On the geometry of geodesics in discrete optimal transport
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Abstract
We consider the space of probability measures on a discrete set \(\mathcal {X}\), endowed with a dynamical optimal transport metric. Given two probability measures supported in a subset \(\mathcal {Y}\subseteq \mathcal {X}\), it is natural to ask whether they can be connected by a constant speed geodesic with support in \(\mathcal {Y}\) at all times. Our main result answers this question affirmatively, under a suitable geometric condition on \(\mathcal {Y}\) introduced in this paper. The proof relies on an extension result for subsolutions to discrete Hamilton–Jacobi equations, which is of independent interest.
Mathematics Subject Classification
49Q20 53C211 Introduction
The metric \(W_2\) plays a special role in the theory, as it is the crucial object in the gradient flow formulation of dissipative PDE (starting from [11, 20]) and in the synthetic theory of Ricci curvature [14, 22], which builds on McCann’s discovery that several important functionals enjoy convexity properties along \(W_2\)geodesics [16].
In spite of the robustness of the optimal transport theory, it is well known that if the underlying space is discrete, \(W_2\) has several undesirable properties that hamper its usefulness. In particular, if \(\mathcal {X}\) is discrete, the metric space \((\mathcal {P}(\mathcal {X}), W_2)\) does not contain any nontrivial geodesics.
To circumvent this problem, several authors introduced discrete dynamical transport metrics \(\mathcal {W}\), based on discrete versions of the Benamou–Brenier formulation of optimal transport [2, 15, 17]. These metrics have been intensively studied in recent years; in particular, gradient flow formulations have been obtained for nonlinear evolution equations [6, 19], and a discrete theory of Ricci curvature has been developed based on geodesic convexity of entropy functionals along discrete optimal transport [5, 18]. Such Ricci curvature bounds have subsequently been obtained in various discrete probabilistic models [4, 7, 8].
In spite of the relevance of the notion of geodesic convexity, geometric properties of \(\mathcal {W}\)geodesics are currently poorly understood. The aim of this paper is to obtain results of this type. We focus on the issue of locality of geodesics in the space of probability measures.
More precisely, let \((\mathcal {X}, \mathsf {d})\) be a metric space, and consider a geodesic metric \(\mathsf {D}\) on (a subset of) the space of Borel probability measures \(\mathcal {P}(\mathcal {X})\). We say that a subset \(\mathcal {Y}\subseteq \mathcal {X}\) has the weak locality property if any pair of probability measures \(\mu _0, \mu _1 \in \mathcal {P}(\mathcal {X})\) supported in \(\mathcal {Y}\) can be connected by a geodesic that is supported in \(\mathcal {Y}\) at all times. The notion of strong locality is defined by requiring this property to hold for any geodesic connecting \(\mu _0\) and \(\mu _1\). If any pair of measures can be connected by a unique geodesic, the notions of weak and strong locality coincide, but this property is currently unknown for discrete dynamical transport metrics.
If \((\mathcal {X}, \mathsf {d})\) is a geodesic metric space, and \(\mathsf {D}\) is the Kantorovich metric \(W_p\) for some \(1 \le p < \infty \), it is well known that a subset \(\mathcal {Y}\) has the weak (resp. strong) locality property if and only if it is weakly (resp. strongly) geodesically convex. This follows from the fact that geodesics in \((\mathcal {P}_p(\mathcal {X}), W_p)\) are supported on geodesics in \((\mathcal {X}, \mathsf {d})\); cf. [12] for a precise formulation of this result in a general setting.
Interestingly, the issue of locality in the discrete setting [with a discrete dynamical transport metric \(\mathcal {W}\) on \(\mathcal {P}(\mathcal {X})\) instead of \(W_p\)] turns out to be much more delicate. For example, if one considers the complete graph on a threepoint set \(K_3\), then any geodesic connecting two Dirac masses transports a nontrivial part of the mass via the third point. Hence, twopoint subsets of \(K_3\) do not have the locality property. This is shown in Sect. 6 of this paper.
