{"id":"ab68f222-4511-46c7-aab5-6954c78f472e","arxiv_id":"2506.22085","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The most likely evolution of interacting particles between two densities is a Markov diffusion solving the hydrodynamic variational problem, with new current-constrained variants included.","lead":"This paper extends the Schrödinger problem, which asks for the most likely way a diffusing system moves between observed states, to interacting particle systems by working with the full path-level empirical measure instead of just density and current. It shows the optimal transition is a Markov diffusion driven by a gradient field, and it adds new versions in which the average particle current is fixed.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The main theorem is conditional on the QRV functional being the true empirical-measure LDP; outside SSEP d≥3 (and the two other cited models) that premise is asserted, not proved, so the general interacting-particle Schrödinger interpretation is not yet established.","rationale":"The reader's weakest_assumption is exactly the one I would stress: the QRV rate function is not proven for general interacting systems. My pass did not uncover an additional internal flaw in the proof of Theorem 4.2 itself: given F of the form (3.8), the inequality and lifting argument are correct, and the use of the canonical equations is stated as an external result [12]. The paper is candid about the missing general LDP, and the conditional verdict is therefore appropriate. I would keep the verdict at CONDITIONAL (no change) rather than reject, because the proof is structurally sound on its own terms and the unproved premise is explicitly flagged in Section 3. The one additional emphasis is that Section 5's Theorem 5.3 and Lemma 5.4 are also stated without proofs; however those are secondary to the central Theorem 4.2 and would not by themselves change the main verdict.","tokens_in":24409,"tokens_out":6144,"duration_ms":71297,"concrete_test":"Verify the process-level LDP behind Lemma 3.1 for the zero-range process by checking whether [16] (or its methods) proves a full empirical-measure LDP with rate function Ent(R|Q(ρ)) as in (3.10), rather than only a hydrodynamic density/current LDP. If the full LDP is not proven, attempt the same check for the partial-reflection Brownian case of [36] and for short-range interacting Brownians; any model whose rate function differs from (3.8) disproves the 'general validity' claim and forces Theorem 4.2 to be restricted to SSEP d≥3.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is Theorem 4.2: equality of HSP and MSP values, Markovianity of MSP minimizers, and identification of the minimizer as the diffusion (4.11). Within the paper, the equality argument is clean: from (3.8), F(R)=I((R_t),J_R)+Ent(R|P(R))≥I, so VT≥VT, and any HSP minimizer can be lifted to a Markovian R with the same I, giving equality. The gap is upstream: the assertion that F in (3.8) is the large-deviation rate function of the empirical measure R^ℓ is proved in [32] only for the symmetric simple exclusion process in d≥3, with partial-reflection Brownians [36] and zero-range [16] as additional cases. The paper states \"should however be of general validity\" but gives no proof for general stochastic lattice gases or interacting Brownians. If a proposed model's empirical-measure LDP has a different rate function, then the MSP infimum V_T does not describe the most probable constrained path of that particle system: the equality V_T=V_T would be an identity about an artificial functional, and the characterization of the optimal measure as the diffusion (4.11) would not be the physical minimizer. Since the paper's advertised scope is general interacting systems, this unproved transfer is load-bearing.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper formulates Schrödinger problems for interacting particle systems in the hydrodynamic scaling limit, replacing the independent-particle relative-entropy cost with the Quastel–Rezakhanlou–Varadhan (QRV) rate function F in Eq. (3.8) for the empirical measure. It introduces the hydrodynamic Schrödinger problem (HSP) for the density/current rate function I and the measure Schrödinger problem (MSP) for F, both with prescribed initial and final densities. Theorem 4.2 claims equality of the two optimal values, characterizes the MSP minimizer as the law of a time-inhomogeneous Markov diffusion with generator (4.11), and proves conversely that every MSP minimizer is Markovian with marginals solving the hydrodynamic problem. The paper also introduces current-constrained versions (MSPC and MSPDC) and states analogous results in Theorem 5.3 and Lemma 5.4, together with a time-reversal / Born-formula interpretation. Special cases are worked out for zero-range processes and mean-field interacting