arXiv subjects

Alessandro Pinzi

Publications and source records attributed to Alessandro Pinzi.

6 recordsLinked to original sources

arXiv2026

Optimal transport between laws of random probability measures and the strict Monge problem

We consider an optimal transport problem between laws of random probability measures: given a base cost function, we build the associated OT cost between probability measures that in turn we use to define the OT cost between probability measures over probability measures. This setting admits a finer reformulation in terms of laws of random couplings, which retain more information than ordinary couplings. One of the main contributions of the paper is the characterization of the optimal ones in terms of Kantorovich potentials. Similarly, we also introduce the strict Monge problem, whose admissible competitors are more restrictive than in the usual Monge formulation. In this setting, we will give sufficient conditions under which the value of this problem is the same as the one considered above, in the spirit of the result by A. Pratelli. Then, for $p>1$, when the underlying cost is the distance to the power $p$ in a strictly convex Banach space, we will give sufficient conditions under which the optimal random coupling is unique and induced by a solution of the strict Monge problem, resembling the Brenier theorem.

math.OC↗

arXiv2026

A direct method for doubly nonlinear equations via convexification in spaces of measures and duality

Existence of solutions to doubly nonlinear equations in reflexive Banach spaces is established by resorting to a global-in-time variational approach inspired by De Giorgi's principle, which characterizes the associated flows as null-minimizers of a suitable energy-dissipation functional defined on trajectories. In contrast to the celebrated minimizing movements scheme, the proposed strategy does not rely on any time-discretization or iterative constructions. Instead, it provides a direct method based on the relaxation of the problem in spaces of measures, constrained by the continuity equation: in this procedure, no gap is introduced due to the Ambrosio's superposition principle. Within this weak convex framework, the validity of the null-minimization property is recovered through two further steps. First, a careful application of the Von Neumann minimax theorem yields an identification of the dual problem as a supremum over the set of smooth and bounded cylinder functions, solving an Hamilton-Jacobi-type inequality. Secondly, a suitable "backward boundedness" property of solutions to such Hamilton-Jacobi system gives a proper bound of the dual problem, ensuring that the minimum value of the original functional is actually zero. The proposed strategy naturally extends to non-autonomous equations, encompassing time- and space-dependent dissipation potentials and time-dependent potential energies.

math.AP↗

arXiv2025

A study of the metric measure space of probability measures via a purely atomic superposition principle

We study the continuity equation on the metric measure space $(\mathcal{P}_p(X),W_p,Q)$, when $X$ is either the Euclidean space or a compact, oriented, and boundaryless Riemannian manifold, for some suitable reference measure $Q \in \mathcal{P}_p(\mathcal{P}_p(X))$, which by construction are concentrated over purely atomic measures. In fact, we consider the equation $\partial_t M_t +\operatorname{div}_{\mathcal{P}}(b_t M_t) = 0$, where $(M_t)_{t\in[0,T]} \subset\mathcal{P}(\mathcal{P}(X))$ and $b:[0,T]\times X \times \mathcal{P}(X) \to TX$, assuming that $M_t\ll Q$ for all $t\in[0,T]$, to then show when the purely atomic property is inherited by the liftings of the curve $M_t$ given by the nested superposition principle. On the Euclidean space, the main assumption is that the $r$-capacity of the diagonal $\Delta \subset \mathbb{R}^d \times \mathbb{R}^d$ is zero with respect to $\nu \otimes \nu$, where $\nu$ is the barycenter of the reference measure $Q$. We will give sufficient conditions to ensure it, and in particular, thanks to the Nash embedding theorem, this analysis will allow us to extent the main results from the Euclidean space to Riemannian manifolds. Finally, we exploit this atomic superposition principle to show the lack of the Sobolev-to-Lipschitz property and the Poincar\'e inequality for functions in $W^{1,p}(\mathcal{P}_p(X),W_p,Q)$. Then, we complete the analysis showing that, however, the $L^2$-Wasserstein space endowed with suitable reference measure $Q$, satisfies a Bakry--\'Emery curvature condition.

math.MG↗

arXiv2025

Nested superposition principle for random measures and the geometry of the Wasserstein on Wasserstein space