Based on this observation one may conjecture that any nontrivial \(\mathcal {W}\)geodesic has support on the whole graph. However, we show that this is not the case. In fact, the main contribution of this paper is the introduction of a geometric condition for subsets \(\mathcal {Y}\subseteq \mathcal {X}\) (the retraction property), that is shown to be sufficient for locality; see Theorem 4.11. The retraction property is easy to check in concrete examples, as is shown in Sect. 4.
As an application of our main result, we show that if \(\mathcal {X}\) is any subset of the grid \(\mathbb {Z}^d\) with the usual graph structure, and \(\mathcal {Y}\subseteq \mathcal {X}\) is a hyperrectangle, then any pair of measures supported in \(\mathcal {Y}\) can be connected by a geodesic supported in \(\mathcal {Y}\). In particular, this property holds for measures supported on subsets of lines, or kdimensional hyperplanes of dimension less than d. Let us also mention that discrete Ricci curvature bounds in the sense of [5, 18] are inherited by subsets with the retraction property; see Corollary 4.12.
A key ingredient in the proof of our main result is a duality result for the discrete transport metric \(\mathcal {W}\), which was recently obtained by Gangbo, Li, and Mou (under slightly more restrictive conditions on the transition rates) [9]. We interpret this result (Theorem 3.4 below) in terms of subsolutions of a discrete Hamilton–Jacobi equation and present a different proof based on Fenchel–Rockafellar duality. We then show that subsolutions of the Hamilton–Jacobi equation on a subset \(\mathcal {Y}\subseteq \mathcal {X}\) can be extended to the full space \(\mathcal {X}\), provided that \(\mathcal {Y}\) has the retraction property; cf. Theorem 4.10. Our main theorem is then a straightforward consequence of this result.
Structure of the paper In Sect. 2 we collect the necessary preliminaries on discrete transport metrics. Section 3 contains the dual formulation of the transport problem in terms of Hamilton–Jacobi subsolutions. In Sect. 4 we introduce the retraction property, we show the extension result for subsolutions to the Hamilton–Jacobi equation (Theorem 4.10), and we prove the main result on weak locality of subsets with the retraction property (Theorem 4.11). In Sect. 5 we show that the strong locality property holds for Markov chains with “dead ends”. Finally, it is shown in Sect. 6 that geodesics between Dirac measures on the triangle have full support.
2 The discrete transport distance
In this section we briefly recall the definition and basic properties of the discrete transport distance constructed in [2, 15, 17].
A Markov chain induces a graph on the vertex set \(\mathcal {X}\), whose edge set is given by \(\mathcal {E}= \{ (x,y) \in \mathcal {X}\times \mathcal {X}: Q(x,y) > 0 \}\). We write \(x \sim y\) iff \(Q(x,y) > 0\). The assumption that Q is irreducible corresponds to the graph \((\mathcal {X},\mathcal {E})\) being connected. The detailed balance condition implies that the graph is undirected.
In order to define the discrete transport distance on the set \(\mathcal {P}(\mathcal {X})\) of probability measures on \(\mathcal {X}\), we introduce the following objects.
Definition 2.1
 (i)
\(\mu :[0,T]\rightarrow \mathbb {R}^\mathcal {X}\) is continuous;
 (ii)
\(V:[0,T]\rightarrow \mathbb {R}^{\mathcal {X}\times \mathcal {X}}\) is locally integrable;
 (iii)
\(\mu _t\in \mathcal {P}(\mathcal {X})\) for all \(t\in [0,T]\);
 (iv)the continuity equation holds in the sense of distributions:$$\begin{aligned} \frac{\mathrm {d}}{\mathrm {d}t}\mu _t(x) + \frac{1}{2}\sum _{y\in \mathcal {X}}\left( V_t(x,y)  V_t(y,x) \right) = 0 \quad \text { for all } x\in \mathcal {X}. \end{aligned}$$(2.1)
Definition 2.2
(Admissible mean) An admissible mean is a continuous function \(\Lambda : \mathbb {R}_+ \times \mathbb {R}_+ \rightarrow \mathbb {R}_+\) that is \(C^\infty \) on \((0,\infty ) \times (0,\infty )\), symmetric, positively 1homogeneous, nondecreasing in each of its variables, jointly concave, and normalised, i.e., \(\Lambda (1,1) = 1\).
