Publications

Articles in Refereed Journals

  • D.R.M. Renger, Flux large deviations of independent and reacting particle systems, with implications for macroscopic fluctuation theory, Journal of Statistical Physics, (2018), published online on 10.07.2018, DOI 10.1007/s10955-018-2083-0 .
    Abstract
    We consider a system of independent particles on a finite state space, and prove a dynamic large-deviation principle for the empirical measure-empirical 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 large-deviation principle. We then show how to use this result to generalise a number of principles from Macroscopic Fluctuation Theory to the finite-space setting.

  • D.R.M. Renger, P. Koltai , From large deviations to transport semidistances: Coherence analysis for finite Lagrangian data, Journal of Nonlinear Science, (2018), published online on 04.06.2018, DOI 10.1007/s00332-018-9471-0 .
    Abstract
    Onsager's 1931 `reciprocity relations' result connects microscopic time-reversibility with a symmetry property of corresponding macroscopic evolution equations. Among the many consequences is a variational characterization of the macroscopic evolution equation as a gradient-flow, steepest-ascent, or maximal-entropy-production equation. Onsager's original theorem is limited to close-to-equilibrium situations, with a Gaussian invariant measure and a linear macroscopic evolution. In this paper we generalize this result beyond these limitations, and show how the microscopic time-reversibility leads to natural generalized symmetry conditions, which take the form of generalized gradient flows.

  • M. Biskup, R. Fukushima, W. König, Eigenvalue fluctuations for lattice Anderson Hamiltonians: Unbounded potentials, Interdisciplinary Information Sciences, (2018), published online on 29.06.2018.
    Abstract
    We consider random Schrödinger operators with Dirichlet boundary conditions outside lattice approximations of a smooth Euclidean domain and study the behavior of its lowest-lying 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

  • F. Flegel, Localization of the principal Dirichlet eigenvector in the heavy-tailed random conductance model, Electronic Journal of Probability, 23 (2018), pp. 1--43, DOI doi:10.1214/18-EJP160 .
    Abstract
    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, Borel-Cantelli arguments, the Rayleigh-Ritz formula, results from percolation theory, and path arguments.

  • W. Wagner, A random walk model for the Schrödinger equation, Mathematics and Computers in Simulation, 143 (2018), pp. 138--148, DOI 10.1016/j.matcom.2016.07.012 .
    Abstract
    A random walk model for the spatially discretized time-dependent 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 complex-valued 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 Gibbs-non-Gibbs transition in the fuzzy Potts models with a Kac-type interaction, Bernoulli. Official Journal of the Bernoulli Society for Mathematical Statistics and Probability, 23 (2017), pp. 2808--2827.
    Abstract
    We investigate the Gibbs properties of the fuzzy Potts model on the $d$-dimensional torus with Kac interaction. We use a variational approach for profiles inspired by that of Fernández, den Hollander and Martínez citeFeHoMa14 for their study of the Gibbs-non-Gibbs transitions of a dynamical Kac-Ising model on the torus. As our main result, we show that the mean-field thresholds dividing Gibbsian from non-Gibbsian behavior are sharp in the fuzzy Kac-Potts model. On the way to this result we prove a large deviation principle for color profiles with diluted total mass densities and use monotocity arguments

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

  • W. van Zuijlen, Large deviations of continuous regular conditional probabilities, Journal of Theoretical Probability, published online on 27.12.2016., DOI 10.1007/s10959-016-0733-1 .

