We study an energy defined through an optimal transport problem between the uniform density on a set and the uniform density on the complement of that set. This quantity captures how much the set is concentrated, and we prove that the maximizers coincide with a ball in many relevant cases. This is achieved by showing the monotonicity of that functional with respect to a certain symmetrization procedure. This talk is based on a joint work with Almut Burchard and Ihsan Topaloglu.

The task of defining and computing the background state of the atmosphere is of great importance to meteorologists trying to find trends in climate data. We interpret McIntyre’s definition of a Modified Lagrangian Mean state in the language of optimal transport. Drawing on this robust mathematical framework, we prove existence and uniqueness of such states, and compute them from real data using semi-discrete optimal transport. In this talk, I will present this definition, its properties, and results of subsequent data analysis. This is joint work with John Methven (University of Reading), Thilo Stier (University of Göttingen), David Bourne (Heriot-Watt University), and Mike Cullen (UK Met Office – retired).

Lipschitz transport maps between two measures are useful tools for transferring analytical properties, such as functional inequalities. The most well-known result in this field is Caffarelli’s contraction theorem, which shows that the optimal transport map from a Gaussian to a uniformly log-concave measure is globally Lipschitz. Note that the transfer of analytical properties does not depend on the optimality of the transport map. This is why several works have established Lipschitz bounds for other transport maps, such as those derived from diffusion processes, as introduced by Kim and Milman. Here, we use the control interpretation of the driving transport vector field inducing the transport map and a coupling strategy to obtain Lipschitz bounds for this map between asymptotically log-concave measures and their Lipschitz perturbations. This talk is based on a joint work with Giovanni Conforti.

The fitting problem is a classical challenge in mathematical finance, concerned with finding martingales that satisfy marginal constraints. Building on the Bass solution to the Skorokhod embedding problem and on optimal transport, Backhoff-Beiglböck-Huesmann-Källblad introduced a solution to the two-marginal problem: the stretched Brownian motion. Recently, Conze and Henry-Labordère (for dimension d=1) and Joseph-Loeper-Obloj (for general d) proposed a fixed-point algorithm, called the martingale Sinkhorn algorithm, for its computation. We analyze this fixed-point iteration scheme and prove its convergence, establishing in particular a linear rate of convergence for d=1

The Random Euclidean Matching is the problem of finding the best matching between two sets of \( N \) points, i.e., of computing the cost function \[ C(N) := \min_{\pi} \sum_{i=1}^{N} |X_i - Y_{\pi_i}|^p, \] where \( \pi \) is a permutation of \(\{1, \dots, N\}\), \(\{X_i, Y_i\}_{i=1}^{N}\) are i.i.d. random variables in \( \mathbb{R}^d \) and \( p \geq 1 \) is the exponent. Denoting by \( E \) the expected value, one is interested in estimating, for large \( N \), \( E[C(N)] \), that is, the expectation of the cost function. It has been understood by several authors that this is an optimal transport problem. In this talk, we focus on the case of \(\{X_i, Y_i\}_{i=1}^{N}\) i.i.d. on \( \mathbb{R}^d \) with the Gaussian probability distribution, for dimension \( d \geq 2 \).

It has been conjectured by Liero, Mielke, and Savaré that the Hellinger-Kantorovich distance can be expressed as the metric infimal convolution of the Hellinger and the Wasserstein distances. This statement is quite clear, at least intuitively, if one compares the dynamical representations of the three distances. However, no rigorous proof had yet been provided. We first discuss the infimal convolution between two generic distances, highlighting the difficulties that may arise: in particular, finiteness and triangle inequality may fail. We then sketch the proof of the main result. To prove it, we study with the tools of Unbalanced Optimal Transport the so-called Marginal Entropy-Transport problem that arises as a single minimization step in the definition of infimal convolution (joint work with Nicolò De Ponti and Giacomo Enrico Sodini).

We prove that the space of (non-negative and finite Borel) measures on a complete Riemannian manifold endowed with the Hellinger-Kantorovich distance is universally infinitesimally Hilbertian and that the class of cylinder functions is dense in energy. We endow the aforementioned metric space with its canonical reference measure (the multiplicative infinite-dimensional Lebesgue measure) and we identify its Dirichlet form with the Cheeger energy of the metric-measure space. This work has been done in collaboration with Lorenzo Dello Schiavo.

Optimal transport provides a natural way to compare probability distributions. Agueh and Carlier in particular introduced a model for computing barycenters with respect to the Wasserstein distance that found numerous applications in data science and image processing. This is conceived as a Fréchet mean, namely a variational problem where one minimizes the weighted sum of the distances squared from N given measures. I will discuss a generalization of this model to the case where one allows negative weights, so as to extrapolate rather than interpolate. In the two-measures case this provides a way to extend geodesics. The difficulty comes from the lack of standard convexity, contrary to the barycenter model of Agueh and Carlier. Surprisingly, in the case with only one positive weight the problem enjoys some hidden convexity properties which allow to rewrite it in equivalent convex formulations and to fully characterize (unique) minimizers. In particular, the problem can be recast as an instance of weak (multimarginal) optimal transport. One can take advantage of this formulation to devise an efficient approach to approximate the solution on point clouds, via entropic regularization and a variant of the Sinkhorn algorithm. In the case with more than one positive weight this structure seems to be lost, but it is possible to work with a suitable relaxation of the problem in order to recover these favorable properties. This talk is based on joint works with Thomas Gallouët and Andrea Natale.