Publications
Articles in Refereed Journals

W. Wagner, A random walk model for the Schrödinger equation, Mathematics and Computers in Simulation, 143 (2018), pp. 138148, DOI 10.1016/j.matcom.2016.07.012 .
Abstract
A random walk model for the spatially discretized timedependent Schrödinger equation is constructed. The model consists of a class of piecewise deterministic Markov processes. The states of the processes are characterized by a position and a complexvalued weight. Jumps occur both on the spatial grid and in the space of weights. Between the jumps, the weights change according to deterministic rules. The main result is that certain functionals of the processes satisfy the Schrödinger equation. 
B. Jahnel, Ch. Külske, Sharp thresholds for GibbsnonGibbs transition in the fuzzy Potts models with a Kactype interaction, Bernoulli. Official Journal of the Bernoulli Society for Mathematical Statistics and Probability, 23 (2017), pp. 28082827.
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 GibbsnonGibbs transitions of a dynamical KacIsing model on the torus. As our main result, we show that the meanfield thresholds dividing Gibbsian from nonGibbsian behavior are sharp in the fuzzy KacPotts 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. 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. 
W. van Zuijlen, Large deviations of continuous regular conditional probabilities, Journal of Theoretical Probability, published online on 27.12.2016., DOI 10.1007/s1095901607331 .

M. Biskup, W. König, R. Soares Dos Santos, Mass concentration and aging in the parabolic Anderson model with doublyexponential tails, Probability Theory and Related Fields, pp. published online on 27.05.2017, urlhttps://doi.org/10.1007/s004400170777x, DOI 10.1007/s004400170777x .
Abstract
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. 
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] 
CH. Hirsch, B. Jahnel, P. Keeler, R.I.A. Patterson, Large deviations in relayaugmented wireless networks, Queueing Systems. Theory and Applications, pp. published online on 28.10.2017, urlhttps://doi.org/10.1007/s1113401795559, DOI 10.1007/s1113401795559 .
Abstract
We analyze a model of relayaugmented cellular wireless networks. The network users, who move according to a general mobility model based on a Poisson point process of continuous trajectories in a bounded domain, try to communicate with a base station located at the origin. Messages can be sent either directly or indirectly by relaying over a second user. We show that in a scenario of an increasing number of users, the probability that an atypically high number of users experiences bad quality of service over a certain amount of time, decays at an exponential speed. This speed is characterized via a constrained entropy minimization problem. Further, we provide simulation results indicating that solutions of this problem are potentially nonunique due to symmetry breaking. Also two general sources for bad quality of service can be detected, which we refer to as isolation and screening. 
K.F. Lee, M. Dosta, A. 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.

A. VAN Rooij, W. van Zuijlen, Bochner integrals in ordered vector spaces, Positivity. An International Mathematics Journal Devoted to Theory and Applications of Positivity, 21 (2017), pp. 10891113.

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.

B. Jahnel, Ch. Külske, The WidomRowlinson model under spin flip: Immediate loss and sharp recovery of quasilocality, The Annals of Applied Probability, 27 (2017), pp. 38453892, DOI 10.1214/17AAP1298 .
Abstract
We consider the continuum WidomRowlinson model under independent spinflip dynamics and investigate whether and when the timeevolved point process has an (almost) quasilocal specification (Gibbsproperty of the timeevolved measure). Our study provides a first analysis of a GibbsnonGibbs transition for point particles in Euclidean space. We find a picture of loss and recovery, in which even more regularity is lost faster than it is for timeevolved spin models on lattices. We show immediate loss of quasilocality in the percolation regime, with full measure of discontinuity points for any specification. For the colorasymmetric percolating model, there is a transition from this nona.s. quasilocal regime back to an everywhere Gibbsian regime. At the sharp reentrance time t_{G }> 0 the model is a.s. quasilocal. For the colorsymmetric model there is no reentrance. On the constructive side, for all t > t_{G } , we provide everywhere quasilocal specifications for the timeevolved measures and give precise exponential estimates on the influence of boundary conditions. 
W. König, Ch. Mukherjee, Meanfield 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. 22142228, DOI 10.1214/16AIHP788 .
Abstract
We study the transformed path measure arising from the selfinteraction of a threedimensional 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 [DV83P] 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 Höldercontinuity of the Coulomb functional of the occupation measure with exponentially small deviation probabilities and invoking the largedeviation theory developed in [MV14] to a certain shiftinvariant functional of the occupation measures. 
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. Mielke, R.I.A. Patterson, M.A. Peletier, D.R.M. Renger, Nonequilibrium thermodynamical principles for chemical reactions with massaction kinetics, SIAM Journal on Applied Mathematics, 77 (2017), pp. 15621585, DOI 10.1137/16M1102240 .
Abstract
We study stochastic interacting particle systems that model chemical reaction networks on the micro scale, converging to the macroscopic Reaction Rate Equation. One abstraction level higher, we study the ensemble of such particle systems, converging to the corresponding Liouville transport equation. For both systems, we calculate the corresponding large deviations and show that under the condition of detailed balance, the large deviations induce a nonlinear relation between thermodynamic fluxes and free energy driving force. 
R.I.A. Patterson, S. Simonella, W. Wagner, A kinetic equation for the distribution of interaction clusters in rarefied gases, Journal of Statistical Physics, 169 (2017), pp. 126167.
Preprints, Reports, Technical Reports

