{"id":"dbd1a3e1-d2bc-488f-8ea0-c1253f84eaee","arxiv_id":"2501.15605","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Singularities of potential energy functionals for weak KAM solutions propagate along a minimizing-movement curve in Wasserstein space, and the curve solves a continuity equation with the minimal-energy gradient selection.","lead":"This paper studies where singularities (non-differentiability points) of optimal transport costs and weak KAM solutions spread over time in the space of probability measures. It proves that once mass touches the cut locus, the propagated measures stay singular, and it builds a curve of measures that solves a transport continuity equation with a non-smooth velocity field.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Global singularity propagation hinges on unproved inputs from [29]: the limit (50) that identifies the velocity field in (45), and the cut-time monotonicity used in Corollary 5.2(2).","rationale":"The reader's weakest-assumption analysis correctly identifies the load-bearing dependence on external inputs: the limit (50) and the cut-time monotonicity from [29], together with Albano's theorem, are precisely what convert the local construction into the global propagation claims. My read does not contradict that assessment, and I did not find an independent internal contradiction. The most delicate point is the use of (50) in Theorem 5.5: the entire construction of the limiting flow via minimizing movements needs the velocity field H_p(·, p#_φ(·)) to be identified at points of Sing(φ), because the theorem's purpose is to show that singular mass remains singular. If [29, Prop 4.2] only supplies the limit at differentiability points, the argument has a genuine gap at exactly the points it needs to track. The same fragility applies to Corollary 5.2(2), where [29, Thm 5.7] is invoked to propagate the cut-locus condition through time. The concrete test proposed above would settle whether the external input has the required uniformity. Since this concern is already the basis of the reader's CONDITIONAL verdict, the appropriate recommendation is to keep that verdict: the paper should either prove the needed companion-preprint statements or state and verify their hypotheses explicitly before the propagation results can be accepted as self-contained.","tokens_in":50505,"tokens_out":14182,"duration_ms":142193,"concrete_test":"Analytical check: restate [29, Prop 4.2] as a standalone lemma with full hypotheses and attempt to prove it for every φ ∈ SCL(T^m) at every x ∈ T^m, especially at x ∈ Sing(φ), using only the envelope formula T^+_t φ(x) = max_y {φ(y) - A_t(x,y)} and strict convexity of H. If the proof requires D^+φ(x) to be a singleton near x, or only gives convergence for almost every x, or only along a subsequence of t → 0+, then (50) is not strong enough for the partition limit in Theorem 5.5 and the vector field in (45) is not justified on Sing(u).","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central assertion that singularities propagate globally is assembled in Theorem 5.5 and Corollary 5.2. The bridge from the discrete minimizing-movement scheme (48) to the limiting continuity equation (45) is the identification of the limiting velocity with H_p(x, p#_φ(x)); that identification is made through equation (50), imported from [29, Prop 4.2]. The manuscript does not state the hypotheses under which (50) holds: whether it requires φ ∈ C^{1,1}, whether it holds at every x ∈ T^m, or only at differentiability points, and whether the convergence is uniform enough to pass through the partition limit in (49). Since the goal is to propagate mass supported on Sing(u), the behavior of the limit exactly at singular points is the case that must be controlled; if (50) only holds off Sing(u), then (45) is not justified on the set that Corollary 5.2 is about. Similarly, Corollary 5.2(2) relies on [29, Thm 5.7] for the inequality τ_u(ξ*(t)) ≤ τ_u(ξ*(0)) along the limiting random curve; the hypotheses of that theorem, including the precise meaning of strict singular characteristic and the admissible starting times, are not restated or verified for the curve ξ* constructed in Theorem 5.5. Finally, the application of Albano's theorem [1, Thm 1.2] requires ξ* to be a generalized characteristic in the exact sense of that theorem; the manuscript verifies the required inclusion only formally through the equalities in (51)-(52). None of these three ingredients is proved in this paper, so the global propagation claims stand or fall with unpublished companion-preprint results whose hypotheses may be stricter than those assumed here.