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Scaling limits of discrete optimal transport

We consider dynamical transport metrics for probability measures on discretisations of a bounded convex domain in R. These metrics are natural discrete counterparts to the Kantorovich metric W2, defined using a Benamou–Brenier type formula. Under mild assumptions we prove an asymptotic upper bound for the discrete transport metric WT in terms of W2, as the size of the mesh T tends to 0. However, we show that the corresponding lower bound may fail in general, even on certain one-dimensional and symmetric two-dimensional meshes. In addition, we show that the asymptotic lower bound holds under an isotropy assumption on the mesh, which turns out to be essentially necessary. This assumption is satisfied, e.g., for tilings by convex regular polygons, and it implies Gromov–Hausdorff convergence of the transport metric.


Introduction
Over the last decades, optimal transport has become a vibrant research area at the interface of analysis, probability, and geometry. A central object is the 2-Kantorovich distance W 2 (often called 2-Wasserstein metric), which plays a major role in non-smooth geometry and analysis, and in the theory of dissipative PDE. We refer to the monographs [1,28,30] for an overview of the theory and its applications.
More recently, discrete dynamical transport metrics have been introduced in the context of Markov chains [22], reaction-diffusion systems [24] and discrete Fokker-Planck equations [7]. These metrics are natural discrete counterparts to W 2 in several ways: they have been used to obtain a gradient flow formulation for discrete evolution equations [11,23], and to develop a discrete theory of Ricci curvature that leads to various functional inequalities for discrete systems [10,15,25]. The geometry of geodesics for these metrics is currently actively studied, both from an analytic point of view [12,17], and through numerical methods [13,29].
It is natural to ask whether the discrete transport metrics converge to W 2 under suitable assumptions. The first result of this type has been obtained in [19]. The authors approximated the continuous torus T d by the discrete torus T d N = (Z/N Z) d , and endowed the space of probability measures P(T d N ) with the discrete transport metric W N . The main result in [19] asserts that, under a natural rescaling, the metric spaces (P(T d N ), W N ) converge to the L 2 -Kantorovich space (P(T d ), W 2 ) in the sense of Gromov-Hausdorff as N → ∞.
A different convergence result was subsequently obtained by Garcia Trillos [18]. Given a set X N consisting of N distinct points in T d , Garcia Trillos considers the graph obtained by connecting all pairs of points that lie at distance less than ε, for a suitable ε depending on N . Under appropriate conditions on the uniformity of the point set, it is shown in [18] that the discrete transport metric converges to W 2 , provided that ε = ε(N ) decays sufficiently slow. While the result of [18] covers a wide range of settings, the latter assumption typically implies that the number of neighbours of a point in X N tends to ∞ as N → ∞; in particular, the result of [19] is not contained in [18].
The aim of this paper is to investigate Gromov-Hausdorff convergence for transport metrics on unstructured discretisations of a bounded convex domain Ω ⊆ R d . While our setting is a different from [18], it corresponds in terms of scaling to the limiting regime in which the results of [18] fail to apply. for µ 0 , µ 1 ∈ P(Ω). Here, the infimum is taken among all absolutely continuous curves (µ t ) t∈ [0,1] in P(Ω) connecting µ 0 and µ 1 , and A * (µ, w) = sup We discretise the domain Ω using a finite volume discretisation, closely following the setup from [16]. An admissible mesh consists of a partition T of Ω into sets K with nonempty and convex interior, together with a family of distinct points {x K } K∈T such that x K ∈ K for all K ∈ T . We write (K|L) = K ∩ L to denote the flat convex surface with (d − 1)-dimensional Hausdorff measure |(K|L)|. We make the geometric assumption that the vector x K − x L is orthogonal to (K|L) if K and L are neighbouring cells. In addition, we impose some mild regularity conditions on the mesh; see Definition 2.11 for the notion of ζ-regularity that is imposed in the sequel. We write [T ] := max K∈T diam(K) to denote the mesh size of T .
The discrete transport metric on P(T ) is defined in terms of a discrete Benamou-Brenier formula: for m 0 , m 1 ∈ P(T ), we set where the action functionals are defined using natural discrete counterparts to (1.1): Here, the transmission coefficients S KL are defined by S KL = |(K|L)| |x K −x L | . This choice ensures the formal consistency of the discrete and the continuous definitions; cf. Remark 2.14 below for a verification at the level of the associated Dirichlet forms.
The functions θ KL : R + × R + → R + are admissible means, i.e., θ KL is a continuous function that is C ∞ on (0, ∞) × (0, ∞), positively 1-homogeneous, jointly concave, and normalised (i.e., θ(1, 1) = 1). Furthermore, we impose the symmetry condition θ KL (s, t) = θ LK (t, s). A common choice is the logarithmic mean θ KL (s, t) :=´1 0 s 1−p t p dp for all K ∼ L, which naturally arises in the gradient flow structure of the discrete heat equation [7,22,24]. We write θ = (θ KL ) to denote the collection of mean functions in the definition of W T , and suppress the dependence of W T on θ in the notation. The freedom to choose these mean functions is due to the discreteness of the problem. We will see that a careful choice is crucial in the sequel.
Various convergence results are known in this setting. In particular, we refer to [16,Theorem 4.2] for a convergence result for the discrete heat equation to the continuous heat equation. In one spatial dimension, evolutionary Γ-convergence has been proved for the entropic gradient flow structure of the discrete heat flow with respect to the transport distance W T [9]. For gradient flow approximations to nonlinear parabolic problems, convergence results have been obtained as well; see, e.g., [5].

