RIMS Kôkyûroku Bessatsu
B79 (2020), 1–17

Analysis of Metastable Behavior via Solutions ofPoisson Equations
By

Insuk Seo∗

Abstract
We herein review the recent progress on the study of metastability based on the analysis
of solutions of Poisson equations related to the generators of the underlying metastable dynamics. This review paper is based on the joint work with Claudio Landim [24] and Fraydoun
Rezakhanlou [26].

§ 1.

Quantitative Analysis of Metastable Behavior

Metastable behavior is a ubiquitous phenomenon exhibited by random dynamics
in a low-temperature regime. To concretely describe this behavior, we first introduce aclassic model known as the small random perturbation of dynamical systems (SRPDS,
see [13] for an extensive discussion). For a smooth potential function U : Rd → R andsmall parameter ϵ > 0, we consider a stochastic diﬀerential equation given by(1.1)

dy ϵ (t) = −∇U (y ϵ (t))dt +√2ϵ dwt ; t ≥ 0 ,where (wt )t≥0 represents the standard d-dimensional Brownian motion. If U has several
local minima as in Figure 1, the stochastic process (y ϵ (t))t≥0 exhibits the metastable behavior. To describe this behavior, we consider the zero-temperature dynamics describedby the following ordinary diﬀerential equation:(1.2)

dy(t) = −∇U (y(t))dt ; t ≥ 0 .

Received December 26, 2018. Revised October 15, 2019.
2010 Mathematics Subject Classification(s): 60J60; 60F99.
Key Words: Non-reversible diﬀusions; Potential theory; Metastability; Dirichlet principle; Thomson principle; Eyring–Kramers formula.
This work is supported by the National Research Foundation of Korea (NRF), grant funded by the
Korea government (MSIT) (No. 2018R1C1B6006896 and No. 2017R1A5A1015626).
∗ Department of Mathematical Sciences and R.I.M., Seoul National University, 27-212, Gwanak-Ro
1, Gwanak-Gu, Seoul 08826, Republic of Korea.
c 2020 Research Institute for Mathematical Sciences, Kyoto University. All rights reserved.
⃝

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Insuk Seo

Figure 1. Example of potential functions: (Left) Double-well potential, (Right) Generalpotential with several global minima
Then, we can observe that the local minima of the potential function U is the stableequilibria of the dynamics (y(t))t≥0 . Namely, the dynamics starting at a domain of
attraction of a local minimum m1 of U converges to m1 exponentially fast. We canregard (1.1) as a small random perturbation of the deterministic dynamical system(1.2), provided that ϵ > 0 is small enough; hence, one can expect a similar behavior;
the random process (y ϵ (t))t≥0 starting at a neighborhood of the local minimum m1 of
U will converge to m1 .
This estimates is locally true; however, if we consider the global picture, a crucial
diﬀerence arises because of the randomness induced by the Brownian motion: if we
wait for a suﬃciently long time, then the process will move from the neighborhood of
m1 to that of another local minimum, e.g., m2 (see Figure 1-(Right)). We now callthe neighborhoods of each local minimum as the metastable set, and the transitions
among these sets explained above are called the metastable transition. We expectthat this metastable transition occurs repeatedly in a suitable time scale, and is an
example of metastable behavior. This type of behavior is exhibited by numerous models,e.g., interacting particle systems such as the zero-range processes [1, 4, 18, 27], simple
inclusion processes [6, 16], and ferromagnetic systems such as the Curie--Weiss model[5], Ising model [11] and, Potts model [23, 25].
To explain the primary questions in this study, we only focus on the SRPDS (1.1)
in this introductory section. In the SRPDS, we can consider two cases as below, andthe primary concerns are slightly diﬀerent for each case.
• Case 1: U contains several local minima but only one global minimum as in Figure 1-(Left). In this case, the process starting from any point first stabilizes at aneighborhood of the local minima. Subsequently, after a suﬃciently long time, it
performs a metastable transition toward the neighborhood of the global minimum,
and remains therein for a longer time scale than the metastable transition time.

