arXiv subjects

Filippo Quattrocchi

Publications and source records attributed to Filippo Quattrocchi.

6 recordsLinked to original sources

arXiv2026

Nonlinear kinetic Fokker-Planck equations as gradient flows of the free energy

We consider nonhomogeneous kinetic equations that involve a free transport operator and a diffusion of porous medium type acting on velocities. The main novelty is a gradient flow interpretation of dynamics driven by an interplay of conservative and dissipative effects. We rely on a notion of discrepancy adapted to a phase space of positions and velocities, built upon second-order characteristics obeying Newton's laws. The equation appears as the steepest descent of the free energy functional. We also prove that approximate solutions constructed with an implicit Euler scheme converge to a solution of the kinetic equation. Thus, we generalise to a family of nonlinear kinetic equations the celebrated JKO scheme in mass transport theory. Most of our results are new even in the case of the linear Vlasov-Fokker-Planck equation.

math.AP↗

arXiv2025

Kinetic Optimal Transport (OTIKIN) -- Part 1: Second-Order Discrepancies Between Probability Measures

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 $\mathsf d$ between two probability distributions in position and velocity states, which is reminiscent of the $2$-Wasserstein distance, but of second-order nature. We construct $\mathsf d$ 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 $\mathsf d$. 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 $\mathsf d$ 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 $\mathsf d$, and identify solutions to Vlasov's equations with curves of measures satisfying a certain $\mathsf d$-absolute continuity condition. One consequence is an explicit formula for the $\mathsf d$-derivative of such curves.

math.AP↗

arXiv2024

Asymptotics for Optimal Empirical Quantization of Measures

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 $1\le p<d$, we establish asymptotic upper and lower bounds as $n \to \infty$ on the rescaled minimal error that have the same, explicit dependency on $\mu$. In some instances, we prove that the rescaled minimal error has a limit. These include general measures in dimension $d = 2$ with $1 \le p < 2$, and uniform measures in arbitrary dimension with $1 \le p < d$. For some uniform measures, we prove the limit existence for $p \ge d$ as well. For a class of compactly supported measures with H\"older densities, we determine the convergence speed of the minimal error for every $p \ge 1$. Furthermore, we establish a new Pierce-type (i.e., nonasymptotic) upper estimate of the minimal error when $1 \le p < d$. In the initial sections, we survey the state of the art and draw connections with similar problems, such as classical and random quantization.

math.PR↗

arXiv2024

Variational structures for the Fokker--Planck equation with general Dirichlet boundary conditions

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 -- strictly 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.

math.AP↗

arXiv2023

Discrete-to-continuum limits of optimal transport with linear growth on periodic graphs

We prove discrete-to-continuum convergence for dynamical optimal transport on $\mathbb{Z}^d$-periodic graphs with energy density having linear growth at infinity. This result provides an answer to a problem left open by Gladbach, Kopfer, Maas, and Portinale (Calc Var Partial Differential Equations 62(5), 2023), where the convergence behaviour of discrete boundary-value dynamical transport problems is proved under the stronger assumption of superlinear growth. Our result extends the known literature to some important classes of examples, such as scaling limits of 1-Wasserstein transport problems. Similarly to what happens in the quadratic case, the geometry of the graph plays a crucial role in the structure of the limit cost function, as we discuss in the final part of this work, which includes some visual representations.

math.OC↗

arXiv2023

Polychromatic Partitions, Kingman Theory, and Ewens Sampling Formula

A polychromatic partition is a partition of a set of colored (marked) elements in which each block is recorded only by the number of elements of each color in that block. We introduce a notion of polychromatic partition structure, i.e. a family of random polychromatic partitions consistent under deletion of elements uniformly at random, simultaneously extending (standard) partition structures, multipartition structures, and partition structures of partially exchangeable type. As an example thereof, we define a polychromatic analogue of the celebrated Ewens Sampling Formula, proving polychromatic versions of Kingman's characterization and of Hoppe's urn model. As our main result, we give a complete characterization of polychromatic partition structures, proving a triple affine homeomorphism of Bauer simplices among: polychromatic partition structures, harmonic densities for the polychromatic Hoppe urn, and probability measures on a polychromatic Kingman simplex. Additionally, we prove that this new simplex is homeomorphic to the Martin boundary of the polychromatic Hoppe urn inside its Martin compactification. We discuss various applications including closed non-recursive non-iterative expressions for multivariate moments of several random measures in terms of cycle index polynomials whose monomials are indexed by polychromatic partitions.

math.PR↗