@unpublished{MeleardRoelly2013, author = {M{\´e}l{\´e}ard, Sylvie and Roelly, Sylvie}, title = {Evolutive two-level population process and large population approximations}, issn = {2193-6943}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-64604}, year = {2013}, abstract = {We are interested in modeling the Darwinian evolution of a population described by two levels of biological parameters: individuals characterized by an heritable phenotypic trait submitted to mutation and natural selection and cells in these individuals influencing their ability to consume resources and to reproduce. Our models are rooted in the microscopic description of a random (discrete) population of individuals characterized by one or several adaptive traits and cells characterized by their type. The population is modeled as a stochastic point process whose generator captures the probabilistic dynamics over continuous time of birth, mutation and death for individuals and birth and death for cells. The interaction between individuals (resp. between cells) is described by a competition between individual traits (resp. between cell types). We are looking for tractable large population approximations. By combining various scalings on population size, birth and death rates and mutation step, the single microscopic model is shown to lead to contrasting nonlinear macroscopic limits of different nature: deterministic approximations, in the form of ordinary, integro- or partial differential equations, or probabilistic ones, like stochastic partial differential equations or superprocesses.}, language = {en} } @unpublished{MeleardRoelly2011, author = {M{\´e}l{\´e}ard, Sylvie and Roelly, Sylvie}, title = {A host-parasite multilevel interacting process and continuous approximations}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-51694}, year = {2011}, abstract = {We are interested in modeling some two-level population dynamics, resulting from the interplay of ecological interactions and phenotypic variation of individuals (or hosts) and the evolution of cells (or parasites) of two types living in these individuals. The ecological parameters of the individual dynamics depend on the number of cells of each type contained by the individual and the cell dynamics depends on the trait of the invaded individual. Our models are rooted in the microscopic description of a random (discrete) population of individuals characterized by one or several adaptive traits and cells characterized by their type. The population is modeled as a stochastic point process whose generator captures the probabilistic dynamics over continuous time of birth, mutation and death for individuals and birth and death for cells. The interaction between individuals (resp. between cells) is described by a competition between individual traits (resp. between cell types). We look for tractable large population approximations. By combining various scalings on population size, birth and death rates and mutation step, the single microscopic model is shown to lead to contrasting nonlinear macroscopic limits of different nature: deterministic approximations, in the form of ordinary, integro- or partial differential equations, or probabilistic ones, like stochastic partial differential equations or superprocesses. The study of the long time behavior of these processes seems very hard and we only develop some simple cases enlightening the difficulties involved.}, language = {en} } @article{ConfortiKosenkovaRoelly2019, author = {Conforti, Giovanni and Kosenkova, Tetiana and Roelly, Sylvie}, title = {Conditioned Point Processes with Application to Levy Bridges}, series = {Journal of theoretical probability}, volume = {32}, journal = {Journal of theoretical probability}, number = {4}, publisher = {Springer}, address = {New York}, issn = {0894-9840}, doi = {10.1007/s10959-018-0863-8}, pages = {2111 -- 2134}, year = {2019}, abstract = {Our first result concerns a characterization by means of a functional equation of Poisson point processes conditioned by the value of their first moment. It leads to a generalized version of Mecke's formula. En passant, it also allows us to gain quantitative results about stochastic domination for Poisson point processes under linear constraints. Since bridges of a pure jump L{\´e}vy process in Rd with a height a can be interpreted as a Poisson point process on space-time conditioned by pinning its first moment to a, our approach allows us to characterize bridges of L{\´e}vy processes by means of a functional equation. The latter result has two direct applications: First, we obtain a constructive and simple way to sample L{\´e}vy bridge dynamics; second, it allows us to estimate the number of jumps for such bridges. We finally show that our method remains valid for linearly perturbed L{\´e}vy processes like periodic Ornstein-Uhlenbeck processes driven by L{\´e}vy noise.}, language = {en} } @unpublished{Roelly2013, author = {Roelly, Sylvie}, title = {Reciprocal processes : a stochastic analysis approach}, issn = {2193-6943}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-64588}, year = {2013}, abstract = {Reciprocal processes, whose concept can be traced back to E. Schr{\"o}dinger, form a class of stochastic processes constructed as mixture of bridges, that satisfy a time Markov field property. We discuss here a new unifying approach to characterize several types of reciprocal processes via duality formulae on path spaces: The case of reciprocal processes with continuous paths associated to Brownian diffusions and the case of pure jump reciprocal processes associated to counting processes are treated. This presentation is based on joint works with M. Thieullen, R. Murr and C. L{\´e}onard.}, language = {en} } @unpublished{RoellyVallois2016, author = {Roelly, Sylvie and Vallois, Pierre}, title = {Convoluted Brownian motion}, volume = {5}, number = {9}, publisher = {Universit{\"a}tsverlag Potsdam}, address = {Potsdam}, issn = {2193-6943}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-96339}, pages = {37}, year = {2016}, abstract = {In this paper we analyse semimartingale properties of a class of Gaussian periodic processes, called convoluted Brownian motions, obtained by convolution between a deterministic function and a Brownian motion. A classical example in this class is the periodic Ornstein-Uhlenbeck process. We compute their characteristics and show that in general, they are neither Markovian nor satisfy a time-Markov field property. Nevertheless, by enlargement of filtration and/or addition of a one-dimensional component, one can in some case recover the Markovianity. We treat exhaustively the case of the bidimensional trigonometric convoluted Brownian motion and the higher-dimensional monomial convoluted Brownian motion.