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Institute
Portal Wissen = Glauben
(2014)
Menschen wollen wissen, was wirklich ist. Kinder lassen sich gern eine Geschichte erzählen, aber spätestens 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 Träume erforschen, tun wir es in der Absicht, die realen sprachlichen Strukturen bzw. die neurologischen Faktoren von bloß vermuteten zu unterscheiden. Im Idealfall können wir Ergebnisse präsentieren, die von anderen logisch nachvollzogen und empirisch wiederholbar sind. Meistens geht das aber nicht. Wir können 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 Ausführungen anderer zu vertrauen, wenn wir wissen wollen, wo es zum Bahnhof geht oder ob es in Ulan Bator schön ist. Deshalb haben wir uns daran gewöhnt, anderen Glauben zu schenken, vom Freund bis zum Tagesschausprecher. Das ist kein kindliches Verhalten, sondern eine Notwendigkeit. Freilich ist das riskant, denn alle anderen könnten uns – wie in der „Truman- Show“ – anlügen. 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.
Für Religiöse ist das der Glaube. Glaube als Unterstellung, dass alle menschlichen Beziehungen erst dann wirklich, ernst und über Zweifel erhaben sind, wenn sie sich vor den Augen Gottes wissen. Erst vor ihm ist etwas als es selbst und nicht nur „für 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 überirdischen 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 Gesprächen nicht nur wissen, was ich weiß, sondern, was ich glaube. Als Religionswissenschaftler und gleichzeitig gläubiger Katholik sitze ich zwischen den Stühlen: Einerseits ist es als Professor meine Aufgabe, alles zu bezweifeln, d.h. jeden religiösen Text auf seine historischen Kontexte und soziologischen Funktionen zurückzuführen. Andererseits hält der Christ in mir bestimmte religiöse Dokumente – in meinem Fall die Bibel – zwar für 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 Verhältnis zum Glauben nach: Wie steht Wissenschaft zum Glauben – ob religiös oder nicht? Wo bringt Wissenschaft Dinge ans Licht, die wir kaum glauben mögen oder uns (wieder) glauben lassen? Was passiert, wenn Forschung irrige Annahmen oder Mythen aufklärt? Ist Wissenschaft in der Lage, Dingen auf den Grund zu gehen, die zwar überzeugend, aber unerklärbar sind? Wie kann sie selbst glaubwürdig bleiben und sich dennoch weiterentwickeln?
In den Beiträgen dieser „Portal Wissen“ scheinen diese Fragen immer wieder auf. Sie bilden ein vielfältiges, spannendes und auch überraschendes Bild der Forschungsprojekte und der Wissenschaftler an der Universität Potsdam. Glauben Sie mir, es erwartet Sie eine anregende Lektüre!
Prof. Dr. Johann Hafner
Professor für Religionswissenschaft mit dem Schwerpunkt Christentum
Dekan der Philosophischen Fakultät
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.
Convoluted Brownian motion
(2016)
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
We consider a system of infinitely many hard balls in R<sup>d 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.
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.
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.
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
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.