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 10 - 14 MAR, 2014   YOUNG EUROPEAN PROBABILISTS    (YEP XI) Workshop on “Mass transport in analysis and probability”

The Young European Probabilists Meeting 2014 will be the 11th of a series of successful yearly workshops at Eurandom, Eindhoven. Other than most previous YEP workshops the workshop of 2014 will have an intradisciplinary flavour, focussing on problems at the interface of probability and analysis. The aim of the workshop is to bring together young researchers from probability and analysis, and to provide a forum for the exchange of ideas and a starting point for future intradisciplinary collaborations.

 SUMMARY SPEAKERS PROGRAMME REGISTRATION ABSTRACTS PARTICIPANTS

SUMMARY

Mass transport is concerned with the transport of mass between prescribed distributions at minimal cost. Although dating back to work of Monge in the 18th century, this problem has recently seen a major boost of activity with many new ideas emerging. A particular focal point of this activity is the interface of analysis (conservation laws, gradient flows) and probability (large deviations, allocation problems, coupling). In our workshop we aim to bring together young researchers from both areas, provide a forum for the exchange of ideas and a starting point for future intradisciplinary collaborations.

The workshop consists of four mini-courses by eminent researchers in the field, augmented with 16 talks of young researchers who will talk about their own research topic. There will be an extended minicourse by Nicola Gigli (Université de Nice), who will give an introduction to optimal transport from an analytic point of view. Further there will be three shorter minicourses by eminent researchers introducing their recent work in the area, by Mathias Beiglböck (University of Vienna and Universität Bonn), Jan Maas (Universität Bonn), and Gero Friesecke (Technische Universität München).

ORGANISERS

 Peter Mörters University of Bath Michiel Renger WIAS Berlin Max von Renesse Universität Leipzig

Mini-courses

 Mathias Beiglböck Universität Wien and Universität Bonn Gero Friesecke Technische Universität München Nicola Gigli Université de Nice Jan Maas Universität Bonn

Invited speakers (confirmed)

 Giovanni Bonaschi Technische Universiteit Eindhoven Gioia Carinci Università degli studi di Modena e Reggio Emilia Fabio Cavalletti Rheinisch-Westfälische Technische Hochschule Aachen Bertrand Cloez Université de Toulouse Simone Di Marino Scuola Normale Superiore di Pisa Yan Dolinsky Hebrew University of Jerusalem Matthias Erbar Scuola Normale Superiore di Pisa Max Fathi Université Pierre et Marie Curie - Paris VI Martin Huesmann Universität Bonn Emanuel Indrei Carnegie Mellon University Pittsburgh Nicolas Juillet Université de Strasbourg Richard Kraaij Technische Universiteit Delft Andrea Mondino ETH Zürich Harald Oberhauser Oxford-Man Institute of Quantitative Finance Xiaolu Tan CEREMADE Université Paris-Dauphine IX Christoph Thäle Ruhr-Universität Bochum

Monday March 10

 09.00 - 09.50 Registration & coffee 09.50 - 10.00 Opening 10.00 - 11.00 Nicola Gigli I 11.00 - 12.00 Gero Friesecke I Optimal transport with Coulomb cost: theory and applications to electronic structure of atoms and molecules 12.00 - 14.00 Lunch 14.00 - 15.00 Nicola Gigli II 15.00 - 16.00 Gero Friesecke II Optimal transport with Coulomb cost: theory and applications to electronic structure of atoms and molecules 16.00 - 17.00 Martin Huesmann Optimal transport between random measures 17.00 - 18.00 Christoph Thäle Functional Poisson approximation and optimal transport 18.00 Reception

Tuesday March 11

 09.00 - 10.00 Mathias Beiglböck I Optimal Transport, Martingales, and Skorokhod embedding 10.00 - 11.00 Gero Friesecke III Optimal transport with Coulomb cost: theory and applications to electronic structure of atoms and molecules 11.00 - 12.00 Nicola Gigli III 12.00 - 14.00 Lunch 14.00 - 15.00 Jan  Maas I Optimal transport in discrete and quantum systems 15.00 - 16.00 Giovanni Bonaschi Equivalence of gradient flows and entropy solutions for singular nonlocal interaction equations in 1D 16.00 - 17.00 Simone Di Marino Multimarginal optimal transportation: the one dimensional symmetric case 18.30 Dinner

