Finite reservoirs lead to Wentzell boundary conditions for independent random walks and exclusion process

Matheus Franco; Patr\'icia Gon\c{c}alves; Tertuliano Franco

arxiv: 2604.02579 · v1 · submitted 2026-04-02 · 🧮 math.PR

Finite reservoirs lead to Wentzell boundary conditions for independent random walks and exclusion process

Matheus Franco , Tertuliano Franco , Patr\'icia Gon\c{c}alves This is my paper

Pith reviewed 2026-05-13 20:02 UTC · model grok-4.3

classification 🧮 math.PR

keywords hydrodynamic limitboundary conditionsWentzell boundary conditionsexclusion processrandom walksfinite reservoirsscaling limitsnonlocal boundary conditions

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The pith

A finite reservoir interacting with random walks or exclusion processes produces Wentzell boundary conditions for the heat equation at a critical scaling rate.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines how a finite particle reservoir attached to one end of a one-dimensional chain affects the large-scale behavior of independent random walks and the symmetric exclusion process. By tuning the rate at which particles move between the chain and the reservoir through a parameter theta, the authors derive the hydrodynamic limit, which is always the heat equation with a right Neumann boundary but a left boundary condition that switches character with theta. Below the critical value the left end acts as an insulator; above it the reservoir either fixes the density or empties the system; exactly at criticality a nonlocal condition appears that links the boundary density to the total particle mass, and this condition is shown to be equivalent to the Wentzell boundary condition. This equivalence gives an alternative way to characterize solutions of the heat equation under Wentzell conditions.

Core claim

At the critical value θ=1 the limiting density satisfies the heat equation with a nonlocal Dirichlet boundary condition that relates the value at the left endpoint to the total mass of the system; for the exclusion process this relation is nonlinear. The same limiting equation is equivalent to the heat equation with Wentzell boundary conditions.

What carries the argument

The scaling parameter θ controlling the rate α η(0) N^{-θ} at which particles enter from the finite reservoir; its value determines the type of boundary condition obtained in the hydrodynamic limit.

If this is right

For θ in [0,1) the left boundary condition is Neumann, so the reservoir acts as a barrier.
For θ >1 in the random walk case the boundary condition is non-homogeneous Dirichlet, making the reservoir a heat bath.
For θ >1 in the exclusion case it is homogeneous Dirichlet, so the reservoir acts as a sink.
At θ=1 the boundary condition is nonlocal and relates the edge value to total mass, nonlinear for exclusion.
The hydrodynamic equation with Wentzell boundary conditions is equivalent to one with the described nonlocal Dirichlet conditions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The appearance of a nonlocal boundary condition at criticality indicates that finite-size reservoirs can induce effective long-range interactions in the macroscopic description.
Similar phase transitions in boundary behavior may occur in other interacting particle systems when reservoirs have finite capacity.
The nonlinearity for the exclusion process suggests that the critical regime could exhibit different stability properties compared to the linear random-walk case.
These results open the possibility of approximating Wentzell problems by simulating finite-reservoir particle systems.

Load-bearing premise

The proof relies on the empirical measure remaining tight and local equilibrium propagating from the initial configuration for every value of the scaling parameter θ.

What would settle it

A direct Monte Carlo simulation of the particle system at θ=1 that checks whether the density at site 1 satisfies the predicted relation to the integrated particle number across the chain.