R.I.A. Patterson, D.R.M. Renger, Large deviations of reaction fluxes, Preprint no. 2491, WIAS, Berlin, 2018, DOI 10.20347/WIAS.PREPRINT.2491 .
Abstract, PDF (304 kByte)
We study a system of interacting particles that randomly react to form new particles. The reaction flux is the rescaled number of reactions that take place in a time interval. We prove a dynamic largedeviation principle for the reaction fluxes under general assumptions that include massaction kinetics. This result immediately implies the dynamic large deviations for the empirical concentration. 
G. Botirov, B. Jahnel, Phase transitions for a model with uncountable spin space on the Cayley tree: The general case, Preprint no. 2490, WIAS, Berlin, 2018, DOI 10.20347/WIAS.PREPRINT.2490 .
Abstract, PDF (188 kByte)
In this paper we complete the analysis of a statistical mechanics model on Cayley trees of any degree, started in [EsHaRo12, EsRo10, BoEsRo13, JaKuBo14, Bo17]. The potential is of nearestneighbor type and the local state space is compact but uncountable. Based on the system parameters we prove existence of a critical value θ_{ c } such that for θ≤θ _{ c } there is a unique translationinvariant splitting Gibbs measure. For θ _{ c } < θ there is a phase transition with exactly three translationinvariant splitting Gibbs measures. The proof rests on an analysis of fixed points of an associated nonlinear Hammerstein integral operator for the boundary laws. 
C. Bartsch, V. John, R.I.A. Patterson, Simulations of an ASA flow crystallizer with a coupled stochasticdeterministic approach, Preprint no. 2483, WIAS, Berlin, 2018, DOI 10.20347/WIAS.PREPRINT.2483 .
Abstract, PDF (378 kByte)
A coupled solver for population balance systems is presented, where the flow, temperature, and concentration equations are solved with finite element methods, and the particle size distribution is simulated with a stochastic simulation algorithm, a socalled kinetic MonteCarlo method. This novel approach is applied for the simulation of an axisymmetric model of a tubular flow crystallizer. The numerical results are compared with experimental data. 
P. Nelson, R. Soares Dos Santos, Brownian motion in attenuated or renormalized inversesquare Poisson potential, Preprint no. 2482, WIAS, Berlin, 2018, DOI 10.20347/WIAS.PREPRINT.2482 .
Abstract, PDF (461 kByte)
We consider the parabolic Anderson problem with random potentials having inversesquare singularities around the points of a standard Poisson point process in ℝ ^{d}, d ≥3. The potentials we consider are obtained via superposition of translations over the points of the Poisson point process of a kernel 𝔎 behaving as 𝔎 (x)≈ Θ x ^{2} near the origin, where Θ ∈(0,(d2)^{2}/16]. In order to make sense of the corresponding path integrals, we require the potential to be either attenuated (meaning that 𝔎 is integrable at infinity) or, when d=3, renormalized, as introduced by Chen and Kulik in [8]. Our main results include existence and largetime asymptotics of nonnegative solutions via FeynmanKac representation. In particular, we settle for the renormalized potential in d=3 the problem with critical parameter Θ = 1/16, left open by Chen and Rosinski in [9]. 
E. Cali, T. EnNajjari, N.N. Gafur, Ch. Christian Hirsch, B. Jahnel, R.I.A. Patterson, Percolation for D2D networks on street systems, Preprint no. 2479, WIAS, Berlin, 2018, DOI 10.20347/WIAS.PREPRINT.2479 .
Abstract, PDF (988 kByte)
We study fundamental characteristics for the connectivity of multihop D2D networks. Devices are randomly distributed on street systems and are able to communicate with each other whenever their separation is smaller than some connectivity threshold. We model the street systems as PoissonVoronoi or PoissonDelaunay tessellations with varying street lengths. We interpret the existence of adequate D2D connectivity as percolation of the underlying random graph. We derive and compare approximations for the critical deviceintensity for percolation, the percolation probability and the graph distance. Our results show that for urban areas, the Poisson Boolean Model gives a very good approximation, while for rural areas, the percolation probability stays far from 1 even far above the percolation threshold. 
F. Flegel, Eigenvector localization in the heavytailed random conductance model, Preprint no. 2472, WIAS, Berlin, 2018, DOI 10.20347/WIAS.PREPRINT.2472 .
Abstract, PDF (292 kByte)
We generalize our former localization result about the principal Dirichlet eigenvector of the i.i.d. heavytailed random conductance Laplacian to the first k eigenvectors. We overcome the complication that the higher eigenvectors have fluctuating signs by invoking the BauerFike theorem to show that the kth eigenvector is close to the principal eigenvector of an auxiliary spectral problem. 
P. Keeler, B. Jahnel, O. Maye, D. Aschenbach, M. Brzozowski, Disruptive events in highdensity cellular networks, Preprint no. 2469, WIAS, Berlin, 2018, DOI 10.20347/WIAS.PREPRINT.2469 .
Abstract, PDF (2524 kByte)