Analysis of Metastable Behavior via Solutions of Poisson Equations

3

Therefore, a primary object to be investigated is this metastable transition timefrom a local minimum to the global minimum. For instance, its expected value andasymptotic law are the primary concern. The robust methodology answering thisquestion in a quantitative manner is the potential-theoretic approach developed by
Bovier et al. in [7, 8]. This methodology is explained in Section 2.1.
• Case 2: U contains multiple global minima as in Figure 1-(Right). As explained
previously, a process starting from a neighborhood of a minimum will be stabilized
in this set. However, after a suﬃciently long time, we observed the metastabletransition, and expect that this transition occurs repeatedly. Therefore, describing
such hopping dynamics in a rigorous manner is an important problem from themathematical perspective. In particular, one expects to demonstrate that a scaling
limit of this process converges to a Markov chain among the metastable sets, in asuitable sense. In particular, if the underlying metastable dynamics is a Markov
process on a finite set, a robust methodology for proving this scaling limit, which iscalled martingale approach, has been established by Beltran and Landim in [2, 3].
We explain this approach in Section 2.2.
Our new methodology introduced in Section 3 can be regarded as another approachto analyze Case 2, and provides concrete answers for some questions that cannot beanswered by both of the approaches explained previously. In particular, we can conduct
a rigorous analysis explained in Case 2, when the underlying dynamics is a diﬀusion
process that is not a process on a finite set. We refer to [24, 26] for details. In this review
paper, instead of focusing on this specific model, we attempt to deliver our general ideaon this new approach by skipping the technical details. In Section 2, we review theprevious approaches; in Section 3, we explain our alternative approach.
§ 2.

Review on Previous Approaches

We shall consider a family of Markov processes {(xϵ (t))t≥0 : ϵ > 0} on Ω. The sets
E1 , . . . , EK represent the metastable set, i.e., the process xϵ (t) starting from a point
in a set Ei remains in this set suﬃciently long when ϵ is small enough. Subsequently,
after a suﬃciently long time, the process exhibits a transition to another metastable set.
We write S = {1, 2, . . . , K} and let µϵ be the invariant measure of the process xϵ (t).
We denote by Pϵx the law of process xϵ (t) starting from x ∈ Ω, and denote by Eϵx thecorresponding expectation.
Remark. The sets Ω and E1 , . . . , EK may depend on ϵ. Instead of the family of
Markov processes parameterized by ϵ, one can consider the sequence of Markov processes{(xN (t))t≥0 : N ∈ N}. The approaches explained below can be applied to this sequence
as well. Instead of the ϵ ↓ 0 limit, we should consider the N ↑ ∞ limit in this case.

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Insuk Seo

§ 2.1.

Potential-theoretic approach

We first consider the question suggested in Case 1 of Section 1, namely, the estimation of the metastable transition time from a metastable set to others. To formulate
this question concretely, we first introduce some notations. For A ⊂ Ω, we denote by∪
τA the hitting time of the set A. We write E := i∈S Ei and Ĕi = E \ Ei . We pick a
point xi ∈ Ei arbitrarily. Subsequently, the primary concern is the estimation of themean transition time Exi [τĔi ]. This quantity corresponds to the escape time from a
local minimum, and is crucially related to the mixing time and the spectral gap (see[10]).
It has been observed in [8] that, if the process xϵ (t) is reversible with respect tothe invariant measure µϵ , then the mean transition time is closely related to a potential theoretic notion known as the capacity. For disjoint subsets A and B of Ω, theequilibrium potential between A and B with respect to the process xϵ (t) is a function
hϵA, B : Ω → [0, 1] defined byhϵA, B (x) = Pϵx [τA < τB ] .
Subsequently, the capacity between A and B is defined by∫capϵ (A, B) =hϵA, B (−Lϵ hϵA, B )dµϵ ,Ωwhere Lϵ is the generator corresponding to the process xϵ (t). The crucial observationis that, under the circumstances of metastability, the following holds:(2.1)

Exi [τĔi ] ≃µϵ (Ei )capϵ (Ei , Ĕi ).

