Exclusion processes with non-reversible boundary: hydrodynamics and large deviations

Beatriz Salvador; Claudio Landim; Jo\~ao Pedro Mangi

REVIEW 4 major objections 3 minor 26 references

Exclusion processes with non-reversible boundary: hydrodynamics and large deviations

T0 review · 4 major / 3 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read This paper proves a dynamical large deviations principle for exclusion processes whose boundary reservoirs inject and remove particles through general, possibly non-reversible rates.

desk verdict The hydrodynamic limit is real and mostly well proved; the advertised dynamical LDP is narrower than claimed, since the concavity assumption is load-bearing and the proof leans on deferred results. read the letter →

arxiv 2411.17653 v1 pith:DAQWCR3J submitted 2024-11-26 math.PR math-phmath.MP

Claudio Landim , João Pedro Mangi , Beatriz Salvador This is my paper · ORCID

classification math.PRmath-phmath.MP MSC 60K3560F1082C22

keywords hydrodynamiclimitdynamicallargedeviationsexclusionprocessnon-reversibleboundaryRobinconditionsmildcontactreservoirs

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 treats a one-dimensional symmetric exclusion process in weak contact with reservoirs at both ends, where the boundary dynamics can create and destroy several particles at a time through arbitrary rates. It establishes two scaling results: the empirical density obeys a hydrodynamic limit given by the heat equation with nonlinear Robin boundary conditions, and, under a concavity condition on the boundary rates, the full trajectory law satisfies a large deviations principle with speed N and an explicit rate function. The significance is that the boundary and bulk stationary states need not agree, so the usual Bernoulli reference measures are not stationary for the full process; the paper shows the hydrodynamics and large deviations still hold, including cases where the hydrodynamic equation has several stationary profiles. This opens the way to studying metastability and transition times between stable density profiles.

What carries the argument

The load-bearing objects are the expected boundary creation and destruction rates B_k(α) and D_{-k}(α), defined as expectations under Bernoulli product measures of density α of the rates at which the boundary adds or removes exactly k particles. Their difference gives the boundary flux F_±, and their exponential generating function b_±(α,M) supplies the boundary cost in the rate function. The argument is carried by a variational decomposition of the rate function into a bulk part, expressed as a weighted Sobolev norm of a current, plus a boundary part given by a Legendre transform of the b_± terms, together with an I-density approximation showing that any finite-cost path can be smoothed. Concavity of B_k and D_{-k} is used to make the rate function convex with compact level sets, and a uniqueness theorem for the nonlinear Robin initial-value problem, including its tilted analogue, closes the hydrodynamic and large-deviation proofs.

What would settle it

Compute B_1''(α) for a three-site boundary window with parameters a0,a1,a2 outside the Appendix B relation, for example with a2 much smaller than a+2b; the second derivative becomes positive somewhere, so concavity fails. To decide the theorem itself, one would look for such non-concave rates together with a smooth initial profile for which the rate function I_{[0,T]} stops being lower semicontinuous or has non-compact level sets, making the large-deviations upper and lower bounds impossible.

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Extended reading notes

Core claim

On the paper's own terms, the central discovery is that the non-reversible boundary dynamics contribute to the dynamical large-deviation rate function through the boundary functionals b±(α,M), and that the whole rate functional is a good rate function whenever the expected k-particle creation and destruction rates B_k and D_{-k} are concave. In that regime the tilted dynamics have a hydrodynamic limit, and the lower bound is obtained by showing that every finite-cost trajectory can be approximated by smooth trajectories in a class of well-behaved paths. The companion hydrodynamic limit theorem identifies the macroscopic density as the unique weak solution of the heat equation with nonlinear Robin boundary conditions, with boundary fluxes given by the expectations F_-(ρ(t,0)) and F_+(ρ(t,1)). A concrete model, called the Exclusion l3 model, is exhibited for which the hydrodynamic equation has a unique time-dependent solution but several distinct stationary solutions.

Load-bearing premise

The proof depends on the expected rates at which the boundary adds or removes k particles, B_k(α) and D_{-k}(α), being concave functions of the local density α on [0,1]; if a natural family of rates violates this assumption, the large deviations principle as stated may fail.

