Pith. sign in

REVIEW 1 major objections 5 minor 50 references

Stochastic homogenisation of nonlinear minimum-cost flow problems

T0 review · 1 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The paper proves that nonlinear minimum-cost flow problems on stationary random graphs converge, after rescaling, to a continuum divergence-constrained variational problem with an explicit effective energy density.

desk verdict Strong stochastic homogenisation theorem with a genuinely new uniform-flow construction; the sigma-additivity assumption is a real limitation but the paper is honest about it and deserves refereeing. read the letter →

arxiv 2412.05217 v3 pith:OQ3P2JW4 submitted 2024-12-06 math.AP math.OC

Peter Gladbach , Jan Maas , Lorenzo Portinale This is my paper · ORCID
classification math.APmath.OC MSC 35B2749J4549Q2205C21
keywords stochastichomogenisationminimum-costflowGamma-convergencerandomgraphsdiscrete-to-continuumlimitsoptimaltransportdiv-quasiconvexitymulti-speciesflows
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Nonlinear minimum-cost flow problems ask for the cheapest way to route prescribed amounts of mass through a graph, with possibly nonconvex costs on edges. The paper's main theorem states that, for a large class of stationary random graphs in $\mathbb{R}^d$, these problems have a well-defined large-scale limit: after rescaling, the discrete functionals $\Gamma$-converge almost surely to a continuum functional in which a random effective cost density is minimised among vector fields with prescribed divergence. The effective density is built from a variational cell formula, and its construction is the main technical achievement because the graph has no periodic structure. A consequence is convergence of the minimal costs and of approximate minimisers, covering multi-species flows and the Wasserstein distances induced by random edge lengths.

What carries the argument

The central objects are the discrete uniform-flow operator $R$ — a bounded linear map sending each constant tensor $j$ to a divergence-free discrete field $Rj$ whose embedded rescaling converges to $j\,\mathcal{L}^d$ — and the associated cell formula $f_{\omega,\varepsilon,R}(j,A)$, defined as the minimal energy of a divergence-free discrete field matching $R_\varepsilon j$ near $\partial A$. The homogenised density $f_{\omega,\mathrm{hom}}(j)$ is the $\varepsilon\to 0$ limit of the cell energy per unit volume, shown to exist by subadditivity and the subadditive ergodic theorem and to be independent of $R$. The lower bound is carried by correctors for the discrete divergence equation and by the blow-up of divergence measures: the measure differentiation theorem shows that local rescalings of the flux converge to tangent measures that are constant-density and divergence-free, so the cellular inequality $f_{\omega,\mathrm{hom}}(j_0) \le$ local energy density can be tested on near-representatives.

What would settle it

Compute the cell energies $f_{\varepsilon,R}(j,Q)$ on a stationary random graph satisfying (G1)–(G3) for two different admissible uniform-flow operators $R$; if the limits differ for some $j$, the claimed independence of the effective density from $R$ fails. More directly, simulate the rescaled minimum-cost flow problem on a random Voronoi tessellation with $m_\varepsilon$ approaching a smooth divergence field and compare the limiting minimal energy with the integral of $f_{\mathrm{hom}}$; a mismatch contradicts the $\Gamma$-convergence statement.

Watch

Extended reading notes

Core claim

Under the paper's standing assumptions — a stationary random graph satisfying (G1)–(G3) (no large gaps, quantitative connectedness, bounded edge lengths) and a stationary random energy satisfying (F1)–(F3) (Lipschitz, linear growth, $\sigma$-additivity) — Theorem 2.8 asserts that for any sequence of flux data $m_\varepsilon$ converging to $\mu$ in the Kantorovich–Rubinstein norm, the constrained flow functionals $F_{\omega,\varepsilon}(\cdot|m_\varepsilon)$ $\Gamma$-converge almost surely to $F_{\omega,\mathrm{hom}}(\cdot|\mu)$. The limit is an integral functional over vector-valued measures with a density $f_{\omega,\mathrm{hom}}$ that is stationary, lower semicontinuous, div-quasiconvex and of linear growth, and the minimal values converge to the minimal value of the limit. The effective density is obtained as $f_{\omega,\mathrm{hom}}(j) = \lim_{\varepsilon\to 0} f_{\omega,\varepsilon,R}(j,Q)/\mathcal{L}^d(Q)$, where $f_{\omega,\varepsilon,R}$ is the infimum of the energy over divergence-free discrete fields that coincide with a chosen uniform-flow operator $R$ near the boundary of the unit cube; the result shows the limit is independent of $R$. The proof combines existence of uniform flows on graphs satisfying (G1)–(G2), correctors for the discrete divergence equation, and a blow-up lower bound using tangent measures of divergence measures.