The asymptotic limit of µϵ (Ei ) of ϵ ↓ 0 is typically not diﬃcult to obtain. The nontrivial part is to estimate the capacity capϵ (Ei , Ĕi ). If the process xϵ (t) is reversible
with respect to µϵ , it can be performed via the Dirichlet and Thomson principles thatprovide the upper and lower bounds for the capacity, respectively.
A successful application of this methodology is the Eyring-Kramers formula for
the SRPDS (1.1). For this model, Ei can be regarded as a small O(1) neighborhood(or O(ϵ1/2+α ), α > 0, neighborhood) of each local minimum. To deliver the primary
result in a concrete and simple form, let us temporarily assume that U is a double-wellpotential as in Figure 1-(Left): two local minima m1 and m2 exist, and a saddle point
σ exists between them. The main result of [8] shows that, if the Hessians ∇2 U at
m1 , m2 , and σ are non-degenerate, and if (∇2 U )(σ) has a unique negative eigenvalue

Analysis of Metastable Behavior via Solutions of Poisson Equations

5−λσ , then under some minor technical assumptions, the following holds:√{}
2π − det(∇2 U )(σ)
U (σ) − U (m1 )ϵ(2.2)
Em1 [τE2 ] ≃exp.
λσdet(∇2 U )(m1 )ϵIt has also been verified that the normalized mean transition time τE2 /Eϵm1 [τE2 ] converges to a mean 1 exponential random variable as ϵ ↓ 0. We remark that, in a general U ,a similar but slightly more complicated expression for the quantity of the form Eϵm1 [τĔ1 ]can be obtained similarlyThis approach is extremely robust for the reversible case, as demonstrated by numerous successful applications for a wide scope of models. Instead of enumerating theseexamples, we refer to the monograph [9] for the comprehensive discussions of this streamof studies.
A shortcoming of this original method is that its application is limited to the casewhen the process xϵ (t) is reversible. Recently, studies of the non-reversible case has
shown significant improvements. First, Beltran and Landim [2, 3] found a formula
corresponding to (2.1) that holds without the reversibility assumption. This formula isfairly similar to (2.1), and hence the sharp estimation of capacity is required as well.
This is another diﬃculty because the classical Dirichlet and Thomson principles hold
only for the reversible case. However, recently, two variational principles generalizingthese principles to the non-reversible case has been obtained. Gaudillière and Landimin [14] found a generalization of the Dirichlet principle, and Slowik in [29] found that of
the Thomson principle. It was developed for the discrete Markov process setting, but
had been generalized to the continuous diﬀusion setting in [20] as well. These principles
are more diﬃcult to use than the case of reversible counterparts, because they are
double variational principles in complicated spaces. However, Landim in [18] used thisprinciple creatively to investigate the metastable behavior of the non-reversible zerorange process. It was the first study that presented quantitative results in the study ofthe metastable behavior of non-reversible processes. More recently, a manual for using
this package of new machinery has been developed in [22] by Landim and the author ofthis article. This manual has been used in [23], [27], and [20] to perform the quantitative
analysis on metastability.
§ 2.2.

Martingale approach

Although the potential-theoretic approach of metastability is a strong method foranalyzing the metastable random process, it cannot answer the question suggested inCase 2 of Section 1. More precisely, if there are multiple metastable sets of the same
depth, we can expect that a properly defined rescaled process should behave like a
Markov chain on these metastable sets, but the sharp asymptotics obtained by the6Insuk Seo

potential-theoretic approach cannot deduce this type of result. In this subsection, we
explain the martingale approach established by Beltran and Landim in [2, 3, 17]. To
explain this approach, we start by introducing several notations.
All the notations defined above are maintained. Recall that there are K metastablesets E1 , . . . , EK for the process xϵ (t). We define S = {1, 2, . . . , K}. Now we shall definea so-called trace process of xϵ (t) on E. Stating heuristically, this is a process obtained
from xϵ (t) by turning oﬀ the clock when the process is not in E. To define this object
rigorously, we define the following:∫ t1{xϵ (s) ∈ E} ds ; t ∈ [0, ∞) .