Statement of the main results
The goal of this paper is to analyse the limiting behaviour of the discrete transport metrics W T as [T ] → 0. To formulate the main results we introduce the canonical projection operator P T : P(Ω) → P(T ) given by for µ ∈ P(Ω) and K ∈ T . (1.2) Our first main result establishes a one-sided asymptotic estimate for the discrete transport metric in great generality. W T (P T µ 0 , P T µ 1 ) ≤ W 2 (µ 0 , µ 1 ) .
In view of the Gromov-Hausdorff convergence results from [19] and [18], one might expect that a corresponding asymptotic lower bound for W T in terms of W 2 holds as well. However, convergence can fail, even in very structured situations, as the following example shows.
Example (A; a one-dimensional period mesh). For N ∈ N and r ∈ (0, 1 2 ), we consider a one-dimensional discretisation T r,N of the unit interval [0, 1], obtained by alternatingly concatenating intervals of length r N and 1−r N . The next result shows that W T can not be bounded from below by W 2 as [T ] → 0.
Proposition 1.2 (Counterexample to the lower bound for W T ). Let Ω = (0, 1) and let T r,N be as in Example (A) above. Fix an admissible symmetric mean θ, and consider the transport metric W T defined by setting θ KL = θ for all K, L ∈ T . Then there exist probability measures µ 0 , µ 1 ∈ P(Ω) such that, for each fixed r ∈ (0, 1 2 ), We emphasize that, in the setting of the counterexample, evolutionary Γ-convergence has been proved for the entropic gradient flow structure of the discrete heat flow with respect to the transport distance W T ; cf. [9].
The idea behind the one-dimensional counterexample is that an "unreasonably cheap" discrete transport can be constructed by introducing microscopic oscillations in the discrete density in such a way that most of the mass is assigned to small cells. We refer to Section 5 for more details.
In view of Proposition 1.2 it is natural to look for additional geometric assumptions on the mesh under which an asymptotic lower bound for W T in terms of W 2 can be obtained. A weight function on T is a mapping λ : T × T → [0, 1] satisfying λ KL + λ LK = 1 for all K, L ∈ T . The following definition plays a central role in our investigations.

Definition 1.3 (Asymptotic isotropy). A family of admissible meshes {T } is said to be
The asymptotic isotropy condition puts a strong geometric constraint on a family of meshes, although it will be shown in Section 5 that isotropy always holds on average on a macroscopic scale.
Remark (Center-of-mass condition). In examples it is often convenient to verify the asymptotic isotropy condition by checking the following stronger condition: we say that T satifies the center-of-mass condition with weight function λ if for any pair of neighbouring cells K, L ∈ T . As λ KL +λ LK = 1, this condition asserts that the center of mass of the interface (K|L) lies on the line segment connecting x K and x L . In the literature on finite volume methods this property is known as superadmissibility of the mesh; see [14,Lemma 2.1].
For all interior cells K ∈ T , we claim that the center-of-mass condition yields the asymptotic isotropy condition (1.3) with equality and η T (K) = 0. To see this, let n = n(x) be the outward unit normal on ∂K, and note that The claim follows by summation over L.
For boundary cells, the asymptotic isotropy condition holds if ∂K ∩ ∂Ω consists of planar interfaces for which it is possible to add "fake points" outside of Ω in such a way that the center-of-mass condition holds. This will be the case in several examples below.
Example (A; revisited). Clearly, the center-of-mass condition holds in every one-dimensional mesh for an appropriate weight function λ. For the one-dimensional periodic lattice T r,N in Example (A), it is immediately checked that λ KL = r if K is small, and Example (B; Two dimensional meshes). The center-of-mass condition holds for the regular hexagonal lattice (B1) and the equilateral triangular lattice (B2) in dimension 2, if the points x K are placed at the center of mass of the cells. In these examples we have λ KL = 1 2 for all K and L. The hexagonal lattice (B1) is truncated in such a way that all interior interface have equal size. More generally, it is immediate to see that the center-of-mass condition holds for any tiling of the plane by convex regular polygons; cf. [21,Chapter 2] for many examples.
The center-of-mass condition is clearly satisfied for rectangular grids in any dimension, if the points x K are placed at the center of the cells. The weights λ KL will depend on the size of the rectangles. It is possible to put the points x K at different positions, as is done in (B3). In that case, the center-of-mass condition is violated, but the isotropy condition holds.
Another example for which the center-of-mass condition holds is shown in (B4). The value of the weights λ KL is determined by the length ratio of the edges.
The center-of-mass condition fails for the lattice in (B5). Indeed, to satisfy this condition, the points x K would have to be placed at the boundary of the cells, in a way that violates our assumption that the points x K are all distinct. This would lead to infinite transmission coefficients S KL . The isotropy condition fails to hold as well, as will be discussed in Section 5.
In each of the examples (B1)-(B4) it is readily checked that the isotropy condition also holds for boundary cells, by introducing suitable fake points outside of the domain.
The mesh in (B6) is not admissible, as the line segments connecting the points x K are not orthogonal to the cell interfaces.
Our next main result provides an asymptotic lower bound for W T in terms of W 2 under the assumption that the meshes {T } satisfy the asymptotic isotropy condition, and the means θ T KL are carefully chosen to reflect this condition. A mean function (θ KL ) is said to be compatible with a weight function (λ KL ) if, for any K, L ∈ T and all a, b ≥ 0, we have or equivalently, ∂ 1 θ KL (1, 1) = λ KL for any K, L; cf. Section 2.2 for a more extensive discussion.
Remark. As discussed above, the assumptions of the theorem are satisfied in the Examples (A) and (B1)-(B4).
In Section 6 we will prove slightly stronger versions of Theorems 1.1 and 1.4, that provide uniform error bounds in terms of µ 0 and µ 1 . As a consequence, we obtain the following result.
Finally, we will show that the asymptotic isotropy condition is essentially necessary in Theorem 1.4 and Corollary 1.5. More precisely, the following result shows that the asymptotic lower bound for W T fails to hold if the asymptotic isotropy condition is locally violated at all scales. In this sense the asymptotic isotropy condition for {T } is essentially equivalent to Gromov-Hausdorff convergence of W T to W 2 . The result relies crucially on the smoothness of the mean functions θ KL . In particular, the result does not apply to θ KL (a, b) = min{a, b}. W T (P T µ 0 , P T µ 1 ) < W 2 (µ 0 , µ 1 ) .
The condition (1.5) can easily be verified in the setting of Examples (A) and (B5); cf. Section 5 for details.

