Diffusionsmodelle der Statistischen Physik

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

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

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

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

M. Fradon, J. Kern, S. Rœlly, A. Zass, Diffusion dynamics for an infinite system of two-type spheres and the associated depletion effect, Stochastic Processes and their Applications, 171 (2024), 104319, DOI 10.1016/j.spa.2024.104319 .
Abstract
We consider a random diffusion dynamics for an infinite system of hard spheres of two different sizes evolving in ℝ^d, its reversible probability measure, and its projection on the subset of the large spheres. The main feature is the occurrence of an attractive short-range dynamical interaction --- known in the physics literature as a depletion interaction -- between the large spheres, which is induced by the hidden presence of the small ones. By considering the asymptotic limit for such a system when the density of the particles is high, we also obtain a constructive dynamical approach to the famous discrete geometry problem of maximisation of the contact number of n identical spheres in ℝ^d. As support material, we propose numerical simulations in the form of movies.

P. Bella, M. Kniely, Regularity of random elliptic operators with degenerate coefficients and applications to stochastic homogenization, Stochastic Partial Differential Equations. Analysis and Computations, published online on 27.02.2024, DOI https://doi.org/10.1007/s40072-023-00322-9 .
Abstract
We consider degenerate elliptic equations of second order in divergence form with a symmetric random coefficient field a. Extending the work of the first author, Fehrman, and Otto [Ann. Appl. Probab. 28 (2018), no. 3, 1379-1422], who established the large-scale regularity of a-harmonic functions in a degenerate situation, we provide stretched exponential moments for the minimal radius describing the minimal scale for this regularity. As an application to stochastic homogenization, we partially generalize results by Gloria, Neukamm, and Otto [Anal. PDE 14 (2021), no. 8, 2497-2537] on the growth of the corrector, the decay of its gradient, and a quantitative two-scale expansion to the degenerate setting. On a technical level, we demand the ensemble of coefficient fields to be stationary and subject to a spectral gap inequality, and we impose moment bounds on the coefficient field and its inverse. We also introduce the ellipticity radius, which encodes the minimal scale where these moments are close to their positive expectation value.

A. Mielke, S. Schindler, Convergence to self-similar profiles in reaction-diffusion systems, SIAM Journal on Mathematical Analysis, 56 (2024), pp. 7108--7135, DOI 10.1137/23M1564298 .
Abstract
We study a reaction-diffusion system on the real line, where the reactions of the species are given by one reversible reaction pair satisfying the mass-action law. We describe different positive limits at both sides of infinityand investigate the long-time behavior. Rescaling space and time according to the parabolic scaling, we show that solutions converge exponentially to a similarity profile when the scaled time goes to infinity. In the original variables, these profiles correspond to asymptotically self-similar behavior describing the phenomenon of diffusive mixing of the different states at infinity.Our method provides global exponential convergence for all initial states with finite relative entropy. For the case with equal stoichiometric coefficients, we can allow for self-similar profiles with arbitrary equilibrated states,while in the other case we need to assume that the two states atinfinity are sufficiently close such that the self-similar profile is relative flat.

K. Hopf, Singularities in $L^1$-supercritical Fokker--Planck equations: A qualitative analysis, Annales de l'Institut Henri Poincare. Analyse Non Lineaire, 41 (2024), pp. 357--403, DOI 10.4171/AIHPC/85 .
Abstract
A class of nonlinear Fokker--Planck equations with superlinear drift is investigated in the L¹-supercritical regime, which exhibits a finite critical mass. The equations have a formal Wasserstein-like gradient-flow structure with a convex mobility and a free energy functional whose minimising measure has a singular component if above the critical mass. Singularities and concentrations also arise in the evolutionary problem and their finite-time appearance constitutes a primary technical difficulty. This paper aims at a global-in-time qualitative analysis -- the main focus being on isotropic solutions, in which case the unique minimiser of the free energy will be shown to be the global attractor. A key step in the analysis consists in properly controlling the singularity profiles during the evolution. Our study covers the 3D Kaniadakis--Quarati model for Bose--Einstein particles, and thus provides a first rigorous result on the continuation beyond blow-up and long-time asymptotic behaviour for this model.

A. Mielke, Non-equilibrium steady states as saddle points and EDP-convergence for slow-fast gradient systems, Journal of Mathematical Physics, 64 (2023), pp. 123502/1-- 123502/27, DOI 10.1063/5.0149910 .
Abstract
The theory of slow-fast gradient systems leads in a natural way to non-equilibrium steady states, because on the slow time scale the fast subsystem stays in steady states that are driven by the interaction with the slow system. Using the theory of convergence of gradient systems in the sense of the energy-dissipation principle shows that there is a natural characterization of these non-equilibrium steady states as saddle points of a Lagrangian where the slow variables are fixed. We give applications to slow-fast reaction-diffusion systems based on the so-called cosh-type gradient structure for reactions. It is shown that two binary reaction give rise to a ternary reaction with a state-dependent reaction coefficient. Moreover, we show that a reaction-diffusion equation with a thin membrane-like layer convergences to a transmission condition, where the formerly quadratic dissipation potential for diffusion convergences to a cosh-type dissipation potential for the transmission in the membrane limit.

A. Mielke, On two coupled degenerate parabolic equations motivated by thermodynamics, Journal of Nonlinear Science, 33 (2023), pp. 42/1--42/55, DOI 10.1007/s00332-023-09892-3 .
Abstract
We discuss a system of two coupled parabolic equations that have degenerate diffusion constants depending on the energy-like variable. The dissipation of the velocity-like variable is fed as a source term into the energy equation leading to conservation of the total energy. The motivation of studying this system comes from Prandtl's and Kolmogorov's one and two-equation models for turbulence, where the energy-like variable is the mean turbulent kinetic energy. Because of the degeneracies there are solutions with time-dependent support like in the porous medium equation, which is contained in our system as a special case. The motion of the free boundary may be driven by either self-diffusion of the energy-like variable or by dissipation of the velocity-like variable. The cross-over of these two phenomena is exemplified for the associated planar traveling fronts. We provide existence of suitably defined weak and very weak solutions. After providing a thermodynamically motivated gradient structure we also establish convergence into steady state for bounded domains and provide a conjecture on the asymptotically self-similar behavior of the solutions in R^d for large times.

