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March 30 / April 02 2015
Random Motion in Random Media
Realizations of onedimensional random walk trajectories in static and dynamic random environments, to the left and to the right, respectively. Both environments have two possible states corresponding to grey and white colours in the background of these pictures. The random walks have a local drift to the right/left on the grey/white states of the underlying environments. In this example, the dynamic random environment is given by a Simple Symmetric Exclusion process (SSE).
SUMMARY The study of random motion in random media has been the object of intensive
mathematical research over the last forty year. The main motivations originally
came from applied fields such as condensed matter, biochemistry and more
recently computer science. In the last decade, significant progress have been
made and new techniques and research lines have been introduced. This workshop
brings together leading researchers from different mathematical schools. The
primary goal is to offer an environment fostering the exchange of the most
recent ideas, and to highlight the current state of the art in this field. The ORGANISERS
LIST OF INVITED SPEAKERS* = to be confirmed
MONDAY, March 30
TUESDAY, March 31
WEDNESDAY, April 1
THURSDAY, April 2
***************************************************************************************************************************************** ABSTRACTS Erich Baur Random walks in random environment in the perturbative regime We consider nonballistic random walks in an i.i.d. random environment when the strength of the disorder is small. We present a result of Bolthausen and Zeitouni (2007) and a result with Erwin Bolthausen (2013) on exit distributions of such walks in dimensions three and higher. If time permits, we also look at mean sojourn times in large balls and discuss an invariance principle for walks which are balanced in one fixed coordinate direction. Francis Comets Maximum likelihood estimation for random walk in a parametric random environment We consider the parametric estimation of the environment distribution on the basis of a long observation of a single realization of a onedimensional random walk in this environment. This is a natural model for DNA sequencing. We will review asymptotic results for the maximum likelihood estimation of the environment. In the recurrent case, some parameters recieve an information proportional to the duration n of observation, while some other get information of order log^2(n). JeanDominique Deuschel Heat Kernel Estimate for symmetric random walks with degenerate weights We establish GaussianType upper bounds on the heat
kernel for a continuoustime random walk on a graph with unbounded weights under
an ergodicity assumption. Dmitry Dolgopyat Local Limit Theorem for sums independent bounded random variables We prove a local limit the sums of independent random variables satisfying appropriate tightness assumptions. In particular, the local limit theorem holds if the summands are uniformly bounded. Alessandra Faggionato Perturbations of Markov processes and applications We consider a Markov process obtained as a
timeindependent perturbation of a given stationary Markov process satisfying
Poincar\`e inequality. By perturbative arguments one can derive the existence of
an asymptotic steady state, a DysonPhilipps expansion of the expected values
of observables in the steady state, quenched invariance principles for additive
functionals (possibly with degenerate diffusion matrix). We then apply this
general results to random walks in a dynamical random environment, with
special emphasis to tracer particles in glassy systems, discussing special
symmetries for the asymptotic velocity, criteria leading to nondegenerate
diffusion and qualitative properties of the environment viewed from the
particle. Finally, we mention some further progresses concerning timeperiodic
perturbations, linear response and Nyquist formula. Nina Gantert Random walks among random conductances: Einstein relation and monotonicity of the speed We consider a random walk among iid, bounded
conductances and prove the Einstein relation which relates its diffusivity to
the derivative at zero of the velocity of a biased random walk among the same
random conductances. We show that the velocity is differentiable as a function
of the bias, and that it is in general not increasing. Ilya Goldsheid CLT for random walks in random environments on a strip . Milton Jara Hydrodynamic limits and random walks in random environments We explain how to use hydrodynamic limits to obtain
information about random walks in dynamic random environments. In particular we
obtain a law of large numbers and a large deviations principle for a slow
particle moving on a dynamic random environment given by a symmetric exclusion