Remark.
On a technical level, our method of proof offers the advantage that the maps for which Gromov-Hausdorff convergence is proved are the canonical projections P T , rather than regularised versions of these maps, as in [19]. Another advantage is that we do not require any regularisation argument at the discrete level, as was done both in [19] and [18]. All regularisation arguments are done at the continuous level. In particular, we do not require any lower Ricci curvature bounds for the Markov chain at the discrete level, which would be quite restrictive.

Structure of the paper
In Section 2 we recall some known facts about the Kantorovich distance W 2 and the heat flow on convex bounded domains. Furthermore we introduce ζ-regular meshes and the associated discrete transport distance W T . Section 3 contains several useful coarse a priori bounds for discrete optimal transport, and in Section 4 we obtain finite volume estimates for the action functionals A T and A * T . The failure of Gromov-Hausdorff convergence to W 2 (Proposition 1.2 and Theorem 1.6) is established in Section 5. Finally, Section 6 contains the proof of the lower bound (Theorem 1.1), the upper bound (Theorem 1.4), and the Gromov-Hausdorff convergence result (Corollary 1.5).

The Kantorovich metric
Let (X , d) be a Polish space, and let P(X ) be the set of Borel probability measures on X . The class of Borel measures on X is denoted M + (X ), and the class of signed Borel measures with mass 0, by M 0 (X ).
Let Ω ⊆ R d be a convex bounded open set. Let D = C ∞ c (R d ) be the space of test functions and let D be the space of distributions. In this paper we will make use of the dynamical formulation of the metric W 2 , which is given in terms of the action functional A : M + (Ω) × C 1 (Ω) → R and its Legendre dual A * : The Benamou-Brenier formula [4] asserts that Here, AC(µ 0 , µ 1 ) denotes the class of all W 2 -absolutely continuous curves (µ t ) t∈[0,1] in P(Ω) connecting µ 0 and µ 1 . For fixed δ > 0, it will be useful to introduce the set P δ (Ω) ⊆ P(Ω) consisting of all µ = u dx such that u : Ω → R is Lipschitz continuous with Lip(u) ≤ 1 δ and min x∈Ω u(x) ≥ δ. If µ ∈ P δ (Ω) and w ∈ L 2 (Ω), the maximiser in the definition of A * (µ, w) is given by the unique distributional solution φ ∈ H 1 (Ω) to the elliptic problem − div(u∇φ) = w in Ω Let (H a ) a≥0 be the heat semigroup associated to the Neumann Laplacian ∆ on Ω. Since Ω is convex, Ω is a CD(0, d) space in the sense of Bakry-Émery and Lott-Villani-Sturm. In particular, the heat semigroup satisfies the Bakry-Émery gradient estimate for all sufficiently smooth functions φ : Ω → R, and the local logarithmic Sobolev inequality for all a > 0 and x, y ∈ Ω, with constants c, C > 0 depending on Ω; see [8, Theorems 3.2.9 and 5.6.1]. The following result asserts that the heat semigroups maps P(Ω) into P δ (Ω).
Lemma 2.1. For all a > 0 there exists a constant δ > 0, depending on Ω and a, such that for any µ ∈ P(Ω) the density u a of H a µ satisfies u a (x) ≥ δ for all x ∈ Ω and Lip(u a ) ≤ 1 δ .
Proof. The lower bound follows immediately from the Gaussian lower bound in (2.6).
To prove the Lipschitz bound, we use the pointwise gradient inequality 2a|∇H a φ| 2 ≤ H a (φ 2 ) , which follows from the CD(0, d) property of Ω; cf. [3,Theorem 4.7.2]. Together with (2.6) we obtain for some C < ∞ depending on a. This implies the result.
We collect some known properties of the Kantorovich metric that will be useful in the sequel. Let us remark that the convexity of the domain Ω is crucial for Part (ii) in the following lemma, as its proof relies on gradient estimates in the sense of Bakry and Emery (see, e.g., [2, Theorem 3.17] and [3]).
Proof. Assertion (i) follows immediately from the definitions. The first part of Assertion (ii) is a consequence of the Bakry-Émery gradient estimate (2.4) and the selfadjointness of H a . The second part follows immediately, and the inequality (2.9) holds by the Benamou-Brenier formula (2.3).
To show Assertion (iii), note first that dγ(x, y) = h a (x, y) dµ(x) dy is a coupling between µ and H a µ. Thus using (2.6) we obtain (2.10) If a ≤ diam(Ω) 2 , we have 1 ∨ a d/2 ≤ C for some constant C depending on Ω, so we may absorb the factor 1 ∨ a d/2 into the constant. If a ≥ diam(Ω) 2 , we trivially have

