On Landscape Functions Associated with Transport Paths

In this paper, we introduce a multiple-sources version of the landscape function which was originally introduced by Santambrogio in [10]. More precisely, we study landscape functions associated with a transport path between two atomic measures of equal mass. We also study p-harmonic functions on a directed graph for nonpositive p. We show an equivalence relation between landscape functions associated with an α-transport path and p-harmonic functions on the underlying graph of the transport path for p = α/(α − 1), which is the conjugate of α. Furthermore, we prove the Lipschitz continuity of a landscape function associated with an optimal transport path on each of its connected components.