@phdthesis{20563, abstract = {The theory of optimal transport provides an elegant and powerful description of many evolution equations as gradient flows. The primary objective of this thesis is to adapt and extend the theory to deal with important equations that are not covered by the classical framework, specifically boundary value problems and kinetic equations. Additionally, we establish new results in periodic homogenization for discrete dynamical optimal transport and in quantization of measures. Section 1.1 serves as an invitation to the classical theory of optimal transport, including the main definitions and a selection of well-established theorems. Sections 1.2-1.5 introduce the main results of this thesis, outline the motivations, and review the current state of the art. In Chapter 2, we consider the Fokker–Planck equation on a bounded set with positive Dirichlet boundary conditions. We construct a time-discrete scheme involving a modification of the Wasserstein distance and, under weak assumptions, prove its convergence to a solution of this boundary value problem. In dimension 1, we show that this solution is a gradient flow in a suitable space of measures. Chapter 3 presents joint work with Giovanni Brigati and Jan Maas. We introduce a new theory of optimal transport to describe and study particle systems at the mesoscopic scale. We prove adapted versions of some fundamental theorems, including the Benamou–Brenier formula and the identification of absolutely continuous curves of measures. Chapter 4 presents joint work with Lorenzo Portinale. We prove convergence of dynamical transportation functionals on periodic graphs in the large-scale limit when the cost functional is asymptotically linear. Additionally, we show that discrete 1-Wasserstein distances converge to 1-Wasserstein distances constructed from crystalline norms on R d . Chapter 5 concerns optimal empirical quantization: the problem of approximating a measure by the sum of n equally weighted Dirac deltas, so as to minimize the error in the p-Wasserstein distance. Our main result is an analog of Zador’s theorem, providing asymptotic bounds for the minimal error as n tends to infinity. }, author = {Quattrocchi, Filippo}, issn = {2663-337X}, keywords = {optimal transport, kinetic equations, boundary value problems, quantization, gradient flows, homogenization}, pages = {240}, publisher = {Institute of Science and Technology Austria}, title = {{Optimal transport methods for kinetic equations, boundary value problems, and discretization of measures}}, doi = {10.15479/AT-ISTA-20563}, year = {2025}, } @unpublished{20569, abstract = {This is the first part of a general description in terms of mass transport for time-evolving interacting particles systems, at a mesoscopic level. Beyond kinetic theory, our framework naturally applies in biology, computer vision, and engineering. The central object of our study is a new discrepancy d between two probability distributions in position and velocity states, which is reminiscent of the 2-Wasserstein distance, but of second-order nature. We construct d in two steps. First, we optimise over transport plans. The cost function is given by the minimal acceleration between two coupled states on a fixed time horizon T. Second, we further optimise over the time horizon T > 0. We prove the existence of optimal transport plans and maps, and study two time-continuous characterisations of d. One is given in terms of dynamical transport plans. The other one -- in the spirit of the Benamou--Brenier formula -- is formulated as the minimisation of an action of the acceleration field, constrained by Vlasov's equations. Equivalence of static and dynamical formulations of d holds true. While part of this result can be derived from recent, parallel developments in optimal control between measures, we give an original proof relying on two new ingredients: Galilean regularisation of Vlasov's equations and a kinetic Monge--Mather shortening principle. Finally, we establish a first-order differential calculus in the geometry induced by d, and identify solutions to Vlasov's equations with curves of measures satisfying a certain d-absolute continuity condition. One consequence is an explicit formula for the d-derivative of such curves.}, author = {Brigati, Giovanni and Maas, Jan and Quattrocchi, Filippo}, booktitle = {arXiv}, keywords = {optimal transport, kinetic theory, second-order discrepancy, Vlasov equation, Wasserstein splines.}, title = {{Kinetic Optimal Transport (OTIKIN) -- Part 1: Second-order discrepancies between probability measures}}, doi = {10.48550/arXiv.2502.15665}, year = {2025}, } @article{14703, abstract = {We present a discretization of the dynamic optimal transport problem for which we can obtain the convergence rate for the value of the transport cost to its continuous value when the temporal and spatial stepsize vanish. This convergence result does not require any regularity assumption on the measures, though experiments suggest that the rate is not sharp. Via an analysis of the duality gap we also obtain the convergence rates for the gradient of the optimal potentials and the velocity field under mild regularity assumptions. To obtain such rates we discretize the dual formulation of the dynamic optimal transport problem and use the mature literature related to the error due to discretizing the Hamilton-Jacobi equation.}, author = {Ishida, Sadashige and Lavenant, Hugo}, issn = {1615-3383}, journal = {Foundations of Computational Mathematics}, keywords = {Optimal transport, Hamilton-Jacobi equation, convex optimization}, publisher = {Springer Nature}, title = {{Quantitative convergence of a discretization of dynamic optimal transport using the dual formulation}}, doi = {10.1007/s10208-024-09686-3}, year = {2024}, } @unpublished{20571, abstract = {We prove the convergence of a modified Jordan--Kinderlehrer--Otto scheme to a solution to the Fokker--Planck equation in $\Omega \Subset \mathbb{R}^d$ with general, positive and temporally constant, Dirichlet boundary conditions. We work under mild assumptions on the domain, the drift, and the initial datum. In the special case where $\Omega$ is an interval in $\mathbb{R}^1$, we prove that such a solution is a gradient flow -- curve of maximal slope -- within a suitable space of measures, endowed with a modified Wasserstein distance. Our discrete scheme and modified distance draw inspiration from contributions by A. Figalli and N. Gigli [J. Math. Pures Appl. 94, (2010), pp. 107--130], and J. Morales [J. Math. Pures Appl. 112, (2018), pp. 41--88] on an optimal-transport approach to evolution equations with Dirichlet boundary conditions. Similarly to these works, we allow the mass to flow from/to the boundary $\partial \Omega$ throughout the evolution. However, our leading idea is to also keep track of the mass at the boundary by working with measures defined on the whole closure $\overline \Omega$. The driving functional is a modification of the classical relative entropy that also makes use of the information at the boundary. As an intermediate result, when $\Omega$ is an interval in $\mathbb{R}^1$, we find a formula for the descending slope of this geodesically nonconvex functional. }, author = {Quattrocchi, Filippo}, booktitle = {arXiv}, keywords = {gradient flows, Jordan–Kinderlehrer–Otto scheme, curves of maximal slope, optimal transport, Dirichlet boundary conditions, Fokker–Planck equation}, title = {{Variational structures for the Fokker-Planck equation with general Dirichlet boundary conditions}}, doi = {10.48550/arXiv.2403.07803}, year = {2024}, } @unpublished{20570, abstract = {We investigate the minimal error in approximating a general probability measure $\mu$ on $\mathbb{R}^d$ by the uniform measure on a finite set with prescribed cardinality $n$. The error is measured in the $p$-Wasserstein distance. In particular, when $1\le p