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops a theory of singularities for potential energy functionals on the space of compactly supported probability measures and studies their propagation in optimal transport. For a semiconcave function φ, the authors characterize the singular points of the functional φ(·) as measures charging Sing(φ), establish regularity properties of the dynamical cost functional C_t, and introduce random Lax-Oleinik operators that solve a Hamilton-Jacobi equation in Wasserstein space. The main results are: if u is a weak KAM solution and μ is a cut point of u(·), then the forward maximizing curve ν_{u,μ}(t) lies in S(u(·)) for small t (Theorem 5.4); for any φ ∈ SCL(T^m) and μ0 ∈ P(T^m), there is a Lipschitz curve μ(·) solving the continuity equation ∂_t μ + div(H_p(·, p#_φ(·)) μ)=0 (Theorem 5.5); and when φ=u is a weak KAM solution, singular support and cut-locus membership propagate forward in time (Corollary 5.2). The paper also constructs an irregular Lagrangian semiflow on the cut locus and discusses its non-regularity with respect to DiPerna-Lions theory.","tokens_in":50812,"tokens_out":11851,"duration_ms":105014,"significance":"If the central assertions hold, the paper establishes a measure-valued propagation theory for singularities in optimal transport after the formation of singularities, a regime where classical regular Lagrangian flows do not apply. The construction is genuinely variational: the target curve in Theorem 5.5 is constructed, not assumed, and the velocity field is identified explicitly through p#_φ without fitted constants. The paper also gives useful structural results on generalized differentials of potential energy functionals and on random Lax-Oleinik operators. However, several load-bearing ingredients are imported from unpublished preprints by the same authors, and the hypotheses under which those ingredients hold are not stated or verified in this manuscript. The propagation claims are therefore conditional on external inputs that are not established here.","major_comments":[{"comment":"The identification lim_{t→0+} DT^+_t φ(x) = p#_φ(x) is imported from [29, Prop. 4.2] without stating the hypotheses under which it holds. The proof then passes this limit through an integral in s and through weak-* limits of measures to obtain (51)-(52). It is not specified whether (50) holds for every x ∈ T^m or only on a differentiability set, whether the convergence is uniform or dominated, and whether it holds at the singular points of φ. Since Corollary 5.2 concerns exactly the mass supported on Sing(u), the behavior of the limit on Sing(u) is the case that must be controlled; if (50) holds only off Sing(u), the equality in (51)-(52) and the ODE for ξ* are not justified on the set relevant to the propagation claim.","section":"§5.3, Theorem 5.5, Eq. (50)"},{"comment":"Corollary 5.2(2) invokes [29, Thm 5.7] to assert τ_u(ξ*(t)) ≤ τ_u(ξ*(0)) along the limiting random curve ξ*, but the hypotheses of that theorem—in particular the precise definition of strict singular characteristic and the admissible starting times—are not restated or verified for the curve constructed in Theorem 5.5. Lemma 5.5 supplies only a one-sided variational characterization, and it is not shown that ξ* satisfies the hypotheses of [29, Thm 5.7] for every t ∈ [0,T], including times after the classical cut time. Without this verification, the global cut-locus propagation assertion in Corollary 5.2(2) is unsupported.","section":"§5.3, Corollary 5.2(2)"},{"comment":"Corollary 5.2(1) applies Albano's theorem [1, Thm 1.2] to conclude that ξ(0,ω) ∈ Sing(u) implies ξ*(t,ω) ∈ Sing(u). The manuscript does not state Albano's definition of generalized characteristic nor verify that the random curve ξ* constructed in Theorem 5.5 is a generalized characteristic in that exact sense. The only verification is through the equalities in (51)-(52), which identify a strict singular characteristic in the sense of Lemma 5.5 (taken from [29]), not necessarily the notion used in [1]. Since this step is the bridge from the constructed velocity field to persistence of Sing(u), it is load-bearing and needs a precise statement and