}, language = {en} } @book{ChampagnatRoelly2007, author = {Champagnat, Nicolas and Roelly, Sylvie}, title = {Multitype Dawson-Watanabe superprocesses conditioned by remote survival}, series = {Preprint / Universit{\"a}t Potsdam, Institut f{\"u}r Mathematik, Mathematische Statistik un}, journal = {Preprint / Universit{\"a}t Potsdam, Institut f{\"u}r Mathematik, Mathematische Statistik un}, publisher = {Univ.}, address = {Potsdam}, issn = {1613-3307}, pages = {39 S.}, year = {2007}, language = {en} } @unpublished{RoellyFradon2006, author = {Roelly, Sylvie and Fradon, Myriam}, title = {Infinite system of Brownian balls : equilibrium measures are canonical Gibbs}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-6720}, year = {2006}, abstract = {We consider a system of infinitely many hard balls in Rd undergoing Brownian motions and submitted to a smooth pair potential. It is modelized by an infinite-dimensional stochastic differential equation with a local time term. We prove that the set of all equilibrium measures, solution of a detailed balance equation, coincides with the set of canonical Gibbs measures associated to the hard core potential added to the smooth interaction potential.}, language = {en} } @unpublished{CattiauxFradonKuliketal.2013, author = {Cattiaux, Patrick and Fradon, Myriam and Kulik, Alexei Michajlovič and Roelly, Sylvie}, title = {Long time behavior of stochastic hard ball systems}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-68388}, year = {2013}, abstract = {We study the long time behavior of a system of two or three Brownian hard balls living in the Euclidean space of dimension at least two, submitted to a mutual attraction and to elastic collisions.}, language = {en} } @unpublished{ConfortiRoelly2015, author = {Conforti, Giovanni and Roelly, Sylvie}, title = {Reciprocal class of random walks on an Abelian group}, volume = {4}, number = {1}, publisher = {Universit{\"a}tsverlag Potsdam}, address = {Potsdam}, issn = {2193-6943}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-72604}, pages = {22}, year = {2015}, abstract = {Processes having the same bridges as a given reference Markov process constitute its reciprocal class. In this paper we study the reciprocal class of a continuous time random walk with values in a countable Abelian group, we compute explicitly its reciprocal characteristics and we present an integral characterization of it. Our main tool is a new iterated version of the celebrated Mecke's formula from the point process theory, which allows us to study, as transformation on the path space, the addition of random loops. Thanks to the lattice structure of the set of loops, we even obtain a sharp characterization. At the end, we discuss several examples to illustrate the richness of reciprocal classes. We observe how their structure depends on the algebraic properties of the underlying group.}, language = {en} } @misc{RoellySortais2004, author = {Roelly, Sylvie and Sortais, Michel}, title = {Space-time asymptotics of an infinite-dimensional diffusion having a long- range memory}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-6700}, year = {2004}, abstract = {We develop a cluster expansion in space-time for an infinite-dimensional system of interacting diffusions where the drift term of each diffusion depends on the whole past of the trajectory; these interacting diffusions arise when considering the Langevin dynamics of a ferromagnetic system submitted to a disordered external magnetic field.}, language = {en} } @misc{RoellyThieullen2005, author = {Roelly, Sylvie and Thieullen, Mich{\`e}le}, title = {Duality formula for the bridges of a Brownian diffusion : application to gradient drifts}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-6710}, year = {2005}, abstract = {In this paper, we consider families of time Markov fields (or reciprocal classes) which have the same bridges as a Brownian diffusion. We characterize each class as the set of solutions of an integration by parts formula on the space of continuous paths C[0; 1]; R-d) Our techniques provide a characterization of gradient diffusions by a duality formula and, in case of reversibility, a generalization of a result of Kolmogorov.}, language = {en} } @misc{RoellyDaiPra2004, author = {Roelly, Sylvie and Dai Pra, Paolo}, title = {An existence result for infinite-dimensional Brownian diffusions with non- regular and non Markovian drift}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-6684}, year = {2004}, abstract = {We prove in this paper an existence result for infinite-dimensional stationary interactive Brownian diffusions. The interaction is supposed to be small in the norm ||.||∞ but otherwise is very general, being possibly non-regular and non-Markovian. Our method consists in using the characterization of such diffusions as space-time Gibbs fields so that we construct them by space-time cluster expansions in the small coupling parameter.}, language = {en} } @article{DombrowskyUndRoelly2019, author = {Dombrowsky, Charlotte and Und, Myriam Fradon and Roelly, Sylvie}, title = {Packungen aus Kreisscheiben}, series = {Elemente der Mathematik}, volume = {74}, journal = {Elemente der Mathematik}, number = {2}, publisher = {EMS Publ.}, address = {Z{\"u}rich}, issn = {0013-6018}, doi = {10.4171/EM/381}, pages = {45 -- 62}, year = {2019}, abstract = {Der englische Seefahrer Sir Walter Raleigh fragte sich einst, wie er in seinem Schiffsladeraum moeglichst viele Kanonenkugeln stapeln koennte. Johannes Kepler entwickelte daraufhin 1611 eine Vermutung ueber die optimale Anordnung der Kugeln. Diese Vermutung sollte sich als eine der haertesten mathematischen Nuesse der Geschichte erweisen. Selbst in der Ebene sind dichteste Packungen kongruenter Kreise eine Herausforderung. 