  • M. Biskup, W. König, R. Soares Dos Santos, Mass concentration and aging in the parabolic Anderson model with doubly-exponential tails, Probability Theory and Related Fields, 171 (2018), pp. 251--331 (published online on 27.05.2017), DOI 10.1007/s00440-017-0777-x .
    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, Mean-field interaction of Brownian occupation measures. II: A rigorous construction of the Pekar process, Communications on Pure and Applied Mathematics, 70 (2017), pp. 1598--1629.
    Abstract
    We consider mean-field interactions corresponding to Gibbs measures on interacting Brownian paths in three dimensions. The interaction is self-attractive and is given by a singular Coulomb potential. The logarithmic asymptotics of the partition function for this model were identified in the 1980s by Donsker and Varadhan [DV83] in terms of the Pekar variational formula, which coincides with the behavior of the partition function corresponding to the polaron problem under strong coupling. Based on this, Spohn ([Sp87]) made a heuristic observation that the strong coupling behavior of the polaron path measure, on certain time scales, should resemble a process, named as the itPekar process, whose distribution could somehow be guessed from the limiting asymptotic behavior of the mean-field measures under interest, whose rigorous analysis remained open. The present paper is devoted to a precise analysis of these mean-field path measures and convergence of the normalized occupation measures towards an explicit mixture of the maximizers of the Pekar variational problem. This leads to a rigorous construction of the aforementioned Pekar process and hence, is a contribution to the understanding of the “mean-field approximation" of the polaron problem on the level of path measures. The method of our proof is based on the compact large deviation theory developed in [MV14], its extension to the uniform strong metric for the singular Coulomb interaction carried out in [KM15], as well as an idea inspired by a itpartial path exchange argument appearing in [BS97]

  • CH. Hirsch, B. Jahnel, P. Keeler, R.I.A. Patterson, Large deviations in relay-augmented wireless networks, Queueing Systems. Theory and Applications, pp. published online on 28.10.2017, urlhttps://doi.org/10.1007/s11134-017-9555-9, DOI 10.1007/s11134-017-9555-9 .
    Abstract
    We analyze a model of relay-augmented 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 non-unique due to symmetry breaking. Also two general sources for bad quality of service can be detected, which we refer to as isolation and screening.

  • CH. Hirsch, B. Jahnel, P. Keeler, R.I.A. Patterson, Traffic flow densities in large transport networks, Advances in Applied Probability, 49 (2017), pp. 1091--1115, DOI 10.1017/apr.2017.35 .
    Abstract
    We consider transport networks with nodes scattered at random in a large domain. At certain local rates, the nodes generate traffic flowing according to some navigation scheme in a given direction. In the thermodynamic limit of a growing domain, we present an asymptotic formula expressing the local traffic flow density at any given location in the domain in terms of three fundamental characteristics of the underlying network: the spatial intensity of the nodes together with their traffic generation rates, and of the links induced by the navigation. This formula holds for a general class of navigations satisfying a link-density and a sub-ballisticity condition. As a specific example, we verify these conditions for navigations arising from a directed spanning tree on a Poisson point process with inhomogeneous intensity function.

  • K.F. Lee, M. Dosta, A. Mc Guire, S. Mosbach, W. Wagner, S. Heinrich, M. Kraft, Development of a multi-compartment population balance model for high-shear wet granulation with discrete element method, Comput. Chem. Engng., 99 (2017), pp. 171--184.

  • 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. 1089--1113.

  • O. Gün, A. Yilmaz, The stochastic encounter-mating model, Acta Applicandae Mathematicae. An International Survey Journal on Applying Mathematics and Mathematical Applications, 148 (2017), pp. 71--102.

  • B. Jahnel, Ch. Külske, The Widom--Rowlinson model under spin flip: Immediate loss and sharp recovery of quasilocality, The Annals of Applied Probability, 27 (2017), pp. 3845--3892, DOI 10.1214/17-AAP1298 .
    Abstract
    We consider the continuum Widom-Rowlinson model under independent spin-flip dynamics and investigate whether and when the time-evolved point process has an (almost) quasilocal specification (Gibbs-property of the time-evolved measure). Our study provides a first analysis of a Gibbs-non-Gibbs 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 time-evolved 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 color-asymmetric percolating model, there is a transition from this non-a.s. quasilocal regime back to an everywhere Gibbsian regime. At the sharp reentrance time tG > 0 the model is a.s. quasilocal. For the colorsymmetric model there is no reentrance. On the constructive side, for all t > tG , we provide everywhere quasilocal specifications for the time-evolved measures and give precise exponential estimates on the influence of boundary conditions.

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

  • W. König, (Book review:) Firas Rassoul-Agha and Timo Seppäläinen: A Course on Large Deviations with an Introduction to Gibbs Measures, Jahresbericht der Deutschen Mathematiker-Vereinigung, 119 (2017), pp. 63--67.

