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Winter School“Stochastic Analysis of SpatiallyExtended Models”March 23-27, 2015TU Darmstadt
Short Courses Invited Speakers OrganizationZhan Shi Massimiliano Gubinelli Volker BetzHendrik Weber Simon C. Harris Tadahisa Funaki
Nicola Kistler Matthias MeinersNicolas Perkowski
March 19, 2015
Contents
1 General Information 21.1 Accommodation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Registration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Lecture Hall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.4 Map & Points of Interest . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.5 Public Transportation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.6 Food & Beverage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.7 Conference Dinner . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.8 Free Afternoon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.9 Contact Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
Acknowledgements 5
2 List of Talks 132.1 Short Courses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.2 Invited Speakers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.3 Further Speakers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3 Participants 20
1
1 General Information
1.1 Accommodation
The participants are recommended to stay in one of the following hotels, locatedin walking distance (5 - 15 minutes) to the lecture venue.
• WELCOME HOTEL DARMSTADTKarolinenplatz 464289 DarmstadtTel: +49-6151-3914-0Fax: +49-6151-3914-444
• HOTEL AtlantaKasinostraße 12964293 DarmstadtTel: +49-6151-1789-0Fax: +49-6151-1789-66
• HOTEL BockshautKirchstraße 7-964283 Darmstadt-InnenstadtTel: +49-6151-99670Fax: +49-6151-99 67-29
For directions please see the map in Section 1.4.
1.2 Registration
On Monday morning, starting from 8:00, registration is possible in the lobby of thelecture hall.
1.3 Lecture Hall
Location: Technische Universität Darmstadt. The registration and all lectures willtake place in building S2|04, Hochschulstraße 8 | 64289 Darmstadt in lecture hallS2|04/213. In the lecture hall, there are 2 large and 4 small blackboards, anoverhead projector and a beamer.
2
1.4 Map & Points of Interest
The map can be found on the last page.
1.5 Public Transportation
The closest bus and tram stops to the venue of the workshop are Schloss (trams:S2, S3, S9) and Willy-Brandt-Platz (trams: S4, S5, S6, S7, S8). Both stops arewithin 10 minutes walking distance to the lecture hall.
1.6 Food & Beverage
Cheap and plain food can be purchased at the TU Darmstadt Refectory-Canteen,building S1|11 (see map), Monday to Friday 11:15 to 14:00. Additionally thereare lots of good restaurants and bistros near TU Darmstadt. Please dial 0049 6151preceding the number given below.
Name Address Phone Cuisine Opening Hours
Ratskeller Marktplatz 8 26444 German 10:00 - 24:00Pizzeria da Nino Alexanderstr. 29 24220 Italian 12:00 - 23:00
Haroun’s Friedensplatz 6 23487 Oriental 11:00 - 01:00Vis à Vis Furhmannstr. 2 9670806 Bistro 10:00 - 15:30
Central Station Carree 809460 Bistro 12:00 - 14:30Ristorante Sardegna Kahlertstraße 1 23029 Italian 11:30 - 15:00
1.7 Conference Dinner
On Thursday, March 26th there will be a conference dinner at the RistoranteSardegna, Kahlertstraße 1, 64293 Darmstadt, Telefon: 06151 / 23029.
1.8 Free Afternoon
On Wednesday, March 25th there will be a free afternoon.
1.9 Contact Information
If you have any questions concerning the workshop, please feel free to contact oneof the local organizers or the technical support:
• Prof. Dr. Volker BetzOffice: S2-15, Room 340Phone: +49 (0) 6151 - 16 2288
1.4 Map & Points of Interest 3
• Jun.-Prof. Dr. Matthias MeinersOffice: S2-15, Room 351Phone: +49 (0) 6151 - 16 76199
• Office DepartmentOffice: S2-15, Room 339Phone:+49 (0) 6151 - 16 5343 / or 3638
4 1 General Information
Acknowledgements
Financial support by the ESF and the DFG German Research Foundation, the De-partment of Mathematics at Technische Universität Darmstadt, and the Universityof Münster (via the dissertation prize 2009) is gratefully acknowledged.