Definition 2.3
It has been shown in [5] that minimisers exist in the minimisation problem above. Any minimal curve \((\mu _t)_{t\in [0,1]}\) is a constant speed geodesic, i.e., it satisfies \(\mathcal {W}(\mu _s,\mu _t)=ts\mathcal {W}(\mu _0,\mu _1)\) for all \(s,t\in [0,1]\).
Remark 2.4
Without loss of generality we may assume in the minimisation (2.3) that V is antisymmetric, i.e., \(V_t(x,y)= V_t(y,x)\). In fact, for each \(U \in \mathbb {R}\), the quantity \(V(x,y)^2+V(y,x)^2\) is minimised among all choices of V(x, y), V(y, x) such that \(V(x,y)  V(y,x) = U\) by choosing \(V(y,x) = V(x,y) = U/2\).
3 Duality for discrete optimal transport
We present a dual formulation for the discrete transport distance which can be seen as a discrete analogue of the Kantorovich duality. This result has recently been proved in [9] using different methods; cf. Proposition 3.10 and Theorems 5.10 and 7.4 in that paper. Note that the result in [9] is stated under slightly stronger assumptions on the transition rates. In our notation, it is assumed there that \(Q(x,y) = Q(y,x)\) and \(\pi \) is constant. The slightly greater generality here does not cause additional difficulties.
Definition 3.1
Remark 3.2
Hamilton–Jacobi subsolutions obey a simple scaling relation: given \(\phi \in \mathsf {HJ}_\mathcal {X}^T\) and \(\lambda >0\), set \(\phi ^\lambda _t:=\lambda \phi _{\lambda t}\). It is immediate to check that \(\phi ^\lambda \in \mathsf {HJ}_\mathcal {X}^{\lambda T}\).
Remark 3.3
Informally, (3.1) may be seen as a onesided discrete version of the Hamilton–Jacobi equation \(\partial _t \phi + \frac{1}{2}\nabla \phi ^2 = 0\). Note however that the dependence on \(\mu \) in (3.1) is nonlinear, which prevents us from formulating the inequality pointwise in terms of \(\phi \) only. This is a crucial difference between the discrete and the continuous setting, and a source of several difficulties.
In the continuous setting, a full treatment of Hamilton–Jacobi equations relies on the theory of viscosity solutions [3], but this concept will not play any role in our discrete setting. Let us also mention that Hamilton–Jacobi equations have been studied in the setting of metric length spaces [10, 13] as well as on graphs [21]. Our discrete notion of Hamilton–Jacobi subsolution is different from the one studied in [21].
Theorem 3.4
Theorem 3.5
Proof of Theorem 3.4
Let us first note that, by the convexity of the constraint (3.1), any \(\phi \in \mathsf {HJ}^1_\mathcal {X}\) can be approximated uniformly by \(C^1\) functions satisfying (3.1) by convolution (after scaling the function to a slightly larger interval \([\,\delta ,1+\delta ]\) via Remark 3.2). Therefore, the final part of the theorem follows.
4 Locality of optimal curves
In this section we investigate locality properties for discrete transport geodesics. More precisely, we study the following question: Given two probability measures supported in a subset \(\mathcal {Y}\) of a state space \(\mathcal {X}\), is there an optimal curve connecting them that is supported in \(\mathcal {Y}\)? The crucial tool to analyse this question is the dual characterisation of the transport problem given in the previous section. We prove two types of results.
Firstly, we show that the question can be answered affirmatively, under a simple condition (the retraction property of the subgraph \(\mathcal {Y}\)), which will be introduced below. This property ensures that any competitor in the dual problem on the subgraph can be extended to a competitor on the full graph. We present several examples where this property is satisfied. Later, in Sect. 6, we will show that locality may fail if the retraction property is not satisfied.
We start by introducing the retraction property and we give several examples. To increase readability, we often write subscripts instead of parentheses, e.g., \(Q_{xy} = Q(x,y)\).