Wednesday March 12

 09.00 - 10.00 Jan Maas II Optimal transport in discrete and quantum systems 10.00 - 11.00 Mathias Beiglböck II Optimal Transport, Martingales, and Skorokhod embedding 11.00 - 12.00 Fabio Cavalletti Decomposition of Wasserstein geodesics 12.00 - 14.00 Lunch 14.00 - 15.00 Emanuel Indrei 15.00 - 16.00 Harald Oberhauser On the Skorokhod embedding problem 16.00 - 17.00 Xiaolu Tan Martingale transport and peacocks

Thursday March 13

 09.00 - 10.00 Mathias Beiglböck III Optimal Transport, Martingales, and Skorokhod embedding 10.00 - 11.00 Jan Maas III Optimal transport in discrete and quantum systems 11.00 - 12.00 Gioia Carinci Mass transport via current reservoirs: a microscopic model for a free boundary problem 12.00 - 14.00 Lunch 14.00 - 15.00 Andrea Mondino 15.00 - 16.00 Matthias Erbar 16.00 - 17.00 Nicolas Juillet An optimal transport problem for two measures in the convex order

Friday March 14

(note: change of location - "Dorgelo" room in Traverse building)

 09.00 - 10.00 Max Fathi Quantitative rates of convergence to the hydrodynamic limit 10.00 - 11.00 Richard Kraaij A Lagrangian formalism for large deviations of Feller processes 11.00 - 12.00 Bertrand Cloez Wasserstein curvature of Markov processes 12.00 - 13.00 Yan Dolinsky Hedging of Game Options under Model Uncertainty in Discrete Time 13.00 - 14.30 Lunch

ABSTRACTS

Mathias Beiglböck (mini course)

Optimal Transport, Martingales, and Skorokhod embedding

We will explain a recently discovered connection between Optimal Transport and the areas of model independence / martingale inequalities in probability. This link has a number of fruitful consequences. For instance, the duality theorem from optimal transport leads to new super-replication results. Optimality criteria from the theory of mass transport can be translated to the martingale setup and allow to characterize minimizing/maximizing models in finance. Moreover, the transport viewpoint provides new insights to the inequalities of Doob / Burkholder-Davis-Gundy and the classical Skorokhod embedding problem.

Giovanni Bonaschi

Equivalence of gradient flows and entropy solutions for singular nonlocal interaction equations in 1D

Recently many systems have been shown to be gradient flows w.r.t. the Wasserstein metric. Also the notion of Entropy solution of a Burgers-type scalar conservation law has been studied extensively. Though the solutions might be different in this talk I present a specific  case  wherein the solutions to the systems are equivalent.

PRESENTATION

Gioia Carinci

Stationary non equilibrium states are characterized by the presence of steady currents

flowing through the system and a basic question in statistical mechanics is to understand their structure. Many papers have been devoted to the subject in the context of stochastic interacting particle systems. Usually current density is produced by fixing two different densities at the boundary. We want instead to implement the mass transport by introducing current reservoirs that produce a given current by sending in particles from the left at some rate and taking out particles from the rightmost occupied site at same rate. The removal mechanism is therefore of topological rather than metric nature, since the determination of the rightmost occupied site requires a knowledge of the entire configuration. This prevents from using correlation functions techniques. I will discuss recent results obtained in the study of these topics, whose final purpose is to provide a particle version of a free boundary-type problem.

PRESENTATION

Fabio Cavalletti

Decomposition of Wasserstein geodesics

We present a decomposition result for geodesics of the L2-Wasserstein space over a metric measure space (X, d, m) verifying a local synthetic Ricci curva- ture lower bound, CDloc (K, N ) in brief, for some K and N . We then present two applications. The first one is to the globalization problem, that is if (X, d, m) verifies CDloc (K, N ) then (X, d, m) also enjoys CD(K, N ). We obtain globalization for a well prepared family of optimal transport. The second one is to L1 transport problems in RCD(K, N ). Existence of optimal maps is proved via some fine analysis on the set of branching geodesics.