Load-bearing premise

The energy must be exactly additive over disjoint regions (condition F3); the cell formula, the subadditivity of the cell energies, and the upper-bound gluing argument all use this, and the paper only notes that a milder almost-additivity condition would probably suffice without proving it.

Editorial extensions

If this is right

  • When costs are 1-homogeneous edge lengths, the rescaled graph 1-Wasserstein distances converge almost surely to a homogenised distance induced by the norm $f_{\omega,\mathrm{hom}}$; under ergodicity the limiting norm is deterministic.
  • The $\Gamma$-convergence and compactness imply convergence of minimal costs and of approximate minimisers: any cluster point of embedded discrete minimisers minimises the continuum functional, and a unique continuum minimiser attracts the discrete ones.
  • Multi-species flows, where measures take values in a finite-dimensional vector space, are covered, so the continuum limit applies to multi-commodity transportation on random networks.
  • The limiting energy density is automatically div-quasiconvex and of linear growth, matching the expected structure of variational limits under divergence constraints.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The paper's Remark 2.3 suggests that $\sigma$-additivity (F3) can be relaxed to almost-additivity for well-separated sets; once proved, the method should extend to mildly nonlocal graph energies whose nonlocality fades in the large-scale limit.
  • Applied to iid edge weights on $\mathbb{Z}^d$ with homogeneous linear costs, the result implies convergence of rescaled graph Wasserstein distances to a deterministic norm; identifying that norm with the first-passage percolation time constant would link the homogenisation limit to the shape theorem, but that identification is not proved here.
  • A natural next step is quantitative homogenisation: for graphs with finite range of dependence, rates for the cell formula could be obtained from concentration or variance bounds on the subadditive limit, going beyond the paper's almost-sure statement.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

1 major / 5 minor

Summary. The paper proves an almost-sure Γ-convergence theorem, Theorem 2.8, for rescaled nonlinear minimum-cost flow functionals on stationary random graphs satisfying the geometric assumptions (G1)–(G3). For stationary random energies satisfying Lipschitz, linear-growth, and σ-additivity assumptions (F1)–(F3), and for prescribed discrete fluxes converging in the gKR topology, the rescaled functionals converge to a continuum divergence-constrained integral functional whose energy density is defined by a variational multi-cell formula built from uniform-flow operators. The proof combines deterministic discrete calculus, corrector estimates (Proposition 7.4), a blow-up argument in the style of Fonseca–Müller, and tangent-measure analysis of divergence measures. The upper bound is developed in Section 9 and the lower bound in Section 10, with quantitative non-asymptotic estimates in Propositions 10.1 and 10.3.

Significance. If the result stands, this is a substantial advance over previous periodic-graph homogenisation results: it removes periodicity, treats multi-species flows, allows nonconvex Lipschitz costs, and gives an explicit stationary effective density independent of the chosen uniform-flow operator. The paper is largely self-contained and contains several genuinely useful tools, notably the construction of uniform flows on stationary graphs (Proposition 5.4), the discrete corrector with quantitative gKR bounds (Proposition 7.4), and the strip-based singular lower bound (Proposition 10.3). There is no fitted parameter and no circularity: the effective density is defined by an independent cell formula and then shown to be the Γ-limit. The authors also honestly state the main structural limitation, the σ-additivity assumption F3, in Remark 2.3, where they note that only an almost-additive version would be needed for some applications but is not proved.

