Optimal transport from a point-like source

We present a dynamical interpretation of the Monge-Kanto-rovich theory in a stationary regime. This new principle, akin to the Fermat principle of geometrical optics, captures the geodesic character of many distribution networks such as plant roots, river basins and the physiological transportation network of metabolites in living systems. Our general continuum framework allows us to map a previously proposed phenomenological principle into a stationary Monge optimization principle in the Kantorovich relaxed format.