Tϵ (t) =0

It measures the amount of time that the process xϵ (·) has spent on the set E up to timet. Subsequently, we define the generalized inverse of Tϵ (·) as(2.3)

Sϵ (t) = sup {s : Tϵ (s) ≤ t} .

Finally, the trace process (xϵ (t))t≥0 is defined byxϵ (t) = xϵ (Sϵ (t)) .
One can check that this process is a Markov process on E, with possible long jumps along
the boundary ∂E = ∪i∈S ∂Ei . We denote by χA : Ω → {0, 1} the indicator function on
A ⊂ Ω, and define the projection function Ψ : E → {1, 2, . . . , K} asΨ(x) =

K∑i χEi (x) .

i=1

Finally, we define the projected process as xϵ (t) = Ψ(xϵ (t)) which is a random process
on S. With this package of notations, our primary question can be stated as follows.
Question.
Can we prove that a scaling limit of the process (xϵ (t))t≥0 converges
to a Markov chain (x(t))t≥0 on S?
To answer this question, it would be ideal if we can determine a priori predictionof the correct time scale θϵ and of the candidate for the limiting Markov chain x(t). We
would like to stress that these can be inferred from the results for the mean transition
time obtained by the potential-theoretic approach explained in the previous subsection.
We denote by Pϵπ the law of the Markov process xϵ (t) with a starting measure πon Ω, and by Qi the law of Markov chain x(t) starting at i ∈ S. Finally, we denote by()
Qϵπ the law of the rescaled projected process xϵ (θϵ t) t≥0 under Pπ . The main theorem
can be formulated as follows:Analysis of Metastable Behavior via Solutions of Poisson Equations

7

Theorem 2.1.
Fix i ∈ S. For ϵ > 0, let πϵ be a probability measure on Ωconcentrated on Ei . Then, Qϵπϵ converges to Qi as ϵ tends to 0.
The martingale approach established by Beltran and Landim in [2, 3, 17] provides
a robust methodology to prove this theorem, especially when the process xϵ (t) is a
Markov process on a discrete set. This approach reduces the proof of Theorem 2.1
to the estimates of the so-called mean-jump rate between valleys. To state this more
precisely, denote by rϵ : E × E → R the jump rate of the trace process xϵ (t), which is a
Markov process on E. Subsequently, the mean-jump rate between two metastable sets
Ei and Ej is defined as(2.4)

rϵ (i, j) =1µϵ (Ei )∑µϵ (x)rϵ (x, y) .

x∈Ei , y∈Ej

Denote by r : S × S → R the jump rate of the Markov chain (x(t))t≥0 . The main
result of Beltran and Landim can be summarized as follows: up to several technicalrequirements, proving(2.5)

lim θϵ rϵ (i, j) = r(i, j)

ϵ→0is enough to demonstrate Theorem 2.1. Such an implication has been verified by relating
several martingale problems creatively. Although the rate rϵ , i.e., the jump rate of the
trace process, is not an easy notion to manage, Beltran and Landim observed that its
weighted average rϵ has a rather simple expression in terms of the potential-theoretic
notions, such as the capacity. For instance, if the process xϵ (t) is reversible, we have
(2.6)

]
1[
rϵ (i, j) =capϵ (Ei , Ĕi ) + capϵ (Ej , Ĕj ) − capϵ (Ei ∪ Ej , Ĕi ∩ Ĕj ) .
2

