The mathematical theory of homogenization deals with the rigorous derivation of effective models from partial differential equations with rapidly-oscillating coefficients. In this thesis we deal with modeling and homogenization of random heterogeneous media. Namely, we obtain stochastic homogenization results for certain evolutionary gradient systems. In particular, we derive continuum effective models from discrete networks consisting of elasto-plastic springs with random coefficients in the setting of evolutionary rate-independent systems. Also, we treat a discrete counterpart of gradient plasticity. The second type of problems that we consider are gradient flows. Specifically, we study continuum gradient flows driven by λ-convex energy functionals. In stochastic homogenization the derived deterministic effective equations are typically hardly-accessible for standard numerical methods. For this reason, we study approximation schemes for the effective equations that we obtain, which are well-suited for numerical analysis. For the sake of a simple treatment of these problems, we introduce a general procedure for stochastic homogenization – the stochastic unfolding method. This method presents a stochastic counterpart of the well-established periodic unfolding procedure which is well-suited for homogenization of media with periodic microstructure. The stochastic unfolding method is convenient for the treatment of equations driven by integral functionals with random integrands. The advantage of this strategy in regard to other methods in homogenization is its simplicity and the elementary analysis that mostly relies on basic functional analysis concepts, which makes it an easily accessible method for a wide audience. In particular, we develop this strategy in the setting that is suited for problems involving discrete-to-continuum transition as well as for equations defined on a continuum physical space. We believe that the stochastic unfolding method may also be useful for problems outside of the scope of this work.

Diese Dissertation analysiert BSDEs und PDEs mit singulären Endbedingungen, welche in Problemen der optimalen Portfolioliquidierung auftreten. In den vergangenen Jahren haben Portfolioliquidierungsprobleme in der Literatur zur Finanzmathematik große Aufmerksamkeit erhalten. Ihre wichtigste Eigenschaft ist die singuläre Endbedingung der durch die Liquidierungsbedingung induzierten Wertfunktion, welche eine singuläre Endbedingung der zugehörigen BSDE oder PDE impliziert. Diese Arbeit besteht aus drei Kapiteln. Das erste Kapitel analysiert ein Portfolioliquidierungsproblem für mehrere Wertpapiere mit sofortigem und anhaltendem Preiseinfluss und stochastischer Resilienz. Wir zeigen, dass die Wertfunktion durch eine mehrdimensionale BSRDE mit singulärer Endbedingung beschrieben werden kann. Wir weisen die Existenz einer Lösung dieser BSRDE nach und zeigen, dass diese durch eine Folge von Lösungen von BSRDEs mit endlicher und wachsender Endbedingung approximiert werden kann. Eine neue a priori-Abschätzung für die approximierenden BSRDEs wird für den Nachweis hergeleitet. Das zweite Kapitel betrachtet ein Portfolioliquidierungsproblem mit unbeschränkten Kostenkoeffizienten. Wir weisen die Existenz einer eindeutigen nichtnegativen Viskositätslösung der HJB-Gleichung nach. Das Existenzresultat basiert auf einem neuartigen Vergleichsprinzip für semi-stetige Viskositätssub-/-superlösungen für singuläre PDEs. Stetigkeit der Viskositätslösung ist hinreichend für das Verifikationsargument. Im dritten Kapitel untersuchen wir ein optimales Liquidierungsproblem unter Mehrdeutigkeit der Parameter des Preiseinflusses. In diesem Fall kann die Wertfunktion durch die Lösung einer semilinearen PDE mit superlinearem Gradienten beschrieben werden. Zuerst zeigen wir die Existenz einer Viskositätslösung indem wir unser Vergleichsprinzip für singuläre PDEs erweitern. Sodann weisen wir die Regularität mit einer asymptotischen Entwicklung der Lösung am Endzeitpunkt nach. / This dissertation analyzes BSDEs and PDEs with singular terminal condition arising in models of optimal portfolio liquidation. Portfolio liquidation problems have received considerable attention in the financial mathematics literature in recent years. Their main characteristic is the singular terminal condition of the value function induced by the liquidation constraint, which translates into a singular terminal state constraint on the associated BSDE or PDE. The dissertation consists of three chapters. The first chapter analyzes a multi-asset portfolio liquidation problem with instantaneous and persistent price impact and stochastic resilience. We show that the value function can be described by a multi-dimensional BSRDE with a singular terminal condition. We prove the existence of a solution to this BSRDE and show that it can be approximated by a sequence of the solutions to BSRDEs with finite increasing terminal condition. A novel a priori estimate for the approximating BSRDEs is established for the verification argument. The second chapter considers a portfolio liquidation problem with unbounded cost coefficients. We establish the existence of a unique nonnegative continuous viscosity solution to the HJB equation. The existence result is based on a novel comparison principle for semi-continuous viscosity sub-/supersolutions for singular PDEs. Continuity of the viscosity solution is enough to carry out the verification argument. The third chapter studies an optimal liquidation problem under ambiguity with respect to price impact parameters. In this case the value function can be characterized by the solution to a semilinear PDE with superlinear gradient. We first prove the existence of a solution in the viscosity sense by extending our comparison principle for singular PDEs. Higher regularity is then established using an asymptotic expansion of the solution at the terminal time.