Discrete transport metrics
We briefly recall the definition of the discrete transport metrics introduced in [7,22,24]. Let X be a finite set, and let R : X × X → R + be a non-negative function. We interpret R(x, y) as the transition rate from x to y for a continuous-time Markov chain on X , which we assume to be irreducible. Under this assumption, there exists a unique invariant probability measure π ∈ P(X ). We assume that π satisfies the detailed balance condition: To define the discrete transport metric we need to choose a family of admissible means in the sense of the following definition. Note that these assumptions slightly differ from [10, Assumption 2.1].
We collect some known properties of admissible means in the following result.
Lemma 2.4. Let θ be an admissible mean.
with equality if a = c and b = d.
. Using the concavity, 1-homogeneity and normalisation of θ, The case a < b can be treated analogously, and the case a = b follows immediately by 1-homogeneity and normalisation.
We refer to [10, Lemma 2.2] for a proof of (ii).
It will be useful to associate a number λ (θ) to an admissible mean θ that quantifies its asymmetry.
Definition 2.8 (Mean and weight functions). Let X be a finite set.
(i) A mean function is a family of admissible means θ = (θ xy ) x,y∈X satisfying the symmetry condition θ xy (a, b) = θ yx (b, a) for all x, y ∈ X and a, b ≥ 0.
It follows from (2.12) that the induced weight function is indeed a weight function.
We are now in a position to define the discrete transport metrics. Given a Markov chain (X , R, π) and a mean function θ, the discrete transport metric is defined using discrete analogues of (2.2). The action functional A : P(X ) × R X → R and its dual For m 0 , m 1 ∈ P(X ), the associated transport metric is given by Here, AC(m 0 , m 1 ) denotes the class of all curves (m t ) t∈[0,1] ⊆ P(X ) connecting m 0 and m 1 , with the property that t → m t (x) is absolutely continuous for all x ∈ X .
Remark 2.9. In most of the previous papers dealing with discrete dynamical transport metrics, θ xy has been chosen to be independent of x and y. In particular, to obtain a gradient flow structure for Markov chains [7,22,24], one chooses θ xy to be the logarithmic mean for all x, y. We will see in this paper that the additional freedom can be important to obtain Gromov-Hausdorff convergence. In a noncommutative setting, a similar generalisation has been useful in the setting of Lindblad equations with a nontracial invariant state [6,26].
We will occasionally use an equivalent formulation for W given by Here, CE(m 0 , m 1 ) denotes the class of pairs (m, V ) satisfying • the continuity equation holds in the sense of distributions: Moreover, where the convex and lower semicontinuous function K : otherwise . (2.16) Note that A * can be obtained from K by minimising over all solutions to the continuity equation:

Admissible and ζ-regular meshes
Following [16, Section 3.1.2], we introduce the notion of an admissible mesh.
, a pairwise disjoint covering) of Ω into sets (called cells) with nonempty and convex interior.
Note that all interior cells K are polytopes. Throughout the paper we will use the following notation: -|K| denotes the Lebesgue measure of a cell K ∈ T .
whenever K, L ∈ T are nearest neigbours. An admissible mesh T is called ζ-regular for ζ ∈ (0, 1] conditions hold: Every Voronoi tesselation yields an admissible mesh. Another example is obtained by slicing Ω multiple times in the cardinal directions. In this case there are several degrees of freedom for placing the points {x K } K∈T , so these points are not necessarily uniquely determined by T . We refer to [16] for more information.
In the following result we collect some basic geometric properties of ζ-regular meshes that will be useful in the sequel.
Lemma 2.12. Let T be a ζ-regular mesh of Ω for some ζ ∈ (0, 1]. Then, there exist a constant C < ∞ depending only on Ω and ζ such that the following assertions hold: (i) The number of nearest neighbours of any cell is bounded by C.
(ii) Any pair of cells K, L ∈ T can be connected (for some N ≥ 0) by a path (2.17) (iii) For all K ∼ L we have the following estimates: (2.20) Proof. (i): Write h = [T ] for brevity. By the inner ball condition we have Hence, Assertion (i) follows by comparing volumes.
In turn, each of the cells K i is contained in the cylinder of radius [T ], whose central axis is obtained by extending by a distance [T ] in both directions.
The number of disjoint balls of radius r that can be packed into a cylinder of length s and radius R is bounded by where C depends on ζ and d. As , this yields the desired result.
Since x K − x L ⊥ (K|L), the volume of the bipyramid spanned by (K|L), x K and x L is given by 1 d |(K|L)|d KL . Using the inner ball condition we obtain which proves the upper bound in (2.19). To prove the corresponding lower bound, note that ζh ≤ d KL , hence by ζ-regularity, The inequality (2.20) follows from (2.19).
The ζ-regularity condition allows us to control the constants in several useful inequalities. Most notably, we will use the Poincaré inequality [27] and the trace inequalityˆ∂ both of which are valid for all K ∈ T and all φ ∈ H 1 (K) with´K φ dx = 0; cf. [20, Theorem 1.5.1.10]. The constant C < ∞ depends only on ζ and the dimension d. For the convenience of the reader we record a simple consequence that we will use below.

