{"id":"43fb12d5-9f16-4293-b23f-6314d1cf75da","arxiv_id":"2411.14936","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Free massive particle systems, singular-drift Dean-Kawasaki equations, Wasserstein diffusions, and metric-measure Brownian motions are identified as a single process for any ultracontractive reversible diffusion.","lead":"Four different mathematical descriptions of random clouds of particles, infinite massive-particle systems, Dean-Kawasaki equations, Wasserstein diffusions, and metric-measure Brownian motions, are shown to describe the same process. The result holds for very general ambient spaces and any diffusion with a well-behaved heat kernel, not just the Laplacian.","discovery_kind":"unification","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1.7 overstates uniqueness: Corollary 4.8 only covers initial laws equivalent to Qπ with π∈P(T∘), not finite equal-mass atomic data; finite-n uniqueness for general ultracontractive L rests on [102] and its Bakry–Émery assumption.","rationale":"The reader's weakest_assumption (qpp) is indeed a key condition, and its failure in dimension one is honestly discussed in the paper. However, the more load-bearing gap for the central uniqueness claim concerns the class of initial data. Corollary 4.8, the paper's main uniqueness result, only applies to Qπ-substationary processes whose initial law is equivalent to Qπ and whose mass distribution lies in T∘. Finite equal-mass initial data are singular with respect to Qπ and have zero mass tails, so they fall outside this theorem. The finite-particle uniqueness in Theorem 1.7 is therefore not a consequence of the Dirichlet-form machinery developed in §§3–4; it must be imported from [102], which carries a Bakry–Émery curvature assumption that the paper does not impose on L. This is an internal gap rather than a disagreement with consensus: if the construction in Proposition 3.10 cannot be extended to zero-mass tails, the finite-n case is simply unproved at the advertised level of generality. Since the same verdict, CONDITIONAL, remains appropriate, I keep the reader's verdict unchanged but flag a sharper caveat than qpp alone. The formal square-root identification (4.11) also supports the reader's note that SPDE well-posedness is really martingale-problem well-posedness, but the initial-data gap is the more concrete and load-bearing issue.","tokens_in":68004,"tokens_out":7881,"duration_ms":88368,"concrete_test":"Check whether Proposition 3.10's essential self-adjointness argument can be rerun for s=(1/n,...,1/n,0,0,...): try to construct, for each k≤n, a separator ϕ_k∈R0 with ϕ_k(1/n)=1 and ϕ_k vanishing on {0} and on all other atom masses, despite R0 functions being supported away from 0. If this construction fails, verify whether [102, Thm. 2.2] applies to an ultracontractive recurrent diffusion satisfying Assumption 1.1 but not a Bakry–Émery lower bound; if it does not, Theorem 1.7's finite-n uniqueness is unproved and the theorem should be restricted to Qπ-equivalent initial laws or supplemented with the missing curvature assumption.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's own uniqueness theorem, Corollary 4.8, is confined to Qπ-substationary processes with initial law equivalent to Qπ and to mass distributions π∈P(T∘), i.e. infinitely many strictly positive masses. The second part of Theorem 1.7, and the abstract's claim of uniqueness for the Dean–Kawasaki SPDE with finite equal-mass initial data, therefore does not follow from the paper's Dirichlet-form argument. The finite-particle case is only covered by Theorem 4.2, quoted from [102], which requires a standard Markov triple with a Bakry–Émery Ricci-curvature lower bound, an assumption not present in Assumption 1.1 or Theorem 1.7. Lemma 4.5 establishes only the forward implication from the new shadow martingale problem (4.4) to [102]'s problem (mp)^n_L; the converse is deferred to Theorem 4.7, which does not apply to singular finite-atomic initial laws. Thus the advertised 'unique for every diffusive recurrent Markov generator with spectral gap and ultracontractive semigroup' is not established for the finite-particle case; the statement needs either a matching curvature hypothesis, an extension of the T∘ machinery to finite zero-mass tails, or a separate proof. Additionally, the identification of (1.10) with the shadow martingale problem remains formal via the square-root rule (4.11), so the SPDE statement inherits that heuristic step.