Editorial extensions

If this is right

The density of the process converges to the unique weak solution of ∂tρ=Δρ with nonlinear Robin boundary conditions ∇ρ_t(0)=-F_-(ρ_t(0)) and ∇ρ_t(1)=F_+(ρ_t(1)).
Typical fluctuations away from this hydrodynamic path are exponentially rare, with speed N and cost given by the explicit rate function I_{[0,T]}(π|γ).
When the boundary rates admit more than one stationary density profile, the stationary large-deviations rate function has at least two critical points, signaling metastable behavior of the particle system.
The rate function splits into separate bulk and boundary costs, so the macroscopic cost of creating a density anomaly through the reservoirs can be compared directly with the cost of transporting density through the interior.
If the concavity assumption holds, the same proof covers the full class of mild-contact boundary dynamics, bringing the general non-reversible setting in line with the previously understood reversible and Dirichlet-style cases.

Reading between the lines

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

One could test whether the concavity of B_k and D_{-k} is equivalent to convexity of a boundary free energy; if so, the rate function's good properties would follow from a more physical structural condition rather than a case-by-case check.
The multiple stationary solutions of the Exclusion l3 model suggest that the dynamical LDP is a natural first ingredient for an Eyring–Kramers type estimate of transition times between stable profiles, a question the paper does not address.
The authors' stated program of first taking the diffusive limit and then letting the reservoir intensity grow would turn this weak-contact LDP into a Γ-convergence route toward the rate function of strongly coupled reservoirs; the present theorem supplies the inner limit of that program.
For rates that violate concavity, the theorem as stated does not apply, but the rate functional may still be a valid LDP rate function in a non-convex form; the variational proof here would need a different convexity argument to cover such cases.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Two things to know. First, the hydrodynamic limit (Theorem 2.2) looks right, and a lot of the proof is actually in the paper: tightness, replacement lemmas, energy bounds, and a real uniqueness theorem (Theorem C.1) for nonlinear Robin boundary conditions. That part deserves a careful read and will be useful. Second, the LDP (Theorem 2.6), which the abstract advertises, is not fully established as written. The concavity assumption on B_k and D_-k is an assumption, not a consequence of the standing hypotheses, and several critical upper-bound and approximation steps are deferred to [9,12] or simply left to the reader. The paper is honest about some of this, but the gap between "very general rates" and "concave rates" is real and should be stated plainly in the abstract.

What is genuinely new: the l-window boundary with arbitrary rates, the resulting nonlinear Robin hydrodynamic equation, the uniqueness theorem for that PDE, and an example with multiple stationary profiles but a unique evolution. That is a clear step beyond [4,11,22] and beyond the linear Robin case in [12]. No parameter fitting or normalization forces the results, and the self-citation pattern is not a problem when the cited results are the ones actually doing the work.

The main soft spot is exactly what the stress-test flag says. The l=2 example works: allow one left-boundary creation at rate c(1-eta(2 epsilon_N)), set the other boundary rates to zero, keep Kawasaki exchange inside the window. The dynamics is irreducible, and B_1(alpha) = c(1-alpha)^2, which is strictly convex. So Theorem 2.6 does not cover a natural non-reversible boundary family of exactly the kind the paper claims to treat. The paper should either characterize the admissible class, prove concavity from structural conditions on R±, or explicitly restrict the LDP statement to concave rates.

There is also a smaller technical issue in Lemma 4.27: the bound (4.36) with |a|^l is not the right growth for a Legendre transform of e^{-|x|}; the correct form should involve |a| log|a|. The domination step needs repair before the dominated convergence argument goes through. This is likely fixable, but as printed it is a gap.

Who is this for? Researchers working on boundary-driven exclusion processes and macroscopic fluctuation theory. The hydrodynamic limit alone is worth a serious referee. The LDP needs substantial revision: clean up the concavity hypothesis, fill or explicitly import the deferred estimates, and correct the Lemma 4.27 bound. I would send it to peer review, but I would expect major revisions rather than acceptance in the present form.

Referee Report

4 major / 3 minor

Summary. The paper studies a one-dimensional symmetric simple exclusion process in mild contact with boundary reservoirs through windows of fixed size l with general non-reversible rates R±. The main results are a hydrodynamic limit (Theorem 2.2) identifying the empirical density with the unique weak solution of the heat equation with nonlinear Robin boundary conditions (2.10), and a dynamical large deviations principle (Theorem 2.6) for the empirical measure with speed N and a variational rate function I[0,T](·|γ), under a concavity assumption on the mean creation/destruction rates B_k, D_{-k} and a C^{2+β} initial profile. An example (Exclusion l3) shows multiple stationary profiles while the evolution equation is unique.