1892 und 1910 veroeffentlichte Axel Thue (kritisierte) Beweise, dass die hexagonale Kreispackung optimal sei. Erst 1940 lieferte Laszlo Fejes Toth schliesslich einen wasserdichten Beweis fuer diese Tatsache. Eine Variante des Problems verlangt, Packungen mit endlich vielen kongruenten Kugeln zu finden, die eine gewisse quadratische Energie minimieren: Diese spannende geometrische Aufgabe wurde 1967 von Toth gestellt. Sie ist auch heute noch nicht vollstaendig gelaest. In diesem Beitrag schlagen die Autorinnen eine originelle wahrscheinlichkeitstheoretische Methode vor, um in der Ebene N{\"a}herungen der L{\"o}sung zu konstruieren.}, language = {de} } @article{ConfortiPraRoelly2015, author = {Conforti, Giovanni and Pra, Paolo Dai and Roelly, Sylvie}, title = {Reciprocal Class of Jump Processes}, series = {Journal of theoretical probability}, volume = {30}, journal = {Journal of theoretical probability}, publisher = {Springer}, address = {New York}, issn = {0894-9840}, doi = {10.1007/s10959-015-0655-3}, pages = {551 -- 580}, year = {2015}, abstract = {Processes having the same bridges as a given reference Markov process constitute its reciprocal class. In this paper we study the reciprocal class of compound Poisson processes whose jumps belong to a finite set . 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 plays a crucial role in the design of the transformations, and we use tools from discrete geometry to obtain an optimal characterization. We deduce explicit conditions for two Markov jump processes to belong to the same class. Finally, we provide a natural interpretation of the invariants as short-time asymptotics for the probability that the reference process makes a cycle around its current state.}, language = {en} } @article{RœllyZass2020, author = {Rœlly, Sylvie and Zass, Alexander}, title = {Marked Gibbs point processes with unbounded interaction}, series = {Journal of statistical physics}, volume = {179}, journal = {Journal of statistical physics}, number = {4}, publisher = {Springer}, address = {New York}, issn = {0022-4715}, doi = {10.1007/s10955-020-02559-3}, pages = {972 -- 996}, year = {2020}, abstract = {We construct marked Gibbs point processes in R-d under quite general assumptions. Firstly, we allow for interaction functionals that may be unbounded and whose range is not assumed to be uniformly bounded. Indeed, our typical interaction admits an a.s. finite but random range. Secondly, the random marks-attached to the locations in R-d-belong to a general normed space G. They are not bounded, but their law should admit a super-exponential moment. The approach used here relies on the so-called entropy method and large-deviation tools in order to prove tightness of a family of finite-volume Gibbs point processes. An application to infinite-dimensional interacting diffusions is also presented.}, language = {en} } @article{DereudreMazzonettoRoelly2017, author = {Dereudre, David and Mazzonetto, Sara and Roelly, Sylvie}, title = {Exact simulation of Brownian diffusions with drift admitting jumps}, series = {SIAM journal on scientific computing}, volume = {39}, journal = {SIAM journal on scientific computing}, number = {3}, publisher = {Society for Industrial and Applied Mathematics}, address = {Philadelphia}, issn = {1064-8275}, doi = {10.1137/16M107699X}, pages = {A711 -- A740}, year = {2017}, abstract = {In this paper, using an algorithm based on the retrospective rejection sampling scheme introduced in [A. Beskos, O. Papaspiliopoulos, and G. O. Roberts,Methodol. Comput. Appl. Probab., 10 (2008), pp. 85-104] and [P. Etore and M. Martinez, ESAIM Probab.Stat., 18 (2014), pp. 686-702], we propose an exact simulation of a Brownian di ff usion whose drift admits several jumps. We treat explicitly and extensively the case of two jumps, providing numerical simulations. Our main contribution is to manage the technical di ffi culty due to the presence of t w o jumps thanks to a new explicit expression of the transition density of the skew Brownian motion with two semipermeable barriers and a constant drift.}, language = {en} } @article{FradonRoelly2010, author = {Fradon, Myriam and Roelly, Sylvie}, title = {Infinitely many Brownian globules with Brownian radii}, issn = {0219-4937}, doi = {10.1142/S021949371000311x}, year = {2010}, abstract = {We consider an infinite system of non-overlapping globules undergoing Brownian motions in R-3. The term globules means that the objects we are dealing with are spherical, but with a radius which is random and time-dependent. The dynamics is modelized by an infinite-dimensional stochastic differential equation with local time. Existence and uniqueness of a strong solution is proven for such an equation with fixed deterministic initial condition. We also find a class of reversible measures.}, language = {en} } @book{KellerRoellyValleriana2011, author = {Keller, Peter and Roelly, Sylvie and Valleriana, Angelo}, title = {On Time Duality for Markov Chains with Asborbing Boundardies}, series = {Preprint / Universit{\"a}t Potsdam, Institut f{\"u}r Mathematik, Mathematische Statistik un}, journal = {Preprint / Universit{\"a}t Potsdam, Institut f{\"u}r Mathematik, Mathematische Statistik un}, publisher = {Univ.}, address = {Potsdam}, issn = {1613-3307}, pages = {18 S.}, year = {2011}, language = {en} } @article{RoellyRuszel2014, author = {Roelly, Sylvie and Ruszel, W. M.}, title = {Propagation of gibbsianness for infinite-dimensional diffusions with space-time interaction}, series = {Markov processes and related fields}, volume = {20}, journal = {Markov processes and related fields}, number = {4}, publisher = {Polymat}, address = {Moscow}, issn = {1024-2953}, pages = {653 -- 674}, year = {2014}, abstract = {We consider infinite-dimensional diffusions where the interaction between the coordinates has a finite extent both in space and time. In particular, it is not supposed to be smooth or Markov. The initial state of the system is Gibbs, given by a strong summable interaction. If the strongness of this initial interaction is lower than a suitable level, and if the dynamical interaction is bounded from above in a right way, we prove that the law of the diffusion at any time t is a Gibbs measure with absolutely summable interaction. The main tool is a cluster expansion in space uniformly in time of the Girsanov factor coming from the dynamics and exponential ergodicity of the free dynamics to an equilibrium product measure.