Curves of minimal action over metric spaces

Abstract

Given a metric space $X$, we consider a class of action functionals, generalizing those considered in \cite{BBS} and \cite{AS}, which\ud measure the\ud cost of joining two given points $x_0$ and $x_1$, by means of an absolutely continuous curve. In the case $X$ is given by a space\ud of probability measures, we can think of these action functionals as giving the cost of some congested/concentrated mass transfer problem. We focus\ud on the possibility to split the mass in its {\it moving part} and its part that (in some sense) has\ud already reached its final destination: we consider new action functionals, taking into account only the contribution of the moving part

Infinite horizon value functions in the Wasserstein spaces

“…There is a growing literature on value functions in the space of probability measures [7,9,10,11,14] and more generally in metric spaces [3,4,6,12,15]. However, the bulk of these efforts have been restricted to optimization problems on a finite time horizon; as mentioned, we consider model infinite horizon problems in this paper.…”

A Benamou–Brenier Approach to Branched Transport

“…A first attempt to obtain a dynamical formulation of branched transportation through curves of measures was made in [10], and later refined in [11,12]: in these papers the starting point is the geodesic formulation of the Wasserstein distance, where the length functional is modified considering an energy of the type…”

“…A tentative to perform some modifications in the functionals 3 defined on curves of measures so as to obtain equivalence with the other models has been made in [11], where on the other hand some quite involved distinction between moving mass and still mass has been done. In all of these models, the branched transportation is studied avoiding the Benamou-Brenier approach consisting in the minimization of a suitable cost F(ρ, q) under the constraint of the continuity equation ∂ t ρ + div x q = 0, that we believe is the most natural for this kind of problems.…”

A Benamou-Brenier approach to branched transport

“…These models are not satisfactory yet, because they are in general not equivalent to those by Gilbert, Xia or Bernot-Caselles-Morel. A tentative to perform some modifications in the functionals defined on curves of measures so as to obtain equivalence with the other models has been made in [11], where on the other hand some quite involved distinction between moving mass and still mass has been done. In all of these models, the branched transportation is studied avoiding the Benamou-Brenier approach consisting in the minimization of a suitable cost F(ρ, q) under the constraint of the continuity equation ∂ t ρ + div x q = 0, that we believe is the most natural for this kind of problems.…”