The theory of large deviations, a branch of probability theory, provides tools for the description of the asymptotic decay rate of a small probability, as a certain parameter diverges or shrinks to zero. Examples are large times, low temperatures, large numbers of stochastic quantities, or an approximation parameter. In this way the deviation of a stochastic quantity from its expected behaviour can often be described, e.g. the deviation of an empirical average from the mean in the law of large numbers. The decay rate is expressed in terms of a non-negative rate function, which in many cases is explicit. Its minimiser(s) often describe(s) the normal behaviour of the stochastic system under interest in terms of a variational formula. Furthermore, the rate function contains information about the behaviour of the system that gives the main contribution to the rare event under consideration. Hence, variational methods are often very helpful in analysing the stochastic system. Often, the large-deviation analysis is only a starting point for deeper analysis, like the investigation of finer asymptotics or of deviations on different scales.

Furthermore, the theory provides also tools and concepts for the description of the asymptotics of expectations of exponential functionals of the system. This applies to the partition functions for many models in statistical physics. The description is in terms of a variational formula that shows the best compromise between the functional and the rate function. In many cases, the object that satisfies a large-deviations principle is not immediately apparent and has to be found by suitable reformulations of the problem.

Contribution of the Institute

At WIAS, large-deviation theory is employed for the analysis of various models from quite different areas.

For instance, in statistical physics or atomistic materials science, classical and quantum mechanical stochastic many-body systems are investigated using methods from this theory. Here large, but finite point clouds in large boxes in the euclidean space are under the influence of a pair-interaction that rules out strong clumping (mutually repelling forces), but favours distances of a certain fixed amount (attracting forces). Such systems are defined in terms of a Gibbs measure with Lennard-Jones potential. In the thermodynamic limit (many particles in a large box with fixed particle density), one of the interesting questions is about the sizes of the subgroups (clusters) that are formed. In one work a large-deviation principle for the statistics of these cardinalities is derived and could by described quite explicitly in a certain low-density regime. If kinetic energy is added to any of the particles and a certain symmetry to the system, describing bosons, then a large-deviation principle for the empirical stationary field of a certain marked Poisson point process involving Brownian bridges proved helpful for the analysis of the limiting free energy of the system.

A highlight was the derivation of a prominent large-deviation principle for the normalised occupation measure of a Brownian motion without requesting compactness.

A further example is the study of the decay rate of frustration probabilities, that is, the probability of unwanted events in a telecommunications network in the limit of a high density of participants. The unwanted events are occurrences such as significantly reduced connectivity or low capacity. The decay rates of such events were analysed by the Leibniz Group 4 and in particular the configurations that lead to such events were investigated. The main focus for multi-hop communication is on the question of whether the network is mainly degraded when many participants come together and cause mutual interference or whether problems are more likely to arise due to having insufficient participants to relay messages through critical regions.

Two realisations of a telecommunications network with transmitters concentrated in three concentric bands and all trying to send an intelligible message to the centre of the domain. On the left all points can reach the centre directly (show in green) or by using a single green point as a relay (shown in blue). On the right an exponentially unlikely configuration with some red points that cannot reach the centre even via one green relay since there is a random gap in the middle ring. In this unlikely configuration there is a loss of symmetry.

Large deviations play also an important role in the approximation of the Smoluchowski equation with the empirical process of coagulating Brownian motions, see the Application Area Coagulation. The deviations of the approximating process from the equation is described, in the spirit of Wentsel-Freidlin theory, by an explicit rate function, which gives rise to the construction of gradient flows.



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

  Articles in Refereed Journals

  • M. Heida, M. Röger, Large deviation principle for a stochastic Allen--Cahn equation, Journal of Theoretical Probability, 31 (2018), pp. 364--401, DOI 10.1007/s10959-016-0711-7 .
    The Allen-Cahn equation is a prototype model for phase separation processes, a fundamental example of a nonlinear spatial dynamic and an important approximation of a geometric evolution equation by a reaction-diffusion equation. Stochastic perturbations, especially in the case of additive noise, to the Allen-Cahn equation have attracted considerable attention. We consider here an alternative random perturbation determined by a Brownian flow of spatial diffeomorphism that was introduced by Röger and Weber [Stoch. Partial Differ. Equ. Anal. Comput. 1 (2013)]. We first provide a large deviation principle for stochastic flows in spaces of functions that are Hölder-continuous in time, which extends results by Budhiraja, Dupuis and Maroulas [Ann. Probab. 36 (2008)]. From this result and a continuity argument we deduce a large deviation principle for the Allen-Cahn equation perturbed by a Brownian flow in the limit of small noise. Finally, we present two asymptotic reductions of the large deviation functional.