A subset \(\mathcal {Y}\subseteq \mathcal {X}\) is said to be connected if any two distinct points \(y, y' \in \mathcal {Y}\) can be connected by a path \(\{y_i\}_{i=0}^n \subseteq \mathcal {Y}\) satisfying \(y_0 = y\), \(y_n = y'\), and \(Q(y_{i1}, y_i) > 0\) for \(i=1,\ldots ,n\).
Definition 4.1
 (R1)
\(T(y)=y\) for all \(y\in \mathcal {Y}\);
 (R2)For all \(y,y' \in \mathcal {Y}\) with \(y\ne y'\), and all \(x \in T^{1}(y)\), we have$$\begin{aligned} \sum _{x'\in T^{1}(y')} Q(x,x') \le Q(y,y'). \end{aligned}$$
Remark 4.2
 \((R1')\)

\(T(y) = y\) for all \(y \in \mathcal {Y}\);
 \((R2')\)

If \(x \sim x'\), then \(T(x) = T(x')\) or \(T(x) \sim T(x')\);
 \((R3')\)

If \(x_1' \sim x\), \(x_2' \sim x\), and \(T(x_1') = T(x_2')\) for some \(x_1' \ne x_2'\), then \(T(x) = T(x_1')\).
Definition 4.3
(Restriction) The restriction of a Markov triple \((\mathcal {X},Q,\pi )\) to a connected subset \(\mathcal {Y}\subseteq \mathcal {X}\) is the Markov triple \((\mathcal {Y},Q_{\mathcal {Y}},\pi _{\mathcal {Y}})\), where \(Q_{\mathcal {Y}}\) is the restriction of Q to \(\mathcal {Y}\times \mathcal {Y}\), and \(\pi _{\mathcal {Y}}\) is the normalised restriction of \(\pi \) to \(\mathcal {Y}\).
Connectedness of \(\mathcal {Y}\) implies that the Markov triple \((\mathcal {Y},Q_{\mathcal {Y}},\pi _{\mathcal {Y}})\) is irreducible, and the detailed balance relation is obviously inherited. The following result implies that if \(\mathcal {Y}\) has the retraction property as a subset of \(\mathcal {X}\), it also has this property as a subset of any set \(\mathcal {X}'\) with \(\mathcal {Y}\subseteq \mathcal {X}' \subseteq \mathcal {X}\).
Lemma 4.4
Let \((\mathcal {X}, Q, \pi )\) be a Markov triple and \(\mathcal {Y}\subseteq \mathcal {X}' \subseteq \mathcal {X}\). If \(T : \mathcal {X}\rightarrow \mathcal {Y}\) is a retraction, then its restriction \(T_{\mathcal {X}'} : \mathcal {X}' \rightarrow \mathcal {Y}\) is a retraction as well.
Proof
This follows immediately from the definition. \(\square \)
We present some examples of sets with the retraction property.
Example 4.5
Example 4.6
(Grid) Consider \(\mathbb {Z}^d\) with the usual graph structure given by \(Q_{xy}=1\) if \(xy=1\) and \(Q_{xy}=0\) otherwise. Let \(\mathcal {Y}\subseteq \mathbb {Z}^d\) be a nonempty subset of the form \(\mathcal {Y}= \mathcal {R}\cap \mathbb {Z}^d\), where \(\mathcal {R}= \prod _{j=1}^d [a_j,b_j]\) is a hyperrectangle, and let \(\mathcal {X}\) be a connected subgraph of \(\mathbb {Z}^d\) containing \(\mathcal {Y}\). We claim that \(\mathcal {Y}\) has the retraction property. Indeed, it is readily checked that a retraction from \(\mathcal {X}\) to \(\mathcal {Y}\) can be obtained by mapping \(x \in \mathcal {X}\) to the point in \(\mathcal {Y}\) that is closest to x with respect to the Euclidean distance.