Significance. The hydrodynamic limit part is supported by a fairly detailed proof, including tightness, replacement lemmas, energy estimates, and a uniqueness theorem for nonlinear Robin problems in Appendix C; this is a genuine contribution. The LDP, if fully established, would be a significant step toward macroscopic fluctuation theory for non-reversible boundary-driven systems with nonlinear boundary conditions. However, the LDP proof has major omissions and at least one incorrect estimate, and the concavity hypothesis is not benign; the advertised 'very general rates' is narrower than what Theorem 2.6 actually covers. The paper is therefore not yet ready for publication, but the core ideas are promising.

major comments (4)

[Section 2.4, Theorem 2.6 and Remark 2.7] The concavity of B_k and D_{-k} in (2.18) is a real restriction and does not follow from the standing assumptions. Consider l=2 with left-window rates that create a particle at epsilon_N from the empty state 00 and from the state 01 with rates 2c and c, respectively, and that also have positive destruction rates (e.g., 10->00, 01->00, 11->10, 11->01) so that the boundary-plus-bulk dynamics in the window is irreducible. Then B_1(alpha)=2c(1-alpha)^2 + c alpha(1-alpha) = c(2 - 3 alpha + alpha^2), whose second derivative is 2c>0, so B_1 is strictly convex. This shows that the admissible class is strictly larger than the concave class, and the paper does not characterize which natural rates satisfy concavity. The authors should either prove concavity under natural conditions, characterize the concave class, or explicitly state in the abstract and introduction that the LDP is restricted to the concave case.
[Section 4.3.2, Theorem 4.12; Section 4.4.1, Theorem 4.14] The energy estimate for the upper bound (Theorem 4.12) is stated with the remark 'The details are left to the reader,' and the hydrodynamic limit of the tilted process (Theorem 4.14) is dispatched with 'We leave the details to the reader.' Both results are essential: Theorem 4.12 controls the compactness of level sets and the upper bound, and Theorem 4.14 is the basis of the entropy identity (4.23) for the lower bound. The paper must provide full proofs or a precise reduction to [9] and [12] that accounts for the nonlinear Robin boundary terms introduced here.
[Section 4.4.2, Lemma 4.27, inequality (4.36)] The estimate for Phi^{-}_t(a) is incorrect as printed. The displayed bound sup_x {a x + C(e^{-|x|l} - 1)} is +infinity for a>0, since the linear term dominates as x tends to infinity. Consequently, the bound (4.37) is not justified, and the dominated convergence argument for I^(2) is not established. The correct growth of the Legendre transform of the boundary cost is of order |a| log |a| (or a log a), not |a|^l log(|a|^l). This is a load-bearing step in the proof that Pi_4 is I-dense (Theorem 4.18).
[Section 3.2, equations (3.5)-(3.10)] The assertion after (3.10) that P_{nu_N}(M^H_N(t)=0 for all t and all H)=1 is false: M^H_N(t) defined in (3.5) is a martingale, not the zero process. The proof later uses this claim to conclude that the term in (3.18) vanishes. The correct argument would use the previously stated L^2 convergence of the martingale, lim_N E[M^H_N(t)^2]=0, together with Doob's inequality. As written, the hydrodynamic limit proof has a gap, though it appears fixable.

minor comments (3)

[Equation (2.18)] The definition D_{-k}(alpha)=B_k(alpha) cannot be correct, since it would identify destruction with creation; the formula for b± in (2.16)-(2.17) and the example in Appendix B use different coefficients for creation and destruction. It should be D_{-k}(alpha)=E_{nu_alpha}[sum_xi R±(eta,xi) 1_{sum xi = sum eta - k}]. Please correct this typo.
[Section 2.2] In the definition of C^{n,m}_0(Omega_T), the phrase 'that below to C^{n,m}' should read 'that belong to C^{n,m}'.
[Section 4.4.2, proof of Lemma 4.27] The sentence describing the attained supremum, 'x = (-1)/BD a>0 1/l log(|a|/(Cl))', is garbled and should be replaced by a clean formula such as x = (1/l) log(a/(Cl)) for a>0, with the corresponding expression for a<0.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the LDP is proved from the dynamics under an explicit concavity hypothesis; the technical self-citations are to independent prior works.

