RIMS Kôkyûroku Bessatsu
B79 (2020), 149–162

Hydrostatic limit for exclusion process with slow
boundary revisited
By

Kenkichi Tsunoda ∗Abstract
We revisit in this short article the hydrostatic limit for the exclusion process with slow
boundary. The original proof of this result relies on estimates of the correlation functions. We
achieve the same result based on analysis of two diﬀerent time scales, which do not need any
information about the correlation functions.

§ 1.

Introduction

We study in this article the limiting behavior of the empirical measure under thestationary state, called hydrostatic limit, for the exclusion process with slow boundary.
This model has been introduced in R. Baldasso, O. Menezes, A. Neumann and R. R.
Souza [1], and can be described as follows. Let N ∈ N be a scaling parameter. Eachparticle in the bulk {1, . . . , N − 1} behaves as an independent simple random walk with
exclusive constraints. The terminology slow boundary means that particles are createdor annihilated at the boundary, at a rate proportional to N −θ for some θ ≥ 0. It has
been established in Baldasso et al. [1] that the following phase transition occurs: theboundary condition of the hydrodynamic equation is governed by Dirichlet boundary if
θ < 1, Robin boundary if θ = 1 and Neumann boundary if θ > 1, respectively. We omit
to introduce more detailed description and a historical background of this model here,
see Baldasso et al. [1] and references therein.
The purpose of this article is to introduce another proof of the hydrostatic limit,
stated in Theorem 2.2. It is worth mentioning that our proof does not use any information about the correlation functions, while the original one strongly relies on estimates ofReceived February 17, 2019. Revised May 7, 2019.
2010 Mathematics Subject Classification(s): Primary 60K35, secondary 82C22
Key Words: Exclusion process with slow boundary, Hydrodynamic limit, Hydrostatic limit
Supported by JSPS KAKENHI, Grant-in-Aid for Early-Career Scientists 18K13426∗ Department of Mathematics, Osaka University, Osaka, 560-0043, Japan.
e-mail: k-tsunoda@math.sci.osaka-u.ac.jp.
c 2020 Research Institute for Mathematical Sciences, Kyoto University. All rights reserved.
⃝

150

Kenkichi Tsunoda

the correlation functions. Although our proof can be applied to other particle systems,
we concentrate on the exclusion process with slow boundary in this article to make thepresentation simplest.
The original method we follow in this article has been introduced in Farfan, Landim
and Mourragui [3], to show the hydrostatic limit for the boundary gradient driven
symmetric exclusion process. This method has been generalized to the case of thereaction-diﬀusion model in Landim and Tsunoda [10]. In fact, Landim and Tsunoda’s
method is robust enough to imply Theorem 2.2 for θ ≤ 1, see Section 3. However, theresult established in Section 3 is not enough to deduce Theorem 2.2 for θ > 1. This issue
will be examined in the first paragraph of Section 4. To complete the proof of Theorem
2.2, we further develop Landim and Tsunoda’s method in the case θ > 1, where theboundary condition of the hydrodynamic equation is governed by Neumann boundary
conditions. The method developed in Section 4, which is a main contribution of thisarticle, seems somewhat new and may be of interest in other contexts.
We remark on several papers related to this work, but only on papers published after
Baldasso et al. [1]. The main motivation of this work is based on recent developments on
stationary nonequilibrium states. See Bertini et al [2] for this subject. The equilibriumand non-equilibrium fluctuations for the exclusion process with slow boundary andrelated models have been investigated in a series of studies by T. Franco, P. Gonçalves
and A. Neumann and their collaborators: [5, 4, 7]. The large deviation for the exclusion
process with a slow bond is examined in T. Franco and A. Neumann [6].
This article is organized as follows. In Section 2, we introduce the exclusion processwith slow boundary precisely. We also examine results on the hydrodynamic and hydrostatic limit, established in Baldasso et al. [1], in Subsections 2.2, 2.3, respectively. The
original proof of the hydrostatic limit is also examined in Subsection 2.3. In Sections
3, 4, we study the diﬀusive time scale or a certain sub-diﬀusive time scale, and deduce
Theorem 2.2 for θ ≤ 1 and for θ > 1, respectively.
§ 2.