Example 4.7
(2Point space) Assume that Q takes values in \(\{0, 1\}\) and let \(x, y \in \mathcal {X}\) with \(Q_{xy} = 1\). A disjoint decomposition \(\mathcal {X}=A_x\cup A_y\) with \(x\in A_x\) and \(y\in A_y\) is called an xy cut. An edge \((u,v) \in \mathcal {E}\) is a cross if \(u \in A_x\) and \(v\in A_y\). The subset \(\{x,y\}\) has the retraction property if and only if there exists an xy cut such that no distinct crosses share a point. The correspondence between xy cuts with this property and retractions is given by \(T^{1}(x)=A_x\), \(T^{1}(y)=A_y\) (Fig. 2).
Example 4.8
(Honeycomb lattice) Let \((\mathcal {X},\mathcal {E})\) be a connected subgraph of the honeycomb lattice and define transition rates by setting \(Q_{xy} = 1\) if \((x,y)\in \mathcal {E}\) and zero otherwise. Then each fundamental cell \(\mathcal {Y}=\{y_1,\dots ,y_6\}\) (see Fig. 3) has the retraction property. Indeed, to obtain a retraction of \(\mathcal {X}\) onto \(\mathcal {Y}\), we partition the plane into 6 sectors separated by rays that originate at the centre of \(\mathcal {Y}\) and intersect the midpoints of the sides of \(\mathcal {Y}\) orthogonally. A retraction is then obtained by mapping each \(x \in \mathcal {X}\) to the unique \(y \in \mathcal {Y}\) that belongs to the same sector (cf. Fig. 3).
Example 4.9
(Trees) Assume that the graph \((\mathcal {X}, \mathcal {E})\) is a tree, i.e., it does not contain a cycle. Every subtree \(\mathcal {Y}\) of \(\mathcal {X}\) has the retraction property, and a retraction can be constructed as follows: Fix a vertex \(y \in \mathcal {Y}\). Since \(\mathcal {X}\) is a tree, for every \(x\in \mathcal {X}\) there is a unique path \(\gamma \) without selfintersections connecting x and y. The map assigning to x the first point where the path \(\gamma \) meets \(\mathcal {Y}\) is a retraction of \(\mathcal {X}\) onto \(\mathcal {Y}\). Note that the retraction property depends only on the graph \((\mathcal {X}, \mathcal {E})\) and not on the choice of the transition rates Q (as long as they give rise to the same graph).
Theorem 4.10
(Extension of Hamilton–Jacobi subsolutions) Let \((\mathcal {X}, Q, \pi )\) be a Markov triple, and let \(\mathcal {Y}\) be a connected subset of \(\mathcal {X}\). If \(\mathcal {Y}\) has the retraction property, then every Hamilton–Jacobi subsolution on \(\mathcal {Y}\) can be extended to a Hamilton–Jacobi subsolution on \(\mathcal {X}\).
Proof
The following result shows that any pair of measures supported in a set \(\mathcal {Y}\) with the retraction property can be connected by a geodesic supported in \(\mathcal {Y}\).
Theorem 4.11
(Weak locality under the retraction property) Let \((\mathcal {X}, Q, \pi )\) be a Markov triple, and let \(\mathcal {Y}\) be a subset of \(\mathcal {X}\) with the retraction property. For all \(\mu ^0,\mu ^1 \in \mathcal {P}(\mathcal {X})\) with support in \(\mathcal {Y}\) there exists a minimising \(\mathcal {W}\)geodesic \((\mu _t)_{t \in [0,1]} \subseteq \mathcal {P}(\mathcal {X})\) connecting \(\mu ^0\) and \(\mu ^1\) such that \(\mu _t\) has support in \(\mathcal {Y}\) for all \(t\in [0,1]\).
In fact, we will show that any \(\mathcal {W}_\mathcal {Y}\)geodesic \((\mu _t)_t \subseteq \mathcal {P}(\mathcal {Y})\) is also a \(\mathcal {W}_\mathcal {X}\)geodesic when regarded as a curve in \(\mathcal {P}(\mathcal {X})\).
