Dario Trevisan

Well-posedness of Lagrangian flows and continuity equations in metric measure spaces

We establish, in a rather general setting, an analogue of DiPerna-Lions theory on wellposedness of flows of ODE’s associated to Sobolev vector fields. Key results are a wellposedness result for the continuity equation associated to suitably defined Sobolev vector fields, via a commutator estimate, and an abstract superposition principle in (possibly extended) metric measure spaces, via an embedding into R 1 . When specialized to the setting of Euclidean or infinite dimensional (e.g. Gaussian) spaces, large parts of previously known results are recovered at once. Moreover, the class of RCD(K,∞) metric measure spaces introduced in [AGS11b] and object of extensive recent research fits into our framework. Therefore we provide, for the first time, wellposedness results for ODE’s under low regularity assumptions on the velocity and in a nonsmooth context.

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–Ampère equation.

The One-Mode Quantum-Limited Gaussian Attenuator and Amplifier Have GaussianMaximizers

We determine the

Lecture notes on the DiPerna-Lions theory in abstract measure spaces

p\rightarrow q

Gaussian optimizers for entropic inequalities in quantum information

p→q norms of the Gaussian one-mode quantum-limited attenuator and amplifier and prove that they are achieved by Gaussian states, extending to noncommutative probability the seminal theorem “Gaussian kernels have only Gaussian maximizers” (Lieb in Invent Math 102(1):179–208, 1990). The quantum-limited attenuator and amplifier are the building blocks of quantum Gaussian channels, which play a key role in quantum communication theory since they model in the quantum regime the attenuation and the noise affecting any electromagnetic signal. Our result is crucial to prove the longstanding conjecture stating that Gaussian input states minimize the output entropy of one-mode phase-covariant quantum Gaussian channels for fixed input entropy. Our proof technique is based on a new noncommutative logarithmic Sobolev inequality, and it can be used to determine the

Three superposition principles: Currents, continuity equations and curves of measures☆

p\rightarrow q