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2 List of Talks
2.1 Short Courses
Zhan ShiMini course: Branching Brownian motion
Université Paris VI , France
Abstract: This is an elementary introduction to branching Brownian motion andbranching random walks. The simple but useful tool of the spinal decompositionis presented. Some asymptotic properties of extreme values are studied. Time per-mitting, I will also make discussions on a few related models, such as branchingprocesses in the presence of a selection criterion, or biased random walks on trees.
13
Hendrik WeberMini course on regularity structures and SPDEs
University of Warwick, United Kingdom
Abstract: In this series of lectures I will report on recent progress in the theoryof stochastic PDEs. One main aim is to explain some ideas behind the theory of"Regularity structures" developed recently by M. Hairer. This theory gives a wayto study well-posedness for a class of stochastic PDEs that could not be treatedpreviously. Prominent examples include the KPZ equation as well as the dynamicΦ4
3 model.Such equations were treated previously as perturbative expansions. Roughly
speaking the theory of regularity structures provides a way to truncate this expan-sion after finitely many terms and to solve a fixed point problem for the "remain-der". The key ingredient is a new notion of "regularity" which is based on the termsof this expansion.
I will also discuss how these ideas can be used to study scaling limits of interact-ing particle systems.
14 2 List of Talks
2.2 Invited Speakers
Massimiliano GubinelliSingular Stochastic PDEs and paracontrolled distributions
Université Paris Dauphine, France
Abstract: Non-linear evolution problems perturbed by singular noise sources arisenaturally as scaling limits of certain microscopic evolutions or homogenisationproblems. The parabolic anderson model, the Kardar-Parisi-Zhang equation andthe stochastic quantisation equation are examples of such systems. Solving (oreven giving a meaning to) these equations require a detailed understanding of thepropagation of the stochastic perturbations via the non-linear evolution. I will ex-plain how ideas and tools from harmonic analysis can be useful in this analysis andin the related problem of studying the convergence of small scale models to theirscaling limits.
Simon C. HarrisBranching Brownian motion with killing
University of Bath, United Kingdom
Abstract: We will consider the survival near criticality in a branching Brownianmotion model with killing. In particular, we will discuss the asymptotic survivalprobability of BBM with killing at the boundaries of a strip as the strip’s widthdecreases down to criticality, making use of a probabilistic decomposition of thebranching process into a ‘blue tree’ of immortal particles which is dressed with redtrees that each eventually become extinct. (Based on joint work with M.Hesse andA.Kyprianou)
2.2 Invited Speakers 15
Nicola KistlerA multiscale refinement of the 2nd moment method
Goethe-Universität Frankfurt, Germany
Abstract: The 2nd moment method is arguably the only universal tool to addressthe question of the extremes of large combinatorial structures. More often than not,however, a plain application of the method falls short - this is typically a manifes-tation of severe underlying correlations. I will present a refinement of the methodwhich is particularly efficient in order to tackle the extremes of correlated randomfields where multiple scales can be identified. The refinement relies on elemen-tary steps only, and applies equally well to a number of models, such as branchingBrownian motion, the 2-dim Gaussian free field, the cover time by planar Brownianmotion, etc. The framework also provides a good starting point towards a rigoroustreatment of the extremes of the characteristic polynomial of CUE matrices and theRiemann zeta-function along the critical line. Based on works with L.-P. Arguin(CUNY), D. Belius (NYU) and A. Bovier (BONN).
Nicolas PerkowskiAn invariance principle for the 2d parabolic Anderson model with small potential
Humboldt Universität zu Berlin, Germany
Abstract: We show that the two-dimensional lattice parabolic Anderson model(PAM) with small potential converges under parabolic rescaling weakly to thecontinuum PAM, universally for all centered i.i.d. potentials with sufficiently manymoments. The proof is based on paracontrolled distributions and extends to certainnonlinear generalizations of PAM. Joint work with Khalil Chouk and Jan Gairing.