Model and main result

We introduce in this section the exclusion process with slow boundary and state
the hydrodynamic and hydrostatic limit for this particle system. We constantly refer
the reader to Baldasso et al. [1] as most of the statements in this section borrow fromthe ones of Baldasso et al. [1].
§ 2.1.

Exclusion process with slow boundary

For each N ∈ N, let IN be the one-dimensional discrete interval {1, . . . , N − 1}.
Elements of IN are represented by the letters x, y and z, while an element of the continuum interval [0, 1] is represented by the letter u. Denote the configuration space byHydrostatic limit for exclusion process with slow boundary revisited151ΩN = {0, 1}IN , and its element, called configuration, by η = {η(x) : x ∈ IN }. For
each x ∈ IN , η(x) represents the number of particles sitting at site x, in other words,
η(x) = 1 if there is a particle at site x, η(x) = 0 otherwise. For a configuration η ∈ ΩN ,let η x,y and η x be the configurations obtained from η by exchanging the occupation
variables η(x) and η(y), by flipping the occupation variable η(x), respectively:1 − η(x) if z = x ,η(y) if z = x ,xx,yη (z) =
η (z) = η(x) if z = y ,η(z)if z ̸= x .
η(z) otherwise ,We introduce the exclusion process with slow boundary, which is a Markov process
on ΩN whose generator is given byβ
LN = LN,0 + Lα
N,b + LN,b ,with some fixed α, β ∈ (0, 1). In the previous formula, LN,0 stands for the generator of
the symmetric simple exclusion process in IN , that is, LN,0 acts on functions f : ΩN → R
as
LN,0 f (η) =

N
−2
∑

[
]
f (η x,x+1 ) − f (η) .

x=1
β
On the other hand, Lα
N,b and LN,b correspond to the dynamics at the left and rightboundary, respectively, which act functions f : ΩN → R as
[
]
−θ
Lα
rα (η) f (η 1 ) − f (η) ,N,b f (η) = cN[]
LβN,b f (η) = cN −θ rβ (η) f (η N −1 ) − f (η) ,whererα (η) = α [1 − η(1)] + (1 − α)η(1) ,rβ (η) = β [1 − η(N − 1)] + (1 − β)η(N − 1) ,with some fixed c > 0 and θ ≥ 0.
Denote by νρN the product Bernoulli measure on ΩN with density ρ ∈ [0, 1]. It is
well known that νρN is symmetric with respect to LN,0 for any ρ ∈ [0, 1]. Since rα and rβ
are chosen to satisfy the detailed balance conditions with respect to ναN and νβN , theseβmeasures are symmetric with respect to Lα
N,b and LN,b , respectively. However, it is also
well known that the Bernoulli measures are not invariant with respect to LN unlessα = β. Since the cardinality of the state space ΩN is finite and the Markov process
corresponding to LN is irreducible, there exists a unique stationary state, denoted byµss
N , which is invariant under the dynamics.

152

Kenkichi Tsunoda§ 2.2.

