TY - THES A1 - Wallenta, Daniel T1 - Sequences of compact curvature T1 - Sequenzen mit kompakter Krümmung N2 - By perturbing the differential of a (cochain-)complex by "small" operators, one obtains what is referred to as quasicomplexes, i.e. a sequence whose curvature is not equal to zero in general. In this situation the cohomology is no longer defined. Note that it depends on the structure of the underlying spaces whether or not an operator is "small." This leads to a magical mix of perturbation and regularisation theory. In the general setting of Hilbert spaces compact operators are "small." In order to develop this theory, many elements of diverse mathematical disciplines, such as functional analysis, differential geometry, partial differential equation, homological algebra and topology have to be combined. All essential basics are summarised in the first chapter of this thesis. This contains classical elements of index theory, such as Fredholm operators, elliptic pseudodifferential operators and characteristic classes. Moreover we study the de Rham complex and introduce Sobolev spaces of arbitrary order as well as the concept of operator ideals. In the second chapter, the abstract theory of (Fredholm) quasicomplexes of Hilbert spaces will be developed. From the very beginning we will consider quasicomplexes with curvature in an ideal class. We introduce the Euler characteristic, the cone of a quasiendomorphism and the Lefschetz number. In particular, we generalise Euler's identity, which will allow us to develop the Lefschetz theory on nonseparable Hilbert spaces. Finally, in the third chapter the abstract theory will be applied to elliptic quasicomplexes with pseudodifferential operators of arbitrary order. We will show that the Atiyah-Singer index formula holds true for those objects and, as an example, we will compute the Euler characteristic of the connection quasicomplex. In addition to this we introduce geometric quasiendomorphisms and prove a generalisation of the Lefschetz fixed point theorem of Atiyah and Bott. N2 - Die Theorie der Sequenzen mit kompakter Krümmung, sogenannter Quasikomplexe, ist eine Verallgemeinerung der Theorie der Fredholm Komplexe. Um ein Verständnis für (Quasi-)Komplexe zu gewinnen, müssen Inhalte aus verschiedenen Teilgebieten der Mathematik kombiniert werden. Alle hierfür wesentlichen Grundlagen sind im ersten Kapitel dieser Dissertation zusammengefasst. Dies betrifft unter anderem gewisse Elemente der Funktionalanalysis und der Differentialgeometrie, sowie die Theorie der klassischen Pseudodifferentialoperatoren. Im zweiten Kapitel wird anschließend die abstrakte Theorie der Quasikomplexe und zugehöriger Quasimorphismen im Kontext der Funktionalanalysis entwickelt. Dabei werden verschiedene Typen von Quasikomplexen und Quasimorphismen klassifiziert, deren Eigenschaften analysiert und Beispiele betrachtet. Ein zentraler Punkt hierbei ist die Lösung des Problems, für welche dieser Objekte sich eine besondere charakteristische Zahl, die sogenannte Lefschetz-Zahl, definieren lässt. Die dargestellten Resultate zeigen, dass die in dieser Arbeit gegebene Definition eine natürliche Erweiterung der klassischen Lefschetz-Zahl darstellt. Abschließend wird die entwickelte Theorie im dritten Kapitel auf elliptische Quasikomplexe von Pseudodifferentialoperatoren angewendet. Dabei werden insbesondere Verallgemeinerungen der berühmten Atiyah-Singer-Index-Formel und des Lefschetz-Fixpunkt-Theorems von Atiyah and Bott bewiesen. KW - Index Theorie KW - Fredholm Komplexe KW - Elliptische Komplexe KW - Index theory KW - Elliptic complexes KW - Fredholm complexes Y1 - 2015 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-87489 ER - TY - THES A1 - Di Gesù, Giacomo T1 - Semiclassical spectral analysis of discrete Witten Laplacians T1 - Semiklassische Spektraltheorie von diskreten Witten-Laplace-Operatoren N2 - 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. N2 - In dieser Arbeit wird auf dem n-dimensionalen Gitter der ganzen Zahlen ein Analogon des Witten-Laplace-Operatoren eingeführt. Nach geeigneter Skalierung des Gitters und des Operatoren analysieren wir den Tunneleffekt zwischen verschiedenen Potentialtöpfen und erhalten vollständige Aymptotiken für das tiefliegende Spektrum. Der Beweis (nach Methoden, die von B. Helffer, M. Klein und F. Nier im Falle des kontinuierlichen Witten-Laplace-Operatoren entwickelt wurden) basiert auf der Konstruktion eines diskreten Witten-Komplexes und der Analyse des zugehörigen Witten-Laplace-Operatoren auf 1-Formen. Das Resultat kann im Kontext von metastabilen Markov Prozessen auf dem Gitter reformuliert werden und ermöglicht scharfe Aussagen über metastabile Austrittszeiten. KW - Semiklassische Spektralasymptotik KW - Metastabilität KW - diskreter Witten-Laplace-Operator KW - Eyring-Kramers Formel KW - Tunneleffekt KW - semiclassical spectral asymptotics KW - metastability KW - low-lying eignvalues KW - discrete Witten complex KW - rescaled lattice Y1 - 2012 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-65286 ER - TY - THES A1 - Hohberger, Horst T1 - Semiclassical asymptotics for the scattering amplitude in the presence of focal points at infinity T1 - Semiklassische Asymptotik der Streuamplitude bei unendlich fernen Fokalpunkten N2 - We consider scattering in $\R^n$, $n\ge 2$, described by the Schr\"odinger operator $P(h)=-h^2\Delta+V$, where $V$ is a short-range potential. With the aid of Maslov theory, we give a geometrical formula for the semiclassical asymptotics as $h\to 0$ of the scattering amplitude $f(\omega_-,\omega_+;\lambda,h)$ $\omega_+\neq\omega_-$) which remains valid in the presence of focal points at infinity (caustics). Crucial for this analysis are precise estimates on the asymptotics of the classical phase trajectories and the relationship between caustics in euclidean phase space and caustics at infinity. N2 - Wir betrachten Streuung in $\R^n$, $n\ge 2$, beschrieben durch den Schr\"odinger operator $P(h)=-h^2\Delta+V$, wo $V$ ein kurzreichweitiges Potential ist. Mit Hilfe von Maslov Theorie erhalten wir eine geometrische Formel fuer die semiklassische Asymptotik ($h\to 0$) der Streuamplitude $f(\omega_-,\omega_+;\lambda,h)$ ($\omega_+\neq\omega_-$) welche auch bei Vorhandensein von Fokalpunkten bei Unendlich (Kaustiken) gueltig bleibt. KW - Mathematik KW - Physik KW - Streutheorie KW - Streuamplitude KW - Semiklassik KW - mathematics KW - physics KW - scattering theory KW - semiclassics KW - scattering amplitude Y1 - 2006 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-11574 ER - TY - JOUR A1 - Sukiasyan, Hayk A1 - Melkonyan, Tatev T1 - Semi-recursive algorithm of piecewise linear approximation of two-dimensional function by the method of worst segment dividing JF - Lectures in pure and applied mathematics KW - random point processes KW - statistical mechanics KW - stochastic analysis Y1 - 2020 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-471982 SN - 978-3-86956-485-2 SN - 2199-4951 SN - 2199-496X IS - 6 SP - 35 EP - 44 PB - Universitätsverlag Potsdam CY - Potsdam ER - TY - INPR A1 - Kiselev, Oleg M. A1 - Tarkhanov, Nikolai Nikolaevich T1 - Scattering of autoresonance trajectories upon a separatrix N2 - We study asymptotic properties of solutions to the primary resonance equation with large amplitude on a long time interval. T3 - Preprints des Instituts für Mathematik der Universität Potsdam - 1 (2012) 2 Y1 - 2012 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-56880 ER - TY - INPR A1 - Tarkhanov, Nikolai Nikolaevich T1 - Root functions of elliptic boundary problems in domains with conic points of the boundary N2 - We prove the completeness of the system of eigen and associated functions (i.e., root functions) of an elliptic boundary value problem in a domain whose boundary is a smooth surface away from a finite number of points, each of them possesses a neighbourhood where the boundary is a conical surface. T3 - Preprint - (2005) 07 Y1 - 2005 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-29812 ER - TY - INPR A1 - Böckmann, Christine A1 - Biele, Jens A1 - Neuber, Roland A1 - Niebsch, Jenny T1 - Retrieval of multimodal aerosol size distribution by inversion of multiwavelength data N2 - The ill-posed problem of aerosol size distribution determination from a small number of backscatter and extinction measurements was solved successfully with a mollifier method which is advantageous since the ill-posed part is performed on exactly given quantities, the points r where n(r) is evaluated may be freely selected. A new twodimensional model for the troposphere is proposed. T3 - NLD Preprints - 38 KW - Multiwavelength