proof.","section":"§5.3, Corollary 5.2(1)"},{"comment":"The proof of Theorem 3.3(3) chooses λ_K such that K ⊂ B(x, λ_K t) for all x ∈ K, which forces λ_K ≥ diam(K)/t. Hence the constant C_{λ_K} from Proposition 2.8 depends on t, and the stated conclusion that there exists C_K > 0 depending only on K with semiconcavity constant C_K/t uniformly for t ∈ (0,1) is not justified. A corrected statement should either allow the constant to depend on t or prove the required uniformity separately.","section":"§3.3, Theorem 3.3(3)"}],"minor_comments":[{"comment":"There is a typo in the abstract: 'potenti al' should be 'potential'.","section":"Abstract"},{"comment":"In the proof of Theorem 5.4, the displayed equality 'u(x) − A_t(x,y) = T^+_t u(x)' should read 'u(y) − A_t(x,y) = T^+_t u(x)' for γ^t-almost every (x,y).","section":"§5.2, Theorem 5.4 proof"},{"comment":"In Proposition 4.1(3), the second identity appears to have a sign error: it should be P^+_t ∘ P^-_s φ = T^+_t ∘ T^-_s φ(·), not T^-_t ∘ T^-_s φ(·).","section":"§4.1, Proposition 4.1(3)"},{"comment":"Corollary 5.2 contains a duplicated comma in 'is a viscosity solution to (HJs), ,'.","section":"§5.3, Corollary 5.2"},{"comment":"The paper does not discuss the measurability of the selection p#_φ(x) = argmin{H(x,p): p ∈ D^+φ(x)}; since the continuity equation (45) is integrated against μ(t), a Borel measurable selection should be justified, for instance via a measurable selection theorem.","section":"§5.3, Theorem 5.5"}],"recommendation":"major_revision","confidential_remarks":"The central results of this paper depend heavily on the unpublished preprints [29] and [37] by the same authors, particularly for the limit (50) and the cut-time monotonicity used in Corollary 5.2. The refereeing process should ensure that those preprints are available and that their hypotheses exactly match the statements used here. The paper does not include proofs of these imported inputs, which makes the current version conditional. I would encourage the editors to request that the authors either include the necessary statements and verifications in the paper or provide a detailed appendix deriving them."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this paper has two or three genuinely new theorems and the main idea is sound, but the global propagation part is not self-contained. The singular-point characterization (Thm 3.2) and the cut-locus propagation along the intrinsic characteristic (Thm 5.4) are real contributions, and the minimizing-movement construction in Thm 5.5 is a serious variational argument that does not assume what it proves. I found no circularity and no fitted parameters.\n\nThe soft spots are concentrated where the paper leans on [29] and [37]. The step that identifies the limiting velocity in the continuity equation (45) is (50), imported from [29, Prop 4.2]. The paper does not state the hypotheses: whether the convergence holds at every x, including the singular points that Cor 5.2 is about, or only off Sing(u). That matters because the goal is to propagate mass supported on Sing(u). Same for Cor 5.2(2): the cut-time monotonicity τ_u(ξ*(t)) ≤ τ_u(ξ*(0)) comes from [29, Thm 5.7], and it is not checked that the curve ξ* built in Thm 5.5 satisfies that theorem's definition of strict singular characteristic. The use of Albano's theorem [1] is also more formal than verified: equalities (51)-(52) show an inequality and then assert equality via Lemma 5.4, but the exact sense in which ξ* is a generalized characteristic in Albano's meaning is not pinned down. These gaps are not fatal if the companion results hold under the stated assumptions, but as written the global claims stand or fall with unpublished preprints.\n\nThere is also a smaller, concrete issue: Theorem 3.3(3) as stated is not supported by its proof. The constant should depend on t in a specific way, and the proof's choice of λ_K with K ⊂ B(x,λ_K t) for all x ∈ K needs t bounded away from zero; as written, t ∈ (0,1) is not enough. This is fixable but needs rewriting.\n\nThe typos and sign slips are numerous enough to slow a reader down, but not load-bearing. The citation pattern is mostly fine: self-citations are to the companion papers that actually do the heavy lifting, which is exactly why the missing hypotheses hurt.