Proof
Let \((\mu _t)_t\) be a minimising geodesic in \(\mathcal {P}(\mathcal {Y})\) satisfying the continuity Eq. (2.1) with momentum vector field \((V_t)_t\). Consider the extension to \(\mathcal {X}\) defined by \(\bar{\mu }_t(x)=0\) if \(x\notin \mathcal {Y}\) and \(\bar{V}_t(x,x')=0\) if \(x\notin \mathcal {Y}\) or \(x'\notin \mathcal {Y}\). Clearly, \((\bar{\mu }_t,\bar{V}_t)_t\) has the same action as \((\mu _t,V_t)_t\).
Corollary 4.12
Let \((\mathcal {X}, Q, \pi )\) be a Markov triple, and let \(\mathcal {Y}\) be a subset of \(\mathcal {X}\) with the retraction property. If \({{\,\mathrm{Ric}\,}}(\mathcal {X}, Q, \pi ) \ge \kappa \) for some \(\kappa \in \mathbb {R}\), then \({{\,\mathrm{Ric}\,}}(\mathcal {Y}, Q_\mathcal {Y}, \pi _\mathcal {Y}) \ge \kappa \) as well.
Proof
Take \(\mu _0, \mu _1 \in \mathcal {P}(\mathcal {Y})\), and let \((\mu _t)_t\) be a \(\mathcal {W}_\mathcal {Y}\)geodesic connecting them. By Theorem 4.11, \((\mu _t)_t\) is also a geodesic in \(\mathcal {P}(\mathcal {X})\). Since \({{\,\mathrm{Ric}\,}}(\mathcal {X}, Q, \pi ) \ge \kappa \), it follows that \(t \mapsto {{\,\mathrm{Ent}\,}}(\mu _t\pi )\) is \(\kappa \)convex. As \({{\,\mathrm{Ent}\,}}_{\pi _\mathcal {Y}}(\mu _t) = {{\,\mathrm{Ent}\,}}_{\pi }(\mu _t) + \log (\pi (\mathcal {Y}))\) and \(\mathcal {W}_\mathcal {Y}(\mu _0, \mu _1) = \mathcal {W}_\mathcal {X}(\mu _0, \mu _1)\), we infer that \(t \mapsto {{\,\mathrm{Ent}\,}}_{\pi _\mathcal {Y}}(\mu _t)\) is \(\kappa \)convex as well, which yields the result. \(\square \)
5 Optimal transport avoids dead ends
In this section we prove the intuitively natural statement that optimal curves do not transport mass into “dead ends”. We formalise this concept by considering the gluing of two Markov triples along a vertex.
Definition 5.1
Definition 5.2
(Dead end) Let \((\mathcal {X}, Q, \pi )\) be a Markov triple, and let \(\mathcal {X}_1, \mathcal {X}_2 \subseteq \mathcal {X}\). We say that \(\mathcal {X}_2\) is a dead end for \(\mathcal {X}_1\) (and vice versa) if the intersection of \(\mathcal {X}_1\) and \(\mathcal {X}_2\) contains exactly one point (denoted “\(*\)”), and moreover, \(Q(x,y) = Q(y,x) = 0\) whenever \(x \in \mathcal {X}_1'\) and \(y \in \mathcal {X}_2'\). Here, we write \(\mathcal {X}_i' = \mathcal {X}_i \backslash \{ *\}\).
Remark 5.3
The notions of dead end and gluing of Markov triples are compatible in the following sense: Let \((\mathcal {X}, Q, \pi )\) be a Markov triple, and suppose that \(\mathcal {X}_2 \subseteq \mathcal {X}\) is a dead end for \(\mathcal {X}_1 \subseteq \mathcal {X}\) with intersection point \(*\). Then one recovers \((\mathcal {X}, Q, \pi )\) by gluing together the restrictions of \(\mathcal {X}\) to \(\mathcal {X}_1\) and \(\mathcal {X}_2\) at \(*\).
Proposition 5.4
Let \((\mathcal {X}_1,Q_1,\pi _1)\) and \((\mathcal {X}_2,Q_2,\pi _2)\) be Markov triples, and let \((\mathcal {X}, Q, \pi )\) be the Markov triple obtained by gluing the triples at \(x_1 \in \mathcal {X}_1\) and \(x_2 \in \mathcal {X}_2\). Then \(\mathcal {X}_1\) and \(\mathcal {X}_2\) have the retraction property as subsets of \(\mathcal {X}\).