LIDAR KW - aerosol size distribution KW - ill-posed problem KW - inversion KW - mollifier method KW - coated and absorbing aerosols Y1 - 1997 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-14360 ER - TY - INPR A1 - Klein, Markus A1 - Zitt, Pierre-André T1 - Resonances for a diffusion with small noise N2 - We study resonances for the generator of a diffusion with small noise in R(d) : L = -∈∆ + ∇F * ∇, when the potential F grows slowly at infinity (typically as a square root of the norm). The case when F grows fast is well known, and under suitable conditions one can show that there exists a family of exponentially small eigenvalues, related to the wells of F. We show that, for an F with a slow growth, the spectrum is R+, but we can find a family of resonances whose real parts behave as the eigenvalues of the "quick growth" case, and whose imaginary parts are small. T3 - Mathematische Statistik und Wahrscheinlichkeitstheorie : Preprint - 2008, 02 Y1 - 2008 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-49448 ER - TY - INPR A1 - Gil, Juan B. A1 - Krainer, Thomas A1 - Mendoza, Gerardo A. T1 - Resolvents of elliptic cone operators N2 - We prove the existence of sectors of minimal growth for general closed extensions of elliptic cone operators under natural ellipticity conditions. This is achieved by the construction of a suitable parametrix and reduction to the boundary. Special attention is devoted to the clarification of the analytic structure of the resolvent. T3 - Preprint - (2004) 22 Y1 - 2004 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-26820 ER - TY - INPR A1 - Krainer, Thomas T1 - Resolvents of elliptic boundary problems on conic manifolds N2 - We prove the existence of sectors of minimal growth for realizations of boundary value problems on conic manifolds under natural ellipticity conditions. Special attention is devoted to the clarification of the analytic structure of the resolvent. T3 - Preprint - (2005) 03 Y1 - 2005 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-29773 ER - TY - INPR A1 - Bär, Christian T1 - Renormalized integrals and a path integral formula for the heat kernel on a manifold N2 - 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. T3 - Preprints des Instituts für Mathematik der Universität Potsdam - 1(2012)21 KW - Renormalized integral KW - path integral KW - Feynman-Kac formula KW - generalized Laplace operator KW - Riemannian manifold Y1 - 2012 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-60052 ER - TY - INPR A1 - Kytmanov, Aleksandr A1 - Myslivets, Simona A1 - Tarkhanov, Nikolai Nikolaevich T1 - Removable singularities of CR functions on singular boundaries N2 - The problem of analytic representation of integrable CR functions on hypersurfaces with singularities is treated. The nature o singularities does not matter while the set of singularities has surface measure zero. For simple singularities like cuspidal points, edges, corners, etc., also the behaviour of representing analytic functions near singular points is studied. T3 - Preprint - (2000) 18 Y1 - 2000 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25836 ER - TY - GEN A1 - Ginoux, Nicolas T1 - Remarques sur le spectre de l'opérateur de Dirac T1 - Remarks on the spectrum of the Dirac operator N2 - Nous décrivons un nouvelle famille d'exemples d'hypersurfaces de la sphère satisfaisant le cas d'égalité de la majoration extrinsèque de C. Bär de la plus petite valeur propre de l'opérateur de Dirac. N2 - We describe a new family of examples of hypersurfaces in the sphere satisfying the limitingcase in C. Bär's extrinsic upper bound for the smallest eigenvalue of the Dirac operator. KW - 1st Eigenvalue KW - Submanifolds KW - Bounds KW - Space Y1 - 2003 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-5630 ER - TY - BOOK A1 - Zhuchok, Anatolii V. T1 - Relatively free doppelsemigroups N2 - A doppelalgebra is an algebra defined on a vector space with two binary linear associative operations. Doppelalgebras play a prominent role in algebraic K-theory. We consider doppelsemigroups, that is, sets with two binary associative operations satisfying the axioms of a doppelalgebra. Doppelsemigroups are a generalization of semigroups and they have relationships with such algebraic structures as interassociative semigroups, restrictive bisemigroups, dimonoids, and trioids. In the lecture notes numerous examples of doppelsemigroups and of strong doppelsemigroups are given. The independence of axioms of a strong doppelsemigroup is established. A free product in the variety of doppelsemigroups is presented. We also construct a free (strong) doppelsemigroup, a free commutative (strong) doppelsemigroup, a free n-nilpotent (strong) doppelsemigroup, a free n-dinilpotent (strong) doppelsemigroup, and a free left n-dinilpotent doppelsemigroup. Moreover, the least commutative congruence, the least n-nilpotent congruence, the least n-dinilpotent congruence on a free (strong) doppelsemigroup and the least left n-dinilpotent congruence on a free doppelsemigroup are characterized. The book addresses graduate students, post-graduate students, researchers in algebra and interested readers. N2 - Eine Doppelalgebra ist eine auf einem Vektorraum definierte Algebra mit zwei binären linearen assoziativen Operationen. Doppelalgebren spielen eine herausragende Rolle in der algebraischen K-Theorie. Wir betrachten Doppelhalbgruppen, d.h Mengen mit zwei binären assoziativen Operationen, welche die Axiome der Doppelhalbgruppe erfüllen. Doppelhalbgruppen sind Veralgemeinerungen von Halbgruppen und sie stehen in Beziehung zu solchen algebraischen Strukturen wie interassoziative Halbgruppen, restriktive Bihalbgruppen, Dimonoiden und Trioden. In dieser Lecture Notes werden eine Vielzahl von Beispielen für Doppelhalbgruppen und strong Doppelhalbgruppen gegeben. Die Unabhängigkeit der Axiome für Doppelhalbgruppen wird nachgewiesen. Ein freies Produkt in der Varietät der Doppelhalbgruppen wird vorgestellt. Wir konstruieren auch eine freie (kommutative) strong Doppelhalbgruppe, eine freie n-dinilpotent (strong) Doppelhalbgruppe und eine freie Links n-dinilpotent Doppelhalbgruppe. Darüber hinaus werden die kleinste n-nilpotente Kogruenz, die kleinste n-dinilpotente Kongruenz auf der freien (strong) Doppelhalbgruppe und die kleinste n-dinilpotente Kongruenz auf einer freien Doppelhalbgruppe charakterisiert. Das Buch richtet sich an Graduierte, Doktoranden, Forscher in Algebra und interessierte Leser. T3 - Lectures in pure and applied mathematics - 5 KW - doppelsemigroup KW - interassociativity KW - free algebra KW - semigroup KW - congruence Y1 - 2018 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-407719 SN - 978-3-86956-427-2 SN - 2199-4951 SN - 2199-496X IS - 5 PB - Universitätsverlag Potsdam CY - Potsdam ER - TY - INPR A1 - Nazaikinskii, Vladimir A1 - Sternin, Boris T1 - Relative elliptic theory N2 - This paper is a survey of relative elliptic theory (i.e. elliptic theory in the category of smooth embeddings), closely related to the Sobolev problem, first studied by Sternin in the 1960s. We consider both analytic aspects to the theory (the structure of the algebra of morphismus, ellipticity, Fredholm property) and topological aspects (index formulas and Riemann-Roch theorems). We also study the algebra of Green operators arising as a subalgebra of the algebra of morphisms. T3 - Preprint - (2002) 23 KW - Sobolev problem KW - elliptic morphism KW - (co)boundary operator KW - Green operator KW - index KW - Riemann-Roch theorem Y1 - 2002 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-26400 ER - TY - THES A1 - Scharrer, Christian T1 - Relating diameter and mean curvature for varifolds T1 - Relativer Diameter und mittlere Krümmung für Varifaltigkeiten N2 - The main results of this thesis are formulated in a class of surfaces (varifolds) generalizing closed and connected smooth submanifolds of Euclidean space which allows singularities. Given an indecomposable varifold with dimension at least two in some Euclidean space such that the first variation is locally bounded, the total variation is absolutely continuous with respect to the weight measure, the density of the weight measure is at least one outside a set of weight measure zero and the generalized mean curvature is locally summable to a natural power (dimension of the varifold minus one) with respect to the weight measure. The thesis presents an improved estimate of the set where the lower density is small in terms of the one dimensional Hausdorff measure. Moreover, if the support of the weight measure is compact, then the intrinsic diameter with respect to the support of the weight