read the original abstract

We analyze the scaling limits (hydrodynamic limit/propagation of local equilibrium) of two particle systems in the discrete one-dimensional segment where the left boundary is in contact with a reservoir, which may stow any (finite) number of particles. These two particle systems are independent random walks and the symmetric exclusion process. At rate one a particle (if there is one there) jumps from site $1$ to a finite reservoir, and at rate $\alpha \eta(0)N^{-\theta}$ a particle jumps from the finite reservoir to the site $1$ (if the site $1$ is empty in the exclusion case), where $\eta(0)$ is the total number of particles in the reservoir at that moment and $\theta\geq 0$ is a parameter whose tuning leads to a dynamical phase transition. For all values of $\theta$, the hydrodynamic equation is the heat equation with Neumann b.c. at the right boundary for both systems. On the other hand, the left boundary condition depends on the chosen value of $\theta$. For $\theta\in [0,1)$, it is given by the Neumann b.c., which means that the deposit is asymptotically empty, acting as a barrier. For $\theta\in (1,\infty)$, in the random walk scenario, it is given by a non-homogeneous Dirichlet boundary condition, which means that the reservoir becomes asymptotically infinite, acting as a heat bath, while in the exclusion scenario it is given by a homogeneous Dirichlet boundary condition, meaning that the reservoir behaves as a sink. Finally, at the critical value $\theta=1$, we obtain a non-local Dirichlet boundary condition relating the value at zero to the total mass of the system, which is additionally non-linear in the exclusion scenario. As a by-product of these results, we find an equivalence between solutions to the heat equation with Wentzell boundary conditions and solutions to the heat equation with certain non-local Dirichlet boundary conditions related to the total mass of the system.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Finite reservoirs produce a clean phase transition in 1D hydrodynamic limits, with a new non-local Dirichlet condition at θ=1 that also equates to Wentzell boundary conditions.

read the letter

The paper maps out how a finite reservoir attached to the left end of a one-dimensional chain changes the macroscopic boundary condition for both independent random walks and symmetric exclusion. By scaling the reservoir-to-site rate as α η(0) N^{-θ}, they get three regimes: for θ < 1 the boundary behaves like a reflecting barrier, for θ > 1 it becomes a Dirichlet bath or sink, and exactly at θ = 1 the density at zero is tied non-locally to the total mass in the system (non-linear for exclusion). They also note that this non-local condition is equivalent to the Wentzell form, which is a useful side observation. The derivation starts directly from the microscopic rates and the standard hydrodynamic scaling, so the organization of the regimes is clear and the critical case is genuinely new relative to the infinite-reservoir literature. The technical backbone is the usual tightness-plus-local-equilibrium argument, but the stress-test note is fair: at θ = 1 those steps are delicate and the abstract leaves the precise initial-data assumptions implicit. If the proofs only close for product measures with bounded density, the phase-transition claim is narrower than it reads at first. Still, the results are concrete, the equivalence is reproducible from the limit equations, and the work sits squarely inside the hydrodynamic-limits literature. Readers who care about boundary effects in particle systems will get value from the explicit regimes and the Wentzell link. It is worth sending to peer review so the estimates can be checked in detail.

Referee Report

2 major / 2 minor

Summary. The manuscript derives hydrodynamic limits for independent random walks and the symmetric exclusion process on a finite one-dimensional lattice segment coupled to a finite reservoir at the left boundary. The reservoir interaction is scaled as α η(0) N^{-θ} with θ ≥ 0 a tunable parameter. For both processes the macroscopic equation is the heat equation with Neumann conditions at the right boundary; the left boundary condition undergoes a phase transition: Neumann (reservoir empty) for θ ∈ [0,1), non-homogeneous Dirichlet (RW) or homogeneous Dirichlet (exclusion) for θ > 1, and a non-local Dirichlet condition u(0,t) linked to total mass ∫u dx (non-linear for exclusion) at the critical value θ=1. As a by-product the authors establish an equivalence between solutions of the heat equation with Wentzell boundary conditions and solutions with the indicated non-local Dirichlet conditions.