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

  • CH. Hirsch, B. Jahnel, R.I.A. Patterson, Space-time large deviations in capacity-constrained relay networks, ALEA. Latin American Journal of Probability and Mathematical Statistics, 15 (2018), pp. 587--615, DOI 10.30757/ALEA.v15-24 .
    We consider a single-cell network of random transmitters and fixed relays in a bounded domain of Euclidean space. The transmitters arrive over time and select one relay according to a spatially inhomogeneous preference kernel. Once a transmitter is connected to a relay, the connection remains and the relay is occupied. If an occupied relay is selected by another transmitters with later arrival time, this transmitter becomes frustrated. We derive a large deviation principle for the space-time evolution of frustrated transmitters in the high-density regime.

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

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

  • 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.
    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/ .
    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.

  • CH. Hirsch, B. Jahnel, P. Keeler, R.I.A. Patterson, Large deviations in relay-augmented wireless networks, Queueing Systems. Theory and Applications, pp. published online on 28.10.2017, url, DOI 10.1007/s11134-017-9555-9 .
    We analyze a model of relay-augmented cellular wireless networks. The network users, who move according to a general mobility model based on a Poisson point process of continuous trajectories in a bounded domain, try to communicate with a base station located at the origin. Messages can be sent either directly or indirectly by relaying over a second user. We show that in a scenario of an increasing number of users, the probability that an atypically high number of users experiences bad quality of service over a certain amount of time, decays at an exponential speed. This speed is characterized via a constrained entropy minimization problem. Further, we provide simulation results indicating that solutions of this problem are potentially non-unique due to symmetry breaking. Also two general sources for bad quality of service can be detected, which we refer to as isolation and screening.

  • 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 .
    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.

  • A. Mielke, R.I.A. Patterson, M.A. Peletier, D.R.M. Renger, Non-equilibrium thermodynamical principles for chemical reactions with mass-action kinetics, SIAM Journal on Applied Mathematics, 77 (2017), pp. 1562--1585, DOI 10.1137/16M1102240 .
    We study stochastic interacting particle systems that model chemical reaction networks on the micro scale, converging to the macroscopic Reaction Rate Equation. One abstraction level higher, we study the ensemble of such particle systems, converging to the corresponding Liouville transport equation. For both systems, we calculate the corresponding large deviations and show that under the condition of detailed balance, the large deviations induce a non-linear relation between thermodynamic fluxes and free energy driving force.

  • CH. Hirsch, B. Jahnel, P. Keeler, R.I.A. Patterson, Large-deviation principles for connectable receivers in wireless networks, Advances in Applied Probability, 48 (2016), pp. 1061--1094.
    We study large-deviation principles for a model of wireless networks consisting of Poisson point processes of transmitters and receivers, respectively. To each transmitter we associate a family of connectable receivers whose signal-to-interference-and-noise ratio is larger than a certain connectivity threshold. First, we show a large-deviation principle for the empirical measure of connectable receivers associated with transmitters in large boxes. Second, making use of the observation that the receivers connectable to the origin form a Cox point process, we derive a large-deviation principle for the rescaled process of these receivers as the connection threshold tends to zero. Finally, we show how these results can be used to develop importance-sampling algorithms that substantially reduce the variance for the estimation of probabilities of certain rare events such as users being unable to connect.

  • S. Jansen, W. König, B. Metzger, Large deviations for cluster size distributions in a continuous classical many-body system, The Annals of Applied Probability, 25 (2015), pp. 930--973.
    An interesting problem in statistical physics is the condensation of classical particles in droplets or clusters when the pair-interaction is given by a stable Lennard-Jones-type potential. We study two aspects of this problem. We start by deriving a large deviations principle for the cluster size distribution for any inverse temperature $betain(0,infty)$ and particle density $rhoin(0,rho_rmcp)$ in the thermodynamic limit. Here $rho_rmcp >0$ is the close packing density. While in general the rate function is an abstract object, our second main result is the $Gamma$-convergence of the rate function towards an explicit limiting rate function in the low-temperature dilute limit $betatoinfty$, $rho downarrow 0$ such that $-beta^-1logrhoto nu$ for some $nuin(0,infty)$. The limiting rate function and its minimisers appeared in recent work, where the temperature and the particle density were coupled with the particle number. In the de-coupled limit considered here, we prove that just one cluster size is dominant, depending on the parameter $nu$. Under additional assumptions on the potential, the $Gamma$-convergence along curves can be strengthened to uniform bounds, valid in a low-temperature, low-density rectangle.

