinitely divisible point processes are shown to solve a new integration by parts formula if the underlying potential is positive.}, language = {en} } @unpublished{MakhmudovTarkhanov2013, author = {Makhmudov, Olimdjan and Tarkhanov, Nikolai Nikolaevich}, title = {An extremal problem related to analytic continuation}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-63634}, year = {2013}, abstract = {We show that the usual variational formulation of the problem of analytic continuation from an arc on the boundary of a plane domain does not lead to a relaxation of this overdetermined problem. To attain such a relaxation, we bound the domain of the functional, thus changing the Euler equations.}, language = {en} } @unpublished{KellerRoellyValleriani2013, author = {Keller, Peter and Roelly, Sylvie and Valleriani, Angelo}, title = {A quasi-random-walk to model a biological transport process}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-63582}, year = {2013}, abstract = {Transport Molecules play a crucial role for cell viability. Amongst others, linear motors transport cargos along rope-like structures from one location of the cell to another in a stochastic fashion. Thereby each step of the motor, either forwards or backwards, bridges a fixed distance. While moving along the rope the motor can also detach and is lost. We give here a mathematical formalization of such dynamics as a random process which is an extension of Random Walks, to which we add an absorbing state to model the detachment of the motor from the rope. We derive particular properties of such processes that have not been available before. Our results include description of the maximal distance reached from the starting point and the position from which detachment takes place. Finally, we apply our theoretical results to a concrete established model of the transport molecule Kinesin V.}, language = {en} } @unpublished{BagderinaTarkhanov2013, author = {Bagderina, Yulia Yu. and Tarkhanov, Nikolai Nikolaevich}, title = {Differential invariants of a class of Lagrangian systems with two degrees of freedom}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-63129}, year = {2013}, abstract = {We consider systems of Euler-Lagrange equations with two degrees of freedom and with Lagrangian being quadratic in velocities. For this class of equations the generic case of the equivalence problem is solved with respect to point transformations. Using Lie's infinitesimal method we construct a basis of differential invariants and invariant differentiation operators for such systems. We describe certain types of Lagrangian systems in terms of their invariants. The results are illustrated by several examples.}, language = {en} } @unpublished{Keller2013, author = {Keller, Peter}, title = {Mathematical modeling of molecular motors}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-63045}, year = {2013}, abstract = {Amongst the many complex processes taking place in living cells, transport of cargoes across the cytosceleton is fundamental to cell viability and activity. To move cargoes between the different cell parts, cells employ Molecular Motors. The motors operate by transporting cargoes along the so-called cellular micro-tubules, namely rope-like structures that connect, for instance, the cell-nucleus and outer membrane. We introduce a new Markov Chain, the killed Quasi-Random-Walk, for such transport molecules and derive properties like the maximal run length and time. Furthermore we introduce permuted balance, which is a more flexible extension of the ordinary reversibility and introduce the notion of Time Duality, which compares certain passage times pathwise. We give a number of sufficient conditions for Time Duality based on the geometry of the transition graph. Both notions are closely related to properties of the killed Quasi-Random-Walk.}, language = {en} } @unpublished{Murr2012, author = {Murr, R{\"u}diger}, title = {Reciprocal classes of Markov processes : an approach with duality formulae}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-63018}, year = {2012}, abstract = {In this work we are concerned with the characterization of certain classes of stochastic processes via duality formulae. First, we introduce a new formulation of a characterization of processes with independent increments, which is based on an integration by parts formula satisfied by infinitely divisible random vectors. Then we focus on the study of the reciprocal classes of Markov processes. These classes contain all stochastic processes having the same bridges, and thus similar dynamics, as a reference Markov process. We start with a resume of some existing results concerning the reciprocal classes of Brownian diffusions as solutions of duality formulae. As a new contribution, we show that the duality formula satisfied by elements of the reciprocal class of a Brownian diffusion has a physical interpretation as a stochastic Newton equation of motion. In the context of pure jump processes we derive the following new results. We will analyze the reciprocal classes of Markov counting processes and characterize them as a group of stochastic processes satisfying a duality formula. This result is applied to time-reversal of counting processes. We are able to extend some of these results to pure jump processes with different jump-sizes, in particular we are able to compare the reciprocal classes of Markov pure jump processes through a functional equation between the jump-intensities.}, language = {en} } @phdthesis{Nehring2012, author = {Nehring, Benjamin}, title = {Point processes in statistical mechanics : a cluster expansion approach}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-62682}, school = {Universit{\"a}t Potsdam}, year = {2012}, abstract = {A point process is a mechanism, which realizes randomly locally finite point measures. One of the main results of this thesis is an existence theorem for a new class of point processes with a so called signed Levy pseudo measure L, which is an extension of the class of infinitely divisible point processes. The construction approach is a combination of the classical point process theory, as developed by Kerstan, Matthes and Mecke, with the method of cluster expansions from statistical mechanics. Here the starting point is a family of signed Radon measures, which defines on the one hand the Levy pseudo measure L, and on the other hand locally the point process. The relation between L and the process is the following: this point process solves the integral cluster equation determined by L. We show that the results from the classical theory of infinitely divisible point processes carry over in a natural way to the larger class of point processes with a signed Levy pseudo measure. In this way we obtain e.g. a criterium for simplicity and a characterization through the cluster equation, interpreted as an integration by parts formula, for such point processes. Our main result in chapter 3 is a representation theorem for the factorial moment measures of the above point processes. With its help we will identify the permanental