}, language = {en} } @article{CattiauxFradonKuliketal.2016, author = {Cattiaux, Patrick and Fradon, Myriam and Kulik, Alexei M. and Roelly, Sylvie}, title = {Long time behavior of stochastic hard ball systems}, series = {Bernoulli : official journal of the Bernoulli Society for Mathematical Statistics and Probability}, volume = {22}, journal = {Bernoulli : official journal of the Bernoulli Society for Mathematical Statistics and Probability}, publisher = {International Statistical Institute}, address = {Voorburg}, issn = {1350-7265}, doi = {10.3150/14-BEJ672}, pages = {681 -- 710}, year = {2016}, abstract = {We study the long time behavior of a system of n = 2, 3 Brownian hard balls, living in R-d for d >= 2, submitted to a mutual attraction and to elastic collisions.}, language = {en} } @article{ConfortiLeonardMurretal.2015, author = {Conforti, Giovanni and Leonard, Christian and Murr, R{\"u}diger and Roelly, Sylvie}, title = {Bridges of Markov counting processes. Reciprocal classes and duality formulas}, series = {Electronic communications in probability}, volume = {20}, journal = {Electronic communications in probability}, publisher = {Univ. of Washington, Mathematics Dep.}, address = {Seattle}, issn = {1083-589X}, doi = {10.1214/ECP.v20-3697}, pages = {12}, year = {2015}, abstract = {Processes sharing the same bridges are said to belong to the same reciprocal class. In this article we analyze reciprocal classes of Markov counting processes by identifying their reciprocal invariants and we characterize them as the set of counting processes satisfying some duality formula.}, language = {en} } @article{KellerRoellyValleriani2015, author = {Keller, Peter and Roelly, Sylvie and Valleriani, Angelo}, title = {On time duality for Markov Chains}, series = {Stochastic models}, volume = {31}, journal = {Stochastic models}, number = {1}, publisher = {Taylor \& Francis Group}, address = {Philadelphia}, issn = {1532-6349}, doi = {10.1080/15326349.2014.969736}, pages = {98 -- 118}, year = {2015}, abstract = {For an irreducible continuous time Markov chain, we derive the distribution of the first passage time from a given state i to another given state j and the reversed passage time from j to i, each under the condition of no return to the starting point. When these two distributions are identical, we say that i and j are in time duality. We introduce a new condition called permuted balance that generalizes the concept of reversibility and provides sufficient criteria, based on the structure of the transition graph of the Markov chain. Illustrative examples are provided.}, language = {en} } @article{KellerRoellyValleriani2015, author = {Keller, Peter and Roelly, Sylvie and Valleriani, Angelo}, title = {A Quasi Random Walk to Model a Biological Transport Process}, series = {Methodology and computing in applied probability}, volume = {17}, journal = {Methodology and computing in applied probability}, number = {1}, publisher = {Springer}, address = {Dordrecht}, issn = {1387-5841}, doi = {10.1007/s11009-013-9372-5}, pages = {125 -- 137}, year = {2015}, abstract = {Transport molecules play a crucial role for cell viability. Amongst others, linear motors transport cargos along rope-like structures from one location of the cell to another in a stochastic fashion. Thereby each step of the motor, either forwards or backwards, bridges a fixed distance and requires several biochemical transformations, which are modeled as internal states of the motor. While moving along the rope, the motor can also detach and the walk is interrupted. We give here a mathematical formalization of such dynamics as a random process which is an extension of Random Walks, to which we add an absorbing state to model the detachment of the motor from the rope. We derive particular properties of such processes that have not been available before. Our results include description of the maximal distance reached from the starting point and the position from which detachment takes place. Finally, we apply our theoretical results to a concrete established model of the transport molecule Kinesin V.}, language = {en} } @inproceedings{VallerianiRoellyKulik2017, author = {Valleriani, Angelo and Roelly, Sylvie and Kulik, Alexei Michajlovič}, title = {Stochastic processes with applications in the natural sciences}, series = {Lectures in pure and applied mathematics}, booktitle = {Lectures in pure and applied mathematics}, number = {4}, editor = {Roelly, Sylvie and H{\"o}gele, Michael and Rafler, Mathias}, publisher = {Universit{\"a}tsverlag Potsdam}, address = {Potsdam}, isbn = {978-3-86956-414-2}, issn = {2199-4951}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-401802}, pages = {ix, 124}, year = {2017}, abstract = {The interdisciplinary workshop STOCHASTIC PROCESSES WITH APPLICATIONS IN THE NATURAL SCIENCES was held in Bogot{\´a}, 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 Math{\´e}matiques, 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.}, language = {en} } @article{ConfortiRoelly2017, author = {Conforti, Giovanni and Roelly, Sylvie}, title = {Bridge mixtures of random walks on an Abelian group}, series = {Bernoulli : official journal of the Bernoulli Society for Mathematical Statistics and Probability}, volume = {23}, journal = {Bernoulli : official journal of the Bernoulli Society for Mathematical Statistics and Probability}, publisher = {International Statistical Institute}, address = {Voorburg}, issn = {1350-7265}, doi = {10.3150/15-BEJ783}, pages = {1518 -- 1537}, year = {2017}, language = {en} } @misc{ChampagnatRoelly2008, author = {Champagnat, Nicolas and Roelly, Sylvie}, title = {Limit theorems for conditioned multitype Dawson-Watanabe processes and Feller diffusions}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-18610}, year = {2008}, abstract = {A multitype Dawson-Watanabe process is conditioned, in subcritical and critical cases, on non-extinction in the remote future. On every finite time interval, its distribution is absolutely continuous with respect to the law of the unconditioned process. A martingale problem characterization is also given. Several results on the long time behavior of the conditioned mass process - the conditioned multitype Feller branching diffusion - are then proved. The general case is first considered, where the mutation matrix which models the interaction between the types, is irreducible. Several two-type models with decomposable mutation matrices are analyzed too .