J. Fischer, K. Hopf, M. Kniely, A. Mielke, Global existence analysis of energy-reaction-diffusion systems, SIAM Journal on Mathematical Analysis, 54 (2022), pp. 220--267, DOI 10.1137/20M1387237 .
Abstract
We establish global-in-time existence results for thermodynamically consistent reaction-(cross-)diffusion systems coupled to an equation describing heat transfer. Our main interest is to model species-dependent diffusivities, while at the same time ensuring thermodynamic consistency. A key difficulty of the non-isothermal case lies in the intrinsic presence of cross-diffusion type phenomena like the Soret and the Dufour effect: due to the temperature/energy dependence of the thermodynamic equilibria, a nonvanishing temperature gradient may drive a concentration flux even in a situation with constant concentrations; likewise, a nonvanishing concentration gradient may drive a heat flux even in a case of spatially constant temperature. We use time discretisation and regularisation techniques and derive a priori estimates based on a suitable entropy and the associated entropy production. Renormalised solutions are used in cases where non-integrable diffusion fluxes or reaction terms appear.

Z. Mokhtari, R.I.A. Patterson, F. Höfling, Spontaneous trail formation in populations of auto-chemotactic walkers, New Journal of Physics, 24 (2022), pp. 013012/1--013012/11, DOI 10.1088/1367-2630/ac43ec .
Abstract
We study the formation of trails in populations of self-propelled agents that make oriented deposits of pheromones and also sense such deposits to which they then respond with gradual changes of their direction of motion. Based on extensive off-lattice computer simulations aiming at the scale of insects, e.g., ants, we identify a number of emerging stationary patterns and obtain qualitatively the non-equilibrium state diagram of the model, spanned by the strength of the agent--pheromone interaction and the number density of the population. In particular, we demonstrate the spontaneous formation of persistent, macroscopic trails, and highlight some behaviour that is consistent with a dynamic phase transition. This includes a characterisation of the mass of system-spanning trails as a potential order parameter. We also propose a dynamic model for a few macroscopic observables, including the sub-population size of trail-following agents, which captures the early phase of trail formation.

K. Hopf, M. Burger, On multi-species diffusion with size exclusion, Nonlinear Analysis. Theory, Methods & Applications. An International Multidisciplinary Journal. Series A: Theory and Methods, 224 (2022), pp. 113092/1--113092/27, DOI 10.1016/j.na.2022.113092 .
Abstract
We revisit a classical continuum model for the diffusion of multiple species with size-exclusion constraint, which leads to a degenerate nonlinear cross-diffusion system. The purpose of this article is twofold: first, it aims at a systematic study of the question of existence of weak solutions and their long-time asymptotic behaviour. Second, it provides a weak-strong stability estimate for a wide range of coefficients, which had been missing so far. In order to achieve the results mentioned above, we exploit the formal gradient-flow structure of the model with respect to a logarithmic entropy, which leads to best estimates in the full-interaction case, where all cross-diffusion coefficients are non-zero. Those are crucial to obtain the minimal Sobolev regularity needed for a weak-strong stability result. For meaningful cases when some of the coefficients vanish, we provide a novel existence result based on approximation by the full-interaction case.

K. Hopf, Weak-strong uniqueness for energy-reaction-diffusion systems, Mathematical Models & Methods in Applied Sciences, 21 (2022), pp. 1015--1069, DOI 10.1142/S0218202522500233 .
Abstract
We establish weak-strong uniqueness and stability properties of renormalised solutions to a class of energy-reaction-diffusion systems, which genuinely feature cross-diffusion effects. The systems considered are motivated by thermodynamically consistent models, and their formal entropy structure allows us to use as a key tool a suitably adjusted relative entropy method. Weak-strong uniqueness is obtained for general entropy-dissipating reactions without growth restrictions, and certain models with a non-integrable diffusive flux. The results also apply to a class of (isoenergetic) reaction-cross-diffusion systems.

M. Heida, M. Kantner, A. Stephan, Consistency and convergence for a family of finite volume discretizations of the Fokker--Planck operator, ESAIM: Mathematical Modelling and Numerical Analysis, 55 (2021), pp. 3017--3042, DOI 10.1051/m2an/2021078 .
Abstract
We introduce a family of various finite volume discretization schemes for the Fokker--Planck operator, which are characterized by different weight functions on the edges. This family particularly includes the well-established Scharfetter--Gummel discretization as well as the recently developed square-root approximation (SQRA) scheme. We motivate this family of discretizations both from the numerical and the modeling point of view and provide a uniform consistency and error analysis. Our main results state that the convergence order primarily depends on the quality of the mesh and in second place on the quality of the weights. We show by numerical experiments that for small gradients the choice of the optimal representative of the discretization family is highly non-trivial while for large gradients the Scharfetter--Gummel scheme stands out compared to the others.

T. Orenshtein, Rough invariance principle for delayed regenerative processes, Electronic Communications in Probability, 26 (2021), pp. 37/1--37/13, DOI 10.1214/21-ECP406 .
Abstract
We derive an invariance principle for the lift to the rough path topology of stochastic processes with delayed regenerative increments under an optimal moment condition. An interesting feature of the result is the emergence of area anomaly, a correction term in the second level of the limiting rough path which is identified as the average stochastic area on a regeneration interval. A few applications include random walks in random environment and additive functionals of recurrent Markov chains. The result is formulated in the p-variation settings, where a rough Donsker Theorem is available under the second moment condition. The key renewal theorem is applied to obtain an optimal moment condition.

A. Stephan, EDP-convergence for a linear reaction-diffusion system with fast reversible reaction, Calculus of Variations and Partial Differential Equations, 60 (2021), pp. 226/1--226/35, DOI 10.1007/s00526-021-02089-0 .
Abstract
We perform a fast-reaction limit for a linear reaction-diffusion system consisting of two diffusion equations coupled by a linear reaction. We understand the linear reaction-diffusion system as a gradient flow of the free energy in the space of probability measures equipped with a geometric structure, which contains the Wasserstein metric for the diffusion part and cosh-type functions for the reaction part. The fast-reaction limit is done on the level of the gradient structure by proving EDP-convergence with tilting. The limit gradient system induces a diffusion system with Lagrange multipliers on the linear slow-manifold. Moreover, the limit gradient system can be equivalently described by a coarse-grained gradient system, which induces a diffusion equation with a mixed diffusion constant for the coarse-grained slow variable.

O. Lopusanschi, T. Orenshtein, Ballistic random walks in random environment as rough paths: Convergence and area anomaly, ALEA. Latin American Journal of Probability and Mathematical Statistics, 18 (2021), pp. 945--962, DOI 10.30757/ALEA.v18-34 .
Abstract
Annealed functional CLT in the rough path topology is proved for the standard class of ballistic random walks in random environment. Moreover, the `area anomaly', i.e. a deterministic linear correction for the second level iterated integral of the rescaled path, is identified in terms of a stochastic area on a regeneration interval. The main theorem is formulated in more general settings, namely for any discrete process with uniformly bounded increments which admits a regeneration structure where the regeneration times have finite moments. Here the largest finite moment translates into the degree of regularity of the rough path topology. In particular, the convergence holds in the alpha-Hölder rough path topology for all alpha<1/2 whenever all moments are finite, which is the case for the class of ballistic random walks in random environment. The latter may be compared to a special class of random walks in Dirichlet environments for which the regularity alpha<1/2 is bounded away from 1/2, explicitly in terms of the corresponding trap parameter.

