arXiv subjects
Publications and source records attributed to Luigi Ambrosio.
We consider the class of integer rectifiable currents without boundary satisfying a positivity condition. We establish that these currents can be written as a linear superposition of graphs of finitely many functions with bounded variation.
Nonnegative measure-valued solutions of the continuity equation are uniquely determined by their initial condition, if the characteristic ODE associated to the velocity field has a unique solution. In this paper we give a partial extension of this result to signed measure-valued solutions, under a quantitative two-sided Osgood condition on the velocity field. Our results extend those obtained for log-Lipschitz vector fields by Bahouri and Chemin.
In this paper we extend the DiPerna-Lions theory on ODEs with Sobolev vector fields to the setting of abstract Wiener spaces.
We study Markov processes associated with stochastic differential equations, whose non-linearities are gradients of convex functionals. We prove a general result of existence of such Markov processes and a priori estimates on the transition probabilities. The main result is the following stability property: if the associated invariant measures converge weakly, then the Markov processes converge in law. The proofs are based on the interpretation of a Fokker-Planck equation as the steepest descent flow of the relative Entropy functional in the space of probability measures, endowed with the Wasserstein distance. Applications include stochastic partial differential equations and convergence of equilibrium fluctuations for a class of random interfaces.
The geodesic problem in Wasserstein spaces with a metric perturbed by a conformal factor is considered, and necessary optimality conditions are estabilished in a case where this conformal factor favours the spreading of the probability measure along the curve. These conditions have the form of a system of PDEs of the kind of the compressible Euler equations. Moreover, self-similar solutions to this system are discussed.
In the first part of the paper we briefly decribe the classical problem, raised by Monge in 1781, of optimal transportation of mass. We discuss also Kantorovich's weak solution of the problem, which leads to general existence results, to a dual formulation, and to necessary and sufficient optimality conditions. In the second part we describe some recent progress on the problem of the existence of optimal transport maps. We show that in several cases optimal transport maps can be obtained by a singular perturbation technique based on the theory of $Γ$-convergence, which yields as a byproduct existence and stability results for classical Monge solutions.