major comments (1)
  1. [Remark 2.10, Lemma 8.8(4), Proposition 10.3] The rank condition is stated incorrectly and, taken literally, excludes the scalar case from the singular lower-bound argument. With n = dim V, the assertion rank(dν/d|ν|) ≤ n−1 in Remark 2.10 and Lemma 8.8(4) is false for scalar V: a non-zero singular density is a rank-one vector, so rank = 1 > 0 = n−1. The same hypothesis appears in Proposition 10.3, where it is used to define k = dim ker j. Since the singular part of the lower bound in Section 10.2 applies Proposition 10.3 to j = dξ/d|ξ|, the proof as written does not cover scalar flows, which are a central case of Theorem 2.8. The structural condition needed is that j has a non-trivial kernel, i.e. rank(j) ≤ d−1 (equivalently rank(j) ≤ min{n,d−1}), matching the wave cone of the divergence operator. Please correct the rank bound consistently in Remark 2.10, Lemma 8.8(4), and Proposition 10.3, and verify that the strip construction in Proposition 10.3 goes through with k = dim ker j in this corrected range.
minor comments (5)
  1. [Theorem 2.8] In the statement of the main theorem, µ is described as an element of M(U) at the end of the first paragraph, although the preceding sentence correctly places it in M(U;V); the target space should be M(U;V).
  2. [Lemma 8.5] The final sentence of the statement contains a duplicated and incomplete phrase: 'Finally, for L^d-a.e. x0 ∈ supp(|\nu|) \ E, we have For L^d-a.e. x0 ∈ R^d we have Tan_C(ν,x0) = {jL^d}'. This should be rewritten as a single sentence.
  3. [Proposition 7.4, proof] The proof refers to 'the measure νxy in (10.2)' and to 'Remark 7.6', but (10.2) belongs to Section 10 and there is no Remark 7.6. These references should point to the corresponding construction in the proof of Proposition 7.6 or to an explicit equation in Section 7.
  4. [Lemma 9.1, Step 1] The measure νρ is defined twice in the same paragraph, once before the construction of the deformation Φρ and once after it; the first definition should be removed or the second definition labeled as the final one.
  5. [Section 10.1, 'Approximation 1'] The sentence 'Lemma 6.8, implies that, as ε → 0, f_{ε/δ,R}(j0, Q1(x0/δ)) → f_hom(j0) vaguely' describes convergence of real numbers, so the adverb 'vaguely' is inappropriate and should be deleted.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the homogenised density is defined by an independent cell formula and proven to be the Gamma-limit; self-citations are not load-bearing.

full rationale

The derivation chain is self-contained. The homogenised density fhom is defined in Definition 6.7 through the cell formula fε,R(j,A) (Definition 6.4), an infimum over representatives built from the explicitly constructed uniform-flow operator R (Proposition 5.4). The Γ-limit in Theorem 2.8 is not an input to this definition. The upper bound (Section 9) constructs recovery sequences from cell minimisers and proves limsup ≤ integral of fhom, while the lower bound (Section 10) proves the reverse inequality via blow-up arguments and the quantitative estimates in Propositions 10.1 and 10.3. No parameter is fitted to the target limit, and the limit functional is not used to define fhom. Self-citations to [GK*23] occur for elementary objects such as the unit flux through a path (Definition 7.1, Lemma 7.2) and standard facts on the Kantorovich–Rubinstein norm; these are either verified directly in the text or are simple, parameter-free statements and they do not carry the main argument. Remark 2.3 honestly identifies σ-additivity (F3) as a structural assumption and notes that only almost-additivity would suffice; this is a stated scope restriction, not a circular step. The possible rank-bound concern in Remark 2.10 and Lemma 8.8(4) is a correctness issue independent of circularity, since it does not make any conclusion equal to an input by construction.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The paper adds no fitted parameters and no new physical entities; the homogenised density is a variational limit, and all structural constants are hypotheses. The only auxiliary object is the uniform-flow operator R, which is explicitly constructed in Proposition 5.4 and therefore not an axiom.

