Du 4 au 5 juillet 2019
IRMA
ENTCOTP19 : Eight or Nine Talks on Contemporary Optimal Transport Problems
During the two days we wish to bring together people working on these particular facets of Optimal Transport.
Organizers : Nicolas Juillet (local organizer) ; Mathias Beiglböck (co-organizer) ; Jessica Maurer-Spoerk, Astrid Kollros (administration)
Speakers :
- Julio BACKHOFF VERAGUAS (Wien)
- Thierry CHAMPION (Toulon)
- Simone DI MARINO (Pisa)
- Nikita GLADKOV (Moskvá)
- Nathaël GOZLAN (Paris)
- Benjamin JOURDAIN (Champs-sur-Marne)
- Anna KAUSAMO (Jyväskylä)
- Victor KLEPTSYN (Rennes)
- Nizar TOUZI (Palaiseau)
Venue : Salle de conférences, IRMA, University of Strasbourg.
For more details, see the official website.
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Jeudi 4 juillet 2019
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13:00
Nizar Touzi, Palaiseau
Continuous time Principal Agent and optimal planning
Motivated by the approach introduced by Sanninkov to solve principal-agent problems, we provide a solution approach which allows to address a wider range of problems. The key argument uses a representation result from the theory of backward stochastic differential equations. This methodology extends to the mean field game version of the problem, and provides a connexion with the P.-L. Lions optimal planning problem. -
14:00
Victor Kleptsyn, Rennes
A counter-example to the Cantelli conjecture
Take two Gaussian independent random variables X and Y, both N(0,1). The Cantelli conjecture addresses non-linear combinations of the form Z= X+f(X)*Y, where f is a non-negative function. It states that if Z is Gaussian, f should be constant almost everywhere. In a joint work with Aline Kurtzmann, we have constructed a (measurable) counter-example to this conjecture, with a construction that uses a « Brownian » variation of a transport. This construction will be the subject of my talk. -
15:30
Anna Kausamo, Jyväsklä
The Monge problem in multi-marginal optimal mass transportation
In this talk I will introduce the concept of Multi-Marginal Optimal Mass Transportation (MOT) with the emphasis on repulsive cost functions. Then I will outline the Monge problem, discuss it's difficulty in the MOT setting, and present some nonexistence results that are joint work Augusto Gerolin and Tapio Rajala. -
16:30
Nathaël Gozlan, Paris
Weak optimal transport and applications to Caffarelli contraction theorem
The talk will deal with a variant of the optimal transport problem first considered in a joint paper with C. Roberto, P-M Samson and P. Tetali, where elementary mass transports are penalized through their barycenters. The talk will in particular focus on a recent result obtained in collaboration with N. Juillet describing optimal transport plans for the quadratic barycentric cost. A direct corollary of this result gives a new necessary and sufficient condition for the Brenier map to be 1-Lipschitz. Finally we will present a recent work in collaboration with M. Fathi and M. Prodhomme, where this contractivity criterion is used to give a new proof of the Caffarelli contraction theorem, telling that any probability measure having a log-concave density with respect to the standard Gaussian measure is a contraction of it. -
Vendredi 5 juillet 2019
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08:30
Thierry Champion, Toulon
Optimal transport planning with a non linear cost
In this talk, I consider optimal transport problems that involve non-linear transportation costs which favour optimal plans non associated to a single valued transport map. I will describe some results concerning this type of problem (existence, duality principle, optimality conditions) and focus on specific examples in a finite dimensional compact setting. I will consider in particular the case where the cost involves the opposite of the variance or the indicator of a constraint on the barycenter of p (martingale transport). This is from a joined work with J.J. Alibert and G. Bouchitté. -
09:30
Nikita Gladkov, Moscou
Monge-Kantorovich problem for n-dimensional measures with fixed k-dimensional marginals
The classical Monge-Kantorovich (transportation) problem deals with measures on a product of two spaces with two independent fixed marginals. Its natural generalization (multimarginal Monge-Kantorovich problem) deals with the products of n spaces X_1, ..., X_n with n independent marginals. We study the Monge-Kantorovich problem on X_1 \times X_2 ... \times ... X_n with fixed projections onto the products of X_{i_1} , ... X_{i_k} for all k-tuples of indices (k -
10:50
Marcel Nutz, New York
Fine Properties of the Optimal Skorokhod Embedding Problem
We study the problem of stopping a Brownian motion at a given distribution $\nu$ while optimizing a reward function that depends on the (possibly randomized) stopping time and the Brownian motion. Our first result establishes that the set $T(\nu)$ of stopping times embedding $\nu$ is weakly dense in the set $R(\nu)$ of randomized embeddings. In particular, the optimal Skorokhod embedding problem over $T(\nu)$ has the same value as the relaxed one over $R(\nu)$ when the reward function is semicontinuous, which parallels a fundamental result about Monge maps and Kantorovich couplings in optimal transport. A second part studies the dual optimization in the sense of linear programming. While existence of a dual solution failed in previous formulations, we introduce a relaxation of the dual problem and establish existence of solutions as well as absence of a duality gap, even for irregular reward functions. This leads to a monotonicity principle which complements the key theorem of Beiglbock, Cox and Huesmann. These results can be applied to characterize the geometry of optimal embeddings through a variational condition. (Joint work with Mathias Beiglbock and Florian Stebegg) -
13:10
Benjamin Jourdain, Champs-sur-Marne
The inverse transform martingale coupling
We exhibit a new martingale coupling between two probability measures μ and ν in convex order on the real line. This coupling is explicit in terms of the integrals of the positive and negative parts of the difference between the quantile functions of μ and ν. The integral of |y−x| with respect to this coupling is smaller than twice the Wasserstein distance with index one between μ and ν. When the comonotonous coupling between μ and ν is given by a map T, it minimizes the integral of |y−T(x)| among all martingales couplings. -
14:10
Simone Di Marino, Pise
Seidl conjecture in Density Functional Theory: results and counterexamples
The Seidl conjecture in Density Functional Theory is the equivalent of the Monge Ansatz for the classical optimal transport problem with the cost c(x,y)=|x−y|, in the multimarginal case with the Coulomb cost. We provide positive results in the one dimensional case as well as both positive and negative results in the radial 2-dimensional case.