J.-D. Deuschel, T. Orenshtein, N. Perkowski, Additive functionals as rough paths, The Annals of Probability, 49 (2021), pp. 1450--1479, DOI 10.1214/20-AOP1488 .
Abstract
We consider additive functionals of stationary Markov processes and show that under Kipnis--Varadhan type conditions they converge in rough path topology to a Stratonovich Brownian motion, with a correction to the Lévy area that can be described in terms of the asymmetry (non-reversibility) of the underlying Markov process. We apply this abstract result to three model problems: First we study random walks with random conductances under the annealed law. If we consider the Itô rough path, then we see a correction to the iterated integrals even though the underlying Markov process is reversible. If we consider the Stratonovich rough path, then there is no correction. The second example is a non-reversible Ornstein-Uhlenbeck process, while the last example is a diffusion in a periodic environment. As a technical step we prove an estimate for the p-variation of stochastic integrals with respect to martingales that can be viewed as an extension of the rough path Burkholder-Davis-Gundy inequality for local martingale rough paths of [FV08], [CF19] and [FZ18] to the case where only the integrator is a local martingale.

A. Mielke, A. Montefusco, M.A. Peletier, Exploring families of energy-dissipation landscapes via tilting: Three types of EDP convergence, Continuum Mechanics and Thermodynamics, 33 (2021), pp. 611--637, DOI 10.1007/s00161-020-00932-x .
Abstract
This paper revolves around a subtle distinction between two concepts: passing to the limit in a family of gradient systems, on one hand, and deriving effective kinetic relations on the other. The two concepts are strongly related, and in many examples they even appear to be the same. Our main contributions are to show that they are different, to show that well-known techniques developed for the former may give incorrect results for the latter, and to introduce new tools to remedy this. The approach is based on the Energy-Dissipation Principle that provides a variational formulation to gradient-flow equations that allows one to apply techniques from Γ-convergence of functional on states and functionals on trajectories.

A. Mielke, M.A. Peletier, A. Stephan, EDP-convergence for nonlinear fast-slow reaction systems with detailed balance, Nonlinearity, 34 (2021), pp. 5762--5798, DOI 10.1088/1361-6544/ac0a8a .
Abstract
We consider nonlinear reaction systems satisfying mass-action kinetics with slow and fast reactions. It is known that the fast-reaction-rate limit can be described by an ODE with Lagrange multipliers and a set of nonlinear constraints that ask the fast reactions to be in equilibrium. Our aim is to study the limiting gradient structure which is available if the reaction system satisfies the detailed-balance condition. The gradient structure on the set of concentration vectors is given in terms of the relative Boltzmann entropy and a cosh-type dissipation potential. We show that a limiting or effective gradient structure can be rigorously derived via EDP convergence, i.e. convergence in the sense of the Energy-Dissipation Principle for gradient flows. In general, the effective entropy will no longer be of Boltzmann type and the reactions will no longer satisfy mass-action kinetics.

O. Butkovsky, A. Kulik, M. Scheutzow, Generalized couplings and ergodic rates for SPDEs and other Markov models, The Annals of Applied Probability, 30 (2020), pp. 1--39, DOI 10.1214/19-AAP1485 .
Abstract
We establish verifiable general sufficient conditions for exponential or subexponential ergodicity of Markov processes that may lack the strong Feller property. We apply the obtained results to show exponential ergodicity of a variety of nonlinear stochastic partial differential equations with additive forcing, including 2D stochastic Navier-Stokes equations. Our main tool is a new version of the generalized coupling method.

O. Butkovsky, M. Scheutzow, Couplings via comparison principle and exponential ergodicity of SPDEs in the hypoelliptic setting, Communications in Mathematical Physics, 379 (2020), pp. 1001--1034, DOI 10.1007/s00220-020-03834-w .
Abstract
We develop a general framework for studying ergodicity of order-preserving Markov semigroups. We establish natural and in a certain sense optimal conditions for existence and uniqueness of the invariant measure and exponential convergence of transition probabilities of an order-preserving Markov process. As an application, we show exponential ergodicity and exponentially fast synchronization-by-noise of the stochastic reaction?diffusion equation in the hypoelliptic setting. This refines and complements corresponding results of Hairer and Mattingly (Electron J Probab 16:658?738, 2011).

D. Belomestny, J.G.M. Schoenmakers, Optimal stopping of McKean--Vlasov diffusions via regression on particle systems, SIAM Journal on Control and Optimization, 58 (2020), pp. 529--550, DOI 10.1137/18M1195590 .
Abstract
In this note we consider the problem of using regression on interacting particles to compute conditional expectations for McKean-Vlasov SDEs. We prove general result on convergence of linear regression algorithms and establish the corresponding rates of convergence. Application to optimal stopping and variance reduction are considered.

J.A. Carrillo, K. Hopf, M.-Th. Wolfram, Numerical study of Bose--Einstein condensation in the Kaniadakis--Quarati model for bosons, Kinetic and Related Models, 13 (2020), pp. 507--529, DOI 10.3934/krm.2020017 .
Abstract
Kaniadakis and Quarati (1994) proposed a Fokker--Planck equation with quadratic drift as a PDE model for the dynamics of bosons in the spatially homogeneous setting. It is an open question whether this equation has solutions exhibiting condensates in finite time. The main analytical challenge lies in the continuation of exploding solutions beyond their first blow-up time while having a linear diffusion term. We present a thoroughly validated time-implicit numerical scheme capable of simulating solutions for arbitrarily large time, and thus enabling a numerical study of the condensation process in the Kaniadakis--Quarati model. We show strong numerical evidence that above the critical mass rotationally symmetric solutions of the Kaniadakis--Quarati model in 3D form a condensate in finite time and converge in entropy to the unique minimiser of the natural entropy functional at an exponential rate. Our simulations further indicate that the spatial blow-up profile near the origin follows a universal power law and that transient condensates can occur for sufficiently concentrated initial data.