PRESENTATION

Bertrand Cloez

Wasserstein curvature of Markov processes

The Wasserstein curvature has been introduced recently by Joulin, Ollivier and Sammer, in order to have a better understanding of Markov processes with jumps, and of the geometry of discrete space. In this talk, we introduce this notion of curvature and give some examples. As application, we give some results on the long time behavior of Markov processes with random switching.

PRESENTATION

Simone Di Marino

Multimarginal optimal transportation: the one dimensional symmetric case

After the introduction of the physical problem that inspires the symmetric multimarginal problem (strongly interacting DFT), there will be an introduction to classical (2 marginal) optimal trasport problem in
1-dimension. Then the talk will focus on the multimarginal problem in one dimension for a symmetric, repulsive convex cost function, showing the existence of an optimal map also in this case. The seminar
is based on a work done in collaboration with Maria Colombo and Luigi
De Pascale.

PRESENTATION

Yan Dolinsky

Hedging of Game Options under Model Uncertainty in Discrete Time

We introduce a setup of model uncertainty in discrete time. In this setup we derive dual expressions for the super-replication prices of game options. We show that the super-replication price is equal to the supremum over a special (non dominated) set of martingale measures of the corresponding Dynkin game values. This type of results is also new for American options.

Matthias Erbar

Curvature effects for infinite particle systems via optimal transport

This talk is concerned with the geometry of configuration spaces, the natural state spaces of infinite particle systems. We consider a system of independent Brownian particles with drift on a manifold M. The configuration space Conf(M) of locally finite counting measures on M inherits a differentiable stucture from the base space in such a way that the natural diffusion on this infinite-dim. Riemannian manifold is the particle system. An invariant measure is given by the Poisson point process on M.
We are interested in curvature properties of the manifold Conf(M). More precisely, we will show that various manifestations of lower bounded (Ricci-) curvature extend from the base space to the configuration space. This includes gradient estimates for heat semigroup and convexity of the entropy along L^2 transport geodesics. In particular, configuration spaces present a natural new example of infinite-dimensional metric measure spaces with curvature bounds in the sense of Lott--Villani and Sturm.

PRESENTATION

Max Fathi

Quantitative rates of convergence to the hydrodynamic limit

Deriving a hydrodynamic limit consists in rigorously obtaining some macroscopic equation (for example, the heat equation) as the scaling limit of a large system of interacting particles. In 2009, Grunewald, Otto, Villani and Westdickenberg proposed a new method to obtain quantitative rates of convergence to the hydrodynamic limit in Wasserstein distance for random systems. In this talk, I will present this method, and various extensions obtained in joint works with M. H. Duong and G. Menz.

PRESENTATION

Gero Friesecke (mini course)

Optimal transport with Coulomb cost: theory and applications to electronic structure of atoms and molecules

Density functional theory (DFT) is a computationally feasible electronic structure model which simplifies full quantum mechanics and for which Walter Kohn received a Nobel prize in 1998. In the semiclassical limit, DFT reduces to a multi-marginal optimal transport problem with Coulomb cost 1/|x-y| [1]. Considerable insight into the limit problem had been built up, prior to our work, by physicists (Seidl, Perdew, Levy, Gori-Giorgi, Savin), who essentially developed a considerable amount of optimal transport theory without knowing they were doing optimal transport. The goal of my minicourse is three-fold

(i) to explain the connection electronic structure of molecules--optimal transport and compare physicist's and OT theory approaches. For instance the Gangbo-McCann formula for the optimal map in terms of the Kantorovich potential is arrived at in an intriguingly simple way by physicists.

(ii) to discuss what is known rigorously about the limit problem, including
-- justification of the formal semiclassical limit [1]
-- qualitative theory of OT problems with Coulomb cost, including the question whether ''Kantorovich minimizers'' must be ''Monge minimizers'' (yes for 2 particles, sometimes yes sometimes no for N particles, no for infinitely many particles) [1,2]
-- exactly soluble cases (N=2 in 1d (for a nice generalization to higher N see DiMarino's talk); N=2 with radial density; N arbitrary but density restricted to two sites) [1, 2]

(iii) to discuss the limit N to infinity, in which - remarkably and totally unlike in the case of positive power costs - the highly correlated optimal N-point densities weakly converge to INDEPENDENT densities [3].

