Contribution of the Institute
After 30 years of silence, a significant progress was made in the understanding of the meanfield polaron model, after Mukherjee and Varadhan found a ingenious strategy to prove a largedeviation principle for a prominent object, the normalised occupation measure of a Brownian motion, without assuming compactness. On base of this, the most interesting path properties of the polaron could be resolved in subsequent works by Bolthausen, König and Mukherjee, and a new swing came into the possibilities to analyse also the path properties of the original model.The mathematical description of condensation in the Bose gas using probabilitytheory methods is based on a representation of the interacting particles by means of a Poisson point process of Brownian bridges. In a work by Adams, King and Collevecchio, for the first time to characterization was presented for the free energy of the infinitelylarge system in certain parameter regimes, as a variation problem via stationary point processes. The methodology used opens up the basic possibility of recognizing the condensate as the absence of a minimizer. Moreover, the addition of interlacement processes can be used to describe the condensate itself.
At the heart of the BoseEinstein condensation stands the repulsive nature of particles in the Bose gas, which are intertwined into loops of different lengths. To better understand these interactions and the resulting phasetransition phenomena, we analyze reduced models of interacting random paths and polygons. In this field we build on a series of recent results from Taggi, Betz et al. for selfavoiding random polygons and random permutations.
Publications
Monographs

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. 271299, (Chapter Published), DOI 10.1007/9783030126612_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. 161178, (Chapter Published).
Articles in Refereed Journals

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. 139, DOI 10.1214/19AAP1485 .
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 NavierStokes 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. 10011034, DOI 10.1007/s0022002003834w .
Abstract
We develop a general framework for studying ergodicity of orderpreserving 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 orderpreserving Markov process. As an application, we show exponential ergodicity and exponentially fast synchronizationbynoise 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 McKeanVlasov diffusions via regression on particle systems, SIAM Journal on Control and Optimization, 58 (2020), pp. 529550, DOI 10.1137/18M1195590 .
Abstract
In this note we consider the problem of using regression on interacting particles to compute conditional expectations for McKeanVlasov 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 BoseEinstein condensation in the KaniadakisQuarati model for bosons, Kinetic and Related Models, 13 (2020), pp. 507529, DOI 10.3934/krm.2020017 .
Abstract
Kaniadakis and Quarati (1994) proposed a FokkerPlanck 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 blowup time while having a linear diffusion term. We present a thoroughly validated timeimplicit numerical scheme capable of simulating solutions for arbitrarily large time, and thus enabling a numerical study of the condensation process in the KaniadakisQuarati model. We show strong numerical evidence that above the critical mass rotationally symmetric solutions of the KaniadakisQuarati 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 blowup 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 FokkerPlanck model with superlinear drift, Advances in Mathematics, 360 (2020), pp. 106883/1106883/66, DOI 10.1016/j.aim.2019.106883 .
Abstract
We consider a class of FokkerPlanck equations with linear diffusion and superlineardrift enjoying a formal Wassersteinlike gradient flow structure with convex mobility function. In the driftdominant 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 onedimensional case, is based on a reformulation of the problem in terms of the pseudoinverse distribution function. Motivated by the structure of the equation in the new variables, we establish a general framework for globalintime 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, blowup behaviour, formation of condensates (i.e. Dirac measures at zero) and longtime asymptotics are investigated. As a consequence, in the masssupercritical case,solutions will blow up in L^{∞} in finite time andunderstood in a generalised, measure sensethey 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. 27782807, DOI 10.1016/j.spa.2019.08.001 .