J.A. Carrillo, K. Hopf, J.L. Rodrigo, On the singularity formation and relaxation to equilibrium in 1D Fokker--Planck model with superlinear drift, Advances in Mathematics, 360 (2020), pp. 106883/1--106883/66, DOI 10.1016/j.aim.2019.106883 .
Abstract
We consider a class of Fokker--Planck equations with linear diffusion and superlineardrift enjoying a formal Wasserstein-like gradient flow structure with convex mobility function. In the drift-dominant regime, the equations have a finite critical mass above which the measure minimising the associated entropy functional displays a singular component. Our approach, which addresses the one-dimensional case, is based on a reformulation of the problem in terms of the pseudo-inverse distribution function. Motivated by the structure of the equation in the new variables, we establish a general framework for global-in-time existence, uniqueness and regularity of monotonic viscosity solutions to a class of nonlinear degenerate (resp. singular) parabolic equations, using as a key tool comparison principles and maximum arguments. We then focus on a specific equation and study in more detail the regularity and dynamics of solutions. In particular, blow-up behaviour, formation of condensates (i.e. Dirac measures at zero) and long-time asymptotics are investigated. As a consequence, in the mass-supercritical case,solutions will blow up in L^∞ in finite time and--understood in a generalised, measure sense--they will eventually have condensate. We further show that the singular part of the measure solution does in general interact with the density and that condensates can be transient. The equations considered are motivated by a model for bosons introduced by Kaniadakis and Quarati (1994), which has a similar entropy structure and a critical mass if d≥3.

J.--D. Deuschel, T. Orenshtein, Scaling limit of wetting models in 1+1 dimensions pinned to a shrinking strip, Stochastic Processes and their Applications, 130 (2020), pp. 2778--2807, DOI 10.1016/j.spa.2019.08.001 .

A. Mielke, A. Stephan, Coarse-graining via EDP-convergence for linear fast-slow reaction systems, Mathematical Models & Methods in Applied Sciences, 30 (2020), pp. 1765--1807, DOI 10.1142/S0218202520500360 .
Abstract
We consider linear reaction systems with slow and fast reactions, which can be interpreted as master equations or Kolmogorov forward equations for Markov processes on a finite state space. We investigate their limit behavior if the fast reaction rates tend to infinity, which leads to a coarse-grained model where the fast reactions create microscopically equilibrated clusters, while the exchange mass between the clusters occurs on the slow time scale. Assuming detailed balance the reaction system can be written as a gradient flow with respect to the relative entropy. Focusing on the physically relevant cosh-type gradient structure we show how an effective limit gradient structure can be rigorously derived and that the coarse-grained equation again has a cosh-type gradient structure. We obtain the strongest version of convergence in the sense of the Energy-Dissipation Principle (EDP), namely EDP-convergence with tilting.

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

F. Flegel, M. Heida, The fractional p-Laplacian emerging from homogenization of the random conductance model with degenerate ergodic weights and unbounded-range jumps, Calculus of Variations and Partial Differential Equations, 59 (2020), pp. 8/1--8/39 (published online on 28.11.2019), DOI 10.1007/s00526-019-1663-4 .
Abstract
We study a general class of discrete p-Laplace operators in the random conductance model with long-range jumps and ergodic weights. Using a variational formulation of the problem, we show that under the assumption of bounded first moments and a suitable lower moment condition on the weights, the homogenized limit operator is a fractional p-Laplace operator. Under strengthened lower moment conditions, we can apply our insights also to the spectral homogenization of the discrete Lapalace operator to the continuous fractional Laplace operator.

C. Bartsch, V. Wiedmeyer, Z. Lakdawala, R.I.A. Patterson, A. Voigt, K. Sundmacher, V. John, Stochastic-deterministic population balance modeling and simulation of a fluidized bed crystallizer experiment, Chemical Engineering Sciences, 208 (2019), pp. 115102/1--115102/14, DOI 10.1016/j.ces.2019.07.020 .

B. Jahnel, Ch. Külske, Attractor properties for irreversible and reversible interacting particle systems, Communications in Mathematical Physics, 366 (2019), pp. 139--172, DOI 10.1007/s00220-019-03352-4 .
Abstract
We consider translation-invariant interacting particle systems on the lattice with finite local state space admitting at least one Gibbs measure as a time-stationary measure. The dynamics can be irreversible but should satisfy some mild non-degeneracy conditions. We prove that weak limit points of any trajectory of translation-invariant measures, satisfying a non-nullness condition, are Gibbs states for the same specification as the time-stationary measure. This is done under the additional assumption that zero entropy loss of the limiting measure w.r.t. the time-stationary measure implies that they are Gibbs measures for the same specification.We also give an alternate version of the last condition such that the non-nullness requirement can be dropped. For dynamics admitting a reversible Gibbs measure the alternative condition can be verified, which yields the attractor property for such dynamics. This generalizes convergence results using relative entropy techniques to a large class of dynamics including irreversible and non-ergodic ones. We use this to show synchronization for the rotation dynamics exhibited in citeJaKu12 possibly at low temperature, and possibly non-reversible. We assume the additional regularity properties on the dynamics: 1 There is at least one stationary measure which is a Gibbs measure. 2 Zero loss of relative entropy density under dynamics implies the Gibbs property.

B. Jahnel, Ch. Külske, Gibbsian representation for point processes via hyperedge potentials, Journal of Theoretical Probability, 34 (2021), pp. 391--417 (published online on 03.11.2019, urlhttps://doi.org/10.1007/s10959-019-00960-7), DOI 10.1007/s10959-019-00960-7 .
Abstract
We consider marked point processes on the d-dimensional euclidean space, defined in terms of a quasilocal specification based on marked Poisson point processes. We investigate the possibility of constructing uniformly absolutely convergent Hamiltonians in terms of hyperedge potentials in the sense of Georgii [2]. These potentials are natural generalizations of physical multibody potentials which are useful in models of stochastic geometry.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

G. Nika, B. Vernescu, Micro-geometry effects on the nonlinear effective yield strength response of magnetorheological fluids, in: Emerging Problems in the Homogenization of Partial Differential Equations, P. Donato, M. Luna-Laynez, eds., 10 of SEMA SIMAI Springer Series, Springer, Cham, 2021, pp. 1--16, DOI 10.1007/978-3-030-62030-1_1 .
Abstract
We use the novel constitutive model in [15], derived using the homogenization method, to investigate the effect particle chain microstructures have on the properties of the magnetorheological fluid. The model allows to compute the constitutive coefficients for different geometries. Different geometrical realizations of chains can significantly change the magnetorheological effect of the suspension. Numerical simulations suggest that particle size is also important as the increase of the overall particle surface area can lead to a decrease of the overall magnetorheological effect while keeping the volume fraction constant.

F. DEN Hollander, W. König, R. Soares Dos Santos, The parabolic Anderson model on a Galton--Watson tree, in: In and Out of Equilibrium 3: Celebrating Vladas Sidovaricius, M.E. Vares, R. Fernandez, L.R. Fontes, C.M. Newman, eds., 77 of Progress in Probability, Birkhäuser, 2021, pp. 591--635, DOI 10.1007/978-3-030-60754-8_25 .
Abstract
We study the long-time asymptotics of the total mass of the solution to the parabolic Anderson model ( PAM) on a supercritical Galton-Watson random tree with bounded degrees. We identify the second-order contribution to this asymptotics in terms of a variational formula that gives information about the local structure of the region where the solution is concentrated. The analysis behind this formula suggests that, under mild conditions on the model parameters, concentration takes place on a tree with minimal degree. Our approach can be applied to finite locally tree-like random graphs, in a coupled limit where both time and graph size tend to infinity. As an example, we consider the configuration model or, more precisely, the uniform simple random graph with a prescribed degree sequence.