Stochastic geometry models are used to study wireless networks, particularly cellular phone networks, but most of the research focuses on the typical user, often ignoring atypical events, which can be highly disruptive and of interest to network operators. We examine atypical events when a unexpected large proportion of users are disconnected or connected by proposing a hybrid approach based on ray launching simulation and point process theory. This work is motivated by recent results [12] using large deviations theory applied to the signaltointerference ratio. This theory provides a tool for the stochastic analysis of atypical but disruptive events, particularly when the density of transmitters is high. For a section of a European city, we introduce a new stochastic model of a single network cell that uses ray launching data generated with the open source RaLaNS package, giving deterministic path loss values. We collect statistics on the fraction of (dis)connected users in the uplink, and observe that the probability of an unexpected large proportion of disconnected users decreases exponentially when the transmitter density increases. This observation implies that denser networks become more stable in the sense that the probability of the fraction of (dis)connected users deviating from its mean, is exponentially small. We also empirically obtain and illustrate the density of users for network configurations in the disruptive event, which highlights the fact that such bottleneck behaviour not only stems from too many users at the cell boundary, but also from the nearfar effect of many users in the immediate vicinity of the base station. We discuss the implications of these findings and outline possible future research directions. 
W. König, A. Tóbiás, Routeing properties in a Gibbsian model for highly dense multihop networks, Preprint no. 2466, WIAS, Berlin, 2017, DOI 10.20347/WIAS.PREPRINT.2466 .
Abstract, PDF (683 kByte)
We investigate a probabilistic model for routeing in a multihop adhoc communication network, where each user sends a message to the base station. Messages travel in hops via the other users, used as relays. Their trajectories are chosen at random according to a Gibbs distribution that favours trajectories with low interference, measured in terms of sum of the signaltointerference ratios for all the hops, and collections of trajectories with little total congestion, measured in terms of the number of pairs of hops arriving at each relay. This model was introduced in our earlier paper [KT17], where we expressed, in the highdensity limit, the distribution of the optimal trajectories as the minimizer of a characteristic variational formula. In the present work, in the special case in which congestion is not penalized, we derive qualitative properties of this minimizer. We encounter and quantify emerging typical pictures in analytic terms in three extreme regimes. We analyze the typical number of hops and the typical length of a hop, and the deviation of the trajectory from the straight line in two regimes, (1) in the limit of a large communication area and large distances, and (2) in the limit of a strong interference weight. In both regimes, the typical trajectory turns out to quickly approach a straight line, in regime (1) with equallysized hops. Surprisingly, in regime (1), the typical length of a hop diverges logarithmically as the distance of the transmitter to the base station diverges. We further analyze the local and global repulsive effect of (3) a densely populated area on the trajectories. Our findings are illustrated by numerical examples. We also discuss a gametheoretic relation of our Gibbsian model with a joint optimization of message trajectories opposite to a selfish optimization, in case congestion is also penalized 
CH. Hirsch, B. Jahnel, Large deviations for the capacity in dynamic spatial relay networks, Preprint no. 2463, WIAS, Berlin, 2017, DOI 10.20347/WIAS.PREPRINT.2463 .
Abstract, PDF (296 kByte)
We derive a large deviation principle for the spacetime evolution of users in a relay network that are unable to connect due to capacity constraints. The users are distributed according to a Poisson point process with increasing intensity in a bounded domain, whereas the relays are positioned deterministically with given limiting density. The preceding work on capacity for relay networks by the authors describes the highly simplified setting where users can only enter but not leave the system. In the present manuscript we study the more realistic situation where users leave the system after a random transmission time. For this we extend the point process techniques developed in the preceding work thereby showing that they are not limited to settings with strong monotonicity properties. 
R. Kraaij, F. Redig, W. van Zuijlen, A HamiltonJacobi point of view on meanfield GibbsnonGibbs transitions, Preprint no. 2461, WIAS, Berlin, 2017, DOI 10.20347/WIAS.PREPRINT.2461 .
Abstract, PDF (694 kByte)