A. Mielke, A. Stephan, Coarsegraining via EDPconvergence for linear fastslow reaction systems, Mathematical Models & Methods in Applied Sciences, 30 (2020), pp. 17651807, 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 coarsegrained 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 coshtype gradient structure we show how an effective limit gradient structure can be rigorously derived and that the coarsegrained equation again has a coshtype gradient structure. We obtain the strongest version of convergence in the sense of the EnergyDissipation Principle (EDP), namely EDPconvergence with tilting. 
F. Flegel, M. Heida, M. Slowik, Homogenization theory for the random conductance model with degenerate ergodic weights and unboundedrange jumps, Annales de l'Institut Henri Poincare. Probabilites et Statistiques, 55 (2019), pp. 12261257, DOI 10.1214/18AIHP917 .
Abstract
We study homogenization properties of the discrete Laplace operator with random conductances on a large domain in Zd. More precisely, we prove almostsure homogenization of the discrete Poisson equation and of the top of the Dirichlet spectrum. We assume that the conductances are stationary, ergodic and nearestneighbor conductances are positive. In contrast to earlier results, we do not require uniform ellipticity but certain integrability conditions on the lower and upper tails of the conductances. We further allow jumps of arbitrary length. Without the longrange connections, the integrability condition on the lower tail is optimal for spectral homogenization. It coincides with a necessary condition for the validity of a local central limit theorem for the random walk among random conductances. As an application of spectral homogenization, we prove a quenched large deviation principle for thenormalized and rescaled local times of the random walk in a growing box. Our proofs are based on a compactness result for the Laplacian's Dirichlet energy, Poincaré inequalities, Moser iteration and twoscale convergence 
F. Flegel, M. Heida, The fractional pLaplacian emerging from homogenization of the random conductance model with degenerate ergodic weights and unboundedrange jumps, Calculus of Variations and Partial Differential Equations, 59 (2020), pp. 8/18/39 (published online on 28.11.2019), DOI 10.1007/s0052601916634 .
Abstract
We study a general class of discrete pLaplace operators in the random conductance model with longrange 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 pLaplace 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, Stochasticdeterministic population balance modeling and simulation of a fluidized bed crystallizer experiment, Chemical Engineering Sciences, 208 (2019), pp. 115102/1115102/14, 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. 139172, DOI 10.1007/s00220019033524 .
Abstract
We consider translationinvariant interacting particle systems on the lattice with finite local state space admitting at least one Gibbs measure as a timestationary measure. The dynamics can be irreversible but should satisfy some mild nondegeneracy conditions. We prove that weak limit points of any trajectory of translationinvariant measures, satisfying a nonnullness condition, are Gibbs states for the same specification as the timestationary measure. This is done under the additional assumption that zero entropy loss of the limiting measure w.r.t. the timestationary measure implies that they are Gibbs measures for the same specification.We also give an alternate version of the last condition such that the nonnullness 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 nonergodic ones. We use this to show synchronization for the rotation dynamics exhibited in citeJaKu12 possibly at low temperature, and possibly nonreversible. 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, pp. published online on 03.11.2019, urlhttps://doi.org/10.1007/s10959019009607, DOI 10.1007/s10959019009607 .
Abstract
We consider marked point processes on the ddimensional euclidean space, defined in terms of a quasilocal specification based on marked Poisson point processes. We investigate the possibility of constructing uniformly absolutely convergent Hamiltonians in terms of hyperedge potentials in the sense of Georgii [2]. These potentials are natural generalizations of physical multibody potentials which are useful in models of stochastic geometry. 
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. 596/1596/26, 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 OnsagerMachlup 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 McKeanVlasov equations, SIAM Journal on Numerical Analysis, 56 (2018), pp. 31693195, DOI 10.1137/17M1111024 .
Abstract
We propose a novel projectionbased particle method for solving McKeanVlasov stochastic differential equations. Our approach is based on a projectiontype estimation of the marginal density of the solution in each time step. The projectionbased 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 McKeanVlasov equations with affine drift. The performance of the proposed algorithm is illustrated by several numerical examples. 