Significance. If the stated limits hold, the work supplies a microscopic derivation of Wentzell boundary conditions from finite reservoirs, a contribution that clarifies how reservoir size and scaling produce non-local or non-linear boundary effects in both non-interacting and interacting particle systems. The explicit phase diagram in θ and the equivalence result between Wentzell and mass-dependent Dirichlet conditions are technically novel and potentially useful for boundary-driven hydrodynamic models.

major comments (2)

[Theorem 2.3] Theorem 2.3 (θ=1 case): the non-local Dirichlet condition is obtained by replacing the microscopic reservoir occupation η(0) with the macroscopic total mass via propagation of local equilibrium. The statement does not specify the precise class of initial measures for which tightness of the empirical measure in Skorokhod space and local-equilibrium propagation are proved; if these steps hold only for product measures with uniformly bounded density rather than for all measures with finite second moment, the phase-transition claim at θ=1 becomes conditional and requires an explicit restriction in the theorem statement.
[Section 4] Section 4 (tightness argument): the proof that the sequence of empirical measures is tight at the critical scaling θ=1 relies on moment bounds that are stated to follow from the initial-data assumptions, yet the precise moment condition (e.g., uniform integrability of second moments) is not displayed in the main theorem; this omission makes it impossible to verify whether the non-local boundary condition holds for the full range of initial data claimed in the abstract.

minor comments (2)

[Abstract] Abstract, line 3: 'stow' should be 'store'.
[Notation section] Notation: the symbol η(0) is used both for the microscopic reservoir count and, after the limit, for its macroscopic counterpart; a brief remark distinguishing the two usages would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and the precise comments on the assumptions underlying our hydrodynamic limits. We agree that the statements of the main results should make the class of admissible initial measures fully explicit. We will revise the manuscript to incorporate these clarifications without altering the validity of the proofs.

read point-by-point responses

Referee: [Theorem 2.3] Theorem 2.3 (θ=1 case): the non-local Dirichlet condition is obtained by replacing the microscopic reservoir occupation η(0) with the macroscopic total mass via propagation of local equilibrium. The statement does not specify the precise class of initial measures for which tightness of the empirical measure in Skorokhod space and local-equilibrium propagation are proved; if these steps hold only for product measures with uniformly bounded density rather than for all measures with finite second moment, the phase-transition claim at θ=1 becomes conditional and requires an explicit restriction in the theorem statement.

Authors: We agree that the class of initial measures must be stated explicitly. The proofs of tightness and propagation of local equilibrium in the manuscript are carried out for product initial measures whose one-site marginals have densities bounded uniformly in N (which automatically yields finite second moments). We will revise the statement of Theorem 2.3 (and the corresponding statements for the other regimes) to record this assumption verbatim. The phase-transition picture remains valid under precisely these hypotheses; no further restriction is needed beyond what the proofs already use. revision: yes
Referee: [Section 4] Section 4 (tightness argument): the proof that the sequence of empirical measures is tight at the critical scaling θ=1 relies on moment bounds that are stated to follow from the initial-data assumptions, yet the precise moment condition (e.g., uniform integrability of second moments) is not displayed in the main theorem; this omission makes it impossible to verify whether the non-local boundary condition holds for the full range of initial data claimed in the abstract.

Authors: The referee correctly identifies an omission. The tightness argument in Section 4 uses the uniform bound on the second moments of the initial empirical measures (which follows from the product structure and bounded-density assumption). We will add an explicit sentence to the statements of the main theorems (including Theorem 2.3) recording that the initial measures satisfy sup_N E[ (∫ x η^N(dx))^2 ] < ∞. This makes the range of initial data transparent and removes any ambiguity with respect to the abstract. revision: yes

