Pith. sign in

REVIEW 3 major objections 5 minor 39 references

Macroscopic fluctuation theory from a Lagrangian viewpoint and the Schr\"odinger problem

T0 review · 3 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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

Pith's one-line read 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.

desk verdict New measure-level Schrödinger formulation for interacting diffusions; the equality result is clean, but the advertised generality rests on an unproved large-deviations premise. read the letter →

arxiv 2506.22085 v1 pith:4SLMFZV4 submitted 2025-06-27 math.PR cond-mat.stat-mech

L. Bertini , D. Gabrielli , G. Jona-Lasinio This is my paper · ORCID
classification math.PRcond-mat.stat-mech MSC 60F1060K3582C22
keywords Schrödingerproblemmacroscopicfluctuationtheorylargedeviationsempiricalmeasureinteractingparticlesystemshydrodynamiclimitself-diffusioncurrentfluctuations
topics The Measurement Problem
open problems The Measurement Problem
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

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.

What carries the argument

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.

What would settle it

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.

Watch

Extended reading notes

Core claim

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 $E^*$.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

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

  • 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.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

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.

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 (3)
  1. [Section 3, Eq. (3.8)] 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.
  2. [Section 5, Theorem 5.3 and Lemma 5.4] 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.
  3. [Section 4, Theorem 4.2 (converse) and Eq. (4.3)] 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.
minor comments (5)
  1. [Section 4, proof of Theorem 4.2] 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.
  2. [Section 4, Definition 4.1] 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.
  3. [Section 4] 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.
  4. [Section 5, Theorem 5.3] The acronym '(HSPCD)' appears in Theorem 5.3 while the definition uses '(HSPDC)'; please make the notation consistent throughout the section.
  5. [Section 4, 'Independent particles'] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the HSP/MSP equality is established by a genuine two-sided variational argument, and the QRV rate-function premise is an explicit external, though conditional, input.

full rationale

The paper's central identity, Theorem 4.2, is not circular. From the definition (3.8), F(R) = I((R_t), J_R) + Ent(R|P(R)) >= I((R_t), J_R) >= V_T, giving V_T >= V_T; the reverse inequality is obtained by lifting a minimizer of (HSP') to the Markovian measure P(R*) with the same single-time marginals and expected current, for which the entropy term vanishes, so F(P(R*)) = I(...) = V_T. This is a genuine two-sided variational argument rather than an equality by definition: the existence and uniqueness of the diffusion P(R) with prescribed marginals, current, and diffusion coefficient D_s(rho) is a nontrivial ingredient, argued in Section 2 and supported by the external large-deviation results of [32] and related papers. No parameter is fitted and no target result is used as an input. The main caveat is upstream: the identification of F in (3.8) as the empirical-measure large-deviation rate function is proved in [32] only for the symmetric simple exclusion process in d>=3, with the zero-range and partial-reflection Brownian cases supported by [16] and [36], and the paper explicitly states the general validity assertion ('should however be of general validity', Section 3). That is a limitation on the scope of the theorem, not a circular step: if the premise fails for some model, the MSP value and minimizer characterization do not describe the physical empirical measure, but the variational equivalence V_T = V_T remains a theorem about the functionals as defined. Self-citations such as [8] and [16] provide context or published mathematical results and are not the load-bearing reduction; no quoted step reduces to an unverified self-citation chain. Hence the circularity score is 0.

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

No fitted parameters appear in this paper: the transport coefficients σ, Dh, Ds are inputs inherited from the microscopic models, and the conjugate fields H*, E* are variational multipliers rather than free parameters. The central claim rests on imported large-deviation theorems and on the canonical-equation characterization from [12]. The auxiliary measures P(R) and Q(ρ) are constructed objects, not new physical entities, and no independent empirical evidence is claimed for them.