Discrete optimal transport on admissible meshes
Given an admissible mesh T of Ω we consider an irreducible Markov chain on T with transition rates and R(K, L) = 0 otherwise.
Remark 2.14 (Formal consistency). This choice of the transition rates R(K, L) is motivated by the following formal consistency computation for the Dirichlet energy associated to our problem. Let U be the uniform probability measure on Ω. For a smooth function φ : Ω → R, the continuous action functional satisfies Let U T ∈ P(T ) be the canonical discretisation of U given by U T = P T U, and let ψ T : T → R be given by ψ T (K) = φ(x K ). Then the latter expression is indeed of the form A(U T , ψ T ) defined in (2.13), provided that the coefficients R(K, L) are defined by (2.25).
It is immediate to check that the unique invariant probability measure π on T is given π(K) = |K| |Ω| . Moreover, the detailed balance condition π(K)R(K, L) = π(L)R(L, K) holds.
For this Markov chain, the discrete action functional A T : P(T ) × R T → R and its dual A * T : P(T ) × R T → R ∪ {+∞} defined in (2.13) are given by The main object of study in this paper is the associated transport metric, defined for m 0 , m 1 ∈ P(T ) by

A priori estimates
In this section we prove the necessary a priori estimates. Throughout this section, we fix a convex bounded open set Ω ⊆ R d and a ζ-regular mesh T for some ζ ∈ (0, 1]. Moreover, we use the convention that the constants C appearing in this section (which are allowed to change from line to line) may depend on Ω and ζ, but not on other properties of T .
Lemma 3.1. There exists a constant C < ∞ such that for any K, L ∈ T with K ∼ L, Proof. It follows from [10, Lemma 2.13] that for some universal constant c < ∞. The claim follows by ζ-regularity in view of (2.19).
To compare discrete and continuous measures, we use the canonical projection operator P T defined in (1.2). The associated embedding operator Q T : P(T ) → P(Ω) is given by where U K denotes the uniform probability measure on K. Note that P T • Q T is the identity on P(T ). The following lemma quantifies how close Q T • P T is to the identity on P(Ω).
Lemma 3.2 (Consistency). For all µ ∈ P(Ω) we have Proof. For U ∈ T , let γ K ∈ P(Ω × Ω) be any coupling betweenμ K , the normalised restriction of µ to K, and U K , the uniform probability measure on K. It then follows that γ := K∈T µ(K)γ K belongs to Γ(µ, Q T P T µ). Consequently, The following result provides a coarse bound for the discrete distance W T in terms of the Kantorovich distance W 2 . (3.1) Proof. Let γ ∈ Γ o (Qm 0 , Qm 1 ) be an optimal plan for W 2 , and set γ KL := γ(K × L) for brevity. Observe that m 0 = K,L γ KL δ K and m 1 = K,L γ KL δ L . Therefore, by convexity of the squared distance (cf. [10, Proposition 2.11]) we have For K, L ∈ T , take a connecting path {K i } N i=0 ⊆ T with K 0 = K and K N = L satisfying the ζ-regularity estimate (2.17). Using Lemma 3.1 we obtain For all x ∈ K and y ∈ L we have d 2 . Combining these estimates, the result follows.
The following lemma provides a coarse lower bound for the discrete dual action functional in terms of its continuous counterpart. Proof. Let σ ∈ M 0 (T ), and let V : T × T → R be an anti-symmetric momentum vector field solving the discrete continuity equation for x ∈ (K|L) with K ∼ L, and g(x) = 0 for x ∈ ∂Ω. It then follows that σ(K) = −´∂ K g dS. Therefore, denoting the outward unit normal on ∂K by n, the Neumann problem

(3.4)
Since´K φ K dx = 0, the right-hand side can be bounded in terms of ∇φ K L 2 (K) using the Poincaré inequality (2.21) and the trace inequality (2.22). Moreover, using the ζregularity inequalities (2.18) (2.19), and the bound on the number of neighbours from Lemma 2.12, we obtain |∂K| diam(K) ≤ C|K|. Consequently, Using the ζ-regularity inequality (2.18) we obtain, and therefore,ˆK Let us now define the vector field Φ ∈ L 2 (Ω; R d ) by Φ(x) = ∇φ K (x) for x ∈ K. By (3.3) and the anti-symmetry of the Neumann boundary values, one has Q T σ + div Φ = 0 (3.5) in the sense of distributions, and Φ · n = 0 on ∂Ω. Take now m ∈ P(T ). Writing ρ(K) = m(K) |K| , we obtain where θ harm (a, b) = 2ab a+b is the harmonic mean. We would like to obtain a similar estimate involving the means θ KL , but as ρ is in general not bounded away from 0, we cannot bound the harmonic mean by a multiple of θ KL . To remedy this issue, we perform an additional regularisation step. Consider the functionρ : T → R + given byρ(K) = ρ(K) + L∼K ρ(L), and setm(K) =ρ(K)|K|. In view of (2.7), Lemma 3.5 below, and (2.8), we obtain, for a = [T ], We stress that to obtain the first inequality, the choice a = [T ] is crucial; cf. Remark 3.6. Moreover, applying (3.6) tom, we obtain .
. Therefore, combining the estimates above, we obtain Taking the infimum over all V satisfying (3.2), the result follows.

