TY - INPR A1 - Fedchenko, Dmitry A1 - Tarkhanov, Nikolai Nikolaevich T1 - An index formula for Toeplitz operators N2 - 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. T3 - Preprints des Instituts für Mathematik der Universität Potsdam - 3(2014)12 KW - Toeplitz operators KW - Fredholm property KW - index Y1 - 2014 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-72499 SN - 2193-6943 VL - 3 IS - 12 PB - Universitätsverlag Potsdam CY - Potsdam ER - TY - INPR A1 - Sultanov, Oskar A1 - Kalyakin, Leonid A1 - Tarkhanov, Nikolai Nikolaevich T1 - Elliptic perturbations of dynamical systems with a proper node N2 - 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. T3 - Preprints des Instituts für Mathematik der Universität Potsdam - 3 (2014) 4 KW - dynamical system KW - singular perturbation KW - asymptotic methods Y1 - 2014 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-70460 SN - 2193-6943 VL - 3 IS - 4 PB - Universitätsverlag Potsdam CY - Potsdam ER - TY - INPR A1 - Conforti, Giovanni A1 - Léonard, Christian A1 - Murr, Rüdiger A1 - Roelly, Sylvie T1 - Bridges of Markov counting processes : reciprocal classes and duality formulas N2 - 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. T3 - Preprints des Instituts für Mathematik der Universität Potsdam - 3 (2014) 9 KW - counting process KW - bridge KW - reciprocal class KW - duality formula Y1 - 2014 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-71855 SN - 2193-6943 VL - 3 IS - 9 PB - Universitätsverlag Potsdam CY - Potsdam ER - TY - INPR A1 - Flandoli, Franco A1 - Högele, Michael T1 - A solution selection problem with small stable perturbations N2 - 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. T3 - Preprints des Instituts für Mathematik der Universität Potsdam - 3 (2014) 8 KW - stochastic differential equations KW - singular drifts KW - zero-noise limit KW - Peano phenomena KW - non-uniqueness Y1 - 2014 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-71205 SN - 2193-6943 VL - 3 IS - 8 PB - Universitätsverlag Potsdam CY - Potsdam ER - TY - INPR A1 - Makhmudov, Olimdjan A1 - Tarkhanov, Nikolai Nikolaevich T1 - The first mixed problem for the nonstationary Lamé system N2 - We find an adequate interpretation of the Lamé operator within the framework of elliptic complexes and study the first mixed problem for the nonstationary Lamé system. T3 - Preprints des Instituts für Mathematik der Universität Potsdam - 3(2014)10 KW - Lamé system KW - evolution equation KW - first boundary value problem Y1 - 2014 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-71923 SN - 2193-6943 VL - 3 IS - 10 PB - Universitätsverlag Potsdam CY - Potsdam ER - TY - INPR A1 - Gairing, Jan A1 - Högele, Michael A1 - Kosenkova, Tetiana A1 - Kulik, Alexei Michajlovič T1 - On the calibration of Lévy driven time series with coupling distances : an application in paleoclimate N2 - 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 Lévy 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 Lévy 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 Lévy component for some tail index greater than 2. T3 - Preprints des Instituts für Mathematik der Universität Potsdam - 3 (2014) 2 KW - time series with heavy tails KW - index of stability KW - goodness-of-fit KW - empirical Wasserstein distance Y1 - 2014 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-69781 SN - 2193-6943 VL - 3 IS - 2 PB - Universitätsverlag Potsdam CY - Potsdam ER - TY - INPR A1 - Aizenberg, Lev A. A1 - Tarkhanov, Nikolai Nikolaevich T1 - An integral formula for the number of lattice points in a domain N2 - 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. T3 - Preprints des Instituts für Mathematik der Universität Potsdam - 3 (2014) 3 KW - logarithmic residue KW - lattice point Y1 - 2014 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-70453 SN - 2193-6943 VL - 3 IS - 3 PB - Universitätsverlag Potsdam CY - Potsdam ER - TY - INPR A1 - Högele, Michael A1 - Pavlyukevich, Ilya T1 - Metastability of Morse-Smale dynamical systems perturbed by heavy-tailed Lévy type noise N2 - 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 Lévy type noise of small intensity ε > 0. Specifically we consider perturbations leading to a Itô, 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. T3 - Preprints des Instituts für Mathematik der Universität Potsdam - 3 (2014) 5 KW - hyperbolic dynamical system KW - Morse-Smale property KW - stable limit cycle KW - small noise asymptotic KW - multiplicative noise Y1 - 2014 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-70639 SN - 2193-6943 VL - 3 IS - 5 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 - INPR A1 - Dyachenko, Evgueniya A1 - Tarkhanov, Nikolai Nikolaevich T1 - Singular perturbations of elliptic operators N2 - 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'. T3 - Preprints des Instituts für Mathematik der Universität Potsdam - 3 (2014) 1 KW - singular perturbation KW - pseudodifferential operator KW - ellipticity with parameter KW - regularization KW - asymptotics Y1 - 2014 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-69502 SN - 2193-6943 VL - 3 IS - 1 PB - Universitätsverlag Potsdam CY - Potsdam ER -