}, language = {en} } @unpublished{ConfortiLeonardMurretal.2014, author = {Conforti, Giovanni and L{\´e}onard, Christian and Murr, R{\"u}diger and Roelly, Sylvie}, title = {Bridges of Markov counting processes : reciprocal classes and duality formulas}, volume = {3}, number = {9}, publisher = {Universit{\"a}tsverlag Potsdam}, address = {Potsdam}, issn = {2193-6943}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-71855}, pages = {12}, year = {2014}, abstract = {Processes having the same bridges are said to belong to the same reciprocal class. In this article we analyze reciprocal classes of Markov counting processes by identifying their reciprocal invariants and we characterize them as the set of counting processes satisfying some duality formula.}, language = {en} } @unpublished{FradonRoelly2005, author = {Fradon, Myriam and Roelly, Sylvie}, title = {Infinite system of Brownian balls with interaction : the non-reversible case}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-51546}, year = {2005}, abstract = {We consider an infinite system of hard balls in Rd undergoing Brownian motions and submitted to a smooth pair potential. It is modelized by an infinite- dimensional Stochastic Differential Equation with an infinite-dimensional local time term. Existence and uniqueness of a strong solution is proven for such an equation with fixed deterministic initial condition. We also show that Gibbs measures are reversible measures.}, language = {en} } @misc{ImkellerRoelly2007, author = {Imkeller, Peter and Roelly, Sylvie}, title = {Die Wiederentdeckung eines Mathematikers: Wolfgang D{\"o}blin}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-16397}, year = {2007}, abstract = {"Considerons une particule mobile se mouvant aleatoirement sur la droite (ou sur un segment de droite). Supposons qu'il existe une probabilite F(x,y;s,t) bien definie pour que la particule se trouvant a l'instant s dans la position x se trouve a l'instant t (> s) a gauche de y, probabilite independante du mouvement anterieur de la particule...." Mit diesen Worten beginnt eines der ber{\"u}hmtesten mathematischen Manuskripte des letzten Jahrhunderts. Es stammt vom Soldaten Wolfgang D{\"o}blin, Sohn des deutschen Schriftstellers Alfred D{\"o}blin, und tr{\"a}gt den Titel "Sur l'equation de Kolmogoroff". Seine Ver{\"o}ffentlichung verbindet sich mit einer unglaublichen Geschichte. Wolfgang D{\"o}blin, stationiert mit seiner Einheit in den Ardennen im Winter 1939/1940, arbeitete an diesem Manuskript. Er entschloss sich, es als versiegeltes Manuskript an die Academie des Sciences in Paris zu schicken. Aber er kehrte nie aus diesem Krieg zur{\"u}ck. Sein Manuskript blieb 60 Jahre unter Verschluss im Archiv, und wurde erst im Jahre 2000 ge{\"o}ffnet. Wie weit D{\"o}blin damit seiner Zeit voraus war, wurde erkannt, nachdem es von Bernard Bru und Marc Yor ausgewertet worden war. Im ersten Satz umschreibt W. D{\"o}blin gleichzeitig das Programm des Manuskripts: "Wir betrachten ein bewegliches Teilchen, das sich zuf{\"a}llig auf der Geraden (oder einem Teil davon) bewegt." Er widmet sich damit der Aufgabe, die Fundamente eines Gebiets zu legen, das wir heute als stochastische Analysis bezeichnen.}, language = {de} } @unpublished{ChampagnatRoelly2007, author = {Champagnat, Nicolas and Roelly, Sylvie}, title = {Limit theorems for conditioned multitype Dawson-Watanabe processes}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-49426}, year = {2007}, abstract = {A multitype Dawson-Watanabe process is conditioned, in subcritical and critical cases, on non-extinction in the remote future. On every nite time interval, its distribution law is absolutely continuous with respect to the law of the unconditioned process. A martingale problem characterization is also given. The explicit form of the Laplace functional of the conditioned process is used to obtain several results on the long time behaviour of the mass of the conditioned and unconditioned processes. The general case is considered first, where the mutation matrix which modelizes the interaction between the types, is irreducible. Several two-type models with decomposable mutation matrices are also analysed.}, language = {en} } @unpublished{FradonRoelly2005, author = {Fradon, Myriam and Roelly, Sylvie}, title = {Infinite system of Brownian Balls: Equilibrium measures are canonical Gibbs}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-51594}, year = {2005}, abstract = {We consider a system of infinitely many hard balls in Rd undergoing Brownian motions and submitted to a smooth pair potential. It is modelized by an infinite-dimensional Stochastic Differential Equation with a local time term. We prove that the set of all equilibrium measures, solution of a Detailed Balance Equation, coincides with the set of canonical Gibbs measures associated to the hard core potential added to the smooth interaction potential.}, language = {en} } @unpublished{Roelly2010, author = {Roelly, Sylvie}, title = {Unas propiedades basicas de procesos de ramificaci{\´o}n : Lectures held at ICIMAF La Habana, Cuba, 2009 and 2010}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-49620}, year = {2010}, abstract = {Aus dem Inhalt: 1. Unas propiedades de los procesos de Bienaym{\´e}-Galton-Watson de tiempo dis- creto (BGW) 2. Unas propiedades del proceso BGW de tiempo continuo 3. Limites de procesos de BGW cuando la poblaci{\´o}n es numerosa}, language = {mul} } @unpublished{FradonRoelly2009, author = {Fradon, Myriam and Roelly, Sylvie}, title = {Infinitely many Brownian globules with Brownian radii}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-49552}, year = {2009}, abstract = {We consider an infinite system of non overlaping globules undergoing Brownian motions in R3. The term globules means that the objects we are dealing with are spherical, but with a radius which is random and time-dependent. The dynamics is modelized by an infinitedimensional Stochastic Differential Equation with local time. Existence and uniqueness of a strong solution is proven for such an equation with fixed deterministic initial condition. We also find a class of reversible measures.