assumptions (7)
  • standard math Kingman's subadditive ergodic theorem (as in [LiM02, Theorem 4.1]) is used to define f_hom in Definition 6.7.
    Ensures the cell-problem limit exists almost surely for every j.
  • standard math Besicovitch differentiation theorem for vector measures (Proposition 8.1, cited from [AmM92, AFP00]).
    Provides the pointwise densities used in the lower-bound blow-up argument.
  • standard math Fonseca-Muller blow-up method ([FoM92]) as adapted to the divergence constraint.
    Underlies the lower-bound proof in Section 10.
  • domain assumption Graph assumptions (G1)-(G3): no large gaps, quantitative connectedness, bounded edge lengths.
    Used to construct uniform-flow operators (Prop 5.4) and correctors (Prop 7.4); excludes Poisson point process, as acknowledged.
  • domain assumption Energy assumptions (F1)-(F3): Lipschitz locality, linear growth, sigma-additivity.
    F1-F2 give compactness and continuity of rescaled energies; F3 is used for subadditivity of the cell formula and the gluing argument. F3 is the most restrictive; Remark 2.3 discusses a weaker version not covered.
  • domain assumption Stationarity of the graph and energy with respect to Z^d translations.
    Needed for ergodic theorems producing an almost-sure deterministic or stationary limit.
  • standard math Vectorial Kantorovich-Rubinstein duality T1 = gKR norm (Ciosmak [Cio21, Theorem 2]) is used in Proposition 7.4.
    Provides the bound on correctors in terms of the gKR distance.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Stochastic homogenisation of nonlinear minimum-cost flow problems." pith.science (2026). https://pith.science/paper/OQ3P2JW4

@misc{pith2026241205217,
  author       = {Pith},
  title        = {Pith review of: Stochastic homogenisation of nonlinear minimum-cost flow problems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OQ3P2JW4}},
  note         = {Machine review of arXiv:2412.05217}
}
abstract

This paper deals with the large-scale behaviour of nonlinear minimum-cost flow problems on random graphs. In such problems, a random nonlinear cost functional is minimised among all flows (discrete vector-fields) with a prescribed net flux through each vertex. On a stationary random graph embedded in $\mathbb{R}^d$, our main result asserts that these problems converge, in the large-scale limit, to a continuous minimisation problem where an effective cost functional is minimised among all vector fields with prescribed divergence. Our main result is formulated using $\Gamma$-convergence and applies to multi-species problems. The proof employs the blow-up technique by Fonseca and M\"uller in a discrete setting. One of the main challenges overcome is the construction of the homogenised energy density on random graphs without a periodic structure.

Figures

Figures reproduced from arXiv: 2412.05217 by the authors.

Figure 1
Figure 1. In red (resp. in orange), a concatenation of paths on (X , E) of the form Pz,1 (resp. Pz,2) with z ∈ Z 2 . Along the red paths Rj = je2, whilst along the orange path Rj = je1 [PITH_FULL_IMAGE:figures/full_fig_p019_1.png] view at source ↗
Figure 2
Figure 2. A two-scale decomposition of the paths in the direction ed. Highlighted the subcomponent given by JεP m z′ . In light orange, a repre￾sentation of the set Rz ′m . On the mesoscopic scale δ ≫ ε, the path JεP m z′ is approximately the segment joining z ′ m and z ′ m+1, hence ιεJεP m z′ has an approximate average orientation given by ed, see (5.4). Indeed, noticing that ´ dιε(JεPz,i ) = ε(xz+ei − xz), we have the teles… view at source ↗
Figure 3
Figure 3. The set of R-representatives of j in A must coincide with Rj on all the edges highlighted in green, which at distance of order 1 from ∂A. For (ε, R) representatives, the boundary conditions must be satisfied for edges at distance of order ε from ∂A. In other words, the set Repε,R(j, A) contains all the divergence-free discrete vector fields which coincide with the uniform flow Rεj at distance ε from the boundary ∂A … view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: Construction of good paths. For every i, we use Assumption (G1) and find a point xi ∈ Xε such that xi ∈ BεR1 (zi) (we set x0 := x and xm := y). Note that by traingle inequality, we must necessarily have |xi+1 − xi | ≤ 2εR1, for every i = 0, . . . , m − 1. Therefore, by…
Figure 5
Figure 5. Figure 5: A representation of the three layers involved in the proof of Proposition 10.1: a microscopic scale ε (edge length), a mesoscopic scale 1 ≫ η ≫ ε, representing the size of the strip where we apply a cutoff func￾tion to fix the boundary conditions, which must be chosen …
Figure 6
Figure 6. Figure 6: An oriented strip with sides paralell to ker j and (ker j) ⊥. In a similar spirit as for the absolutely continuous part, the next proposition shows a quantitative error estimates in terms of how far a discrete flux is from being a competitor of the cell formula, when w…
Figure 7
Figure 7. Figure 7: Around a singular point, we perform a cutoff procedure close to the boundary of the oriented strip Rα. On the sides parallel to ker j, we choose a mesoscopic scale η ≫ ε, whilst for the faces paralell to (ker j) ⊥ we pick a mesoscopic scale η ′ ≫ ε, chosen in such a wa…

