Etliche Modelle der Statistischen Physik beinhalten zufällige Pfade mit Interaktionen vielfältiger Natur. Ein erstes Beispiel ist ein zuälliger Pfad in einem zufälligen Potential (auch das parabolische Anderson-Modell genannt, siehe das mathematische Forschungsthema Spektra zufälliger Operatoren). Ein zweites Beispiel ist das Polaron-Modell einer selbstinteragierenden dreidimensionalen Brown'schen Bewegung und seine Mean-Field-Variante. Weitere Modelle basieren zum Beispiel aus einer Kollektion von mehreren eindimensionalen Pfaden, die sich gegenseitig nie überholen (nichtkollidierende Partikel) oder auch Irrfahrten in zufälligen Umgebungen oder eine große Familie unabhängiger Brown'scher Bewegungen, die bei Kontakt zufällig koagulieren (siehe das Anwendungsthema Koagulation). Ferner wird der Einfluss von positiver Temperatur oft mit Hilfe von Modellen von interagierenden Brown'schen Bewegungen modelliert, etwa beim Bose-Gas (siehe das mathematische Forschungsthema Wechselwirkende stochastische Systeme und das Anwendungsthema Partikelbasierte Modellierung in den Naturwissenschaften).

In die Thematik der Diffusionsmodelle fallen auch viele andere Gebiete, die am WIAS Bedeutung haben, wie etwa interagierende stochastische Differentialgleichungssysteme in der Modellierung von Ladungsvorgängen in Batterien (siehe das Anwendungsthema Thermodynamische Modelle für elektrochemische Systeme) oder Partikelmodelle von interagierenden stochastischen Pfaden in turbulenten Umgebungen bei der Untersuchung von Populationsbilanzsystemen (siehe das Anwendungsthema Numerische Verfahren zur Simulation von Populationsbilanzsystemen). Häufig basieren die untersuchten Modelle auf empirischen Daten.

### Beitrag des Instituts

Im Verständnis des Polaron-Modells wurden einige Fortschritte erzielt in Arbeiten von Bolthausen, König, Mukherjee und Varadhan. Der Startpunkt dieser Neuigkeiten (nach einem Stillstand von fast 30 Jahren) war eine neue und sehr flexible Technik der Herleitung eines Prinzips großer Abweichungen für die normierten Aufenthaltsmasse einer Brown'schen Bewegung, die Mukherjee und Varadhan fanden. Damit ist es nun möglich, das Mean-Field-Modell für das Polaron pfadweise exakt zu beschreiben, und die Möglichkeit ist eröffnet, auch das originale Polaron-Modell damit pfadweise zu analysieren.

Die mathematische Beschreibung der Kondensation im Bose-Gas mit Hilfe von Methoden der Wahrscheinlichkeitstheorie fußt auf einer Darstellung der interagierender Partikel mittels eines Punktprozesses von Brown'schen Brücken. In einer Arbeit von Adams, König und Collevecchio gelang es erstmals, die freie Energie des unendlich großen Systems in bestimmten Parameter-Regimen als Variationsproblem über stationäre Punkt Prozesse zu charakterisieren. Die verwendete Methodik eröffnet die prinzipielle Möglichkeit, das Kondensat als Nichtexistenz eines Minimierers zu erkennen. Darüber hinaus kann die Hinzunahme von Interlacement Prozessen dazu genutzt werden, das Kondensat selbst zu beschreiben.

Im Kern der Bose-Einstein-Kondensation steht die Abstoßungseigenschaft von Partikeln im Bose Gas, welche zu Schlingen unterschiedlicher Länge verflochten sind. Um diese Interaktionen und die daraus resultierenden Phasenübergangs-Phänomene besser zu verstehen, analysieren wir reduzierte Modelle von interagierenden Zufallswegen und Polygonen. In diesem Feld bauen wir auf ein Reihe von jüngeren Resultaten von Taggi, Betz et.al. für selbst-abstoßende zufällige Polygone und zufällige Permutationen auf. Fig. 1 - Dichotomie im spektralen Verhalten des Laplacians mit zufälligen i.i.d. Leitfähigkeiten.

## Publikationen

### Monografien

• H. Neidhardt, A. Stephan, V.A. Zagrebnov, Chapter 13: Trotter Product Formula and Linear Evolution Equations on Hilbert Spaces, in: Analysis and Operator Theory, Th.M. Rassias , V.A. Zagrebnov , eds., 146 of Springer Optimization and Its Applications, Springer, Cham, 2019, pp. 271--299, (Chapter Published), DOI 10.1007/978-3-030-12661-2_13 .

• H.-Chr. Kaiser, D. Knees, A. Mielke, J. Rehberg, E. Rocca, M. Thomas, E. Valdinoci, eds., PDE 2015: Theory and Applications of Partial Differential Equations, 10 of Discrete and Continuous Dynamical Systems -- Series S, American Institute of Mathematical Science, Springfield, 2017, iv+933 pages, (Collection Published).