We study the loss, recovery, and preservation of differentiability of timedependent large deviation rate functions. This study is motivated by meanfield GibbsnonGibbs transitions. The gradient of the ratefunction evolves according to a Hamiltonian flow. This Hamiltonian flow is used to analyze the regularity of the time dependent rate function, both for Glauber dynamics for the CurieWeiss model and Brownian dynamics in a potential. We hereby create a unifying framework for the treatment of meanfield GibbsnonGibbs transitions, based on Hamiltonian dynamics and viscosity solutions of HamiltonJacobi equations. 
CH. Hirsch, B. Jahnel, E. Cali, Continuum percolation for Cox point processes, Preprint no. 2445, WIAS, Berlin, 2017, DOI 10.20347/WIAS.PREPRINT.2445 .
Abstract, PDF (438 kByte)
We investigate continuum percolation for Cox point processes, that is, Poisson point processes driven by random intensity measures. First, we derive sufficient conditions for the existence of nontrivial sub and supercritical percolation regimes based on the notion of stabilization. Second, we give asymptotic expressions for the percolation probability in largeradius, highdensity and coupled regimes. In some regimes, we find universality, whereas in others, a sensitive dependence on the underlying random intensity measure survives. 
M. Biskup, R. Fukushima, W. König, Eigenvalue fluctuations for lattice Anderson Hamiltonians: Unbounded potentials, Preprint no. 2439, WIAS, Berlin, 2017, DOI 10.20347/WIAS.PREPRINT.2439 .
Abstract, PDF (264 kByte)
We consider random Schrödinger operators with Dirichlet boundary conditions outside lattice approximations of a smooth Euclidean domain and study the behavior of its lowestlying eigenvalues in the limit when the lattice spacing tends to zero. Under a suitable moment assumption on the random potential and regularity of the spatial dependence of its mean, we prove that the eigenvalues of the random operator converge to those of a deterministic Schrödinger operator. Assuming also regularity of the variance, the fluctuation of the random eigenvalues around their mean are shown to obey a multivariate central limit theorem. This extends the authors' recent work where similar conclusions have been obtained for bounded random potentials. endabstract 
O. Blondel, M.R. Hilário, R. Soares Dos Santos, V. Sidoravicius, A. Teixeira, Random walk on random walks: Higher dimensions, Preprint no. 2435, WIAS, Berlin, 2017, DOI 10.20347/WIAS.PREPRINT.2435 .
Abstract, PDF (389 kByte)
We study the evolution of a random walker on a conservative dynamic random environment composed of independent particles performing simple symmetric random walks, generalizing results of [16] to higher dimensions and more general transition kernels without the assumption of uniform ellipticity or nearestneighbour jumps. Specifically, we obtain a strong law of large numbers, a functional central limit theorem and large deviation estimates for the position of the random walker under the annealed law in a high density regime. The main obstacle is the intrinsic lack of monotonicity in higherdimensional, nonnearest neighbour settings. Here we develop more general renormalization and renewal schemes that allow us to overcome this issue. As a second application of our methods, we provide an alternative proof of the ballistic behaviour of the front of (the discretetime version of) the infection model introduced in [23]. 
O. Blondel, M.R. Hilário, R. Soares Dos Santos, V. Sidoravicius, A. Teixeira, Random walk on random walks: Low densities, Preprint no. 2434, WIAS, Berlin, 2017, DOI 10.20347/WIAS.PREPRINT.2434 .
Abstract, PDF (356 kByte)
We consider a random walker in a dynamic random environment given by a system of independent simple symmetric random walks. We obtain ballisticity results under two types of perturbations: low particle density, and strong local drift on particles. Surprisingly, the random walker may behave very differently depending on whether the underlying environment particles perform lazy or nonlazy random walks, which is related to a notion of permeability of the system. We also provide a strong law of large numbers, a functional central limit theorem and large deviation bounds under an ellipticity condition. 
S. Muirhead, R. Pymar, R. Soares Dos Santos, The BouchaudAnderson model with doubleexponential potential, Preprint no. 2433, WIAS, Berlin, 2017, DOI 10.20347/WIAS.PREPRINT.2433 .
Abstract, PDF (459 kByte)
The BouchaudAnderson model (BAM) is a generalisation of the parabolic Anderson model (PAM) in which the driving simple random walk is replaced by a random walk in an inhomogeneous trapping landscape; the BAM reduces to the PAM in the case of constant traps. In this paper we study the BAM with doubleexponential potential. We prove the complete localisation of the model whenever the distribution of the traps is unbounded. This may be contrasted with the case of constant traps (i.e. the PAM), for which it is known that complete localisation fails. This shows that the presence of an inhomogeneous trapping landscape may cause a system of branching particles to exhibit qualitatively distinct concentration behaviour. 
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
Talks, Poster