16 2 List of Talks
2.3 Further Speakers
André de Oliveira GomesFirst Exit Times for Jump Diffusions
Humboldt-Universität zu Berlin, Germany
Abstract: Lévy noises (or jump noises) are used to describe huge oscillations suchas the sudden fresh-water release by huge icebergs, so called Heinrich events, orlarge fluctuations of the solar radiation steered by huge fluid outbursts on the sur-face of the sun. In climate dynamics, it is being effectively used dynamical systemsperturbed by jump noises to capture abrupt changes of temperature in very smalltime scales in comparison with the time horizon of the study and the comprehen-sion of the first exit time problem is crucial to understand the occurrence of thoseabrupt changes of temperatures. In this setting the big jumps of the jump noiseperturbation are used to describe the rapid catastrophic climate changes occurredin the Earth’s northern hemisphere (the so called Daansgard-Oeschger events). It isour intention to describe the problem of the First Exit Time for dynamical systemsperturbed in low intensity with jump noise. The problem of the First Exit Timecan be viewed as the problem of understanding probabilistically (saying somethingabout the expected value and the law) of the first time a certain stochastic processleaves a pre-established domain. It is known that a dynamical system never leavesthe domain of attraction of a stable state after some time and it is remarkable thatthe excitation of the same dynamical system even in low intensity by a source ofrandomness ables the particle described by it to leave the same domain. With thiskind of stochastic perturbation, it becomes interesting the problem of transition be-tween the domains of attraction of the stable equilibria of the deterministic system.So the stable states become meta-stable and the stochastic perturbation determinestheir asymptotic dynamical properties. We will present a survey of results for theFirst Exit Times Problem for Jump Diffusions, doing remarks with ongoing workon this subject and confronting with the classical results for the continuous sourceof perturbation captured by White Noise (the so called Freidlin-Wentzell theory).After presenting the results available in one dimension, we will address the prob-lem in d-dimensions, in privileged directions, using techniques of separating thenoise according to a given threshold for the sizes of the jumps and secondly wewill address the most interesting problem of a Lévy isotropic noise with exponen-tially lighted jumps, using techniques the Weak convergence Approach to LargeDeviations Theory.
2.3 Further Speakers 17
Dai NoboriguchiA kinetic formulation for stochastic scalar conservation laws with boundary
conditionsWaseda University, Japan
Abstract: In this talk we consider the first order stochastic scalar conservation lawof the following type
du+ div(A(u))d t = Φ(u)dW (t) in (0, T )× D,
with the initial condition
u(0, ·) = u0(·) on D,
and the formal boundary condtion
“u= ub” in (0, T )× ∂ D.
Here D ⊂ Rd is a bounded domain with a Lipschitz boundary ∂ D, T > 0, and W isa one-dimensional Brownian motion defined on a stochastic basis (Ω,F , Ft,P).
We introduce definition of kinetic solutions characterized by kinetic formula-tions with the boundary defect measures. In such a solution, we obtain results ofuniqueness and existence.
Martin RedmannSPA applied to SPDEs
Max-Planck-Institut für Dynamik komplexer technischer Systeme Magdeburg,Germany
In this talk we consider a controlled infinite dimensional system equipped with anoutput equation:
dX (t) = [AX (t) + Bu(t)] d t + NX (t−)dM(t), (2.1)
Y (t) = CX (t), t ≥ 0,
where A generates a contraction semigroup, B ∈ L(Rm, H), C ∈ L(H,Rp), N ∈L(H, H), M is a scalar Lévy process and H is a Hilbert space. Using a finite dimen-sional approximation of (2.1) and the singular perturbation approximation (SPA),we obtain a system
dX (t) = [AX (t) +Bu(t)] d t +N X (t−)dM(t), (2.2)
Y (t) =CX (t), t ≥ 0,
18 2 List of Talks
withA ∈ Rr×r ,B ∈ Rr×m, C ∈ Rp×r , N ∈ Rr×r having a small state dimension r.Of course, the system (2.2) should be chosen in a meaningful way such that Y andY are close in a certain norm. To see this we provide an error bound for the SPAand present an example to illustrate the quality of the method.