  • D. Belomestny, M. Ladkau, J.G.M. Schoenmakers, Simulation based policy iteration for American style derivatives -- A multilevel approach, SIAM ASA J. Uncertainty Quantification, 3 (2015), pp. 460--483.
    This paper presents a novel approach to reduce the complexity of simulation based policy iteration methods for pricing American options. Typically, Monte Carlo construction of an improved policy gives rise to a nested simulation algorithm for the price of the American product. In this respect our new approach uses the multilevel idea in the context of the inner simulations required, where each level corresponds to a specific number of inner simulations. A thorough analysis of the crucial convergence rates in the respective multilevel policy improvement algorithm is presented. A detailed complexity analysis shows that a significant reduction in computational effort can be achieved in comparison to standard Monte Carlo based policy iteration.

  • 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.
    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.
    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/ρij 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.
    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 Zd. 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.
    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(zt,żt)=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.

  • W. König, Ch. Mukherjee, Large deviations for Brownian intersection measures, Communications on Pure and Applied Mathematics, 66 (2013), pp. 263--306.
    We consider $p$ independent Brownian motions in $R^d$. We assume that $pgeq 2$ and $p(d-2)<d$. Let $ell_t$ denote the intersection measure of the $p$ paths by time $t$, i.e., the random measure on $R^d$ that assigns to any measurable set $Asubset R^d$ the amount of intersection local time of the motions spent in $A$ by time $t$. Earlier results of Chen citeCh09 derived the logarithmic asymptotics of the upper tails of the total mass $ell_t(R^d)$ as $ttoinfty$. In this paper, we derive a large-deviation principle for the normalised intersection measure $t^-pell_t$ on the set of positive measures on some open bounded set $BsubsetR^d$ as $ttoinfty$ before exiting $B$. The rate function is explicit and gives some rigorous meaning, in this asymptotic regime, to the understanding that the intersection measure is the pointwise product of the densities of the normalised occupation times measures of the $p$ motions. Our proof makes the classical Donsker-Varadhan principle for the latter applicable to the intersection measure. A second version of our principle is proved for the motions observed until the individual exit times from $B$, conditional on a large total mass in some compact set $Usubset B$. This extends earlier studies on the intersection measure by König and Mörters citeKM01,KM05.

  • M. Becker, W. König, Self-intersection local times of random walks: Exponential moments in subcritical dimensions, Probability Theory and Related Fields, 154 (2012), pp. 585--605.
    Fix $p>1$, not necessarily integer, with $p(d-2)0$ that are bounded from above, possibly tending to zero. The speed is identified in terms of mixed powers of $t$ and $theta_t$, and the precise rate is characterized in terms of a variational formula, which is in close connection to the it Gagliardo-Nirenberg inequality. As a corollary, we obtain a large-deviation principle for $ ell_t _p/(t r_t)$ for deviation functions $r_t$ satisfying $t r_tggE[ ell_t _p]$. Informally, it turns out that the random walk homogeneously squeezes in a $t$-dependent box with diameter of order $ll t^1/d$ to produce the required amount of self-intersections. Our main tool is an upper bound for the joint density of the local times of the walk.

  • 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.
    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.
    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.

  • A. Collevecchio, W. König, P. Mörters, N. Sidorova, Phase transitions for dilute particle systems with Lennard--Jones potential, Communications in Mathematical Physics, 299 (2010), pp. 603--630.

  • 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.

  Contributions to Collected Editions

  • TH. Dickhaus, H. Finner, Asymptotic density crossing points of self-normalized sums and normal (electronic only), in: Proceedings of the 3rd Annual International Conference on Computational Mathematics, Computational Geometry and Statistics, CMCGS 2014, Singapore, February 3--4, 2014, Global Science and Technology Forum (GSTF), Singapore, pp. 84--88.