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

50 extracted references · 40 canonical work pages

  1. [1]

    write newline

    " write newline "" before.all 'output.state := FUNCTION fin.entry add.period write newline FUNCTION new.block output.state before.all = 'skip after.block 'output.state := if FUNCTION new.sentence output.state after.block = 'skip output.state before.all = 'skip after.sentence 'output.state := if if FUNCTION not #0 #1 if FUNCTION and 'skip pop #0 if FUNCTIO...

    work page

  2. [2]

    Alicandro, M

    R. Alicandro, M. Cicalese, and A. Gloria . Integral representation results for energies defined on stochastic lattices and application to nonlinear elasticity. Arch. Ration. Mech. Anal. , 200(3), 881--943, 2011

    work page 2011

  3. [3]

    Alicandro, M

    R. Alicandro, M. Cicalese, and M. Ruf . Domain formation in magnetic polymer composites: an approach via stochastic homogenization. Arch. Ration. Mech. Anal. , 218(2), 945--984, 2015

    work page 2015

  4. [4]

    Auffinger, M

    A. Auffinger, M. Damron, and J. Hanson . 50 years of first-passage percolation , volume 68 of University Lecture Series . American Mathematical Society, Providence, RI, 2017

    work page 2017

  5. [5]

    Ambrosio, N

    L. Ambrosio, N. Fusco, and D. Pallara . Functions of bounded variation and free discontinuity problems . Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York, 2000

    2000

  6. [6]

    M. A. Akcoglu and U. Krengel . Ergodic theorems for superadditive processes. 1981

    work page 1981

  7. [7]

    G. Alberti . Rank one property for derivatives of functions with bounded variation. Proc. Roy. Soc. Edinburgh Sect. A , 123(2), 239--274, 1993

    work page 1993

  8. [8]

    Ambrosio and G

    L. Ambrosio and G. D. Maso . On the relaxation in BV ( ; R ^m ) of quasi-convex integrals. Journal of Functional Analysis , 109(1), 76--97, 1992

    work page 1992

Show all 50 references
  1. [9]

    Ba\' a, M

    M. Ba\' a, M. Chermisi, J. Matias, and P. M. Santos . Lower semicontinuity and relaxation of signed functionals with linear growth in the context of A -quasiconvexity. Calc. Var. Partial Differential Equations , 47(3-4), 465--498, 2013

    2013

  2. [10]

    Beckmann

    M. Beckmann . A continuous model of transportation. Econometrica , 20, 643--660, 1952

    1952

  3. [11]

    Bouchitt\' e , I

    G. Bouchitt\' e , I. Fonseca, G. Leoni, and L. Mascarenhas . A global method for relaxation in W^ 1,p and in SBV _p . Arch. Ration. Mech. Anal. , 165(3), 187--242, 2002

    2002

  4. [12]

    Braides, I

    A. Braides, I. Fonseca, and G. Leoni . A-quasiconvexity: relaxation and homogenization. ESAIM: Control, Optimisation and Calculus of Variations , 5, 539--577, 2000

    2000

  5. [13]

    Braides, M

    A. Braides, M. Maslennikov, and L. Sigalotti . Homogenization by blow-up. Appl. Anal. , 87(12), 1341--1356, 2008

    2008

  6. [14]

    V. I. Bogachev . Measure theory. V ol. I , II . Springer-Verlag, Berlin, 2007

    2007

  7. [15]