K. Hopf, Global existence analysis of energy-reaction-diffusion systems, in: Report 29: Variational Methods for Evolution (hybrid meeting), A. Mielke, M. Peletier, D. Slepcev, eds., 17 of Oberwolfach Reports, European Mathematical Society Publishing House, Zurich, 2021, pp. 1418--1421, DOI 10.4171/OWR/2020/29 .

W. König, Branching random walks in random environment, in: Probabilistic Structures in Evolution, E. Baake, A. Wakolbinger, eds., Probabilistic Structures in Evolution, EMS Series of Congress Reports, European Mathematical Society Publishing House, 2021, pp. 23--41, DOI 10.4171/ECR/17-1/2 .

E. Bolthausen, W. König, Ch. Mukherjee, Self-repellent Brownian bridges in an interacting Bose gas, Preprint no. 3110, WIAS, Berlin, 2024, DOI 10.20347/WIAS.PREPRINT.3110 .
Abstract, PDF (478 kByte)
We consider a model of d-dimensional interacting quantum Bose gas, expressed in terms of an ensemble of interacting Brownian bridges in a large box and undergoing the influence of all the interactions between the legs of each of the Brownian bridges. We study the thermodynamic limit of the system and give an explicit formula for the limiting free energy and a necessary and sufficient criterion for the occurrence of a condensation phase transition. For d ≥ 5 and sufficiently small interaction, we prove that the condensate phase is not empty. The ideas of proof rely on the similarity of the interaction to that of the self-repellent random walk, and build on a lace expansion method conducive to treating paths undergoing mutual repellence within each bridge.

A. Mielke, M.A. Peletier, J. Zimmer, Deriving a GENERIC system from a Hamiltonian system, Preprint no. 3108, WIAS, Berlin, 2024, DOI 10.20347/WIAS.PREPRINT.3108 .
Abstract, PDF (651 kByte)
We reconsider the fundamental problem of coarse-graining infinite-dimensional Hamiltonian dynamics to obtain a macroscopic system which includes dissipative mechanisms. In particular, we study the thermodynamical implications concerning Hamiltonians, energy, and entropy and the induced geometric structures such as Poisson and Onsager brackets (symplectic and dissipative brackets). We start from a general finite-dimensional Hamiltonian system that is coupled linearly to an infinite-dimensional heat bath with linear dynamics. The latter is assumed to admit a compression to a finite-dimensional dissipative semigroup (i.e., the heat bath is a dilation of the semigroup) describing the dissipative evolution of new macroscopic variables. Already in the finite-energy case (zero-temperature heat bath) we obtain the so-called GENERIC structure (General Equations for Non-Equilibrium Reversible Irreversibe Coupling), with conserved energy, nondecreasing entropy, a new Poisson structure, and an Onsager operator describing the dissipation. However, their origin is not obvious at this stage. After extending the system in a natural way to the case of positive temperature, giving a heat bath with infinite energy, the compression property leads to an exact multivariate Ornstein--Uhlenbeck process that drives the rest of the system. Thus, we are able to identify a conserved energy, an entropy, and an Onsager operator (involving the Green-Kubo formalism) which indeed provide a GENERIC structure for the macroscopic system.

A. Stephan, Trotter-type formula for operator semigroups on product spaces, Preprint no. 3030, WIAS, Berlin, 2023, DOI 10.20347/WIAS.PREPRINT.3030 .
Abstract, PDF (252 kByte)
We consider a Trotter-type-product formula for approximating the solution of a linear abstract Cauchy problem (given by a strongly continuous semigroup), where the underlying Banach space is a product of two spaces. In contrast to the classical Trotter-product formula, the approximation is given by freezing subsequently the components of each subspace. After deriving necessary stability estimates for the approximation, which immediately provide convergence in the natural strong topology, we investigate convergence in the operator norm. The main result shows that an almost optimal convergence rate can be established if the dominant operator generates a holomorphic semigroup and the off-diagonal coupling operators are bounded.

W. König, N. Pétrélis, R. Soares Dos Santos, W. van Zuijlen, Weakly self-avoiding walk in a Pareto-distributed random potential, Preprint no. 3023, WIAS, Berlin, 2023, DOI 10.20347/WIAS.PREPRINT.3023 .
Abstract, PDF (604 kByte)
We investigate a model of continuous-time simple random walk paths in ℤ ^d undergoing two competing interactions: an attractive one towards the large values of a random potential, and a self-repellent one in the spirit of the well-known weakly self-avoiding random walk. We take the potential to be i.i.d. Pareto-distributed with parameter α > d, and we tune the strength of the interactions in such a way that they both contribute on the same scale as t → ∞. Our main results are (1) the identification of the logarithmic asymptotics of the partition function of the model in terms of a random variational formula, and, (2) the identification of the path behaviour that gives the overwhelming contribution to the partition function for α > 2d: the random-walk path follows an optimal trajectory that visits each of a finite number of random lattice sites for a positive random fraction of time. We prove a law of large numbers for this behaviour, i.e., that all other path behaviours give strictly less contribution to the partition function.The joint distribution of the variational problem and of the optimal path can be expressed in terms of a limiting Poisson point process arising by a rescaling of the random potential. The latter convergence is in distribution?and is in the spirit of a standard extreme-value setting for a rescaling of an i.i.d. potential in large boxes, like in KLMS09.

R. Bazaes, A. Mielke, Ch. Mukherjee, Stochastic homogenization of Hamilton--Jacobi--Bellman equations on continuum percolation clusters, Preprint no. 2955, WIAS, Berlin, 2022, DOI 10.20347/WIAS.PREPRINT.2955 .
Abstract, PDF (598 kByte)
We prove homogenization properties of random Hamilton--Jacobi--Bellman (HJB) equations on continuum percolation clusters, almost surely w.r.t. the law of the environment when the origin belongs to the unbounded component in the continuum. Here, the viscosity term carries a degenerate matrix, the Hamiltonian is convex and coercive w.r.t. the degenerate matrix and the underlying environment is non-elliptic and its law is non-stationary w.r.t. the translation group. We do not assume uniform ellipticity inside the percolation cluster, nor any finite-range dependence (i.i.d.) assumption on the percolation models and the effective Hamiltonian admits a variational formula which reflects some key properties of percolation. The proof is inspired by a method of Kosygina--Rezakhanlou--Varadhan developed for the case of HJB equations with constant viscosity and uniformly coercive Hamiltonian in a stationary, ergodic and elliptic random environment. In the non-stationary and non-elliptic set up, we leverage the coercivity property of the underlying Hamiltonian as well as a relative entropy structure (both being intrinsic properties of HJB, in any framework) and make use of the random geometry of continuum percolation.