Hydrodynamic limit

It has been investigated in Baldasso et al. [1] that the boundary condition of thehydrodynamic equation depends on the parameter θ. More precisely, the hydrodynamic
behavior of the exclusion process with slow boundary is described as follows. Assume
for a while that the macroscopic density at time 0 is given by a measurable function
ρ0 : [0, 1] → [0, 1]. For any θ ≥ 0, the system in the bulk evolves according to the heat
equation in (0, 1):∂ ρ(t, u) = ∂ 2 ρ(t, u) ,tu
ρ(0, u) = ρ0 (u) ,where ρ(t, u) stands for the macroscopic density at time t ≥ 0 and position u ∈ [0, 1]. In
the case θ < 1, the boundary condition is governed by Dirichlet boundary conditions:ρ(t, 0) = α ,ρ(t, 1) = β .
In the case θ = 1, the boundary condition is governed by Robin boundary conditions:∂ ρ(t, 0) = c [ρ(t, 0) − α] ,u∂u ρ(t, 1) = c [β − ρ(t, 1)] .
In the case θ > 1, the boundary condition is governed by Neumann boundary conditions:∂ ρ(t, 0) = 0 ,u∂u ρ(t, 1) = 0 .
We do not review precise definitions of weak solutions to these Cauchy problems here,
see [1, Subsection 2.3] for them. For each θ ≥ 0, denote these Cauchy problems by(HDE)θ .
For each N ∈ N, denote by {StN : t ≥ 0} the semigroup associated to the Markov
process generated by N 2 LN and by µN a given initial distribution. Note that the
distribution of the process at time t is given by µN StN .
The following result has been established in Baldasso et al. [1].
Theorem 2.1 (Hydrodynamic limit).
Assume that the initial distribution µN is
associated to a measurable function ρ0 : [0, 1] → [0, 1]. Namely, it holds for any δ > 0
and continuous function H : [0, 1] → R that
)
(
∫ 1
∑
1
H(x/N )η(x) −H(u)ρ0 (u)du ≥ δ = 0 .
lim µN η :
N →∞
N −1
0
x∈IN

Hydrostatic limit for exclusion process with slow boundary revisited

153

Then, for any t ≥ 0, δ > 0 and continuous function H : [0, 1] → R, we have
(
)
∫ 1
∑
1
lim µN StN η :H(x/N )η(x) −H(u)ρ(t, u)du ≥ δ = 0 .
N →∞
N −1
0
x∈IN

where ρ = ρ(θ) : [0, ∞) × [0, 1] → [0, 1] stands for the unique weak solution to (HDE)θ .
§ 2.3.

Hydrostatic limit

We examine in this subsection the hydrostatic limit established in Baldasso et al.
[1] and outline their proof to clarify the diﬀerence between their approach and ours. The
hydrodynamic limit describes the dynamical behavior of the empirical measure while
the hydrostatic limit states the law of large numbers for the empirical measure under
the stationary state µss
N.
For each θ ≥ 0, let ρθ : [0, 1] → [0, 1] be the function defined byρD (u) = (β − α)u + α ,if θ < 1 ,c(β − α)β−αu+α+, if θ = 1 ,ρθ (u) = ρR (u) =
2
+
c
2
+
c




ρN (u) = β + α ,if θ > 1 .
2
Note that, for each θ ≥ 0, ρθ is a stationary solution to (HDE)θ .
The following result has been established in Baldasso et al. [1].
Theorem 2.2 (Hydrostatic limit).
For any δ > 0 and continuous function H :[0, 1] → R, we have
(
)
∫ 1
∑
1
H(x/N )η(x) −lim µssη:
H(u)ρθ (u)du ≥ δ = 0 .
N
N →∞
N −1
0
x∈IN

We here outline the proof given in Baldasso et al. [1]. Their proof is summarized
as follows. For x, y ∈ IN , let ρN (x) and φN (x, y) be the mean of η(x) and the two-point
correlation function of η(x), η(y) under the stationary state µss
N , respectively:∫Nη(x)µss
ρ (x) =N (dη) ,
ΩN
∫
[
][
]
N
φ (x, y) =η(x) − ρN (x) η(y) − ρN (y) µss
N (dη) .
ΩN

Since µss
N is invariant with respect to LN , for each x ∈ IN , we have∫
LN η(x)µss
N (dη) = 0 .
ΩN

154

Kenkichi Tsunoda

Computing the left-hand side, we can obtain a system of linear equations for {ρN (x) :x ∈ IN }, see the proof of [1, Lemma 3.1] for this system. Since this system is linear, it
is not diﬃcult to obtain the explicit formula(2.1)

