DI MARINO S.

We analyze some parabolic PDEs with different drift terms which are gradient flows in the Wasserstein space and consider the corresponding discrete-in-time JKO scheme. We prove with optimal transport techniques how to control the LP and L1 norms of the iterated solutions in terms of the previous norms, essentially recovering well-known results...

We study the entropic regularization of the optimal transport problem in dimension 1 when the cost function is the distance c (x, y) = | y - x |.The selected plan at the limit is, among those which are optimal for the nonpenalized problem, the most "diffuse" one on the zones where it may have a density.

We provide new characterizations of Sobolev ad BV spaces in doubling and Poincaré metric spaces in the spirit of the Bourgain–Brezis–Mironescu and Nguyen limit formulas holding in domains of R^N .

A standard question arising in optimal transport theory is whether the Monge problem and the Kantorovich relaxation have the same infimum; the positive answer means that we can pass to the relaxed problem without loss of information. In the classical case with two marginals, this happens when the cost is positive, continuous, and possibly...

This paper exploit the equivalence between the Schrödinger Bridge problem (Léonard in J Funct Anal 262:1879–1920, 2012; Nelson in Phys Rev 150:1079, 1966; Schrödinger in Über die umkehrung der naturgesetze. Verlag Akademie der wissenschaften in kommission bei Walter de Gruyter u, Company, 1931) and the entropy penalized optimal transport...

In this paper, we present some basic uniqueness results for evolution equations under density constraints. First, we develop a rigorous proof of a well-known result (among specialists) in the case where the spontaneous velocity field satisfies a monotonicity assumption: we prove the uniqueness of a solution for first-order systems modeling...

We construct a regular random projection of a metric space onto a closed doubling subset and use it to linearly extend Lipschitz and C1 functions. This way we prove more directly a result by Lee and Naor [5] and we generalize the C1 extension theorem by Whitney [8] to Banach spaces.

We provide a quick proof of the following known result: the Sobolev space associated with the Euclidean space, endowed with the Euclidean distance and an arbitrary Radon measure, is Hilbert. Our new approach relies upon the properties of the Alberti–Marchese decomposability bundle. As a consequence of our arguments, we also prove that if the...

The intent of this short note is to extend real valued Lipschitz functions on metric spaces, while locally preserving the asymptotic Lipschitz constant. We then apply this results to give a simple and direct proof of the fact that Sobolev spaces on metric measure spaces defined with a relaxation approach à la Cheeger are invariant under...

We study the discretisation of generalised Wasserstein distances with nonlinear mobilities on the real line via suitable discrete metrics on the cone of N ordered particles, a setting which naturally appears in the framework of deterministic particle approximation of partial differential equations. In particular, we provide a Gamma-convergence...

We propose and analyze a natural extension of the Moreau sweeping process: given a family of moving convex sets (C(t))t, we look for the evolution of a probability density Pt, constrained to be supported on C(t). We describe in detail three cases: in the first, particles do not interact with each other and stay at rest unless pushed by the...

Lower semi-continuity results for polyconvex functionals of the calculus of variations along sequences of maps u: Ω ⊂ Rn → Rm in W1,m, 2 ≤ m ≤ n, weakly converging in W 1,m-1, are established. In addition, for m = n + 1, we also consider the autonomous case for weakly converging maps in W 1,n-1.

We investigate a number of formal properties of the adiabatic strictly-correlated electrons (SCE) functional, relevant for time-dependent potentials and for kernels in linear response time-dependent density functional theory. Among the former, we focus on the compliance to constraints of exact many-body theories, such as the generalised...