A. Stephan, H. Stephan, Positivity and polynomial decay of energies for square-field operators, Preprint no. 2901, WIAS, Berlin, 2021, DOI 10.20347/WIAS.PREPRINT.2901 .
Abstract, PDF (328 kByte)
We show that for a general Markov generator the associated square-field (or carré du champs) operator and all their iterations are positive. The proof is based on an interpolation between the operators involving the generator and their semigroups, and an interplay between positivity and convexity on Banach lattices. Positivity of the square-field operators allows to define a hierarchy of quadratic and positive energy functionals which decay to zero along solutions of the corresponding evolution equation. Assuming that the Markov generator satisfies an operator-theoretic normality condition, the sequence of energies is log-convex. In particular, this implies polynomial decay in time for the energy functionals along solutions.

A. Stephan, Coarse-graining and reconstruction for Markov matrices, Preprint no. 2891, WIAS, Berlin, 2021, DOI 10.20347/WIAS.PREPRINT.2891 .
Abstract, PDF (248 kByte)
We present a coarse-graining (or model order reduction) procedure for stochastic matrices by clustering. The method is consistent with the natural structure of Markov theory, preserving positivity and mass, and does not rely on any tools from Hilbert space theory. The reconstruction is provided by a generalized Penrose-Moore inverse of the coarse-graining operator incorporating the inhomogeneous invariant measure of the Markov matrix. As we show, the method provides coarse-graining and reconstruction also on the level of tensor spaces, which is consistent with the notion of an incidence matrix and quotient graphs, and, moreover, allows to coarse-grain and reconstruct fluxes. Furthermore, we investigate the connection with functional inequalities and Poincaré-type constants.

M. Heida, B. Jahnel, A.D. Vu, Stochastic homogenization on irregularly perforated domains, Preprint no. 2880, WIAS, Berlin, 2021, DOI 10.20347/WIAS.PREPRINT.2880 .
Abstract, PDF (668 kByte)
We study stochastic homogenization of a quasilinear parabolic PDE with nonlinear microscopic Robin conditions on a perforated domain. The focus of our work lies on the underlying geometry that does not allow standard homogenization techniques to be applied directly. Instead we prove homogenization on a regularized geometry and demonstrate afterwards that the form of the homogenized equation is independent from the regularization. Then we pass to the regularization limit to obtain the anticipated limit equation. Furthermore, we show that Boolean models of Poisson point processes are covered by our approach.

G. Nika, An existence result for a class of nonlinear magnetorheological composites, Preprint no. 2804, WIAS, Berlin, 2021, DOI 10.20347/WIAS.PREPRINT.2804 .
Abstract, PDF (257 kByte)
We prove existence of a weak solution for a nonlinear, multi-physics, multi-scale problem of magnetorheological suspensions introduced in Nika & Vernescu (Z. Angew. Math. Phys., 71(1):1--19, '20). The hybrid model couples the Stokes' equation with the quasi-static Maxwell's equations through the Lorentz force and the Maxwell stress tensor. The proof of existence is based on: i) the augmented variational formulation of Maxwell's equations, ii) the definition of a new function space for the magnetic induction and the proof of a Poincaré type inequality, iii) the Altman--Shinbrot fixed point theorem when the magnetic Reynold's number, R_m, is small.

W. König, Branching random walks in random environment: A survey, Preprint no. 2779, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2779 .
Abstract, PDF (253 kByte)
We consider branching particle processes on discrete structures like the hypercube in a random fitness landscape (i.e., random branching/killing rates). The main question is about the location where the main part of the population sits at a late time, if the state space is large. For answering this, we take the expectation with respect to the migration (mutation) and the branching/killing (selection) mechanisms, for fixed rates. This is intimately connected with the parabolic Anderson model, the heat equation with random potential, a model that is of interest in mathematical physics because of the observed prominent effect of intermittency (local concentration of the mass of the solution in small islands). We present several advances in the investigation of this effect, also related to questions inspired from biology.

J.-D. Deuschel, T. Orenshtein, G.R. Moreno Flores, Aging for the stationary Kardar--Parisi--Zhang equation and related models, Preprint no. 2763, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2763 .
Abstract, PDF (368 kByte)
We study the aging property for stationary models in the KPZ universality class. In particular, we show aging for the stationary KPZ fixed point, the Cole-Hopf solution to the stationary KPZ equation, the height function of the stationary TASEP, last-passage percolation with boundary conditions and stationary directed polymers in the intermediate disorder regime. All of these models are shown to display a universal aging behavior characterized by the rate of decay of their correlations. As a comparison, we show aging for models in the Edwards-Wilkinson universality class where a different decay exponent is obtained. A key ingredient to our proofs is a characteristic of space-time stationarity - covariance-to-variance reduction - which allows to deduce the asymptotic behavior of the correlations of two space-time points by the one of the variances at one point. We formulate several open problems.

J.A. Griepentrog, On regularity, positivity and long-time behavior of solutions to an evolution system of nonlocally interacting particles, Preprint no. 1932, WIAS, Berlin, 2014, DOI 10.20347/WIAS.PREPRINT.1932 .
Abstract, PDF (1279 kByte)
An analytical model for multicomponent systems of nonlocally interacting particles is presented. Its derivation is based on the principle of minimization of free energy under the constraint of conservation of particle number and justified by methods established in statistical mechanics. In contrast to the classical Cahn-Hilliard theory with higher order terms, the nonlocal theory leads to an evolution system of second order parabolic equations for the particle densities, weakly coupled by nonlinear and nonlocal drift terms, and state equations which involve both chemical and interaction potential differences. Applying fixed-point arguments and comparison principles we prove the existence of variational solutions in suitable Hilbert spaces for evolution systems. Moreover, using maximal regularity for nonsmooth parabolic boundary value problems in Sobolev-Morrey spaces and comparison principles, we show uniqueness, global regularity and uniform positivity of solutions under minimal assumptions on the regularity of interaction. Applying a refined version of the Łojasiewicz-Simon gradient inequality, this paves the way to the convergence of solutions to equilibrium states. We conclude our considerations with the presentation of simulation results for a phase separation process in ternary systems.

A. Lamacz, S. Neukamm, F. Otto, Moment bounds for the corrector in stochastic homogenization of a percolation model, Preprint no. 1836, WIAS, Berlin, 2013, DOI 10.20347/WIAS.PREPRINT.1836 .
Abstract, PDF (472 kByte)
We study the corrector equation in stochastic homogenization for a simplified Bernoulli percolation model on Z^d, d > 2. The model is obtained from the classical Bernoulli bond percolation by conditioning all bonds parallel to the first coordinate direction to be open. As a main result we prove (in fact for a slightly more general model) that stationary correctors exist and that all finite moments of the corrector are bounded. This extends a previous result by Gloria and the third author, where uniformly elliptic conductances are treated, to the degenerate case. Our argument is based on estimates on the gradient of the elliptic Green's function.