respective determinantal point processes, which belong to the classes of Boson respective Fermion processes. As a by-product we obtain a representation of the (reduced) Palm kernels of infinitely divisible point processes. In chapter 4 we see how the existence theorem enables us to construct (infinitely extended) Gibbs, quantum-Bose and polymer processes. The so called polymer processes seem to be constructed here for the first time. In the last part of this thesis we prove that the family of cluster equations has certain stability properties with respect to the transformation of its solutions. At first this will be used to show how large the class of solutions of such equations is, and secondly to establish the cluster theorem of Kerstan, Matthes and Mecke in our setting. With its help we are able to enlarge the class of Polya processes to the so called branching Polya processes. The last sections of this work are about thinning and splitting of point processes. One main result is that the classes of Boson and Fermion processes remain closed under thinning. We use the results on thinning to identify a subclass of point processes with a signed Levy pseudo measure as doubly stochastic Poisson processes. We also pose the following question: Assume you observe a realization of a thinned point process. What is the distribution of deleted points? Surprisingly, the Papangelou kernel of the thinning, besides a constant factor, is given by the intensity measure of this conditional probability, called splitting kernel.}, language = {en} } @phdthesis{Murr2012, author = {Murr, R{\"u}diger}, title = {Reciprocal classes of Markov processes : an approach with duality formulae}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-62091}, school = {Universit{\"a}t Potsdam}, year = {2012}, abstract = {This work is concerned with the characterization of certain classes of stochastic processes via duality formulae. In particular we consider reciprocal processes with jumps, a subject up to now neglected in the literature. In the first part we introduce a new formulation of a characterization of processes with independent increments. This characterization is based on a duality formula satisfied by processes with infinitely divisible increments, in particular L{\´e}vy processes, which is well known in Malliavin calculus. We obtain two new methods to prove this duality formula, which are not based on the chaos decomposition of the space of square-integrable function- als. One of these methods uses a formula of partial integration that characterizes infinitely divisible random vectors. In this context, our characterization is a generalization of Stein's lemma for Gaussian random variables and Chen's lemma for Poisson random variables. The generality of our approach permits us to derive a characterization of infinitely divisible random measures. The second part of this work focuses on the study of the reciprocal classes of Markov processes with and without jumps and their characterization. We start with a resume of already existing results concerning the reciprocal classes of Brownian diffusions as solutions of duality formulae. As a new contribution, we show that the duality formula satisfied by elements of the reciprocal class of a Brownian diffusion has a physical interpretation as a stochastic Newton equation of motion. Thus we are able to connect the results of characterizations via duality formulae with the theory of stochastic mechanics by our interpretation, and to stochastic optimal control theory by the mathematical approach. As an application we are able to prove an invariance property of the reciprocal class of a Brownian diffusion under time reversal. In the context of pure jump processes we derive the following new results. We describe the reciprocal classes of Markov counting processes, also called unit jump processes, and obtain a characterization of the associated reciprocal class via a duality formula. This formula contains as key terms a stochastic derivative, a compensated stochastic integral and an invariant of the reciprocal class. Moreover we present an interpretation of the characterization of a reciprocal class in the context of stochastic optimal control of unit jump processes. As a further application we show that the reciprocal class of a Markov counting process has an invariance property under time reversal. Some of these results are extendable to the setting of pure jump processes, that is, we admit different jump-sizes. In particular, we show that the reciprocal classes of Markov jump processes can be compared using reciprocal invariants. A characterization of the reciprocal class of compound Poisson processes via a duality formula is possible under the assumption that the jump-sizes of the process are incommensurable.}, language = {en} } @unpublished{AntonioukKiselevStepanenkoetal.2012, author = {Antoniouk, Alexandra Viktorivna and Kiselev, Oleg and Stepanenko, Vitaly and Tarkhanov, Nikolai Nikolaevich}, title = {Asymptotic solutions of the Dirichlet problem for the heat equation at a characteristic point}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-61987}, year = {2012}, abstract = {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.}, language = {en} } @unpublished{AlsaedyTarkhanov2012, author = {Alsaedy, Ammar and Tarkhanov, Nikolai Nikolaevich}, title = {The method of Fischer-Riesz equations for elliptic boundary value problems}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-61792}, year = {2012}, abstract = {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.}, language = {en} } @unpublished{DyachenkoTarkhanov2012, author = {Dyachenko, Evgueniya and Tarkhanov, Nikolai Nikolaevich}, title = {Degeneration of boundary layer at singular points}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-60135}, year = {2012}, abstract = {We study the Dirichlet problem in a bounded plane domain for the heat equation with small parameter multiplying the derivative in t. The behaviour of solution at characteristic points of the boundary is of special interest. The behaviour is well understood if a characteristic line is tangent to the boundary with contact degree at least 2. We allow the boundary to not only have contact of degree less than 2 with a characteristic line but also a cuspidal singularity at a characteristic point. We construct an asymptotic solution of the problem near the characteristic point to describe how the boundary layer degenerates.}, language = {en} } @unpublished{Baer2012, author = {B{\"a}r, Christian}, title = {Some properties of solutions to weakly hypoelliptic equations}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-60064}, year = {2012}, abstract = {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.}, language = {en} } @unpublished{Baer2012, author = {B{\"a}r, Christian}, title = {Renormalized integrals and a path integral formula for the heat kernel on a manifold}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-60052}, year = {2012}, abstract = {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.