Stephen TateVirial Expansion Bounds from Tree Partition Schemes (joint work with Mr S.
Ramawadh)Imperial College London, United Kingdom
Abstract: The virial expansion is the power series development of pressure in termsof density. The coefficients of this expansion is usually written in terms of weightedtwo-connected graphs. In this work, we prove an alternative description of thevirial coefficients in terms of weighted tree graphs. This is done through the al-gebraic relationship between the virial coefficients and the cluster coefficients andpartition schemes for weighted connected graphs. The advantage of a tree parti-tion scheme is the ability to make easier estimates on the integrals that need to becomputed and it requires significantly fewer calculations.
Jonas M. TölleStability and rescaling of singular nonlinear SPDEs with nonlocal drift, joint work
with Benjamin Gess, Chicago/BielefeldUniversität Bielefeld , Germany
Abstract: We are studying SPDEs with nonlinear, singular drift term which is givenby a convolution type nonlocal p-Laplacian integral operator. We obtain existenceand uniqueness of solutions in the sense of stochastic variational inequalities. Foradditive Gaussian forcing, under rescaling of the kernels, we prove the convergenceof solutions and the RDS to the local SPDE. We also obtain convergence of invariantdistributions of the associated stochastic flows. Possibly multi-valued SPDEs withlocal, highly singular drift term are briefly discussed.
2.3 Further Speakers 19
3 Participants
Aurzada, Frank Fachbereich Mathematik, Technische Universität Darmstadt, Schlossgartenstr. 7,64289 Darmstadt, Germany, [email protected]
Betz, Volker Fachbereich Mathematik, Technische Universität Darmstadt, Schlossgartenstr. 7,64289 Darmstadt, Germany, [email protected]
Birkner, Matthias Johannes-Gutenberg-Universität Mainz,Institut für Mathematik, Staudingerweg9, 55099 Mainz, Germany, [email protected]
Bott, Ann-Kathrin Fachbereich Mathematik, Technische Universität Darmstadt, Schlossgartenstr. 7,64289 Darmstadt, Germany, [email protected]
Cannizzaro, Giuseppe Institut für Mathematik, Technische Universität Berlin, Straße des 17. Juni136, 10623 Berlin, Germany, [email protected]
Dalinger, Alexander Fachbereich Mathematik, Technische Universität Darmstadt, Schlossgartenstr.7, 64289 Darmstadt, Germany, [email protected]
Depperschmidt, Andrej Abteilung für Mathematische Stochastik, Universität Freiburg, Eckerstr. 1,79104 Freiburg i. Br., Germany, [email protected]
de Oliveira Gomes, André Institut für Mathematik, Humboldt University zu Berlin, Unter den Lin-den 6 10099 Berlin„ Germany, [email protected]
Funaki, Tadahisa Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1, Komaba,Meguro-ku, Tokyo, 153-8914, Japan, [email protected]
Gubinelli, Massimiliano Université Paris Dauphine, Place du Maréchal de Lattre de Tassigny,75775 PARIS Cedex 16, France, [email protected]
Hammami, Mohamed Ali Department of Mathematics, Faculty of Sciences of Sfax, , Rte Soukra BP1171, Sfax 3000, Tunisia, [email protected]
Harris, Simon Colin Department of Mathematical Sciences, University of Bath, 4W.3.36 Bath, BA27AY, United Kingdom, [email protected]
Hoshino, Masato The University of Tokyo, 3-8-1 Komaba Meguro-ku, Tokyo 153-8914, Japan,[email protected]
Jeribi, Aref Department of Mathematics, Faculty of Sciences of Sfax, , Rte Soukra BP 1171, Sfax3000, Tunisia, [email protected]
