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 $\alpha $-transport path and $ 
p$-harmonic functions on the underlying graph of the transport path for $ 
p=\alpha /(\alpha -1)$, which is the conjugate of $\alpha $. 
Furthermore, we prove the Lipschitz continuity of a landscape 
function associated with an optimal transport path on each of its 
connected components.

In particular, in [10], Santambrogio introduced a concept called "the landscape function" associated with an optimal irrigation pattern from a Dirac mass to another probability measure µ.This concept is motivated from the study of river basins, and represents the elevation of the landscape.We refer to the introduction of [10] for a detailed discussion about the geophysics background on this topic.A main result of [10] is the Hölder regularity of the landscape function when the targeting measure µ has a density bounded from below by a positive constant and its support satisfies the "type A" condition.These hypotheses on the targeting measure µ imply that the measure µ is Ahlfors regular from below in the ambient Euclidean space.Later in [5], Brancolini and Solimini extended the Hölder continuity of the 1684 QINGLAN XIA landscape function to general Ahlfors regular measures from below.In the context of ramified optimal transportation, the landscape function at a point can be viewed as the marginal transportation cost from the point to the source.Analogous ideas of the landscape function have been used in applications such as [20] in modeling diffusion-limited aggregation driven by optimal transportation and [21] about the ramified optimal allocation problem in economics.
In the works [5,10], it was assumed that the initial measure is a Dirac mass (i.e. a single common source).In this article, we generalize the concept of landscape function by allowing multiple sources rather than a single source.For simplicity, we will only consider transport paths between atomic measures in this article.The continuous version will be left for future exploration.
We organized this paper as follows: First, we introduce preliminary notations and concepts about graph theory and ramified optimal transportation in section 2. In particular, we introduce the concept of the incidence matrix of a directed graph and see how to relate it with concepts in ramified optimal transportation.In section 3, we generalize the concept of landscape functions associated with a transport path between atomic measures.By means of the incidence matrix, landscape functions can be solved using linear algebra.Then, we also explore some properties of landscape functions.In section 4, we consider p-harmonic functions defined on a directed graph for p ≤ 0. These p-harmonic functions minimize the corresponding p-energy with a Dirichlet boundary condition.In section 5, we showed an equivalence relation between an α-landscape function associated with a transport path and p-harmonic function on the underlying graph of the transport path when the parameters p ≤ 0 and 0 ≤ α < 1 are Hölder conjugate.In section 6, we studied properties of a landscape function associated with an optimal transport path.In particular, we showed Lipschitz continuity of any landscape function associated with an optimal transport path on each connected component of the transport path.