• W. König, The Parabolic Anderson Model -- Random Walks in Random Potential, Pathways in Mathematics, Birkhäuser, Basel, 2016, xi+192 pages, (Monograph Published).

• J. Diehl, P. Friz, H. Mai , H. Oberhauser, S. Riedel, W. Stannat, Chapter 8: Robustness in Stochastic Filtering and Maximum Likelihood Estimation for SDEs, in: Extraction of Quantifiable Information from Complex Systems, S. Dahlke, W. Dahmen, M. Griebel, W. Hackbusch, K. Ritter, R. Schneider, Ch. Schwab, H. Yserentant, eds., 102 of Lecture Notes in Computational Science and Engineering, Springer International Publishing Switzerland, Cham, 2014, pp. 161--178, (Chapter Published).

### Artikel in Referierten Journalen

• F. Flegel, M. Heida, M. Slowik, Homogenization theory for the random conductance model with degenerate ergodic weights and unbounded-range jumps, Annales de l'Institut Henri Poincare. Probabilites et Statistiques, 55 (2019), pp. 1226--1257, DOI 10.1214/18-AIHP917 .
Abstract
We study homogenization properties of the discrete Laplace operator with random conductances on a large domain in Zd. More precisely, we prove almost-sure homogenization of the discrete Poisson equation and of the top of the Dirichlet spectrum. We assume that the conductances are stationary, ergodic and nearest-neighbor conductances are positive. In contrast to earlier results, we do not require uniform ellipticity but certain integrability conditions on the lower and upper tails of the conductances. We further allow jumps of arbitrary length. Without the long-range connections, the integrability condition on the lower tail is optimal for spectral homogenization. It coincides with a necessary condition for the validity of a local central limit theorem for the random walk among random conductances. As an application of spectral homogenization, we prove a quenched large deviation principle for thenormalized and rescaled local times of the random walk in a growing box. Our proofs are based on a compactness result for the Laplacian's Dirichlet energy, Poincaré inequalities, Moser iteration and two-scale convergence

• F. Flegel, M. Heida, The fractional p-Laplacian emerging from homogenization of the random conductance model with degenerate ergodic weights and unbounded-range jumps, Calculus of Variations and Partial Differential Equations, Published online on 28.11.2019, DOI 10.1007/s00526-019-1663-4 .
Abstract
We study a general class of discrete p-Laplace operators in the random conductance model with long-range jumps and ergodic weights. Using a variational formulation of the problem, we show that under the assumption of bounded first moments and a suitable lower moment condition on the weights, the homogenized limit operator is a fractional p-Laplace operator. Under strengthened lower moment conditions, we can apply our insights also to the spectral homogenization of the discrete Lapalace operator to the continuous fractional Laplace operator.

• C. Bartsch, V. Wiedmeyer, Z. Lakdawala, R.I.A. Patterson, A. Voigt, K. Sundmacher, V. John, Stochastic-deterministic population balance modeling and simulation of a fluidized bed crystallizer experiment, Chemical Engineering Sciences, 208 (2019), 115102, DOI 10.1016/j.ces.2019.07.020 .

• B. Jahnel, Ch. Külske, Gibbsian representation for point processes via hyperedge potentials, Journal of Theoretical Probability, published online on 03.11.2019 (2019), pp. 1--27, DOI 10.1007/s10959-019-00960-7 .
Abstract
We consider marked point processes on the d-dimensional euclidean space, defined in terms of a quasilocal specification based on marked Poisson point processes. We investigate the possibility of constructing uniformly absolutely convergent Hamiltonians in terms of hyperedge potentials in the sense of Georgii . These potentials are natural generalizations of physical multibody potentials which are useful in models of stochastic geometry.

• D.R.M. Renger, Gradient and GENERIC systems in the space of fluxes, applied to reacting particle systems, Entropy. An International and Interdisciplinary Journal of Entropy and Information Studies, 20 (2018), pp. 596/1--596/26, DOI 10.3390/e20080596 .
Abstract
In a previous work we devised a framework to derive generalised gradient systems for an evolution equation from the large deviations of an underlying microscopic system, in the spirit of the Onsager-Machlup relations. Of particular interest is the case where the microscopic system consists of random particles, and the macroscopic quantity is the empirical measure or concentration. In this work we take the particle flux as the macroscopic quantity, which is related to the concentration via a continuity equation. By a similar argument the large deviations can induce a generalised gradient or Generic system in the space of fluxes. In a general setting we study how flux gradient or generic systems are related to gradient systems of concentrations. The arguments are explained by the example of reacting particle systems, which is later expanded to include spatial diffusion as well.