M. Biskup, M. Salvi, T. Wolff, A central limit theorem for the effective conductance: I. Linear boundary data and small ellipticity contrasts, Preprint no. 1739, WIAS, Berlin, 2012, DOI 10.20347/WIAS.PREPRINT.1739 .
Abstract, Postscript (1269 kByte), PDF (348 kByte)
We consider resistor networks on $Z^d$ where each nearest-neighbor edge is assigned a non-negative random conductance. Given a finite set with a prescribed boundary condition, the effective conductance is the minimum of the Dirichlet energy over functions that agree with the boundary values. For shift-ergodic conductances, linear (Dirichlet) boundary conditions and square boxes, the effective conductance scaled by the volume of the box is known to converge to a deterministic limit as the box-size tends to infinity. Here we prove that, for i.i.d. conductances with a small ellipticity contrast, also a (non-degenerate) central limit theorem holds. The proof is based on the corrector method and the Martingale Central Limit Theorem; a key integrability condition is furnished by the Meyers estimate. More general domains, boundary conditions and arbitrary ellipticity contrasts are to be addressed in a subsequent paper.

H. Stephan, A mathematical framework for general classical systems and time irreversibility as its consequence, Preprint no. 1629, WIAS, Berlin, 2011, DOI 10.20347/WIAS.PREPRINT.1629 .
Abstract, Postscript (3232 kByte), PDF (431 kByte)
It is well known that important models in statistical physics like the Fokker-Planck equation satisfy an H-theorem, i.e., have a decreasing Lyapunov function (or increasing entropy). This illustrates a symmetry break in time and reflects the second law of thermodynamics. In this paper, we show that any physically reasonable classical system has to have this property. For this purpose, we develop an abstract mathematical framework based on the theory of compact topological spaces and convex analysis. Precisely, we show:
1) Any statistical state space can be described as the convex hull of the image of the canonical embedding of the bidual space of its deterministic state space (a compact topological Hausdorff space).
2) The change of any statistical state is effected by the adjoint of a Markov operator acting in the space of observables.
3) Any Markov operator satisfies a wide class of inequalities, generated by arbitrary convex functions. As a corollary, these inequalities imply a time monotone behavior of the solution of the corresponding evolution equations.
Moreover, due to the general abstract setting, the proof of the underlying inequalities is very simple and therefore illustrates, where time symmetry breaks: A model is time reversible for any states if and only if the corresponding Markov operator is a deterministic one with dense range.
In addition, the proposed framework provides information about the structure of microscopic evolution equations, the choice of the best function spaces for their analysis and the derivation of macroscopic evolution equations.

A. Mielke, On the stability of NESS in gradient systems with ports, Gradient Flows face-to-face 4, September 9 - 12, 2024, Technische Universität München, Raitenhaslach, September 10, 2024.

A.D. Vu, Discrete contact process in random environment, Mathematics of Random Systems: Summer School 2023, September 11 - 15, 2023, Kyoto University, Research Institute for Mathematical Sciences, Kyoto, Japan, September 15, 2023.

A.D. Vu, Percolation on the Manhattan grid, Stochastic Processes and Related Fields, Kyoto, Japan, September 4 - 8, 2023.

A. Zass, Diffusion dynamics for an system of two-type speres and the associated depletion effect, Workshop MathMicS 2023: Mathematics and microscopic theory for random Soft Matter systems, February 13 - 15, 2023, Heinrich-Heine-Universität Düsseldorf, Institut für Theoretische Physik II - Soft Matter, February 14, 2023.

M. Kniely, A thermodynamically correct framework for electro-energy-reaction-diffusion systems, 22nd ECMI Conference on Industrial and Applied Mathematics, June 26 - 30, 2023, Wrocław University of Science and Technology, Poland, June 30, 2023.

M. Kniely, On a thermodynamically consistent electro-energy-reaction-diffusion system, 93rd Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2023), Session 14 ``Applied Analysis'', May 30 - June 2, 2023, Technische Universität Dresden, June 1, 2023.

E. Magnanini, Spatial coagulation and gelation, SPP2265-Reviewer-Kolloquium, Köln, August 29, 2023.

B. Jahnel, Continuum percolation in random environment, Oberseminar zur Stochastik, Otto-von-Guericke-Universität Magdeburg, Fakultät für Mathematik, June 22, 2023.

B. Jahnel, Percolation, Oberseminar, Technische Universität Braunschweig, Institut für Mathematische Stochastik, November 8, 2023.

W. König, Survey of 1st phase of the SPP2265, SPP2265-Reviewer-Kolloquium, August 29, 2023, Deutsches Zentrum für Luft- und Raumfahrt, Köln, August 29, 2023.

A. Stephan, Positivity and polynomial decay of energies for square-field operators, Variational and Geometric Structures for Evolution, October 9 - 13, 2023, Centro Internazionale per la Ricerca Matematica (CIRM), Levico Terme, Italy, October 13, 2023.

A. Stephan, Fast-slow chemical reaction systems: Gradient systems and EDP-convergence, Oberseminar Dynamics, Technische Universität München, Department of Mathematics, April 17, 2023.

A. Stephan, On time-splitting methods for gradient flows with two dissipation mechanisms, In Search of Model Structures for Non-equilibrium Systems, April 24 - 28, 2023, Westfälische Wilhelms-Universität Münster, April 28, 2023.

A. Stephan, EDP-convergence for a linear reaction-diffusion systems with fast reversible reaction (online talk), SIAM Conference on Analysis of Partial Differential Equations (PD22) (Online Event), Minisymposium MS11: ``Bridging Gradient Flows, Hypocoercivity and Reaction-Diffusion Systems'', March 14 - 18, 2022, March 14, 2022.

M. Kniely, Degenerate random elliptic operators: Regularity aspects and stochastic homogenization, Annual Workshop of the GAMM Activity Group ``Analysis of PDEs" 2022, October 5 - 7, 2022, Institute of Science and Technology Austria (ISTA), Klosterneuburg, Austria, October 6, 2022.

M. Kniely, Global renormalized solutions and equilibration of reaction-diffusion systems with nonlinear diffusion (online talk), SIAM Conference on Analysis of Partial Differential Equations (PD22) (Online Event), Minisymposium ``Bridging Gradient Flows, Hypocoercivity and Reaction-Diffusion Systems'', March 14 - 18, 2022, March 14, 2022.

M. Kniely, Global solutions to a class of energy-reaction-diffusion systems, Conference on Differential Equations and Their Applications (EQUADIFF 15), Minisymposium NAA-03 ``Evolution Differential Equations with Application to Physics and Biology'', July 11 - 15, 2022, Masaryk University, Brno, Czech Republic, July 12, 2022.