ρN (x) = aN x + bN ,x ∈ IN ,

where
( θ
)
c(β − α)NaN =and bN = α + aN
−1 .
2N θ + c(N − 2)c[][]
A similar computation for η(x) − ρN (x) η(y) − ρN (y) together with some coupling argument permits us to obtain the estimate
(2.2)

max

0<x<y<N

φN (x, y) ≤CNθ + N,for some constant C > 0. Theorem 2.2 easily follows from (2.1), (2.2) and standard
arguments based on the Chebyshev inequality.
We conclude this subsection mentioning a few comments on the proof. For θ ≤ 1,note that the function ρθ is the unique stationary solution to (HDE)θ . This fact together
with Proposition 3.1 below implies Theorem 2.2 immediately. However, for θ > 1, the setof stationary solutions to (HDE)θ is not a singleton, since the corresponding Neumann
Laplacian on [0, 1] has the eigenvalue 0 in its spectrum. Therefore, the concentration
result, given in Proposition 3.1, does not imply Theorem 2.2 for θ > 1. To overcome
this diﬃculty, besides Proposition 3.1, we need another characterization of the density(α + β)/2 amoung [0, 1], which is a limiting density in the case θ > 1. Indeed, we will
see that (α + β)/2 can be characterized as a unique attractor of the integral equation(4.4). This is what we will investigate in Section 4.
An approach based on the estimates for the correlation functions is very useful forseveral problems if available, see for instance [5, 4, 7]. However, this approach can not beapplied to almost all interacting systems, even so-called gradient systems. Compared tothis approach, the method developed in this paper is robust enough to be applicable toa gradient particle system (should be possible for a non-gradient system). For instance,
one can obtain similar results for the setting of Farfan et al. [3] with slow boundary.
§ 3.

The diﬀusive time scale N 2

We investigate in this section the diﬀusive time scale, to analyze the empiricalmeasure under µss
N . As examined in Section 1, we shall follow the method developedin Landim and Tsunoda [10], to prove some concentration result, stated in Proposition

Hydrostatic limit for exclusion process with slow boundary revisited155

3.1. As a direct consequence of Proposition 3.1, which is a main result of this section,
we shall prove Theorem 2.2 for θ ≤ 1.
Let M+ be the set of all Borel measures on [0, 1], whose total mass is bounded
above by 1. M+ is endowed with the weak topology, which is metrizable and becomes
a compact Polish space. Denote its metric by d, see for instance [10, Subsection 2.2]
for the definition of d. For a masure π ∈ M+ and a function H : [0, 1] → R, denoteby ⟨π, H⟩ the integral of H with respect to π whenever it has a meaning. For functions
H1 , H2 : [0, 1] → R, we also denote by ⟨H1 , H2 ⟩ the L2 -inner product with respect to
the Lebesgue measure du whenever it has a meaning.
For a configuration η ∈ ΩN , define the empirical measure byπN (du) = π(η, du) =∑1
η(x)δx/N (du) ,
N −1
x∈IN

where δu stands for the point mass at u ∈ [0, 1]. Recall the definition of the stationary
state µss
N , introduced at the last paragraph of Subsection 2.1. Define the probability
measure PN on M+ by−1
PN = µss.
N ◦ (πN )For each θ ≥ 0, let Eθ be the set of all measures π(du) = ρ(u)du in M+ whose
density is a stationary solution to (HDE)θ . It is easy to see that Eθ coincides with
{ρD (u)du} for θ < 1, {ρR (u)du} for θ = 1 and {ϱdu : ϱ ∈ [0, 1]} for θ > 1, respectively.
Following the proof of [10, Theorem 2.2], we can prove the following proposition:
Proposition 3.1.
The sequence of measures {PN }N ∈N asymptotically concentrates on the set Eθ . Namely, for any δ > 0, we have)(lim PN π ∈ M+ : inf d(π, π) ≥ δ = 0 .
N →∞