full rationale

The derivation chain is not circular. The central large-deviations theorem (Theorem 2.6) is proved from the Markov generator via the exponential martingale, the tilted-process hydrodynamic limit (Theorem 4.14), and the entropy identity (Lemma 4.15); the rate function I[0,T](.|gamma) is defined variationally as a supremum of J_{T,H}, not as a quantity fitted to the LDP. The concavity assumption on B_k and D_{-k} is an explicit hypothesis used to prove convexity and compact level sets (Theorem 4.5) and the I-density approximation (Lemma 4.22); it is not derived from the theorem it supports, and the skeptic's convex-rate example only shows that Theorem 2.6's scope is narrower than 'very general', which is a correctness/scope concern, not circularity. Several technical lemmas are delegated to prior works [9] and [12], which share an author, but those are peer-reviewed, parameter-free mathematical results with stated assumptions, and the present paper proves its genuinely novel ingredient, uniqueness for the nonlinear Robin IVP, in Appendix C rather than importing it by citation. No fitted parameter is renamed as a prediction, no uniqueness theorem is invoked to forbid alternatives, and no ansatz is smuggled in as an external fact. The overall score is therefore 0.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The model has no fitted parameters; the central results rest on the concavity assumption and on standard PDE and entropy tools. The main unproved dependencies are the deferred proofs in the LDP section and Proposition C.6 in Appendix C.

assumptions (4)

domain assumption B_k and D_{-k} are concave for each k (creation/destruction rate expectations under Bernoulli measures).
Assumed in Theorem 2.6 for the LDP; used to prove convexity and compact level sets of the rate function (Theorem 4.5) and in the I-density approximation (Lemma 4.22).
domain assumption The boundary dynamics on each window Sigma^pm_l, together with the restricted Kawasaki dynamics, is irreducible.
Stated in Section 2.1; ensures the full process is irreducible and has a unique stationary measure, foundational for the entropy method.
standard math Standard parabolic regularity and maximum principle results from Ladyzenskaja et al. [7] and Protter-Weinberger [20] for the nonlinear Robin IVP.
Used in Appendix C (Lemma C.9 smooth solution, Lemma C.10 maximum principle) to establish qualitative properties of the hydrodynamic solution.
domain assumption Equivalence of weak and mild solutions for the linearized problem (Proposition C.6), adapted from [4] without proof.
Load-bearing for the uniqueness proof of the nonlinear Robin IVP (Theorem C.1); if the adaptation fails, uniqueness and hence both main theorems are in question.

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Cite this review

Pith. "Pith review of Exclusion processes with non-reversible boundary: hydrodynamics and large deviations." pith.science (2026). https://pith.science/paper/DAQWCR3J

@misc{pith2026241117653,
  author       = {Pith},
  title        = {Pith review of: Exclusion processes with non-reversible boundary: hydrodynamics and large deviations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DAQWCR3J}},
  note         = {Machine review of arXiv:2411.17653}
}

read the original abstract

We consider a one-dimensional exclusion dynamics in mild contact with boundary reservoirs. In the diffusive scale, the particles' density evolves as the solution of the heat equation with non-linear Robin boundary conditions. For appropriate choices of the boundary rates, these partial differential equations have more than one stationary solution. We prove the dynamical large deviations principle.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

26 extracted references · 26 canonical work pages

[12]

and Neumann, A

Franco, T., Gon¸ calves, P., Landim, C. and Neumann, A. ( 2022) Dynamical large deviations for the boundary driven symmetric exclusion process with Robin bou ndary conditions, ALEA Lat. Am. J. Probab. Math. Stat., no. 19 (2), pp. 1497–1546

work page 2022
[9]

Landim, C

Bertini, L. Landim, C. and Mourragui, M. (2009) Dynamica l large deviations for the boundary driven weakly asymmetric exclusion process, The Annals of P robability, 37 (6), pp. 2357–2403

work page 2009
[1]

and Landim, C

Kipnis, C. and Landim, C. (1999) Scaling Limits of Intera cting Particle Systems, Springer

work page 1999
[2]