Proof
Define \(T : \mathcal {X}\rightarrow \mathcal {X}_1\) by \(T(x) = x\) for \(x \in \mathcal {X}_1\) and \(T(x) = *\) for \(x \in \mathcal {X}_2'\). One verifies that T indeed defines a retraction by distinguishing cases. \(\square \)
In view of Theorem 4.11, the previous result implies that any two measures \(\mu _0, \mu _1\) supported in (the image of) \(\mathcal {X}_1\) can be connected by a geodesic that is supported in \(\mathcal {X}_1\) for all times; i.e., weak locality holds. We will now show that in fact strong locality holds: any geodesic connecting \(\mu _0\) and \(\mu _1\) has to be supported in \(\mathcal {X}_1\).
Theorem 5.5
Let \((\mathcal {X}_1,Q_1,\pi _1)\) and \((\mathcal {X}_2,Q_2,\pi _2)\) be Markov triples, and let \((\mathcal {X}, Q, \pi )\) be the Markov triple obtained by gluing the triples at \(x_1 \in \mathcal {X}_1\) and \(x_2 \in \mathcal {X}_2\). If \((\mu _t)_{t\in [0,1]}\) is a geodesic in \((\mathcal {P}(\mathcal {X}),\mathcal {W})\) with \({\text {supp}}\mu _0, {\text {supp}}\mu _1\subseteq \mathcal {X}_1\), then \({\text {supp}}\mu _t\subseteq \mathcal {X}_1\) for all \(t\in [0,1]\).
Proof
6 Nonlocality of optimal transport on the triangle
Consider a Markov triple \((\mathcal {X}, Q, \pi )\) and a connected subset \(\mathcal {Y}\subseteq \mathcal {X}\). In this section we show that locality of geodesics in \(\mathcal {P}(\mathcal {Y})\) may fail if \(\mathcal {Y}\) does not have the retraction property. We consider the simplest possible setting, where \((\mathcal {X}, Q, \pi )\) corresponds to simple random walk on a triangle, and \(\mathcal {Y}\subseteq \mathcal {X}\) is a twopoint set. We show that the canonical lift of a geodesic between Dirac measures on the twopoint space is not an optimal curve in \(\mathcal {P}(\mathcal {X})\), by constructing a competitor that transports mass along all edges.
Throughout this section we make the following additional assumption on the mean \(\Lambda \).
Assumption 6.1
Clearly, this assumption is satisfied for the arithmetic, geometric, and logarithmic means, but not for the harmonic mean.
The main result of this section relies on the following lemma concerning the variation of the action functional on cycles of arbitrary length.
Lemma 6.2
Proof
Now we can prove the nonlocality result.
Theorem 6.3
Let \((\mathcal {X},Q,\pi )\) be a Markov triple with \(\mathcal {X}=\{1,2,3\}\) and such that \(Q(x,y) > 0\) for all \(x\ne y\). Let \((\mu _t)_{t \in [0,1]}\) be a \(\mathcal {W}\)geodesic connecting \(\mu _0 = \delta _1\) to \(\mu _1 = \delta _2\). Then, \(\mu _t(3)>0\) for some \(0<t<1\).
As \(\mu _t(3)>0\) for some \(0<t<1\), the result implies that mass is transported along the edges (1, 3) and (3, 2).
Proof
Notes
Acknowledgements
Open access funding provided by Institute of Science and Technology (IST Austria). Matthias Erbar gratefully acknowledges support by the German Research Foundation through the Hausdorff Centre for Mathematics and the Collaborative Research Centre 1060, The Mathematics of Emergent Effects. Jan Maas gratefully acknowledges support by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant Agreement No. 716117), and by the Austrian Science Fund (FWF), Project SFB F65. Melchior Wirth gratefully acknowledges financial support by the German Academic Scholarship Foundation (Studienstiftung des deutschen Volkes) and by the German Research Foundation (DFG) via RTG 1523. We thank an anonymous referee for suggesting to include the statement of Corollary 4.12 in the paper.
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