π∈Eθ

The proof of this proposition is consisting of two main ingredients, as examined in
the first paragraph of [10, Section 3]: the macroscopic density of the system is describedby a hydrodynamic limit, and for any initial profile the solution of the hydrodynamic
equation converges to some stationary solution as time goes to infinity. Indeed, theexclusion process with slow boundary and its hydrodynamic equation satisfy these two
properties for any θ ≥ 0. Invoking these properties, the proof of Proposition 3.1 iscompletely same as the one of [10, Theorem 2.2] and thus is omitted.
Note that Eθ is a singleton for each θ ≤ 1: Eθ = {ρD (u)du} or {ρR (u)du}. Theorem
2.2 for θ ≤ 1 follows from Proposition 3.1 immediately.
Proof of Theorem 2.2 for θ ≤ 1. From Proposition 3.1 and the fact that Eθ isa singleton for each θ ≤ 1, the empirical measure πN under µss
N converges to ρθ (u)du156

Kenkichi Tsunoda

as N → ∞ in probability. Therefore, for any continuous function H : [0, 1] → R, therandom variable ⟨πN , H⟩ under µss
N converges to ⟨ρθ , H⟩ as N → ∞ in probability,
which completes the proof of Theorem 2.2 for θ ≤ 1.

§ 4.

The sub-diﬀusive time scale N 1+θ

In the rest of this paper, we always treat with the case θ > 1. Since the solution
to the heat equation with 0-Neumann boundary conditions conserves the total mass,
the total number of particles in IN can not evolve under the diﬀusive time scale. Thisis exactly caused by the presence of slow boundary. However, at the process level, we
can observe exchange of particles at a rate proportional to N −θ through the boundary.
Therefore, to observe the correct evolution of the total number of particles in IN , weneed to introduce another time scale, which should be longer than the diﬀusive timescale. As understood in the computations below, the correct speeded up factor (or thetime scale) is given by N 1+θ , which is in fact longer than the diﬀusive time scale in thecase θ > 1.
For the sake of the previous paragraph, let {ηtN : t ≥ 0} be the Markov processN
generated by N 1+θ LN with the initial distribution µss
N . For each t ≥ 0, dente by mt
the averaged density defined by
mN
t =

∑
1
ηtN (x) .
N −1
x∈IN

By the reason examined in the previous paragraph, mNt does not evolve under the diﬀu2
sive time scale N . On the other hand, as we will see later, mN
t evolves macroscopically1+θ
under the time scale N.
For each T > 0, let D([0, T ], R) be the set of all càdlàg trajectories m· : [0, T ] → R,
endowed with the Skorokhod topology. For each N ∈ N, let QN = QN,T be the
distribution of {mN· } on D([0, T ], R).
Our approach to study the sequence {QN }N ∈N is based on a standard machinery
used in the study of hydrodynamic limit. We first show the relative compactness ofthe sequence {QN }N ∈N and characterize its all limit points. This is the content of
Propositions 4.1, 4.4 below, respectively.
We start with the relative compactness of the sequence {QN }N ∈N .
Proposition 4.1.

The sequence {mN· }N ∈N is relatively compact in D([0, T ], R).

Proof. Fix T > 0. It is enough to show that the sequence {mN· }N ∈N is relatively
compact in D([0, T ], R). For this purpose, introduce the function GN (η) = G(η) =Hydrostatic limit for exclusion process with slow boundary revisited(N − 1)−1157∑x∈IN η(x) and the corresponding Dynkin’s martingale:

∫ t
(4.1)

MtN

=

G(ηtN ) − G(η0N ) − N 1+θ

(LN G)(ηsN )ds ,

t≥0.