}, language = {en} } @unpublished{LeonardRoellyZambrini2013, author = {L{\´e}onard, Christian and Roelly, Sylvie and Zambrini, Jean-Claude}, title = {Temporal symmetry of some classes of stochastic processes}, issn = {2193-6943}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-64599}, year = {2013}, abstract = {In this article we analyse the structure of Markov processes and reciprocal processes to underline their time symmetrical properties, and to compare them. Our originality consists in adopting a unifying approach of reciprocal processes, independently of special frameworks in which the theory was developped till now (diffusions, or pure jump processes). This leads to some new results, too.}, language = {en} } @unpublished{KellerRoellyValleriani2013, author = {Keller, Peter and Roelly, Sylvie and Valleriani, Angelo}, title = {A quasi-random-walk to model a biological transport process}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-63582}, year = {2013}, abstract = {Transport Molecules play a crucial role for cell viability. Amongst others, linear motors transport cargos along rope-like structures from one location of the cell to another in a stochastic fashion. Thereby each step of the motor, either forwards or backwards, bridges a fixed distance. While moving along the rope the motor can also detach and is lost. We give here a mathematical formalization of such dynamics as a random process which is an extension of Random Walks, to which we add an absorbing state to model the detachment of the motor from the rope. We derive particular properties of such processes that have not been available before. Our results include description of the maximal distance reached from the starting point and the position from which detachment takes place. Finally, we apply our theoretical results to a concrete established model of the transport molecule Kinesin V.}, language = {en} } @unpublished{FradonRoelly2005, author = {Fradon, Myriam and Roelly, Sylvie}, title = {Brownian Hard Balls submitted to an infinite rangeinteraction with slow decay}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-49379}, year = {2005}, abstract = {We consider an infinite system of hard balls in Rd undergoing Brownian motions and submitted to a pair potential with infinite range and quasi polynomial decay. It is modelized by an infinite-dimensional Stochastic Differential Equation with an infinite-dimensional local time term. Existence and uniqueness of a strong solution is proven for such an equation with deterministic initial condition. We also show that the set of all equilibrium measures, solution of a Detailed Balance Equation, coincides with the set of canonical Gibbs measures associated to the hard core potential.}, language = {en} } @misc{ZimmermannSophiaDoetmannetal.2014, author = {Zimmermann, Matthias and Sophia, Rost and D{\"o}tmann, Eik and Kampe, Heike and G{\"o}rlich, Petra and S{\"u}tterlin, Sabine and Eckardt, Barbara and Horn-Conrad, Antje and Schwaibold, Julia and J{\"a}ger, Sophie and Sophia, Rost and Mangelsdorf, Birgit and Roelly, Sylvie}, title = {Portal Wissen = Glauben}, number = {01/2014}, organization = {Universit{\"a}t Potsdam, Referat f{\"u}r Presse- und {\"O}ffentlichkeitsarbeit}, issn = {2194-4237}, doi = {10.25932/publishup-44083}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-440830}, pages = {99}, year = {2014}, abstract = {Menschen wollen wissen, was wirklich ist. Kinder lassen sich gern eine Geschichte erz{\"a}hlen, aber sp{\"a}testens mit vier Jahren fragten meine, ob diese Geschichte so passiert sei oder nur erfunden. Das setzt sich fort: Auch unsere wissenschaftliche Neugier wird vom Interesse befeuert herauszufinden, was wirklich ist. Selbst dort, wo wir poetische Texte oder Tr{\"a}ume erforschen, tun wir es in der Absicht, die realen sprachlichen Strukturen bzw. die neurologischen Faktoren von bloß vermuteten zu unterscheiden. Im Idealfall k{\"o}nnen wir Ergebnisse pr{\"a}sentieren, die von anderen logisch nachvollzogen und empirisch wiederholbar sind. Meistens geht das aber nicht. Wir k{\"o}nnen nicht jedes Buch lesen und nicht in jedes Mikroskop schauen, nicht einmal innerhalb der eigenen Disziplin. Wie viel mehr sind wir in der Lebenswelt darauf angewiesen, den Ausf{\"u}hrungen anderer zu vertrauen, wenn wir wissen wollen, wo es zum Bahnhof geht oder ob es in Ulan Bator sch{\"o}n ist. Deshalb haben wir uns daran gew{\"o}hnt, anderen Glauben zu schenken, vom Freund bis zum Tagesschausprecher. Das ist kein kindliches Verhalten, sondern eine Notwendigkeit. Freilich ist das riskant, denn alle anderen k{\"o}nnten uns - wie in der „Truman- Show" - anl{\"u}gen. In der Wirklichkeit wissen wir uns erst dann, wenn wir unser Selbstbewusstsein verlassen und akzeptieren, dass wir erstens nicht nur Objekte, sondern Subjekte im Bewusstsein von anderen sind, und zweitens, dass alle unsere dialogischen Beziehungen noch einmal von einem Dritten betrachtet werden, der nicht Teil dieser Welt ist. F{\"u}r Religi{\"o}se ist das der Glaube. Glaube als Unterstellung, dass alle menschlichen Beziehungen erst dann wirklich, ernst und {\"u}ber Zweifel erhaben sind, wenn sie sich vor den Augen Gottes wissen. Erst vor ihm ist etwas als es selbst und nicht nur „f{\"u}r mich" oder „unter uns". Daher unterscheidet die biblische Sprache drei Formen des Glaubens: die Beziehung zur Ding-Welt („glauben, dass"), die Beziehung zur Subjekt-Welt („jemandem glauben") und die Annahme einer subjekthaften {\"u}berirdischen Wirklichkeit („glauben an"). Wissenschaftstheoretisch gesehen ist Glaube also eine Totalhypothese. Glaube ist nicht das Gegenteil von Wissen, sondern der Versuch, Wirklichkeit vor dem Zweifel zu retten, indem man die fragile empirische Welt als Ausdruck einer stabilen transzendenten Welt begreift. Oft wollen Studierende in Gespr{\"a}chen nicht nur wissen, was ich weiß, sondern, was ich glaube. Als Religionswissenschaftler und gleichzeitig gl{\"a}ubiger Katholik sitze ich zwischen den St{\"u}hlen: Einerseits ist es als Professor meine Aufgabe, alles zu bezweifeln, d.h. jeden religi{\"o}sen Text auf seine historischen Kontexte und soziologischen Funktionen zur{\"u}ckzuf{\"u}hren. Andererseits h{\"a}lt der Christ in mir bestimmte religi{\"o}se Dokumente - in meinem Fall die Bibel - zwar f{\"u}r einen interpretierbaren, aber doch irreversiblen, offenbarten Text, der vom Ursprung der Wirklichkeit handelt. Werktags ist das Neue Testament eine antike Schriftensammlung neben vielen anderen, am Sonntag ist es die