We study the geometric structure of the space of random measures $\mathcal{P}_p(\mathcal{P}_p(X))$, endowed with the Wasserstein on Wasserstein metric, where $(X, d)$ is a complete separable metric space. In this setting, we prove a metric superposition principle, in the spirit of the result by S. Lisini, that will allow us to recover important geometric features of the space. When $X$ is $\mathbb{R}^d$, we study the differential structure of $\mathcal{P}_p(\mathcal{P}_p(\mathbb{R}^d))$ in analogy with the simpler Wasserstein space $\mathcal{P}_p(\mathbb{R}^d)$. We show that continuity equations for random measures involving the abstract concept of derivation acting on cylinder functions can be more conveniently described by suitable non-local vector fields $b:[0,T]\times \mathbb{R}^d \times \mathcal{P}(\mathbb{R}^d) \to \mathbb{R}^d$. In this way, we can: 1) characterize the absolutely continuous curves on the Wasserstein on Wasserstein space; 2) define and characterize its tangent bundle; 3) prove a superposition principle for the solutions to the standard non-local continuity equation in terms of solutions of interacting particle systems.

math.FA↗

arXiv2025

First order equation on random measures as superposition of weak solutions to the McKean-Vlasov equation

The goal of this paper is to define an evolution equation for a curve of random probability measures $(M_t)_{t\in[0,T]}\subset \mathcal{P}(\mathcal{P}(\mathbb{R}^d))$ associated to a non-local drift $b:[0,T]\times\mathbb{R}^d \times \mathcal{P}(\mathbb{R}^d) \to \mathbb{R}^d$ and a non-local diffusion term $a:[0,T]\times \mathbb{R}^d \times \mathcal{P}(\mathbb{R}^d) \to \operatorname{Sym}_+(\mathbb{R}^{d\times d})$. Then, we show that any solution to such an equation on random measures can be lifted twice: to a superposition of solutions to a non-linear Kolmogorov-Fokker-Planck equation and to a superposition of weak solutions to the McKean-Vlasov equations. Finally, we exploit this nested superposition result to show how existence and uniqueness can be transferred from the equation on random measures to the associated non-linear Kolmogorov-Fokker-Planck equation and to the McKean-Vlasov equation, assuming uniqueness of the linearized version of KFP. As a tool, we will introduce integral metrics over the spaces of probability measures $\mathcal{P}(\mathbb{R}^d)$ in duality with smooth functions, including a weighted second-order metric.

math.AP↗

arXiv2025

Totally convex functions, $L^2$-Optimal transport for laws of random measures, and solution to the Monge problem

We study the Optimal Transport problem for laws of random measures in the Kantorovich-Wasserstein space $\mathcal{P}_2(\mathcal{P}_2(\mathrm{H}))$, associated with a Hilbert space $\mathrm{H}$ (with finite or infinite dimension) and for the corresponding quadratic cost induced by the squared Wasserstein metric in $\\mathcal{P}_2(\mathrm{H}).$ Despite the lack of smoothness of the cost, the fact that the space $\mathcal{P}_2(\mathrm{H})$ is not Hilbertian, and the curvature distortion induced by the underlying Wasserstein metric, we will show how to recover at the level of random measures in $\mathcal{P}_2(\mathcal{P}_2(\mathrm{H}))$ the same deep and powerful results linking Euclidean Optimal Transport problems in $\mathcal{P}_2(\mathrm{H})$ and convex analysis. Our approach relies on the notion of totally convex functionals, on their total subdifferentials, and their Lagrangian liftings in the space square integrable $\mathrm{H}$-valued maps $L^2(\mathrm{Q},\mathbb{M};\mathrm{H}).$ With these tools, we identify a natural class of regular measures in $\mathcal{P}_2(\mathcal{P}_2(\mathrm{H}))$ for which the Monge formulation of the OT problem has a unique solution and we will show that this class includes relevant examples of measures with full support in $\mathcal{P}_2(\mathrm{H})$ arising from the push-forward transformation of nondegenerate Gaussian measures in $L^2(\mathrm{Q},\mathbb{M};\mathrm{H}).$

math.FA↗