Cancer as a disease causes about 8.8 million deaths worldwide per year, a number that will largely increase in the next decades. Although the cellular processes involved in tumor emergence are more and more understood, the implications of specific changes at the cellular scale on tumor emergence at the tissue scale remain elusive. Main reasons for this lack of understanding are that the cellular processes are often hardly observable especially in the early phase of tumor development and that the interplay between cellular and tissue scale is difficult to deduce. Cell-based mathematical models provide a valuable tool to investigate in which way observable phenomena on the tissue scale develop by cellular processes. The implications of these models can elucidate underlying mechanisms and generate quantitative predictions that can be experimentally validated. In this thesis, we infer the role of genetic and phenotypic cell changes on tumor development with the help of cell-based Markov chain models which are calibrated by tissue-scale data. In the first part, we utilize data on the diagnosed fractions of benign and malignant tumor subtypes to unravel the consequences of genetic cell changes on tumor development. We introduce extensions of Moran models to investigate two specific biological questions. First, we evaluate the tumor regression behavior of pilocytic astrocytoma which represents the most common brain tumor in children and young adults. We formulate a Moran model with two absorbing states representing different subtypes of this tumor, derive the absorption probabilities in these states and calculate the tumor regression probability within the model. This analysis allows to predict the chance for tumor regression in dependency of the remaining tumor size and implies a different clinical resection strategy for pilocytic astrocytoma compared to other brain tumors. Second, we shed light on the hardly observable early cellular dynamics of tumor development and its consequences on the emergence of different tumor subtypes on the tissue scale. For this purpose, we utilize spatial and non-spatial Moran models with two absorbing states which describe both benign and malignant tumor subtypes and estimate lower and upper bounds for the range of cellular competition in different tissues. Our results suggest the existence of small and tissue-specific tumor-originating niches in which the fate of tumor development is decided long before a tumor manifests. These findings might help to identify the tumor-originating cell types for different cancer types. From a theoretical point of view, the novel analytical results regarding the absorption behavior of our extended Moran models contribute to a better understanding of this model class and have several applications also beyond the scope of this thesis. The second part is devoted to the investigation of the role of phenotypic plasticity of cancer cells in tumor development. In order to understand how phenotypic heterogeneity in tumors arises we describe cell state changes by a Markov chain model. This model allows to quantify the cell state transitions leading to the observed heterogeneity from experimental tissue-scale data on the evolution of cell state proportions. In order to bridge the gap between mathematical modeling and the analysis of such data, we developed an R package called CellTrans which is freely available. This package automatizes the whole process of mathematical modeling and can be utilized to (i) infer the transition probabilities between different cell states, (ii) predict cell line compositions at a certain time, (iii) predict equilibrium cell state compositions and (iv) estimate the time needed to reach this equilibrium. We utilize publicly available data on the evolution of cell compositions to demonstrate the applicability of CellTrans. Moreover, we apply CellTrans to investigate the observed cellular phenotypic heterogeneity in glioblastoma. For this purpose, we use data on the evolution of glioblastoma cell line compositions to infer to which extent the heterogeneity in these tumors can be explained by hierarchical phenotypic transitions. We also demonstrate in which way our newly developed R package can be utilized to analyze the influence of different micro-environmental conditions on cell state proportions. Summarized, this thesis contributes to gain a better understanding of the consequences of both genetic and phenotypic cell changes on tumor development with the help of Markov chain models which are motivated by the specific underlying biological questions. Moreover, the analysis of the novel Moran models provides new theoretical results, in particular regarding the absorption behavior of the underlying stochastic processes.