measure is estimated in terms of the generalized mean curvature. This estimate is in analogy to the diameter control for closed connected manifolds smoothly immersed in some Euclidean space of Peter Topping. Previously, it was not known whether the hypothesis in this thesis implies that two points in the support of the weight measure have finite geodesic distance. N2 - Die wichtigsten Ergebnisse dieser Arbeit sind formuliert für eine Klasse von Oberflächen (Varifaltigkeiten), welche geschlossene glatte Untermannigfaltigkeiten des euklidischen Raums verallgemeinern und Singularitäten erlauben. Gegeben sei eine mindestens zwei-dimensionale unzerlegbare Varifaltigkeit im euklidischen Raum, sodass die erste Variation lokal beschränkt ist, die totale Variation absolut stetig bezüglich dem Gewichtsmaß ist, die Dichte des Gewichtsmaßes außerhalb einer Nullmenge mindesten eins ist, und die verallgemeinerte mittlere Krümmung bezüglich dem Gewichtsmaß lokal summierbar zu einer natürlichen Potenz (Dimension der Varifaltigkeit minus eins) ist. Es wird die Menge, wo die untere Dichte klein ist, durch das ein-dimensionale Hausdorff-Maß abgeschätzt. Das Ergebnis ist eine neue, stark verbesserte untere Dichte-Schranke. Ist der Träger des Gewichtsmaßes kompakt, so wird der intrinsische Diameter des Trägers des Gewichtsmaßes abgeschätzt durch ein Integral der verallgemeinerten mittleren Krümmung. Diese Ungleichung ist analog zu der Ungleichung von Peter Topping für geschlossene zusammenhängende Mannigfaltigkeit, welche durch eine glatte Immersion in den euklidischen Raum eingebettet sind. Bisher war nicht bekannt, dass die oben genannten Annahmen an die Varifaltigkeit implizieren, dass der geodätische Abstand zweier Punkte aus dem Träger des Gewichtsmaßes endlich ist. KW - varifold KW - rectifiable varifold KW - indecomposable varifold KW - first variation KW - mean curvature KW - isoperimetric inequality KW - density of a measure KW - geodesic distance KW - intrinsic diameter KW - Varifaltigkeit KW - rektifizierbare Varifaltigkeit KW - unzerlegbare Varifaltigkeit KW - erste Variation KW - mittlere Krümmung KW - isoperimetrische Ungleichung KW - Dichte eines Maßes KW - geodätischer Abstand KW - intrinsischer Diameter Y1 - 2016 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-97013 ER - TY - INPR A1 - Makhmudov, O. I. A1 - Niyozov, I. E. T1 - Regularization of the Cauchy Problem for the System of Elasticity Theory in R up (m) N2 - In this paper we consider the regularization of the Cauchy problem for a system of second order differential equations with constant coefficients. T3 - Preprint - (2005) 22 KW - the Cauchy problem KW - Lame system KW - elliptic system KW - ill-posed problem KW - Carleman matrix KW - regularization KW - Laplace equation Y1 - 2005 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-29983 ER - TY - INPR A1 - Schulze, Bert-Wolfgang A1 - Shlapunov, Alexander A1 - Tarkhanov, Nikolai Nikolaevich T1 - Regularisation of mixed boundary problems N2 - We show an application of the spectral theorem in constructing approximate solutions of mixed boundary value problems for elliptic equations. T3 - Preprint - (1999) 09 Y1 - 1999 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25454 ER - TY - INPR A1 - Harutjunjan, G. A1 - Schulze, Bert-Wolfgang T1 - Reduction of orders in boundary value problems without the transmission property N2 - Given an algebra of pseudo-differential operators on a manifold, an elliptic element is said to be a reduction of orders, if it induces isomorphisms of Sobolev spaces with a corresponding shift of smoothness. Reductions of orders on a manifold with boundary refer to boundary value problems. We consider smooth symbols and ellipticity without additional boundary conditions which is the relevant case on a manifold with boundary. Starting from a class of symbols that has been investigated before for integer orders in boundary value problems with the transmission property we study operators of arbitrary real orders that play a similar role for operators without the transmission property. Moreover, we show that order reducing symbols have the Volterra property and are parabolic of anisotropy 1; analogous relations are formulated for arbitrary anisotropies. We finally investigate parameter-dependent operators, apply a kernel cut-off construction with respect to the parameter and show that corresponding holomorphic operator-valued Mellin symbols reduce orders in weighted Sobolev spaces on a cone with boundary. T3 - Preprint - (2002) 03 KW - Boundary value problems KW - elliptic operators KW - order reduction KW - Volterra symbols Y1 - 2002 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-26220 ER - TY - THES A1 - Demircioglu, Aydin T1 - Reconstruction of deligne classes and cocycles T1 - Rekonstruktion von Deligne Klassen und Kozykeln N2 - In der vorliegenden Arbeit verallgemeinern wir im Wesentlichen zwei Theoreme von Mackaay-Picken und Picken (2002, 2004). Im ihrem Artikel zeigen Mackaay und Picken,dass es eine bijektive Korrespodenz zwischen Deligne 2-Klassen $\xi \in \check{H}^2(M, \mathcal{D}^2)$ und Holonomie Abbildungen von der zweiten dünnen Homotopiegruppe $\pi_2^2(M)$ in die abelsche Gruppe $U(1)$ gibt. Im zweiten Artikel wird eine Verallgemeinerung dieses Theorems bewiesen: Picken zeigt, dass es eine Bijektion gibt zwischen Deligne 2-Kozykeln und gewissen 2-dimensionalen topologischen Quantenfeldtheorien. In dieser Arbeit zeigen wir, dass diese beiden Theoreme in allen Dimensionen gelten.Wir betrachten zunächst den Holonomie Fall und können mittels simplizialen Methoden nachweisen, dass die Gruppe der glatten Deligne $d$-Klassen isomorph ist zu der Gruppe der glatten Holonomie Abbildungen von der $d$-ten dünnen Homotopiegruppe $\pi_d^d(M)$ nach $U(1)$, sofern $M$ eine $(d-1)$-zusammenhängende Mannigfaltigkeit ist. Wir vergleichen dieses Resultat mit einem Satz von Gajer (1999). Gajer zeigte, dass jede Deligne $d$-Klasse durch eine andere Klasse von Holonomie-Abbildungen rekonstruiert werden kann, die aber nicht nur Holonomien entlang von Sphären, sondern auch entlang von allgemeinen $d$-Mannigfaltigkeiten in $M$ enthält. Dieser Zugang benötigt dann aber nicht, dass $M$ hoch-zusammenhängend ist. Wir zeigen, dass im Falle von flachen Deligne $d$-Klassen unser Rekonstruktionstheorem sich von Gajers unterscheidet, sofern $M$ nicht als $(d-1)$, sondern nur als $(d-2)$-zusammenhängend angenommen wird. Stiefel Mannigfaltigkeiten besitzen genau diese Eigenschaft, und wendet man unser Theorem auf diese an und vergleicht das Resultat mit dem von Gajer, so zeigt sich, dass es zuviele Deligne Klassen rekonstruiert. Dies bedeutet, dass unser Rekonstruktionsthreorem ohne die Zusatzbedingungen an die Mannigfaltigkeit M nicht auskommt, d.h. unsere Rekonstruktion benötigt zwar weniger Informationen über die Holonomie entlang von d-dimensionalen Mannigfaltigkeiten, aber dafür muss M auch $(d-1)$-zusammenhängend angenommen werden. Wir zeigen dann, dass auch das zweite Theorem verallgemeinert werden kann: Indem wir das Konzept einer Picken topologischen Quantenfeldtheorie in beliebigen Dimensionen einführen, können wir nachweisen, dass jeder Deligne $d$-Kozykel eine solche $d$-dimensionale Feldtheorie mit zwei besonderen Eigenschaften, der dünnen Invarianz und der Glattheit, induziert. Wir beweisen, dass jede $d$-dimensionale topologische Quantenfeldtheorie nach Picken mit diesen zwei Eigenschaften auch eine Deligne $d$-Klasse definiert und prüfen nach, dass diese Konstruktion sowohl surjektiv als auch injektiv ist. Demzufolge sind beide Gruppen isomorph. N2 - In this thesis we mainly generalize two theorems from Mackaay-Picken and Picken (2002, 2004). In the first paper, Mackaay and Picken show that there is a bijective correspondence between Deligne 2-classes $\xi \in \check{H}^2(M,\mathcal{D}^2)$ and holonomy maps from the second thin-homotopy group $\pi_2^2(M)$ to $U(1)$. In the second one, a generalization of this theorem to manifolds with boundaries is given: Picken shows that there is a bijection between Deligne 2-cocycles and a certain variant of 2-dimensional topological quantum field theories. In this thesis we show that these two theorems hold in every dimension. We consider first the holonomy case, and by using simplicial methods we can prove that the group of smooth Deligne $d$-classes is isomorphic to the group of smooth holonomy maps from the $d^{th}$ thin-homotopy group $\pi_d^d(M)$ to $U(1)$, if $M$ is $(d-1)$-connected. We contrast this with a result of Gajer (1999). Gajer showed that Deligne $d$-classes can be reconstructed by a different class of holonomy maps, which not only include