assumptions (7)
  • domain assumption The QRV functional F(R) in (3.8) is the large deviation rate function for the empirical measure in the hydrodynamic scaling limit for general interacting particle systems.
    Assumed when defining (MSP), (MSPC), and (MSPDC). The proof in [32] is only for the symmetric simple exclusion process in d≥3; the paper states the statement should be of general validity (Section 3, 'Large deviations for the empirical measure').
  • domain assumption The hydrodynamic rate function I(π,J) in (3.2)-(3.3), with mobility σ and diffusion Dh satisfying the Einstein relation (3.1), is valid for the models considered.
    This is the standard Macroscopic Fluctuation Theory input from [8] and related work; used throughout Sections 4 and 5 and in the construction of the field E from (ρ,j).
  • domain assumption Minimizers of (HSP') satisfy the canonical equations (4.3) with the Hamiltonian (4.2), as established in [12].
    Invoked in Theorem 4.2 to construct the Markovian minimizer R*; the paper explicitly refers to [12] for the proof of this statement.
  • domain assumption For a finite-cost path measure R, there is a unique time-inhomogeneous diffusion P(R) with the single-time marginals of R, expected current jR, and diffusion coefficient Ds(ρ_t); its generator is (3.7).
    This uniqueness underpins the entropy term Ent(R|P(R)) in F and the Markovian lift used in Theorem 4.2; it is argued from the discussion after (2.12), not proven for general R.
  • standard math The time-reversal of a diffusion is given by the Haussmann-Pardoux theory [18].
    Used in Lemma 4.4 and Lemma 5.4 to derive the backward generators (4.13) and (5.16).
  • domain assumption For mean-field interacting Brownians, the large deviation rate function for the empirical measure is the one from [2]; Lemma 3.2 shows the QRV recipe reproduces it.
    The paper uses [2] as the external benchmark for the mean-field case rather than proving the LDP itself.
  • domain assumption The variational problems (HSP'), (HSPC), and (HSPDC) admit minimizers; transport coefficients are smooth and the free energy is strictly convex; non-uniqueness is allowed.
    Stated in Section 4 before Definition 4.1; needed for the minimizer characterizations and the canonical equations to be well-posed.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Macroscopic fluctuation theory from a Lagrangian viewpoint and the Schr\"odinger problem." pith.science (2026). https://pith.science/paper/4SLMFZV4

@misc{pith2026250622085,
  author       = {Pith},
  title        = {Pith review of: Macroscopic fluctuation theory from a Lagrangian viewpoint and the Schr\"odinger problem},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4SLMFZV4}},
  note         = {Machine review of arXiv:2506.22085}
}
read the original abstract

We formulate the Schr\"odinger problem for interacting particle systems in the hydrodynamical regime thus extending the standard setting of independent particles. This involves the large deviations rate function for the empirical measure which is in fact a richer observable than the hydrodynamic observables density and current. In the case in which the constraints are the initial and final density, we characterize the optimal measure for the Schr\"odinger problem. We also introduce versions of the Schr\"odinger problem in which the constraints are related to the current and analyze the corresponding optimal measures.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

39 extracted references · 39 canonical work pages

  1. [12]

    Chiarini, G

    A. Chiarini, G. Conforti, L. Tamanini; Schr¨ odinger problem for lattice gases: a heuris- tic point of view. In ‘Geometric science of information’, 891–899. Lecture Notes in Comput. Sci. 12829. Springer 2021

    work page 2021

  2. [32]

    Quastel, F

    J. Quastel, F. Rezakhanlou, S.R.S.Varadhan; Large deviations for the symmetric sim- ple exclusion process in dimensions d ≥ 3. Probab. Theory Related Fields 113, 1–84 (1999)

    work page 1999

  3. [36]

    Seo; Large-deviation principle for interacting Brownian motions

    I. Seo; Large-deviation principle for interacting Brownian motions. Commun. Pure Appl. Math. 70, 203–288 (2017)

    work page 2017

  4. [16]

    Gabrielli, G

    D. Gabrielli, G. Jona-Lasinio, C. Landim; Onsager reciprocity relations without mi- croscopic reversibility. Phys. Rev. Lett. 77, 1202–1205 (1996)

    work page 1996

  5. [1]

    Arratia; The motion of a tagged particle in the simple symmetric exclusion system on Z

    R. Arratia; The motion of a tagged particle in the simple symmetric exclusion system on Z. Ann. Probab. 11, 362–373 (1983)

    work page 1983

  6. [2]

    Backhoff, G

    J. Backhoff, G. Conforti, I. Gentil, C. L´ eonard; The mean field Schr¨ odinger prob- lem: ergodic behavior, entropy estimates and functional inequalities. Probab. Theory Related Fields 178, 475–530 (2020)

    work page 2020

  7. [3]