Brownians.","tokens_in":24559,"tokens_out":12306,"duration_ms":136664,"significance":"If the results hold, they give a clean structural reduction: under endpoint constraints on hydrodynamic observables, the path-measure Schrödinger problem for interacting particles collapses to the hydrodynamic variational problem, and non-Markovian empirical measures are never optimal. The proof of Theorem 4.2 is short and, conditional on its inputs, internally coherent; no parameters are fitted, and the minimizer is given by an explicit generator. The paper also connects the results to the Hamilton structure of macroscopic fluctuation theory and to integrable-system transformations, which is valuable. The caveat is that the advertised scope is considerably wider than the class of models for which the QRV functional is known to be the true large-deviation rate function; the significance is therefore conditional on an unproved premise.","major_comments":[{"comment":"The functional F is introduced as the large-deviation rate function for the empirical measure R^ℓ, but the text immediately notes that the proof is carried out only for the symmetric simple exclusion process in d≥3 ([32]), with partial-reflection Brownians ([36]) and zero-range ([16]) as further cases, and that the result 'should however be of general validity.' Since F in (3.8) is the objective of the MSP in Definition 4.1, all results in Sections 4 and 5 (Theorem 4.2, Corollary 4.3, Theorem 5.3) are conditional on an unproved LDP for general stochastic lattice gases or interacting Brownians. If a given model has a different rate function, the equality V_T=V_T and the identification of R* as the most probable constrained path are statements about an artificial functional rather than about the particle system. The paper should either prove the LDP for the claimed generality, restrict the statements to the models where the LDP is known, or explicitly formulate the results as conditional on this assumption.","section":"Section 3, Eq. (3.8)"},{"comment":"Both statements are announced without proof, with the sentence 'achieved by the same arguments in Theorem 4.2 and it is therefore omitted.' This is not sufficient for the advertised current-constrained Schrödinger problems: the constraint (5.1) is a time-average constraint rather than an endpoint condition, and the relevant Hamilton equations (5.12) are vector-valued and allow non-gradient momenta, so the reduction is not literally identical to the proof of Theorem 4.2. The omitted arguments are load-bearing for the claims Φ_T=φ_T and Ψ_T=ψ_T and for the characterization of the minimizer; the paper should include complete proofs or precise statements of the reduction.","section":"Section 5, Theorem 5.3 and Lemma 5.4"},{"comment":"The converse half of Theorem 4.2 and the identification E* = ∇H* rest on the assertion that any minimizer of (HSP') satisfies the canonical equations (4.3). The paper cites [12] for this fact, but [12] is described in its own title as a heuristic point of view. If the canonical-equations characterization is not proved rigorously there, then the paper should prove it or state it as an explicit assumption; as written, the structure of the MSP minimizer is not fully established.","section":"Section 4, Theorem 4.2 (converse) and Eq. (4.3)"}],"minor_comments":[{"comment":"The sentence 'V_T ≥ V_T with equality if and only if R is equal to P(R)' is imprecise: equality for a given R requires both Ent(R|P(R))=0 and I((R_t),J_R)=V_T, not just Markovianity.","section":"Section 4, proof of Theorem 4.2"},{"comment":"The role of I_in is not specified for arbitrary µ0; if the initial particle distribution is deterministic, I_in(µ0) is infinite off the prescribed density, and the statement that this term 'can be dropped' needs a precise convention for the initial sampling.","section":"Section 4, Definition 4.1"},{"comment":"The notation for the two values V_T and V_T is visually almost identical in the text; please use clearly distinct symbols for the hydrodynamic and measure Schrödinger problems.","section":"Section 4"},{"comment":"The acronym '(HSPCD)' appears in Theorem 5.3 while the definition uses '(HSPDC)'; please make the notation consistent throughout the section.","section":"Section 5, Theorem 5.3"},{"comment":"The unnumbered formula before (4.5) is referred to as 'Formula (4.5) is (3.10) with Q(ρ)=W(ρ0)', but the displayed equation (4.5) is F(R)=Ent(R|W(ρin)); please correct the cross-reference.","section":"Section 4, 'Independent particles'"}],"recommendation":"major_revision","confidential_remarks":"The main reason for major revision is not a hidden error but a mismatch between the paper's advertised generality and its proven premises. The authors explicitly acknowledge the heuristic nature of several steps; if the journal is willing to publish conceptual, conditional results in macroscopic fluctuation theory, a clearly labeled conditional theorem would be acceptable after revision. I would also ask the editor to consider whether the omitted proofs in Section 5 are consistent with the journal's standards for original research articles."