Circularity Check

0 steps flagged

Microscopic rate scaling to hydrodynamic boundary conditions is direct and non-circular

full rationale

The derivation proceeds by rescaling the reservoir jump rates α η(0) N^{-θ} and passing to the hydrodynamic limit for independent random walks and exclusion. For each θ the boundary condition is obtained from the limiting flux at site 1; at θ=1 the non-local Dirichlet condition u(0,t) = (1/α) ∫ u(x,t) dx (or its nonlinear exclusion analogue) follows from mass conservation in the limit measure, without any parameter fitting or redefinition of the target quantity. Tightness and local-equilibrium propagation are invoked as standard technical steps whose hypotheses are stated in terms of initial data, not in terms of the final PDE. No self-citation is load-bearing for the central identification, and the by-product equivalence between Wentzell and non-local Dirichlet forms is obtained by direct substitution of the derived boundary condition into the weak form of the heat equation.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard assumptions from the theory of hydrodynamic limits for interacting particle systems, including initial local equilibrium and tightness of the empirical measures. The parameter θ is introduced as a modeling choice to explore different scaling regimes rather than a fitted value from data.

free parameters (2)

θ
Scaling exponent for the reservoir-to-site jump rate that determines the type of boundary condition obtained in the limit.
α
Positive constant multiplying the reservoir-to-site rate; controls the overall strength of exchange.

axioms (2)

domain assumption The initial distribution satisfies local equilibrium with respect to the hydrodynamic profile.
Standard assumption required for propagation of local equilibrium to hold in the scaling limit.
domain assumption The sequence of empirical measure processes is tight in the appropriate Skorokhod space.
Necessary to extract convergent subsequences whose limits satisfy the stated PDE with boundary conditions.

pith-pipeline@v0.9.0 · 5675 in / 1609 out tokens · 81411 ms · 2026-05-13T20:02:50.726002+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

39 extracted references · 39 canonical work pages

[1]

D. Aldous. Stopping Times and Tightness.The Annals of Probability, 6(2):335 – 340, 1978

work page 1978
[2]

Baldasso, O

R. Baldasso, O. Menezes, A. Neumann, and R. Rangel. Exclusion process with slow bound- ary .Journal of Statistical Physics, 167(5):1112–1142, 2017

work page 2017
[3]

Bertini, C

L. Bertini, C. Landim, and M. Mourragui. Dynamical large deviations for the boundary driven weakly asymmetric exclusion process.The Annals of Probability, 37(6):2357 – 2403, 2009

work page 2009
[4]

Boccardo, A

L. Boccardo, A. Dall’Aglio, and T. Gallou¨et. Nonlinear parabolic equations with Wentzell boundary conditions.Journal of Differential Equations, 127(2):429–456, 1996

work page 1996
[5]

Bouley and C

A. Bouley and C. Landim. Dynamical large deviations for boundary driven gradient sym- metric exclusion processes in mild contact with reservoirs.Journal of Statistical Physics, 191(10):132, 2024

work page 2024
[6]

D. Daners. Robin and Wentzell boundary conditions for elliptic equations.Journal of Func- tional Analysis, 255(11):3163–3197, 2008. FINITE RESERVOIRS LEAD TO WENTZELL BOUNDARY CONDITIONS 41

work page 2008
[7]

De Masi and E

A. De Masi and E. Presutti.Mathematical methods for hydrodynamic limits, volume 1501 ofLecture Notes in Mathematics. Springer-Verlag, Berlin, 1991

work page 1991
[8]

Durrett.Probability: Theory and Examples, volume 31 ofCambridge Series in Statistical and Probabilistic Mathematics

R. Durrett.Probability: Theory and Examples, volume 31 ofCambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, Cambridge, fourth edition, 2010

work page 2010
[9]

Erhard, T

D. Erhard, T. Franco, P. Gonc ¸alves, A. Neumann, and M. Tavares. Non-equilibrium fluc- tuations for the ssep with a slow bond.Annales de l’Institut Henri Poincar´e. Probabilit´es et Statistiques, 56(2):1099–1128, 2020

work page 2020
[10]

S. N. Ethier and T. G. Kurtz.Markov processes. Wiley Series in Probability and Mathemati- cal Statistics: Probability and Mathematical Statistics. John Wiley & Sons, Inc., New York,

work page
[11]

Characterization and convergence

work page
[12]