holonomies along spheres, but also along general $d$-manifolds in $M$. This approach does not require the manifold $M$ to be $(d-1)$-connected. We show that in the case of flat Deligne $d$-classes, our result differs from Gajers, if $M$ is not $(d-1)$-connected, but only $(d-2)$-connected. Stiefel manifolds do have this property, and if one applies our theorem to these and compare the result with that of Gajers theorem, it is revealed that our theorem reconstructs too many Deligne classes. This means, that our reconstruction theorem cannot live without the extra assumption on the manifold $M$, that is our reconstruction needs less informations about the holonomy of $d$-manifolds in $M$ at the price of assuming $M$ to be $(d-1)$-connected. We continue to show, that also the second theorem can be generalized: By introducing the concept of Picken-type topological quantum field theory in arbitrary dimensions, we can show that every Deligne $d$-cocycle induces such a $d$-dimensional field theory with two special properties, namely thin-invariance and smoothness. We show that any $d$-dimensional topological quantum field theory with these two properties gives rise to a Deligne $d$-cocycle and verify that this construction is surjective and injective, that is both groups are isomorphic. KW - Holonomie KW - Hauptfaserbündel KW - Gerben KW - Deligne Kohomologie KW - Globale Differentialgeometrie KW - Holonomy KW - Prinicipal Fibre Bundles KW - Gerbes KW - Deligne Cohomology KW - Global Differentialgeometry Y1 - 2007 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-13755 ER - TY - INPR A1 - Roelly, Sylvie T1 - Reciprocal processes : a stochastic analysis approach N2 - Reciprocal processes, whose concept can be traced back to E. Schrödinger, 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. Léonard. T3 - Preprints des Instituts für Mathematik der Universität Potsdam - 2 (2013) 6 KW - Reciprocal process KW - Brownian bridge KW - Poisson bridge KW - duality formula Y1 - 2013 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-64588 SN - 2193-6943 ER - TY - THES A1 - Murr, Rüdiger T1 - Reciprocal classes of Markov processes : an approach with duality formulae T1 - Reziproke Klassen von Markov Prozessen : ein Ansatz mit Dualitätsformeln N2 - 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 Lévy 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. N2 - Diese Arbeit befasst sich mit der Charakterisierung von Klassen stochastischer Prozesse durch Dualitätsformeln. Es wird insbesondere der in der Literatur bisher unbehandelte Fall reziproker Klassen stochastischer Prozesse mit Sprungen untersucht. Im ersten Teil stellen wir eine neue Formulierung einer Charakterisierung von Prozessen mit unabhängigen Zuwächsen vor. Diese basiert auf der aus dem Malliavinkalkül bekannten Dualitätsformel für Prozesse mit unendlich oft teilbaren Zuwächsen. Wir präsentieren zusätzlich zwei neue Beweismethoden dieser Dualitätsformel, die nicht auf der Chaoszerlegung des Raumes quadratintegrabler Funktionale beruhen. Eine dieser Methoden basiert auf einer partiellen Integrationsformel fur unendlich oft teilbare Zufallsvektoren. In diesem Rahmen ist unsere Charakterisierung eine Verallgemeinerung des Lemma fur Gaußsche Zufallsvariablen von Stein und des Lemma fur Zufallsvariablen mit Poissonverteilung von Chen. Die Allgemeinheit dieser Methode erlaubt uns durch einen ähnlichen Zugang die Charakterisierung unendlich oft teilbarer Zufallsmaße. Im zweiten Teil der Arbeit konzentrieren wir uns auf die Charakterisierung reziproker Klassen ausgewählter Markovprozesse durch Dualitätsformeln. Wir beginnen mit einer Zusammenfassung bereits existierender Ergebnisse zu den reziproken Klassen Brownscher Bewegungen mit Drift. Es ist uns möglich die Charakterisierung solcher reziproken Klassen durch eine Dualitätsformel physikalisch umzudeuten in eine Newtonsche Gleichung. Damit gelingt uns ein Brückenschlag zwischen derartigen Charakterisierungsergebnissen und der Theorie stochastischer Mechanik durch den Interpretationsansatz, sowie der Theorie stochastischer optimaler Steuerung durch den mathematischen Ansatz. Unter Verwendung der Charakterisierung reziproker Klassen durch Dualitätsformeln beweisen wir weiterhin eine Invarianzeigenschaft der reziproken Klasse Browscher Bewegungen mit Drift unter Zeitumkehrung. Es gelingt uns weiterhin neue Resultate im Rahmen reiner Sprungprozesse zu beweisen. Wir beschreiben reziproke Klassen Markovscher Zählprozesse, d.h. Sprungprozesse mit Sprunghöhe eins, und erhalten eine Charakterisierung der reziproken Klasse vermöge einer Dualitätsformel. Diese beinhaltet als Schlüsselterme eine stochastische Ableitung nach den Sprungzeiten, ein kompensiertes stochastisches Integral und eine Invariante der reziproken Klasse. Wir präsentieren außerdem eine Interpretation der Charakterisierung einer reziproken Klasse im Rahmen der stochastischen Steuerungstheorie. Als weitere Anwendung beweisen wir eine Invarianzeigenschaft der reziproken Klasse Markovscher Zählprozesse unter Zeitumkehrung. Einige dieser Ergebnisse werden fur reine Sprungprozesse mit unterschiedlichen Sprunghöhen verallgemeinert. Insbesondere zeigen wir, dass die reziproken Klassen Markovscher Sprungprozesse vermöge reziproker Invarianten unterschieden werden können. Eine Charakterisierung der reziproken Klasse zusammengesetzter Poissonprozesse durch eine Dualitätsformel gelingt unter der Annahme inkommensurabler Sprunghöhen. KW - unendliche Teilbarkeit KW - Dualitätsformeln KW - reziproke Klassen KW - Zählprozesse KW - stochastische Mechanik KW - infinite divisibility KW - duality formulae KW - reciprocal class KW - counting process KW - stochastic mechanics Y1 - 2012 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-62091 ER - TY - INPR A1 - Murr, Rüdiger T1 - Reciprocal classes of Markov processes : an approach with duality formulae N2 - 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. T3 - Preprints des Instituts für Mathematik der Universität Potsdam - 1(2012)26 KW - Duality formula KW - reciprocal class KW - Levy process KW - infinite divisibility KW - counting process KW - Malliavin calculus Y1 - 2012 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-63018 ER - TY - THES A1 - Conforti, Giovanni T1 - Reciprocal classes of continuous time Markov Chains T1 - Reziproke Klassen zeitkontinuierlicher Markov-Ketten N2 - In this thesis we study reciprocal classes of Markov chains. Given a continuous time Markov chain on a countable state space, acting as reference dynamics, the associated reciprocal class is the set of all probability measures on path space that can be written as a mixture of its bridges. These processes possess a conditional independence property that generalizes the Markov property, and evolved from an idea of Schrödinger, who wanted to obtain a probabilistic interpretation of quantum mechanics. Associated to a reciprocal class is a set of reciprocal characteristics, which are space-time functions that determine the reciprocal class. We compute explicitly these characteristics, and divide them into two main families: arc characteristics and cycle characteristics. As a byproduct, we obtain an explicit criterion to check when two different Markov chains share their bridges. Starting from the characteristics we offer two different descriptions of the reciprocal class, including its non-Markov probabilities. The first one is based on a pathwise approach and the second one on short time asymptotic. With the first approach one produces a family of functional equations whose only solutions are precisely the elements of the reciprocal class. These equations are integration by parts on path space associated with derivative operators which perturb the paths by mean of the addition of random loops. Several geometrical tools are employed to construct such formulas. The problem of obtaining sharp characterizations is also considered, showing some interesting connections with discrete geometry. Examples of such formulas are given in the framework of counting processes and random walks on Abelian groups, where the set of loops has a group structure. In addition to this global description, we propose a second approach by looking at the short time behavior of a reciprocal process. In the same way as the Markov property and short time expansions of transition probabilities characterize Markov chains, we show that a reciprocal class is characterized by imposing the reciprocal property and two families of short time expansions for the bridges. Such local approach is suitable to study reciprocal processes on general countable graphs. As application of our characterization, we