}, language = {en} } @unpublished{PfaeffleStephan2012, author = {Pf{\"a}ffle, Frank and Stephan, Christoph A.}, title = {Chiral asymmetry and the spectral action}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-60046}, year = {2012}, abstract = {We consider orthogonal connections with arbitrary torsion on compact Riemannian manifolds. For the induced Dirac operators, twisted Dirac operators and Dirac operators of Chamseddine-Connes type we compute the spectral action. In addition to the Einstein-Hilbert action and the bosonic part of the Standard Model Lagrangian we find the Holst term from Loop Quantum Gravity, a coupling of the Holst term to the scalar curvature and a prediction for the value of the Barbero-Immirzi parameter.}, language = {en} } @unpublished{PfaeffleStephan2012, author = {Pf{\"a}ffle, Frank and Stephan, Christoph A.}, title = {The Holst action by the spectral action principle}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-60032}, year = {2012}, abstract = {We investigate the Holst action for closed Riemannian 4-manifolds with orthogonal connections. For connections whose torsion has zero Cartan type component we show that the Holst action can be recovered from the heat asymptotics for the natural Dirac operator acting on left-handed spinor fields.}, language = {en} } @unpublished{BaerBallmann2012, author = {B{\"a}r, Christian and Ballmann, Werner}, title = {Boundary value problems for elliptic differential operators of first order}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-60023}, year = {2012}, abstract = {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.}, language = {en} } @unpublished{BaerPfaeffle2012, author = {B{\"a}r, Christian and Pf{\"a}ffle, Frank}, title = {Wiener measures on Riemannian manifolds and the Feynman-Kac formula}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-59998}, year = {2012}, abstract = {This is an introduction to Wiener measure and the Feynman-Kac formula on general Riemannian manifolds for Riemannian geometers with little or no background in stochastics. We explain the construction of Wiener measure based on the heat kernel in full detail and we prove the Feynman-Kac formula for Schr{\"o}dinger operators with bounded potentials. We also consider normal Riemannian coverings and show that projecting and lifting of paths are inverse operations which respect the Wiener measure.}, language = {en} } @unpublished{PfaeffleStephan2012, author = {Pf{\"a}ffle, Frank and Stephan, Christoph A.}, title = {On gravity, torsion and the spectral action principle}, issn = {2193-6943}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-59989}, year = {2012}, abstract = {We consider compact Riemannian spin manifolds without boundary equipped with orthogonal connections. We investigate the induced Dirac operators and the associated commutative spectral triples. In case of dimension four and totally anti-symmetric torsion we compute the Chamseddine-Connes spectral action, deduce the equations of motions and discuss critical points.}, language = {en} } @unpublished{BaerGinoux2012, author = {B{\"a}r, Christian and Ginoux, Nicolas}, title = {Classical and quantum fields on Lorentzian manifolds}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-59973}, year = {2012}, abstract = {We construct bosonic and fermionic locally covariant quantum fields theories on curved backgrounds for large classes of fields. We investigate the quantum field and n-point functions induced by suitable states.}, language = {en} } @phdthesis{Hanisch2011, author = {Hanisch, Florian}, title = {Variational problems on supermanifolds}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-59757}, school = {Universit{\"a}t Potsdam}, year = {2011}, abstract = {In this thesis, we discuss the formulation of variational problems on supermanifolds. Supermanifolds incorporate bosonic as well as fermionic degrees of freedom. Fermionic fields take values in the odd part of an appropriate Grassmann algebra and are thus showing an anticommutative behaviour. However, a systematic treatment of these Grassmann parameters requires a description of spaces as functors, e.g. from the category of Grassmann algberas into the category of sets (or topological spaces, manifolds). After an introduction to the general ideas of this approach, we use it to give a description of the resulting supermanifolds of fields/maps. We show that each map is uniquely characterized by a family of differential operators of appropriate order. Moreover, we demonstrate that each of this maps is uniquely characterized by its component fields, i.e. by the coefficients in a Taylor expansion w.r.t. the odd coordinates. In general, the component fields are only locally defined. We present a way how to circumvent this limitation. In fact, by enlarging the supermanifold in question, we show that it is possible to work with globally defined components. We eventually use this formalism to study variational problems. More precisely, we study a super version of the geodesic and a generalization of harmonic maps to supermanifolds. Equations of motion are derived from an energy functional and we show how to decompose them into components. Finally, in special cases, we can prove the existence of critical points by reducing the problem to equations from ordinary geometric analysis. After solving these component equations, it is possible to show that their solutions give rise to critical points in the functor spaces of fields.}, language = {en} } @unpublished{Nehring2012, author = {Nehring, Benjamin}, title = {Construction of point processes for classical and quantum gases}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-59648}, year = {2012}, abstract = {We propose a new construction of point processes, which generalizes the class of infinitely divisible point processes. Examples are the quantum Boson and Fermion gases as well as the classical Gibbs point processes, where the interaction is given by a stable and regular pair potential.}, language = {en} } @unpublished{RattanaBoeckmann2012, author = {Rattana, Amornrat and B{\"o}ckmann, Christine}, title = {Matrix methods for computing Eigenvalues of Sturm-Liouville problems of order four}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-59279}, year = {2012}, abstract = {This paper examines and develops matrix methods to approximate the eigenvalues of a fourth order Sturm-Liouville problem subjected to a kind of fixed boundary conditions, furthermore, it extends the matrix methods for a kind of general boundary conditions. The idea of the methods comes from finite difference and Numerov's method as well as boundary value methods for second order regular Sturm-Liouville problems. Moreover, the determination of the correction term formulas of the matrix methods are investigated in order to obtain better approximations of the problem with fixed boundary conditions since the exact eigenvalues for q = 0 are known in this case. Finally, some numerical examples are illustrated.