0
N
N
It follows from the definition of mNt that G(ηt ) = mt . Since the total number ofparticles in IN is conserved by LN,0 , we have LN,0 G = 0. One the other hand, as LαN,bβand LN,b act only at the left and right boundary, respectively, we haveβ(LαN,b + LN,b )G(η) ==cN θ (N − 1){rα (η) [1 − 2η(1)] + rβ (η) [1 − 2η(N − 1)]}c
[α + β − η(1) − η(N − 1)] .
N θ (N − 1)

Therefore (4.1) can be rewritten as
(4.2)

mN
t

∫ t
[
]
cN
=
α + β − ηsN (1) − ηsN (N − 1) ds
N −1 0∫ t[]
NN= m0 + Mt + c
α + β − ηsN (1) − ηsN (N − 1) ds + O(N −1 ) ,N
mN
0 + Mt +0

where big O notation stands for the Bachman-Landu notation.
N
Note that the sequence {mN0 }N ∈N is relatively compact since m0 takes values in[0, 1] for any N ∈ N. On the other hand, in view of Aldous’s criterion, cf. [8, page51, Proposition 4.1.6], we can obtain the relative compactness of the integral term in
the last line of (4.2). Therefore, to conclude the proof, it is enough to show that thesequence {M·N }N ∈N is relatively compact in D([0, T ], R).
Indeed, it follows from a straightforward computation that the quadratic variationof MtN is given by∫ tcN
ηsN (1) − α + ηsN (N − 1) − β ds = O(N −1 ) .
(4.3)
2(N − 1) 0This formula together with the standard argument as in the proof of [8, page 55, Theorem 4.2.1] gives the relative compactness for the sequence {M·N }N ∈N , which completes
the proof of Proposition 4.1.
It follows from (4.3) that the martingale term MtN vanishes in the limit N → ∞.
Therefore, if we can replace ηsN (1) + ηsN (N − 1) by 2mN
s in (4.2), we can obtain thefollowing integral equation in the limit:
∫ t
(4.4)
mt = m0 + c[α + β − 2ms ] ds , t ≥ 0 .
0This replacement can not be achieved in the diﬀusive time scale since the relaxation
time, which is the inverse of the spectral gap, of the exclusion process inside a box with

158

Kenkichi Tsunodaside length ℓ is of order ℓ2 . However, such a replacement should be achieved in the timescale N 1+θ . This is the idea hidden in the proof of Lemma 4.2, so-called replacement
lemma.
Before starting the proof of the replacement lemma, we introduce some notationand estimates, which will be used in the proof of the replacement lemma.
For two probability measures µ, ν on ΩN , let HN (µ|ν) be the relative entropy of µwith respect to ν:{∫}∫f
HN (µ|ν) = supf dµ − loge dν ,fΩNΩN

where the supremum is carried over all functions f : ΩN → R. It is well known that∫dµdµ
HN (µ|ν) =
log
dν ,
dν
ΩN dν
if µ is absolutely continuous with respect to ν, HN (µ|ν) = ∞, otherwise. Since there is
at most one particle per site, there exists a constant C0 = C0 (α) > 0 such that
HN (µ|ναN ) ≤ C0 N ,(4.5)

for any probability measure µ on ΩN .
∫A function f : ΩN → [0, ∞) is said to be a density if f dναN = 1. For any density
f , define the Dirichlet form with respect to ναN by
N −2 ∫
[√
]2
√
1 ∑
N
DN,0 (f ; νa ) =f (η x,x+1 ) − f (η) dναN .
2 x=1 ΩN
It has been established in the proof of [1, Lemma 5.9] that there exists a constantCα,β > 0 such that
√ √
Cα,β
(4.6)
,
⟨LN f , f ⟩α ≤ −DN,0 (f ; ναN ) +Nθ
for any density f , where ⟨·, ·⟩α stands for the L2 -inner product with respect to ναN . Since
the actual value of the density of the reference measure is not important, we always fix
it to be α.
From the observation examined after the proof of Proposition 4.1, introduce thefunction V = VN given by V (η) = η(1)+η(N −1)−2G(η), where G has been introducedin the proof of Proposition 4.1.
We are ready to prove the replacement lemma.
Lemma 4.2 (Replacement lemma).
For any t ≥ 0, we have
]
[∫ t
N
N
(4.7)
V (ηs )ds = 0 ,
lim E
N →∞

where EN

0stands for expectation with respect to the process η·N .