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper constructs Dirichlet forms for infinite systems of independent massive particles on locally compact Polish spaces, transfers them to the space of probability measures via an empirical-measure map, and identifies the resulting process with Wasserstein Brownian motions and metric-measure Brownian motions. It also connects the process to a singular-drift Dean–Kawasaki-type SPDE through a shadow martingale problem, and treats singular interactions by Girsanov transforms. The technical core proves closability, essential self-adjointness, quasi-regularity, proper association, and several characterizations of the generator, extending earlier work on Dirichlet–Ferguson diffusions and the Dean–Kawasaki equation.","tokens_in":68290,"tokens_out":7506,"duration_ms":77757,"significance":"If the stated results are taken at face value, the paper is a substantial unifying contribution: it replaces the Laplacian in the Dean–Kawasaki setting by a general diffusive recurrent generator with ultracontractive semigroup, gives explicit generators and martingale problems on Wasserstein space, and provides new Rademacher and Varadhan-type statements. The central technical achievements—essential self-adjointness on a cylinder-function algebra and uniqueness of the associated martingale problem—are proved rather than assumed, and there is no parameter fitting in the construction. The Girsanov treatment of singular interactions is also a genuine extension of earlier C²-regularity frameworks. However, the advertised well-posedness claims for the Dean–Kawasaki SPDE are broader than the theorems actually prove, and the SPDE itself is only interpreted through a formal square-root rule.","major_comments":[{"comment":"The paper never gives a rigorous construction of the SPDE (1.10): the term div(√μ_t ξ) is handled only through the shadow martingale problem (4.4), and the identification is explicitly based on the algebraic rule (√ρ_t f)^2 = ρ_t(f^2), which the paper itself labels formal and says cannot be given a rigorous meaning. Consequently, the claim that μ• is the unique analytically weak martingale solution to (1.10) is not supported by the proofs. The rigorous statement should be uniqueness for the shadow martingale problem (4.4), with (1.10) described as its formal counterpart, unless a genuine weak solution theory for (1.10) is added.","section":"§4.2.2, Eq. (4.11); Theorem 1.7"},{"comment":"Uniqueness for finite equal-mass atomic initial data is not proved in this paper. Corollary 4.8 only covers initial laws equivalent to Qπ with π∈P(T°), i.e. infinitely many strictly positive masses; a finite equal-mass law such as δ_{(1/n)∑δxi} is mutually singular with every such Qπ. Lemma 4.5 proves only the forward implication from the Konarovskyi–Lehmann–von Renesse problem (mp)_L^n to the new shadow problem (cmp)_L, while the converse in Theorem 4.7 requires an initial distribution equivalent to Qπ. The finite-particle uniqueness therefore rests on Theorem 4.2, quoted from [102], which requires a standard Markov triple with a Bakry–Émery Ricci-curvature lower bound, an assumption not present in Assumption 1.1. Thus the advertised uniqueness for every diffusive recurrent generator with spectral gap and ultracontractive semigroup is not established for the finite-particle case; the statement needs either the curvature hypothesis, an extension of the T° machinery to finite zero-mass tails, or a separate proof.","section":"§4.1.3, Corollary 4.8; Theorem 1.7, second paragraph"},{"comment":"The uniqueness that is actually proved is uniqueness 'up to Qπ-equivalence' for Qπ-substationary processes whose initial distribution is equivalent to Qπ, with the additional limitation π∈P(T°). This is weaker than uniqueness from an arbitrary deterministic starting point in P_pa, which is the natural reading of the theorem's wording 'the unique solution is the measure representation'. The distinction is material for the SPDE application, where one would expect at least uniqueness in law for every deterministic atomic initial measure. The statements in Theorem 1.7 and in the abstract should be qualified to match the precise uniqueness notion of Corollary 4.8.","section":"Corollary 4.8; §1.3.5"},{"comment":"The uniqueness argument for the shadow martingale problem uses Assumption 3.16 (qpp), i.e. cap_{1,1}(ΔM)=0, while Theorem 1.7 is stated under the qualitative