Publications
Monographs

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

W. König, J. Sprekels, eds., Karl Weierstraß (18151897): Aspekte seines Lebens und Werkes  Aspects of his Life and Work, Mathematik  Analysis, Springer Spektrum, Wiesbaden, 2016, xv+289 pages, (Collection Published).
Abstract
Der Berliner Mathematiker Karl Weierstraß (18151897) lieferte grundlegende Beiträge zu den mathematischen Fachgebieten der Funktionentheorie, Algebra und Variationsrechnung. Er gilt weltweit als Begründer der mathematisch strengen Beweisführung in der Analysis. Mit seinem Namen verbunden ist zum Beispiel die berühmte EpsilonDeltaDefinition des Begriffs der Stetigkeit reeller Funktionen. Weierstraß? Vorlesungszyklus zur Analysis in Berlin wurde weithin gerühmt und er lehrte teilweise vor 250 Hörern aus ganz Europa; diese starke mathematische Schule prägt bis heute die Mathematik. Aus Anlass seines 200. Geburtstags am 31. Oktober 2015 haben internationale Experten der Mathematik und Mathematikgeschichte diesen Festband zusammengestellt, der einen Einblick in die Bedeutung von Weierstraß? Werk bis zur heutigen Zeit gibt. Die Herausgeber des Buches sind leitende Wissenschaftler am WeierstraßInstitut für Angewandte Analysis und Stochastik in Berlin, die Autoren eminente Mathematikhistoriker. 
W. König, ed., Mathematics and Society, European Mathematical Society Publishing House, Zurich, 2016, 314 pages, (Collection Published).
Abstract
The ubiquity and importance of mathematics in our complex society is generally not in doubt. However, even a scientifically interested layman would be hard pressed to point out aspects of our society where contemporary mathematical research is essential. Most popular examples are f inance, engineering, wheather and industry, but the way mathematics comes into play is widely unknown in the public. And who thinks of application fields like biology, encryption, architecture, or voting systems? This volume comprises a number of success stories of mathematics in our society ? important areas being shaped by cutting edge mathematical research. The authors are eminent mathematicians with a high sense for public presentation, addressing scientifically interested laymen as well as professionals in mathematics and its application disciplines.
Articles in Refereed Journals

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. 135, 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 nonquadratic 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 largedeviation principles. While Markovian diffusion processes lead to classical gradient structures, Poissonian jump processes give rise to coshtype dissipation potentials. A second origin arises via a new form of convergence, that we call EDPconvergence. 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 Gammalimit. 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 reactiondiffusion system. 
E. Bolthausen, W. König, Ch. Mukherjee, Meanfield interaction of Brownian occupation measures. II: A rigorous construction of the Pekar process, Communications on Pure and Applied Mathematics, 70 (2017) pp. 15981629.
Abstract
We consider meanfield interactions corresponding to Gibbs measures on interacting Brownian paths in three dimensions. The interaction is selfattractive 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 meanfield measures under interest, whose rigorous analysis remained open. The present paper is devoted to a precise analysis of these meanfield 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 ``meanfield 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] 
K.F. Lee, M. Dosta, A. caps">Mc Guire, S. Mosbach, W. Wagner, S. Heinrich, M. Kraft, Development of a multicompartment population balance model for highshear wet granulation with discrete element method, Comput. Chem. Engng., 99 (2017) pp. 171184.

O. Gün, A. Yilmaz, The stochastic encountermating model, Acta Applicandae Mathematicae. An International Survey Journal on Applying Mathematics and Mathematical Applications, 148 (2017) pp. 71102.

W. König, (Book review:) Firas RassoulAgha and Timo Seppäläinen: A Course on Large Deviations with an Introduction to Gibbs Measures, Jahresbericht der Deutschen MathematikerVereinigung, 119 (2017) pp. 6367.

A. González Casanova Soberón, N. Kurt, A. Wakolbinger, L. Yuan, An individualbased mathematical model for the Lenski experiment, and the deceleration of the relative fitness, Stochastic Processes and their Applications, 126 (2016) pp. 22112252.

CH. Mukherjee, A. Shamov, O. Zeitouni, Weak and strong disorder for the stochastic heat equation and continuous directed polymers in $dgeq 3$, Electronic Communications in Probability, 21 (2016) pp. 112.