Hydrostatic limit for exclusion process with slow boundary revisited

159

Proof. For any γ > 0, from the entropy inequality and (4.5), we have
[∫ t
]
[
{
}]
∫ t
C0
1
N
N
N
(4.8)
E
V (ηs )ds ≤+log Eα exp γNV (ηs )ds
,
γ
γN
0
0
where Eα stands for expectation with respect to the process starting from the productmeasure ναN . One can get rid of the absolute value in the right-hand side of (4.8) by theelementary inequality e|x| ≤ ex +e−x . Therefore, the estimate for the second term in theright-hand side of (4.8) is reduced to the one without the absolute value. Furthermore,from [1, Lemma 7.3], to conclude the proof, it is enough to show that the following
variational expression vanishes as N → ∞ and γ → ∞:
{
}
√ √
−1 θ
(4.9)
sup γ N ⟨LN f , f ⟩α + ⟨V, f ⟩α ,fwhere the supremum is carried over all densities f .
From (4.6) and Lemma 4.3 below, the supremum (4.9) is bounded above by{}
Cα,β(4.10)+ sup −γ −1 N θ DN,0 (f ; ναN ) + 4N 1/2 DN,0 (f ; ναN )1/2 .
γf
The previous supremum is easily computed and is bounded above by 4γN 1−θ . Since θis larger than 1, the expression (4.10) vanishes as N → ∞ and γ → ∞, which completes
the proof of Lemma 4.2.
The following lemma in fact has been proved in the proof of [9, Lemma 3.1]. However, we give the proof for reader’s convenience.
Lemma 4.3 (Moving particle lemma).

For any density f , we have⟨V, f ⟩α ≤ 4N 1/2 DN,0 (f ; ναN )1/2 .

(4.11)

Proof. Fix a density f . The left-hand side in (4.11) can be written as∑ ∫1(4.12){[η(1) − η(x)] + [η(N − 1) − η(x)]} f (η)ναN (dη) .
N −1
ΩN
x∈IN

In the following argument, we give an estimate for the sum involving η(1) − η(x) only
since the other sum is similar.
For each x ∈ IN , by the change of variables η 7→ η 1,x , the sum involving η(1) − η(x)in (4.12) can be rewritten as
∑ ∫
[
]
1
[η(1) − η(x)] f (η) − f (η 1,x ) ναN (dη)
2(N − 1)x∈IN ΩN
] [√
]
[√
∑ ∫
√
√
1
=
f (η) + f (η 1,x )f (η) − f (η 1,x ) ναN (dη) .
[η(1) − η(x)]2(N − 1)
ΩN
x∈IN

160

Kenkichi Tsunoda

Since there is at most one particle per each site and f is a density, applying the Schwartz
inequality, the last expression is bounded above by(4.13)∑
1(N − 1)

x∈IN

{∫

[√

f (η) −√]2f (η 1,x )}1/2ναN (dη).

ΩN

For each x, y ∈ IN , consider the transformation σ x,y on ΩN defined by σ x,y η =η x,y , η ∈ ΩN . Clearly, σ x,y is ναN -measure preserving and satisfies the relation
σ 1,x = σ 1,2 ◦ σ 2,3 ◦ · · · ◦ σ x−2,x−1 ◦ σ x,x−1 ◦ σ x−1,x−2 ◦ · · · ◦ σ 3,2 ◦ σ 2,1 ,for any x ∈ IN , where the symbol ◦ stands for the composition of transformations. By
adding and subtracting the terms by this sequence into the brackets in (4.13), from the
Cauchy-Schwarz inequality, we have
∫ [√
]2
√
1,x
f (η) − f (η ) ναN (dη) ≤ 4N DN,0 (f ; ναN ) ,ΩNwhich in turn implies the conclusion of Lemma 4.3. Note that the constant 4 in (4.11)
comes from the contribution of the sum involving η(N − 1) − η(x) in (4.12).
We summarize the previous computations as a single proposition, which plays afundamental role in the proof of Theorem 2.2. However, as the proof follows from theformula (4.2) and Lemma 4.2 easily, we omit the proof.
Proposition 4.4.
Let A be the set of all trajectories {mt : t ≥ 0} in D([0, T ], R)
satisfying the integral equation (4.4) with the initial value m0 in [0, 1]. Then, any limit
point Q∗ of the sequence {QN }N ∈N is concentrated on A, namely, Q∗ (A) = 1.
The following lemma states that (α+β)/2 can be characterized as a unique attractorof the integral equation (4.4).
Proposition 4.5.