Thus, the analysis of metastable behavior can be performed by estimating the capacity,
similar as before. This technology has been applied to various reversible models suchas the zero-range process [4], the simple inclusion process [6], and random walks in a
potential field [21].
Furthermore, the result of Beltran and Landim in [3] indicates that the implicationfrom (2.5) to Theorem 2.1 also holds for the non-reversible case. The diﬃculty in the
non-reversible case is that the formula (2.6) for rϵ is no longer valid. Meanwhile, arather complicated formula for rϵ in terms of the so-called collapsed Markov chain has
been found in [3] and [18]. Based on this, Landim in [18] first established an analysis of
the non-reversible metastable process by analyzing the totally asymmetric zero-range
process. Subseqeuntly, a robust way to use this complicated formula to deduce (2.5)
is established in [22]. Recently, this has been applied to various non-reversible models
in [22, 23, 27]. We refer to the review paper [19] by Landim for the comprehensive

8Insuk Seo

discussion of this topic.
Before concluding this review section, we have to emphasize that the martingale
approach has not yet been successfully applied to metastable diﬀusion processes suchas the SRPDS (1.1). Because in this case the trace process becomes a diﬀusion process
on disconnected set E with a long jump along ∂E, which is not a conventional object,the mean-jump rate such as (2.4) is almost impossible to define. Our new methodology(explained in the next section) can be regarded as an entirely diﬀerent approach tometastability, and can be applied to continuous models such as the SRPDS that cannot
be answered by the martingale approach.
§ 3.

Approach via Poisson Equations

We recall all the notations from the previous section. In this section, we explain a
new approach developed in [24, 26]. This approach provides an alternative method toprove Theorem 2.1 by analyzing a Poisson equation, instead of estimating the potentialtheoretic notions such as capacity.
We shall assume that we have predictions of correct time scale θϵ as well as thelimiting Markov chain x(t), as mentioned before. In addition, we assume that
(3.1)

µϵ (Ei ) = (1 + oϵ (1)) ν(i) ; for all i ∈ {1, . . . , K} ,where ν is a measure on S = {1, · · · , K} satisfying
(3.2)

K
∑

ν(i) = 1 ,i=1Therefore, using (3.1) and (3.2), we can verify that ν is the invariant measure of to theMarkov chain x(t), provided that Theorem 2.1 is correct.
Denote by Lϵ and L the generators corresponding to the Markov processes xϵ (t)and x(t), respectively. Define aϵ = (aϵ (i) : i ∈ S) ∈ RS byaϵ (i) =ν(i); i∈S.
µϵ (Ei )By (3.1), we have
(3.3)

aϵ (i) = 1 + oϵ (1) for all i ∈ S .

The following proposition is the primary step in our new approach.
Proposition 3.1.
For all f : S → R, there exists a bounded function ϕϵ = ϕfϵ :Ω → R satisfying the following conditions simultaneously:Analysis of Metastable Behavior via Solutions of Poisson Equations

9

1. The function ϕϵ satisfies the following equation:∑(3.4)
Lϵ ϕϵ = θϵ−1
aϵ (i) (Lf )(i) χEi .
i∈S

2. For all i ∈ S, the following holds:
lim sup |ϕϵ (x) − f (i)| = 0 .

(3.5)

ϵ→0 x∈Ei

We observed that finding a test function ϕfϵ explained in the last proposition implies
Theorem 2.1, up to several minor technical issues. Hence, in the remaining part ofthe current section, we explain the model-independent proof of Theorem 2.1 assuming
Proposition 3.1. The proof of Proposition 3.1 it the only model-dependent part. We
explain a general idea for this part in Section 3.3.
In general, two ingredients are required to complete the proof of the limit theorem
such as Theorem 2.1: the tightness of family (Qϵπϵ )ϵ>0 , and the identification of limit
points of this family. These two crucial ingredients are proven in Sections 3.1 and 3.2,
respectively.
§ 3.1.