CH. Mukherjee, S.R.S. Varadhan, Brownian occupation measures, compactness and large deviations, The Annals of Probability, 44 (2016) pp. 39343964.
Abstract
In proving large deviation estimates, the lower bound for open sets and upper bound for compact sets are essentially local estimates. On the other hand, the upper bound for closed sets is global and compactness of space or an exponential tightness estimate is needed to establish it. In dealing with the occupation measure $L_t(A)=frac1tint_0^t1_A(W_s) d s$ of the $d$ dimensional Brownian motion, which is not positive recurrent, there is no possibility of exponential tightness. The space of probability distributions $mathcal M_1(R^d)$ can be compactified by replacing the usual topology of weak c onvergence by the vague toplogy, where the space is treated as the dual of continuous functions with compact support. This is essentially the one point compactification of $R^d$ by adding a point at $infty$ that results in the compactification of $mathcal M_1(R^d)$ by allowing some mass to escape to the point at $infty$. If one were to use only test functions that are continuous and vanish at $infty$ then the compactification results in the space of subprobability distributions $mathcal M_le 1(R^d)$ by ignoring the mass at $infty$. The main drawback of this compactification is that it ignores the underlying translation invariance. More explicitly, we may be interested in the space of equivalence classes of orbits $widetildemathcal M_1=widetildemathcal M_1(R^d)$ under the action of the translation group $R^d$ on $mathcal M_1(R^d)$. There are problems for which it is natural to compactify this space of orbits. We will provide such a compactification, prove a large deviation principle there and give an application to a relevant problem. 
H. Döring, G. Faraud, W. König, Connection times in large adhoc mobile networks, Bernoulli. Official Journal of the Bernoulli Society for Mathematical Statistics and Probability, 22 (2016) pp. 21432176.
Abstract
We study connectivity properties in a probabilistic model for a large mobile adhoc network. We consider a large number of participants of the system moving randomly, independently and identically distributed in a large domain, with a spacedependent population density of finite, positive order and with a fixed time horizon. Messages are instantly transmitted according to a relay principle, i.e., they are iteratedly forwarded from participant to participant over distances $leq 2R$, with $2R$ the communication radius, until they reach the recipient. In mathematical terms, this is a dynamic continuum percolation model. We consider the connection time of two sample participants, the amount of time over which these two are connected with each other. In the above thermodynamic limit, we find that the connectivity induced by the system can be described in terms of the counterplay of a local, random, and a global, deterministic mechanism, and we give a formula for the limiting behaviour. A prime example of the movement schemes that we consider is the wellknown random waypoint model (RWP). Here we describe the decay rate, in the limit of large time horizons, of the probability that the portion of the connection time is less than the expectation. 
V. Gayrard, O. Gün, Aging in the GREMlike trap model, Markov Processes and Related Fields, 22 (2016) pp. 165202.
Abstract
The GREMlike trap model is a continuous time Markov jump process on the leaves of a finite volume Llevel tree whose transition rates depend on a trapping landscape built on the vertices of the whole tree. We prove that the natural twotime correlation function of the dynamics ages in the infinite volume limit and identify the limiting function. Moreover, we take the limit L→ ∞ of the twotime correlation function of the infinite volume Llevel 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 GREMlike trap model both for finite and infinite levels. 
M. Biskup, R. Fukushima, W. König, Eigenvalue fluctuations for lattice Anderson Hamiltonians, SIAM Journal on Mathematical Analysis, 48 (2016) pp. 26742700.
Abstract
We consider the random Schrödinger operator on a large box in the lattice with a large prefactor in front of the Laplacian part of the operator, which is proportional to the square of the diameter of the box. The random potential is assumed to be independent and bounded; its expectation function and variance function is given in terms of continuous bounded functions on the rescaled box. Our main result is a multivariate central limit theorem for all the simple eigenvalues of this operator, after centering and rescaling. The limiting covariances are expressed in terms of the limiting homogenized eigenvalue problem; more precisely, they are equal to the integral of the product of the squares of the eigenfunctions of that problem times the variance function. 
M. Biskup, W. König, Eigenvalue order statistics for random Schrödinger operators with doublyexponential tails, Communications in Mathematical Physics, 341 (2016) pp. 179218.

J. Blath, A. González Casanova Soberón, N. Kurt, M. WilkeBerenguer, A new coalescent for seedbank models, The Annals of Applied Probability, 26 (2016) pp. 857891.

E. Bouchet , Ch. Sabot, R. Soares Dos Santos, A quenched functional central limit theorem for random walks in random environments under (T)_gamma, Stochastic Processes and their Applications, 126 (2016) pp. 12061225.

A. Chiarini, A. Cipriani, R.S. Hazra, Extremes of some Gaussian random interfaces, Journal of Statistical Physics, 165 (2016) pp. 521544.
Abstract
In this article we give a general criterion for some dependent Gaussian models to belong to maximal domain of attraction of Gumbel, following an application of the SteinChen method studied in citeAGG. We also show the convergence of the associated point process. As an application, we show the conditions are satisfied by some of the wellknown supercritical Gaussian interface models, namely, membrane model, massive and massless discrete Gaussian free field, fractional Gaussian free field. 
A. Chiarini, A. Cipriani, R.S. Hazra, Extremes of the supercritical Gaussian free field, ALEA. Latin American Journal of Probability and Mathematical Statistics, 13 (2016) pp. 711724.
Abstract
We show that the rescaled maximum of the discrete Gaussian Free Field (DGFF) in dimension larger or equal to 3 is in the maximal domain of attraction of the Gumbel distribution. The result holds both for the infinitevolume field as well as the field with zero boundary conditions. We show that these results follow from an interesting application of the SteinChen method from Arratia et al. (1989). 
T. Orenshtein, R. Soares Dos Santos, Zeroone law for directional transience of onedimensional random walks in dynamic random environments, Electronic Communications in Probability, 21 (2016) pp. 15/115/11.
Abstract
We prove the trichotomy between transience to the right, transience to the left and recurrence of onedimensional nearestneighbour random walks in dynamic random environments under fairly general assumptions, namely: stationarity under spacetime translations, ergodicity under spatial translations, and a mild ellipticity condition. In particular, the result applies to general uniformly elliptic models and also to a large class of nonuniformly elliptic cases that are i.i.d. in space and Markovian in time. 
E.K.Y. Yapp, R.I.A. Patterson, J. Akroyd, S. Mosbach, E.M. Adkins, J.H. Miller, M. Kraft, Numerical simulation and parametric sensitivity study of optical band gap in a laminar coflow ethylene diffusion flame, Combustion and Flame, 167 (2016) pp. 320334.