Assumption 1.3 only. Since (qpp) is introduced as a sufficient condition for Assumption 1.3, the theorem should either state (qpp) explicitly or prove the implication; otherwise the uniqueness statement is one assumption short. The distinction is not cosmetic, since the one-dimensional examples in §1.3.4 show that non-uniqueness appears precisely when (qpp) fails.","section":"Theorem 1.7 vs Assumption 3.16"}],"minor_comments":[{"comment":"The line 'Let I := I ×∞' appears to contain a typo: the symbol I on the right cannot be the same set being defined; presumably one side should be the unit interval or a different symbol.","section":"§2.2.1"},{"comment":"The phrase 'existence and uniqueness of solutions' should be qualified as 'uniqueness in law up to Qπ-equivalence under Qπ-equivalent initial laws' to match Corollary 4.8; the current wording suggests a stronger pathwise or initial-value uniqueness that is not proved.","section":"Abstract"},{"comment":"The white-noise construction defines P0 as 'the law of the standard white noise on S′(R)', but P0 is never used afterwards; either remove it or clarify the role of the time component in the product measure P ⊗ P0.","section":"§4.2.2"},{"comment":"The notation Cπ(τ) is introduced with τ in Definition 5.8, but the text later writes τ(n), τ(s_i^{-1}), and 'some τ as above'; ensure that the function parameter and the variable of integration are consistently distinguished.","section":"§1.4.3 and Definition 5.8"}],"recommendation":"major_revision","confidential_remarks":"The paper contains substantial and valuable Dirichlet-form theory, and the central technical results appear sound. The main issue is that the Dean–Kawasaki well-posedness claims in the abstract and in Theorem 1.7 exceed what is proved: the SPDE identification is formal, and the finite equal-mass uniqueness is imported from [102] under a Bakry–Émery assumption not present in this paper's setting. A careful rewriting that states the shadow-martingale-problem result as the rigorous theorem and demotes the SPDE formulation to an interpretation would make the paper publishable. The overstatement should be corrected rather than defended, because it is exactly the advertised headline result."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Bottom line: this is a substantial, mostly correct piece of Dirichlet-form theory, but the uniqueness claim for the Dean–Kawasaki equation advertised in the abstract and Theorem 1.7 is wider than what the paper actually proves. For finite equal-mass initial data the advertised uniqueness is not established by the paper's own argument; it is inherited from [102], which needs a Bakry–Émery curvature condition absent from Assumption 1.1.\n\nThe genuinely new content is the systematic construction of the measure representation of free massive particle systems for an arbitrary ultracontractive reversible diffusion on a locally compact Polish space, plus the associated Dirichlet form, generator, and martingale problem. Essential self-adjointness on the cylinder core is proved, not assumed. That yields an explicit generator on P that resolves the Ren–Wang conjecture for P2(R^d), extends the Dirichlet–Ferguson diffusion beyond closed manifolds, and comes with Rademacher and Varadhan-type statements. The paper is honest about the formal nature of the SPDE identification: the square-root rule (4.11) is heuristic, and the well-posedness is strictly well-posedness of the shadow martingale problem. No parameters are fitted and the citation pattern is appropriate; the dependence on [46] is natural since this is a generalization of it.\n\nThe main soft spot is the overstatement I started with. Corollary 4.8, the paper's own uniqueness theorem, covers only initial laws equivalent to Qπ with π∈P(T∘), i.e. infinitely many strictly positive masses. Finite equal-mass initial data, the case highlighted in Theorem 1.7 and central to the rigidity literature, is handled only through Theorem 4.2 quoted from [102], under a Bakry–Émery Ricci-curvature bound. The paper's Dirichlet-form machinery does not deliver that case. So 'unique for every diffusive recurrent Markov generator with spectral gap and ultracontractive semigroup' is too strong as stated for finite particles. This is a load-bearing overstatement, not a cosmetic one: a reader wanting to use Theorem 1.7 for finite-particle DK on a general generator will not find a proof.