  • A. Mielke, R.I.A. Patterson, M.A. Peletier, D.R.M. Renger, Non-equilibrium thermodynamical principles for chemical reactions with mass-action kinetics, SIAM Journal on Applied Mathematics, 77 (2017), pp. 1562--1585, 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 non-linear 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. 126--167.

Preprints, Reports, Technical Reports

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

  • 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 large-deviation principle for the reaction fluxes under general assumptions that include mass-action 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 nearest-neighbor 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 translation-invariant splitting Gibbs measure. For θ c < θ there is a phase transition with exactly three translation-invariant splitting Gibbs measures. The proof rests on an analysis of fixed points of an associated non-linear Hammerstein integral operator for the boundary laws.

  • C. Bartsch, V. John, R.I.A. Patterson, Simulations of an ASA flow crystallizer with a coupled stochastic-deterministic 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 so-called kinetic Monte-Carlo 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 inverse-square 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 inverse-square 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,(d-2)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 large-time asymptotics of non-negative solutions via Feynman-Kac 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. En-Najjari, 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 multi-hop 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 Poisson-Voronoi or Poisson-Delaunay 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 device-intensity 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 heavy-tailed 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. heavy-tailed random conductance Laplacian to the first k eigenvectors. We overcome the complication that the higher eigenvectors have fluctuating signs by invoking the Bauer-Fike 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 high-density 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 signal-to-interference 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 near-far 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 ad-hoc 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 signal-to-interference 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 high-density 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 equally-sized 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 game-theoretic 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 space-time 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 Hamilton--Jacobi point of view on mean-field Gibbs-non-Gibbs 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 time-dependent large deviation rate functions. This study is motivated by mean-field Gibbs-non-Gibbs transitions. The gradient of the rate-function 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 Curie-Weiss model and Brownian dynamics in a potential. We hereby create a unifying framework for the treatment of mean-field Gibbs-non-Gibbs transitions, based on Hamiltonian dynamics and viscosity solutions of Hamilton-Jacobi 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 non-trivial sub- and super-critical percolation regimes based on the notion of stabilization. Second, we give asymptotic expressions for the percolation probability in large-radius, high-density 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 lowest-lying 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 nearest-neighbour 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 higher-dimensional, non-nearest 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 discrete-time 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 non-lazy 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 Bouchaud--Anderson model with double-exponential potential, Preprint no. 2433, WIAS, Berlin, 2017, DOI 10.20347/WIAS.PREPRINT.2433 .
    Abstract, PDF (459 kByte)
    The Bouchaud--Anderson 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 double-exponential 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 wave-vector). 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 one-dimensional test case. Approximation properties with respect to the grid size and the number of particles are studied. Convergence of a time-splitting scheme to the no-splitting algorithm is demonstrated. The no-splitting 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 d-dimensional euclidean space, defined in terms of a quasilocal specification based on marked Poisson point processes. We investigate the possibility of constructing uniformly absolutely convergent Hamiltonians in terms of hyperedge potentials in the sense of Georgii [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 multi-hop 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 relay-augmented multihop ad-hoc 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 signal-to-interference 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 $Xi$-Wright--Fisher processes with frequency-dependent selection, Preprint no. 2390, WIAS, Berlin, 2017, DOI 10.20347/WIAS.PREPRINT.2390 .
    Abstract, PDF (347 kByte)
    A two-types, discrete-time population model with finite, constant size is constructed, allowing for a general form of frequency-dependent 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 two-types Ξ--Fleming-Viot jump-diffusion process with frequency-dependent selection, and a branching-coalescing 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.

  • F. Flegel, M. Heida, M. Slowik, Homogenization theory for the random conductance model with degenerate ergodic weights and unbounded-range 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 almost-sure homogenization of the discrete Poisson equation and of the top of the Dirichlet spectrum. We assume that the conductances are stationary, ergodic and nearest-neighbor conductances are positive. In contrast to earlier results, we do not require uniform ellipticity but certain integrability conditions on the lower and upper tails of the conductances. We further allow jumps of arbitrary length. Without the long-range connections, the integrability condition on the lower tail is optimal for spectral homogenization. It coincides with a necessary condition for the validity of a local central limit theorem for the random walk among random conductances. As an application of spectral homogenization, we prove a quenched large deviation principle for thenormalized and rescaled local times of the random walk in a growing box. Our proofs are based on a compactness result for the Laplacian's Dirichlet energy, Poincaré inequalities, Moser iteration and two-scale convergence

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, Ruhr-Universität 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, Random walk on random walks, Oberseminar Mathematische Stochastik, Westfälische Wilhelms-Universität Münster, Fachbereich Mathematik und Informatik, Münster, July 4, 2018.