A. Cipriani, A. Feidt, Rates of convergence for extremes of geometric random variables and marked point processes, Extremes. Statistical Theory and Applications in Science, Engineering and Economics, 19 (2016) pp. 105138.
Abstract
We use the SteinChen method to study the extremal behaviour of univariate and bivariate geometric laws. We obtain a rate for the convergence, to the Gumbel distribution, of the law of the maximum of i.i.d. geometric random variables, and show that convergence is faster when approximating by a discretised Gumbel. We similarly find a rate of convergence for the law of maxima of bivariate MarshallOlkin geometric random pairs when approximating by a discrete limit law. We introduce marked point processes of exceedances (MPPEs), both with univariate and bivariate MarshallOlkin geometric variables as marks and determine bounds on the error of the approximation, in an appropriate probability metric, of the law of the MPPE by that of a Poisson process with same mean measure. We then approximate by another Poisson process with an easiertouse mean measure and estimate the error of this additional approximation. This work contains and extends results contained in the second author's PhD thesis under the supervision of Andrew D. Barbour. The thesis is available at http://arxiv.org/abs/1310.2564. 
A. Mielke, M.A. Peletier, D.R.M. Renger, A generalization of Onsager's reciprocity relations to gradient flows with nonlinear mobility, Journal of NonEquilibrium Thermodynamics, 41 (2016) pp. 141149.
Abstract
Onsager's 1931 `reciprocity relations' result connects microscopic timereversibility with a symmetry property of corresponding macroscopic evolution equations. Among the many consequences is a variational characterization of the macroscopic evolution equation as a gradientflow, steepestascent, or maximalentropyproduction equation. Onsager's original theorem is limited to closetoequilibrium 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 timereversibility leads to natural generalized symmetry conditions, which take the form of generalized gradient flows. 
R.I.A. Patterson, S. Simonella, W. Wagner, Kinetic theory of cluster dynamics, Physica D. Nonlinear Phenomena, 335 (2016) pp. 2632.
Abstract
In a Newtonian system with localized interactions the whole set of particles is naturally decomposed into dynamical clusters, dened as nite groups of particles having an in uence on each other's trajectory during a given interval of time. For an ideal gas with shortrange intermolecular force, we provide a description of the cluster size distribution in terms of the reduced Boltzmann density. In the simplied context of Maxwell molecules, we show that a macroscopic fraction of the gas forms a giant component in nite kinetic time. The critical index of this phase transition is in agreement with previous numerical results on the elastic billiard. 
R.I.A. Patterson, Properties of the solutions of delocalised coagulation and inception problems with outflow boundaries, Journal of Evolution Equations, 16 (2016) pp. 261291.
Abstract
Well posedness is established for a family of equations modelling particle populations undergoing delocalised coagulation, advection, inflow and outflow in a externally specified velocity field. Very general particle types are allowed while the spatial domain is a bounded region of $d$dimensional space for which every point lies on exactly one streamline associated with the velocity field. The problem is formulated as a semilinear ODE in the Banach space of bounded measures on particle position and type space. A local Lipschitz property is established in total variation norm for the propagators (generalised semigroups) associated with the problem and used to construct a Picard iteration that establishes local existence and global uniqueness for any initial condition. The unique weak solution is shown further to be a differentiable or at least bounded variation strong solution under smoothness assumptions on the parameters of the coagulation interaction. In the case of one spatial dimension strong differentiability is established even for coagulation parameters with a particular bounded variation structure in space. This one dimensional extension establishes the convergence of the simulation processes studied in [Patterson, Stoch. Anal. Appl. 31, 2013] to a unique and differentiable limit. 
W. Wagner, O. Muscato, A class of stochastic algorithms for the Wigner equation, SIAM Journal on Scientific Computing, 38 (2016) pp. A1483A1507.
Abstract
A class of stochastic algorithms for the numerical treatment of the Wigner equation is introduced. The algorithms are derived using the theory of pure jump processes with a general state space. The class contains several new algorithms as well as some of the algorithms previously considered in the literature. The approximation error and the efficiency of the algorithms are analyzed. Numerical experiments are performed in a benchmark test case, where certain advantages of the new class of algorithms are demonstrated. 
W. Wagner, A random cloud model for the Wigner equation, Kinetic and Related Models, 9 (2016) pp. 217235.
Abstract
A probabilistic model for the Wigner equation is studied. The model is based on a particle system with the time evolution of a piecewise deterministic Markov process. Each particle is characterized by a realvalued weight, a position and a wavevector. The particle position changes continuously, according to the velocity determined by the wavevector. New particles are created randomly and added to the system. The main result is that appropriate functionals of the process satisfy a weak form of the Wigner equation.
Preprints, Reports, Technical Reports

