This work is devoted to the study of rates of convergence of the empirical measures μn = 1 n ∑n k=1 δXk , n ≥ 1, over a sample (Xk)k≥1 of independent identically distributed real-valued random variables towards the common distribution μ in Kantorovich transport distances Wp. The focus is on finite range bounds on the expected Kantorovich distances E(Wp(μn, μ)) or [ E(W p p (μn, μ)) ]1/p in terms of moments and analytic conditions on the measure μ and its distribution function. The study describes a variety of rates, from the standard one 1 √ n to slower rates, and both lower and upperbounds on E(Wp(μn, μ)) for fixed n in various instances. Order statistics, reduction to uniform samples and analysis of beta distributions, inverse distribution functions, logconcavity are main tools in the investigation. Two detailed appendices collect classical and some new facts on inverse distribution functions and beta distributions and their densities necessary to the investigation.

One-Dimensional Empirical Measures, Order Statistics, and Kantorovich Transport Distances

We reformulate the strong-interaction limit of electronic density functional theory in terms of a classical problem with a degenerate minimum. This allows us to clarify many aspects of this limit, and to write a general solution, which is explicitly calculated for spherical densities. We then compare our results with previous approximate solutions and discuss the implications for density functional theory.

/pdf/strictly-correlated-electrons-in-density-functional-theory-a-1tvfz0mifb.pdf

Strictly correlated electrons in density-functional theory: A general formulation with applications to spherical densities

We prove a regularity result for Lagrangian flows of Sobolev vector fields over RCD(K,N) metric measure spaces, regularity is understood with respect to a newly defined quasi-metric built from the Green function of the Laplacian. Its main application is that RCD(K,N) spaces have constant dimension. In this way we generalize to such abstract framework a result proved by Colding-Naber for Ricci limit spaces, introducing ingredients that are new even in the smooth setting.

Constancy of the Dimension for RCD(K,N) Spaces via Regularity of Lagrangian Flows

The aim of this paper is to provide new stability results for sequences of metric measure spaces $(X_i,d_i,m_i)$ convergent in the measured Gromov-Hausdorff sense. By adopting the so-called extrinsic approach of embedding all metric spaces into a common one $(X,d)$, we extend the results of Gigli-Mondino-Savar\'e by providing Mosco convergence of Cheeger's energies and compactness theorems in the whole range of Sobolev spaces $H^{1,p}$, including the space $BV$, and even with a variable exponent $p_i\in [1,\infty]$. In addition, building on the results of Ambrosio-Stra-Trevisan, we provide local convergence results for gradient derivations. We use these tools to improve the spectral stability results, previously known for $p>1$ and for Ricci limit spaces, getting continuity of Cheeger's constant. In the dimensional case $N<\infty$, we improve some rigidity and almost rigidity results by Ketterer and Cavaletti-Mondino. On the basis of the second-order calculus by Gigli, in the class of $RCD(K,\infty)$ spaces we provide stability results for Hessians and $W^{2,2}$ functions and we treat the stability of the Bakry-\'Emery condition $BE(K,N)$ and of ${\bf Ric}\geq KI$, with $K$ and $N$ not necessarily constant.

/pdf/new-stability-results-for-sequences-of-metric-measure-spaces-42w4vgpz6l.pdf

New stability results for sequences of metric measure spaces with uniform Ricci bounds from below

Let X1, . . .,Xn be independent random variables having as common distribution the standard Gaussian measure μ on ℝ2 and let $$ {\mu}_n=\frac{1}{n}\sum \limits_{i=1}^n{\delta}_{X_i} $$
 be the associated empirical measure. We show that $$ \frac{1}{C}\frac{\log n}{n}\le $$
 𝔼 $$ \left({\mathrm{W}}_2^2\left({\mu}_n,\mu \right)\right)\le C\frac{{\left(\log n\right)}^2}{n} $$
 for some numerical constant C > 0, where W2 is the quadratic Kantorovich metric, and conjecture that the left-hand side provides the correct order. The proof is based on the recent PDE and mass transportation approach developed by L. Ambrosio, F. Stra, and D. Trevisan.

On Optimal Matching of Gaussian Samples

We prove asymptotic results for 2-dimensional random matching problems. In particular, we obtain the leading term in the asymptotic expansion of the expected quadratic transportation cost for empirical measures of two samples of independent uniform random variables in the square. Our technique is based on a rigorous formulation of the challenging PDE ansatz by S.\ Caracciolo et al.\ (Phys. Rev. E, {\bf 90} 012118, 2014) that "linearise" the Monge-Amp\`ere equation.

A PDE approach to a 2-dimensional matching problem

We prove asymptotic results for 2-dimensional random matching problems. In particular, we obtain the leading term in the asymptotic expansion of the expected quadratic transportation cost for empirical measures of two samples of independent uniform random variables in the square. Our technique is based on a rigorous formulation of the challenging PDE ansatz by Caracciolo et al. (Phys Rev E 90:012118, 2014) that linearizes the Monge–Ampere equation.

Weak and strong convergence of derivations and stability of flows with respect to MGH convergence

We provide sharp conditions for the finiteness and the continuity of multimarginal optimal transport with repulsive cost, expressed in terms of a suitable concentration property of the measure. To ...

/pdf/continuity-of-multimarginal-optimal-transport-with-repulsive-2s3nj4h70h.pdf

Continuity of Multimarginal Optimal Transport with Repulsive Cost

We disprove a conjecture in Density Functional Theory, relative to multimarginal optimal transport maps with Coulomb cost. We also provide examples of maps satisfying optimality conditions for special classes of data.

Federico Stra

Papers

A PDE approach to a 2-dimensional matching problem

A PDE approach to a 2-dimensional matching problem

Weak and strong convergence of derivations and stability of flows with respect to MGH convergence

Continuity of Multimarginal Optimal Transport with Repulsive Cost

Counterexamples in multimarginal optimal transport with Coulomb cost and spherically symmetric data