B. Jahnel, Ch. Külske, Sharp thresholds for GibbsnonGibbs transition in the fuzzy Potts models with a Kactype interaction, Bernoulli. Official Journal of the Bernoulli Society for Mathematical Statistics and Probability, 23 (2017), pp. 28082827.
Abstract
We investigate the Gibbs properties of the fuzzy Potts model on the $d$dimensional torus with Kac interaction. We use a variational approach for profiles inspired by that of Fernández, den Hollander and Martínez citeFeHoMa14 for their study of the GibbsnonGibbs transitions of a dynamical KacIsing model on the torus. As our main result, we show that the meanfield thresholds dividing Gibbsian from nonGibbsian behavior are sharp in the fuzzy KacPotts model. On the way to this result we prove a large deviation principle for color profiles with diluted total mass densities and use monotocity arguments 
M. Liero, A. Mielke, M.A. Peletier, D.R.M. Renger, On microscopic origins of generalized gradient structures, Discrete and Continuous Dynamical Systems  Series S, 10 (2017), pp. 135, DOI 10.3934/dcdss.2017001 .
Abstract
Classical gradient systems have a linear relation between rates and driving forces. In generalized gradient systems we allow for arbitrary relations derived from general nonquadratic dissipation potentials. This paper describes two natural origins for these structures. A first microscopic origin of generalized gradient structures is given by the theory of largedeviation principles. While Markovian diffusion processes lead to classical gradient structures, Poissonian jump processes give rise to coshtype dissipation potentials. A second origin arises via a new form of convergence, that we call EDPconvergence. Even when starting with classical gradient systems, where the dissipation potential is a quadratic functional of the rate, we may obtain a generalized gradient system in the evolutionary Gammalimit. As examples we treat (i) the limit of a diffusion equation having a thin layer of low diffusivity, which leads to a membrane model, and (ii) the limit of diffusion over a high barrier, which gives a reactiondiffusion system. 
E. Bolthausen, W. König, Ch. Mukherjee, Meanfield interaction of Brownian occupation measures. II: A rigorous construction of the Pekar process, Communications on Pure and Applied Mathematics, 70 (2017), pp. 15981629.
Abstract
We consider meanfield interactions corresponding to Gibbs measures on interacting Brownian paths in three dimensions. The interaction is selfattractive and is given by a singular Coulomb potential. The logarithmic asymptotics of the partition function for this model were identified in the 1980s by Donsker and Varadhan [DV83] in terms of the Pekar variational formula, which coincides with the behavior of the partition function corresponding to the polaron problem under strong coupling. Based on this, Spohn ([Sp87]) made a heuristic observation that the strong coupling behavior of the polaron path measure, on certain time scales, should resemble a process, named as the itPekar process, whose distribution could somehow be guessed from the limiting asymptotic behavior of the meanfield measures under interest, whose rigorous analysis remained open. The present paper is devoted to a precise analysis of these meanfield path measures and convergence of the normalized occupation measures towards an explicit mixture of the maximizers of the Pekar variational problem. This leads to a rigorous construction of the aforementioned Pekar process and hence, is a contribution to the understanding of the “meanfield approximation" of the polaron problem on the level of path measures. The method of our proof is based on the compact large deviation theory developed in [MV14], its extension to the uniform strong metric for the singular Coulomb interaction carried out in [KM15], as well as an idea inspired by a itpartial path exchange argument appearing in [BS97] 
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. 22282269 (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 (klayer, 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, Meanfield interaction of Brownian occupation measures. I: Uniform tube property of the Coulomb functional, Annales de l'Institut Henri Poincare. Probabilites et Statistiques, 53 (2017), pp. 22142228, DOI 10.1214/16AIHP788 .
Abstract