• D. Belomestny, J.G.M. Schoenmakers, Projected particle methods for solving McKean--Vlasov equations, SIAM Journal on Numerical Analysis, 56 (2018), pp. 3169--3195, DOI 10.1137/17M1111024 .
Abstract
We propose a novel projection-based particle method for solving McKean--Vlasov stochastic differential equations. Our approach is based on a projection-type estimation of the marginal density of the solution in each time step. The projection-based particle method leads in many situations to a significant reduction of numerical complexity compared to the widely used kernel density estimation algorithms. We derive strong convergence rates and rates of density estimation. The convergence analysis, particularly in the case of linearly growing coefficients, turns out to be rather challenging and requires some new type of averaging technique. This case is exemplified by explicit solutions to a class of McKean--Vlasov equations with affine drift. The performance of the proposed algorithm is illustrated by several numerical examples.

• B. Jahnel, Ch. Külske, Sharp thresholds for Gibbs-non-Gibbs transition in the fuzzy Potts models with a Kac-type interaction, Bernoulli. Official Journal of the Bernoulli Society for Mathematical Statistics and Probability, 23 (2017), pp. 2808--2827.
Abstract
We investigate the Gibbs properties of the fuzzy Potts model on the \$d\$-dimensional torus with Kac interaction. We use a variational approach for profiles inspired by that of Fernández, den Hollander and Martínez citeFeHoMa14 for their study of the Gibbs-non-Gibbs transitions of a dynamical Kac-Ising model on the torus. As our main result, we show that the mean-field thresholds dividing Gibbsian from non-Gibbsian behavior are sharp in the fuzzy Kac-Potts model. On the way to this result we prove a large deviation principle for color profiles with diluted total mass densities and use monotocity arguments

• M. Liero, A. Mielke, M.A. Peletier, D.R.M. Renger, On microscopic origins of generalized gradient structures, Discrete and Continuous Dynamical Systems -- Series S, 10 (2017), pp. 1--35, DOI 10.3934/dcdss.2017001 .
Abstract
Classical gradient systems have a linear relation between rates and driving forces. In generalized gradient systems we allow for arbitrary relations derived from general non-quadratic dissipation potentials. This paper describes two natural origins for these structures. A first microscopic origin of generalized gradient structures is given by the theory of large-deviation principles. While Markovian diffusion processes lead to classical gradient structures, Poissonian jump processes give rise to cosh-type dissipation potentials. A second origin arises via a new form of convergence, that we call EDP-convergence. Even when starting with classical gradient systems, where the dissipation potential is a quadratic functional of the rate, we may obtain a generalized gradient system in the evolutionary Gamma-limit. As examples we treat (i) the limit of a diffusion equation having a thin layer of low diffusivity, which leads to a membrane model, and (ii) the limit of diffusion over a high barrier, which gives a reaction-diffusion system.

• E. Bolthausen, W. König, Ch. Mukherjee, Mean-field interaction of Brownian occupation measures. II: A rigorous construction of the Pekar process, Communications on Pure and Applied Mathematics, 70 (2017), pp. 1598--1629.
Abstract
We consider mean-field interactions corresponding to Gibbs measures on interacting Brownian paths in three dimensions. The interaction is self-attractive and is given by a singular Coulomb potential. The logarithmic asymptotics of the partition function for this model were identified in the 1980s by Donsker and Varadhan [DV83] in terms of the Pekar variational formula, which coincides with the behavior of the partition function corresponding to the polaron problem under strong coupling. Based on this, Spohn ([Sp87]) made a heuristic observation that the strong coupling behavior of the polaron path measure, on certain time scales, should resemble a process, named as the itPekar process, whose distribution could somehow be guessed from the limiting asymptotic behavior of the mean-field measures under interest, whose rigorous analysis remained open. The present paper is devoted to a precise analysis of these mean-field path measures and convergence of the normalized occupation measures towards an explicit mixture of the maximizers of the Pekar variational problem. This leads to a rigorous construction of the aforementioned Pekar process and hence, is a contribution to the understanding of the “mean-field approximation" of the polaron problem on the level of path measures. The method of our proof is based on the compact large deviation theory developed in [MV14], its extension to the uniform strong metric for the singular Coulomb interaction carried out in [KM15], as well as an idea inspired by a itpartial path exchange argument appearing in [BS97]

• J.-D. Deuschel, P. Friz, M. Maurelli, M. Slowik, The enhanced Sanov theorem and propagation of chaos, Stochastic Processes and their Applications, 128 (2018), pp. 2228--2269 (published online on 21.09.2017), DOI 10.1016/j.spa.2017.09.010 .
Abstract
We establish a Sanov type large deviation principle for an ensemble of interacting Brownian rough paths. As application a large deviations for the (k-layer, enhanced) empirical measure of weakly interacting diffusions is obtained. This in turn implies a propagation of chaos result in a space of rough paths and allows for a robust analysis of the particle system and its McKean?Vlasov type limit, as shown in two corollaries.