Tightness

For convenience, we fix i ∈ S and the sequence of the family of probability measures(πϵ )ϵ>0 concentrated on Ei throughout this subsection. The tightness result required inour context can be stated as the following proposition.
Proposition 3.2.
The family {Qϵπϵ : ϵ ∈ (0, 1]} is tight on D([0, ∞), S). Furthermore, every limit points Q∗ of this family, as ϵ tends to 0, satisfy
Q∗ (x(0) = i) = 1 and Q∗ (x(t) ̸= x(t−)) = 0 for all t > 0 .
It is shown in [24, Sections 7, 8] that the proof of this tightness result is based
entirely on the two estimates stated in Lemmas 3.3 and 3.4. The former verifies thata transition from a metastable set to another one cannot occur in a scale shorter thanθϵ , while the latter demonstrates that in the course of the metastable transition, theprocess does not spend much time outside the metastable sets, namely Ω \ E. We startby proving the former lemma, whose proof depends solely on Proposition 3.1. We simply
write Pϵx := Pϵδx , where δx is a Dirac delta measure at x ∈ Ω.
Lemma 3.3.
(3.6)

It holds that[]
lim lim sup sup Pϵx τĔi ≤ aθϵ = 0 .

a→0

ϵ→0

x∈Ei

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Insuk Seo

Proof. Consider a function f : S → R defined by0 if j = i
f (j) =
1 otherwise .
Denote by ϕϵ = ϕfϵ the function that we obtain in Proposition 3.1 with respect to thisfunction f . For x ∈ Ei , by Itö’s formula and (3.4), one can deduce that[]Eϵx ϕϵ (xϵ (aθϵ ∧ τE\Ei )) − ϕϵ (x)[∫]aθϵ ∧τE\Ei∑=
Eϵx
θϵ−1 aϵ (i) (Lf )(i) χEi (xϵ (s))ds .

(3.7)

0

i∈S

It is noteworthy that for some constant C > 0,∫ aθϵ ∧τE\E
(3.8)

i

θϵ−1 aϵ (i) (Lf )(i) χEi (xϵ (s))ds ≤ aθϵ · θϵ−1 |aϵ (i) (Lf )(i)| ≤ Ca .

0Moreover, since x ∈ Ei , it follows from (3.5) that
(3.9)

|ϕϵ (x)| = |ϕϵ (x) − f (i)| = oϵ (1) .

By combining (3.7), (3.8), and (3.9), we can deduce that
(3.10)][
Eϵx ϕϵ (xϵ (aθϵ ∧ τE\Ei )) ≤ Ca + oϵ (1) .

Next we attempt to bound the last expectation from below. Let δ > 0 be an arbitrarily
small constant. Then, again by (3.5), we have ϕϵ + δ ≥ 0 on Ei and ϕϵ + δ ≥ 1 on
E \ Ei for all suﬃciently small ϵ. Hence, by the maximum principle, ϕϵ + δ ≥ 0 on Ω.
Summing these, we have ϕϵ + δ ≥ χE\Ei . Therefore,
]
]
[
]
[
[
(3.11) Eϵx ϕϵ (xϵ (aθϵ ∧ τE\Ei )) + δ ≥ Eϵx χE\Ei (xϵ (aθϵ ∧ τE\Ei )) = Pϵx τE\Ei < aθϵ .
By (3.10) and (3.11), we deduce[]
lim sup sup Pϵx τE\Ei ≤ aθϵ ≤ Ca + δ
ϵ→0

x∈Ei

Because δ > 0 is arbitrary, the proof is completed.
Next, we discuss the second ingredient. Denote by ∆ = Ω \ E the outside of themetastable sets. Subsequently, by (3.1) and (3.2), we have µϵ (∆) = oϵ (1). In otherwords, the set ∆ is negligible in view of the equilibrium measure. However, the second
ingredient of tightness requires us to show this negligibility of ∆ in a dynamical sense.

Analysis of Metastable Behavior via Solutions of Poisson Equations

11

Hence, we define the excursion time of the process xϵ (t) on the set ∆ up to time t as∫ t∆(t) =

χ∆ (xϵ (s)) ds .
0

We remark that ∆(t) is a notion depending on ϵ although we did not stress this in thenotation. Then, we can formulate the dynamic negligibility of ∆ as follows:
Lemma 3.4.
For any sequence (πϵ )ϵ>0 of probability measures concentrated on
Ei , the following holds:
lim θϵ−1 Eϵπϵ [∆(θϵ t)] = 0 for all t ≥ 0.