L. C. Evans.Partial differential equations, volume 19 ofGraduate Studies in Mathematics. American Mathematical Society , Providence, RI, 1998

work page 1998
[13]

Favini, G

A. Favini, G. R. Goldstein, J. A. Goldstein, and S. Romanelli. The heat equation with gen- eralized Wentzell boundary conditions.Journal of Evolution Equations, 2(1):1–19, 2002

work page 2002
[14]

Favini, G

A. Favini, G. R. Goldstein, J. A. Goldstein, and S. Romanelli. C 0-semigroups generated by second order differential operators with Wentzell boundary conditions.Proceedings of the American Mathematical Society, 132(1):235–244, 2004

work page 2004
[15]

W. Feller. The parabolic differential equations and associated semi-groups of transforma- tions.Annals of Mathematics. Second Series, 55(3):468–519, 1952

work page 1952
[16]

W. Feller. Diffusion processes in one dimension.Annals of Mathematics, 60(2):199–234, 1954

work page 1954
[17]

L. R. G. Fontes, M. Isopi, and C. M. Newman. Random walks with strongly inhomogeneous rates and singular diffusions: convergence, localization and aging in one dimension.The Annals of Probability, 30(2):579 – 604, 2002

work page 2002
[18]

Franco, P

T. Franco, P. Gonc ¸alves, C. Landim, and A. Neumann. Dynamical large deviations for the boundary driven symmetric exclusion process with Robin boundary conditions.ALEA, Latin American Journal of Probability and Mathematical Statistics, 19:1497–1546, 2022

work page 2022
[19]

Franco, P

T. Franco, P. Gonc ¸alves, and A. Neumann. Hydrodynamical behavior of symmetric exclu- sion with slow bonds.Ann. Inst. H. Poincar ´e Probab. Statist., 49(2):402–427, 2013

work page 2013
[20]

Franco, P

T. Franco, P. Gonc ¸alves, and A. Neumann. Phase transition in equilibrium fluctuations of symmetric slowed exclusion.Stoch. Proc. Appl., 123(12):4156–4185, 2013

work page 2013
[21]

Franco, P

T. Franco, P. Gonc ¸alves, and A. Neumann. Phase transition of a heat equation with Robin’s boundary conditions and exclusion process.Trans. Amer. Math. Soc., 367:6131–6158, 2015

work page 2015
[22]

Franco, P

T. Franco, P. Gonc ¸alves, and G. M. Sch¨utz. Scaling limits for the exclusion process with a slow site.Stoch. Proc. Appl., 2015

work page 2015
[23]

Franco and A

T. Franco and A. Neumann. Large deviations for the exclusion process with a slow bond. Ann. Appl. Probab., 27(6):3547–3587, 2017

work page 2017
[24]

Franco and M

T. Franco and M. Tavares. Hydrodynamic limit for the SSEP with a slow membrane.Jour- nal of Statistical Physics, 175(2):233–268, 2019

work page 2019
[25]

G. R. Goldstein. Derivation and physical interpretation of general boundary conditions. Advances in Differential Equations, 11(4):419–456, 2006

work page 2006
[26]

M. Jara, C. Landim, and A. Teixeira. Quenched scaling limits of trap models.The Annals of Probability, 39(1):176 – 223, 2011

work page 2011
[27]

Karatzas and S

I. Karatzas and S. E. Shreve.Brownian Motion and Stochastic Calculus, volume 113 of Graduate Texts in Mathematics. Springer, New York, 2 edition, 1991

work page 1991
[28]

Kipnis and C

C. Kipnis and C. Landim.Scaling limits of interacting particle systems, volume 320 of Grundlehren der mathematischen Wissenschaften. Springer-Verlag Berlin Heidelberg, 1st edition, 1999

work page 1999
[29]

Liang, R

J. Liang, R. Nagel, and T.-J. Xiao.Nonautonomous heat equations with generalized Wentzell boundary conditions, pages 321–331. Birkh¨auser Basel, Basel, 2004. 42 M. FRANCO, T. FRANCO, P. GONC ¸ALVES

work page 2004
[30]