}, language = {en} } @unpublished{AlsaedyTarkhanov2012, author = {Alsaedy, Ammar and Tarkhanov, Nikolai Nikolaevich}, title = {Spectral projection for the dbar-Neumann problem}, issn = {2193-6943}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-58616}, year = {2012}, abstract = {We show that the spectral kernel function of the dbar-Neumann problem on a non-compact strongly pseudoconvex manifold is smooth up to the boundary.}, language = {en} } @unpublished{Tarkhanov2012, author = {Tarkhanov, Nikolai Nikolaevich}, title = {A simple numerical approach to the Riemann hypothesis}, issn = {2193-6943}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-57645}, year = {2012}, abstract = {The Riemann hypothesis is equivalent to the fact the the reciprocal function 1/zeta (s) extends from the interval (1/2,1) to an analytic function in the quarter-strip 1/2 < Re s < 1 and Im s > 0. Function theory allows one to rewrite the condition of analytic continuability in an elegant form amenable to numerical experiments.}, language = {en} } @unpublished{ShlapunovTarkhanov2012, author = {Shlapunov, Alexander and Tarkhanov, Nikolai Nikolaevich}, title = {On completeness of root functions of Sturm-Liouville problems with discontinuous boundary operators}, issn = {2193-6943}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-57759}, year = {2012}, abstract = {We consider a Sturm-Liouville boundary value problem in a bounded domain D of R^n. By this is meant that the differential equation is given by a second order elliptic operator of divergent form in D and the boundary conditions are of Robin type on bD. The first order term of the boundary operator is the oblique derivative whose coefficients bear discontinuities of the first kind. Applying the method of weak perturbation of compact self-adjoint operators and the method of rays of minimal growth, we prove the completeness of root functions related to the boundary value problem in Lebesgue and Sobolev spaces of various types.}, language = {en} } @unpublished{GrudskyTarkhanov2012, author = {Grudsky, Serguey and Tarkhanov, Nikolai Nikolaevich}, title = {Conformal reduction of boundary problems for harmonic functions in a plane domain with strong singularities on the boundary}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-57745}, year = {2012}, abstract = {We consider the Dirichlet, Neumann and Zaremba problems for harmonic functions in a bounded plane domain with nonsmooth boundary. The boundary curve belongs to one of the following three classes: sectorial curves, logarithmic spirals and spirals of power type. To study the problem we apply a familiar method of Vekua-Muskhelishvili which consists in using a conformal mapping of the unit disk onto the domain to pull back the problem to a boundary problem for harmonic functions in the disk. This latter is reduced in turn to a Toeplitz operator equation on the unit circle with symbol bearing discontinuities of second kind. We develop a constructive invertibility theory for Toeplitz operators and thus derive solvability conditions as well as explicit formulas for solutions.}, language = {en} } @unpublished{KoppitzMusunthia2012, author = {Koppitz, J{\"o}rg and Musunthia, Tiwadee}, title = {Maximal subsemigroups containing a particular semigroup}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-57465}, year = {2012}, abstract = {We study maximal subsemigroups of the monoid T(X) of all full transformations on the set X = N of natural numbers containing a given subsemigroup W of T(X), where each element of a given set U is a generator of T(X) modulo W. This note continues the study of maximal subsemigroups of the monoid of all full transformations on an infinite set.}, language = {en} } @unpublished{BlanchardMathe2012, author = {Blanchard, Gilles and Math{\´e}, Peter}, title = {Discrepancy principle for statistical inverse problems with application to conjugate gradient iteration}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-57117}, year = {2012}, abstract = {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.}, language = {en} } @unpublished{KleinLeonardRosenberger2012, author = {Klein, Markus and L{\´e}onard, Christian and Rosenberger, Elke}, title = {Agmon-type estimates for a class of jump processes}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-56995}, year = {2012}, abstract = {In the limit we analyze the generators of families of reversible jump processes in the n-dimensional space associated with a class of symmetric non-local Dirichlet forms and show exponential decay of the eigenfunctions. The exponential rate function is a Finsler distance, given as solution of certain eikonal equation. Fine results are sensitive to the rate functions being twice differentiable or just Lipschitz. Our estimates are similar to the semiclassical Agmon estimates for differential operators of second order. They generalize and strengthen previous results on the lattice.}, language = {en} } @unpublished{KleinRosenberger2012, author = {Klein, Markus and Rosenberger, Elke}, title = {Tunneling for a class of difference operators}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-56989}, year = {2012}, abstract = {We analyze a general class of difference operators containing a multi-well potential and a small parameter. We decouple the wells by introducing certain Dirichlet operators on regions containing only one potential well, and we treat the eigenvalue problem as a small perturbation of these comparison problems. We describe tunneling by a certain interaction matrix similar to the analysis for the Schr{\"o}dinger operator, and estimate the remainder, which is exponentially small and roughly quadratic compared with the interaction matrix.}, language = {en} } @unpublished{KellerRoellyValleriani2012, author = {Keller, Peter and Roelly, Sylvie and Valleriani, Angelo}, title = {On time duality for quasi-birth-and-death processes}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-56973}, year = {2012}, abstract = {We say that (weak/strong) time duality holds for continuous time quasi-birth-and-death-processes if, starting from a fixed level, the first hitting time of the next upper level and the first hitting time of the next lower level have the same distribution. We present here a criterion for time duality in the case where transitions from one level to another have to pass through a given single state, the so-called bottleneck property. We also prove that a weaker form of reversibility called balanced under permutation is sufficient for the time duality to hold. We then discuss the general case.}, language = {en} } @unpublished{TarkhanovWallenta2012, author = {Tarkhanov, Nikolai Nikolaevich and Wallenta, Daniel}, title = {The Lefschetz number of sequences of trace class curvature}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-56969}, year = {2012}, abstract = {For a sequence of Hilbert spaces and continuous linear operators the curvature is defined to be the composition of any two consecutive operators. This is modeled on the de Rham resolution of a connection on a module over an algebra. Of particular interest are those sequences for which the curvature is "small" at each step, e.g., belongs to a fixed operator ideal. In this context we elaborate the theory of Fredholm sequences and show how to introduce the Lefschetz number.