ϵ→0Here, we only provide the proof of Lemma 3.4 when πϵ has a density function withrespect to µϵ for each ϵ > 0, and this density function belongs to Lp (µϵ ) for some p > 1
in a uniform manner, i.e.,∫pdπϵ(3.12)
lim supdµϵ = M < ∞.
ϵ→0
Ω dµϵ
For this case with mild initial distribution, we can deduce a simple proof. For thegeneral case, see the remark after the proof.
Proof of Lemma 3.4 under the assumption (3.12). We fix t ≥ 0 and writeuϵ (x) = θϵ−1 Eϵx [∆(θϵ t)] .
By Fubini’s theorem, we obtain(3.13)[∫]∫∫ θϵ tθϵ t−1 ϵ−1uϵ dµϵ = θϵ Eµϵχ∆ (xϵ (s))ds = θϵ
Pϵµϵ [xϵ (s) ∈ ∆] ds = t µϵ (∆) .
Ω

0

0

dπϵ
so that we can write
We write fϵ = dµϵ∫θϵ−1 Eϵπϵ [∆(θϵ t)] =(3.14)(uϵ fϵ ) dµϵΩNow, we apply Holder’s inequality, trivial bound uϵ ≤ t, (3.13) and (3.12) to the righthand side of the previous display to deduce the following:[∫∫(uϵ fϵ ) dµϵ ≤Ω(3.15)]1/q [∫
uϵ dµϵΩ≤ tµϵ (∆)

]1/p
(uϵ fϵp ) dµϵ

[∫
1/q
Rd

Ω

fϵp dµϵ]1/p< tµϵ (∆)1/q M 1/p ,12Insuk Seo

where q is the conjugate exponent of p satisfying p1 + 1q = 1. Since µϵ (∆) = oϵ (1), we
complete the proof by conditions (3.14) and (3.15).
Remark.

To address the general case, it is suﬃcient to demonstrate thatlim sup θϵ−1 Eϵx [∆(θϵ t)] = 0 .

ϵ→0 x∈Ei

In view of the previous lemma for the special case, it suﬃces to verify thatsupx, y∈Ei( ϵ)
Ex [∆(θϵ t)] − Eϵy [∆(θϵ t)] ≪ θϵ .

When Ω is a discrete set, this can be typically performed by the coupling (see [2, 3]).
In the case of diﬀusion processes in a 1-dimensional torus, the same type of couplingargument can be applied (see [24]). However, the diﬀusion processes such as the SRPDS(1.1) in dimension d ≥ 2, two processes starting from diﬀerent points cannot be coupled
exactly; hence, another argument is required. In [26, Appendix], an argument based on
the large-deviation theory has been introduced. We refer to these listed articles for thedetails of the proof of Lemma 3.4 for general cases.
The proof of Proposition 3.2 from these two lemmas above are routine and follows
from Aldous’ criterion. We refer to [24, Section 7] or [26, Section 5] for more details, but
we herein provide a brief sketch of the proof. First, we should introduce the appropriatebϵ the law ofb ϵ (t) = xϵ (θϵ t) as the accelerated process, and write P
filtration. Write xπϵ0process (bxϵ (t))t≥0 starting from πϵ . Then, denote by {Ft : t ≥ 0} the natural filtrationof D([0, ∞), Ω) (or C([0, ∞), Ω) if (xϵ (t)) is a diﬀusion process) with respect to theb ϵ (·), namely,
process x
Ft0 = σ(bxϵ (s) : s ∈ [0, t]) .
bϵ .
and define {Ft : t ≥ 0} as the usual augmentation of {Ft0 : t ≥ 0} with respect to P
πϵDefine Gt = FSϵ (t) for t ≥ 0, where Sϵ is defined in (2.3). For M > 0, we define TM as
the collection of stopping times with respect to the filtration {Gt }t≥0 bounded by M .
Proof of Proposition 3.2. We start by considering the first statement of the proposition. By Aldous’ criterion, it suﬃces to verify that, for all M > 0,
(3.16)

lim lim sup sup

a0 →0

ϵ→0

sup

τ ∈TM a∈(0, a0 )Pϵπϵ [xϵ (τ + a) ̸= xϵ (τ )] = 0 .