Cyber-physical systems and the Internet of Things raise various challenges concerning the modelling and analysis of large modular systems. Models for such systems typically require uncountable state and action spaces, samplings from continuous distributions, and non-deterministic choices over uncountable many alternatives. In this thesis we fo- cus on a general modelling formalism for stochastic systems called stochastic transition system. We introduce a novel composition operator for stochastic transition systems that is based on couplings of probability measures. Couplings yield a declarative modelling paradigm appropriate for the formalisation of stochastic dependencies that are caused by the interaction of components. Congruence results for our operator with respect to standard notions for simulation and bisimulation are presented for which the challenge is to prove the existence of appropriate couplings. In this context a theory for stochastic transition systems concerning simulation, bisimulation, and trace-distribution relations is developed. We show that under generic Souslin conditions, the simulation preorder is a subset of trace-distribution inclusion and accordingly, bisimulation equivalence is finer than trace-distribution equivalence. We moreover establish characterisations of the simulation preorder and the bisimulation equivalence for a broad subclass of stochastic transition systems in terms of expressive action-based probabilistic logics and show that these characterisations are still maintained by small fragments of these logics, respectively. To treat associated measurability aspects, we rely on methods from descriptive set theory, properties of Souslin sets, as well as prominent measurable-selection principles.:1 Introduction 2 Probability measures on Polish spaces 3 Stochastic transition systems 4 Simulations and trace distributions for Souslin systems 5 Action-based probabilistic temporal logics 6 Parallel composition based on spans and couplings 7 Relations to models from the literature 8 Conclusions 9 Bibliography

A discrete-time stochastic optimal control problem was recently proposed to address the GLOSA (Green Light Optimal Speed Advisory) problem in cases where the next signal switching time is decided in real-time and is therefore uncertain in advance. However, there was an assumption that the traffic signal is initially red and turns to green, which means that only half traffic light cycle was considered. In this work, the aforementioned problem is extended considering a full traffic light cycle, consisting of four phases: a certain green phase, during which the vehicle can freely pass; an uncertain green phase, in which there is a probability that the traffic light will extend its duration or turn to red at any time; a certain red phase during which the vehicle cannot pass; and an uncertain red phase, in which there is a probability that the red signal may be extended or turn to green at any time. It is demonstrated, based on preliminary results, that the proposed SDP (Stochastic Dynamic Programming) approach achieves better average performance, in terms of fuel consumption, compared to the IDM (Intelligent Driver Model), which emulates human-driving behavior.

This paper describes a general approach to the specification the stable regime speed-flow function, for motorways, as a part of the stable regime Stochastic Fundamental Diagram consistent with main assumptions of Transportation Systems Theory. Main original elements are: • Specification of speed-flow functions consistent with travel time function, such as BPR-like functions; • Calibration from disaggregate data, say data from single vehicle trajectories; • Specification of the speed r. v. distribution consistent with those used in RUT for route choice behavior modelling, such as Gamma, Inv-Gamma.