The solution of the integral equation (4.4) is given by
(
)
α+β
α+β
mt =
+ m0 −
e−2ct .
2
2

In particular, mt converges to (α + β)/2 as t → ∞, uniformly in initial values in [0, 1].
The proof of this proposition is elementary, and left to the reader.
We have now all the ingredients to prove Theorem 2.2 for θ > 1.
Proof of Theorem 2.2 for θ > 1. Since M+ is compact, the sequence {PN }N ∈N isrelatively compact. Let P∗ , Q∗ be any limit point of the sequence {PN }N ∈N , {QN }N ∈N ,respectively. Take a subsequence Nk , if necessary, so that the sequences {PNk }k∈N , {QNk }k∈NHydrostatic limit for exclusion process with slow boundary revisited161

converge to P∗ , Q∗ , respectively. Note that Proposition 3.1 shows that P∗ (Eθ ) = 1. Recall the definition of the function ρN : ρN (u) = (α + β)/2, u ∈ [0, 1]. To conclude theproof, it is enough to show that
(4.14)P∗ ({ρN (u)du}) = 1 .

Fix δ > 0. Denote by Oδ the subset of [0, 1] given by
]
[
α+β
α+β
− δ,
+δ ,
Oδ = [0, 1] \2
2and by Oδ the closure of Oδ . From the stationarity of µss
N , we have
(4.15)PN (π : ⟨π, 1⟩ ∈ Oδ ) = µss
N (η : G(η) ∈ Oδ ) = QN (m· : mt ∈ Oδ ) ,for any t ≥ 0, where 1 stands for the constant function 1(u) = 1, u ∈ [0, 1].
Since the application π 7→ ⟨π, 1⟩ is continuous with respect to the weak topology,
and PNk , QNk converge to P∗ , Q∗ weakly, respectively, we haveP∗ (π : ⟨π, 1⟩ ∈ O2δ ) ≤ lim inf PNk (π : ⟨π, 1⟩ ∈ O2δ )k→∞= lim inf QNk (m· : mt ∈ O2δ )k→∞)(≤ lim sup QNk m· : mt ∈ Oδk→∞()
≤ Q∗ m· : mt ∈ Oδ .
We used (4.15) to obtain the second equality and the monotonicity of QNk the third
inequality. For the last inequality, one should pay an attention since the application
m· 7→ mt is not continuous with respect to the Skorokhod topology. However, one canjustify this inequality by the fact that Q∗ is concentrated on continuous trajectories,
see the proof of [10, Theorem 2.2] for a similar argument. Since Oδ does not contain()(α + β)/2, it follows from Propositions 4.4, 4.5 that Q∗ m· : mt ∈ Oδ vanishes if t islarger than −(2c)−1 log δ. Thus, (4.14) has been shown. This completes the proof of
Theorem 2.2 for θ > 1.
Acknowledgements
The author would like to thank Professor Tomoyuki Shirai for giving him an opportunity to attend the international conference Stochastic dynamics out of equilibrium
held in Institut Henri Poincaré from June 12–16, 2017. Most of the proof presented in
Section 4 has been given during this visit. He is also grateful to the anonymous referee
for the comments that have been helpful in revising the paper.

162

Kenkichi Tsunoda

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