We study the transformed path measure arising from the selfinteraction of a threedimensional Brownian motion via an exponential tilt with the Coulomb energy of the occupation measures of the motion by time $t$. The logarithmic asymptotics of the partition function were identified in the 1980s by Donsker and Varadhan [DV83P] in terms of a variational formula. Recently [MV14] a new technique for studying the path measure itself was introduced, which allows for proving that the normalized occupation measure asymptotically concentrates around the set of all maximizers of the formula. In the present paper, we show that likewise the Coulomb functional of the occupation measure concentrates around the set of corresponding Coulomb functionals of the maximizers in the uniform topology. This is a decisive step on the way to a rigorous proof of the convergence of the normalized occupation measures towards an explicit mixture of the maximizers, which will be carried out elsewhere. Our methods rely on deriving Höldercontinuity of the Coulomb functional of the occupation measure with exponentially small deviation probabilities and invoking the largedeviation theory developed in [MV14] to a certain shiftinvariant functional of the occupation measures. 
V. Gayrard, O. Gün, Aging in the GREMlike trap model, Markov Processes and Related Fields, 22 (2016), pp. 165202.
Abstract
The GREMlike trap model is a continuous time Markov jump process on the leaves of a finite volume Llevel tree whose transition rates depend on a trapping landscape built on the vertices of the whole tree. We prove that the natural twotime correlation function of the dynamics ages in the infinite volume limit and identify the limiting function. Moreover, we take the limit L→ ∞ of the twotime correlation function of the infinite volume Llevel tree. The aging behavior of the dynamics is characterized by a collection of clock processes, one for each level of the tree. We show that for any L, the joint law of the clock processes converges. Furthermore, any such limit can be expressed through Neveu's continuous state branching process. Hence, the latter contains all the information needed to describe aging in the GREMlike trap model both for finite and infinite levels. 
A. Mielke, M.A. Peletier, D.R.M. Renger, A generalization of Onsager's reciprocity relations to gradient flows with nonlinear mobility, Journal of NonEquilibrium Thermodynamics, 41 (2016), pp. 141149.
Abstract
Onsager's 1931 `reciprocity relations' result connects microscopic timereversibility with a symmetry property of corresponding macroscopic evolution equations. Among the many consequences is a variational characterization of the macroscopic evolution equation as a gradientflow, steepestascent, or maximalentropyproduction equation. Onsager's original theorem is limited to closetoequilibrium situations, with a Gaussian invariant measure and a linear macroscopic evolution. In this paper we generalize this result beyond these limitations, and show how the microscopic timereversibility leads to natural generalized symmetry conditions, which take the form of generalized gradient flows. 
TH. Koprucki, N. Rotundo, P. Farrell, D.H. Doan, J. Fuhrmann, On thermodynamic consistency of a ScharfetterGummel scheme based on a modified thermal voltage for driftdiffusion equations with diffusion enhancement, Optical and Quantum Electronics, 47 (2015), pp. 13271332.
Abstract
Driven by applications like organic semiconductors there is an increased interest in numerical simulations based on driftdiffusion 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 ScharfetterGummel 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 ScharfetterGummel 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. 112.
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 Gammaconvergence) to the JordanKinderlehrerOtto functional arising in the Wasserstein gradient flow structure of the FokkerPlanck 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. 17261742.
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 longtime asymptotics. We first derive, for fixed time, a formula for the expected population size with fixed offspring distributions, which is reminiscent of a FeynmanKac formula. We choose Weibulltype 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 longtime 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. 591638.
Abstract
We derive an annealed large deviation principle (LDP) for the normalised and rescaled local times of a continuoustime random walk among random conductances (RWRC) in a timedependent, 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 timedependent 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 pth power of the pnorm of the gradient of the square root for some 2d/(d+2) < p < 2. This extends the DonskerVaradhanGärtner 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 nonexit 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 largedeviation principle, with applications to Markov chains and diffusion, Potential Analysis, 41 (2014), pp. 12931325.
Abstract
Motivated by the occurence in rate functions of timedependent largedeviation principles, we study a class of nonnegative 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 entropyWasserstein 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 manyparticle limits, ESAIM. Control, Optimisation and Calculus of Variations, 19 (2013), pp. 11661188.