Offenbarung. Beides kann klar unterschieden werden, aber es ist schwer zu entscheiden, ob das Zweifeln oder das Glauben wirklicher ist. Das vorliegende Heft geht diesem doppelten Verh{\"a}ltnis zum Glauben nach: Wie steht Wissenschaft zum Glauben - ob religi{\"o}s oder nicht? Wo bringt Wissenschaft Dinge ans Licht, die wir kaum glauben m{\"o}gen oder uns (wieder) glauben lassen? Was passiert, wenn Forschung irrige Annahmen oder Mythen aufkl{\"a}rt? Ist Wissenschaft in der Lage, Dingen auf den Grund zu gehen, die zwar {\"u}berzeugend, aber unerkl{\"a}rbar sind? Wie kann sie selbst glaubw{\"u}rdig bleiben und sich dennoch weiterentwickeln? In den Beitr{\"a}gen dieser „Portal Wissen" scheinen diese Fragen immer wieder auf. Sie bilden ein vielf{\"a}ltiges, spannendes und auch {\"u}berraschendes Bild der Forschungsprojekte und der Wissenschaftler an der Universit{\"a}t Potsdam. Glauben Sie mir, es erwartet Sie eine anregende Lekt{\"u}re! Prof. Dr. Johann Hafner Professor f{\"u}r Religionswissenschaft mit dem Schwerpunkt Christentum Dekan der Philosophischen Fakult{\"a}t}, language = {de} } @unpublished{ConfortiDaiPraRoelly2014, author = {Conforti, Giovanni and Dai Pra, Paolo and Roelly, Sylvie}, title = {Reciprocal class of jump processes}, volume = {3}, number = {6}, publisher = {Universit{\"a}tsverlag Potsdam}, address = {Potsdam}, issn = {2193-6943}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-70776}, pages = {30}, year = {2014}, abstract = {Processes having the same bridges as a given reference Markov process constitute its reciprocal class. In this paper we study the reciprocal class of compound Poisson processes whose jumps belong to a finite set A in R^d. We propose a characterization of the reciprocal class as the unique set of probability measures on which a family of time and space transformations induces the same density, expressed in terms of the reciprocal invariants. The geometry of A plays a crucial role in the design of the transformations, and we use tools from discrete geometry to obtain an optimal characterization. We deduce explicit conditions for two Markov jump processes to belong to the same class. Finally, we provide a natural interpretation of the invariants as short-time asymptotics for the probability that the reference process makes a cycle around its current state.}, language = {en} } @unpublished{RoellyRuszel2013, author = {Roelly, Sylvie and Ruszel, Wioletta M.}, title = {Propagation of Gibbsianness for infinite-dimensional diffusions with space-time interaction}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-69014}, year = {2013}, abstract = {We consider infinite-dimensional diffusions where the interaction between the coordinates has a finite extent both in space and time. In particular, it is not supposed to be smooth or Markov. The initial state of the system is Gibbs, given by a strong summable interaction. If the strongness of this initial interaction is lower than a suitable level, and if the dynamical interaction is bounded from above in a right way, we prove that the law of the diffusion at any time t is a Gibbs measure with absolutely summable interaction. The main tool is a cluster expansion in space uniformly in time of the Girsanov factor coming from the dynamics and exponential ergodicity of the free dynamics to an equilibrium product measure.}, language = {en} } @unpublished{KellerRoellyValleriani2012, author = {Keller, Peter and Roelly, Sylvie and Valleriani, Angelo}, title = {On time duality for quasi-birth-and-death processes}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-56973}, year = {2012}, abstract = {We say that (weak/strong) time duality holds for continuous time quasi-birth-and-death-processes if, starting from a fixed level, the first hitting time of the next upper level and the first hitting time of the next lower level have the same distribution. We present here a criterion for time duality in the case where transitions from one level to another have to pass through a given single state, the so-called bottleneck property. We also prove that a weaker form of reversibility called balanced under permutation is sufficient for the time duality to hold. We then discuss the general case.}, language = {en} } @unpublished{RedigRoellyRuszel2009, author = {Redig, Frank and Roelly, Sylvie and Ruszel, Wioletta}, title = {Short-time Gibbsianness for infinite-dimensional diffusions with space-time interaction}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-49514}, year = {2009}, abstract = {We consider a class of infinite-dimensional diffusions where the interaction between the components is both spatial and temporal. We start the system from a Gibbs measure with finiterange uniformly bounded interaction. Under suitable conditions on the drift, we prove that there exists t0 > 0 such that the distribution at time t = t0 is a Gibbs measure with absolutely summable interaction. The main tool is a cluster expansion of both the initial interaction and certain time-reversed Girsanov factors coming from the dynamics.}, language = {de} } @book{DereudreRoelly2004, author = {Dereudre, David and Roelly, Sylvie}, title = {On Gibbsianness of infinite-dimensional diffussions}, series = {Preprint / Universit{\"a}t Potsdam, Institut f{\"u}r Mathematik, Mathematische Statistik un}, journal = {Preprint / Universit{\"a}t Potsdam, Institut f{\"u}r Mathematik, Mathematische Statistik un}, publisher = {Univ.}, address = {Potsdam}, issn = {1613-3307}, pages = {14 S.}, year = {2004}, language = {en} } @unpublished{DereudreMazzonettoRoelly2016, author = {Dereudre, David and Mazzonetto, Sara and Roelly, Sylvie}, title = {Exact simulation of Brownian diffusions with drift admitting jumps}, volume = {5}, number = {7}, publisher = {Universit{\"a}tsverlag Potsdam}, address = {Potsdam}, issn = {2193-6943}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-91049}, pages = {25}, year = {2016}, abstract = {Using an algorithm based on a retrospective rejection sampling scheme, we propose an exact simulation of a Brownian diffusion whose drift admits several jumps. We treat explicitly and extensively the case of two jumps, providing numerical simulations. Our main contribution is to manage the technical difficulty due to the presence of two jumps thanks to a new explicit expression of the transition density of the skew Brownian motion with two semipermeable barriers and a constant drift.