2.
Preliminaries.We first recall the definition of the incidence matrix of a directed graph as given in [1].
consisting of directed edges e j 's.The (vertex-edge) incidence matrix of G, denoted by Q (G), is the J ×K matrix defined as follows.For any where e − j is the starting endpoint of e j and e + j is the ending point of e j .Sometimes, we denote Here are some basic properties of incidence matrix Q (G) that we will use in this article.Proofs of these properties can be found for instance in [1].
Let G be a connected graph on J vertices.Then the column space of Q (G) consists of all vectors x ∈ R J such that J i=1 x i = 0. 3. Columns of Q (G) are linearly independent if and only if G is an acyclic graph.
We now recall some concepts about ramified optimal transportation as in Xia [12].Let X be a compact convex subset of a Euclidean space R m , equipped with the standard Euclidean distance • .Recall that a positive Radon measure a on X is atomic if a is a finite sum of Dirac measures with positive multiplicities.That is for some integer k ≥ 1 and some points are two atomic measures on X of equal mass.A transport path P = {V (P ) , E (P ) , w} from a to b is a directed graph G P = {V (P ) , E (P )} together with a weight function ω : E (P ) → (0, +∞) such that 1. V (P ) ⊆ R m and {x 1 , x 2 , ..., x k } ∪ {y 1 , y 2 , ..., y } ⊆ V (P ); 2. Each directed edge e ∈ E (P ) is a line segment from the starting endpoint e − ∈ V (P ) to the ending endpoint e + ∈ V (P ); 3. The weight function ω : E (P ) → (0, +∞) satisfies a balance equation The directed graph G P = {V (P ) , E (P )} is called the underlying graph of the transport path P .Also, we denote P ath (a, b) as the space of all transport paths from a to b.
For each transport path P ∈ P ath (a, b) as above and any α ∈ [0, 1], the M α cost of P is defined by where length (e) = e + − e − is the Euclidean distance between endpoints e + and e − of edge e. Ramified optimal transportation studies how to find a transport path to minimize the M α cost, i.e., min whose minimizer is called an α−optimal transport path from a to b.
For any α−optimal transport path P = {V (P ) , E (P ) , ω}, the first variation of M α (P ) yields , where e is the unit directional vector of the directed edge e.
According to Xia [12, Proposition 2.1], the underlying graph of an optimal transport path is acyclic, i.e. contains no cycles.Thus, without loss of generality, we will only consider acyclic transport paths in this article.
For any atomic measures a and b on X of equal mass, define the minimum transportation cost as (2.7) As shown in Xia [12], d α is indeed a metric on the space of atomic measures of equal mass.Also, for each λ > 0, it holds that Let P = {V (P ) , E (P ) , ω} be a transport path from a to b with Then in matrix notation, the balance equation ( 2.3) can be expressed as where Q (G P ) denotes the transpose of the incidence matrix Q (G P ) of the underlying graph ) of Proposition 2.1, when G P is acyclic, rows of Q (G P ) are linearly independent.If in addition G P is connected, then by (2) of Proposition 2.1, c lies in the row space of Q (G P ) .In this case, the linear system (2.9) has a unique solution w.
In other words, when G P is acyclic and connected, then w is uniquely determined by the incidence matrix Q (G P ) and c.