}, language = {en} } @unpublished{KiselevTarkhanov2012, author = {Kiselev, Oleg M. and Tarkhanov, Nikolai Nikolaevich}, title = {Scattering of autoresonance trajectories upon a separatrix}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-56880}, year = {2012}, abstract = {We study asymptotic properties of solutions to the primary resonance equation with large amplitude on a long time interval.}, language = {en} } @unpublished{BlanchardDelattreRoquain2012, author = {Blanchard, Gilles and Delattre, Sylvain and Roquain, {\´E}tienne}, title = {Testing over a continuum of null hypotheses}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-56877}, year = {2012}, abstract = {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.}, language = {en} } @book{Liero2010, author = {Liero, Hannelore}, title = {Estimation and testing the effect of covariates in accelerated life time models under censoring}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-52823}, publisher = {Universit{\"a}t Potsdam}, year = {2010}, abstract = {The accelerated lifetime model is considered. To test the influence of the covariate we transform the model in a regression model. Since censoring is allowed this approach leads to a goodness-of-fit problem for regression functions under censoring. So nonparametric estimation of regression functions under censoring is investigated, a limit theorem for a L2-distance is stated and a test procedure is formulated. Finally a Monte Carlo procedure is proposed.}, language = {en} } @book{DereudreRoelly2004, author = {Dereudre, David and Roelly, Sylvie}, title = {On Gibbsianness of infinite-dimensional diffusions}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-52630}, publisher = {Universit{\"a}t Potsdam}, year = {2004}, abstract = {We analyse different Gibbsian properties of interactive Brownian diffusions X indexed by the lattice \$Z^{d} : X = (X_{i}(t), i ∈ Z^{d}, t ∈ [0, T], 0 < T < +∞)\$. In a first part, these processes are characterized as Gibbs states on path spaces of the form \$C([0, T],R)Z^{d}\$. In a second part, we study the Gibbsian character on \$R^{Z}^{d}\$ of \$v^{t}\$, the law at time t of the infinite-dimensional diffusion X(t), when the initial law \$v = v^{0}\$ is Gibbsian.}, language = {en} } @unpublished{Murr2011, author = {Murr, R{\"u}diger}, title = {Characterization of L{\´e}vy Processes by a duality formula and related results}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-43538}, year = {2011}, abstract = {Processes with independent increments are characterized via a duality formula, including Malliavin derivative and difference operators. This result is based on a characterization of infinitely divisible random vectors by a functional equation. A construction of the difference operator by a variational method is introduced and compared to approaches used by other authors for L´evy processes involving the chaos decomposition. Finally we extend our method to characterize infinitely divisible random measures.}, language = {en} } @unpublished{MeleardRoelly2011, author = {M{\´e}l{\´e}ard, Sylvie and Roelly, Sylvie}, title = {A host-parasite multilevel interacting process and continuous approximations}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-51694}, year = {2011}, abstract = {We are interested in modeling some two-level population dynamics, resulting from the interplay of ecological interactions and phenotypic variation of individuals (or hosts) and the evolution of cells (or parasites) of two types living in these individuals. The ecological parameters of the individual dynamics depend on the number of cells of each type contained by the individual and the cell dynamics depends on the trait of the invaded individual. Our models are rooted in the microscopic description of a random (discrete) population of individuals characterized by one or several adaptive traits and cells characterized by their type. The population is modeled as a stochastic point process whose generator captures the probabilistic dynamics over continuous time of birth, mutation and death for individuals and birth and death for cells. The interaction between individuals (resp. between cells) is described by a competition between individual traits (resp. between cell types). We look for tractable large population approximations. By combining various scalings on population size, birth and death rates and mutation step, the single microscopic model is shown to lead to contrasting nonlinear macroscopic limits of different nature: deterministic approximations, in the form of ordinary, integro- or partial differential equations, or probabilistic ones, like stochastic partial differential equations or superprocesses. The study of the long time behavior of these processes seems very hard and we only develop some simple cases enlightening the difficulties involved.}, language = {en} } @unpublished{Penisson2010, author = {P{\´e}nisson, Sophie}, title = {Conditional Limit Theorems for Multitype Branching Processes and Illustration in Epidemiological Risk Analysis}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-49589}, year = {2010}, abstract = {This thesis is concerned with the issue of extinction of populations composed of different types of individuals, and their behavior before extinction and in case of a very late extinction. We approach this question firstly from a strictly probabilistic viewpoint, and secondly from the standpoint of risk analysis related to the extinction of a particular model of population dynamics. In this context we propose several statistical tools. The population size is modeled by a branching process, which is either a continuous-time multitype Bienaym{\´e}-Galton-Watson process (BGWc), or its continuous-state counterpart, the multitype Feller diffsion process. We are interested in different kinds of conditioning on nonextinction, and in the associated equilibrium states. These ways of conditioning have been widely studied in the monotype case. However the literature on multitype processes is much less extensive, and there is no systematic work establishing connections between the results for BGWc processes and those for Feller diffusion processes. In the first part of this thesis, we investigate the behavior of the population before its extinction by conditioning the associated branching process Xt on non-extinction (Xt 6= 0), or more generally on non-extinction in a near future 0 < 1 (Xt+ 0 = 0), and by letting t tend to infinity. We prove the result, new in the multitype framework and for 0 > 0, that this limit exists and is nondegenerate. This re ects a stationary behavior for the dynamics of the population conditioned on non-extinction, and provides a generalization of the so-called Yaglom limit, corresponding to the case 0 = 0. In a second step we study the behavior of the population in case of a very late extinction, obtained as the limit when 0 tends to infinity of the process conditioned by Xt+ 0 = 0. The resulting conditioned process is a known object in the monotype case (sometimes