Gon¸ calves, P. (2019) Hydrodynamics for symmetric exclusion in contact with reservoirs, Stochastic Dynamics Out of Equilibrium, Institut Henri Poincar´ e, Par is, France, 2017, Springer Proceedings in Mathematics and Statistics book series, pp. 137–205

work page 2019
[3]

and Tsunoda, K

Jara, M., Landim, C. and Tsunoda, K. (2021) Derivation of viscous Burgers equations from weakly asymmetric exclusion processes, Ann. Inst. Henri Poincar´ e Probab. Statist, 57 (1), pp. 169–194

work page 2021
[4]

and Gon¸ calves, P

Erignoux, C. and Gon¸ calves, P. and Nahum, G. (2020) Hydr odynamics for SSEP with Non- reversible Slow Boundary Dynamics: Part I, the Critical Reg ime and Beyond, Journal of Statistical Physics, 181, Springer Science and Business Media LLC, 4, pp .1433—1469 EXCLUSION PROCESSES WITH NON-REVERSIBLE BOUNDARY 43

work page 2020
[5]

and Landim, C

Bouley, A. and Landim, C. (2021) Thermodynamics of noneq uilibrium driven diﬀusive systems in weak contact with boundary reservoirs, arxiv preprint 2112 .01091

work page 2021
[6]

Bouley, A, Erignoux, C. and Landim, C (2021) Steady state large deviations for one-dimensional, symmetric exclusion processes in weak contact with reservo irs, to appear in Annales de l’Institut Henri Poincar´ e (B) Probabilit´ es et Statistiques

work page 2021

Show all 26 references

[7]

Ladyzenskaja, O. A. and Solonnikov, V. A. and Ural’ceva, N. N. (1968) Linear and quasilinear equations of parabolic type, Translations of Mathematical Monographs, 23

1968
[8]

and Gon¸ calves, P

Franco, T. and Gon¸ calves, P. and Neumann, A. (2013) Hydr odynamical behaviour of symmetric exclusion with slow bonds, Annales de l’Institut Henri Poin car´ e: Probability and Statistics, 49, 2, pp. 402–427

2013
[10]

and Neumann, A

Franco, T. and Neumann, A. (2017) Large deviations for t he exclusion process with a slow bond, The Annals of Applied Probability, 27, No 6, pp. 3547–3587

2017
[11]

(2018) Hydrodynamic limit of boundary dri ven exclusion processes with nonreversible boundary dynamics, Journal of Statistical Physics, 172, 5, pp

Erignoux, C. (2018) Hydrodynamic limit of boundary dri ven exclusion processes with nonreversible boundary dynamics, Journal of Statistical Physics, 172, 5, pp. 1327–1357, Springer

2018
[13]

(2018 ) Stationary states of boundary driven exclusion processes with nonreversible boundary dynamics

Erignoux, Cl´ ement and Landim, Claudio and Xu, T. (2018 ) Stationary states of boundary driven exclusion processes with nonreversible boundary dynamics . Journal of Statistical Physics, Springer, 171, pp. 599–631

2018
[14]

Bertini, Lorenzo, De Sole, Alberto, Gabrielli, Davide , Jona-Lasinio, Giovanni and Landim, Claudio (2015) Macroscopic ﬂuctuation theory, Reviews of Modern Ph ysics, APS, 87, 2, pp. 593

2015
[15]

Spohn, Herbert (1983) Long range correlations for stoc hastic lattice gases in a non-equilibrium steady state, Journal of Physics A: Mathematical and Genera l, IOP Publishing, 16, 18, pp. 4275

1983
[16]

De Masi, Anna, Presutti, Errico and Tsagkarogiannis, D imitrios and Vares, Maria (2012) Truncated correlations in the stirring process with births and deaths , Electronic Journal of Probability, nstitute of Mathematical Statistics and Bernoulli Society, 17, pp. 1 – 35

2012
[17]

and Gon¸ calves, P

Franceschini, C. and Gon¸ calves, P. and Salvador, B. (2023) Hydrodynamical behavior of generalized symmetric exclusion with open boundary, MPAG, 26, 2, pp. 1–2 3

2023
[18]

and Landim, C

Bertini, L., De Sole, A., Gabrielli, D., Jona-Lasinio, G. and Landim, C. (2003) Large Deviations for the Boundary Driven Symmetric Simple Exclusion Process, Ma thematical Physics, Analysis and Geometry, 6, pp. 231–267

2003
[19]