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, Large deviations for reaction fluxes, Workshop on Transformations and phase transitions, January 29  31, 2018, RuhrUniversität Bochum, Fakultät für Mathematik, Bochum, January 29, 2018.

D.R.M. Renger, Large deviations for reaction fluxes, Università degli Studi dell'Aquila, Dipartimento di Ingegneria e Scienze dell'Informazione e Matematica, L'Aquila, France, January 10, 2018.

R. Soares Dos Santos, Random walk on random walks, Groningen, Netherlands, February 14, 2018.

R. Soares Dos Santos, The parabolic Anderson model with renormalized inversesquare Poisson potential, 13th German Probability and Statistics Days 2018  Freiburger StochastikTage, February 27  March 2, 2018, AlbertLudwigsUniversität Freiburg, Department of Mathematical Stochastics, Freiburg, February 2, 2018.

W. van Zuijlen, A HamiltonJacobi point of view on meanfield GibbsnonGibbs transitions, Workshop on Transformations and phase transitions, January 29  31, 2018, RuhrUniversität Bochum, Fakultät für Mathematik, Bochum, January 30, 2018.

W. van Zuijlen, Meanfield GibbsnonGibbs transition, Spring School, Spin Systems: Discrete and Continuous, March 19  23, 2018, Technische Universität Darmstadt, Fachbereich Mathematik, Darmstadt.

W. van Zuijlen, The principal eigenvalue of the Anderson Hamiltonian in continuous space, 13th German Probability and Statistics Days 2018  Freiburger StochastikTage, February 27  March 2, 2018, AlbertLudwigsUniversität Freiburg, Department of Mathematical Stochastics, Freiburg, February 28, 2018.

W. van Zuijlen, tba, Leiden University, Institute of Mathematics, Leiden, Netherlands.

L. Andreis, Ergodicity of a system of interacting random walks with asymmetric interaction, 13th German Probability and Statistics Days 2018  Freiburger StochastikTage, February 27  March 2, 2018, AlbertLudwigsUniversität Freiburg, Department of Mathematical Stochastics, Freiburg, February 1, 2018.

L. Andreis, Selfsustained periodic behavior in interacting systems, Random Structures in Neuroscience and Biology, March 26  29, 2018, LudwigMaximilians Universität München, Fakultät für Mathematik, Informatik und Statistik, Herrsching, March 26, 2018.