\n\nIn sum: the paper deserves a serious referee. The right outcome is a conditional accept after the authors either prove the needed statements from [29] or state them in appendices, and fix Thm 3.3(3). I would send it to review.","headline":"A plausible and genuinely new singularity-propagation result in Wasserstein space, but the global claims lean on unpublished companion-preprint theorems whose hypotheses are not restated.","tokens_in":51457,"tokens_out":1864,"would_cite":true,"duration_ms":17321,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35D40","49Q22","37Jxx","37Kxx"],"pacs":[],"model":"deepseek-v4-flash","headline":"If a measure ever enters the singular set of a Hamilton-Jacobi potential, the paper proves it stays there forever.","keywords":["optimal transport","potential energy functional","cut locus","propagation of singularities","Hamilton-Jacobi equation","weak KAM theory","Wasserstein space","semiconcave functions"],"falsifier":"If the theorem is right, then for a weak KAM solution $u$ with a nonempty singular set and $\\mu_0=\\delta_{x_0}$ with $x_0\\in\\operatorname{Sing}(u)$, the computed maximizer $\\nu_{u,\\mu}(t)$ must assign full mass to $\\operatorname{Sing}(u)$ for every small $t>0$. A decisive calculation is to take a one-dimensional Tonelli Hamiltonian with a known weak KAM solution having an isolated corner, solve the one-step maximization $u(\\nu)-C^t(\\delta_{x_0},\\nu)$ explicitly, and check the optimizer: if it is ever absolutely continuous with respect to Lebesgue measure, or more generally has $\\nu(\\operatorname{Sing} u)=0$, the propagation claim fails.","tokens_in":50244,"feed_emoji":"🌊","tokens_out":16540,"duration_ms":137807,"temperature":0.7,"pith_summary":"The paper is trying to establish that singularities in optimal transport are permanent: once a measure meets the singular set of a Hamilton-Jacobi potential, the flow keeps it there for all later times. For a weak KAM solution $u$ of $H(x,Du)=c[0]$, it proves that each cut point of the induced potential-energy functional $u(\\cdot)$ is carried by the canonical maximizing movement into singular points of $u(\\cdot)$, and that the resulting curve of measures solves a continuity equation with velocity $H_p(x,p^\\#_u(x))$, where $p^\\#_u(x)$ is the minimal-momentum element of the superdifferential $D^+u(x)$. In measure terms, $\\mu_0(\\operatorname{Sing} u)>0$ forces $\\mu(t)(\\operatorname{Sing} u)>0$ for every $t\\in[0,T]$, and the same forward invariance holds for the cut locus $\\mathcal{C}(u(\\cdot))$. The bridge that makes this tractable is the equivalence: a measure is a singular point of the functional $u(\\cdot)$ exactly when it assigns positive mass to $\\operatorname{Sing}(u)$. If correct, this supplies a global propagation theory for the regime after the cut time, when the classical regular-flow machinery for transport equations no longer applies.","feed_headline":"Singularities in optimal transport propagate forever once they form","feed_subtitle":"Once a probability measure meets the singular set of the potential, the transport keeps it there forever.","key_machinery":"The load-bearing object is the potential-energy functional $u(\\mu)=\\int u\\,d\\mu$ with its localized Frechet superdifferential, together with the characterization $\\mu\\in\\mathcal{S}(u(\\cdot))$ if and only if $\\mu(\\operatorname{Sing} u)>0$. Singularity transport is carried by the minimal-momentum selection $p^\\#_\\phi(x)=\\operatorname{argmin}\\{H(x,p):p\\in D^+\\phi(x)\\}$; the measure curve of Theorem 5.5 is the law of random solutions of $\\dot x=H_p(x,p^\\#_\\phi(x))$, an irregular Lagrangian semiflow on the cut locus because the vector field loses the regularity needed for classical flow theory after the cut time. The construction uses the random Lax-Oleinik operators $P^\\pm_t$, the cut-time identity $T_u(\\mu)=\\inf\\{\\tau_u(x):x\\in\\operatorname{supp}\\mu\\}$, and an energy-dissipation-inequality (EDI) minimizing-movement scheme that extracts the limit curve and the continuity equation from a sequence of discrete maximization steps.","core_discovery":"The central claim is Theorem 5.4 together with Corollary 5.2. Let $u$ be a weak KAM solution of $H(x,Du(x))=c[0]$ on the torus, let $u(\\mu)=\\int u\\,d\\mu$ be the induced potential-energy functional, and let $C^t$ be the dynamical cost built from the fundamental solution. If $\\mu\\in\\mathcal{C}(u(\\cdot))$, then for every $t\\in(0,t_{u,\\mu}]$ the unique maximizer $\\nu_{u,\\mu}(t)=\\operatorname{argmax}\\{u(\\nu)-C^t(\\mu,\\nu)\\}$ is a singular point of $u(\\cdot)$, i.e. $\\nu_{u,\\mu}(t)(\\operatorname{Sing} u)>0$. Moreover, the minimizing-movement curve $\\mu(\\cdot):[0,T]\\to P(\\mathbb{T}^m)$ obtained by letting the time step go to zero satisfies the continuity equation $\\frac{d}{dt}\\mu+\\operatorname{div}(H_p(\\cdot,p^\\#_u(\\cdot))\\mu)=0$, $\\mu(0)=\\mu_0$; if $\\mu_0(\\operatorname{Sing} u)>0$ then $\\mu(t)(\\operatorname{Sing} u)>0$ for all $t$, and if $\\mu_0\\in\\mathcal{C}(u(\\cdot))$ then $\\mu(t)\\in\\mathcal{C}(u(\\cdot))$ for all $t$. In the authors' phrasing, the forward dynamics after the formation of singularities is governed by an irregular Lagrangian semiflow on the cut locus.","pith_inferences":["A natural numerical test is to discretize the maximizing movement with step $h$ for a semiconcave potential with an isolated corner, and measure the mass assigned to $\\operatorname{Sing}(\\phi)$; the theorem predicts that every weak-* accumulation point as $h\\to0$ keeps full singular mass at all positive times.","The vector field $H_p(x,p^\\#_\\phi(x))$ is irregular and uniqueness for the continuity equation is not supplied by classical theory; a question left implicit is whether the minimizing-movement curve is the unique selection solution of the continuity equation.","If the cut-time monotonicity from the companion preprint holds globally, the same mechanism should yield an even stronger statement not isolated in the paper: the semiflow is forward-invariant on the cut locus for every choice of initial measure, not only for measures already known to be cut points."],"forward_implications":["An absolutely continuous measure transported before its cut time stays absolutely continuous, while after the cut time the evolution can make it singular; Theorem 5.2(3) and Remark 5.5 make this transition explicit.","Singularity mass is persistent: if $\\mu_0(\\operatorname{Sing} u)>0$, then $\\mu(t)(\\operatorname{Sing} u)>0$ for every $t\\in[0,T]$, so singular support cannot disappear once present.","The cut locus $\\mathcal{C}(u(\\cdot))$ is forward-invariant: a trajectory starting in the cut locus remains in the cut locus over arbitrary horizons $T$, giving a global semiflow on the cut locus.","For any smooth observable $f$, the right time-derivative of $f(\\mu(t))$ is $\\int\\langle Df(x),H_p(x,p^\\#_\\phi(x))\\rangle\\,d\\mu(t)$, so the irregular flow still has a well-defined generator on smooth observables.","The continuity-equation construction works for every semiconcave $\\phi$ on the torus, not only weak KAM solutions, so the propagation mechanism is tied to semiconcavity rather than to the specific equation."],"supporting_citations":[{"why":"Supplies the small-time limit identifying $p^\\#_\\varphi$ and the cut-time monotonicity along strict singular characteristics; both are used to pass from discrete maximization steps to the continuity equation and to Corollary 5.2.","marker":"[29]"},{"why":"Gives the propagation theorem for the singular set along generalized characteristics, invoked in Corollary 5.2(1) to move pointwise singularities to the measure level.","marker":"[1]"},{"why":"Establishes intrinsic singular characteristics and the fact that they stay in the singular set before the cut time, the core mechanism of Theorem 5.4.","marker":"[25]"},{"why":"Provides the weak-KAM cut-locus toolkit, including the $B_u$ function and the identity $T_u(\\mu)=\\inf\\tau_u(x)$ used to characterize cut points.","marker":"[28]"},{"why":"Supplies the classical weak KAM theory: calibrated curves, Lax-Oleinik semigroups, and the fixed-point characterization of weak KAM solutions that the measure-level operators mirror.","marker":"[43]"},{"why":"Establishes displacement interpolation for Tonelli costs, identifying minimizers of $C^t$ with random Euler-Lagrange curves and thus grounding the dynamical cost setup.","marker":"[16]"},{"why":"Provides the Wasserstein gradient-flow toolbox: localized Frechet differentials, semiconcavity of functionals, and the minimizing-movement/EDI construction used in Theorem 5.5.","marker":"[9]"},{"why":"Sets up semiconcavity and localized differentials for functionals on compactly supported probability measures, the framework in which the singularity equivalence for potential energies is proved.","marker":"[21]"}],"fun_headline_variants":["Singularities in optimal transport are permanent once formed","Transport singularities never disappear after they appear","In optimal