M.A. Peletier, M. Renger, M. Veneroni, Variational formulation of the FokkerPlanck equation with decay: A particle approach, Communications in Contemporary Mathematics, 15 (2013), pp. 1350017/11350017/43.

S. Adams, A. Collevecchio, W. König, A variational formula for the free energy of an interacting manyparticle system, The Annals of Probability, 39 (2011), pp. 683728.
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 wellknown FeynmanKac 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 largedeviations analysis of the stationary empirical field. The resulting variational formula ranges over random shiftinvariant 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 BoseEinstein 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. 499522.
Abstract
We examine the nonexit 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 stretchedexponential 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. 286295.

G. Grüninger, W. König, Potential confinement property in the parabolic Anderson model, Annales de l'Institut Henri Poincare. Probabilites et Statistiques, 45 (2009), pp. 840863.

W. König, H. Lacoin, P. Mörters, N. Sidorova, A two cities theorem for the parabolic Anderson model, The Annals of Probability, 37 (2009), pp. 347392.
Contributions to Collected Editions

F. DEN Hollander, W. König, R. Soares Dos Santos, The parabolic Anderson model on a GaltonWatson tree, in: In and out of euilibrium 3: Celebrating Vladas Sidovaricius, M.E. Vares, R. Fernandez, L.R. Fontes, C.M. Newman, eds., 77 of Progress in Probability, Birkhäuser, 2021, pp. X, 590, DOI 10.1007/9783030607548 .
Abstract
We study the longtime asymptotics of the total mass of the solution to the parabolic Anderson model ( PAM) on a supercritical GaltonWatson random tree with bounded degrees. We identify the secondorder 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 treelike 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.
Preprints, Reports, Technical Reports

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, multiphysics, multiscale problem of magnetorheological suspensions introduced in Nika & Vernescu (Z. Angew. Math. Phys., 71(1):119, '20). The hybrid model couples the Stokes' equation with the quasistatic 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 AltmanShinbrot fixed point theorem when the magnetic Reynold's number, R_{m}, is small. 
A. Mielke, M.A. Peletier, A. Stephan, EDPconvergence for nonlinear fastslow reaction systems with detailed balance, Preprint no. 2781, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2781 .
Abstract, PDF (897 kByte)
We consider nonlinear reaction systems satisfying massaction kinetics with slow and fast reactions. It is known that the fastreactionrate 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 detailedbalance condition. The gradient structure on the set of concentration vectors is given in terms of the relative Boltzmann entropy and a coshtype 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 EnergyDissipation Principle for gradient flows. In general, the effective entropy will no longer be of Boltzmann type and the reactions will no longer satisfy massaction kinetics. 
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 KardarParisiZhang 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 ColeHopf solution to the stationary KPZ equation, the height function of the stationary TASEP, lastpassage 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 EdwardsWilkinson universality class where a different decay exponent is obtained. A key ingredient to our proofs is a characteristic of spacetime stationarity  covariancetovariance reduction  which allows to deduce the asymptotic behavior of the correlations of two spacetime points by the one of the variances at one point. We formulate several open problems. 
J.D. Deuschel, T. Orenshtein, N. Perkowski, Additive functionals as rough paths, Preprint no. 2685, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2685 .
Abstract, PDF (335 kByte)
We consider additive functionals of stationary Markov processes and show that under KipnisVaradhan 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 (nonreversibility) 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 nonreversible OrnsteinUhlenbeck process, while the last example is a diffusion in a periodic environment. As a technical step we prove an estimate for the pvariation of stochastic integrals with respect to martingales that can be viewed as an extension of the rough path BurkholderDavisGundy inequality for local martingale rough paths of [FV08], [CF19] and [FZ18] to the case where only the integrator is a local martingale. 
M. Heida, M. Kantner, A. Stephan, Consistency and convergence for a family of finite volume discretizations of the FokkerPlanck operator, Preprint no. 2684, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2684 .
Abstract, PDF (2719 kByte)
We introduce a family of various finite volume discretization schemes for the FokkerPlanck operator, which are characterized by different weight functions on the edges. This family particularly includes the wellestablished ScharfetterGummel discretization as well as the recently developed squareroot 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 nontrivial while for large gradients the ScharfetterGummel scheme stands out compared to the others. 
G. Nika, B. Vernescu, Microgeometry effects on the nonlinear effective yield strength response of magnetorheological fluids, Preprint no. 2673, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2673 .
Abstract, PDF (1608 kByte)
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. 
A. Mielke, A. Montefusco, M.A. Peletier, Exploring families of energydissipation landscapes via tilting  Three types of EDP convergence, Preprint no. 2668, WIAS, Berlin, 2019, DOI 10.20347/WIAS.PREPRINT.2668 .
Abstract, PDF (372 kByte)
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 wellknown 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 EnergyDissipation Principle that provides a variational formulation to gradientflow equations that allows one to apply techniques from Γconvergence of functional on states and functionals on trajectories.
Talks, Poster

T. Orenshtein, Aging for the O'ConellYor model in intermediate disorder (online talk), Joint Israeli Probability Seminar (Online Event), Israel, November 17, 2020.