Donsker, M. D. and Varadhan, S. R. S. (1989) Large deviat ions from a hydrodynamic scaling limit, Comm. Pure Appl. Math., 42, pp. 243–270

1989
[20]

M. H. Protter and H. F. Weinberger (1984) Maximum princi ples in diﬀerential equations, Springer- Verlag, New York

1984
[21]

Bouley, Ang` ele (2024) Grandes d´ eviations dynamiques du processus de l’exclusion en contact faible avec des r´ eservoirs, PhD thesis at ’Universit´ e de Rouen Normandie

2024
[22]

Erignoux, Cl´ ement and Gon¸ calves, Patricia and Nahum, Gabriel (2019) Hydrodynamics for SSEP with non-reversible slow boundary dynamics: Part II, below the critical regime, arXiv preprint arXiv:1912.09842

2019 arXiv
[23]

Farfan, Jonathan, Landim, Claudio and Tsunoda, Kenkic hi (2019) Static large deviations for a reaction–diﬀusion model, Probability Theory and Related F ields, Springer, 174, pp. 49–101

2019
[24]

Faris, William G and Jona-Lasinio, Giovanni (1982) Lar ge ﬂuctuations for a nonlinear heat equation with noise, Journal of Physics A: Mathematical and General, IOP Publishing, 15, 10, pp. 3025

1982
[25]

Landim, C, Olla, S and Volchan, SB (1998) Driven tracer p article in one dimensional symmetric simple exclusion, Communications in mathematical physics , 192, Springer, pp. 287–307

1998
[26]

and Tsunoda, K

Landim, C. and Tsunoda, K. (2018) Hydrostatics and dyna mical large deviations for a reaction- diﬀusion model. 44 C. LANDIM, J. MANGI, AND B. SAL V ADOR Instituto de matem ´atica pura e aplicada, Estrada Dona Castorina 110, J. Botani co, 22460 Rio de Janeiro, Brazil Email addr...

2018

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [12]

and Neumann, A

Franco, T., Gon¸ calves, P., Landim, C. and Neumann, A. ( 2022) Dynamical large deviations for the boundary driven symmetric exclusion process with Robin bou ndary conditions, ALEA Lat. Am. J. Probab. Math. Stat., no. 19 (2), pp. 1497–1546

work page 2022

[2] [9]

Landim, C

Bertini, L. Landim, C. and Mourragui, M. (2009) Dynamica l large deviations for the boundary driven weakly asymmetric exclusion process, The Annals of P robability, 37 (6), pp. 2357–2403

work page 2009

[3] [1]

and Landim, C

Kipnis, C. and Landim, C. (1999) Scaling Limits of Intera cting Particle Systems, Springer

work page 1999

[4] [2]

Gon¸ calves, P. (2019) Hydrodynamics for symmetric exclusion in contact with reservoirs, Stochastic Dynamics Out of Equilibrium, Institut Henri Poincar´ e, Par is, France, 2017, Springer Proceedings in Mathematics and Statistics book series, pp. 137–205

work page 2019

[5] [3]

and Tsunoda, K

Jara, M., Landim, C. and Tsunoda, K. (2021) Derivation of viscous Burgers equations from weakly asymmetric exclusion processes, Ann. Inst. Henri Poincar´ e Probab. Statist, 57 (1), pp. 169–194

work page 2021

[6] [4]

and Gon¸ calves, P

Erignoux, C. and Gon¸ calves, P. and Nahum, G. (2020) Hydr odynamics for SSEP with Non- reversible Slow Boundary Dynamics: Part I, the Critical Reg ime and Beyond, Journal of Statistical Physics, 181, Springer Science and Business Media LLC, 4, pp .1433—1469 EXCLUSION PROCESSES WITH NON-REVERSIBLE BOUNDARY 43

work page 2020

[7] [5]

and Landim, C

Bouley, A. and Landim, C. (2021) Thermodynamics of noneq uilibrium driven diﬀusive systems in weak contact with boundary reservoirs, arxiv preprint 2112 .01091

work page 2021

[8] [6]

Bouley, A, Erignoux, C. and Landim, C (2021) Steady state large deviations for one-dimensional, symmetric exclusion processes in weak contact with reservo irs, to appear in Annales de l’Institut Henri Poincar´ e (B) Probabilit´ es et Statistiques

work page 2021