transport, singularities propagate for all time","Once a measure hits the singular set, it stays there forever","Cut locus dynamics keep singularities alive in transport"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole construction stands or falls on three facts proved outside the paper: the small-time limit that selects the minimal-momentum element $p^\\#_\\varphi$ from the superdifferential, the inequality $\\tau_u(\\xi(t))\\le\\tau_u(\\xi(0))$ along strict singular characteristics, and the theorem that singularities propagate along generalized characteristics; if any of these holds only under hypotheses stricter than the ones stated, the continuity equation and the propagation conclusions do not follow.","fun_headline_variants_meta":{"raw":{"variants":["Singularities in optimal transport are permanent once formed","Transport singularities never disappear after they appear","In optimal transport, singularities propagate for all time","Once a measure hits the singular set, it stays there forever","Cut locus dynamics keep singularities alive in transport"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000207,"raw_usage":{"total_tokens":1437,"prompt_tokens":1022,"completion_tokens":415,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":638,"completion_tokens_details":{"reasoning_tokens":340}},"tokens_in":638,"tokens_out":415,"duration_ms":4036,"temperature":1.0,"reasoning_tokens":340,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T14:08:22.553832+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"If the theorem is right, then for a weak KAM solution $u$ with a nonempty singular set and $\\mu_0=\\delta_{x_0}$ with $x_0\\in\\operatorname{Sing}(u)$, the computed maximizer $\\nu_{u,\\mu}(t)$ must assign full mass to $\\operatorname{Sing}(u)$ for every small $t>0$. A decisive calculation is to take a one-dimensional Tonelli Hamiltonian with a known weak KAM solution having an isolated corner, solve the one-step maximization $u(\\nu)-C^t(\\delta_{x_0},\\nu)$ explicitly, and check the optimizer: if it is ever absolutely continuous with respect to Lebesgue measure, or more generally has $\\nu(\\operatorname{Sing} u)=0$, the propagation claim fails.","supporting_citations":[{"cited_title":"V ariational construction of generalized charac- teristics and maximal slope curve","cited_arxiv_id":null,"evidence_quote":"Supplies the small-time limit identifying $p^\\#_\\varphi$ and the cut-time monotonicity along strict singular characteristics; both are used to pass from discrete maximization steps to the continuity equation and to Corollary 5.2."},{"cited_title":"Propagation of singularities for solutio ns of Hamilton-Jacobi equations","cited_arxiv_id":null,"evidence_quote":"Gives the propagation theorem for the singular set along generalized characteristics, invoked in Corollary 5.2(1) to move pointwise singularities to the measure level."},{"cited_title":"Generalized charact eristics and Lax-Oleinik operators: global theory","cited_arxiv_id":null,"evidence_quote":"Establishes intrinsic singular characteristics and the fact that they stay in the singular set before the cut time, the core mechanism of Theorem 5.4."},{"cited_title":"Topolo gical and control theoretic properties of Hamilton– Jacobi equations via Lax-Oleinik commutators","cited_arxiv_id":null,"evidence_quote":"Provides the weak-KAM cut-locus toolkit, including the $B_u$ function and the identity $T_u(\\mu)=\\inf\\tau_u(x)$ used to characterize cut points."},{"cited_title":"Optimal mass transportation and Mather theory","cited_arxiv_id":null,"evidence_quote":"Establishes displacement interpolation for Tonelli costs, identifying minimizers of $C^t$ with random Euler-Lagrange curves and thus grounding the dynamical cost setup."},{"cited_title":"Gradient flows in metric spaces and in the space of proba- bility measures","cited_arxiv_id":null,"evidence_quote":"Provides the Wasserstein gradient-flow toolbox: localized Frechet differentials, semiconcavity of functionals, and the minimizing-movement/EDI construction used in Theorem 5.5."},{"cited_title":"Semiconcavity an d sensitivity analysis in mean-field optimal control and applications","cited_arxiv_id":null,"evidence_quote":"Sets up semiconcavity and localized differentials for functionals on compactly supported probability measures, the framework in which the singularity equivalence for potential energies is proved."}],"review_version":1}