• W. König, Ch. Mukherjee, Mean-field interaction of Brownian occupation measures. I: Uniform tube property of the Coulomb functional, Annales de l'Institut Henri Poincare. Probabilites et Statistiques, 53 (2017), pp. 2214--2228, DOI 10.1214/16-AIHP788 .
Abstract
We study the transformed path measure arising from the self-interaction of a three-dimensional Brownian motion via an exponential tilt with the Coulomb energy of the occupation measures of the motion by time \$t\$. The logarithmic asymptotics of the partition function were identified in the 1980s by Donsker and Varadhan [DV83-P] in terms of a variational formula. Recently [MV14] a new technique for studying the path measure itself was introduced, which allows for proving that the normalized occupation measure asymptotically concentrates around the set of all maximizers of the formula. In the present paper, we show that likewise the Coulomb functional of the occupation measure concentrates around the set of corresponding Coulomb functionals of the maximizers in the uniform topology. This is a decisive step on the way to a rigorous proof of the convergence of the normalized occupation measures towards an explicit mixture of the maximizers, which will be carried out elsewhere. Our methods rely on deriving Hölder-continuity of the Coulomb functional of the occupation measure with exponentially small deviation probabilities and invoking the large-deviation theory developed in [MV14] to a certain shift-invariant functional of the occupation measures.

• V. Gayrard, O. Gün, Aging in the GREM-like trap model, Markov Processes and Related Fields, 22 (2016), pp. 165--202.
Abstract
The GREM-like trap model is a continuous time Markov jump process on the leaves of a finite volume L-level tree whose transition rates depend on a trapping landscape built on the vertices of the whole tree. We prove that the natural two-time correlation function of the dynamics ages in the infinite volume limit and identify the limiting function. Moreover, we take the limit L→ ∞ of the two-time correlation function of the infinite volume L-level tree. The aging behavior of the dynamics is characterized by a collection of clock processes, one for each level of the tree. We show that for any L, the joint law of the clock processes converges. Furthermore, any such limit can be expressed through Neveu's continuous state branching process. Hence, the latter contains all the information needed to describe aging in the GREM-like trap model both for finite and infinite levels.

• A. Mielke, M.A. Peletier, D.R.M. Renger, A generalization of Onsager's reciprocity relations to gradient flows with nonlinear mobility, Journal of Non-Equilibrium Thermodynamics, 41 (2016), pp. 141--149.
Abstract
Onsager's 1931 `reciprocity relations' result connects microscopic time-reversibility with a symmetry property of corresponding macroscopic evolution equations. Among the many consequences is a variational characterization of the macroscopic evolution equation as a gradient-flow, steepest-ascent, or maximal-entropy-production equation. Onsager's original theorem is limited to close-to-equilibrium situations, with a Gaussian invariant measure and a linear macroscopic evolution. In this paper we generalize this result beyond these limitations, and show how the microscopic time-reversibility leads to natural generalized symmetry conditions, which take the form of generalized gradient flows.

• TH. Koprucki, N. Rotundo, P. Farrell, D.H. Doan, J. Fuhrmann, On thermodynamic consistency of a Scharfetter--Gummel scheme based on a modified thermal voltage for drift-diffusion equations with diffusion enhancement, Optical and Quantum Electronics, 47 (2015), pp. 1327--1332.
Abstract
Driven by applications like organic semiconductors there is an increased interest in numerical simulations based on drift-diffusion models with arbitrary statistical distribution functions. This requires numerical schemes that preserve qualitative properties of the solutions, such as positivity of densities, dissipativity and consistency with thermodynamic equilibrium. An extension of the Scharfetter-Gummel scheme guaranteeing consistency with thermodynamic equilibrium is studied. It is derived by replacing the thermal voltage with an averaged diffusion enhancement for which we provide a new explicit formula. This approach avoids solving the costly local nonlinear equations defining the current for generalized Scharfetter-Gummel schemes.

• M. Erbar, J. Maas, D.R.M. Renger, From large deviations to Wasserstein gradient flows in multiple dimensions, Electronic Communications in Probability, 20 (2015), pp. 1--12.
Abstract
We study the large deviation rate functional for the empirical distribution of independent Brownian particles with drift. In one dimension, it has been shown by Adams, Dirr, Peletier and Zimmer [ADPZ11] that this functional is asymptotically equivalent (in the sense of Gamma-convergence) to the Jordan-Kinderlehrer-Otto functional arising in the Wasserstein gradient flow structure of the Fokker-Planck equation. In higher dimensions, part of this statement (the lower bound) has been recently proved by Duong, Laschos and Renger, but the upper bound remained open, since the proof in [DLR13] relies on regularity properties of optimal transport maps that are restricted to one dimension. In this note we present a new proof of the upper bound, thereby generalising the result of [ADPZ11] to arbitrary dimensions.