7)
holds, with η(a) = a −2 + (a −1 log a) ∨ 0. Proof. Sincem ≥ m, the first inequality follows from the positivity of H a , so it remains to prove the second inequality.
To prove the second inequality, note that We claim that there exists a constant C < ∞ such that for x, y, z ∈ Ω. Let T KL be a (not necessarily optimal) transport map between the uniform probability measures on neighbouring cells K and L. As |T KL (z) − z| ≤ 2[T ] for z ∈ K, the claim yields Since |L| ≤ C|K| by the ζ-regularity estimate (2.20), and since #{L : L K} ≤ C by Lemma 2.12, we obtain which yields the result. It remains to prove the claim (3.8). For this purpose, note that by the heat kernel bounds (2.6) there exist Ω-dependent constants c, C > 0 such that, for all a > 0, sup For any smooth function φ : Ω → (0, M ] with M ∈ (0, ∞), the local logarithmic Sobolev inequality (2.5) implies that Applying this inequality with φ = h a/2 (x, ·), we obtain using the semigroup property, which implies (3.8).

Finite volume estimates for discrete optimal transport
The goal of this section is to show that the dual action functional A * T for the discrete transport problems is a good approximation to its continuous counterpart A * . This will be shown in Proposition 4.6.
To obtain this result, we first show an error estimate for a discrete elliptic problem, in the spirit of [16,Theorem 3.5]. In our application, we think of w as beingμ t at some fixed time t, so that the elliptic equation below is the continuity equation at time t.
As in Section 3, we fix a convex bounded open set Ω ⊆ R d and a ζ-regular mesh T for some ζ ∈ (0, 1]. H 1 -error bound). Let w ∈ L 2 (Ω) and µ = u dx ∈ P δ (Ω) be given, and let φ ∈ H 2 (Ω) be the unique variational solution to Then there exists a constant C < ∞ depending only on δ, Ω, and ζ, such that

Proposition 4.1 (Weighted
Remark 4.2. The existence of a unique variational solution φ to the Neumann problem (4.1) in H 1 (Ω) follows from the Lax-Milgram theorem. The existence of a unique solution to (4.2) follows from elementary linear algebra, cf. [22]. In both cases, uniqueness holds up to an additive constant.
where n denotes the outward unit normal on ∂K. We define  Observe that To estimate the first term on the right-hand side of (4.6), we note that the function u is Lipschitz since µ ∈ P δ (Ω), and the mean θ KL is Lipschitz on [δ, ∞) 2 . Using this observation followed by Lemma 2.13, we obtain for some constant C < ∞ depending on Ω, δ and ζ.
To estimate the second integral in (4.6) for neigbouring cells K ∼ L, write Using the trace inequality (2.22) we have Arguing as in (2.24), we obtain Furthermore, writing B K = B(x K , ζ[T ]) as before, the fundamental theorem of calculus yields where f is a nonnegative function satisfying − B K f (x) dx = 1 and f L ∞ ≤ C < ∞. Therefore, using the Poincaré inequality (2.21), (4.10) Combining the inequalities (4.8), (4.9) and (4.10), we obtain, using the ζ-regularity once more, Together with (4.7), the latter estimate yields Thus, using (4.5) we find In the following result it suffices to assume that T is admissible. We do not need to require ζ-regularity. |K| ≥ δ > 0 for all K ∈ T , we have Proof. This is a straightforward modification of the proof in [16,Lemma 3.7]. Define φ : Ω → R by φ = K χ K ψ(K) and set ρ(K, L) = θ KL (ρ(K), ρ(L)). We need to show that For K ∼ L and x, y ∈ R d , put χ (K|L) (x, y) = 1 if x, y belong to Ω, (K|L) intersects the straight line segment connecting x and y, and (y − x) · (x L − x K ) > 0. Otherwise, we set As Ω is convex, we obtain for a.e. x, y ∈ Ω, Note that c K,L;y−x > 0 whenever χ (K|L) (x, y) > 0. Using this fact, the Cauchy-Schwarz inequality yields For fixed x and y, let K 0 , . . . , K N be the subsequent cells intersecting the line segment 1] as t ranges from 0 to 1. By definition, χ (K|L) (x, y) vanishes, unless (K|L) = (K i−1 |K i ) for some i = 1, . . . , N . We thus have where R = diam(Ω). Let B R denote the ball of radius R around the origin. Using a change of variables, we observe that for some Ω-dependent constant C < ∞, which completes the proof. Now we are ready to prove the main result of this section.
To bound the first term, let φ(K) = − K φ dx, and observe that, using Lemma 2.13 and the Poincaré inequality, Therefore, the first term can be bounded by To estimate the second term, we use Proposition 4.5 and Proposition 4.1 to obtain . Combining these estimates yields the result.

Counterexamples to Gromov-Hausdorff convergence
In this section we show that if the asymptotic isotropy condition fails sufficiently often, then the discrete transport metric W T does not converge to the 2-Kantorovich metric W 2 , in spite of the fact that the discrete heat flow converges to the continuous heat flow; see, e.g., [16,Theorem 4.2]. In fact, in the one-dimensional example below, even evolutionary Γ-convergence has been proved for the entropic gradient flow structure of the discrete heat flow with respect to the transport distance W T ; cf. [9].