3.
Landscape functions associated with a transport path.Suppose the initial source is a single Dirac mass δ S and the targeting measure µ is a probability measure on X.In [10], Santambrogio introduced the landscape function z (x).It is the marginal transportation cost of the mass from the initial source S to the point x.A main result of [10] is the Hölder regularity of the landscape function when the targeting measure µ has a density bounded from below by a positive constant and its support satisfies the "type A" condition.Let a = δ S and b be an atomic measure on X.Let P = {V (P ) , E (P ) , w} be an acyclic transport path from a to b.Then for any x in the support of P , there is a unique polyhedral curve γ x on P from the initial source S to x.In this case, Santambrogio's landscape function z (x) associated with P can be simply written as where H 1 represents the 1−dimensional Hausdorff measure, and θ is a function on P defined as follows: for any point p on the support of P , set w(e), if p is in the interior of some edge e ∈ E (P ) 1, if p = S w(e), if p is the ending endpoint e + for some e ∈ E (P ) which represents the mass flowing through p.
In general, an atomic measure a is not necessarily a single source but of the form in (2.2) that contains multiple-sources.In this section, we will study a multiplesources version of landscape functions associated with a transport path.Definition 3.1.Let P = {V (P ) , E (P ) , ω} be an acyclic transport path from a to b with and 0 Using matrix notations the system (3.2) of linear equations on Z becomes is the incidence matrix of the underlying graph G P of P and If G P has g connected components, then by (1) of Proposition 2.1, the rank of Q (G P ) is J − g.Since G P is acyclic, by (3) of Proposition 2.1, columns of Q (G P ) are linearly independent.As a result, the solution space of the system (3.3) has dimension g.In particular, for a connected transport path P , its α-landscape function is unique up to a constant.This agrees with the motivation that the landscape function represents the elevation of the landscape.Let Then, the incidence matrix of G P is By calculating the reduced row echelon form corresponding to the system (2.9), the weight function w solving (2.9) is given by The lengths of edges are given by the vector Now, we can find landscape functions associated with P by solving the system (3.3).For instance, when α = 0.5,  In particular, if P is an α-optimal transport path from a to b, then Let Z be an α-landscape function associated with a transport path P ∈ P ath (a, b).For each edge e ∈ E (P ), define Now, we will prove an estimate on the landscape function.Such an estimate will be used later in Proposition 6.3 to show the Lipschitz continuity of landscape function on an optimal transport path.This result is analogous to Theorem 3.7 in [5] by Brancolini and Solimini.Proposition 3.5.Let Z be an α-landscape function associated with an acyclic transport path P ∈ P ath (a, b).Then, for any x, y ∈ V (P ) on the same connected components of G P , it holds that for any with | | ≤ min e⊆γ ω (e), where γ is the unique polyhedral curve on G P from x to y and P + γ is an acyclic transport path from a + δ x to b + δ y .
Proof.Since P is acyclic, there exists a unique polyhedral curve γ on P from x to y.This polyhedral curve γ is represented by a list of vertices {v for some edge e i in E (P ) for each i = 1, 2, • • • , k, where if e i has the same direction with γ −1, if e i has the opposite direction with γ .
By definition, the α-landscape function Z satisfies for each i.Using the inequality That is, (ω for each i and Remark 3.6.By linear extension of a landscape function Z on each edge e in E (G), we can view Z as a continuous and "edge-wise" linear function on the support of the transport path P .The above proof still work and the inequality (3.5) holds for all x, y in the support of P .
4. p-harmonic functions on directed graphs.Let G be an acyclic directed graph with a vertex set V (G) and an edge set E (G) of directed edges.
Definition 4.1.For any two vertices v, ṽ ∈ V (G), we say v ≺ ṽ if there exists a list of edges {e i1, e i2 , , and e − i1 = v, e + i k = ṽ.Note that since G is acyclic, the partial order ≺ is well-defined on V (G).In particular, e − ≺ e + for any edge e ∈ E (G).Definition 4.2.Let Ṽ be a subset of V (G).A function u : Ṽ → R is monotone increasing with respect to G if for any x, y ∈ Ṽ with x ≺ y, it holds that u (x) < u (y).
Clearly, when Ṽ = V (G), a function u : V (G) → R is monotone increasing with respect to G if and only if for any edge e ∈ E (G), Let F G be the family of all monotone increasing functions u : V (G) → R with respect to G.For instance, any landscape function Z associated with an acyclic transport path P is monotone increasing with respect to the underlying graph G P .
For each u ∈ F G and p ≤ 0, the p-energy of u is given by where for each e ∈ E (G) with unit directional vector e, We are interested in minimizing E p (u) among u ∈ F G with a Dirichlet boundary condition.among all continuous real valued functions u on the support of G such that u is differentiable and monotone increasing on each directed edge e in E (G).By considering the first variation of Ẽp , the minimum of each e |∇u| p dH 1 (x) is achieved when u is linear on e.Thus, it is sufficient to restrict u to be linear on each edge when considering the minimization problem (4.3).The corresponding energy Ẽp for an edge-wise linear function u is E p (u) in the form of (4.2).
Example 4.4.Without the monotone increasing condition (4.1), a minimizer of E p may fail to exist when p < 0. For instance, let G be a directed graph with We consider the minimization problem: minimize p → 0 as n → ∞ when p < 0. This indicates that the infimum of E p (u) with u ∈ {u : V (G) → R with u (−1) = 0 and u (1) = 1} is zero.Thus, without the condition (4.1), the minimizer of E p does not exist in this case.Nevertheless, with the monotone increasing condition (4.1), the problem has a unique minimizer given by u * (−1) = 0, u * (0) = 1/2 and u * (1) = 1.The minimum value is Let ∂G be a subset of V (G) such that v ∈ V (G) : either there is no edge e ∈ E (G) with e + = v or there is no edge e ∈ E (G) with e − = v ⊆ ∂G ⊆ V (G) .
In other words, ∂G contains all source and sink vertices in V (G), and may contain some other vertices.We view ∂G as the boundary set of V (G).Without loss of generality, we may assume that For a given function u 0 : ∂G → R, we consider the map for each u ∈ U .This map g is injective, and we denote Ω ⊆ R I as the image of U under the map g.
Lemma 4.5.The set Ω is a nonempty convex domain in R I if and only if u 0 : ∂G → R is monotone increasing with respect to G as in definition 4.2.
Proof.For any function u 0 : ∂G → R, since u ∈ F G is determined by a system of linear inequalities in the form of (4.1), Ω is a convex domain in R I if it is nonempty.We will show that Ω is nonempty if and only if u 0 is monotone increasing with respect to G.
If Ω is nonempty, then U is nonempty.For any u ∈ U , by definition, u ∈ F G and u = u 0 on ∂G.Since u : V (G) → R is monotone increasing, its restriction u 0 : ∂G → R is also monotone increasing with respect to G.
On the other hand, suppose u 0 is monotone increasing with respect to G. We construct a function u in U as follows.We first define u = u 0 on ∂G.If ∂G = V (G), then the partially ordered set (V (G) \ ∂G, ≺) has a minimal element x min which is not greater than any other element in V (G) \ ∂G.Note that since G is acyclic and ∂G contains all source and sink vertices of V (G), both {x ∈ ∂G, x ≺ x min } and {y ∈ ∂G, x min ≺ y} are nonempty sets.If x ∈ ∂G with x ≺ x min and y ∈ ∂G with x min ≺ y, then x ≺ y.By the monotonicity of u 0 , it follows that u 0 (x) < u 0 (y).Thus, max {u 0 (x) : x ∈ ∂G, x ≺ x min } < min {u 0 (y) : y ∈ ∂G, x min ≺ y} .
Define u (x min ) to be a number with Now, u is defined on ∂G ∪ {x min }.Continue the above process by treating ∂G ∪ {x min } as the new ∂G, we eventually define u on the set V (G).By the construction process, this function u is monotone increasing with respect to G and u = u 0 on the original ∂G.Thus, u ∈ U and U is nonempty.