    Bodineau, B

    T. Bodineau, B. Derrida; Current fluctuations in nonequilibrium diffusive systems: an additivity principle . Phys. Rev. Lett. 92, 180601 (2004)

    work page 2004

  8. [4]

    Bodineau, B

    T. Bodineau, B. Derrida; Distribution of current in nonequilibrium diffusive systems and phase transitions. Phys. Rev. E 72, 066110 (2005)

    work page 2005

Show all 39 references
  1. [5]

    Bertini, A

    L. Bertini, A. De Sole, D. Gabrielli, G. Jona Lasinio, C. Landim; Non equilibrium current fluctuations in stochastic lattice gases J. Stat. Phys. 123, 237–276 (2006)

    2006

  2. [6]

    Bertini, A

    L. Bertini, A. De Sole, D. Gabrielli, G. Jona Lasinio, C. Landim; Large deviations of the empirical current in interacting particle systems. Theory Probab. Appl. 51, 2–27 (2007)

    2007

  3. [7]

    Bertini, A

    L. Bertini, A. De Sole, D. Gabrielli, G. Jona Lasinio, C. Landim; Action functional and quasi-potential for the Burgers equation in a bounded interval. Comm. Pure Appl. Math. 64, 649–696 (2011)

    2011

  4. [8]

    Bertini, A

    L. Bertini, A. De Sole, D. Gabrielli, G. Jona Lasinio, C. Landim; Macroscopic fluctu- ation theory. Rev. Mod. Phys. 87, 593–636 (2015)

    2015

  5. [9]

    Bertini, D

    L. Bertini, D. Gabrielli, C. Landim; Concurrent Donsker-Varadhan and hydrodynam- ical large deviations. Ann. Probab. 51, 1289–1341 (2023). 26 L. BERTINI, D. GABRIELLI, AND G. JONA-LASINIO

    2023

  6. [10]

    Bettelheim, N.R

    E. Bettelheim, N.R. Smith, B. Meerson; Inverse scattering method solves the problem of full statistics of nonstationary heat transfer in the Kipnis-Marchioro-Presutti model. Phys. Rev. Lett. 128, Paper No. 130602 (2022)

    2022

  7. [11]

    Budhiraja, M

    A. Budhiraja, M. Conroy; Asymptotic behavior of stochastic currents under large deviation scaling with mean field interaction and vanishing noise. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 23, 1749–1805 (2022)

    2022

  8. [13]

    Dawson, J

    D.A. Dawson, J. G¨ artner;Large deviations from the McKean-Vlasov limit for weakly interacting diffusions. Stochastics 20, 247–308 (1987)

    1987

  9. [14]

    Flandoli, M

    F. Flandoli, M. Gubinelli, M. Giaquinta, V.M. Tortorelli; Stochastic currents. Sto- chastic Process. Appl. 115, 1583–1601 (2005)

    2005

  10. [15]

    Freidlin, A

    M.I. Freidlin, A. D. Wentzell; Random perturbation of dynamical systems. 2nd edn. Springer, New York, 1998

    1998

  11. [17]

    Griffeath; Additive and cancellative interacting particle systems

    D. Griffeath; Additive and cancellative interacting particle systems. Lecture Notes in Mathematics 724. Springer, Berlin 1979

    1979

  12. [18]

    Haussmann, ´E

    U.G. Haussmann, ´E. Pardoux; Time reversal of diffusions. Ann. Probab. 14, 1188– 1205 (1986)

    1986

  13. [19]

    M.D. Jara, C. Landim, S. Sethuraman; Nonequilibrium fluctuations for a tagged parti- cle in mean-zero one-dimensional zero-range processes.Probab. Theory Related Fields 145, 565–590 (2009)

    2009

  14. [20]

    Kipnis, C

    C. Kipnis, C. Landim; Scaling limits of interacting particle systems. Springer-Verlag, Berlin, 1999

    1999

  15. [21]

    Kipnis, S

    C. Kipnis, S. Olla, S.R.S. Varadhan; Hydrodynamics and large deviation for simple exclusion processes. Comm. Pure Appl. Math. 42, 115–137 (1989)

    1989

  16. [22]

    Kipnis, S.R.S

    C. Kipnis, S.R.S. Varadhan; Central limit theorem for additive functionals of re- versible Markov processes and applications to simple exclusions. Comm. Math. Phys. 104, 1–19 (1986)