[1] C.Cotar, G.F., C.Klueppelberg, CPAM 66, 548-599, 2013 (arXiv 2011)
[2] G.F., Ch.Mendl, B.Pass, C.Cotar, C.Klueppelberg, Journal of Chemical Physics 139, 164109, 2013 (arXiv 2013)
[3] C.Cotar, G.F., B.Pass, arXiv 1307.6540, 2013

LECTURE 1    LECTURE 2    LECTURE 3

Nicola Gigli (mini course)

Spaces with Ricci curvature bounded from below

I will give an overview over the fast expanding theory of nonsmooth spaces with Ricci curvature bounded from below, focussing on their definition, analytical properties and the links with probability theory.

LECTURE I    LECTURE II    LECTURE III

Martin Huesmann

Optimal transport between random measures

We consider couplings between the Lebesgue measure and a simple point process and ask for minimizers of the cost per volume. If the cost per volume is finite there is a unique invariant coupling attaining this minimal cost, called optimal coupling. The optimal coupling is induced by a transportation map which therefore defines a fair allocation rule for the point process. If the cost function is the squared distance the transportation map defines a partition of the Euclidean space into convex polytopes each of which has volume one. Similar results can be obtained for the transportation problem between two jointly invariant random measures.

Emanuel Indrei

A sharp quantitative log-Sobolev inequality

The so-called logarithmic Sobolev inequalities appear in various branches of statistical mechanics, quantum field theory, Riemannian geometry, and partial differential equations. In this talk, we will show
how optimal transportation theory can be used to attack the stability problem for the classical Gaussian log-Sobolev inequality. This is based on joint work with D. Marcon.

PRESENTATION

Nicolas Juillet

An optimal transport problem for two measures in the convex order

We consider two probability measures on R, taken in the convex order, that is, according to a theorem of Strassen such that there exists a two time martingale with these measures as marginals.  Note that one can see the joint law as a transport plan with a martingale constraint. The theorem of Strassen does not designate a special transport plan. In this talk, we present one, introduced by Mathias Beiglb¨ock and the speaker, that we called “curtain” coupling . This transport plan possesses remarkable monotonicity properties that make it the solution of several optimal transport problems. We will also examinate the stability of the curtain
transport plan with respect to the two given marginals.

PRESENTATION

Richard Kraaij

A Lagrangian formalism for large deviations of Feller processes

Dawson and Gaertner (1987) showed that the path of the empirical process of n independent identically distributed diffusion processes satisfy a large deviation principle in n with a rate function expressed as the integral of a 'Lagrangian' cost function.
In this talk, I will generalise their result to a large class of Feller processes on locally compact spaces.

PRESENTATION

Jan Maas (mini course)

Optimal transport in discrete and quantum systems

Optimal transport has become a powerful tool to attack non-smooth problems in analysis and geometry. A key role is played by the 2-Wasserstein metric, which induces a rich geometric structure on the space of probability measures. This structure allows to obtain gradient flow structures for diffusion equations and to exploit geodesic convexity of the entropy. However, in discrete settings the 2-Wasserstein metric degenerates and the theory seems to break down.
In recent years a new class of transport metrics has emerged, which allows to apply ideas from optimal transport to a number of different situations, which had been outside the scope of the existing theory. In this mini-course we give an overview of these developments in the setting of discrete Markov chains, chemical reaction networks, and
open quantum systems.