    Braides and M

    A. Braides and M. Caroccia . Asymptotic behavior of the dirichlet energy on poisson point clouds. Journal of Nonlinear Science , 33(5), 80, 2023

    2023

  8. [16]

    K. J. Ciosmak . Optimal transport of vector measures. Calculus of Variations and Partial Differential Equations , 60(6), 1--22, 2021

    2021

  9. [17]

    Conti, S

    S. Conti, S. M\" u ller, and M. Ortiz . Symmetric div-quasiconvexity and the relaxation of static problems. Arch. Ration. Mech. Anal. , 235(2), 841--880, 2020

    2020

  10. [18]

    Davoli and I

    E. Davoli and I. Fonseca . Homogenization of integral energies under periodically oscillating differential constraints. Calculus of Variations and Partial Differential Equations , 55, 1--60, 2016

    2016

  11. [19]

    Dal Maso and L

    G. Dal Maso and L. Modica . Nonlinear stochastic homogenization. Annali di matematica pura ed applicata , 144, 347--389, 1986

    1986

  12. [20]

    De Lellis

    C. De Lellis . Lecture notes on rectifiable sets, densities, and tangent measures. Preprint , 23, 2006

    2006

  13. [21]

    De Giorgi and G

    E. De Giorgi and G. Letta . Une notion g\'en\'erale de convergence faible pour des fonctions croissantes d'ensemble. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze , 4e s \'e rie, 4(1), 61--99, 1977

    1977

  14. [22]

    De Philippis and F

    G. De Philippis and F. Rindler . On the structure of A -free measures and applications. Ann. of Math. (2) , 184(3), 1017--1039, 2016

    2016

  15. [23]

    Disser and M

    K. Disser and M. Liero . On gradient structures for M arkov chains and the passage to W asserstein gradient flows. Netw. Heterog. Media , 10(2), 233--253, 2015

    2015

  16. [24]

    Esposito, G

    A. Esposito, G. Heinze, and A. Schlichting . Graph-to-local limit for the nonlocal interaction equation. arXiv:2306.03475 , 2023

    2023 arXiv

  17. [25]

    Esposito, F

    A. Esposito, F. S. Patacchini, A. Schlichting, and D. Slepcev . Nonlocal-interaction equation on graphs: gradient flow structure and continuum limit. Arch. Ration. Mech. Anal. , 240(2), 699--760, 2021

    2021

  18. [26]

    Esposito, F

    A. Esposito, F. S. Patacchini, and A. Schlichting . On a class of nonlocal continuity equations on graphs. European J. Appl. Math. , 35(1), 109--126, 2024

    2024

  19. [27]

    Esposito and L

    A. Esposito and L. Mikol \'a s . On evolution PDE s on co-evolving graphs. arXiv:2310.10350 , 2023

    2023 arXiv

  20. [28]

    Forkert, J

    D. Forkert, J. Maas, and L. Portinale . Evolutionary -convergence of entropic gradient flow structures for F okker- P lanck equations in multiple dimensions. SIAM J. Math. Anal. , 54(4), 4297--4333, 2022

    2022

  21. [29]

    L. R. Ford, Jr. and D. R. Fulkerson . Flows in networks . Princeton University Press, Princeton, N.J., 1962

    1962

  22. [30]

    Fonseca and S

    I. Fonseca and S. M \"u ller . Quasi-convex integrands and lower semicontinuity in L^1 . SIAM J. Math. Anal. , 23(5), 1081--1098, 1992

    1992

  23. [31]

    Fonseca and S

    I. Fonseca and S. M\" u ller . A -quasiconvexity, lower semicontinuity, and Y oung measures. SIAM J. Math. Anal. , 30(6), 1355--1390, 1999

    1999

  24. [32]

    Garc\' a Trillos

    N. Garc\' a Trillos . Gromov- H ausdorff limit of W asserstein spaces on point clouds. Calc. Var. Partial Differential Equations , 59(2), Paper No. 73, 43, 2020

    2020

  25. [33]