A one-dimensional counterexample
We present a one-dimensional example to illustrate the non-convergence to W 2 in the simplest possible setting.
We start with a well-known result on the existence of smooth W 2 -geodesics in the one-dimensional case. For the convenience of the reader we include a direct proof. We write I = [0, 1] for brevity.
Proof. Let F i denote the distribution function of µ i given by F i (x) =´x a u i (y) dy, which is readily seen to be invertible. The unique optimal transport map T between µ 0 and µ 1 is then given by T = F −1 1 • F 0 . By the inverse function theorem, T ∈ C 1 (I) and T (x) ∈ [M −1 , M ] for all x ∈ I, where M > 1 depends on δ. The unique W 2 -geodesic between µ 0 and µ 1 is given by .
The result follows directly from this explicit expression.
For N ∈ N and r ∈ (0, 1 2 ), we consider the 1 N -periodic mesh T r,N of I = [0, 1] from Figure 1, given by The cells in T r,N will be denoted K k for k = 0, . . . , 2N − 1 according to their natural ordering. To make sure that T r,N is a partition of [0, 1], one should add the point 1 to the set K 2N −1 , but this will be irrelevant in what follows. Let x k = r+k 2N be the midpoints of K k , so that d k,k+1 = 1 2N and [T r,N ] = 1−r N . (For notational simplicity we write d k,k+1 instead of d K k ,K k+1 . Similarly, we will P r,N instead of P T r,N etc.) According to (2.25), the transition rates R k,k±1 from cell k to cell k ± 1 are given by with the understanding that q 0,−1 = q N,N +1 = 0.
We fix an admissible mean θ (in the sense of Definition 2.3) that is assumed to be symmetric, i.e., θ(a, b) = θ(b, a), and consider the transport metric W r,N defined by setting θ KL = θ for all K ∼ L. For each fixed r ∈ (0, 1 2 ), the next result implies that the distances W r,N do not Gromov-Hausdorff converge to W 2 . In Section 6 below we will show that Gromov-Hausdorff convergence holds if one takes a different (non-symmetric) mean adapted to the inhomogeneity of the mesh.
Let V : T × T × (0, 1) → R be an anti-symmetric function satisfying the continuity equation (2.14), given bẏ it follows that (m η , V ) solves the continuity equation as well. Therefore, (5.3) yields Combining this bound with (5.4), we infer that there exists a constant c ∈ (0, 1) depending only onδ, such that provided η is chosen sufficiently small depending onδ and r.

Necessity of the asymptotic isotropy condition
Our next aim is to show that for any family of meshes {T } for which the asymptotic isotropy condition fails at every scale, the distance W T is asymptotically strictly smaller than W 2 . We start with a lemma that guarantees the existence of certain smooth W 2 -geodesics that transport mass in a parallel fashion. (i) the continuity equation
The following lemma asserts that, at the macroscopic scale, the isotropy condition holds without any assumption on the mesh. For A ⊆ R d and r > 0, we let B(A, r) := x∈A B(x, r) denote the r-neigbourhood of A. Proof. We consider the cells C KL = C LK ⊆ R d defined by Observe that these sets have pairwise disjoint interiors (up to the symmetry condition C KL = C LK ). Set U − = U \ B(∂U, 4[T ]) and U + = B(U, 4[T ]). It then follows that The result follows, as the area formula yields |C KL | = (v · n KL ) 2 |(K|L)|d KL .
As the right-hand side in the previous result is small, the contribution of the term L∈T ;L⊆U λ KL (v · n KL ) 2 |(K|L)|d KL is equal to |K| on average, up to a microscopically small error. However, it may happen that the isotropy condition fails at the microscopic scale, in the sense that the microscopically small error in (5.6) results from a cancellation of positive and negative contributions of macroscopic size. The following definition makes this intuition precise.
for any open cube V ⊆ U .
Example 5.6 (Anisotropy in one dimension). For r ∈ (0, 1 2 ) and N ≥ 1, consider the onedimensional periodic mesh T r,N from Section 5.1. We fix s ∈ [0, 1] and define a weight function λ on T r,N by setting λ KL = s is K is small, and λ KL = 1 − s if K is large. As large and small cells alternate, this indeed defines a weight function.
Fix an interval V = (a, b) for some 0 < a < b < 1, and set v = 1. For any K ∈ T r,N and [T ] sufficiently small, it follows that Note that for any neighbouring pair K, L, we have S K + S L = 1 N = |K| + |L|. This means that the isotropy condition holds on average, in accordance with Proposition 5.4. However, it follows that Therefore, the local anisotropy condition (5.7) holds whenever r = s. If r = s, we have already seen in the introduction that the asymptotic isotropy condition (and in fact the center-of-mass condition) holds.
An analogous computation shows that We thus find that in accordance to the fact that isotropy holds on average, for any r ∈ (0, 1 2 ). To show that the family {T r,N } N is locally anisotropic, we fix v = (1, 0). Then: which shows that the mesh is everywhere locally anisotropic, for any r ∈ (0, 1 2 ). The following proposition shows that if the mesh is locally {λ T }-anisotropic, and if the mean functions θ T are chosen accordingly, then the discrete transport distances are asymptotically strictly smaller than W 2 . Proof. We fix x 0 ∈ Ω and v ∈ S d−1 . Using Lemma 5.3 we obtain δ > 0, r > 0, and a geodesic (µ t ) t∈[0,1] ⊆ P δ (Ω), solving the continuity equationμ + div(µ∇φ) = 0 where the velocity vector field ∇φ t ∈ L 2 (Ω) satisfies ∇φ t (x) = κv for some κ > 0, for all t ∈ [0, 1] and all x in the ball B = B(x 0 , r). For brevity we writeB = B(x 0 , r 2 ). Fix > 0, and consider the collection of open cubes given by For Q ∈ Q we define T Q := {K ∈ T : K ⊆ Q}, and for K ∈ T Q we set We define the subsets T + Q , T − Q ⊆ T by It follows directly from (5.7) that, for all Q ∈ Q , lim inf Moreover, as µ t ∈ P δ (Ω), we have for some C < ∞ depending on δ. Using this estimate and (5.14) we obtain lim sup which bounds the third term in (5.13).
To treat the second term, we write ). Since sup K,L γ KL ≤ 1 and µ t ∈ P δ (Ω), Proposition 4.1 yields where C depends on Ω, ζ, and δ. Furthermore, for K ∈ T ± Q and L ∼ K, we have ∇φ = κv on K ∪ L, which implies that φ L − φ K = κv · (x L − x K ). Therefore, using the fact that Using (5.10) and (5.11), this identity yields lim inf provided is sufficiently small. Together with (5.16), it follows that lim inf Inserting the three estimates (5.14), (5.15) and (5.17) into (5.13), we obtain lim sup for suitable constants c > 0 and C < ∞. On the other hand, since K∈T Q ν T (K)|K| = 0 for all Q ∈ Q , it follows from Lemma 3.3 that and the same holds at t = 1. In summary, we obtain lim sup for all ε < ε 0 . As > 0 is arbitrary, this yields the result.
Remark 5.10. For the mesh in Figure 4, the construction in the proof of Theorom 5.8 can be somewhat simplified: as discussed in Example 5.7, isotropy fails to hold in the coordinate directions v = ±e 1 , ±e 2 . Picking a W 2 -geodesic transporting mass in direction e 2 in some open ball B, we notice that the discretisation of that geodesic transports no mass over vertical edges in B. The variation π T can then be set to π T = 1 for all N and S cells and π T = −1 for all E and W cells. Along diagonal edges, the change in θ KL is o(ε), whereas for horizontal edges, the change in θ N,S+e 2 is −ε + o(ε). For vertical edges, the change in θ E,W +e 1 is ε + o(ε), which would be costly, but here the momentum vector field V (E, W + e 1 ) vanishes.