Andrea Mondino

Some analytic and geometric properties of infinitesimally Hilbertian metric measure spaces with lower Ricci curvature bounds

Infinitesimally Hilbertian metric measure spaces with lower Ricci curvature bounds, RCD^*(K,N)-spaces for short (where $K \in \R$ stands for the lower bound on the Ricci curvature and $N\in [1, +\infty]$ for the upper bound on the dimension) constitute a natural abstract framework where to study Gromov-Hausdorff limits of Riemannian manifolds with Ricci bounds. After a brief introduction to the topic, in the seminar I will report on some recent geometric and analytic properties of these spaces, in particular I will focus on
- Stability properties: obtained without any a priori control on the dimension and without any compactness assumption of the spaces (joint work with Gigli and Savaré)
- Local-to-Global property without any a priori non branching assumption (joint work with Ambrosio and Savaré)
- Dimensional Bochner inequality (joint work with Ambrosio and Savaré)
- Li-Yau and Harnack type estimates on the heat flow (joint work with Garofalo)
- Study of the local geometry, in particular existence of euclidean tangent cones (joint work with Gigli and Rajala)

PRESENTATION

Harald Oberhauser

On the Skorokhod embedding problem

A classic problem in probability theory is the Skorokhod embedding problem. The question on how to construct solutions that are intuitive, computable and in some sense extremal has recently received more interest. I will speak about recent work that is inspired by classic results due to Chacon, Root, Rost and many others. If time permits I will discuss some connections with the theory of rough paths.

Xiaolu Tan

Martingale transport and peacocks

We extend the martingale version of the one-dimensional Bernier's theorem (Frechet-Hoeffding coupling) established in Beiglbock and Juillet (2012), Henry-Labordere and Touzi (2013) to the infinitely-many marginals case. By approximation technique, we show that for a class of cost/reward function, the solution of the martingale transport problem given infinitely-many marginals is provided by a pure downward jump local Levy model. In particular, it provides a new construction of the martingale peacock process, and a new remarkable example of discontinuous fake Brownian motion. Moreover, we also provide a duality result together with a dual optimizer in explicit form. This is a joint work with P. Henry-Labordere and N. Touzi.

PRESENTATION

Christoph Thäle

Functional Poisson approximation and optimal transport

The arguably most prominent functional limit theorem is Donsker's invariance principle. It asserts that the distribution of the linear interpolation between the points of a suitably re-scaled random walk converges to the Wiener measure on the space of continuous functions on the non-negative real half-line. Besides the Wiener process, there is another fundamental stochastic process, which plays and important role in probability theory and its applications, namely the Poisson process. However, functional limit theorems involving the Poisson process have found much less attention so far. The aim of this talk is to provide a form of a Poisson functional limit theorem, which is well suited for concrete applications to problems arising in stochastic geometry. Moreover, the limit theorem will be quantitative in that we provide estimates for the rate of convergence. The point process distance we use, the so-called Rubinstein distance, is based on the on the concept of an optimal transport between point processes. (This is joint work in progress with Laurent Decreusefond and Matthias Schulte)

PRESENTATION

PRACTICAL INFORMATION

### ●      Venue

Eurandom, 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, (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).
Accessibility TU/e campus and map.

The conference will be held at the Eindhoven Technical University. The TU/e is a relatively young university. It was founded some 50 years ago and is situated in the southern part of The Netherlands in the city of Eindhoven, well known as the hometown of the giant in Electronics, the Philips Company, and the famous football club, PSV Eindhoven. The TU/e intends to be a research driven, design oriented university of technology at an international level, with the primary objective of providing young people with an academic education within the ‘engineering science & technology’ domain.

Participants have to make their own hotel booking. However, they can get a reduced rate if they book our preferred hotel.
Patty Koorn for instructions on how to obtain this special price.

For more hotel options and hotel bookings please see:
Hotels (note: prices listed are "best available"), or the web pages of the Tourist Information Eindhoven.

Some limited funds are available to contribute to local and travel costs.

### ●      Travel

For 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&12

The meeting-room 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 Secretariat

Upon 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 day-to-day running of the conference: registration, issuing certificates and receipts, etc.

### ●      Cancellation

Should you need to cancel your participation, please contact Patty Koorn, the Workshop Officer (deadline for cancellation is 3,2014). There is no registration fee, but when not cancelling in time, there will be a no-show fee of 50 euro.

### ●      Contact

Mrs. Patty Koorn, Workshop Officer, Eurandom/TU Eindhoven, koorn@eurandom.tue.nl