O. Muscato, W. Wagner, A stochastic algorithm without time discretization error for the Wigner equation, Preprint no. 2415, WIAS, Berlin, 2017, DOI 10.20347/WIAS.PREPRINT.2415 .
Abstract, PDF (400 kByte)
Stochastic particle methods for the numerical treatment of the Wigner equation are considered. The approximation properties of these methods depend on several numerical parameters. Such parameters are the number of particles, a time step (if transport and other processes are treated separately) and the grid size (used for the discretization of the position and the wavevector). A stochastic algorithm without time discretization error is introduced. Its derivation is based on the theory of piecewise deterministic Markov processes. Numerical experiments are performed in a onedimensional test case. Approximation properties with respect to the grid size and the number of particles are studied. Convergence of a timesplitting scheme to the nosplitting algorithm is demonstrated. The nosplitting algorithm is shown to be more efficient in terms of computational effort. 
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 ddimensional 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 [2]. These potentials are natural generalizations of physical multibody potentials which are useful in models of stochastic geometry. 
W. König, A. Tóbiás, A Gibbsian model for message routing in highly dense multihop network, Preprint no. 2392, WIAS, Berlin, 2017, DOI 10.20347/WIAS.PREPRINT.2392 .
Abstract, PDF (468 kByte)
We investigate a probabilistic model for routing in relayaugmented multihop adhoc communication networks, where each user sends one message to the base station. Given the (random) user locations, we weigh the family of random, uniformly distributed message trajectories by an exponential probability weight, favouring trajectories with low interference (measured in terms of signaltointerference ratio) and trajectory families with little congestion (measured by how many pairs of hops use the same relay). Under the resulting Gibbs measure, the system targets the best compromise between entropy, interference and congestion for a common welfare, instead of a selfish optimization. We describe the joint routing strategy in terms of the empirical measure of all message trajectories. In the limit of high spatial density of users, we derive the limiting free energy and analyze the optimal strategy, given as the minimizer(s) of a characteristic variational formula. Interestingly, expressing the congestion term requires introducing an additional empirical measure. 
A. González Casanova Soberón, D. Spanò, Duality and fixation in $Xi$WrightFisher processes with frequencydependent selection, Preprint no. 2390, WIAS, Berlin, 2017, DOI 10.20347/WIAS.PREPRINT.2390 .
Abstract, PDF (347 kByte)
A twotypes, discretetime population model with finite, constant size is constructed, allowing for a general form of frequencydependent selection and skewed offspring distribution. Selection is defined based on the idea that individuals first choose a (random) number of emphpotential parents from the previous generation and then, from the selected pool, they inherit the type of the fittest parent. The probability distribution function of the number of potential parents per individual thus parametrises entirely the selection mechanism. Using duality, weak convergence is then proved both for the allele frequency process of the selectively weak type and for the population's ancestral process. The scaling limits are, respectively, a twotypes ΞFlemingViot jumpdiffusion process with frequencydependent selection, and a branchingcoalescing process with general branching and simultaneous multiple collisions. Duality also leads to a characterisation of the probability of extinction of the selectively weak allele, in terms of the ancestral process' ergodic properties. 
D.R.M. Renger, Flux large deviations of independent and reacting particle systems, with implications for macroscopic fluctuation theory, Preprint no. 2375, WIAS, Berlin, 2017, DOI 10.20347/WIAS.PREPRINT.2375 .
Abstract, PDF (424 kByte)
We consider a system of independent particles on a finite state space, and prove a dynamic largedeviation principle for the empirical measureempirical flux pair, taking the specific fluxes rather than net fluxes into account. We prove the large deviations under deterministic initial conditions, and under random initial conditions satisfying a largedeviation principle. We then show how to use this result to generalise a number of principles from Macroscopic Fluctuation Theory to the finitespace setting. 
F. Flegel, M. Heida, M. Slowik, Homogenization theory for the random conductance model with degenerate ergodic weights and unboundedrange jumps, Preprint no. 2371, WIAS, Berlin, 2017, DOI 10.20347/WIAS.PREPRINT.2371 .
Abstract, PDF (598 kByte)
We study homogenization properties of the discrete Laplace operator with random conductances on a large domain in Zd. More precisely, we prove almostsure homogenization of the discrete Poisson equation and of the top of the Dirichlet spectrum. We assume that the conductances are stationary, ergodic and nearestneighbor 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 longrange 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 twoscale convergence 
R.I.A. Patterson, S. Simonella, W. Wagner, A kinetic equation for the distribution of interaction clusters in rarefied gases, Preprint no. 2365, WIAS, Berlin, 2017, DOI 10.20347/WIAS.PREPRINT.2365 .
Abstract, PDF (978 kByte)
We consider a stochastic particle model governed by an arbitrary binary interaction kernel. A kinetic equation for the distribution of interaction clusters is established. Under some additional assumptions a recursive representation of the solution is found. For particular choices of the interaction kernel (including the Boltzmann case) several explicit formulas are obtained. These formulas are confirmed by numerical experiments. The experiments are also used to illustrate various conjectures and open problems. 
N. Berger, Ch. Mukherjee, K. Okamura, Quenched large deviations for simple random walks on percolation clusters including longrange correlations, Preprint no. 2360, WIAS, Berlin, 2016, DOI 10.20347/WIAS.PREPRINT.2360 .
Abstract, PDF (395 kByte)
We prove a quenched large deviation principle (LDP)for a simple random walk on a supercritical percolation cluster (SRWPC) on the lattice.The models under interest include classical Bernoulli bond and site percolation as well as models that exhibit long range correlations, like the random cluster model, the random interlacement and its vacant set and the level sets of the Gaussian free field. Inspired by the methods developed by Kosygina, Rezakhanlou and Varadhan ([KRV06]) for proving quenched LDP for elliptic diffusions with a random drift, and by Yilmaz ([Y08]) and Rosenbluth ([R06]) for similar results regarding elliptic random walks in random environment, we take the point of view of the moving particle and prove a large deviation principle for the quenched distribution of the pair empirical measures if the environment Markov chain in the nonelliptic case of SRWPC. Via a contraction principle, this reduces easily to a quenched LDP for the distribution of the mean velocity of the random walk and both rate functions admit explicit variational formulas. The main approach of our proofs are based on exploiting coercivity properties of the relative entropy in the context of convex variational analysis, combined with input from ergodic theory and invoking geometric properties of the percolation cluster under supercriticality. 
M. Heida, R.I.A. Patterson, D.R.M. Renger, The space of bounded variation with infinitedimensional codomain, Preprint no. 2353, WIAS, Berlin, 2016, DOI 10.20347/WIAS.PREPRINT.2353 .
Abstract, PDF (600 kByte)
We study functions of bounded variation with values in a Banach or in a metric space. We provide several equivalent notions of variations and provide the notion of a time derivative in this abstract setting. We study four distinct topologies on the space of bounded variations and provide some insight into the structure of these topologies. In particular, we study the meaning of convergence, duality and regularity for these topologies and provide some useful compactness criteria, also related to the classical AubinLions theorem. We finally provide some useful applications to stochastic processes. 