\n\nThe non-collision assumption (qpp) excludes one-dimensional driving noises; the paper flags the corresponding non-uniqueness, so that is a stated limitation, not a hidden flaw. Several technical proofs are deferred (quasi-regularity of the Cheeger energy, some transfer arguments), but they look standard rather than circular.\n\nAudience: stochastic analysts working on measure-valued diffusions, Dirichlet forms, and DK-type SPDEs. It deserves a serious referee. The referee should ask for either a corrected Theorem 1.7 or a separate treatment of the finite-particle case; with that fixed, the core constructions should survive.","headline":"A substantial and mostly correct Dirichlet-form unification, but the advertised uniqueness for the Dean–Kawasaki equation with finite equal-mass data is wider than what the paper actually proves.","tokens_in":68810,"tokens_out":3133,"would_cite":true,"duration_ms":30796,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60G57","60H17","60J46","49Q22","70F45"],"pacs":[],"model":"deepseek-v4-flash","headline":"Empirical measures of random-mass particle systems uniquely solve the singular-drift Dean–Kawasaki equation and are Wasserstein Brownian motions.","keywords":["interacting particle systems","Wasserstein diffusions","measure-valued diffusions","marked point processes","Dean–Kawasaki equation","Dirichlet forms","Cheeger energy","singular SPDE"],"falsifier":"Construct two distinct analytically weak martingale solutions to the SPDE $d\\mu_t = \\operatorname{div}(\\sqrt{\\mu_t}\\,\\xi) + \\sum_{x\\in\\mu_t} \\mathsf{L}'\\delta_x\\,dt$ for some diffusion $\\mathsf{L}$ satisfying the non-collision capacity condition (qpp); the paper's Corollary 4.8 asserts uniqueness up to $Q_{\\pi}$-equivalence, so any such pair would refute the central claim. A complementary test: prove uniqueness for the same equation with one-dimensional Brownian motion on the real line, where (qpp) fails, which would show the boundary of the theorem lies elsewhere.","tokens_in":67794,"feed_emoji":"⚛️","tokens_out":15275,"duration_ms":123375,"temperature":0.7,"pith_summary":"This paper aims to establish that four objects studied in separate literatures are one and the same: infinite systems of independently diffusing particles with random masses, solutions of the Dean–Kawasaki equation with singular drift and space-time white noise, Wasserstein diffusions with purely atomic reversible measures, and metric measure Brownian motions arising from Cheeger energies on $L^2$-Wasserstein spaces. The central claim is that for any diffusive Markov generator $\\mathsf{L}$ with spectral gap and ultracontractive semigroup on a locally compact Polish space, the empirical measure process $\\mu_t = \\sum_i s_i \\delta_{X^i_t}$ of a free massive system is a reversible Markov diffusion, is the unique analytically weak martingale solution to the SPDE $d\\mu_t = \\operatorname{div}(\\sqrt{\\mu_t}\\,\\xi) + \\sum_{x\\in\\mu_t} \\mathsf{L}'\\delta_x\\,dt$, and is the Brownian motion of the Wasserstein geometry induced by $\\mathsf{L}$. The paper also derives the explicit generator of this process on the space of probability measures, identifies its Dirichlet form with the Cheeger energy when the ambient space is a weighted Riemannian manifold, and extends the construction to singular repulsive interactions of Riesz and logarithmic type via Girsanov transforms. A fair reader should care because this supplies well-posedness for a class of singular SPDEs previously accessible only through rigidity results, and provides an explicit Laplace operator on Wasserstein space for a large family of reversible measures.","feed_headline":"Massive particle systems are the Dean–Kawasaki equation","feed_subtitle":"Empirical measures of random-mass diffusions uniquely solve a singular SPDE and give Wasserstein space its Brownian motion.","key_machinery":"The central object is the measure representation map $\\operatorname{em}(s,x) = \\sum_i s_i \\delta_{x_i}$, which turns a mass-weighted configuration on a locally compact Polish space $M$ into a purely atomic probability measure; the transfer works through the weak atomic topology $\\tau_a$, which makes $\\operatorname{em}$ a homeomorphism on the off-diagonal configuration space. The load-bearing identity is the explicit generator (3.25), $\\widehat{\\mathsf{L}}(u)(\\eta) = \\int_M \\mathsf{L}^z|_{z=x} u(\\eta + \\eta\\{x\\}\\delta_z - \\eta\\{x\\}\\delta_x)/(\\eta\\{x\\})^2\\,d\\eta(x)$, with its square-field