In dieser Arbeit versuche ich, den neuronalen Code, d. h. die Art und Weise, wie die Nervenzellen des Gehirns Informationen in ihrer Aktivität übertragen und verarbeiten, besser zu verstehen, indem ich die Kodierung von Stimuli in neuronalen Systemen untersuche. Zu diesem Zweck analysiere ich die Veränderungen in der Dynamik von neuronalen Standardmodellen, die im Rahmen der statistischen Physik entwickelt wurden, in Bezug auf Veränder- ungen der Parameter und der Konnektivität bei Vorhandensein bzw. Fehlen eines Reizes. Ich verwende informationstheoretische Maße, um die Fähigkeit neuronaler Populationen, empfangene Informationen durch ihren Output zu übertragen, zu quantifizieren. Die vorgestellten Ergebnisse bauen auf einer Vielzahl früherer Studien über unverbundene und rekurrente neuronale Pop- ulationen auf. Einige dieser Studien heben zwei neuronale Code-Kandidaten hervor, die unterschiedliche Profile der Informationsfilterung aufweisen: einen Integrationscode, der als Tiefpass-Informationsfilter fungiert, und einen Synchroniecode, der als Bandpassfilter fungiert. Das Ziel der vorliegenden Arbeit ist es, die Ergebnisse dieser Studien auf Netzwerke mit einem höheren Konnektivitätsgrad, wie er im Kortex beobachtet wird, auszuweiten. / In this thesis I attempt to better understand the neural code, or the way in which the nerve cells of the brain transmit and process information in their activity, through the investigation of stimulus encoding in neural systems. To this end, I analyze changes in the dynamics of standard neuronal models, de- veloped in the framework of statistical physics, to variations in parameters and connectivity in the presence versus the absence of a stimulus. In conjunction, information theoretical measures are utilized to quantify the ability of neu- ronal populations to transmit received information through their output. The presented results build upon a multitude of previous studies of both uncon- nected and recurrent neural populations. Some of these studies highlight two neural code candidates that have distinct information filtering profiles: an in- tegration code that acts as a low-pass information filter and a synchrony code that acts as a bandpass filter. In the following, synaptic connectivity is added in diverse ways in order to extend results of these studies to networks with a higher level of connectivity, as observed in the cortex.

Sowohl in der neuronalen als auch in der Kalzium Signalübertragung werden Informationen durch kurze Pulse oder Spikes, übertragen. Obwohl beide Systeme grundlegende Eigenschaften der Spike-Erzeugung teilen, wurden Integrate-and-fire (IF)-Modelle bisher nur auf neuronale Systeme angewendet. Diese Modelle bleiben auch dann behandelbar, wenn sie um Prozesse erweitert werden, die in Übereinstimmung mit Experimenten Spike-Zeiten mit korrelierten Interspike-Intervallen (ISI) erzeugen. Die statistische Analyse solcher nicht erneuerbarer Modelle ist Gegenstand dieser Arbeit. Das zweite Kapitel konzentriert sich auf die Berechnung des seriellen Korrelationskoeffizienten (SCC) in neuronalen Systemen. Es wird ein adaptives Modell betrachtet, das durch einen korrelierten Eingangsstrom getrieben wird. Es zeigt sich, dass neben den langsamen Prozessen auch die Dynamik des Modells den SCC bestimmt. Obwohl die Theorie für schwach gestörte IF-Modelle entwickelt wurde, kann sie auch auf stärker gestörte leitfähigkeitsbasierte Modelle angewendet werden und ist damit in der Lage, ein breites Spektrum biophysikalischer Situationen zu beschreiben. Im dritten Kapitel wird ein IF-Modell zur Beschreibung von Kalzium-Spikes formuliert, das die stochastische Freisetzung von Kalzium aus dem endoplasmatischen Retikulum (ER) und dessen Entleerung berücksichtigt. Die beobachtete Zeitskalentrennung zwischen Kalziumfreisetzung und Spikegenerierung motiviert eine Diffusionsnäherung, die eine analytische Behandlung des Modells ermöglicht. Die experimentell beobachtete Transiente, in der sich die ISIs einem stationären Wert annähern, kann durch die Entleerung des ER beschrieben werden. Es wird untersucht, wie die Statistiken der Transienten mit den stationären Intervallkorrelationen zusammenhängen. Es zeigt sich, dass eine stärkere Anpassung der Intervalle und eine kurze Transiente