considered several interesting graphs, such as lattices, planar graphs, the complete graph, and the hypercube. Finally, we obtain some first results about concentration of measure implied by lower bounds on the reciprocal characteristics. N2 - Diese Dissertation behandelt die reziproke zufällige Prozesse mit Sprüngen. Gegeben eine zeitkontinuierliche Markovkette als Referenzdynamik, ist die assoziierte reziproke Klasse die Menge aller Wahrscheinlichkeiten auf dem Pfadraum, die als eine Mischung ihrer Brücken geschrieben werden kann. Reziproke Prozesse zeichnen sich durch eine Form der bedingten Unabhängigkeit aus, die die Markoveigenschaft verallgemeinert. Ursprünglich ist diese Idee auf Schrödinger zurückzuführen, der nach einer probabilistischen Interpretation für die Quantenmechanik suchte. Einer reziproken Klasse wird eine Familie reziproker Charakteristiken assoziiert. Dies sind Raum-Zeit Abbildungen, die die reziproke Klasse eindeutig definieren. Wir berechnen diese Charakteristiken explizit und unterteilen sie in zwei Typen: Bogen-Charakteristiken und Kreis-Charakteristiken. Zusätzlich erhalten wir ein klares Kriterium zur Prüfung wann die Brücken von zwei verschiedenen Markovketten übereinstimmen. Wir beschreiben auf zwei verschiedene Arten reziproken Klasse und berücksichtigen auch ihre nicht-Markov Elemente. Die erste Charakterisierung basiert auf einem pfadweisen Ansatz, während die zweite kurzzeit Asymptotik benutzt. Der erste Ansatz liefert eine Familie funktionaler Gleichungen deren einzige Lösungen die Elemente der reziproken Klasse sind. Die Gleichungen können als partielle Integration auf dem Pfadraum mit einem Ableitungsoperator, der eine St¨orung der Pfade durch zusätzliche zufällige Kreise hervorruft, interpretiert werden. Die Konstruktion dieser Gleichungen benötigt eine geometrische Analyse des Problems. Wir behandeln außerdem die Fragestellung einer scharfen Charakterisierung und zeigen interessante Verbindungen zur diskreten Geometrie. Beispiele, für die wir eine solche Formel finden konnten, sind für Zählprozesse und für Irrfahrte auf abelschen Gruppen, in denen die Menge der Kreise eine Gruppenstruktur erweist. Zusätzlich zu diesem globalen Zugang, erforschen wir eine lokale Beschreibung durch die Analyse des kurzfristigen Verhaltens eines reziproken Prozesses. Analog zur Markoveigenschaft und kurzzeit Entwicklung ihrer Übergangswahrscheinlichkeit Markovketten charakterisieren, zeigen wir, dass eine reziproke Klasse charakterisiert werden kann indem wir ihre reziproke Eigenschaft und zwei Familien von Kurzzeit Entwicklungen der Brücken voraussetzen. Solche lokalen Ansatz ist geeignet, um Sprungprozesse auf allgemeine zählbaren Graphen zu studieren. Als Beispiele unserer Charakterisierung, betrachten wir Gitter, planare Graphen, komplette Graphen und die Hyperwürfel. Zusätzlich präsentieren wir erste Ergebnisse über Maßenkonzentration eines reziproken Prozesses, als Konsequenz unterer Schranken seiner Charakteristiken. N2 - In questa tesi si studiano le classi reciproche delle catene di Markov. Data una catena di Markov a tempo continuo su uno spazio numerabile, che svolge il ruolo di dinamica di riferimento, la sua classe reciproca é costituita da tutte le leggi sullo spazio dei cammini che si possono scrivere come un miscuglio dei ponti della legge di riferimento. Questi processi stocastici godono di una propriet`a di independenza condizionale che generalizza la proprietá di Markov ed é ispirata ad un’idea avuta da Schrödinger nel tentativo di derivare un’interpretazione stocastica della meccanica quantistica. A ciascuna classe reciproca é associato un insieme di caratteristiche reciproche. Una caratteristica reciproca é una proprietá della dinamica di riferimento che viene trasmessa a tutti gli elementi della classe, e viene espressa matematicamente da un opportuna combinazione di funzionali del generatore della catena di riferimento. Nella tesi, le caratteristiche vengono calcolate esplicitamente e suddivise in due famiglie principali: le caratteristiche di arco e le caratteristice di ciclo. Come sottoprodotto, otteniamo un criterio esplicito per decidere quando due catene di Markov hanno gli stessi ponti. A partire dalle caratteristiche reciproche, vengono proposte due caratterizzazioni della classe reciproca, compresi i suoi elementi non Markoviani. La prima é basata su un approccio traiettoriale, mentre la seconda si basa sul comportamento asintotico locale dei processi reciproci. Utilizzando il primo approccio, si ottiene una famiglia di equazioni funzionali che ammette come soluzioni tutti e soli gli elementi della classe reciproca. Queste equazioni sono integrazioni per parti sullo spazio dei cammini associate ad operatori differenziali che perturbano le traiettorie del processo canonico con l’aggiunta di loops casuali. Nella costruzione di queste equazioni si impiegano tecniche di geometria discreta, stabilendo un interessante collegamento con risultati recenti in questo campo. Le caratterizzazioni ottenute sono ottimali, in quanto impiegano un numero minimo di equazioni per descrivere la classe. Con questo metodo vengono studiate le classi reciproche di processi di conteggio, di camminate aleatorie su gruppi Abeliani, dove l’insieme dei cicli gode anch’esso di una struttura di gruppo. Il secondo approccio, di natura locale, si basa su stime asintotiche in tempo corto. É ben noto come una catena di Markov sia caratterizzata dal fatto di possedere la propriet`a di Markov e dal comportamento in tempo corto delle probabilitá di transizione. In questa tesi mostriamo che una classe reciproca é caratterizzata dalla propriet`a reciproca, e da due famiglie di stime asintotiche per i ponti del processo. Questo approccio locale permette di analizzare le classi reciproche di passeggiate aleatorie su grafi generali. Come applicazione dei risultati teorici, consideriamo i lattici, i grafi planari, il grafo completo, e l’ipercubo discreto. Infine, otteniamo delle stime di concentrazione della misura e sul comportamento globale dei ponti, sotto l’ipotesi di un limite inferiore per le caratteristiche reciproche. KW - reciprocal characteristics KW - random walks on graphs KW - reziproke Invarianten KW - reziproke Klassen KW - Schrödinger Problem KW - partielle Integration auf dem Pfadraum KW - Irrfahrten auf Graphen Y1 - 2015 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-82255 ER - TY - INPR A1 - Conforti, Giovanni T1 - Reciprocal classes of continuous time Markov Chains N2 - 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. T3 - Preprints des Instituts für Mathematik der Universität Potsdam - 4 (2015) 8 KW - random walks on graphs KW - bridges of random walks KW - reciprocal characteristics KW - Schrödinger problem KW - integration by parts on path space Y1 - 2015 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-78234 SN - 2193-6943 VL - 4 IS - 8 PB - Universitätsverlag Potsdam CY - Potsdam ER - TY - INPR A1 - Conforti, Giovanni A1 - Roelly, Sylvie T1 - Reciprocal class of random walks on an Abelian group N2 - 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. T3 - Preprints des Instituts für Mathematik der Universität Potsdam - 4 (2015) 1 KW - reciprocal class KW - stochastic bridge KW - random walk on Abelian group Y1 - 2015 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-72604 SN - 2193-6943 VL - 4 IS - 1 PB - Universitätsverlag Potsdam CY - Potsdam ER - TY - INPR A1 - Conforti, Giovanni A1 - Dai Pra, Paolo A1 - Roelly, Sylvie T1 - Reciprocal class of jump processes N2 - 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. T3 - Preprints des Instituts für Mathematik der Universität Potsdam - 3 (2014) 6 KW - reciprocal processes KW - stochastic bridges KW - jump processes KW - compound Poisson processes Y1 - 2014 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-70776 SN - 2193-6943 VL - 3 IS - 6 PB - Universitätsverlag Potsdam CY - Potsdam ER - TY - THES A1 - Fischer, Jens Walter T1 - Random dynamics in collective behavior - consensus, clustering & extinction of populations T1 - Stochastische Dynamiken in kollektivem Verhalten: Konsens, Gruppenbildung, Aussterben von Populationen N2 - The echo chamber model describes the development of groups in heterogeneous social networks. By heterogeneous social network we mean a set of individuals, each of whom represents exactly one opinion. The existing relationships between individuals can then be represented by a graph. The echo chamber model is a time-discrete model which, like a board game, is played in rounds. In each round, an existing relationship is randomly and uniformly selected from the network and the two connected