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

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, CD_loc (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 CD_loc (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

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