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nQuick take: this is a genuine conceptual step forward. Bertini, Gabrielli, and Jona-Lasinio formulate the Schrödinger problem at the level of path measures for interacting diffusions in the hydrodynamic limit, using the QRV rate function for the empirical measure. The core result, Theorem 4.2, shows that for endpoint-density constraints the measure-level problem (MSP) has the same value as the hydrodynamic problem (HSP), and that the minimizer is a time-inhomogeneous Markov diffusion with the explicit generator (4.11). Given the inputs — the QRV functional F, the canonical equations from [12], and existence of minimizers — the proof is short and coherent. The converse direction, that any MSP minimizer is Markovian and projects to an HSP minimizer, is also sound. That is real progress: it connects the Lagrangian path-measure viewpoint to the established MFT action functional and gives a physical realization of the optimal path via a gradient external field.\n\nThe paper is honest about some of its own limits. Section 5 states Theorem 5.3 and Lemma 5.4 without proof, saying the arguments are 'the same'; the text even admits being 'cavalier on the technical details.' That is a fair referee concern, not a hidden flaw. More important is the scope of F itself. The QRV functional is proven to be the empirical-measure large-deviation rate function only for SSEP in d≥3, partial-reflection Brownians, and zero range. For general interacting systems it is asserted, with 'should however be of general validity.' The stress-test note is right on target: without that LDP, the equality VT = VT is an identity about an artificial functional, and the diffusion (4.11) need not be the physical most probable path. The paper flags this, but the advertised scope is general, so the flag should be louder.\n\nI do not think this is fatal. There are no fitted parameters or presupposed conclusions; the equality follows from definitions and a legitimate Markovian lift. The zero-range and mean-field checks support the universality claim as a conjecture. The fixes are straightforward: state Theorem 4.2 as conditional on the known models, add a precise conjecture for general systems, and either prove Theorem 5.3 or label it as a conjecture with the same status.\n\nWho is this for? Researchers in macroscopic fluctuation theory, large deviations for interacting particles, and the Schrödinger problem / entropic optimal transport. It deserves a serious referee. I would send it to review, with a request to tighten the scope and supply or flag the missing proofs.\n\nBest.","headline":"New measure-level Schrödinger formulation for interacting diffusions; the equality result is clean, but the advertised generality rests on an unproved large-deviations premise.","tokens_in":25239,"tokens_out":2680,"would_cite":true,"duration_ms":27842,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60F10","60K35","82C22"],"pacs":[],"model":"deepseek-v4-flash","headline":"For interacting particles, the Schrödinger problem reduces to a hydrodynamic one: the measure-level and density-level costs coincide, and the optimal path is an explicit Markov diffusion.","keywords":["Schrödinger problem","macroscopic fluctuation theory","large deviations","empirical measure","interacting particle systems","hydrodynamic limit","self-diffusion","current fluctuations"],"falsifier":"Compute the empirical-measure large-deviation rate function for the one-dimensional nearest-neighbor symmetric exclusion process, where $D_s=0$; if it is not of the QRV form or the relative-entropy term degenerates, then the equality in Theorem 4.2 and the Markovian-diffusion characterization of minimizers fail for that model.","tokens_in":2047,"feed_emoji":"🎲","tokens_out":2702,"duration_ms":91884,"temperature":0.7,"pith_summary":"This paper asks what is the most probable way an interacting many-particle system in the hydrodynamic scaling limit passes from an initial density to a final density—the Schrödinger problem generalized from independent diffusions to interacting particles. The authors formulate it as a minimization over path measures of the large-deviation rate function for the empirical measure, which records the behavior of individual particle paths and not just the macroscopic density and current. Their main result is that for density constraints this measure-level problem has the same minimal cost as the hydrodynamic problem, and every minimizer is a Markovian path measure: the law of an explicit time-inhomogeneous diffusion whose drift is produced