individuals interact. If the opinions of the individuals involved are sufficiently similar, they continue to move closer together in their opinions, whereas in the case of opinions that are too far apart, they break off their relationship and one of the individuals seeks a new relationship. In this paper we examine the building blocks of this model. We start from the observation that changes in the structure of relationships in the network can be described by a system of interacting particles in a more abstract space. These reflections lead to the definition of a new abstract graph that encompasses all possible relational configurations of the social network. This provides us with the geometric understanding necessary to analyse the dynamic components of the echo chamber model in Part III. As a first step, in Part 7, we leave aside the opinions of the inidividuals and assume that the position of the edges changes with each move as described above, in order to obtain a basic understanding of the underlying dynamics. Using Markov chain theory, we find upper bounds on the speed of convergence of an associated Markov chain to its unique stationary distribution and show that there are mutually identifiable networks that are not apparent in the dynamics under analysis, in the sense that the stationary distribution of the associated Markov chain gives equal weight to these networks. In the reversible cases, we focus in particular on the explicit form of the stationary distribution as well as on the lower bounds of the Cheeger constant to describe the convergence speed. The final result of Section 8, based on absorbing Markov chains, shows that in a reduced version of the echo chamber model, a hierarchical structure of the number of conflicting relations can be identified. We can use this structure to determine an upper bound on the expected absorption time, using a quasi-stationary distribution. This hierarchy of structure also provides a bridge to classical theories of pure death processes. We conclude by showing how future research can exploit this link and by discussing the importance of the results as building blocks for a full theoretical understanding of the echo chamber model. Finally, Part IV presents a published paper on the birth-death process with partial catastrophe. The paper is based on the explicit calculation of the first moment of a catastrophe. This first part is entirely based on an analytical approach to second degree recurrences with linear coefficients. The convergence to 0 of the resulting sequence as well as the speed of convergence are proved. On the other hand, the determination of the upper bounds of the expected value of the population size as well as its variance and the difference between the determined upper bound and the actual value of the expected value. For these results we use almost exclusively the theory of ordinary nonlinear differential equations. N2 - Beziehungen und damit Interaktion sowie Diskussion, aber auch Konflikt und Opposition bilden die Grundbausteine einer jeden Gesellschaft. Häufig wird Kommunikation als der übergreigende Begriff zur Beschreibung interner Strukturen einer Gesellschaft identifiziert. Dabei muss es sich aber nicht um eine Gesellschaft im Sinne von Nationen handeln, sondern kann auch schlicht eine Gruppe von Menschen umfassen, die miteinander strukturiert interagieren, beispielsweise, eine Gruppe von Angestellten, die an einem gemeinsamen Projekt arbeiten, oder die Mitglieder eines sozialen Netzwerks. In dieser Arbeit befassen wir uns mit der mathematischen Beschreibung solcher Prozesse innerhalb von Gruppen und Gesellschaften und legen dabei unseren Fokus auf die Bildung eines Konsens durch Interaktion aber auch die Konsequenzen von Konflikt und das potentielle Aussterben einer Population. Dabei werden zwei Modelle im Fokus des Interesses stehen: Das Echokammer Model sowie eine Erweiterung des Geburts-Todes Prozesses, die die Möglichkeit eines radikalen Abfalls der Populationsgr öße miteinschließt. Wir beginnen mit einer Einführung in Part I und teilen die verbleibende Arbeit in drei Teile auf, wobei sich die ersten beiden technischen Abschnitte, Part II und III, mit einer ausführlichen Analyse der Bausteine des Echokammer Models befassen und im dritten Abschnitt, in Part IV, der erweiterte Geburts- Todes Prozess untersucht wird. Dieser wird im Folgenden als Geburts-Todes Prozess mit teilweiser Katastrophe bezeichnet werden. Das Echokammer Model beschreibt die Entwicklung von Gruppen in zunächst heterogenen sozialen Netzwerken. Unter einem heterogenen sozialen Netzwerk verstehen wir dabei eine Menge von Individuen, von denen jedes exakt eine Meinungen vertritt. Meinungen werden vereinfacht durch Werte in [0, 1] modelliert. Bestehende Beziehungen unter den Individuen können dann durch einen Graphen dargestellt werden. Es handelt sich bei dem Echokammer Modell um ein zeit-diskretes Modell, das entsprechend, ähnlich einem Brettspiel, in Zügen abläuft. In jedem Zug wird zufällig gleichverteilt eine bestehende Beziehung aus dem Netzwerk ausgewählt und die beiden verbundenen Individuen interagieren. Dabei kann es zu zwei verschiedenen Interaktionen kommen. Sind die Meinungen der betroffenen Individuen hinreichend ähnlich, so nähern sie sich weiter in ihren Meinungen an, während sie im Fall von Meinungen, die zu weit von einander liegen, ihre Beziehung auflösen und sich eines der Individuen eine neue Beziehung sucht. 8 In dieser Arbeit untersuchen wir theoretisch die Bausteine dieses Modells. Dabei legen wir die Beobachtung zu Grunde, dass die Veränderungen der Beziehungsstruktur im Netzwerk durch einen System von interagierenden Partikeln auf einem abstrakteren Raum beschrieben werden kann. Dies erlaubt es insbesondere graphentheoretische überlegungen in die Analyse einfließen zu lassen. Diese überlegungen werden ausührlich in Part II diskutiert und führen zur Definition eines neuen, abstrahierten Graphens, der alle möglichen Beziehungskonfigurationen des sozialen Netzwerks umfasst. Dies erlaubt es uns einen ähnlichkeitsbegriff für Beziehungskonfigurationen auf Basis der benachbarten Knoten in besagtem Graphen zu definieren. Dies liefert uns das notwendige geometrische Verständnis um in Part III die dynamischen Komponenten des Echokammer models zu analysieren. Insbesondere fokusieren wir uns dabei auf die Dynamik der Kanten, für die bisher in der Literatur noch keine Ergebnisse existieren. Wir lassen zunächst in Abschnitt 7 die Meinungen der Individuen beiseite und nehmen an, dass die Position der Kanten sich in jedem Zug wie zuvor beschrieben ändert, um eine grundlegendes Verständnis der unterliegenden Dynamik zu erhalten. Unter der Verwendung der Theorie von Markovketten finden wir obere Schranken an die Konvergenzgeschwindigkeit einer assoziierten Markovkette gegen ihre eindeutige stationäre Verteilung und zeigen, dass es Netzwerke gibt, die miteinander identifizierbar und unter der analysierten Dynamik daheingehend ununterscheinbar sind, dass die stationäre Verteilung der assozierten Markovkette diesen Netzwerken dasselbe Gewicht zuordnet. Anschließend beweisen wir eine Reihe von quantitativen Resultaten, die sich insbesondere in Fällen, in denen die assozierte Markovkette reversibel ist, als berechenbar herausstellen. Insbesondere die explizite Form der stationären Verteilung sowie untere Schranken an die Cheeger Konstante zur Beschreibung der Konvergenzgeschwindigkeit stehen dabei im Fokus und werden ausführlich diskutiert. Nach dieser vertieften Analyse des reduzierten Modells, fügen wir die Meinungen unserer Betrachtung wieder hinzu. Das abschließende Result in Abschnitt 8, basierend auf absorbierenden Markovketten, liefert dann, dass in einer reduzierte Version des Echokammer Modells, in dem sich Individuen ähnlicher Meinung nicht annähern, eine hierarchische Struktur der Anzahl der konfliktreichen Beziehung identifiziert werden kann. Dies können wir ausnutzen, um eine obere Schranke an die erwartete Absorptionszeit, unter Zuhilfenahme einer quasi-stationären Verteilung, zu bestimmen. Diese hierarchische Struktur bildet außerdem eine Brücke zu klassischen Theorien von Geburts-Todes und, insbesondere, reinen Todes-Prozessen, für die eine reiche Literatur existiert. Wir zeigen abschließend auf, wie künftige Forschung diese Verbindung ausnutzen kann und diskutieren die Wichtigkeit der Ergbenisse als Bausteine eines vollständigen theoretischen Verständnisses des Echokammer Modells. Part IV stellt abschließend einen veröffentlichten Artikel vor, der sich dem Geburts- Todes Prozess mit teilweiser Katastrophe widmet. Besagter Artikel steht dabei auf zwei Säulen. Zum Einen