mit stärkeren Korrelationen einhergehen. Der Vergleich mit experimentellen Daten bestätigt diese Trends qualitativ. / In both neuronal and calcium signaling, information is transmitted by short pulses, so-called spikes. Although both systems share some basic principles of spike generation, integrate-and-fire (IF) models have so far only been applied to neuronal systems. These models remain analytically tractable even when extended to include processes that lead to the generation of spike times with correlated interspike intervals (ISIs) as observed in experiments. The statistical analysis of such non-renewal models is the subject of this thesis. In the second chapter we focus on the calculation of the serial correlation coefficient (SCC) in neural systems. We consider an adaptive model driven by a correlated input current. We show that in addition to the two slow processes, the dynamics of the model also determines the SCC. Although the theory is developed for weakly perturbed IF models, it can also be applied to more strongly perturbed conductance-based models and is thus able to account for a wide range of biophysical situations. In the third chapter, we formulate an IF model to describe the generation of calcium spikes, taking into account the stochastic release of calcium from the endoplasmic reticulum (ER) and its depletion. The observed time-scale separation between calcium release and spike generation motivates a diffusion approximation that allows an analytical treatment of the model. The experimentally observed transient, during which the ISIs approach a steady state value, can be captured by the depletion of the ER. We study how the transient ISI statistics are related to the stationary interval correlations. We show that a stronger adaptation of the intervals as well as a short transient are associated with stronger interval correlations. Comparison with experimental data qualitatively confirms these trends.

Die Dissertationsschrift thematisiert die Frage nach den Kosten und Nutzen einer Diversifikationsstrategie im europäischen Erdgasmarkt und gliedert sich in neun Kapitel. In einer Vorbetrachtung beschreiben die Kapitel eins bis vier die Ausganglage mit Blick auf Angebots- und Nachfragestrukturen sowie der Gasinfrastruktur. Unsicherheiten in Bezug auf die Entwicklung der Nachfrage, Importverfügbarkeit und Preisniveaus werden diskutiert. In einem analytischen Rahmen wird das Thema Diversifikation in den Kontext der Energiesicherheit eingeordnet. Die Kapitel fünf bis sieben befassen sich mit der Beschreibung und der Analyse des europäischen Gasmarkts. Dafür wird ein lineares Modell, GAMAMOD-EU, entwickelt, welches als stochastische Optimierung den Ausbau der Erdgasinfrastruktur unter Einbezug von drei Unsicherheitsdimensionen in den Jahren 2030 und 2045 abbildet. Zusätzlich werden drei Diversifikationsstrategien in Hinblick auf Infrastrukturentwicklung und Versorgungssicherheit analysiert. In einer Erweiterung wird der Import Grüner Gase in die Betrachtung einbezogen. Kapitel acht stellt das deutsche Gasnetzmodell GAMAMOD-DE mit einer Fallstudie vor, die die Versorgungslage im kalten Winter 2012 nachmodelliert. Im abschließenden Kapitel neun werden die zu Beginn aufgeworfenen Forschungsfragen beantwortet, politische Handlungsempfehlungen gegeben und der weitere Forschungsbedarf skizziert.:Table of Contents List of Figures List of Tables Abbreviations Country Codes Nomenclature: GAMAMOD-EU Nomenclature: GAMAMOD-DE 1 Introduction 2 Uncertainties in Gas Markets 3 Diversification in Gas Markets to Ensure Security of Supply 4 Natural Gas Infrastructure 5 The European Natural Gas Market Model (GAMAMOD-EU) 6 Results on Security of Supply in the European Gas Market 7 Impact of Green Gas Imports on Infrastructure Investments 8 The German Natural Gas Market Model (GAMAMOD-DE) 9 Conclusion and Outlook Laws and Communication Papers References Appendix / The dissertation addresses the question of the costs and benefits of a diversification strategy in the European natural gas market and is divided into nine chapters. In a preliminary analysis, chapters one to four describe the initial situation with regard to supply and demand structures as well as the gas infrastructure. Uncertainties regarding the development of