referred to as Q-process), and has also been studied when Xt is a multitype Feller diffusion process. We investigate the not yet considered case where Xt is a multitype BGWc process and prove the existence of the associated Q-process. In addition, we examine its properties, including the asymptotic ones, and propose several interpretations of the process. Finally, we are interested in interchanging the limits in t and 0, as well as in the not yet studied commutativity of these limits with respect to the high-density-type relationship between BGWc processes and Feller processes. We prove an original and exhaustive list of all possible exchanges of limit (long-time limit in t, increasing delay of extinction 0, diffusion limit). The second part of this work is devoted to the risk analysis related both to the extinction of a population and to its very late extinction. We consider a branching population model (arising notably in the epidemiological context) for which a parameter related to the first moments of the offspring distribution is unknown. We build several estimators adapted to different stages of evolution of the population (phase growth, decay phase, and decay phase when extinction is expected very late), and prove moreover their asymptotic properties (consistency, normality). In particular, we build a least squares estimator adapted to the Q-process, allowing a prediction of the population development in the case of a very late extinction. This would correspond to the best or to the worst-case scenario, depending on whether the population is threatened or invasive. These tools enable us to study the extinction phase of the Bovine Spongiform Encephalopathy epidemic in Great Britain, for which we estimate the infection parameter corresponding to a possible source of horizontal infection persisting after the removal in 1988 of the major route of infection (meat and bone meal). This allows us to predict the evolution of the spread of the disease, including the year of extinction, the number of future cases and the number of infected animals. In particular, we produce a very fine analysis of the evolution of the epidemic in the unlikely event of a very late extinction.}, language = {en} } @unpublished{Kuxhaus2010, author = {Kuxhaus, Olga}, title = {Parametrische Sch{\"a}tzungen von elliptischen Copulafunktionen}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-51681}, year = {2010}, abstract = {Aus dem Inhalt: Inhaltsverzeichnis Abbildungsverzeichnis Tabellenverzeichnis 1 Einleitung und Motivation 2 Multivariate Copulafunktionen 2.1 Einleitung 2.2 Satz von Sklar 2.3 Eigenschaften von Copulafunktionen 3 Abh{\"a}ngigkeitskonzepte 3.1 Lineare Korrelation 3.2 Copulabasierte Abh{\"a}ngigkeitsmaße 3.2.1 Konkordanz 3.2.2 Kendall's und Spearman's 3.2.3 Asymptotische Randabh{\"a}ngigkeit 4 Elliptische Copulaklasse 4.1 Sph{\"a}rische und elliptische Verteilungen 4.2 Normal-Copula 4.3 t-Copula 5 Parametrische Sch{\"a}tzverfahren 5.1 Maximum-Likelihood-Methode 5.1.1 ExakteMaximum-Likelihood-Methode 5.1.2 2-stufige parametrische Maximum-Likelihood-Methode 5.1.3 2-stufige semiparametrische Maximum-Likelihood-Methode 5.2 Momentenmethode 5.3 Kendall's -Momentenmethode 6 Parametersch{\"a}tzungen f{\"u}r Normal- und t-Copula 6.1 Normal-Copula 6.1.1 Maximum-Likelihood-Methode 6.1.2 Momentenmethode 6.1.3 Kendall's Momentenmethode 6.1.4 Spearman's Momentenmethode 6.2 t-Copula 6.2.1 Verfahren 1 (exakte ML-Methode) 6.2.2 Verfahren 2 (2-stufige rekursive ML-Methode) 6.2.3 Verfahren 3 (2-stufige KM-ML-Methode) 6.2.4 Verfahren 4 (3-stufige M-ML-Methode) 7 Simulationen 7.1 Grundlagen 7.2 Parametrischer Fall 7.3 Nichtparametrischer Fall 7.4 Fazit A Programmausschnitt Literaturverzeichnis}, language = {de} } @unpublished{Runge2010, author = {Runge, Antonia}, title = {Modellierung der Lebensdauer von Systemen}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-51674}, year = {2010}, abstract = {Aus dem Inhalt: Einleitung und Zusammenfassung 1 Grundlagen der Lebensdaueranalyse 2 Systemzuverl{\"a}ssigkeit 3 Zensierung 4 Sch{\"a}tzen in nichtparametrischen Modellen 5 Sch{\"a}tzen in parametrischen Modellen 6 Konfidenzintervalle f{\"u}r Parametersch{\"a}tzungen 7 Verteilung einer gemischten Population 8 Kurze Einf{\"u}hrung: Lebensdauer und Belastung 9 Ausblick A R-Quellcode B Symbole und Abk{\"u}rzungen}, language = {de} } @unpublished{Rafler2008, author = {Rafler, Mathias}, title = {Martin-Dynkin Boundaries of the Bose Gas}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-51667}, year = {2008}, abstract = {The Ginibre gas is a Poisson point process defined on a space of loops related to the Feynman-Kac representation of the ideal Bose gas. Here we study thermodynamic limits of different ensembles via Martin-Dynkin boundary technique and show, in which way infinitely long loops occur. This effect is the so-called Bose-Einstein condensation.}, language = {en} } @unpublished{Voss2010, author = {Voss, Carola Regine}, title = {Harness-Prozesse}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-49651}, year = {2010}, abstract = {Harness-Prozesse finden in der Forschung immer mehr Anwendung. Vor allem gewinnen Harness-Prozesse in stetiger Zeit an Bedeutung. Grundlegende Literatur zu diesem Thema ist allerdings wenig vorhanden. In der vorliegenden Arbeit wird die vorhandene Grundlagenliteratur zu Harness-Prozessen in diskreter und stetiger Zeit aufgearbeitet und Beweise ausgef{\"u}hrt, die bisher nur skizziert waren. Ziel dessen ist die Existenz einer Zerlegung von Harness-Prozessen {\"u}ber Z beziehungsweise R+ nachzuweisen.}, language = {de} } @unpublished{Kunze2010, author = {Kunze, Simone}, title = {Das Sammelbilderproblem}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-51646}, year = {2010}, abstract = {Aus dem Inhalt: 1 Einleitung 2 Entwicklung der L{\"o}sungsans{\"a}tze 3 Martingalansatz 4 Markov-Ketten Ansatz 5 Einbettung in Poisson Prozesse 6 Kombinatorische Ans{\"a}tze 7 Zusammenfassung und Ausblick Literaturverzeichnis}, language = {de} } @unpublished{NehringZessin2010, author = {Nehring, Benjamin and Zessin, Hans}, title = {A path integral representation of the moment measures of the general ideal Bose gas}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-49635}, year = {2010}, abstract = {We reconsider the fundamental work of Fichtner ([2]) and exhibit the permanental structure of the ideal Bose gas again, using another approach which combines a characterization of infinitely divisible random measures (due to Kerstan,Kummer and Matthes [5, 6] and Mecke [8, 9]) with a decomposition of the moment measures into its factorial measures due to Krickeberg [4]. To be more precise, we exhibit the moment measures of all orders of the general ideal Bose gas in terms of certain path integrals. This representation can be considered as a point process analogue of the old idea of Symanzik [11] that local times and self-crossings of the Brownian motion can be used as a tool in quantum field theory. Behind the notion of a general ideal Bose gas there is a class of infinitely divisible point processes of all orders with a Levy-measure belonging to some large class of measures containing the one of the classical ideal Bose gas considered by Fichtner. It is well known that the calculation of moments of higher order of point processes are notoriously complicated. See for instance Krickeberg's calculations for the Poisson or the Cox process in [4].