TY - INPR A1 - Gairing, Jan A1 - Högele, Michael A1 - Kosenkova, Tetiana A1 - Kulik, Alexei Michajlovič T1 - Coupling distances between Lévy measures and applications to noise sensitivity of SDE N2 - We introduce the notion of coupling distances on the space of Lévy measures in order to quantify rates of convergence towards a limiting Lévy jump diffusion in terms of its characteristic triplet, in particular in terms of the tail of the Lévy measure. The main result yields an estimate of the Wasserstein-Kantorovich-Rubinstein distance on path space between two Lévy 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. T3 - Preprints des Instituts für Mathematik der Universität Potsdam - 2(2013)16 KW - Lévy diffusion approximation KW - coupling methods KW - Skorokhod' s invariance principle KW - statistical model selection Y1 - 2013 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-68886 ER - TY - INPR A1 - Cattiaux, Patrick A1 - Fradon, Myriam A1 - Kulik, Alexei Michajlovič A1 - Roelly, Sylvie T1 - Long time behavior of stochastic hard ball systems N2 - 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. T3 - Preprints des Instituts für Mathematik der Universität Potsdam - 2(2013)15 KW - Stochastic differential equations KW - hard core interaction KW - reversible measure KW - normal reflection KW - local time Y1 - 2013 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-68388 ER - TY - CHAP A1 - Valleriani, Angelo A1 - Roelly, Sylvie A1 - Kulik, Alexei Michajlovič ED - Roelly, Sylvie ED - Högele, Michael ED - Rafler, Mathias T1 - Stochastic processes with applications in the natural sciences BT - international workshop at Universidad de los Andes, Bogotá, Colombia T2 - Lectures in pure and applied mathematics N2 - The interdisciplinary workshop STOCHASTIC PROCESSES WITH APPLICATIONS IN THE NATURAL SCIENCES was held in Bogotá, at Universidad de los Andes from December 5 to December 9, 2016. It brought together researchers from Colombia, Germany, France, Italy, Ukraine, who communicated recent progress in the mathematical research related to stochastic processes with application in biophysics. The present volume collects three of the four courses held at this meeting by Angelo Valleriani, Sylvie Rœlly and Alexei Kulik. A particular aim of this collection is to inspire young scientists in setting up research goals within the wide scope of fields represented in this volume. Angelo Valleriani, PhD in high energy physics, is group leader of the team "Stochastic processes in complex and biological systems" from the Max-Planck-Institute of Colloids and Interfaces, Potsdam. Sylvie Rœlly, Docteur en Mathématiques, is the head of the chair of Probability at the University of Potsdam. Alexei Kulik, Doctor of Sciences, is a Leading researcher at the Institute of Mathematics of Ukrainian National Academy of Sciences. T3 - Lectures in pure and applied mathematics - 4 KW - macromolecular decay KW - Markov processes KW - branching processes KW - long-time behaviour KW - makromolekularer Zerfall KW - Markovprozesse KW - Verzweigungsprozesse KW - Langzeitverhalten Y1 - 2017 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-401802 SN - 978-3-86956-414-2 SN - 2199-4951 SN - 2199-496X IS - 4 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 - BOOK A1 - Kulik, Alexei Michajlovič ED - Roelly, Sylvie T1 - Introduction to Ergodic rates for Markov chains and processes BT - with applications to limit theorems N2 - 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. T3 - Lectures in pure and applied mathematics - 2 KW - Markov processes KW - Markovprozesse KW - long-time behaviour KW - Langzeitverhalten KW - ergodic rates KW - Konvergenzrate Y1 - 2015 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-79360 SN - 978-3-86956-338-1 SN - 2199-4951 SN - 2199-496X PB - Universitätsverlag Potsdam CY - Potsdam ER -