process. Elena Kosygina Remarks on a homogenization problem for stochastic HJB equations with a nonconvex Hamiltonian In the last few years there appeared several works on stochastic homogenization of levelsetconvex and genuinely nonconvex inviscid HamitonJacobi equations. The progress so far is achieved by using predominantly PDE methods. The main goal of this talk is to attract attention of the probabilistic community to this important problem and suggest some open questions which might look attractive to probabilists. Some of these questions can be formulated in terms of controlled discrete spacetime random walks in random environments. Dasha Loukianova Hidden Markov model for parameter estimation of a random walk in a Markov environment We focus on the parametric estimation of the distribution of a Markov reversible environment from the observation of a single trajectory of a onedimensional nearestneighbor path evolving in this random environment. In the ballistic case, as the length of the path increases, we prove consistency, asymptotic normality and efficiency of the maximum likelihood estimator. Our contribution is twofold: we cast the problem into the one of parameter estimation in a hidden Markov model (HMM) and establish that the bivariate Markov chain underlying this HMM is positive Harris recurrent. We provide different examples of setups in which our results apply, in particular that of DNA unzipping model, and we give a simple synthetic experiment to illustrate those results. JeanChristophe Mourrat Heat kernel bounds for random walks in dynamic random environments I will present a technique based on the ideas of Nash
that enables to show heat kernel bounds for certain random walks in random
environments. Yoann Offret Invariant distributions and scaling limits for some dissusions in timevarying random environments I will introduce a model of timeinhomogeneous Brox’s diffusions, which generalizes the diffusion studied by Brox (1986) in the homogeneous case and those investigated by Gradinaru and Offret (2011) in deterministic media. I will show how these processes are connected, with the help of a suitable scaling transformation, to random perturbations of the Ornstein–Uhlenbeck diffusion process for which I prove quenched and annealed convergences in distribution under weighted total variation norms. I find two kind of stationary probability measures, which are either the standard normal distribution or a quasiinvariant measure, depending on the environment, and which is naturally connected to a underlying random dynamical system. Felix Otto The corrector in stochastic homogenization Stochastic homogenization is the analyst’s view upon invariance principles for random walks in random environments. A key concept is the corrector, which provides the appropriate harmonic coordinates. We have shown that in dimensions larger than 2, a stationary corrector exists (joint work with A. Gloria) and have characterized its covariance structure (joint work with J.C. Mourrat). Under minimal assumptions, the largescale regularity of aharmonic functions as encoded by Liouville principles is better than expected by the De GiorgiNashMoser theory (joint work with A. Gloria, S. Neukamm) Under mild assumptions, higherorder Liouville principles are valid and hint at a complete regularity theory in harmonic coordinates (work with J. Fischer). For both results, it is helpful to not just consider the scalar, but also the vector potential for the corrector (if considered as d independent harmonic vector fields) Jonathon Peterson Hydrodynamic limits for directed traps and systems of independent RWRE We study the evolution of a system of independent random
walks in a common random environment (RWRE). Previously a hydrodynamic limit was
proved in the case where the environment is such that the random walks are
ballistic (i.e., transient with nonzero speed $v_0 \neq 0$). In this case it
was shown that the asymptotic particle density is simply translated
deterministically by the speed $v_0$. In this talk we will consider the more
difficult case of RWRE that are transient but with $v_0=0$. Under the
appropriate spacetime scaling, we prove a hydrodynamic limit for the system of
random walks. The statement of the hydrodynamic limit that we prove is
nonstandard in that the evolution of the asymptotic particle density is given
by the solution of a random rather than a deterministic PDE. The randomness in
the PDE comes from the fact that under the hydrodynamic scaling the effect of
the environment does not ``average out'' and so the specific instance of the
environment chosen actually matters. Firas RassoulAgha The growth model: Busemann functions, shape, geodesics, and other stories We consider the directed lastpassage percolation model on the planar integer lattice with nearestneighbor steps and general i.i.d. weights on the vertices, outside the class of exactly solvable models. Stationary cocycles are constructed for this percolation model from queueing fixed points. These cocycles define solutions to variational formulas that characterize limit shapes and yield new results for Busemann functions, geodesics and the competition interface. This is joint work with Nicos Georgiou and Timo Seppalainen. Silke Rolles Localization for a nonlinear sigma model related to vertex reinforced jump processes We show exponential localization on a strip for a
nonlinear sigma model which was introduced by Zirnbauer. The proof uses a