• O. Gün, W. König, O. Sekulović, Moment asymptotics for multitype branching random walks in random environment, Journal of Theoretical Probability, 28 (2015), pp. 1726--1742.
Abstract
We study a discrete time multitype branching random walk on a finite space with finite set of types. Particles follow a Markov chain on the spatial space whereas offspring distributions are given by a random field that is fixed throughout the evolution of the particles. Our main interest lies in the averaged (annealed) expectation of the population size, and its long-time asymptotics. We first derive, for fixed time, a formula for the expected population size with fixed offspring distributions, which is reminiscent of a Feynman-Kac formula. We choose Weibull-type distributions with parameter 1/ρij for the upper tail of the mean number of j type particles produced by an i type particle. We derive the first two terms of the long-time asymptotics, which are written as two coupled variational formulas, and interpret them in terms of the typical behavior of the system.

• W. König, T. Wolff, Large deviations for the local times of a random walk among random conductances in a growing box, Special issue for Pastur's 75th birthday, Markov Processes and Related Fields, 21 (2015), pp. 591--638.
Abstract
We derive an annealed large deviation principle (LDP) for the normalised and rescaled local times of a continuous-time random walk among random conductances (RWRC) in a time-dependent, growing box in Zd. We work in the interesting case that the conductances are positive, but may assume arbitrarily small values. Thus, the underlying picture of the principle is a joint strategy of small conductance values and large holding times of the walk. The speed and the rate function of our principle are explicit in terms of the lower tails of the conductance distribution as well as the time-dependent size of the box.
An interesting phase transition occurs if the thickness parameter of the conductance tails exceeds a certain threshold: for thicker tails, the random walk spreads out over the entire growing box, for thinner tails it stays confined to some bounded region. In fact, in the first case, the rate function turns out to be equal to the p-th power of the p-norm of the gradient of the square root for some 2d/(d+2) < p < 2. This extends the Donsker-Varadhan-Gärtner rate function for the local times of Brownian motion (with deterministic environment) from p=2 to these values.
As corollaries of our LDP, we derive the logarithmic asymptotics of the non-exit probability of the RWRC from the growing box, and the Lifshitz tails of the generator of the RWRC, the randomised Laplace operator. To contrast with the annealed, not uniformly elliptic case, we also provide an LDP in the quenched setting for conductances that are bounded and bounded away from zero. The main tool here is a spectral homogenisation result, based on a quenched invariance principle for the RWRC.

• A. Mielke, M.A. Peletier, D.R.M. Renger, On the relation between gradient flows and the large-deviation principle, with applications to Markov chains and diffusion, Potential Analysis, 41 (2014), pp. 1293--1325.
Abstract
Motivated by the occurence in rate functions of time-dependent large-deviation principles, we study a class of non-negative functions ℒ that induce a flow, given by ℒ(zt,żt)=0. We derive necessary and sufficient conditions for the unique existence of a generalized gradient structure for the induced flow, as well as explicit formulas for the corresponding driving entropy and dissipation functional. In particular, we show how these conditions can be given a probabilistic interpretation when ℒ is associated to the large deviations of a microscopic particle system. Finally, we illustrate the theory for independent Brownian particles with drift, which leads to the entropy-Wasserstein gradient structure, and for independent Markovian particles on a finite state space, which leads to a previously unknown gradient structure.

• M.H. Duong, V. Laschos, M. Renger, Wasserstein gradient flows from large deviations of many-particle limits, ESAIM. Control, Optimisation and Calculus of Variations, 19 (2013), pp. 1166--1188.

• M.A. Peletier, M. Renger, M. Veneroni, Variational formulation of the Fokker--Planck equation with decay: A particle approach, Communications in Contemporary Mathematics, 15 (2013), pp. 1350017/1--1350017/43.

• S. Adams, A. Collevecchio, W. König, A variational formula for the free energy of an interacting many-particle system, The Annals of Probability, 39 (2011), pp. 683--728.
Abstract
We consider \$N\$ bosons in a box in \$R^d\$ with volume \$N/rho\$ under the influence of a mutually repellent pair potential. The particle density \$rhoin(0,infty)\$ is kept fixed. Our main result is the identification of the limiting free energy, \$f(beta,rho)\$, at positive temperature \$1/beta\$, in terms of an explicit variational formula, for any fixed \$rho\$ if \$beta\$ is sufficiently small, and for any fixed \$beta\$ if \$rho\$ is sufficiently small. The thermodynamic equilibrium is described by the symmetrised trace of \$rm e^-beta Hcal_N\$, where \$Hcal_N\$ denotes the corresponding Hamilton operator. The well-known Feynman-Kac formula reformulates this trace in terms of \$N\$ interacting Brownian bridges. Due to the symmetrisation, the bridges are organised in an ensemble of cycles of various lengths. The novelty of our approach is a description in terms of a marked Poisson point process whose marks are the cycles. This allows for an asymptotic analysis of the system via a large-deviations analysis of the stationary empirical field. The resulting variational formula ranges over random shift-invariant marked point fields and optimizes the sum of the interaction and the relative entropy with respect to the reference process. In our proof of the lower bound for the free energy, we drop all interaction involving lq infinitely longrq cycles, and their possible presence is signalled by a loss of mass of the lq finitely longrq cycles in the variational formula. In the proof of the upper bound, we only keep the mass on the lq finitely longrq cycles. We expect that the precise relationship between these two bounds lies at the heart of Bose-Einstein condensation and intend to analyse it further in future.