A. Cipriani, R.S. Hazra, W.M. Ruszel, The divisible sandpile with heavytailed variables, Preprint no. 2328, WIAS, Berlin, 2016, DOI 10.20347/WIAS.PREPRINT.2328 .
Abstract, PDF (347 kByte)
This work deals with the divisible sandpile model when an initial configuration sampled from a heavytailed distribution. Extending results of Levine et al. (2015) and Cipriani et al. (2016) we determine sufficient conditions for stabilization and nonstabilization on infinite graphs. We determine furthermore that the scaling limit of the odometer on the torus is an αstable random distribution. 
L. Avena, O. Gün, M. Hesse, The parabolic Anderson model on the hypercube, Preprint no. 2319, WIAS, Berlin, 2016, DOI 10.20347/WIAS.PREPRINT.2319 .
Abstract, PDF (240 kByte)
We consider the parabolic Anderson model (PAM) on the ndimensional hypercube with random i.i.d. potentials. We parametrize time by volume and study the solution at the location of the kth largest potential. Our main result is that, for a certain class of potential distributions, the solution exhibits a phase transition: for short time scales it behaves like a system without diffusion, whereas, for long time scales the growth is dictated by the principle eigenvalue and the corresponding eigenfunction of the Anderson operator, for which we give precise asymptotics. Moreover, the transition time depends only on the difference between the largest and kth largest potential. One of our main motivations in this article is to investigate the mutationselection model of population genetics on a random fitness landscape, which is given by the ratio of the solution of PAM to its total mass, with the field corresponding to the fitness landscape. We show that the phase transition of the solution translates to the mutationselection model as follows: a population initially concentrated at the site of the kth best fitness value moves completely to the site of the best fitness on time scales where the transition of growth rates happens. The class of potentials we consider involve the Random Energy Model (REM) of statistical physics which is studied as one of the main examples of a random fitness landscape. 
E. Bolthausen, A. Cipriani, N. Kurt, Exponential decay of covariances for the supercritical membrane model, Preprint no. 2301, WIAS, Berlin, 2016, DOI 10.20347/WIAS.PREPRINT.2301 .
Abstract, PDF (291 kByte)
We consider the membrane model, that is the centered Gaussian field on Zd whose covariance matrix is given by the inverse of the discrete Bilaplacian. We impose a δpinning condition, giving a reward of strength ε for the field to be 0 at any site of the lattice. In this paper we prove that in dimensions d≥5 covariances of the pinned field decay at least exponentially, as opposed to the field without pinning, where the decay is polynomial. The proof is based on estimates for certain discrete weighted norms, a percolation argument and on a Bernoulli domination result. 
M. Biskup, W. König, R. Soares Dos Santos, Mass concentration and aging in the parabolic Anderson model with doublyexponential tails, Preprint no. 2295, WIAS, Berlin, 2016, DOI 10.20347/WIAS.PREPRINT.2295 .
Abstract, PDF (810 kByte)
We study the solutions to the Cauchy problem on the with random potential and localised initial data. Here we consider the random Schr?dinger operator, i.e., the Laplace operator with random field, whose upper tails are doubly exponentially distributed in our case. We prove that, for large times and with large probability, a majority of the total mass of the solution resides in a bounded neighborhood of a site that achieves an optimal compromise between the local Dirichlet eigenvalue of the Anderson Hamiltonian and the distance to the origin. The processes of mass concentration and the rescaled total mass are shown to converge in distribution under suitable scaling of space and time. Aging results are also established. The proof uses the characterization of eigenvalue order statistics for the random Schrödinger operator in large sets recently proved by the first two authors. 
F. Flegel, Localization of the principal Dirichlet eigenvector in the heavytailed random conductance model, Preprint no. 2290, WIAS, Berlin, 2016, DOI 10.20347/WIAS.PREPRINT.2290 .
Abstract, PDF (567 kByte)
We study the asymptotic behavior of the principal eigenvector and eigenvalue of the random conductance Laplacian in a large domain of Zd (d ≥ 2) with zero Dirichlet condition. We assume that the conductances w are positive i.i.d. random variables, which fulfill certain regularity assumptions near zero. If γ = sup q ≥ 0; E [w^q]<∞ <¼, then we show that for almost every environment the principal Dirichlet eigenvector asymptotically concentrates in a single site and the corresponding eigenvalue scales subdiffusively. The threshold γrm c = ¼ is sharp. Indeed, other recent results imply that for γ>¼ the top of the Dirichlet spectrum homogenizes. Our proofs are based on a spatial extreme value analysis of the local speed measure, BorelCantelli arguments, the RayleighRitz formula, results from percolation theory, and path arguments. 
A. Chiarini, A. Cipriani, A note on the Green's function for the transient random walk without killing on the half lattice, orthant and strip, Preprint no. 2289, WIAS, Berlin, 2016, DOI 10.20347/WIAS.PREPRINT.2289 .
Abstract, PDF (765 kByte)
In this note we derive an exact formula for the Green's function of the random walk on different subspaces of the discrete lattice (orthants, including the half space, and the strip) without killing on the boundary in terms of the Green's function of the simple random walk on $Z^d$, $dge 3$. 
R.I.A. Patterson, D.R.M. Renger, Dynamical large deviations of countable reaction networks under a weak reversibility condition, Preprint no. 2273, WIAS, Berlin, 2016, DOI 10.20347/WIAS.PREPRINT.2273 .
Abstract, PDF (343 kByte)
A dynamic large deviations principle for a countable reaction network including coagulationfragmentation models is proved. The rate function is represented as the infimal cost of the reaction fluxes and a minimiser for this variational problem is shown to exist. A weak reversibility condition is used to control the boundary behaviour and to guarantee a representation for the optimal fluxes via a Lagrange multiplier that can be used to construct the changes of measure used in standard tilting arguments. Reflecting the pure jump nature of the approximating processes, their paths are treated as elements of a BV function space. 
A. Cipriani, R.S. Hazra, W.M. Ruszel, Scaling limit of the odometer in divisible sandpiles, Preprint no. 2268, WIAS, Berlin, 2016, DOI 10.20347/WIAS.PREPRINT.2268 .
Abstract, PDF (377 kByte)
In a recent work [LMPU] prove that the odometer function of a divisible sandpile model on a finite graph can be expressed as a shifted discrete bilaplacian Gaussian field. For the discrete torus, they suggest the possibility that the scaling limit of the odometer may be related to the continuum bilaplacian field. In this work we show that in any dimension the rescaled odometer converges to the continuum bilaplacian field on the unit torus. 
E. Bolthausen, A. Cipriani, N. Kurt, Fast decay of covariances under deltapinning in the critical and supercritical membrane model, Preprint no. 2220, WIAS, Berlin, 2016, DOI 10.20347/WIAS.PREPRINT.2220 .
Abstract, PDF (271 kByte)
We consider the membrane model, that is the centered Gaussian field on Z^d whose covariance matrix is given by the inverse of the discrete Bilaplacian. We impose a deltapinning condition, giving a reward of strength epsilon for the field to be 0 at any site of the lattice. In this paper we prove that in dimensions larger than 4 covariances of the pinned field decay at least stretchedexponentially, as opposed to the field without pinning, where the decay is polynomial in dimensions larger than 5 and logarithmic in 4 dimensions. The proof is based on estimates for certain discrete Sobolev norms, and on a Bernoulli domination result.
Talks, Poster