}, language = {en} } @misc{RoellyDereudre2004, author = {Roelly, Sylvie and Dereudre, David}, title = {Propagation of Gibbsiannes for infinite-dimensional gradient Brownian diffusions}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-6918}, year = {2004}, abstract = {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.}, language = {en} } @unpublished{DereudreRoelly2004, author = {Dereudre, David and Roelly, Sylvie}, title = {Propagation of Gibbsianness for infinite-dimensional gradient Brownian diffusions}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-51535}, year = {2004}, abstract = {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.}, language = {en} } @unpublished{DereudreMazzonettoRoelly2015, author = {Dereudre, David and Mazzonetto, Sara and Roelly, Sylvie}, title = {An explicit representation of the transition densities of the skew Brownian motion with drift and two semipermeable barriers}, volume = {4}, number = {9}, publisher = {Universit{\"a}tsverlag Potsdam}, address = {Potsdam}, issn = {2193-6943}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-80613}, pages = {23}, year = {2015}, abstract = {In this paper we obtain an explicit representation of the transition density of the one-dimensional skew Brownian motion with (a constant drift and) two semipermeable barriers. Moreover we propose a rejection method to simulate this density in an exact way.}, language = {en} } @unpublished{DereudreRoelly2014, author = {Dereudre, David and Roelly, Sylvie}, title = {Path-dependent infinite-dimensional SDE with non-regular drift : an existence result}, volume = {3}, number = {11}, publisher = {Universit{\"a}tsverlag Potsdam}, address = {Potsdam}, issn = {2193-6943}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-72084}, pages = {27}, year = {2014}, abstract = {We establish in this paper the existence of weak solutions of infinite-dimensional shift invariant stochastic differential equations driven by a Brownian term. The drift function is very general, in the sense that it is supposed to be neither small or continuous, nor Markov. On the initial law we only assume that it admits a finite specific entropy. Our result strongly improves the previous ones obtained for free dynamics with a small perturbative drift. The originality of our method leads in the use of the specific entropy as a tightness tool and on a description of such stochastic differential equation as solution of a variational problem on the path space.}, language = {en} } @misc{RoellyDereudre2004, author = {Roelly, Sylvie and Dereudre, David}, title = {On Gibbsianness of infinite-dimensional diffusions}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-6692}, year = {2004}, abstract = {The authors analyse different Gibbsian properties of interactive Brownian diffusions X indexed by the d-dimensional lattice. In the first part of the paper, these processes are characterized as Gibbs states on path spaces. In the second part of the paper, they study the Gibbsian character on R^{Z^d} of the law at time t of the infinite-dimensional diffusion X(t), when the initial law is Gibbsian. AMS Classifications: 60G15 , 60G60 , 60H10 , 60J60}, language = {en} } @book{DereudreRoelly2004, author = {Dereudre, David and Roelly, Sylvie}, title = {On Gibbsianness of infinite-dimensional diffusions}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-52630}, publisher = {Universit{\"a}t Potsdam}, year = {2004}, abstract = {We analyse different Gibbsian properties of interactive Brownian diffusions X indexed by the lattice \$Z^{d} : X = (X_{i}(t), i ∈ Z^{d}, t ∈ [0, T], 0 < T < +∞)\$. In a first part, these processes are characterized as Gibbs states on path spaces of the form \$C([0, T],R)Z^{d}\$. In a second part, we study the Gibbsian character on \$R^{Z}^{d}\$ of \$v^{t}\$, the law at time t of the infinite-dimensional diffusion X(t), when the initial law \$v = v^{0}\$ is Gibbsian.}, language = {en} } @article{DereudreRoelly2017, author = {Dereudre, David and Roelly, Sylvie}, title = {Path-dependent infinite-dimensional SDE with non-regular drift}, series = {Annales de l'Institut Henri Poincar{\´e} : B, Probability and statistics}, volume = {53}, journal = {Annales de l'Institut Henri Poincar{\´e} : B, Probability and statistics}, number = {2}, publisher = {Inst. of Mathematical Statistics}, address = {Bethesda}, issn = {0246-0203}, doi = {10.1214/15-AIHP728}, pages = {641 -- 657}, year = {2017}, abstract = {We establish in this paper the existence of weak solutions of infinite-dimensional shift invariant stochastic differential equations driven by a Brownian term. The drift function is very general, in the sense that it is supposed to be neither bounded or continuous, nor Markov. On the initial law we only assume that it admits a finite specific entropy and a finite second moment. The originality of our method leads in the use of the specific entropy as a tightness tool and in the description of such infinite-dimensional stochastic process as solution of a variational problem on the path space. Our result clearly improves previous ones obtained for free dynamics with bounded drift.}, language = {en} } @misc{EhlenFloegeGoebeletal.2023, author = {Ehlen, Tobias and Fl{\"o}ge, Annie and G{\"o}bel, Franziska and Keller, Peter and Rœlly, Sylvie}, title = {{\"U}bungsbuch zur Stochastik}, editor = {Keller, Peter and Rœlly, Sylvie}, publisher = {Universit{\"a}tsverlag Potsdam}, address = {Potsdam}, isbn = {978-3-86956-563-7}, doi = {10.25932/publishup-59593}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-595939}, pages = {306}, year = {2023}, abstract = {Dieses Buch stellt {\"U}bungen zu den Grundbegriffen und Grunds{\"a}tzen der Stochastik und ihre L{\"o}sungen zur Verf{\"u}gung. So wie man Tonleitern in der Musik trainiert, so berechnet man {\"U}bungsaufgaben in der Mathematik. In diesem Sinne soll dieses {\"U}bungsbuch vor allem als Vorlage dienen f{\"u}r das eigenst{\"a}ndige, eigenverantwortliche Lernen und {\"U}ben. Die Sch{\"o}nheit und Einzigartigkeit der Wahrscheinlichkeitstheorie besteht darin, dass sie eine Vielzahl von realen Ph{\"a}nomenen modellieren kann. Daher findet man hier Aufgaben mit Verbindungen zur Geometrie, zu Gl{\"u}cksspielen, zur Versicherungsmathematik, zur Demographie und vielen anderen Themen.}, language = {de} }