}, language = {en} } @unpublished{Roelly2010, author = {Roelly, Sylvie}, title = {Unas propiedades basicas de procesos de ramificaci{\´o}n : Lectures held at ICIMAF La Habana, Cuba, 2009 and 2010}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-49620}, year = {2010}, abstract = {Aus dem Inhalt: 1. Unas propiedades de los procesos de Bienaym{\´e}-Galton-Watson de tiempo dis- creto (BGW) 2. Unas propiedades del proceso BGW de tiempo continuo 3. Limites de procesos de BGW cuando la poblaci{\´o}n es numerosa}, language = {mul} } @unpublished{Zessin2010, author = {Zessin, Hans}, title = {Classical Symmetric Point Processes : Lectures held at ICIMAF, La Habana, Cuba, 2010}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-49619}, year = {2010}, abstract = {The aim of these lectures is a reformulation and generalization of the fundamental investigations of Alexander Bach [2, 3] on the concept of probability in the work of Boltzmann [6] in the language of modern point process theory. The dominating point of view here is its subordination under the disintegration theory of Krickeberg [14]. This enables us to make Bach's consideration much more transparent. Moreover the point process formulation turns out to be the natural framework for the applications to quantum mechanical models.}, language = {en} } @unpublished{Siegert2010, author = {Siegert, Sabine}, title = {Das Sankt-Petersburg-Paradoxon}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-49595}, year = {2010}, abstract = {Aus dem Inhalt: 1 Einleitung 2 Historische L{\"o}sungsans{\"a}tze 3 Martingal-Ansatz 4 Markovketten-Ansatz 5 Asymptotische Interpretationen 6 Bezug zur Praxis 7 R{\´e}sum{\´e} Anhang Literaturverzeichnis}, language = {de} } @unpublished{Penisson2010, author = {P{\´e}nisson, Sophie}, title = {Estimation of the infection parameter in the different phases of an epidemic modeled by a branching process}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-49607}, year = {2010}, abstract = {The aim of this paper is to build and compare estimators of the infection parameter in the different phases of an epidemic (growth and extinction phases). The epidemic is modeled by a Markovian process of order d > 1 (allowing non-Markovian life spans), and can be written as a multitype branching process. We propose three estimators suitable for the different classes of criticality of the process, in particular for the subcritical case corresponding to the extinction phase. We prove their consistency and asymptotic normality for two asymptotics, when the number of ancestors (resp. number of generations) tends to infinity. We illustrate the asymptotic properties with simulated examples, and finally use our estimators to study the infection intensity in the extinction phase of the BSE epidemic in Great-Britain.}, language = {en} } @unpublished{LaeuterRamadan2010, author = {L{\"a}uter, Henning and Ramadan, Ayad}, title = {Modeling and Scaling of Categorical Data}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-49572}, year = {2010}, abstract = {Estimation and testing of distributions in metric spaces are well known. R.A. Fisher, J. Neyman, W. Cochran and M. Bartlett achieved essential results on the statistical analysis of categorical data. In the last 40 years many other statisticians found important results in this field. Often data sets contain categorical data, e.g. levels of factors or names. There does not exist any ordering or any distance between these categories. At each level there are measured some metric or categorical values. We introduce a new method of scaling based on statistical decisions. For this we define empirical probabilities for the original observations and find a class of distributions in a metric space where these empirical probabilities can be found as approximations for equivalently defined probabilities. With this method we identify probabilities connected with the categorical data and probabilities in metric spaces. Here we get a mapping from the levels of factors or names into points of a metric space. This mapping yields the scale for the categorical data. From the statistical point of view we use multivariate statistical methods, we calculate maximum likelihood estimations and compare different approaches for scaling.}, language = {de} } @unpublished{LaeuterRamadan2010, author = {L{\"a}uter, Henning and Ramadan, Ayad}, title = {Statistical Scaling of Categorical Data}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-49566}, year = {2010}, abstract = {Estimation and testing of distributions in metric spaces are well known. R.A. Fisher, J. Neyman, W. Cochran and M. Bartlett achieved essential results on the statistical analysis of categorical data. In the last 40 years many other statisticians found important results in this field. Often data sets contain categorical data, e.g. levels of factors or names. There does not exist any ordering or any distance between these categories. At each level there are measured some metric or categorical values. We introduce a new method of scaling based on statistical decisions. For this we define empirical probabilities for the original observations and find a class of distributions in a metric space where these empirical probabilities can be found as approximations for equivalently defined probabilities. With this method we identify probabilities connected with the categorical data and probabilities in metric spaces. Here we get a mapping from the levels of factors or names into points of a metric space. This mapping yields the scale for the categorical data. From the statistical point of view we use multivariate statistical methods, we calculate maximum likelihood estimations and compare different approaches for scaling.}, language = {en} } @unpublished{FradonRoelly2009, author = {Fradon, Myriam and Roelly, Sylvie}, title = {Infinitely many Brownian globules with Brownian radii}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-49552}, year = {2009}, abstract = {We consider an infinite system of non overlaping globules undergoing Brownian motions in R3. The term globules means that the objects we are dealing with are spherical, but with a radius which is random and time-dependent. The dynamics is modelized by an infinitedimensional Stochastic Differential Equation with local time. Existence and uniqueness of a strong solution is proven for such an equation with fixed deterministic initial condition. We also find a class of reversible measures.