F. Flegel, Spectral homogenization vs. localization in the barrier model, Symposium anläßlich des 60. Geburtstags von Igor Sokolov, Bernstein Center for Computational Neuroscience Berlin, HumboldtUniversität zu Berlin, February 26, 2018.

F. Flegel, Spectral homogenization vs. localization in the random conductance model, Workshop `` Interplay of Analysis and Probability in Applied Mathematics'', February 11  17, 2018, Mathematisches Forschungsinstitut Oberwolfach, Oberwolfach, February 12, 2018.

F. Flegel, Spectral homogenization vs. localization in the random conductance model, Seminar Angewandte Analysis, MaxPlanckInstitut für Mathematik in den Naturwissenschaften, Leipzig, March 9, 2018.

C. Bartsch, V. John, R.I.A. Patterson, A new mixed stochasticdeterministic simulation approach to particle populations in fluid flows, 6th International Conference on Population Balance Modelling (PBM2018), Belgium, May 7  9, 2018.

B. Jahnel, Attractor properties for irreversible and reversible interacting particle systems, Random Structures in Neuroscience and Biology, March 26  29, 2018, LudwigMaximilians Universität München, Fakultät für Mathematik, Informatik und Statistik, Herrsching, March 29, 2018.

B. Jahnel, Continuum percolation for Cox point processes, 13th German Probability and Statistics Days 2018  Freiburger StochastikTage, February 27  March 2, 2018, AlbertLudwigsUniversität Freiburg, Department of Mathematical Stochastics, Freiburg, February 28, 2018.

B. Jahnel, Continuum percolation for Cox point processes, Universität Osnabrück, Fachbereich Mathematik / Informatik, February 1, 2018.

B. Jahnel, Dynamical GibbsnonGibbs transitions for continuous spin models, DFGAIMS Workshop on Evolutionary processes on networks, March 20  24, 2018, African Institute of Mathematical Sciences (AIMS), Kigali, Rwanda, March 21, 2018.

B. Jahnel, Gibbsian representation for point processes via hyperedge potentials, Workshop on Transformations and phase transitions, January 29  31, 2018, RuhrUniversität Bochum, Fakultät für Mathematik, Bochum, January 30, 2018.

W. König, Random message routing in highly dense multihop networks, DFGAIMS Workshop on Evolutionary processes on networks, March 20  24, 2018, African Institute of Mathematical Sciences (AIMS), Kigali, Rwanda, March 21, 2018.

W. König, The principal part of the spectrum of random Schrödinger operators in large boxes, RheinMain Kolloquium Stochastik, GoetheUniversität Frankfurt am Main, Institut für Mathematik, Frankfurt am Main, January 26, 2018.

R.I.A. Patterson, Large deviations for reaction fluxes, Workshop `` Interplay of Analysis and Probability in Applied Mathematics'', February 11  17, 2018, Mathematisches Forschungsinstitut Oberwolfach, Oberwolfach, February 15, 2018.

D.R.M. Renger, Gradient flows and GENERIC in flux space, Workshop ``Variational Methods for Evolution'', November 12  18, 2017, Mathematisches Forschungsinstitut Oberwolfach, November 16, 2017.

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, China, May 8, 2017.

A. González Casanova Soberón, Modeling selection via multiple parents, Annual Colloquium SPP 1590, October 4  6, 2017, AlbertLudwigsUniversität Freiburg, Fakultät für Mathematik und Physik, October 6, 2017.

A. González Casanova Soberón, Modelling selection via multiple parents, Seminar Probability, National Autonomous University of Mexico, Mexico City, February 23, 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, Modelling the Lenski experiment, 19th ÖMG Congress and Annual DMV Meeting, Section S16 ``Mathematics in the Science and Technology'', September 11  15, 2017, Austrian Mathematical Society (ÖMG) and Deutsche MathematikerVereinigung (DMV), ParisLodron University of Salzburg, Austria, September 14, 2017.

A. González Casanova Soberón, The ancestral efficiency graph, Spatial Models in Population Genetics, September 6  8, 2017, University of Bath, Department of Mathematical Sciences, UK, September 6, 2017.

A. González Casanova Soberón, The discrete ancestral selection graph, Seminar, Center for Interdisciplinary Research in Biology, Stochastic Models for the Inference of Life Evolution SMILE, Paris, France, October 23, 2017.

B. Jahnel, Fabrics of dreams, Seminar am Ökonomischen Institut, Johannes Gutenberg Universität Mainz, Ökonomisches Institut, April 26, 2017.

