Transport of measures on networks

Abstract

In this paper we formulate a theory of measure-valued linear transport equations on networks. The building block of our approach is the initial/boundary-value problem for the measure-valued linear transport equation on a bounded interval, which is the prototype of an arc of the network. For this problem we give an explicit representation formula of the solution, which also considers the total mass flowing out of the interval. Then we construct the global solution on the network by gluing all the measure-valued solutions on the arcs by means of appropriate distribution rules at the vertexes. The measure-valued approach makes our framework suitable to deal with multiscale flows on networks, with the microscopic and macroscopic phases represented by Lebesgue-singular and Lebesgue-absolutely continuous measures, respectively, in time and space

Generalized dynamic programming principle and sparse mean-field control problems

“…This microscopic sparsity constraint allows also to select a (possibly not unique) probabilistic representation for feasible trajectories, since it can be used to prescribe the paths to be followed by the microscopic particles. This makes it particulary suitable for applications in irrigation problems or dynamics set on networks [8,9]. We call this constraint the L ∞ -extended curve-based feasibility condition.…”

Measure-valued solutions to nonlocal transport equations on networks

“…The aim of this section is twofold. In the first part, we briefly review the results for the linear problem in [4], since they are an important tool for developing the theory of the nonlinear problem via an approximation procedure. Hence, we give a new representation formula for the measurevalued solution of the linear problem (afterwards extended also to the nonlinear problem), which generalizes the well-known push-forward formula to the network setting.…”

“…In order to prove the representation formula (3.9) we preliminarily recall a characterization of the traces of the solution m of (3.1) on the fibers e j × {t} and {x i } × [0, t], where x i = π j (L j ), in terms of the transport of the initial and boundary data inside e j (see [4]).…”

“…In order to extend the measure-valued approach to networks, in [4] it was studied the linear transport equation…”

The current method for stationary mean-field games on networks

“…Also, there is now a vast literature on HJ equations on networks [SC13], [CMS13], [ACCT13], [CM13], [CCM18]. Stationary first-order HJ equations on networks were examined in [SS18] and [ISM17] and the transport of measures on networks in [CDMT17]. Our MFG model can be seen as a generalization of the Wardrop equilibrium [WW52], where the routes between a target and a destination are used in such a way that travel time is independent on the choice of path.…”