  • R. Soares Dos Santos, The parabolic Anderson model with renormalized inverse-square Poisson potential, 13th German Probability and Statistics Days 2018 -- Freiburger Stochastik-Tage, February 27 - March 2, 2018, Albert-Ludwigs-Universität Freiburg, Department of Mathematical Stochastics, Freiburg, February 2, 2018.

  • W. van Zuijlen, A Hamilton-Jacobi point of view on mean-field Gibbs-non-Gibbs transitions, Workshop on Transformations and phase transitions, January 29 - 31, 2018, Ruhr-Universität Bochum, Fakultät für Mathematik, Bochum, January 30, 2018.

  • W. van Zuijlen, Eigenvalues of the Anderson Hamilitonian with white noise potential in 2d, Leiden University, Institute of Mathematics, Leiden, Netherlands, May 1, 2018.

  • W. van Zuijlen, Mean-field Gibbs-non-Gibbs 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 Stochastik-Tage, February 27 - March 2, 2018, Albert-Ludwigs-Universität Freiburg, Department of Mathematical Stochastics, Freiburg, February 28, 2018.

  • L. Andreis, Ergodicity of a system of interacting random walks with asymmetric interaction, 13th German Probability and Statistics Days 2018 -- Freiburger Stochastik-Tage, February 27 - March 2, 2018, Albert-Ludwigs-Universität Freiburg, Department of Mathematical Stochastics, Freiburg, February 1, 2018.

  • L. Andreis, Self-sustained periodic behavior in interacting systems, Random Structures in Neuroscience and Biology, March 26 - 29, 2018, Ludwig--Maximilians Universität München, Fakultät für Mathematik, Informatik und Statistik, Herrsching, March 26, 2018.

  • F. Flegel, Localization vs. homogenization in the random conductance model, Forschungsseminar Analysis, Technische Universität Chemnitz, Fakultät für Mathematik, Chemnitz, June 6, 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, Humboldt-Universität 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, Max-Planck-Institut für Mathematik in den Naturwissenschaften, Leipzig, March 9, 2018.

  • F. Flegel, Spectral localization vs. homogenization in the random conductance model, Seminar des SFB/CRC 1060 Bonn, Rheinische Friedrich-Wilhelms-Universität Bonn, Mathematisch-Naturwissenschaftlichen Fakultät, Bonn, June 12, 2018.

  • F. Flegel, Spectral localization vs. homogenization in the random conductance model, Oberseminar Stochastik, Universität zu Köln, Mathematisches Institut, Köln, June 14, 2018.

  • F. Flegel, Spectral localization vs. homogenization in the random conductance model, Oberseminar Wahrscheinlichkeitstheorie, Ludwigs-Maximilians-Universität München, Fakultät für Mathematik, Informatik und Statistik, München, July 9, 2018.

  • C. Bartsch, V. John, R.I.A. Patterson, A new mixed stochastic-deterministic 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, Ludwig--Maximilians 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 Stochastik-Tage, February 27 - March 2, 2018, Albert-Ludwigs-Universität 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 Gibbs-non-Gibbs transitions for continuous spin models, DFG-AIMS 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, Ruhr-Universität Bochum, Fakultät für Mathematik, Bochum, January 30, 2018.

  • B. Jahnel, Telecommunication models in random environments, BIMoS Day : The mathematics of quantum information, May 23, 2018, Technische Universität Berlin,, Berlin, May 23, 2018.

  • B. Jahnel, Telecommunication models in random environments, BIMOS DAY, Technische Universität Berlin, May 23, 2018.