K. Hopf, Relative entropies and stability in strongly coupled parabolic systems (online talk), SIAM Conference on Analysis of Partial Differential Equations (PD22) (Online Event), Minisymposium ``Variational Evolution: Analysis and Multi-Scale Aspects'', March 14 - 18, 2022, March 16, 2022.

K. Hopf, The Cauchy problem for a cross-diffusion system with incomplete diffusion, Annual Workshop of the GAMM Activity Group ``Analysis of PDEs'' 2022, October 5 - 7, 2022, Institute of Science and Technology Austria (ISTA), Klosterneuburg, October 5, 2022.

TH. Eiter, On the resolvent problems associated with rotating viscous flow, DMV Annual Meeting 2022, Section 09 ``Applied Analysis and Partial Differential Equations", September 12 - 16, 2022, Freie Universität Berlin, September 14, 2022.

TH. Eiter, On uniform resolvent estimates associated with time-periodic rotating viscous flow, Mathematical Fluid Mechanics in 2022 (Hybrid Event), August 22 - 26, 2022, Czech Academy of Sciences, Prague, Czech Republic, August 24, 2022.

A. Mielke, Convergence to thermodynamic equilibrium in a degenerate parabolic system, DMV Annual Meeting 2022, Section 09 ``Applied Analysis and Partial Differential Equations'', September 12 - 16, 2022, Freie Universität Berlin, September 13, 2022.

A. Mielke, Gamma convergence for evolutionary problems: Using EDP convergence for deriving nontrivial kinetic relations, Calculus of Variations. Back to Carthage, May 16 - 20, 2022, Carthage, Tunisia, May 18, 2022.

A. Stephan, EDP-convergence for a linear reaction-diffusion system with fast reversible reaction, Mathematical Models for Biological Multiscale Systems (Hybrid Event), September 12 - 14, 2022, WIAS Berlin, September 12, 2022.

A. Stephan, EDP-convergence for gradient systems and applications to fast-slow chemical reaction systems, Block Course ``Multiscale Problems and Homogenization'' at Freie Universität Berlin from Nov. 10 to Dec. 15, 2022, Berlin Mathematical School & Berlin Mathematics Research Center MATH+, November 24, 2022.

A. Stephan, Coarse-graining via EDP-convergence for linear fast-slow reaction-diffusion systems (online talk), 91st Annual Meeting of the International Association of Applied Mathematics and Mechanics (Online Event), Section S14 ``Applied Analysis'', March 15 - 19, 2021, Universität Kassel, March 17, 2021.

A. Mielke, A. Stephan, Effective models for materials and interfaces with multiple scales, CRC 1114 Conference 2021 (Online Event), March 1 - 3, 2021.

T. Orenshtein, Aging for the O'Conell--Yor model in intermediate disorder (online talk), Joint Israeli Probability Seminar (Online Event), Technion, Haifa, November 17, 2020.

T. Orenshtein, Aging for the stationary KPZ equation, The 3rd Haifa Probability School. Workshop on Random Geometry and Stochastic Analysis, February 24 - 28, 2020, Technion Israel Institute of Technology, Haifa, February 24, 2020.

T. Orenshtein, Aging for the stationary KPZ equation (online talk), Bernoulli-IMS One World Symposium 2020 (Online Event), August 24 - 28, 2020, A virtual one week symposium on Probability and Mathematical Statistics, August 27, 2020.

T. Orenshtein, Aging for the stationary KPZ equation (online talk), 13th Annual ERC Berlin--Oxford Young Researchers Meeting on Applied Stochastic Analysis (Online Event), June 8 - 10, 2020, WIAS Berlin, June 10, 2020.

T. Orenshtein, Aging in Edwards--Wilkinson and KPZ universality classes (online talk), Probability, Stochastic Analysis and Statistics Seminar (Online Event), University of Pisa, Italy, October 27, 2020.

A. Stephan, Coarse-graining via EDP-convergence for linear fast-slow reaction systems, Seminar ``Applied Analysis'', Eindhoven University of Technology, Centre for Analysis, Scientific Computing, and Applications -- Mathematics and Computer Science, Netherlands, January 20, 2020.

A. Stephan, EDP-convergence for nonlinear fast-slow reactions, Workshop ``Variational Methods for Evolution'', September 13 - 19, 2020, Mathematisches Forschungsinstitut Oberwolfach, September 18, 2020.

A. Stephan, On gradient flows and gradient systems (online talk), CRC 1114 PhD Seminar (Online Event), Freie Universität Berlin, November 11, 2020.

A. Stephan, On gradient systems and applications to interacting particle systems (online talk), CRC 1114 PhD Seminar (Online Event), Freie Universität Berlin, November 25, 2020.

A. Stephan, Coarse-graining for gradient systems with applications to reaction systems (online talk), Thematic Einstein Semester on Energy-based Mathematical Methods for Reactive Multiphase Flows: Student Compact Course ``Variational Methods for Fluids and Solids'' (Online Event), October 12 - 23, 2020, WIAS Berlin, October 15, 2020.

A. Stephan, EDP-convergence for nonlinear fast-slow reaction systems (online talk), Annual Workshop of the GAMM Activity Group on Analysis of PDEs (Online Event), September 30 - October 2, 2020, Institute of Science and Technology Austria (IST Austria), Klosterneuburg, October 1, 2020.

K. Hopf, Global existence analysis of energy-reaction-diffusion systems, Workshop ``Variational Methods for Evolution'', September 13 - 19, 2020, Mathematisches Forschungsinstitut Oberwolfach, September 15, 2020.

A. Mielke, Gradient systems and evolutionary Gamma-convergence (online talk), Oberseminar ``Mathematik in den Naturwissenschaften'' (Online Event), Julius-Maximilians-Universität Würzburg, June 5, 2020.

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

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

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

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

A. Stephan, On evolution semigroups and Trotter product operator-norm estimates, Operator Theory and Krein Spaces, December 19 - 22, 2019, Technische Universität Wien, Austria, December 20, 2019.

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

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

L. Taggi, Critical density in activated random walks, Horowitz Seminar on Probability, Ergodic Theory and Dynamical Systems, Tel Aviv University, School of Mathematical Sciences, Israel, May 20, 2019.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

O. Butkovsky, L. Mytnik, Weak uniqueness for singular stochastic equations, Preprint no. arXiv:2405.13780, Cornell University, 2024, DOI 10.48550/arXiv.2405.13780 .

J.-D. Deuschel, P. Friz, M. Maurelli, M. Slowik, The enhanced Sanov theorem and propagation of chaos, Preprint no. arxiv:1602.08043, Cornell University Library, arXiv.org, 2016.