    Gigli and J

    N. Gigli and J. Maas . Gromov- H ausdorff convergence of discrete transportation metrics. SIAM J. Math. Anal. , 45(2), 879--899, 2013

    2013

  26. [34]

    Gladbach, E

    P. Gladbach, E. Kopfer, J. Maas, and L. Portinale . Homogenisation of one-dimensional discrete optimal transport. Journal de Math \'e matiques Pures et Appliqu \'e es , 139, 204 -- 234, 2020

    2020

  27. [35]

    Gladbach, E

    P. Gladbach, E. Kopfer, J. Maas, and L. Portinale . Homogenisation of dynamical optimal transport on periodic graphs. Calc. Var. Partial Differential Equations , 62(5), Paper No. 143, 75, 2023

    2023

  28. [36]

    Gladbach and E

    P. Gladbach and E. Kopfer . Stochastic homogenization of dynamical discrete optimal transport. arXiv:2411.04157 , 2024

    2024 arXiv

  29. [37]

    Hraivoronska and O

    A. Hraivoronska and O. Tse . Diffusive limit of random walks on tessellations via generalized gradient flows. SIAM J. Math. Anal. , 55(4), 2948--2995, 2023

    2023

  30. [38]

    Hraivoronska, A

    A. Hraivoronska, A. Schlichting, and O. Tse . Variational convergence of the S charfetter-- G ummel scheme to the aggregation-diffusion equation and vanishing diffusion limit. Numer. Math. , 156(6), 2221--2292, 2024

    2024

  31. [39]

    Ishida and H

    S. Ishida and H. Lavenant . Quantitative convergence of a discretization of dynamic optimal transport using the dual formulation. Foundations of Computational Mathematics , pages 1--36, 2024

    2024

  32. [40]

    J. F. C. Kingman . Subadditive ergodic theory. Ann. Probability , 1, 883--909, 1973

    1973

  33. [41]

    Kirszbraun

    M. Kirszbraun . \"U ber die zusammenziehende und L ipschitzsche T ransformationen. Fundamenta Mathematicae , 22(1), 77--108, 1934

    1934

  34. [42]

    Lavenant

    H. Lavenant . Unconditional convergence for discretizations of dynamical optimal transport. Mathematics of Computation , 90(328), 739--786, 2021

    2021

  35. [43]

    Licht and G

    C. Licht and G. Michaille . Global-local subadditive ergodic theorems and application to homogenization in elasticity. Ann. Math. Blaise Pascal , 9(1), 21--62, 2002

    2002

  36. [44]

    Matias, M

    J. Matias, M. Morandotti, and P. M. Santos . Homogenization of functionals with linear growth in the context of A -quasiconvexity. Applied Mathematics & Optimization , 72(3), 523--547, 2015

    2015

  37. [45]

    C. B. Morrey, Jr. Quasi-convexity and the lower semicontinuity of multiple integrals. Pacific Journal of Mathematics , 2(1), 25 -- 53, 1952

    1952

  38. [46]

    a ffner, and A. Schl\

    S. Neukamm, M. Sch\" a ffner, and A. Schl\" o merkemper . Stochastic homogenization of nonconvex discrete energies with degenerate growth. SIAM J. Math. Anal. , 49(3), 1761--1809, 2017

    2017

  39. [47]

    Portinale and F

    L. Portinale and F. Quattrocchi . Discrete-to-continuum limits of optimal transport with linear growth on periodic graphs. European Journal of Applied Mathematics , pages 1--29, 2024

    2024

  40. [48]

    Ruf and C

    M. Ruf and C. I. Zeppieri . Stochastic homogenization of degenerate integral functionals with linear growth. Calc. Var. Partial Differential Equations , 62(4), Paper No. 138, 2023

    2023

  41. [49]

    Santambrogio

    F. Santambrogio . Optimal transport for applied mathematicians , volume 87 of Progress in Nonlinear Differential Equations and their Applications . Birkh\" a user/Springer, Cham, 2015

    2015

  42. [50]

    Slep c ev and A

    D. Slep c ev and A. Warren . Nonlocal W asserstein distance: Metric and asymptotic properties. Calculus of Variations and Partial Differential Equations , 62(9), 238, 2023

    2023

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.