}, language = {en} } @unpublished{Penisson2009, author = {P{\´e}nisson, Sophie}, title = {Continuous-time multitype branching processes conditioned on very late extinction}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-49548}, year = {2009}, abstract = {Multitype branching processes and Feller diffusion processes are conditioned on very late extinction. The conditioned laws are expressed as Doob h-transforms of the unconditioned laws, and an interpretation of the conditioned paths for the branching process is given, via the immortal particle. We study different limits for the conditioned process (increasing delay of extinction, long-time behavior, scaling limit) and provide an exhaustive list of exchangeability results.}, language = {en} } @unpublished{Rafler2009, author = {Rafler, Mathias}, title = {Gaussian loop- and polya processes : a point process approach}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-51638}, year = {2009}, abstract = {Zuf{\"a}llige Punktprozesse beschreiben eine (zuf{\"a}llige) zeitliche Abfolge von Ereignissen oder eine (zuf{\"a}llige) r{\"a}umliche Anordnung von Objekten. Deren wichtigster Vertreter ist der Poissonprozess. Der Poissonprozess zum Intensit{\"a}tsmaß, das Lebesgue-Maß ordnet jedem Gebiet sein Volumen zu, erzeugt lokal, d.h in einem beschr{\"a}nkten Gebiet B, gerade eine mit dem Volumen von B poissonverteilte Anzahl von Punkten, die identisch und unabh{\"a}ngig voneinander in B plaziert werden; im Mittel ist diese Anzahl (B). Ersetzt man durch ein Vielfaches a, so wird diese Anzahl mit dem a-fachen Mittelwert erzeugt. Poissonprozesse, die im gesamten Raum unendlich viele Punkte realisieren, enthalten bereits in einer einzigen Stichprobe gen{\"u}gend Informationen, um Statistik betreiben zu k{\"o}nnen: Bedingt man lokal bzgl. der Anzahl der Teilchen einer Stichprobe, so fragt man nach allen Punktprozessen, die eine solche Beobachtung h{\"a}tten liefern k{\"o}nnen. Diese sind Limespunktprozesse zu dieser Beobachtung. Kommt mehr als einer in Frage, spricht man von einem Phasen{\"u}bergang. Da die Menge dieser Limespunktprozesse konvex ist, fragt man nach deren Extremalpunkten, dem Rand. Im ersten Teil wird ein Poissonprozess f{\"u}r ein physikalisches Teilchenmodell f{\"u}r Bosonen konstruiert. Dieses erzeugt sogenannte Loops, das sind geschlossene Polygonz{\"u}ge, die dadurch charakterisiert sind, dass man an einem Ort mit einem Punkt startet, den mit einem normalverteilten Schritt l{\"a}uft und dabei nach einer gegebenen, aber zuf{\"a}lligen Anzahl von Schritten zum Ausgangspunkt zur{\"u}ckkehrt. F{\"u}r verschiedene Beobachtungen von Stichproben werden zugeh{\"o}rige Limespunktprozesse diskutiert. Diese Beobachtungen umfassen etwa das Z{\"a}hlen der Loops gem{\"a}aß ihrer L{\"a}nge, das Z{\"a}hlen der Loops insgesamt, oder das Z{\"a}hlen der von den Loops gemachten Schritte. Jede Wahl zieht eine charakteristische Struktur der invarianten Punktprozesse nach sich. In allen hiesigen F{\"a}llen wird ein charakteristischer Phasen{\"u}bergang gezeigt und Extremalpunkte werden als spezielle Poissonprozesse identifiziert. Insbesondere wird gezeigt, wie die Wahl der Beobachtung die L{\"a}nge der Loops beeinflusst. Geometrische Eigenschaften dieser Poissonprozesse sind der Gegenstand des zweiten Teils der Arbeit. Die Technik der Palmschen Verteilungen eines Punktprozesses erlaubt es, unter den unendlich vielen Loops einer Realisierung den typischen Loop herauszupicken, dessen Geometrie dann untersucht wird. Eigenschaften sind unter anderem die euklidische L{\"a}nge eines Schrittes oder, nimmt man mehrere aufeinander folgende Schritte, das Volumen des von ihnen definierten Simplex. Weiterhin wird gezeigt, dass der Schwerpunkt eines typischen Loops normalverteilt ist mit einer festen Varianz. Der dritte und letzte Teil befasst sich mit der Konstruktion, den Eigenschaften und der Statistik eines neuartigen Punktprozesses, der Polyascher Summenprozess genannt wird. Seine Konstruktion verallgemeinert das Prinzip der Polyaschen Urne: Im Gegensatz zum Poissonprozess, der alle Punkte unabh{\"a}ngig und vor allem identisch verteilt, werden hier die Punkte nacheinander derart verteilt, dass der Ort, an dem ein Punkt plaziert wird, eine Belohnung auf die Wahrscheinlichkeit bekommt, nach der nachfolgende Punkte verteilt werden. Auf diese Weise baut der Polyasche Summenprozess "T{\"u}rmchen", indem sich verschiedene Punkte am selben Ort stapeln. Es wird gezeigt, dass dennoch grundlegende Eigenschaften mit denjenigen des Poissonprozesses {\"u}bereinstimmen, dazu geh{\"o}ren unendliche Teilbarkeit sowie Unabh{\"a}ngigkeit der Zuw{\"a}chse. Zudem werden sein Laplace-Funktional sowie seine Palmsche Verteilung bestimmt. Letztere zeigt, dass die H{\"o}he der T{\"u}rmchen gerade geometrisch verteilt ist. Abschließend werden wiederum Statistiken, nun f{\"u}r den Summenprozess, diskutiert. Je nach Art der Beobachtung von der Stichprobe, etwa Anzahl, Gesamth{\"o}he der T{\"u}rmchen oder beides, gibt es in jedem der drei F{\"a}lle charakteristische Limespunktprozesse und es stellt sich heraus, dass die zugeh{\"o}rigen Extremalverteilungen wiederum Polyasche Summenprozesse sind.}, language = {en} } @unpublished{RedigRoellyRuszel2009, author = {Redig, Frank and Roelly, Sylvie and Ruszel, Wioletta}, title = {Short-time Gibbsianness for infinite-dimensional diffusions with space-time interaction}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-49514}, year = {2009}, abstract = {We consider a class of infinite-dimensional diffusions where the interaction between the components is both spatial and temporal. We start the system from a Gibbs measure with finiterange uniformly bounded interaction. Under suitable conditions on the drift, we prove that there exists t0 > 0 such that the distribution at time t = t0 is a Gibbs measure with absolutely summable interaction. The main tool is a cluster expansion of both the initial interaction and certain time-reversed Girsanov factors coming from the dynamics.}, language = {de} } @unpublished{LouisRouquier2009, author = {Louis, Pierre-Yves and Rouquier, Jean-Baptiste}, title = {Time-to-Coalescence for interacting particle systems : parallel versus sequential updating}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-49454}, year = {2009}, abstract = {Studying the influence of the updating scheme for MCMC algorithm on spatially extended models is a well known problem. For discrete-time interacting particle systems we study through simulations the effectiveness of a synchronous updating scheme versus the usual sequential one. We compare the speed of convergence of the associated Markov chains from the point of view of the time-to-coalescence arising in the coupling-from-the-past algorithm. Unlike the intuition, the synchronous updating scheme is not always the best one. The distribution of the time-to-coalescence for these spatially extended models is studied too.}, language = {en} } @unpublished{Keller2009, author = {Keller, Peter}, title = {Erzeugung gleichverteilter Stichproben von Lozenge-Teilungen mittels Kopplung von Markovketten}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-49506}, year = {2009}, abstract = {Aus dem Inhalt: 1 Einleitung 2 Eigenschaften der Lozengeteilungen 3 Coupling From The Past (CFTP) 4 Simulation von uniform verteilten Lozengeteilungen 5 Programmlisting und Diskussion der Implementierung 6 Ausblick A Anhang}, language = {de} }