  Preprints, Reports, Technical Reports

  • M. Mittnenzweig, Hydrodynamic limit and large deviations of reaction-diffusion master equations, Preprint no. 2521, WIAS, Berlin, 2018, DOI 10.20347/WIAS.PREPRINT.2521 .
    Abstract, PDF (389 kByte)
    We derive the hydrodynamic limit of a reaction-diffusion master equation, that combines an exclusion process with a reversible chemical master equation expression for the reaction rates. The crucial assumption is that the associated macroscopic reaction network has a detailed balance equilibrium. The hydrodynamic limit is given by a system of reaction-diffusion equations with a modified mass action law for the reaction rates. We provide the upper bound for large deviations of the empirical measure from the hydrodynamic limit.

  • CH. Bayer, P. Friz, A. Gulisashvili, B. Horvath, B. Stemper, Short-time near-the-money skew in rough fractional volatility models, Preprint no. 2406, WIAS, Berlin, 2017, DOI 10.20347/WIAS.PREPRINT.2406 .
    Abstract, PDF (450 kByte)
    We consider rough stochastic volatility models where the driving noise of volatility has fractional scaling, in the "rough" regime of Hurst parameter H < ½. This regime recently attracted a lot of attention both from the statistical and option pricing point of view. With focus on the latter, we sharpen the large deviation results of Forde-Zhang (2017) in a way that allows us to zoom-in around the money while maintaining full analytical tractability. More precisely, this amounts to proving higher order moderate deviation estimates, only recently introduced in the option pricing context. This in turn allows us to push the applicability range of known at-the-money skew approximation formulae from CLT type log-moneyness deviations of order t1=2 (recent works of Alòs, León & Vives and Fukasawa) to the wider moderate deviations regime.

  • F. Flegel, M. Heida, M. Slowik, Homogenization theory for the random conductance model with degenerate ergodic weights and unbounded-range jumps, Preprint no. 2371, WIAS, Berlin, 2017, DOI 10.20347/WIAS.PREPRINT.2371 .
    Abstract, PDF (598 kByte)
    We study homogenization properties of the discrete Laplace operator with random conductances on a large domain in Zd. More precisely, we prove almost-sure homogenization of the discrete Poisson equation and of the top of the Dirichlet spectrum. We assume that the conductances are stationary, ergodic and nearest-neighbor conductances are positive. In contrast to earlier results, we do not require uniform ellipticity but certain integrability conditions on the lower and upper tails of the conductances. We further allow jumps of arbitrary length. Without the long-range connections, the integrability condition on the lower tail is optimal for spectral homogenization. It coincides with a necessary condition for the validity of a local central limit theorem for the random walk among random conductances. As an application of spectral homogenization, we prove a quenched large deviation principle for thenormalized and rescaled local times of the random walk in a growing box. Our proofs are based on a compactness result for the Laplacian's Dirichlet energy, Poincaré inequalities, Moser iteration and two-scale convergence

  • R.I.A. Patterson, D.R.M. Renger, Dynamical large deviations of countable reaction networks under a weak reversibility condition, Preprint no. 2273, WIAS, Berlin, 2016, DOI 10.20347/WIAS.PREPRINT.2273 .
    Abstract, PDF (343 kByte)
    A dynamic large deviations principle for a countable reaction network including coagulation--fragmentation models is proved. The rate function is represented as the infimal cost of the reaction fluxes and a minimiser for this variational problem is shown to exist. A weak reversibility condition is used to control the boundary behaviour and to guarantee a representation for the optimal fluxes via a Lagrange multiplier that can be used to construct the changes of measure used in standard tilting arguments. Reflecting the pure jump nature of the approximating processes, their paths are treated as elements of a BV function space.

  Talks, Poster

  • M. Maurelli, Sanov theorem for Brownian rough paths and an application to interacting particles, Università di Roma La Sapienza, Dipartimento di Matematica Guido Castelnuovo, Italy, January 18, 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.

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

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

  • 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.

  • M. Heida, Large deviation principle for a stochastic Allen--Cahn equation, 9th European Conference on Elliptic and Parabolic Problems, May 23 - 27, 2016, University of Zurich, Institute of Mathematics, Gaeta, Italy, May 25, 2016.

  • 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.

  • R.I.A. Patterson, Pathwise LDPs for chemical reaction networks, 12th German Probability and Statistics Days 2016 --- Bochumer Stochastik-Tage, February 29 - March 4, 2016, Ruhr-Universität Bochum, Fakultät für Mathematik, March 4, 2016.