demand, import availability and price levels are discussed. In an analytical framework, the topic of diversification is placed in the context of energy security. Chapters five to seven deal with the description and analysis of the European gas market. For this purpose, a linear model, GAMAMOD-EU, is developed, which maps the expansion of the natural gas infrastructure as a stochastic optimisation, taking into account three uncertainty dimensions in the years 2030 and 2045. In addition, three diversification strategies are analysed with regard to infrastructure development and security of supply. In an extension, the import of green gases is included in the analysis. Chapter eight presents the German gas grid model GAMAMOD-DE with a case study, which models the supply situation in the cold winter of 2012. In the concluding chapter nine, the research questions raised at the beginning are answered, political recommendations for action are given and the need for further research is outlined.:Table of Contents List of Figures List of Tables Abbreviations Country Codes Nomenclature: GAMAMOD-EU Nomenclature: GAMAMOD-DE 1 Introduction 2 Uncertainties in Gas Markets 3 Diversification in Gas Markets to Ensure Security of Supply 4 Natural Gas Infrastructure 5 The European Natural Gas Market Model (GAMAMOD-EU) 6 Results on Security of Supply in the European Gas Market 7 Impact of Green Gas Imports on Infrastructure Investments 8 The German Natural Gas Market Model (GAMAMOD-DE) 9 Conclusion and Outlook Laws and Communication Papers References Appendix

For life to thrive, its fundamental units, i.e., the cells, need to reliably and robustly fulfill their function. However, cellular operability is challenged by the appearance of biological noise in the concentration of proteins and other cell components. This noise arises due to spontaneous fluctuations that are inherent to all chemical reactions. For small (mesoscopic) systems, like cells, these fluctuations can be significant and disturb cellular functions. Cells evolved mechanisms to control and reduce their internal noise. One way to reduce noise in eukaryotic cells is to exploit their internal structure and restrict noise to a particular organelle, thus reducing the noise in the rest of the cell. In recent years it was shown that many cell organelles could be formed by phase separation without the need for a membrane. Thus, it was suggested that phase separation could reduce concentration noise in cells. However, until now, any systematic investigation linking essential aspects of phase separation and concentration noise in cells has been lacking. This motivates the study of fluctuations in mesoscopic phase-separating systems. This thesis develops a generic theoretical model based on a thermodynamic description of phase separation. We consider a binary mixture that can phase separate into two phases - a liquid droplet surrounded by a phase, which we refer to as continuous phase. We merge this description with methods of stochastic chemical reactions in order to account for the active turnover of phase-separating material and, thus, for the non-equilibrium nature of living cells. The resulting framework allows us to study fluctuations due to chemical turnover and phase separation in and out of equilibrium of phase separation. We use this framework to investigate how a phase-separating system can reduce concentration noise for different reaction networks. We find that phase separation can reduce concentration noise in active mesoscopic systems like cells in both phases. When turnover dynamics are slow, concentration noise in the dilute phase can be lowered to the level of Poissonian fluctuations. For the dense phase, we find that noise can fall below the Poissonian threshold. When turnover rates become faster such that the system deviates from the equilibrium configuration, the noise reduction by phase separation becomes less efficient. We test our model on experimental data of an engineered protein expressed in living cells. We find a good agreement between the data and theory and demonstrate that phase separation is a viable mechanism for noise reduction in living cells. Thus, phase separation might play an essential part in ensuring the reliable control of cellular functions.