Now we consider the following minimization problem with Dirichlet boundary condition:
Problem.Suppose u 0 : ∂G → R is monotone increasing with respect to G, and Equivalently, we want to minimize among all (u 1 , u 2 , • • • , u I ) in the nonempty convex domain Ω, where Lemma 4.6.Suppose u 0 : ∂G → R is monotone increasing with respect to G, and p < 0.Then, the function f given in (4.4) is strictly convex in the nonempty convex domain Ω.
] be the incidence matrix of G.The rows and the columns of and K is the total number of edges in E (G).Now, for each edge e ∈ E (G), by (4.1), Thus, for .
Therefore, the partial derivative |∇u (e)| p length (e) Moreover, for any i, j ∈ {1, 2, • • • , I}, the second order partial derivative As a result, the Hessian matrix of f is where Q is the I × K matrix consisting of the first I rows of the incidence matrix Q (G) of G, and R is a diagonal K × K matrix whose diagonal entries are .
When p < 0, luckily p (p − 1) > 0 and the Hessian matrix Hess (f ) is positive definite.Thus, f is strictly convex on the nonempty convex domain Ω.
To find critical points of f in Ω, we set ∂f ∂ui = 0.That is, i.e. for any v ∈ V (G) \∂G.
A solution to the p−Laplace equation (4.7) in U is called a p−harmonic function on the graph G.By the strict convexity of f , any p−harmonic function u is an E pminimizer in F G with respect to its boundary datum.