Kistler, Nicola Institut für Mathematik, Goethe-Universität Frankfurt, Robert-Mayer-Str. 10, 60325Frankfurt, Germany, [email protected]
Kohler, Michael Fachbereich Mathematik, Technische Universität Darmstadt, Schlossgartenstr. 7,64289 Darmstadt, Germany, [email protected]
Kreiß, Alexander Fachbereich Mathematik, Technische Universität Darmstadt, Schlossgartenstr. 7,64289 Darmstadt, Germany, [email protected]
Kristl, Lisa Fachbereich Mathematik, Technische Universität Darmstadt, Schlossgartenstr. 7, 64289Darmstadt, Germany, [email protected]
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Lübbers, Jan-Erik Fachbereich Mathematik, Technische Universität Darmstadt, Schlossgartenstr. 7,64289 Darmstadt, Germany, [email protected]
Maiwald, Martin Institut für Mathematische Statistik, Universität Münster, Orléans-Ring 10, 48149Münster, Germany, [email protected]
Meiners, Matthias Fachbereich Mathematik, Technische Universität Darmstadt, Schlossgartenstr. 7,64289 Darmstadt, Germany, [email protected]
Mönch, Christian Fachbereich Mathematik, Technische Universität Darmstadt, Schlossgartenstr. 7,64289 Darmstadt, Germany, [email protected]
Müller, Florian Fachbereich Mathematik, Technische Universität Darmstadt, Schlossgartenstr. 7,64289 Darmstadt, Germany, [email protected]
Noboriguchi, Dai Waseda University, 1-6-1 Nishi-Waseda,Shinjuku-ku, Tokyo, 169-8050, Japan,[email protected]
Perkowski, Nicolas Institut für Mathematik, HU Berlin, Unter den Linden 6, 10099 Berlin, Ger-many, [email protected]
Redmann, Martin Max-Planck-Institut für Dynamik komplexer technischer Systeme Magdeburg,SandtorstraSSe 1, 39106 Magdeburg, Germany, [email protected]
Saal, Martin Fachbereich Mathematik, Technische Universität Darmstadt, Schlossgartenstr. 7,64289 Darmstadt, Germany, [email protected]
Schäfer, Helge Fachbereich Mathematik, Technische Universität Darmstadt, Schlossgartenstr. 7,64289 Darmstadt, Germany, [email protected]
Shi, Zhan Université Paris VI, Laboratoire de Probabilités et Modèles aléatoires, 4 place Jussieu,75252 Paris cedex 05, France, [email protected]
Sivak, Iryna Ecole Polytechnique, Route de Saclay, 91128 Palaiseau Cedex, France,[email protected]
Tate, Stephen James Department of Mathematics, Imperial College London, South KensingtonCampus, London SW7 2AZ, United Kingdom, [email protected]
Tent, Reinhard Fachbereich Mathematik, Technische Universität Darmstadt, Schlossgartenstr. 7,64289 Darmstadt, Germany, [email protected]
Tölle, Jonas M. Faculty of Mathematics, Universität Bielefeld, Postfach 100131, 33501 Bielefeld,Germany, [email protected]
Tsatsoulis, Pavlos Mathematics Department, The University of Warwick, Zeeman Building, Coven-try, CV4 7AL, United Kingdom, [email protected]
von der Lühe, Katharina Department of Mathematics, Universität Bielefeld, Universitätstr. 25,33615 Bielefeld, Germany, [email protected]
Walter, Stefan Fachbereich Mathematik, Technische Universität Darmstadt, Schlossgartenstr. 7,64289 Darmstadt, Germany, [email protected]
Weber, Hendrik Mathematics Institute, University of Warwick, Zeeman Building, Coventry CV47AL, United Kingdom, [email protected]
Wresch, Lukas Fakultät für Mathematik, Universität Bielefeld, Universitätsstraße 25, 33615 Biele-feld, Germany, [email protected]
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Figure 3.1: Citymap
Points of interest:
• Hotel Atlanta
• Lecture Hall
• Welcome Hotel
• Canteen
• Darmstadt City (Carree)
• Mathildenhöhe
• Schloss
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