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 S. Caracciolo et al. (Phys. Rev. E, 90 012118, 2014) that "linearise" the Monge-Ampère equation. * Scuola Normale Superiore, luigi.ambrosio@sns.it.

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 ∞ . 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 [AGS14b] 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.

We investigate well-posedness for martingale solutions of stochastic differential equations, under low regularity assumptions on their coefficients, widely extending the results first obtained by A. Figalli in [19]. Our main results are: a very general equivalence between different descriptions for multidimensional diffusion processes, such as Fokker-Planck equations and martingale problems, under minimal regularity and integrability assumptions; and new existence and uniqueness results for diffusions having weakly differentiable coefficients, by means of energy estimates and commutator inequalities. Our approach relies upon techniques recently developed jointly with L. Ambrosio in [6], to address well-posedness for ordinary differential equations in metric measure spaces: in particular, we employ in a systematic way new representations for commutators between smoothing operators and diffusion generators

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An ordering between the quantum states emerging from a single mode gauge-covariant bosonic Gaussian channel is proven. Specifically, we show that within the set of input density matrices with the same given spectrum, the element passive with respect to the Fock basis (i.e. diagonal with decreasing eigenvalues) produces an output which majorizes all the other outputs emerging from the same set. When applied to pure input states, our finding includes as a special case the result of A. Mari, et al., Nat. Comm. 5, 3826 (2014) which implies that the output associated to the vacuum majorizes the others. arXiv:1511.00293v2 [quant-ph]