counterpart, expressing the infinitesimal motion of mass by displacing an atom at $x$ along the driving diffusion $\\mathsf{L}$. The quantitative non-collision assumption (qpp), $\\operatorname{cap}_{1,1}(\\Delta M) = 0$, guarantees that $\\operatorname{em}$ is a quasi-homeomorphism between the infinite-product Dirichlet form and the form on probability measures, transferring the Markov property and yielding essential self-adjointness of the generator, hence uniqueness of the martingale problem. On weighted Riemannian manifolds this form equals the Cheeger energy of $(P_2,W_2,Q_{\\pi,\\nu})$, identifying the associated diffusion as the Wasserstein Brownian motion.","core_discovery":"The core discovery is the identification of the measure representation of a free massive particle system with two previously distinct objects. Under the assumptions that the driving noise $W$ is a $\\nu$-reversible irreducible recurrent Hunt process with ultracontractive semigroup and that the diagonal of the ambient space has zero capacity for the product form (Assumption 3.16, (qpp)), the empirical measure process $\\mu_\\bullet = \\sum_i s_i \\delta_{X^i_\\bullet}$ is properly associated with an explicitly constructed quasi-regular Dirichlet form on the space of probability measures, is unique in law for its martingale problem, and is a diffusion precisely when $W$ is. When $M$ is a complete weighted Riemannian manifold and $W$ is the drifted Laplace–Beltrami diffusion, the form coincides with the Cheeger energy $\\operatorname{Ch}_{W_2,Q_{\\pi,\\nu}}$ of the $L^2$-Wasserstein space $(P_2,W_2)$, its generator is the operator (1.5)/(3.25) that moves mass by $\\mathsf{L}$ at each atom weighted by local mass proportions, the Rademacher theorem holds for $W_2$-Lipschitz functions, and the semigroup satisfies a one-sided Varadhan short-time estimate. Finally, under the same assumptions, $\\mu_\\bullet$ is the unique analytically weak martingale solution of the SPDE $d\\mu_t = \\operatorname{div}(\\sqrt{\\mu_t}\\,\\xi) + \\sum_{x\\in\\mu_t} \\mathsf{L}'\\delta_x\\,dt$, which for equal-mass finite systems reduces to the $\\mathsf{L}$-driven Dean–Kawasaki equation. The paper therefore establishes that free massive particle systems, singular-drift Dean–Kawasaki solutions, Wasserstein diffusions with atomic reversible measures, and metric measure Brownian motions are the same object.","pith_inferences":["Beyond the paper: because the singular drift acts as a boundary term forcing atomicity, any truncated or colored-noise approximation of the white noise will produce a different effective drift; this may explain the gap between regularized Dean–Kawasaki models and the rigid white-noise result.","Beyond the paper: the same construction should extend to jump noises once a rigorous Hilbert-module divergence is available, with an $\\alpha$-stable driver as the natural test case; the paper leaves this open.","Beyond the paper: the non-ergodicity when masses are random suggests that sharp Varadhan-type upper bounds should be sought on each ergodic component (fixed mass sequence), where the invariant measure is a product of delta masses.","Beyond the paper: the capacity dichotomy (qpp vs npp) likely governs well-posedness of other measure-valued SPDEs with singular drift, not only Dean–Kawasaki equations."],"forward_implications":["The singular-drift Dean–Kawasaki SPDE is well-posed in the sense of weak existence and uniqueness for every diffusive recurrent generator $\\mathsf{L}$ with spectral gap and ultracontractive semigroup on a locally compact Polish space; its unique solution is the empirical measure of the free massive system driven by $\\mathsf{L}$.","On every complete weighted Riemannian manifold satisfying the assumptions, the measure $Q_{\\pi,\\nu}$ on $(P_2,W_2)$ admits a reversible Brownian motion with explicit generator, and its Dirichlet form is the Cheeger energy; this includes the Dirichlet–Ferguson diffusion and the conjectured Laplace operator on $P_2(\\mathbb{R}^d)$ for Gaussian weights.","The Rademacher theorem holds for $(P_2,W_2,Q_{\\pi,\\nu})$: every $W_2$-Lipschitz function is Fréchet differentiable $Q_{\\pi,\\nu}$-almost everywhere, with its Otto gradient norm bounded by its Lipschitz constant.","The heat semigroup of this Wasserstein Brownian motion obeys a one-sided integral Varadhan short-time estimate; when masses are random the opposite inequality fails because the process is not ergodic.","Interacting