deformation argument and a transfer operator approach. Christophe Sabot Diffusive properties of Random walks in Random Dirichlet Environment Dirichlet random environment is a special case of iid random environment
where at each site transition probabilities are chosen according to a Dirichlet
law. Its annealed law is that of a directed edge reinforced random walk. One
remarkable feature of Dirichlet environments is a property of statistical
invariance by time reversal that was used in particular to prove the existence
of an absolutely continuous invariant measure Daisuke Shiraishi Looperased random walk in three dimensions We prove the existence of the growth exponent for looperased random walk (LERW) in three dimensions. Moreover, we establish exponential tail bounds for the length of 3D LERW. If time permits, we also explain the relation between simple random walk, LERW and loop soup. Augusto Teixeira Random walk on random walks During this talk we will consider a motion on a
dynamical random environment composed of independent random walks. More
precisely, at time zero we place an i.i.d. number of particles at every site of
Z with Poisson(u) distribution. Then we let these particles perform independent,
discrete time random walks. On top of these particles, we place a random walker
X_n that moves to nearest neighbors with two jump distributions: one is used
when X_n sits above an environment particle and the other when it lies on an
empty site. The interest on this model comes from the conservation of particles
in this environment. This leads to long range correlations and requires new
ideas for its analysis. In this talk we will present a LLN and a CLT for such
walker, that holds under certain regimes of the model. Florian Völlering RWs in dynamic RE: A more detailed look at the environment By looking at possible trajectories which lead to the current position of the random walk we gain insight into the distribution of the environment surrounding the random walk. In particular equivalence of the stationary measure of the environment and of the environmentseenfromtherandomwalk and control on the density. Furthermore this approach helps to understand a more complicated model where the dynamic environment itself lives in a static random medium. PRACTICAL INFORMATION ● VenueEurandom, Mathematics and Computer Science Dept, TU Eindhoven, Den Dolech 2, 5612 AZ EINDHOVEN, The Netherlands
Eurandom is located on the campus of
Eindhoven University of
Technology, in the
Metaforum building
(4th floor) (about
the building). The university is
located at 10 minutes walking distance from Eindhoven main railway station (take
the exit north side and walk towards the tall building on the right with the
sign TU/e).
● RegistrationRegistration is free, but compulsory for speakers and participants. Please follow the link: REGISTRATION PAGE
● AccommodationFor invited participants, we will take care of accommodation. Other attendees will have to make their own arrangements. We have a preferred hotel, which can be booked at special rates. Please email Patty Koorn for instructions on how to make use of this special offer. For other hotels around the university, please see: Hotels (please note: prices listed are "best available"). More hotel options can be found on the webpages of the Tourist Information Eindhoven, Postbus 7, 5600 AA Eindhoven.
● TravelFor those arriving by plane, there is a convenient direct train connection between Amsterdam Schiphol airport and Eindhoven. This trip will take about one and a half hour. For more detailed information, please consult the NS travel information pages or see Eurandom web page location. Many low cost carriers also fly to Eindhoven Airport. There is a bus connection to the Eindhoven central railway station from the airport. (Bus route number 401) For details on departure times consult http://www.9292ov.nl The University can be reached easily by car from the highways leading to Eindhoven (for details, see our route descriptions or consult our map with highway connections.
● Conference facilities : Conference room, Metaforum Building MF11&12The meetingroom is equipped with a data projector, an overhead projector, a projection screen and a blackboard. Please note that speakers and participants making an oral presentation are kindly requested to bring their own laptop or their presentation on a memory stick.
● Conference SecretariatUpon arrival, participants should register with the workshop officer, and collect their name badges. The workshop officer will be present for the duration of the conference, taking care of the administrative aspects and the daytoday running of the conference: registration, issuing certificates and receipts, etc.
● CancellationShould you need to cancel your participation, please contact Patty Koorn, the Workshop Officer. There is no registration fee, but should you need to cancel your participation after January 2, 2014, we will be obliged to charge a noshow fee of 30 euro.
● ContactMrs. Patty Koorn, Workshop Officer, Eurandom/TU Eindhoven, koorn@eurandom.tue.nl SPONSORSThe organisers acknowledge the financial support/sponsorship of:
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