B. Jahnel, The WidomRowlinson model under spin flip: Immediate loss and sharp recovery of quasilocality, Oberseminar Wahrscheinlichkeitstheorie, LudwigMaximiliansUniversität München, Fakultät für Mathematik, Informatik und Statistik, February 13, 2017.

B. Jahnel, The WidomRowlinson model under spin flip: Immediate loss and sharp recovery of quasilocality, Westfälische WilhelmsUniversität Münster, Fachbereich Mathematik und Informatik, January 18, 2017.

B. Jahnel, The WidomRowlinson model under spin flip: Immediate loss and sharp recovery of quasilocality, Oberseminar Stochastik, Johannes Gutenberg Universität Mainz, Institut für Mathematik, April 25, 2017.

B. Jahnel, The WidomRowlinson model under spin flip: immediate loss and sharp recovery of quasilocality, Université du Luxembourg, Faculté des Sciences, de la Technologie et de la Communication (FSTC), Luxembourg, March 3, 2017.

P. Keeler, Optimizing spatial throughput in devicetodevice networks, Applied Probability @ The Rock  An International Workshop celebrating Phil Pollett's 60th Birthday, April 17  21, 2017, University of Adelaide, School of Mathematical Sciences, Uluru, Australia, April 20, 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 coagulationadvection simulations, ClausthalGöttingen International Workshop on 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, Large deviations and gradient flows, Spring School 2017: From Particle Dynamics to Gradient Flows, February 27  March 3, 2017, Technische Universität Kaiserslautern, Fachbereich Mathematik, March 1, 2017.

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

R. Soares Dos Santos, Complete localisation in the BouchaudAnderson model, Leiden University, Institute of Mathematics, Netherlands, May 9, 2017.

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

R. Soares Dos Santos, Eigenvalue order statistics of random Schrödinger operators and applications to the parabolic Anderson 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 MathematikerVereinigung (DMV), ParisLodron University of Salzburg, Austria, September 12, 2017.

R. Soares Dos Santos, Random walk on random walks, Mathematical Probability Seminar, New York University Shanghai, China, March 21, 2017.

A. Wapenhans, Data mobility in adhoc networks: Vulnerability & security, Telecom Orange Paris, France, November 17, 2017.

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

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

W. van Zuijlen, The principal eigenvalue of the Anderson Hamiltonian in continuous space, BerlinLeipzig Workshop in Analysis and Stochastics, November 29  December 1, 2017, MaxPlanckInstitut für Mathematik in den Naturwissenschaften, Leipzig, November 29, 2017.

L. Andreis, McKeanVlasov limits, propagation of chaos and longtime behavior of some mean field interacting particle systems, Verteidigung Dissertation, November 15  20, 2017, Università degli Studi di Padova, Dipartimento di Matematica Pura ed Applicata, Padova, Italy, November 16, 2017.

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 MathematikerVereinigung (DMV), ParisLodron University of Salzburg, Austria, September 12, 2017.

F. Flegel, Spectral localization vs. homogenization in the random conductance model, BerlinLeipzig Workshop in Analysis and Stochastics, November 29  December 1, 2017, MaxPlanckInstitut für Mathematik in den Naturwissenschaften, Leipzig, November 29, 2017.

B. Jahnel, Continuum percolation for Cox processes, Seminar, Ruhr Universität Bochum, Fakultät für Mathematik, October 27, 2017.

B. Jahnel, Continuum percolation theory applied to Device to Device, Telecom Orange Paris, France, November 17, 2017.

B. Jahnel, Gibbsian representation for point processes via hyperedge potentials, Workshop on Stochastic Analysis and Random Fields, Second Haifa Probability School, December 18  22, 2017, Technion Israel Institute of Technology, Haifa, Israel, December 18, 2017.

B. Jahnel, Large deviations in relayaugmented wireless networks, Sharif University of Technology Tehran, Mathematical Sciences Department, Teheran, Iran, September 17, 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.

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

W. König, Clustersize distributions in a classical manybody system, BerlinLeipzig Workshop in Analysis and Stochastics, November 29  December 1, 2017, MaxPlanckInstitut für Mathematik in den Naturwissenschaften, Leipzig, November 29, 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, August 29, 2017.