A. González Casanova Soberón, Branching processes with interactions and their relation to population genetics, The 3rd Workshop on branching processes and related topics, May 8  12, 2017, Beijing Normal University, School of Mathematical Sciences, Beijing, China, May 8, 2017.

A. González Casanova Soberón, Modelling selection via multiple parents, Probability Seminar, University of Oxford, Mathematical Institute, UK, January 24, 2017.

A. González Casanova Soberón, Seminar Probability, National Autonomous University of Mexico, UNAM, Mexico city, Mexico, February 23, 2017.

CH. Mukherjee, Asymptotic behavior of the meanfield polaron, Probability and Mathematical Physics Seminar, Courant Institute of Mathematical Sciences, Department of Mathematics, New York, USA, March 20, 2017.

R.I.A. Patterson, Confidence Intervals for Coagulation? Advection Simulations, ClausthalGöttingenInternational Workshop ``Simulation Science'', April 27  28, 2017, GeorgAugustUniversität Göttingen, Institut für Informatik, April 28, 2017.

D.R.M. Renger, Banachvalued functions of bounded variation, Oberseminar Analysis, Universität Regensburg, Fakultät für Mathematik, July 28, 2017.

D.R.M. Renger, Was sind und was sollen die Zahlen, Tag der Mathematik, Universität Regensburg, Fakultät für Mathematik, July 28, 2017.

D.R.M. Renger, Large deviations and gradient flows, Spring School 2017: From Particle Dynamics to Gradient Flows, February 27  March 3, 2017, Technische Universität Kaiserslautern, Fachbereich Mathematik, Kaiserslautern, March 1, 2017.

R. Soares Dos Santos, Complete localisation in the BouchaudAnderson model, Leiden University, Institute of Mathematics, Leiden, Netherlands, May 10, 2017.

R. Soares Dos Santos, Concentration de masse dans le modèle parabolique d'Anderson, Séminaire de Probabilités,, April 10  12, 2017, Université de Grenoble, Institut Fourier, Laboratoire des Mathematiques, Grenoble, France, April 11, 2017.

R. Soares Dos Santos, Random walk on random walks, Mathematical Probability Seminar, Shanghai New York University, Shanghai, China, March 31, 2017.

R. Dos Santos, Mass concentration in the parabolic Anderson model, Université Claude Bernard Lyon 1, Institut Camille Jordan, Lyon, France, February 2, 2017.

W. van Zuijlen, Meanfield GibbsnonGibbs transitions, Mark Kac Seminar, Utrecht University, Mathematical Institute, Netherlands, February 3, 2017.

B. Jahnel, Stochastic geometry in telecommunications, Summer School 2017: Probabilistic and statistical methods for networks, August 21  September 1, 2017, Technische Universität Berlin, Berlin Mathematical School, Berlin.

W. König, A variational formula for an interacting manybody system, Probability Seminar, University of California, Los Angeles, Department of Mathematics, Los Angeles, USA, January 19, 2017.

W. König, Connectivity in large mobile adhoc networks, Summer School 2017: Probabilistic and statistical methods for networks, August 21  September 1, 2017, Technische Universität Berlin, Berlin Mathematical School, Berlin, August 29, 2017.

W. König, Moment asymptotics of branching random walks in random environment, Modern perspective of branching in probability, September 26  29, 2017, Westfälische WilhelmsUniversität Münster, Fachbereich Mathematik und Informatik, Münster.

W. König, The principal part of the spectrum of a random Schrödinger operator in a large box, Mathematisches Kolloquium der Technischen Universität Dormund, Oberseminar Stochastik und Analysis, May 15  16, 2017, Technische Universität Dormund, May 15, 2017.

A. Pandey, Meshfree method for fluctuating hydrodynamics, International Conference on Advances in Scientific Computing, November 28  30, 2016, Indian Institute of Technology, Department of Mathematics, Madras, November 30, 2016.

A. González Casanova Soberón, An individual based model for the Lenski experiment, 1st Leibniz MMS Days, January 27  29, 2016, WIAS Berlin, Berlin, January 27, 2016.

A. González Casanova Soberón, An individual based model for the Lenski experiment, 1st Leibniz MMS Days, WIAS Berlin, Berlin, January 27, 2016.

A. González Casanova Soberón, Fixation in a Xi coalescent model with selection, Probability seminar, University of Warwick, Mathematics Institute, Warwick, UK, November 30, 2016.

A. González Casanova Soberón, Fixation in a Xi coalescent with selection, Miniworkshop on Probabilistic Models in Evolutionary Biology, November 24  25, 2016, GeorgAugustUniversität Göttingen, Institut für Mathematische Stochastik, November 25, 2016.