• W. König, P. Schmid, Brownian motion in a truncated Weyl chamber, Markov Processes and Related Fields, 17 (2011), pp. 499--522.
Abstract
We examine the non-exit probability of a multidimensional Brownian motion from a growing truncated Weyl chamber. Different regimes are identified according to the growth speed, ranging from polynomial decay over stretched-exponential to exponential decay. Furthermore we derive associated large deviation principles for the empirical measure of the properly rescaled and transformed Brownian motion as the dimension grows to infinity. Our main tool is an explicit eigenvalue expansion for the transition probabilities before exiting the truncated Weyl chamber.

• W. König, P. Schmid, Random walks conditioned to stay in Weyl chambers of type C and D, Electronic Communications in Probability, (2010), pp. 286--295.

• G. Grüninger, W. König, Potential confinement property in the parabolic Anderson model, Annales de l'Institut Henri Poincare. Probabilites et Statistiques, 45 (2009), pp. 840--863.

• W. König, H. Lacoin, P. Mörters, N. Sidorova, A two cities theorem for the parabolic Anderson model, The Annals of Probability, 37 (2009), pp. 347--392.

### Preprints, Reports, Technical Reports

• A. Mielke, A. Stephan, Coarse-graining via EDP-convergence for linear fast-slow reaction systems, Preprint no. 2643, WIAS, Berlin, 2019, DOI 10.20347/WIAS.PREPRINT.2643 .
Abstract, PDF (426 kByte)
We consider linear reaction systems with slow and fast reactions, which can be interpreted as master equations or Kolmogorov forward equations for Markov processes on a finite state space. We investigate their limit behavior if the fast reaction rates tend to infinity, which leads to a coarse-grained model where the fast reactions create microscopically equilibrated clusters, while the exchange mass between the clusters occurs on the slow time scale. Assuming detailed balance the reaction system can be written as a gradient flow with respect to the relative entropy. Focusing on the physically relevant cosh-type gradient structure we show how an effective limit gradient structure can be rigorously derived and that the coarse-grained equation again has a cosh-type gradient structure. We obtain the strongest version of convergence in the sense of the Energy-Dissipation Principle (EDP), namely EDP-convergence with tilting.

• D.R.M. Renger, Gradient and Generic systems in the space of fluxes, applied to reacting particle systems, Preprint no. 2516, WIAS, Berlin, 2018, DOI 10.20347/WIAS.PREPRINT.2516 .
Abstract, PDF (392 kByte)
In a previous work we devised a framework to derive generalised gradient systems for an evolution equation from the large deviations of an underlying microscopic system, in the spirit of the Onsager-Machlup relations. Of particular interest is the case where the microscopic system consists of random particles, and the macroscopic quantity is the empirical measure or concentration. In this work we take the particle flux as the macroscopic quantity, which is related to the concentration via a continuity equation. By a similar argument the large deviations can induce a generalised gradient or Generic system in the space of fluxes. In a general setting we study how flux gradient or generic systems are related to gradient systems of concentrations. The arguments are explained by the example of reacting particle systems, which is later expanded to include spatial diffusion as well.

• D. Belomestny, J.G.M. Schoenmakers, Regression on particle systems connected to mean-field SDEs with applications, Preprint no. 2464, WIAS, Berlin, 2017, DOI 10.20347/WIAS.PREPRINT.2464 .
Abstract, PDF (268 kByte)
In this note we consider the problem of using regression on interacting particles to compute conditional expectations for McKean-Vlasov SDEs. We prove general result on convergence of linear regression algorithms and establish the corresponding rates of convergence. Application to optimal stopping and variance reduction are considered.

• B. Jahnel, Ch. Külske, Gibbsian representation for point processes via hyperedge potentials, Preprint no. 2414, WIAS, Berlin, 2017, DOI 10.20347/WIAS.PREPRINT.2414 .
Abstract, PDF (288 kByte)
We consider marked point processes on the d-dimensional euclidean space, defined in terms of a quasilocal specification based on marked Poisson point processes. We investigate the possibility of constructing uniformly absolutely convergent Hamiltonians in terms of hyperedge potentials in the sense of Georgii . These potentials are natural generalizations of physical multibody potentials which are useful in models of stochastic geometry.