by a gradient external field. This gives a pathwise, Lagrangian picture of macroscopic fluctuations and extends the classical dual-potential and Born-formula structure to interacting systems. The same reduction is proved for versions with constraints on expected current.","feed_headline":"Interacting Schrödinger problem solved by a Markov diffusion","feed_subtitle":"In the hydrodynamic limit, measure-level and density-level costs coincide, and the minimizer is explicit.","key_machinery":"The engine is the Quastel–Rezakhanlou–Varadhan functional $F(R)=I\\big((R_t)_{t\\in[0,T]},J^R\\big)+\\mathrm{Ent}\\big(R\\,\\big|\\,P(R)\\big)$ on the space of empirical path measures. Here $I$ is the hydrodynamic large-deviation rate for density and current, $J^R$ is the expected stochastic current of $R$, and $P(R)$ is the unique Markov diffusion with the same single-time marginals and expected current as $R$ and with diffusion coefficient given by the self-diffusion $D_s(\\rho)$; the relative-entropy term charges exactly the non-Markovian part of $R$. Because $F(R)\\ge I$ with equality only for $R=P(R)$, minimizing $F$ under marginals constraints forces the optimizer to be Markovian, and the proof reduces to showing that the optimal hydrodynamic current has the form $-D_h(\\rho^*)\\nabla\\rho^*+2\\sigma(\\rho^*)\\nabla H^*$. For zero-range and mean-field models $F$ collapses to $\\mathrm{Ent}(R\\,|\\,Q(\\rho))$ plus the initial cost, which yields explicit formulas such as Corollary 4.3.","core_discovery":"In Theorem 4.2 the paper proves that for every $T>0$ and absolutely continuous measures $\\mu_0,\\mu_1$ of equal mass, the hydrodynamic Schrödinger problem (HSP) and the measure Schrödinger problem (MSP) have the same value: the infimum of $I$ over density–current paths equals the infimum of the empirical-measure rate functional $F$ over path measures with those marginals. Moreover, if $\\pi^*$ minimizes the hydrodynamic problem with density $\\rho^*$ and conjugated momentum $H^*$, then a minimizer of the measure problem is $R^*$, the law of the time-inhomogeneous Markov diffusion with generator (4.11), with $E^*=\\nabla H^*$; conversely, any minimizer of (MSP) is Markovian, equals $P(R^*)$, its single-time marginals minimize the hydrodynamic problem (HSP'), and its external field is a gradient. For current-constrained versions, Theorem 5.3 gives the same structure with a generally non-gradient field $E^*$.","pith_inferences":["If the QRV functional is as general as the paper expects, Theorem 4.2 should extend verbatim to all systems with an empirical-measure large-deviation principle, including non-gradient models and higher dimensions; the restriction to $d\\ge3$ appears technical.","The gradient form of the optimal field suggests a measurable signature: in a fluctuating experiment, the most likely path to a rare density profile is accompanied by a conservative drift, so the vorticity of the empirical velocity field vanishes at the optimizer.","The connection with AKNS integrable systems (for exclusion/inclusion and KMP processes) implies that explicit solutions of the interacting Schrödinger problem can be obtained by inverse scattering, giving quantitative predictions for optimal fluctuation paths and their time-reversed duals; this is an extension the paper only sketches.","For current-constrained versions, the non-gradient momentum $B$ can rotate the drift, so the time-averaged current optimum may be realized by a solenoidal external field, opening a route to characterize non-reversible optimal measures beyond the density-constrained case."],"forward_implications":["For any model where $F$ is the empirical-measure rate function, the measure-level and hydrodynamic Schrödinger costs coincide for density constraints and for current constraints, so the simpler hydrodynamic variational problem gives the exact large-deviation cost.","The optimal path measure is a time-inhomogeneous Markov diffusion with generator (4.11); for density constraints it is realized by the microscopic dynamics with a weak time-dependent external field $2\\nabla H^*$, so optimal fluctuations are caused by a gradient field.","Any minimizer of the measure Schrödinger problem is Markovian, and its single-time marginals solve the canonical equations (4.3); the dual potentials of [12] acquire the pathwise meaning of forward and backward momenta linked by $H^*_t+\\hat H_t=f'(\\rho^*_t)$.","For zero-range processes and mean-field Brownians the problem becomes the relative-entropy minimization $\\mathrm{Ent}(R\\,|\\,Q(\\rho))$, which implies explicit Benamou–Brenier-type formulas like Corollary 4.3 and the mean-field analogue.","The same Markovian reduction holds for the current-constrained problems (Theorem 5.3), so the analysis of time-averaged current large deviations can be carried out with the same machinery, including