A. González Casanova Soberón, Modeling the Lenski experiment, Mathematical and Computational Evolutionary Biology, June 12  16, 2016, Le Laboratoire d'Informatique, de Robotique et de Microélectronique de Montpellier (LIRMM), Hameau de l'Etoile, France, June 14, 2016.

A. González Casanova Soberón, The seed bank model, VIII School on Probability and Stochastic Processes, September 12  16, 2016, Centro de Investigación en Matemáticas (CIMAT), Department of Probability and Statistics, Guanajuato, Mexico.

A. González Casanova Soberón, The seedbank coalescent, World Congress in Probability and Statistics, Invited Session ``Stochastic Models of Evolution'', July 11  15, 2016, Fields Institute, Toronto, Canada, July 5, 2016.

CH. Mukherjee, Compactness and large deviations, Probability Seminar, Stanford University, Department of Mathematics and Statistics, USA, November 14, 2016.

CH. Mukherjee, Compactness and large deviations, Mathematisches Kolloquium, Universität Konstanz, Fachbereich Mathematik und Statistik, May 18, 2016.

CH. Mukherjee, Compactness and large deviations, Probability Seminar, University of California at Berkeley, Department of Statistics, USA, October 19, 2016.

CH. Mukherjee, Compactness, large deviations and statistical mechanics, Seminar des Fachbereichs Mathematik und Statistik, Westfälische WilhelmsUniversität Münster, Fachbereich Mathematik und Informatik, October 17, 2016.

CH. Mukherjee, Compactness, large deviations and the polaron, Probability Seminar, University of Washington, Department of Mathematics, Seattle, USA, October 31, 2016.

CH. Mukherjee, Compactness, large deviations and the polaron, The University of Arizona, Department of Mathematics, USA, November 2, 2016.

CH. Mukherjee, Compactness, large deviations, and the polaron problem, 12th German Probability and Statistics Days 2016  Bochumer StochastikTage, February 29  March 4, 2016, RuhrUniversität Bochum, Fakultät für Mathematik, March 3, 2016.

CH. Mukherjee, Occupation measures, compactness and large deviations, Young European Probabilists Workshop ``Large Deviations for Interacting Particle Systems and Partial Differential Equations'' (YEP XIII), March 6  11, 2016, European Institute for Statistics, Probability, Stochastic Operations Research and their Applications (EURANDOM), Eindhoven, Netherlands, March 7, 2016.

CH. Mukherjee, On some aspects of large deviations, Mathematics Colloquium, West Virginia University, Department of Mathematics, Morgantown, USA, March 17, 2016.

CH. Mukherjee, Polaron problem, Probability Seminar, University of California at Irvine, Department of Mathematics, USA, October 25, 2016.

CH. Mukherjee, Quenched large deviations for random walks on supercritical percolation clusters, Probability and Mathematical Physics Seminar, Courant Institute, New York, Department of Mathematics, USA, November 4, 2016.

CH. Mukherjee, The polaron problem, Rutgers University, Department of Mathematics, New Brunswick, USA, November 17, 2016.

CH. Mukherjee, Weak/strong disorder for stochastic heat equation, Analysis Seminar, University of California at Berkeley, Department of Mathematics, USA, October 21, 2016.

CH. Mukherjee, Weak/strong disorder for stochastic heat equation, Probability and Mathematical Physics Seminar, University of California at Los Angeles, Department of Mathematics, USA, October 27, 2016.

CH. Mukherjee, Weak/strong disorder for stochastic heat equation, City University of New York, Department of Mathematics, USA, November 8, 2016.

D.R.M. Renger, Functions of bounded variation with an infinitedimensional codomain, Meeting in Applied Mathematics and Calculus of Variations, September 13  16, 2016, Università di Roma ``La Sapienza'', Dipartimento di Matematica ``Guido Castelnuovo'', Italy, September 16, 2016.

D.R.M. Renger, Large deviations for reacting particle systems: The empirical and ensemble process, Young European Probabilists Workshop ``Large Deviations for Interacting Particle Systems and Partial Differential Equations'' (YEP XIII), March 6  11, 2016, European Institute for Statistics, Probability, Stochastic Operations Research and their Applications (EURANDOM), Eindhoven, Netherlands, March 7, 2016.

R. Soares Dos Santos, Random walk on random walks, University College London, Department of Mathematics, London, UK, June 15, 2016.

R. Soares Dos Santos, Random walk on random walks, Seminar, Leiden University, Institute of Mathematics, Netherlands, May 16, 2016.

R. Soares Dos Santos, Random walk on random walks, RheinMain Kolloquium Stochastik, JohannesGutenberg Universität, Institut für Mathematik, Mainz, May 13, 2016.

W. van Zuijlen, Mean field GibbsnonGibbs transitions, 6th BerlinOxford Meeting, December 8  10, 2016, University of Oxford, Mathematics Department, UK, December 9, 2016.

A. Cipriani, Extremes of some Gaussian random interfaces, Seminar Series in Probability and Statistics, Delft University of Technology, Department of Applied Probability, Netherlands, January 21, 2016.

A. Cipriani, The membrane model, seminar, Delft University of Technology, Department of Applied Probability, Netherlands, September 5, 2016.

A. Cipriani, The membrane model, MaxPlanckInstitut für Mathematik in den Naturwissenschaften, Leipzig, September 29, 2016.

F. Flegel, Spectral localization in the random conductance model, 2nd Berlin Dresden Prague Würzburg Workshop on Mathematics of Continuum Mechanics, Technische Universität Dresden, Fachbereich Mathematik, December 5, 2016.

F. Flegel, Spectral localization vs. homogenization in the random conductance model, Summer School 2016, August 21  26, 2016, Research Training Group (RTG) 1845 ``Stochastic Analysis with Applications in Biology, Finance and Physics'', Hejnice, Czech Republic, August 22, 2016.