    1986

  17. [23]

    Komorowski, C

    T. Komorowski, C. Landim, S. Olla; Fluctuations in Markov processes. Time sym- metry and martingale approximation. Springer, Heidelberg, 2012

    2012

  18. [24]

    Krapivsky, K

    P.L. Krapivsky, K. Mallick, T. Sadhu; Tagged particle in single-file diffusion. J. Stat. Phys. 160, 885–925 (2015)

    2015

  19. [25]

    L´ eonard;A survey of the Schr¨ odinger problem and some of its connections with optimal transport

    C. L´ eonard;A survey of the Schr¨ odinger problem and some of its connections with optimal transport. Discrete Contin. Dyn. Syst. 34, 1533–1574 (2014)

    2014

  20. [26]

    Mallick, H

    K. Mallick, H. Moriya, T. Sasamoto; Exact solution of the macroscopic fluctuation theory for the symmetric exclusion process. Phys. Rev. Lett. 129, no. 4, Paper No. 040601, 7 pp (2022)

    2022

  21. [27]

    Mallick, H

    K. Mallick, H. Moriya, T. Sasamoto; Exact solutions to macroscopic fluctuation theory through classical integrable systems. J. Stat. Mech. Theory Exp. 2024, no. 7, Paper No. 074001, 18 pp

    2024

  22. [28]

    Orrieri; Large deviations for interacting particle systems: joint mean-field and small-noise limit

    C. Orrieri; Large deviations for interacting particle systems: joint mean-field and small-noise limit. Electron. J. Probab. 25, Paper No. 111 (2020)

    2020

  23. [29]

    Osada; An invariance principle for Markov processes and Brownian particles with singular interaction

    H. Osada; An invariance principle for Markov processes and Brownian particles with singular interaction. Ann. Inst. H. Poincar´ e Probab. Statist.34, 217–248 (1998)

    1998

  24. [30]

    Osada; Positivity of the self-diffusion matrix of interacting Brownian particles with hard core

    H. Osada; Positivity of the self-diffusion matrix of interacting Brownian particles with hard core. Probab. Theory Related Fields 112, 53–90 (1998)

    1998

  25. [31]

    Quastel; Diffusion of color in the simple exclusion process

    J. Quastel; Diffusion of color in the simple exclusion process. Comm. Pure Appl. Math. 45, 623–679 (1992). MFT AND SCHR ¨ODINGER PROBLEMS 27

    1992

  26. [33]

    Quastel, S.R.S.Varadhan; Diffusion semigroups and diffusion processes correspond- ing to degenerate divergence form operators

    J. Quastel, S.R.S.Varadhan; Diffusion semigroups and diffusion processes correspond- ing to degenerate divergence form operators. Comm. Pure Appl. Math. 50, 667–706 (1997)

    1997

  27. [34]

    Rezakhanlou; Propagation of chaos for symmetric simple exclusions

    F. Rezakhanlou; Propagation of chaos for symmetric simple exclusions. Comm. Pure Appl. Math. 47, 943–957 (1994)

    1994

  28. [35]

    Schr¨ odinger; Sur la th´ eorie relativiste de l’´ electron et l’interpr´ etation de la m´ ecanique quantique

    E. Schr¨ odinger; Sur la th´ eorie relativiste de l’´ electron et l’interpr´ etation de la m´ ecanique quantique. Ann. Inst. H. Poincar´ e2, 269–310 (1932)

    1932

  29. [37]

    Spohn; Large Scale Dynamics of Interacting Particles

    H. Spohn; Large Scale Dynamics of Interacting Particles. Springer-Verlag, Heidelberg 1991

    1991

  30. [38]

    Varadhan; Scaling limits for interacting diffusions

    S.R.S. Varadhan; Scaling limits for interacting diffusions. Comm. Math. Phys. 135, 313–353 (1991)

    1991

  31. [39]

    Varadhan; Entropy, large deviations, and scaling limits

    S.R.S. Varadhan; Entropy, large deviations, and scaling limits. Commun. Pure Appl. Math. 64, 1914–1932 (2013). Lorenzo Bertini Dipartimento di Matematica, Universit`a di Roma ‘La Sapienza’, I-00185 Roma, Italy Email address : bertini@mat.uniroma1.it Davide Gabbrielli DISIM, Un...

    2013

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.