long-time dynamical phase transitions."],"supporting_citations":[{"why":"Supplies the large-deviation rate function F for the empirical measure in the hydrodynamic limit (symmetric simple exclusion, d≥3), the functional on which all MSP results rest.","marker":"[32]"},{"why":"Provides the macroscopic-fluctuation-theory framework: hydrodynamic rate functional I for density and current, the Einstein relation, and the Hamiltonian structure used in the canonical equations.","marker":"[8]"},{"why":"Introduced the hydrodynamic Schrödinger problem for lattice gases (the objective called HSP here) whose solutions the paper extends to the measure level.","marker":"[12]"},{"why":"Gives the standard Schrödinger problem for independent particles, dual potentials, and the Benamou–Brenier formula that the interacting results generalize.","marker":"[25]"},{"why":"Provides the rate function for mean-field interacting Brownians, used to show the QRV recipe reproduces the known mean-field result.","marker":"[2]"},{"why":"Establishes large deviations of the empirical current and the long-time behavior used for the current-constrained problems (HSPC).","marker":"[6]"},{"why":"Defines the diffusion semigroups and generators used to construct the Markovian measure P(R) with given marginals and current.","marker":"[33]"}],"fun_headline_variants":["Interacting Schrödinger problem: measure and density costs coincide","Markov diffusion solves interacting Schrödinger problem","Hydrodynamic limit unifies Schrödinger problems","Measure-level Schrödinger minimizer is Markovian","Current constraints yield non-gradient Schrödinger field"],"cache_read_input_tokens":27136,"weakest_assumption_plain":"The entire argument rests on $F$ being the true large-deviation rate function for the empirical measure of the interacting system, which is proven for the symmetric simple exclusion process in $d\\ge3$ and only expected to hold generally.","fun_headline_variants_meta":{"raw":{"variants":["Interacting Schrödinger problem: measure and density costs coincide","Markov diffusion solves interacting Schrödinger problem","Hydrodynamic limit unifies Schrödinger problems","Measure-level Schrödinger minimizer is Markovian","Current constraints yield non-gradient Schrödinger field"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000673,"raw_usage":{"total_tokens":3012,"prompt_tokens":844,"completion_tokens":2168,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":460,"completion_tokens_details":{"reasoning_tokens":2098}},"tokens_in":460,"tokens_out":2168,"duration_ms":16355,"temperature":1.0,"reasoning_tokens":2098,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T22:12:14.868962+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the empirical-measure large-deviation rate function for the one-dimensional nearest-neighbor symmetric exclusion process, where $D_s=0$; if it is not of the QRV form or the relative-entropy term degenerates, then the equality in Theorem 4.2 and the Markovian-diffusion characterization of minimizers fail for that model.","supporting_citations":[{"cited_title":"Quastel, F","cited_arxiv_id":null,"evidence_quote":"Supplies the large-deviation rate function F for the empirical measure in the hydrodynamic limit (symmetric simple exclusion, d≥3), the functional on which all MSP results rest."},{"cited_title":"Bertini, A","cited_arxiv_id":null,"evidence_quote":"Provides the macroscopic-fluctuation-theory framework: hydrodynamic rate functional I for density and current, the Einstein relation, and the Hamiltonian structure used in the canonical equations."},{"cited_title":"Chiarini, G","cited_arxiv_id":null,"evidence_quote":"Introduced the hydrodynamic Schrödinger problem for lattice gases (the objective called HSP here) whose solutions the paper extends to the measure level."},{"cited_title":"L´ eonard;A survey of the Schr¨ odinger problem and some of its connections with optimal transport","cited_arxiv_id":null,"evidence_quote":"Gives the standard Schrödinger problem for independent particles, dual potentials, and the Benamou–Brenier formula that the interacting results generalize."},{"cited_title":"Backhoff, G","cited_arxiv_id":null,"evidence_quote":"Provides the rate function for mean-field interacting Brownians, used to show the QRV recipe reproduces the known mean-field result."},{"cited_title":"Bertini, A","cited_arxiv_id":null,"evidence_quote":"Establishes large deviations of the empirical current and the long-time behavior used for the current-constrained problems (HSPC)."},{"cited_title":"Quastel, S.R.S.Varadhan; Diffusion semigroups and diffusion processes correspond- ing to degenerate divergence form operators","cited_arxiv_id":null,"evidence_quote":"Defines the diffusion semigroups and generators used to construct the Markovian measure P(R) with given marginals and current."}],"review_version":1}