der expliziten Berechnung des ersten Zeitpunkts einer Katastrophe, wenn die Population zu Beginn der Beobachtung von instabiler Größe ist. KW - Markov chains KW - graph theory KW - complex systems KW - interacting particle systems KW - Markovketten KW - komplexe Systeme KW - Graphentheorie KW - Systeme interagierender Partikel Y1 - 2022 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-553725 ER - TY - INPR A1 - Denk, Robert A1 - Krainer, Thomas T1 - R-Boundedness, pseudodifferential operators, and maximal regularity for some classes of partial differential operators N2 - It is shown that an elliptic scattering operator A on a compact manifold with boundary with operator valued coefficients in the morphisms of a bundle of Banach spaces of class (HT ) and Pisier’s property (α) has maximal regularity (up to a spectral shift), provided that the spectrum of the principal symbol of A on the scattering cotangent bundle avoids the right half-plane. This is accomplished by representing the resolvent in terms of pseudodifferential operators with R-bounded symbols, yielding by an iteration argument the R-boundedness of λ(A − λ)−1 in R(λ)≥ τ for some τ ∈ IR. To this end, elements of a symbolic and operator calculus of pseudodifferential operators with R-bounded symbols are introduced. The significance of this method for proving maximal regularity results for partial differential operators is underscored by considering also a more elementary situation of anisotropic elliptic operators on Rd with operator valued coefficients. T3 - Preprint - (2006) 14 Y1 - 2006 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-30147 ER - TY - INPR A1 - Berman, Gennady A1 - Tarkhanov, Nikolai Nikolaevich T1 - Quantum dynamics in the Fermi-Pasta-Ulam problem N2 - We study the dynamics of four wave interactions in a nonlinear quantum chain of oscillators under the "narrow packet" approximation. We determine the set of times for which the evolution of decay processes is essentially specified by quantum effects. Moreover, we highlight the quantum increment of instability. T3 - Preprint - (2004) 05 Y1 - 2004 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-26695 ER - TY - INPR A1 - Nazaikinskii, Vladimir A1 - Schulze, Bert-Wolfgang A1 - Sternin, Boris A1 - Shatalov, Victor T1 - Quantization of symplectic transformations on manifolds with conical singularities N2 - The structure of symplectic (canonical) transformations on manifolds with conical singularities is established. The operators associated with these transformations are defined in the weight spaces and their properties investigated. T3 - Preprint - (1997) 23 KW - manifolds with conical singularities KW - symplectic (canonical) transformations KW - quantization KW - Mellin transform Y1 - 1997 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25084 ER - TY - INPR A1 - Nazaikinskii, Vladimir A1 - Schulze, Bert-Wolfgang A1 - Sternin, Boris T1 - Quantization methods in differential equations : Part II: Quantization by the method of ordered operators (Noncommutative Analysis) : Chapter 1: Noncommutative Analysis: Main Ideas, Definitions, and Theorems N2 - Content: 0.1 Preliminary Remarks Chapter 1: Noncommutative Analysis: Main Ideas, Definitions, and Theorems 1.1 Functions of One Operator (Functional Calculi) 1.2 Functions of Several Operators 1.3 Main Formulas of Operator Calculus 1.4 Main Tools of Noncommutative Analysis 1.5 Composition Laws and Ordered Representations T3 - Preprint - (2000) 11 Y1 - 2000 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25762 ER - TY - INPR A1 - Nazaikinskii, Vladimir A1 - Schulze, Bert-Wolfgang A1 - Sternin, Boris T1 - Quantization methods in differential equations : Chapter 3: Applications of noncommutative analysis to operator algebras on singular manifolds N2 - Content: Chapter 3: Applications of Noncommutative Analysis to Operator Algebras on Singular Manifolds 3.1 Statement of the problem 3.2 Operators on the Model Cone 3.3 Operators on the Model Cusp of Order k 3.4 An Application to the Construction of Regularizers and Proof of the Finiteness Theorem T3 - Preprint - (2000) 15 Y1 - 2000 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25801 ER - TY - INPR A1 - Nazaikinskii, Vladimir E. A1 - Schulze, Bert-Wolfgang A1 - Sternin, Boris T1 - Quantization methods in differential equations : Chapter 2: Quantization of Lagrangian modules N2 - In this chapter we use the wave packet transform described in Chapter 1 to quantize extended classical states represented by so-called Lagrangian sumbanifolds of the phase space. Functions on a Lagrangian manifold form a module over the ring of classical Hamiltonian functions on the phase space (with respect to pointwise multiplication). The quantization procedure intertwines this multiplication with the action of the corresponding quantum Hamiltonians; hence we speak of quantization of Lagrangian modules. The semiclassical states obtained by this quantization procedure provide asymptotic solutions to differential equations with a small parameter. Locally, such solutions can be represented by WKB elements. Global solutions are given by Maslov's canonical operator [2]; also see, e.g., [3] and the references therein. Here the canonical operator is obtained in the framework of the universal quantization procedure provided by the wave packet transform. This procedure was suggested in [4] (see also the references there) and further developed in [5]; our exposition is in the spirit of these papers. Some further bibliographical remarks can be found in the beginning of Chapter 1. T3 - Preprint - (1999) 22 Y1 - 1999 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25582 ER - TY - INPR A1 - Nazaikinskii, Vladimir A1 - Schulze, Bert-Wolfgang A1 - Sternin, Boris T1 - Quantization methods in differential equations : Chapter 2: Exactly soluble commutation relations (The simplest class of classical mechanics) N2 - Content: Chapter 2: Exactly SolubleCommutation Relations (The Simplest Class of Classical Mechanics) 2.1 Some examples 2.2 Lie commutation relations 2.3 Non-Lie (nonlinear) commutation relations T3 - Preprint - (2000) 14 Y1 - 2000 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25796 ER - TY - INPR A1 - Nazaikinskii, Vladimir A1 - Schulze, Bert-Wolfgang A1 - Sternin, Boris T1 - Quantization methods in differential equations : Chapter 11: Noncommutative analysis and high-frequency asymptotics N2 - Content: Chapter 11: Noncommutative Analysis and High-Frequency Asymptotics 11.1 Statement of the Problem 11.2 Mixed Asymptotics: the General Scheme 11.3 The Asymptotic Solution of Main Problem 11.4 Analysis of the Asymptotic Solution T3 - Preprint - (2000) 20 Y1 - 2000 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25857 ER - TY - INPR A1 - Nazaikinskii, Vladimir A1 - Schulze, Bert-Wolfgang A1 - Sternin, Boris T1 - Quantization and the wave packet transform T3 - Preprint - (1999) 08 Y1 - 1999 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25447 ER - TY - INPR A1 - Savin, Anton A1 - Sternin, Boris T1 - Pseudodifferential subspaces and their applications in elliptic theory N2 - The aim of this paper is to explain the notion of subspace defined by means of pseudodifferential projection and give its applications in elliptic theory. Such subspaces are indispensable in the theory of well-posed boundary value problems for an arbitrary elliptic operator, including the Dirac operator, which has no classical boundary value problems. Pseudodifferential subspaces can be used to compute the fractional part of the spectral Atiyah–Patodi–Singer eta invariant, when it defines a homotopy invariant (Gilkey’s problem). Finally, we explain how pseudodifferential subspaces can be used to give an analytic realization of the topological K-group with finite coefficients in terms of elliptic operators. It turns out that all three applications are based on a theory of elliptic operators on closed manifolds acting in subspaces. T3 - Preprint - (2005) 17 KW - elliptic operator KW - boundary value problem KW - pseudodifferential subspace KW - dimension functional KW - η-invariant KW - index KW - modn-index KW - parity condition Y1 - 2005 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-29937 ER - TY - INPR A1 - Schulze, Bert-Wolfgang A1 - Tarkhanov, Nikolai Nikolaevich T1 - Pseudodifferential operators on manifolds with corners N2 - We describe an algebra of pseudodifferential operators on a manifold with corners. T3 - Preprint - (2000) 13 Y1 - 2000 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25783 ER - TY - INPR A1 - Xiaochun, Liu A1 - Witt, Ingo T1 - Pseudodifferential calculi on the half-line respecting prescribed