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

T. Orenshtein, Aging for the stationary KPZ equation (online talk), BernoulliIMS 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 BerlinOxford Young Researchers Meeting on Applied Stochastic Analysis (Online Event), June 8  10, 2020, Berlin, June 10, 2020.

T. Orenshtein, Aging in EdwardsWilkinson and KPZ universality classes (online talk), Probability, Stochastic Analysis and Statistics Seminar (Online Event), University of Pisa, Italy, October 27, 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, Coarsegraining for gradient systems with applications to reaction systems (online talk), Thematic Einstein Semester: Student Compact Course ``Variational Methods for Fluids and Solids" (Online Event), October 12  23, 2020, WIAS Berlin, October 15, 2020.

A. Stephan, EDPconvergence for nonlinear fastslow 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.

A. Stephan, Coarsegraining via EDPconvergence for linear fastslow 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, EDPconvergence for nonlinear fastslow reactions, Variational Methods for Evolution, September 13  19, 2020, Mathematisches Forschungszentrum Oberwolfach, September 18, 2020.

K. Hopf, Global existence analysis of energyreactiondiffusion systems, Variational Methods for Evolution, September 13  19, 2020, Mathematisches Forschungszentrum Oberwolfach, September 15, 2020.

A. Mielke, Gradient systems and evolutionary Gammaconvergence (online talk), Oberseminar ``Mathematik in den Naturwissenschaften'' (Online Event), JuliusMaximiliansUniversität Würzburg, June 5, 2020.

A. Stephan, EDPconvergence 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, EDPconvergence for linear reactiondiffusion 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 Gammaconvergence for a linear reactiondiffusion system with different time scales, COPDESCWorkshop ``Calculus of Variation and Nonlinear Partial Differential Equations", March 25  28, 2019, Universität Regensburg, March 26, 2019.

A. Stephan, Evolutionary Gammaconvergence for a linear reactiondiffusion 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 operatornorm 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 FokkerPlanck 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 FokkerPlanck 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 NonEquilibrium Thermodynamics, March 5  7, 2018, RheinischWestfälische 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 MathematikerVereinigung (DMV), ParisLodron University of Salzburg, Austria, September 12, 2017.

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

A. Mielke, Perspectives for gradient flows, GAMMWorkshop 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 McKeanVlasov 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 reactiondiffusion 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 reactiondiffusion equation, Mathematisches Kolloquium, Westfälische WilhelmsUniversität, 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, OxfordMan 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 twopoint 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 FokkerPlanck 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 entropydriven 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 entropydriven 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, BerlinOxford Young Researchers Meeting on Applied Stochastic Analysis, July 1  2, 2014, University of Oxford, OxfordMan 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, GermanJapanese Meeting on Stochastic Analysis, September 9  13, 2013, Universität Leipzig, Mathematisches Institut, September 9, 2013.

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

A. Mielke, On the geometry of reactiondiffusion 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 AndersonModell, 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.
External Preprints

J. Fischer , K. Hopf , M. Kniely, A. Mielke, Global existence analysis of energyreactiondiffusion systems, Preprint no. arXiv:2012.03792, Cornell University, 2020.
Abstract
We establish globalintime existence results for thermodynamically consistent reaction(cross)diffusion systems coupled to an equation describing heat transfer. Our main interest is to model speciesdependent diffusivities, while at the same time ensuring thermodynamic consistency. A key difficulty of the nonisothermal case lies in the intrinsic presence of crossdiffusion 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 nonintegrable diffusion fluxes or reaction terms appear. 
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.