F. Flegel, Spectral localization vs. homogenization in the random conductance model, Probability Seminar at UCLA, University of California, Los Angeles, Department of Mathematics, Los Angeles, USA, October 13, 2016.

O. Gün, Fixation times for the mutationselection model on random fitness landscapes, Joint Meeting of the SPP 1590 and 1819, September 28  29, 2016, Universität zu Köln, Köln, September 29, 2016.

W. König, A variational formula for the free energy of an interacting manybody system, Workshop ``Variational Structures and Large Deviations for Interacting Particle Systems and Partial Differential Equations'', March 15  18, 2016, European Institute for Statistics, Probability, Stochastic Operations Research and their Applications (EURANDOM), Eindhoven, Netherlands, March 17, 2016.

W. König, Connection times in large adhoc mobile networks, Workshop on Dynamical Networks and Network Dynamics, January 18  21, 2016, International Centre for Mathematical Science, Edinburgh, UK, January 18, 2016.

W. König, The meanfield polaron model, Workshop on Stochastic Processes in honour of Erwin Bolthausen's 70th birthday, September 14  16, 2016, Universität Zürich, Institut für Mathematik, Switzerland, September 15, 2016.

W. König, The spatially discrete parabolic Anderson model with timedependent potential, ``Guided Tour: Random Media''  Special occasion to celebrate the 60th birthday of Frank den Hollander, December 14  16, 2016, EURANDOM, Eindhoven, Netherlands, December 16, 2016.

R.I.A. Patterson, Monte Carlo simulation of soot, King Abdullah University of Science and Technology (KAUST), Clean Combustion Research Center, Thuwal, Saudi Arabia, January 11, 2016.

R.I.A. Patterson, Pathwise LDPs for chemical reaction networks, 12th German Probability and Statistics Days 2016  Bochumer StochastikTage, February 29  March 4, 2016, RuhrUniversität Bochum, Fakultät für Mathematik, March 4, 2016.

R.I.A. Patterson, Population balance simulation, University of Cambridge, Department for Chemical Engineering and Biotechnology, UK, May 5, 2016.

R.I.A. Patterson, Simulations of flame generated particles, Advances in Uncertainty Quantification Methods, Algorithms and Applications (UQAW 2016), January 5  10, 2016, King Abdullah University of Science and Technology (KAUST), Thuwal, Saudi Arabia, January 5, 2016.
External Preprints

A. González Casanova Soberón, J.C. Pardo, J.L. Perez, Branching processes with interactions: the subcritical cooperative regime, Preprint no. arXiv:1704.04203, , 2017.
Abstract
In this paper, we introduce a particular family of processes with values on the nonnegative integers that model the dynamics of populations where individuals are allow to have different types of inter actions. The types of interactions that we consider include pairwise: competition, annihilation and cooperation; and interaction among several individuals that can be consider as catastrophes. We call such families of processes branching processes with interactions. In particular, we prove that a process in this class has a moment dual which turns out to be a jumpdiffusion that can be thought as the evolution of the frequency of a trait or phenotype. The aim of this paper is to study the long term behaviour of branching processes with interac tions under the assumption that the cooperation parameter satisfies a given condition that we called subcritical cooperative regime. The moment duality property is useful for our purposes. 
A. González Casanova Soberón, N. Kurt, A. Wakolbinger, L. Yuan, An individualbased model for the Lenski experiment, and the deceleration of the relative fitness, Preprint no. arxiv.org:1505.01751, Cornell University Library, arXiv.org, 2016.

J. Blath, A. González Casanova Soberón, N. Kurt, M. WilkeBerenguer, A new coalescent for seedbank models, Preprint no. arxiv.org:1411.4747, Cornell University Library, arXiv.org, 2016.

M. Kraft, W. Wagner, A numerical scheme for the Random Cloud Model, Technical report no. 173, c4ePreprint Series, 2016.

K.F. Lee, M. Dosta, A.D. Mcguire, S. Mosbach, W. Wagner, S. Heinrich, M. Kraft, Development of a multicompartment population balance model for highshear wet granulation with Discrete Element Method, Technical report no. 170, c4ePreprint Series, 2016.
Abstract
This paper presents a multicompartment population balance model for wet granulation coupled with DEM (Discrete Element Method) simulations. Methodologies are developed to extract relevant data from the DEM simulations to inform the population balance model. First, compartmental residence times are calculated for the population balance model from DEM. Then, a suitable collision kernel is chosen for the population balance model based on particleparticle collision frequencies extracted from DEM. It is found t hat the population balance model is able to predict the trends exhibited by the experimental size and porosity distributions by utilising the information provided by the DEM simulations. 
R.I.A. Patterson, S. Simonella, W. Wagner, Kinetic theory of cluster dynamics, Preprint no. arXiv: 1601.05838, Cornell University Libary, arXiv.org, 2016.
Abstract
In a Newtonian system with localized interactions the whole set of particles is naturally decomposed into dynamical clusters, dened as nite groups of particles having an in uence on each other's trajectory during a given interval of time. For an ideal gas with shortrange intermolecular force, we provide a description of the cluster size distribution in terms of the reduced Boltzmann density. In the simplied context of Maxwell molecules, we show that a macroscopic fraction of the gas forms a giant component in nite kinetic time. The critical index of this phase transition is in agreement with previous numerical results on the elastic billiard.
Research Groups
 Partial Differential Equations
 Laser Dynamics
 Numerical Mathematics and Scientific Computing
 Nonlinear Optimization and Inverse Problems
 Interacting Random Systems
 Stochastic Algorithms and Nonparametric Statistics
 Thermodynamic Modeling and Analysis of Phase Transitions
 Nonsmooth Variational Problems and Operator Equations