Gromov-Hausdorff convergence
In this section will prove Theorems 1.1 and 1.4, as well as Corollary 1.5. Let us start by stating the definiton of Gromov-Hausdorff convergence.
Definition 6.1 (Gromov-Hausdorff convergence). We say that a sequence of compact metric spaces (X n , d n ) n≥1 converges in the sense of Gromov-Hausdorff to a compact metric space (X , d), if there exist maps f n : X → X n which are • ε n -isometric, i.e., for all x, y ∈ X , |d n (f n (x), f n (y)) − d(x, y)| ≤ ε n (6.1) and

Upper bound for the discrete transport metric
In this subsection we prove an upper bound for W T , that relies on the finite volume bounds obtained in Section 4. Since these bounds require an ellipticity condition on the densities, we use a regularisation argument involving the heat flow. Lemma 6.3 contains the desired bound for the regularised measures. The regularisation is removed in Theorem 6.4. We emphasise that these results hold under the mere assumption of ζ-regularity, and do not require the additional symmetry assumptions that we will impose in Section 6.2. Lemma 6.3. Fix ζ ∈ (0, 1] and a > 0. There exists a constant C < ∞, depending on Ω, ζ, and a, such that for any ζ-regular mesh T of Ω, and µ 0 , µ 1 ∈ P(Ω) the following estimate holds: Proof. Let (µ t ) t∈[0,1] be a geodesic connecting µ 0 and µ 1 . Take η : R → R as in Lemma 6.2 and define, for δ ∈ (0, 1 2 ), where the η-dependence is absorbed in the constant C. Furthermore, using the convexity of (µ, w) → A * (µ, w) as in Lemma 6.2, and the contraction bound from Lemma 2.2(ii), we obtainˆ1 Since the constant C does not depend on δ, we obtain the desired result by passing to the limit δ → 0 (and absorbing the factor a −d ∨ 1 into the constant C).

Lower bound for the discrete transport metric under isotropy conditions
Since the counterexamples in Section 5 show that W T does not Gromov-Hausdorff converge to W 2 in general, we will impose an additional condition on the mesh. Let I d denote the d × d identity matrix. where sup K∈T |η T (K)| → 0 as [T ] → 0.
The following proposition contains the crucial bounds on the action functionals A T and their duals A * T . To obtain the result, one needs to carefully choose the means θ KL according to the geometry of the mesh.
We say that a sequence of signed measures {w T } T ⊆ M 0 (Ω) converges weakly to a signed measure w if φ, w T → φ, w for all φ ∈ C 0 (Ω). In this case, we write w T w.
Proposition 6.6 (Action bounds). Let {T } be a family of ζ-regular meshes satisfying the asymptotic isotropy condition with weight functions {λ T }, and let {θ T } be a family of weight functions that are compatible with {λ T }. Suppose that m T ∈ P(T ) satisfies Q T m T µ as [T ] → 0 for some µ ∈ P(Ω).
Suppose first that A * (µ, w) is finite. Fix ε > 0 and choose φ ∈ C 1 (Ω) such that Using the claim and (6.8), it follows that Since ε > 0 is arbitrary, the result follows. Suppose next that A * (µ, w) = ∞. Then, for each N > 0, there exists a function φ N ∈ C 1 (Ω) such that Define ψ N T : T → R by ψ N T (K) := φ N (x K ). Using the claim and (6.8) once more, we obtain which implies that lim inf [T ]→0 A * T (m T , σ T ) = ∞.

Proof of the Gromov-Hausdorff convergence
It remains to conclude the proof of one of the main results of this paper.