asymptotic types N2 - Contents: 1. Introduction 2. Preliminaries 3. Basic Elements of the Calculus 4. Further Elements of the Calculus T3 - Preprint - (2002) 06 Y1 - 2002 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-26255 ER - TY - INPR A1 - Schulze, Bert-Wolfgang A1 - Seiler, Jörg T1 - Pseudodifferential boundary value problems with global projection conditions N2 - Contents: Introduction 1 Operators with the transmission property 1.1 Operators on a manifold with boundary 1.2 Conditions with pseudodifferential projections 1.3 Projections and Fredholm families 2 Boundary value problems not requiring the transmission property 2.1 Interior operators 2.2 Edge amplitude functions 2.3 Boundary value problems 3 Operators with global projection conditions 3.1 Construction for boundary symbols 3.2 Ellipticity of boundary value problems with projection data 3.3 Operators of order zero T3 - Preprint - (2002) 04 Y1 - 2002 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-26233 ER - TY - INPR A1 - Lauter, Robert A1 - Seiler, Jörg T1 - Pseudodifferential analysis on manifolds with boundary - a comparison of b-calculus and cone algebra N2 - We establish a relation between two different approaches to a complete pseudodifferential analysis of totally characteristic or Fuchs type operators on compact manifolds with boundary respectively conical singularities: Melrose's (overblown) b-calculus and Schulze's cone algebra. Though quite different in their definition, we show that these two pseudodifferential calculi basically contain the same operators. T3 - Preprint - (1999) 27 Y1 - 1999 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25611 ER - TY - INPR A1 - Fedosov, Boris T1 - Pseudo-differential operators and deformation quantization N2 - Using the Riemannian connection on a compact manifold X, we show that the algebra of classical pseudo-differential operators on X generates a canonical deformation quantization on the cotangent manifold T*X. The corresponding Abelian connection is calculated explicitly in terms of the of the exponential mapping. We prove also that the index theorem for elliptic operators may be obtained as a consequence of the index theorem for deformation quantization. T3 - Preprint - (1999) 32 Y1 - 1999 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25651 ER - TY - INPR A1 - Kapanadze, David A1 - Schulze, Bert-Wolfgang T1 - Pseudo-differential crack theory N2 - Crack problems are regarded as elements in a pseudo-differential algbra, where the two sdes int S± of the crack S are treated as interior boundaries and the boundary Y of the crack as an edge singularity. We employ the pseudo-differential calculus of boundary value problems with the transmission property near int S± and the edge pseudo-differential calculus (in a variant with Douglis-Nirenberg orders) to construct parametrices od elliptic crack problems (with extra trace and potential conditions along Y) and to characterise asymptotics of solutions near Y (expressed in the framework of continuous asymptotics). Our operator algebra with boundary and edge symbols contains new weight and order conventions that are necessary also for the more general calculus on manifolds with boundary and edges. T3 - Preprint - (2000) 09 KW - Crack theory KW - pseudo-differential boundary value problems KW - operator algebras on manifolds with singularities KW - conormal asymptotics Y1 - 2000 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25759 ER - TY - INPR A1 - Schulze, Bert-Wolfgang T1 - Pseudo-differential calculus on manifolds with geometric singularities N2 - Differential and pseudo-differential operators on a manifold with (regular) geometric singularities can be studied within a calculus, inspired by the concept of classical pseudo-differential operators on a C1 manifold. In the singular case the operators form an algebra with a principal symbolic hierarchy σ = (σj)0≤j≤k, with k being the order of the singularity and σk operator-valued for k ≥ 1. The symbols determine ellipticity and the nature of parametrices. It is typical in this theory that, similarly as in boundary value problems (which are special edge problems, where the edge is just the boundary), there are trace, potential and Green operators, associated with the various strata of the configuration. The operators, obtained from the symbols by various quantisations, act in weighted distribution spaces with multiple weights. We outline some essential elements of this calculus, give examples and also comment on new challenges and interesting problems of the recent development. T3 - Preprint - (2006) 20 Y1 - 2006 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-30204 ER - TY - INPR A1 - Dereudre, David A1 - Roelly, Sylvie T1 - Propagation of Gibbsianness for infinite-dimensional gradient Brownian diffusions N2 - We study the (strong-)Gibbsian character on RZd of the law at time t of an infinitedimensional gradient Brownian diffusion , when the initial distribution is Gibbsian. T3 - Mathematische Statistik und Wahrscheinlichkeitstheorie : Preprint - 2004, 06 KW - infinite-dimensional Brownian diffusion KW - Gibbs measure KW - cluster expansion Y1 - 2004 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-51535 ER - TY - INPR A1 - Roelly, Sylvie A1 - Ruszel, Wioletta M. T1 - Propagation of Gibbsianness for infinite-dimensional diffusions with space-time interaction N2 - 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. T3 - Preprints des Instituts für Mathematik der Universität Potsdam - 2(2013)18 KW - infinite-dimensional diffusion KW - cluster expansion KW - non-Markov drift KW - Girsanov formula KW - ultracontractivity Y1 - 2013 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-69014 ER - TY - GEN A1 - Roelly, Sylvie A1 - Dereudre, David T1 - Propagation of Gibbsiannes for infinite-dimensional gradient Brownian diffusions N2 - We study the (strong-)Gibbsian character on R Z d of the law at time t of an infinitedimensional gradient Brownian diffusion , when the initial distribution is Gibbsian. KW - infinite-dimensional Brownian diffusion KW - Gibbs measure KW - cluster expansion Y1 - 2004 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-6918 ER - TY - BOOK A1 - Zass, Alexander A1 - Zagrebnov, Valentin A1 - Sukiasyan, Hayk A1 - Melkonyan, Tatev A1 - Rafler, Mathias A1 - Poghosyan, Suren A1 - Zessin, Hans A1 - Piatnitski, Andrey A1 - Zhizhina, Elena A1 - Pechersky, Eugeny A1 - Pirogov, Sergei A1 - Yambartsev, Anatoly A1 - Mazzonetto, Sara A1 - Lykov, Alexander A1 - Malyshev, Vadim A1 - Khachatryan, Linda A1 - Nahapetian, Boris A1 - Jursenas, Rytis A1 - Jansen, Sabine A1 - Tsagkarogiannis, Dimitrios A1 - Kuna, Tobias A1 - Kolesnikov, Leonid A1 - Hryniv, Ostap A1 - Wallace, Clare A1 - Houdebert, Pierre A1 - Figari, Rodolfo A1 - Teta, Alessandro A1 - Boldrighini, Carlo A1 - Frigio, Sandro A1 - Maponi, Pierluigi A1 - Pellegrinotti, Alessandro A1 - Sinai, Yakov G. ED - Roelly, Sylvie ED - Rafler, Mathias ED - Poghosyan, Suren T1 - Proceedings of the XI international conference stochastic and analytic methods in mathematical physics N2 - The XI international conference Stochastic and Analytic Methods in Mathematical Physics was held in Yerevan 2 – 7 September 2019 and was dedicated to the memory of the great mathematician Robert Adol’fovich Minlos, who passed away in January 2018. The present volume collects a large majority of the contributions presented at the conference on the following domains of contemporary interest: classical and quantum statistical physics, mathematical methods in quantum mechanics, stochastic analysis, applications of point processes in statistical mechanics. The authors are specialists from Armenia, Czech Republic, Denmark, France, Germany, Italy, Japan, Lithuania, Russia, UK and Uzbekistan. A particular aim of this volume is to offer young scientists basic material in order to inspire their future research in the wide fields presented here. T3 - Lectures in pure and applied mathematics - 6 KW - statistical mechanics KW - random point processes KW - stochastic analysis Y1 - 2020 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-459192 SN - 978-3-86956-485-2 SN - 2199-4951 SN - 2199-496X IS - 6 PB - Universitätsverlag Potsdam CY - Potsdam ER - TY - INPR A1 - Kytmanov, Aleksandr A1 - Myslivets, Simona A1 - Tarkhanov, Nikolai Nikolaevich T1 - Power sums of roots of a nonlinear system N2 - For a system of meromorphic functions f = (f1, . . . , fn) in Cn, an explicit formula is given for evaluating negative power sums of the roots of the nonlinear system f(z) = 0. T3 - Preprint - (2004) 18 Y1 - 2004 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-26788 ER -