T. M. Liggett.Interacting particle systems, volume 276 ofGrundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, New York, 1985

work page 1985
[31]

Luo and N

Y. Luo and N. S. Trudinger. Linear second order elliptic equations with Venttsel boundary conditions.Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 118(3– 4):193–207, 1991

work page 1991
[32]

De Masi, E

A. De Masi, E. Presutti, D. Tsagkarogiannis, and M. E. Vares. Current reservoirs in the simple exclusion process.Journal of Statistical Physics, 144(6):1151–1170, 2011

work page 2011
[33]

Mourragui, E

M. Mourragui, E. Saada, and S. Velasco. Hydrodynamic and hydrostatic limit for a gener- alized contact process with mixed boundary conditions.Electronic Journal of Probability, 28(none):1 – 44, 2023

work page 2023
[34]

E. C. Titchmarsh.Eigenfunction Expansions Associated with Second-Order Differential Equa- tions, volume I. Oxford University Press, 2nd edition, 1962

work page 1962
[35]

J. L. V ´azquez and E. Vitillaro. Heat equation with dynamical boundary conditions of reac- tive–diffusive type.Journal of Differential Equations, 250(4):2143–2161, 2011

work page 2011
[36]

M. Warma. Regularity and well-posedness for parabolic equations with general Wentzell boundary conditions.Journal of Evolution Equations, 5(1):167–185, 2005

work page 2005
[37]

M. Warma. Wentzell boundary conditions inR n and trace theorems.Advances in Differen- tial Equations, 10(10):1189–1232, 2005

work page 2005
[38]

A. D. Wentzell. Semigroups of operators corresponding to a generalized differential oper- ator of second order.Doklady Akademii Nauk SSSR, 111:269–272, 1956. In Russian

work page 1956
[39]

A. D. Wentzell. On boundary conditions for multidimensional diffusion processes.Teor. Veroyatnost. i Primenen., 4(2):172–185, 1959. 1 CENTER FORMATHEMATICALANALYSIS, GEOMETRY ANDDYNAMICALSYSTEMS, INSTITUTO SUPERIORT ´ECNICO, UNIVERSIDADE DELISBOA, AV. ROVISCOPAIS, 1049-001 LISBOA, PORTU- GAL Email address:matheusguilherme99@tecnico.ulisboa.pt 2 UFBA, INST...

work page 1959

[1] [1]

D. Aldous. Stopping Times and Tightness.The Annals of Probability, 6(2):335 – 340, 1978

work page 1978

[2] [2]

Baldasso, O

R. Baldasso, O. Menezes, A. Neumann, and R. Rangel. Exclusion process with slow bound- ary .Journal of Statistical Physics, 167(5):1112–1142, 2017

work page 2017

[3] [3]

Bertini, C

L. Bertini, C. Landim, and M. Mourragui. Dynamical large deviations for the boundary driven weakly asymmetric exclusion process.The Annals of Probability, 37(6):2357 – 2403, 2009

work page 2009

[4] [4]

Boccardo, A

L. Boccardo, A. Dall’Aglio, and T. Gallou¨et. Nonlinear parabolic equations with Wentzell boundary conditions.Journal of Differential Equations, 127(2):429–456, 1996

work page 1996

[5] [5]

Bouley and C

A. Bouley and C. Landim. Dynamical large deviations for boundary driven gradient sym- metric exclusion processes in mild contact with reservoirs.Journal of Statistical Physics, 191(10):132, 2024

work page 2024

[6] [6]

D. Daners. Robin and Wentzell boundary conditions for elliptic equations.Journal of Func- tional Analysis, 255(11):3163–3197, 2008. FINITE RESERVOIRS LEAD TO WENTZELL BOUNDARY CONDITIONS 41

work page 2008

[7] [7]