W. König, Intersections of Brownian motions, Workshop ``Peter's Network'', October 31  November 1, 2017, University of Bath, Department of Mathematical Sciences, UK, November 1, 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, September 28, 2017.

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

R.I.A. Patterson, Coagulation  Transport Simulations with Stochastic Particles, CIMWIAS Workshop ``Topics in Applied Analysis and Optimisation'', December 6  8, 2017, University of Lisbon, International Center for Mathematics, Lisboa, Portugal, December 7, 2017.

R.I.A. Patterson, Simulation of particle coagulation and advection, Numerical Methods and Applications of Population Balance Equations, October 13, 2017, GRK 1932, Technische Universität Kaiserslautern, Fachbereich Mathematik, October 13, 2017.
External Preprints

A.D. Mcguire, S. Mosbach, G. Reynolds, R.I.A. Patterson, E.J. Bringley, N.A. Eaves, J. Dreyer, M. Kraft, Analysing the effect of screw configuration using a stochastic twinscrew granulation model, Technical report no. 195, University of Cambridge, c4ePreprint Series, 2018.

A. González Casanova Soberón, J.C. Pardo, J.L. Perez, Branching processes with interactions: The subcritical cooperative regime, Preprint no. arXiv:1704.04203, Cornell University Library, arXiv.org, 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. 
D.R.M. Renger, P. Koltai , From large deviations to transport semidistances: Coherence analysis for finite Lagrangian data, Preprint no. arXiv:1709.02352, Cornell University Library, arXiv.org, 2017.
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. 
J. Blath, E. Buzzoni, A. Casanova Soberón, M.W. Berenguer, The seed bank diffusion, and its relation to the twoisland model, Preprint no. arXiv:1710.08164, Cornell University Library, arXiv.org, 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. 
L. Andreis, A. Asselah, P. Dai Pra , Ergodicity of a system of interacting random walks with asymmetric interaction, Preprint no. arXiv:1702.02754, Cornell University Library, arXiv.org, 2017.
Abstract
We study N interacting random walks on the positive integers. Each particle has drift delta towards infinity, a reflection at the origin, and a drift towards particles with lower positions. This inhomogeneous mean field system is shown to be ergodic only when the interaction is strong enough. We focus on this latter regime, and point out the effect of piles of particles, a phenomenon absent in models of interacting diffusion in continuous space. 
L. Andreis, P. Dai Pra, M. Fischer, McKeanVlasov limit for interacting systems with simultaneous jumps, Preprint no. arXiv:1704.01052, Cornell University Library, arXiv.org, 2017.
Abstract
Motivated by several applications, including neuronal models, we consider the McKeanVlasov limit for meanfield systems of interacting diffusions with simultaneous jumps. We prove propagation of chaos via a coupling technique that involves an intermediate process and that gives a rate of convergence for the W1 Wasserstein distance between the empirical measures of the two systems on the space of trajectories D([0,T],R^d). 
L. Andreis, F. Polito, L. Sacerdote, On a class of timefractional continuousstate branching processes, Preprint no. arXiv:1702.03188, Cornell University Library, arXiv.org, 2017.
Abstract
We propose a class of nonMarkov population models with continu ous or discrete state space via a limiting procedure involving sequences of rescaled and randomly timechanged Galton?Watson processes. The class includes as specific cases the classical continuousstate branching processes and Markov branching processes. Several results such as the expressions of moments and the branching inequality governing the evolution of the process are presented and commented. The gener alized Feller branching diffusion and the fractional Yule process are analyzed in detail as special cases of the general model. 
L. Andreis, D. Tovazzi, Coexistence of stable limit cycles in a generalized CurieWeiss model with dissipation, Preprint no. arXiv:1711.05129, Cornell University Library, arXiv.org, 2017.
Abstract
In this paper, we modify the Langevin dynamics associated to the generalized CurieWeiss model by introducing noisy and dissipative evolution in the interaction potential. We show that, when a zeromean Gaussian is taken as singlesite distribution, the dynamics in the thermodynamic limit can be described by a finite set of ODEs. Depending on the form of the interaction function, the system can have several phase transitions at different critical temperatures. Because of the dissipation effect, not only the magnetization of the systems displays a selfsustained periodic behavior at sufficiently low temperature, but, in certain regimes, any (finite) number of stable limit cycles can exist. We explore some of these peculiarities with explicit examples.
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