  • W. König, Large deviations theory and applications (Mini-course- UoC Forum), Classical and quantum dynamics of interacting particle systems, June 15, 2018, Universität zu Köln, Mathematisches Institut, Köln.

  • W. König, Random message routing in highly dense multi-hop networks, DFG-AIMS 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, Rhein-Main Kolloquium Stochastik, Goethe-Universität Frankfurt am Main, Institut für Mathematik, Frankfurt am Main, January 26, 2018.

  • R.I.A. Patterson, Large deviations for reaction fluxes, Séminaire EDP, Modélisation et Calcul Scientifique (commun ICJ & UMPA), Ecole Normale Superieure de Lyon (CNRS), Lyon, France, July 12, 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, Albert-Ludwigs-Universität 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 Mathematiker-Vereinigung (DMV), Paris-Lodron 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 Widom--Rowlinson model under spin flip: Immediate loss and sharp recovery of quasilocality, Oberseminar Wahrscheinlichkeitstheorie, Ludwig-Maximilians-Universität München, Fakultät für Mathematik, Informatik und Statistik, February 13, 2017.

  • B. Jahnel, The Widom--Rowlinson model under spin flip: Immediate loss and sharp recovery of quasilocality, Westfälische Wilhelms-Universität Münster, Fachbereich Mathematik und Informatik, January 18, 2017.

  • B. Jahnel, The Widom--Rowlinson 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 Widom-Rowlinson 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 device-to-device 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 mean-field 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, Clausthal-Göttingen International Workshop on Simulation Science, April 27 - 28, 2017, Georg-August-Universität Göttingen, Institut für Informatik, April 28, 2017.

  • D.R.M. Renger, Banach-valued 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 Bouchaud--Anderson 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 Mathematiker-Vereinigung (DMV), Paris-Lodron 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 ad-hoc 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, Mean-field Gibbs-non-Gibbs 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, Berlin-Leipzig Workshop in Analysis and Stochastics, November 29 - December 1, 2017, Max-Planck-Institut für Mathematik in den Naturwissenschaften, Leipzig, November 29, 2017.

  • L. Andreis, McKean-Vlasov limits, propagation of chaos and long-time 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 Mathematiker-Vereinigung (DMV), Paris-Lodron University of Salzburg, Austria, September 12, 2017.

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

  • 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 relay-augmented 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 many-body system, Probability Seminar, University of California, Los Angeles, Department of Mathematics, USA, January 19, 2017.

  • W. König, Cluster-size distributions in a classical many-body system, Berlin-Leipzig Workshop in Analysis and Stochastics, November 29 - December 1, 2017, Max-Planck-Institut für Mathematik in den Naturwissenschaften, Leipzig, November 29, 2017.

  • W. König, Connectivity in large mobile ad-hoc 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 Wilhelms-Universität 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, CIM-WIAS 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 twin-screw granulation model, Technical report no. 195, University of Cambridge, c4e-Preprint 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 jump-diffusion 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 time-reversibility with a symmetry property of corresponding macroscopic evolution equations. Among the many consequences is a variational characterization of the macroscopic evolution equation as a gradient-flow, steepest-ascent, or maximal-entropy-production equation. Onsager's original theorem is limited to close-to-equilibrium situations, with a Gaussian invariant measure and a linear macroscopic evolution. In this paper we generalize this result beyond these limitations, and show how the microscopic time-reversibility leads to natural generalized symmetry conditions, which take the form of generalized gradient flows.

  • J. Blath, E. Buzzoni, A. Casanova Soberón, M.W. Berenguer, The seed bank diffusion, and its relation to the two-island 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 jump-diffusion 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, McKean--Vlasov 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 McKean-Vlasov limit for mean-field 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 time-fractional continuous-state branching processes, Preprint no. arXiv:1702.03188, Cornell University Library, arXiv.org, 2017.
    Abstract
    We propose a class of non-Markov population models with continu- ous or discrete state space via a limiting procedure involving sequences of rescaled and randomly time-changed Galton?Watson processes. The class includes as specific cases the classical continuous-state 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 Curie--Weiss 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 Curie-Weiss model by introducing noisy and dissipative evolution in the interaction potential. We show that, when a zero-mean Gaussian is taken as single-site 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 self-sustained 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.