massive systems with repulsive singular pair potentials (Riesz and logarithmic type, and more general Sobolev-class weights) yield unique solutions of the Dean–Kawasaki-type SPDE with interaction drift, via Girsanov transforms that need only local Sobolev regularity of the weight."],"supporting_citations":[{"why":"supplies the absolute-continuity result for infinite-product heat-kernel measures on compact spaces that is extended here to non-compact spaces.","marker":"[13]"},{"why":"provides the canonical infinite-product Dirichlet form construction that this paper generalizes to non-local, non-smooth, and non-compact settings.","marker":"[19]"},{"why":"establishes the Dirichlet–Ferguson diffusion on closed manifolds, the base case that Theorem 1.6 extends to weighted manifolds.","marker":"[46]"},{"why":"introduces the weak atomic topology used to make the empirical-measure map a homeomorphism and to define the natural topology for solutions.","marker":"[65]"},{"why":"proves universal infinitesimal Hilbertianity of Wasserstein space, which underpins the identification of the form with the Cheeger energy.","marker":"[71]"},{"why":"formulates the shadow martingale problem for the free Dean–Kawasaki equation and proves its rigidity for equal-mass finite systems; this is the martingale problem the paper extends.","marker":"[102]"},{"why":"extends rigidity to smooth interactions and supplies the comparison baseline for the Girsanov approach, which here needs only Sobolev regularity.","marker":"[103]"},{"why":"constructs the one-dimensional coalescing-fragmentating Wasserstein dynamics and shows non-uniqueness where the non-collision assumption fails; this marks the boundary of the uniqueness theorem.","marker":"[106]"},{"why":"supplies the standard theory of Dirichlet forms and Hunt processes used throughout to associate the form with a Markov process.","marker":"[117]"},{"why":"introduces the Wasserstein diffusion on $P_2(S^1)$ and the entropic reference measure, the prototypical instance of the geometric Brownian motion.","marker":"[145]"}],"fun_headline_variants":["Massive particles unify Dean-Kawasaki and Wasserstein Brownian motion","Particle systems are Dean-Kawasaki solutions and Wasserstein Brownian motion","Massive particle systems equal Dean-Kawasaki and Wasserstein diffusions","One process: massive particles, Dean-Kawasaki, Wasserstein Brownian motion"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument assumes that pairs of independently diffusing particles never meet, guaranteed by the quantitative condition that the diagonal of the ambient space has zero capacity for the two-particle product Dirichlet form (Assumption 3.16); this condition fails for one-dimensional driving noises, and that is exactly where uniqueness of the Dean–Kawasaki martingale problem is known to break down.","fun_headline_variants_meta":{"raw":{"variants":["Massive particles unify Dean-Kawasaki and Wasserstein Brownian motion","Particle systems are Dean-Kawasaki solutions and Wasserstein Brownian motion","Massive particle systems equal Dean-Kawasaki and Wasserstein diffusions","One process: massive particles, Dean-Kawasaki, Wasserstein Brownian motion"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000789,"raw_usage":{"total_tokens":3607,"prompt_tokens":1205,"completion_tokens":2402,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":821,"completion_tokens_details":{"reasoning_tokens":2328}},"tokens_in":821,"tokens_out":2402,"duration_ms":18270,"temperature":1.0,"reasoning_tokens":2328,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T14:42:29.837366+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Construct two distinct analytically weak martingale solutions to the SPDE $d\\mu_t = \\operatorname{div}(\\sqrt{\\mu_t}\\,\\xi) + \\sum_{x\\in\\mu_t} \\mathsf{L}'\\delta_x\\,dt$ for some diffusion $\\mathsf{L}$ satisfying the non-collision capacity condition (qpp); the paper's Corollary 4.8 asserts uniqueness up to $Q_{\\pi}$-equivalence, so any such pair would refute the central claim. A complementary test: prove uniqueness for the same equation with one-dimensional Brownian motion on the real line, where (qpp) fails, which would show the boundary of the theorem lies elsewhere.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"introduces the weak atomic topology used to make the empirical-measure map a homeomorphism and to define the natural topology for solutions."}],"review_version":1}