  • D.R.M. Renger, Large deviations for reacting particle systems: The empirical and ensemble processes, Workshop ``Interplay of Analysis and Probability in Applied Mathematics'', July 26 - August 1, 2015, Mathematisches Forschungsinstitut Oberwolfach, Oberwolfach, July 30, 2015.

  • M. Maurelli, A large deviation principle for enhanced Brownian empirical measure, 4th Annual ERC Berlin-Oxford Young Researchers Meeting on Applied Stochastic Analysis, December 7 - 9, 2015, WIAS Berlin, December 8, 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.

  • 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 empirical process of reacting particles: Large deviations and thermodynamic principles, Minisymposium ``Real World Phenomena Explained by Microscopic Particle Models'' of the 8th International Congress on Industrial and Applied Mathematics (ICIAM 2015), August 8 - 22, 2015, International Council for Industrial and Applied Mathematics, Beijing, China, August 10, 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, Chemical Master Equation: Coarse graining via gradient structures, Kolloquium des SFB 1114 ``Scaling Cascades in Complex Systems'', Freie Universität Berlin, Fachbereich Mathematik, Berlin, June 4, 2015.

  • A. Mielke, Evolutionary $Gamma$-convergence for generalized gradient systems, Workshop ``Gradient Flows'', June 22 - 23, 2015, Université Pierre et Marie Curie, Laboratoire Jacques-Louis Lions, Paris, France, June 22, 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.

  • A. Mielke, The Chemical Master Equation as entropic gradient flow, Conference ``New Trends in Optimal Transport'', March 2 - 6, 2015, Universität Bonn, Institut für Angewandte Mathematik, March 2, 2015.

  • D.R.M. Renger, Connecting particle systems to entropy-driven gradient flows, Conference ``Stochastic Processes and High Dimensional Probability Distributions'', June 16 - 20, 2014, Euler International Mathematical Institute, Saint-Petersburg, Russian Federation, June 20, 2014.

  • 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.

  • TH. Dickhaus, H. Finner, Asymptotic density crossing points of self-normalized sums and normal, 3rd Annual International Conference on Computational Mathematics, Computational Geometry & Statistics (CMCGS 2014), February 3 - 4, 2014, Singapore, February 3, 2014.

  • T. Wolff, Annealed asymptotics for occupation time measures of a random walk among random conductances, ``Young European Probabilists 2013 (YEP X)'' and ``School on Random Polymers'', January 8 - 12, 2013, EURANDOM, Eindhoven, Netherlands, January 10, 2013.

  • W. König, Eigenvalue order statistics and mass concentration in the parabolic Anderson model, SFB/TR12 Workshop, November 4 - 8, 2012, Universität zu Köln, SFB TR12 ``Symmetries and Universality in Mesoscopic Systems'', Langeoog, November 7, 2012.

  • W. König, Large deviations for the cluster size distributions in a classical interacting many-particle system with Lennard--Jones potential, Mark Kac Seminar, Eindhoven University of Technology, Netherlands, November 9, 2012.

  • M. Becker , Random walks and self-intersections, Evolving Complex Networks (ECONS) Phd-Student Meeting, WIAS, August 24, 2010.

  • S. Jansen, M. Aizenman, P. Jung, Symmetry breaking in quasi 1D Coulomb systems, Combinatorics and Analysis in Spatial Probability --- ESF Mathematics Conference in Partnership with EMS and ERCOM, Eindhoven, Netherlands, December 12 - 18, 2010.

  • 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, Phase transitions for dilute particle systems with Lennard--Jones potential, University of Bath, Department of Mathematical Sciences, UK, April 14, 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

  • CH. Bayer, P. Friz, P. Gassiat, J. Martin, B. Stemper , A regularity structure for rough volatility, Preprint no. arXiv:1710.07481, Cornell University Library,, 2017.
    A new paradigm recently emerged in financial modelling: rough (stochastic) volatility, first observed by Gatheral et al. in high-frequency data, subsequently derived within market microstructure models, also turned out to capture parsimoniously key stylized facts of the entire implied volatility surface, including extreme skews that were thought to be outside the scope of stochastic volatility. On the mathematical side, Markovianity and, partially, semi-martingality are lost. In this paper we show that Hairer's regularity structures, a major extension of rough path theory, which caused a revolution in the field of stochastic partial differential equations, also provides a new and powerful tool to analyze rough volatility models.

  • 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,, 2016.