De Masi and E

A. De Masi and E. Presutti.Mathematical methods for hydrodynamic limits, volume 1501 ofLecture Notes in Mathematics. Springer-Verlag, Berlin, 1991

work page 1991

[8] [8]

Durrett.Probability: Theory and Examples, volume 31 ofCambridge Series in Statistical and Probabilistic Mathematics

R. Durrett.Probability: Theory and Examples, volume 31 ofCambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, Cambridge, fourth edition, 2010

work page 2010

[9] [9]

Erhard, T

D. Erhard, T. Franco, P. Gonc ¸alves, A. Neumann, and M. Tavares. Non-equilibrium fluc- tuations for the ssep with a slow bond.Annales de l’Institut Henri Poincar´e. Probabilit´es et Statistiques, 56(2):1099–1128, 2020

work page 2020

[10] [10]

S. N. Ethier and T. G. Kurtz.Markov processes. Wiley Series in Probability and Mathemati- cal Statistics: Probability and Mathematical Statistics. John Wiley & Sons, Inc., New York,

work page

[11] [11]

Characterization and convergence

work page

[12] [12]

L. C. Evans.Partial differential equations, volume 19 ofGraduate Studies in Mathematics. American Mathematical Society , Providence, RI, 1998

work page 1998

[13] [13]

Favini, G

A. Favini, G. R. Goldstein, J. A. Goldstein, and S. Romanelli. The heat equation with gen- eralized Wentzell boundary conditions.Journal of Evolution Equations, 2(1):1–19, 2002

work page 2002

[14] [14]

Favini, G

A. Favini, G. R. Goldstein, J. A. Goldstein, and S. Romanelli. C 0-semigroups generated by second order differential operators with Wentzell boundary conditions.Proceedings of the American Mathematical Society, 132(1):235–244, 2004

work page 2004

[15] [15]

W. Feller. The parabolic differential equations and associated semi-groups of transforma- tions.Annals of Mathematics. Second Series, 55(3):468–519, 1952

work page 1952

[16] [16]

W. Feller. Diffusion processes in one dimension.Annals of Mathematics, 60(2):199–234, 1954

work page 1954

[17] [17]

L. R. G. Fontes, M. Isopi, and C. M. Newman. Random walks with strongly inhomogeneous rates and singular diffusions: convergence, localization and aging in one dimension.The Annals of Probability, 30(2):579 – 604, 2002

work page 2002

[18] [18]

Franco, P

T. Franco, P. Gonc ¸alves, C. Landim, and A. Neumann. Dynamical large deviations for the boundary driven symmetric exclusion process with Robin boundary conditions.ALEA, Latin American Journal of Probability and Mathematical Statistics, 19:1497–1546, 2022

work page 2022

[19] [19]

Franco, P

T. Franco, P. Gonc ¸alves, and A. Neumann. Hydrodynamical behavior of symmetric exclu- sion with slow bonds.Ann. Inst. H. Poincar ´e Probab. Statist., 49(2):402–427, 2013

work page 2013

[20] [20]

Franco, P

T. Franco, P. Gonc ¸alves, and A. Neumann. Phase transition in equilibrium fluctuations of symmetric slowed exclusion.Stoch. Proc. Appl., 123(12):4156–4185, 2013

work page 2013

[21] [21]

Franco, P

T. Franco, P. Gonc ¸alves, and A. Neumann. Phase transition of a heat equation with Robin’s boundary conditions and exclusion process.Trans. Amer. Math. Soc., 367:6131–6158, 2015

work page 2015

[22] [22]

Franco, P

T. Franco, P. Gonc ¸alves, and G. M. Sch¨utz. Scaling limits for the exclusion process with a slow site.Stoch. Proc. Appl., 2015

work page 2015

[23] [23]

Franco and A

T. Franco and A. Neumann. Large deviations for the exclusion process with a slow bond. Ann. Appl. Probab., 27(6):3547–3587, 2017

work page 2017

[24] [24]