5.
Equivalence between α-landscape functions and p-harmonic functions.In this section, we will show that an α-landscape function associated with a transport path P is equivalent to a p−harmonic function on the underlying graph G P for conjugate parameters α ∈ [0, 1) and p ≤ 0. This shows that u is an α-landscape function associated with the transport path P .
In the following proposition, we have the reverse statement of Proposition 5.1.
Proposition 5.2.Let α ∈ [0, 1) and p = α α−1 ≤ 0 be the conjugate of α.If Z is an α-landscape function associated with an acyclic transport path P , then, Z is a p-harmonic function on the underlying graph G P .Moreover, M α (P ) = E p (Z).
Proof.Let G = G P be the underlying graph of P .Clearly, any α-landscape function Z of P satisfies the monotone increasing condition (4.1) with respect to G. So, We need to show that |∇Z (e)| p length (e) = E p (Z) .
6. Landscape functions associated with optimal transport paths.In this section, we study some properties of landscape functions associated with optimal transport paths.
Proposition 6.1.Suppose P ∈ P ath (a, b) is an α-optimal transport path.For any α-landscape function Z associated with P, it holds that at each vertex v ∈ Using this identity, equation (2.6) yields (6.2).
Remark 6.2.For any transport path P , by (2.3), mass is conserved at each interior vertex of P .When the transport path P is optimal, by (6.1), momentum is also conserved at each interior vertex when viewing ∇Z(e) = w(e) α−1 e as the velocity of a moving particle of mass w(e) on each e.
The following proposition says that any α-landscape function associated with an α−optimal transport path is Lipschitz.Before stating the proposition, we first extend the domain of a landscape function Z : V (P ) → R to the support of P by linear extensions of Z on each edge of P .Proposition 6.3.Suppose P ∈ P ath (a, b) is an α-optimal transport path for some α ∈ (0, 1).Let Z be an α-landscape function associated with P .Then, for any x, y on the same connected components of the support of G P , it holds that Proof.As shown in Figure 2, for any with | | ≤ σ, we set P = P + γ − [x, y] where [x, y] denotes the line segment from x to y.The construction of P here is standard in this kind of problems, see for instance Proposition 3.5 in [16] and Theorem 3.7 in [5].By Definition 2.2, P is also a transport path in P ath(a, b).Since P is optimal, it follows that  In the end of this section, we derive a formula of M α (P ) of an optimal transport path P in terms of boundary values of ∇Z of a landscape function Z.Such a formula might be useful in the future works including the one considering the dual problem of ramified optimal transportation.

Figure 1 .
Figure 1.Landscape functions associated with a transport path.

Proposition 3 . 3 .
For any α−landscape function Z associated with a transport path P ∈ P ath (a, b), it holds that X Zd (b − a) = M α (P ) .

Remark 4 . 3 .
One may also consider the minimization of Ẽp (u) =: v i ∈ V (G) \ ∂G.Using the discrete version of the divergence notation: div V (v) = either e + =v or e − =v V (e) • e for any vector field V : E (G) → R m , equation (4.6) can be expressed as the p−Laplace equation on the graph G : div |∇u| p−2 ∇u (v) = 0 (4.7)

Proposition 5 . 1 . 1 . 1 p− 1
Let p ≤ 0 and α = p p−1 ∈ [0, 1) be the conjugate of p.If u is a p−harmonic function on an acyclic graph G, then u is an α-landscape function associated with an acyclic transport path P ∈ P ath(a P , b P ) with G being its underlying graph, where a P and b P are two atomic measures of equal mass given in (5.1).Moreover, E p (u) = M α (P ).Proof.Suppose u is a p−harmonic function on G.For each edge e ∈ E (G), setω (e) = |∇u (e)| p−1 = u (e + ) − u (e − ) length(e)p−Since u is p−harmonic on G, by (4.6), it follows that e + =v ω (e) = e − =v ω (e) for each v ∈ V (G) \∂G.Now, for each v ∈ V (G), define m (v) = e − =v ω (e) − e + =v ω (e) .Let a P = m(vi)>0 m (v i ) δ vi and b P = m(vi)<0 (−m (v i )) δ vi .(v i )) .This says that the atomic measures a P and b P have the same mass on X. Together with the weight function ω, G becomes a transport path P = {V (G) , E (G) , ω} from a P to b P .Moreover, the transport cost of P is M α (P ) = e∈E(P ) ω (e) α length (e) = e∈E(P ) ω (e) p p−1 length (e) = e∈E(P ) |∇u| p length (e) = E p (u) .That is, the transport cost M α (P ) is the least p−energy E p (u) on G. Furthermore, for each edge e, u e + − u e − = |∇u (e)| length (e) = ω (e) length (e) = ω (e) α−1 length (e) .

Figure 2 .
Figure 2. The construction for the proof of Proposition 6.3.