### Vorträge, Poster

• A. Stephan, EDP-convergence for linear reaction diffusion systems with different time scales, Calculus of Variations on Schiermonnikoog 2019, July 1 - 5, 2019, Utrecht University, Schiermonnikoog, Netherlands, July 2, 2019.

• A. Stephan, EDP-convergence for linear reaction-diffusion systems with different time scales, Winter School ``Gradient Flows and Variational Methods in PDEs'', November 25 - 29, 2019, Universität Ulm, November 29, 2019.

• A. Stephan, Evolutionary Γ-convergence for a linear reaction-diffusion system with different time scales, COPDESC-Workshop ``Calculus of Variation and Nonlinear Partial Differential Equations", March 25 - 28, 2019, Universität Regensburg, March 26, 2019.

• A. Stephan, Evolutionary Γ-convergence for a linear reaction-diffusion system with different time scales, 9th International Congress on Industrial and Applied Mathematics (ICIAM 2019), July 15 - 19, 2019, Universitat de València, Spain, July 16, 2019.

• K. Hopf, On the singularity formation and relaxation to equilibrium in 1D Fokker--Planck model with superlinear drift, Winter School ``Gradient Flows and Variational Methods in PDEs'', November 25 - 29, 2019, Universität Ulm, November 25, 2019.

• K. Hopf, On the singularity formation and relaxation to equilibrium in 1D Fokker--Planck model with superlinear drift, Gradient Flows and Variational Methods in PDEs, November 25 - 29, 2019, Universität Ulm, November 25, 2019.

• D.R.M. Renger, Gradient and GENERIC structures from flux large deviations, POLYPHYS Seminar, Eidgenössische Technische Hochschule Zürich, Department of Materials, Zürich, Switzerland, March 28, 2018.

• D.R.M. Renger, Gradient and GENERIC structures in the space of fluxes, Analysis of Evolutionary and Complex Systems (ALEX2018), September 24 - 28, 2018, WIAS Berlin, September 27, 2018.

• D.R.M. Renger, Gradient and Generic structures in the space of fluxes, Analysis of Evolutionary and Complex Systems (ALEX2018), September 24 - 28, 2018, WIAS Berlin, September 27, 2018.

• A. Mielke, Construction of effective gradient systems via EDP convergence, Workshop on Mathematical Aspects of Non-Equilibrium Thermodynamics, March 5 - 7, 2018, Rheinisch-Westfälische Technische Hochschule Aachen, March 6, 2018.

• A. Mielke, EDP convergence and optimal transport, Workshop ``Optimal Transportation and Applications'', November 12 - 15, 2018, Scuola Normale Superiore, Università di Pisa, Università di Pavia, Pisa, Italy, November 13, 2018.

• F. Flegel, Spectral localization vs. homogenization in the random conductance model, 19th ÖMG Congress and Annual DMV Meeting, Minisymposium M6 ``Spectral and Scattering Problems in Mathematical Physics'', September 11 - 15, 2017, Austrian Mathematical Society (ÖMG) and Deutsche Mathematiker-Vereinigung (DMV), Paris-Lodron University of Salzburg, Austria, September 12, 2017.

• F. Flegel, Spectral localization vs. homogenization in the random conductance model, Berlin-Leipzig Workshop in Analysis and Stochastics, November 29 - December 1, 2017, Max-Planck-Institut für Mathematik in den Naturwissenschaften, Leipzig, November 29, 2017.

• A. Mielke, Perspectives for gradient flows, GAMM-Workshop on Analysis of Partial Differential Equations, September 27 - 29, 2017, Eindhoven University of Technology, Mathematics and Computer Science Department, Netherlands, September 28, 2017.

• J.G.M. Schoenmakers, Projected particle methods for solving McKean--Vlasov SDEs, Dynstoch 2017, April 5 - 7, 2017, Universität Siegen, Department Mathematik, April 6, 2017.

• M. Maurelli, Enhanced Sanov theorem and large deviations for interacting particles, Workshop ``Rough Paths, Regularity Structures and Related Topics'', May 1 - 7, 2016, Mathematisches Forschungsinstitut Oberwolfach, May 5, 2016.

• A. Mielke, Exponential decay into thermodynamical equilibrium for reaction-diffusion systems with detailed balance, Workshop ``Patterns of Dynamics'', July 25 - 29, 2016, Freie Universität Berlin, Fachbereich Mathematik und Informatik, July 28, 2016.

• A. Mielke, Gradient structures and dissipation distances for reaction-diffusion equation, Mathematisches Kolloquium, Westfälische Wilhelms-Universität, Institut für Mathematik, Münster, April 28, 2016.

• B. Jahnel, Classes of nonergodic interacting particle systems with unique invariant measure, Kyoto University, Research Institute for Mathematical Sciences, Kyoto, Japan, November 16, 2015.