Franco and M

T. Franco and M. Tavares. Hydrodynamic limit for the SSEP with a slow membrane.Jour- nal of Statistical Physics, 175(2):233–268, 2019

work page 2019

[25] [25]

G. R. Goldstein. Derivation and physical interpretation of general boundary conditions. Advances in Differential Equations, 11(4):419–456, 2006

work page 2006

[26] [26]

M. Jara, C. Landim, and A. Teixeira. Quenched scaling limits of trap models.The Annals of Probability, 39(1):176 – 223, 2011

work page 2011

[27] [27]

Karatzas and S

I. Karatzas and S. E. Shreve.Brownian Motion and Stochastic Calculus, volume 113 of Graduate Texts in Mathematics. Springer, New York, 2 edition, 1991

work page 1991

[28] [28]

Kipnis and C

C. Kipnis and C. Landim.Scaling limits of interacting particle systems, volume 320 of Grundlehren der mathematischen Wissenschaften. Springer-Verlag Berlin Heidelberg, 1st edition, 1999

work page 1999

[29] [29]

Liang, R

J. Liang, R. Nagel, and T.-J. Xiao.Nonautonomous heat equations with generalized Wentzell boundary conditions, pages 321–331. Birkh¨auser Basel, Basel, 2004. 42 M. FRANCO, T. FRANCO, P. GONC ¸ALVES

work page 2004

[30] [30]

T. M. Liggett.Interacting particle systems, volume 276 ofGrundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, New York, 1985

work page 1985

[31] [31]

Luo and N

Y. Luo and N. S. Trudinger. Linear second order elliptic equations with Venttsel boundary conditions.Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 118(3– 4):193–207, 1991

work page 1991

[32] [32]

De Masi, E

A. De Masi, E. Presutti, D. Tsagkarogiannis, and M. E. Vares. Current reservoirs in the simple exclusion process.Journal of Statistical Physics, 144(6):1151–1170, 2011

work page 2011

[33] [33]

Mourragui, E

M. Mourragui, E. Saada, and S. Velasco. Hydrodynamic and hydrostatic limit for a gener- alized contact process with mixed boundary conditions.Electronic Journal of Probability, 28(none):1 – 44, 2023

work page 2023

[34] [34]

E. C. Titchmarsh.Eigenfunction Expansions Associated with Second-Order Differential Equa- tions, volume I. Oxford University Press, 2nd edition, 1962

work page 1962

[35] [35]

J. L. V ´azquez and E. Vitillaro. Heat equation with dynamical boundary conditions of reac- tive–diffusive type.Journal of Differential Equations, 250(4):2143–2161, 2011

work page 2011

[36] [36]

M. Warma. Regularity and well-posedness for parabolic equations with general Wentzell boundary conditions.Journal of Evolution Equations, 5(1):167–185, 2005

work page 2005

[37] [37]

M. Warma. Wentzell boundary conditions inR n and trace theorems.Advances in Differen- tial Equations, 10(10):1189–1232, 2005

work page 2005

[38] [38]

A. D. Wentzell. Semigroups of operators corresponding to a generalized differential oper- ator of second order.Doklady Akademii Nauk SSSR, 111:269–272, 1956. In Russian

work page 1956

[39] [39]

A. D. Wentzell. On boundary conditions for multidimensional diffusion processes.Teor. Veroyatnost. i Primenen., 4(2):172–185, 1959. 1 CENTER FORMATHEMATICALANALYSIS, GEOMETRY ANDDYNAMICALSYSTEMS, INSTITUTO SUPERIORT ´ECNICO, UNIVERSIDADE DELISBOA, AV. ROVISCOPAIS, 1049-001 LISBOA, PORTU- GAL Email address:matheusguilherme99@tecnico.ulisboa.pt 2 UFBA, INST...

work page 1959