• P. Keeler, When do wireless network signals appear Poisson?, 18th Workshop on Stochastic Geometry, Stereology and Image Analysis, March 22 - 27, 2015, Universität Osnabrück, Lingen, March 24, 2015.

• M. Maurelli, A large deviation principle for interacting particle SDEs via rough paths, 38th Conference on Stochastic Processes and their Applications, July 13 - 17, 2015, University of Oxford, Oxford-Man Institute of Quantitative Finance, UK, July 14, 2015.

• M. Maurelli, Enhanced Sanov theorem for Brownian rough paths and an application to interacting particles, Seminar Stochastic Analysis, Imperial College London, UK, October 20, 2015.

• M. Maurelli, Stochastic 2D Euler equations: A poorly correlated multiplicative noise regularizes the two-point motion, Universität Augsburg, Institut für Mathematik, March 24, 2015.

• D.R.M. Renger, From large deviations to Wasserstein gradient flows in multiple dimensions, Workshop on Gradient Flows, Large Deviations and Applications, November 22 - 29, 2015, EURANDOM, Mathematics and Computer Science Department, Eindhoven, Netherlands, November 23, 2015.

• D.R.M. Renger, The inverse problem: From gradient flows to large deviations, Workshop ``Analytic Approaches to Scaling Limits for Random System'', January 26 - 30, 2015, Universität Bonn, Hausdorff Research Institute for Mathematics, January 26, 2015.

• A. Mielke, The Chemical Master Equation as a discretization of the Fokker--Planck and Liouville equation for chemical reactions, Colloquium of Collaborative Research Center/Transregio ``Discretization in Geometry and Dynamics'', Technische Universität Berlin, Institut für Mathematik, Berlin, February 10, 2015.

• D.R.M. Renger, Connecting particle systems to entropy-driven gradient flows, Conference on Nonlinearity, Transport, Physics, and Patterns, October 6 - 10, 2014, Fields Institute for Research in Mathematical Sciences, Toronto, Canada, October 9, 2014.

• D.R.M. Renger, Connecting particle systems to entropy-driven gradient flows, Oberseminar ``Stochastische und Geometrische Analysis'', Universität Bonn, Institut für Angewandte Mathematik, May 28, 2014.

• H. Mai, Pathwise stability of likelihood estimators for diffusions via rough paths, International Workshop ``Advances in Optimization and Statistics'', May 15 - 16, 2014, Russian Academy of Sciences, Institute of Information Transmission Problems (Kharkevich Institute), Moscow, May 16, 2014.

• H. Mai, Robust drift estimation: Pathwise stability under volatility and noise misspecification, Berlin-Oxford Young Researchers Meeting on Applied Stochastic Analysis, July 1 - 2, 2014, University of Oxford, Oxford-Man Institute of Quantitative Finance, UK, July 2, 2014.

• S. Neukamm, Optimal decay estimate on the semigroup associated with a random walk among random conductances, Dirichlet Forms and Applications, German-Japanese Meeting on Stochastic Analysis, September 9 - 13, 2013, Universität Leipzig, Mathematisches Institut, September 9, 2013.

• A. Mielke, Gradient structures and dissipation distances for reaction-diffusion systems, Workshop ``Material Theory'', December 16 - 20, 2013, Mathematisches Forschungsinstitut Oberwolfach, December 17, 2013.

• A. Mielke, On the geometry of reaction-diffusion systems: Optimal transport versus reaction, Recent Trends in Differential Equations: Analysis and Discretisation Methods, November 7 - 9, 2013, Technische Universität Berlin, Institut für Mathematik, November 9, 2013.

• B. Metzger, The parabolic Anderson model: The asymptotics of the statistical moments and Lifshitz tails revisited, EURANDOM, Eindhoven, Netherlands, December 1, 2010.

• W. König, Die Universalitätsklassen im parabolischen Anderson-Modell, Mathematisches Kolloquium, Technische Universität Darmstadt, Fachbereich Mathematik, July 7, 2010.

• W. König, Ordered random walks, Augsburger Mathematisches Kolloquium, Universität Augsburg, Institut für Mathematik, January 26, 2010.

• W. König, Ordered random walks, Mathematisches Kolloquium der Universität Trier, Fachbereich Mathematik, April 29, 2010.

• W. König, The parabolic Anderson model, XIV Escola Brasileira de Probabilidade, August 2 - 7, 2010, Instituto Nacional de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, Brazil.

### Preprints im Fremdverlag

• J.-D. Deuschel, P. Friz, M. Maurelli, M. Slowik, The enhanced Sanov theorem and propagation of chaos, Preprint no. arxiv:1602.08043, Cornell University Library, arXiv.org, 2016. ## Ansprechpartner ## Beteiligte Gruppen des WIAS ## Mathematischer Kontext ## Projekte/Drittmittel 