By Lemma 3.4, it suﬃces to demonstrate that
lim lim sup sup

a0 →0

ϵ→0

sup

τ ∈TM a∈(0, a0 )Pϵπϵ [xϵ (τ + a) ̸= xϵ (τ ), Sϵ (τ + a) − Sϵ (τ ) ≤ 2a0 ] = 0 .

Analysis of Metastable Behavior via Solutions of Poisson Equations

13

Since xϵ (t) = Ψ(bxϵ (Sϵ (t))), the last probability can be bounded above by
Pϵπϵ [Ψ(bxϵ (Sϵ (τ ) + t) ̸= Ψ(bxϵ (Sϵ (τ ))) for some t ∈ (0, 2a0 ]] .
One can readily demonstrate that S ϵ (τ ) is a stopping time with respect to the filtration{Ft } (see [24, Lemma 7.2]); hence, by the strong Markov property the last probability
is bounded above by[]
sup Pϵx [Ψ(bxϵ (t)) ̸= Ψ(x) for some t ∈ (0, 2a0 ]] = sup Pϵx τĔi ≤ 2a0 θϵ .

x∈Ei

x∈Ei

Thus, the proof of (3.16) is completed by Lemma 3.3.
The assertion Q∗ (x(0) = i) = 1 is trivial. For the last assertion of the proposition,
it suﬃces to prove that
lim lim sup Pϵπϵ [xϵ (t − a) ̸= xϵ (t) for some a ∈ (0, a0 )] = 0 .

a0 →0

ϵ→0

The proof for this is the same as that above.
§ 3.2.

Identification of limit points and the proof of Theorem 2.1

The proof of Theorem 2.1 can be completed by identifying the limit points ofTo this end, fix f ∈ RS , and let ϕϵ = ϕfϵ be the function obtained in
Proposition 3.1. We fix i and πϵ appearing in the statement of Theorem 2.1. Sinceb (t) is θϵ Lϵ , the process (Mϵ (t))t≥0 defined by
the generator of the accelerated process x(Qϵπϵ )ϵ>0 .

∫ t
Mϵ (t) = ϕϵ (bxϵ (t)) − θϵ(Lϵ ϕϵ )(bxϵ (s))ds0b ϵ (Sϵ (t)) =is a martingale with respect to the filtration {Ft } introduced above. Since xxϵ (θϵ t) by definition, we can write∫ t(3.17)Mϵ (Sϵ (t)) = ϕϵ (xϵ (θϵ t)) − θϵ(Lϵ ϕϵ )(xϵ (θϵ s))ds .
0

Since Gt = FSϵ (t) (see [26, Lemma 5.5]), the process (Mϵ (Sϵ (t)))t≥0 is a martingale
with respect to {Gt }. By Proposition 3.1, we haveϕϵ = f ◦ Ψ + oϵ (1) and θϵ Lϵ ϕϵ = (Lf ) ◦ Ψ + oϵ (1) on E .
Since Ψ(xϵ (θϵ t)) = xϵ (t), we can rewrite (3.17) as∫ t
Mϵ (Sϵ (t))) = f (xϵ (t)) −(Lf )(xϵ (s))ds + oϵ (1) .
014Insuk Seo

Hence, if Q∗ is a limit point of the family {Qϵπϵ }ϵ∈(0, 1] , the process (M ∗ (t))t≥0 definedby(3.18)∫ t∗

M (t) = f (x(t)) −(Lf )(x(s))ds0

is a martingale under Q∗ . This completes the characterization of the limit points.
Proof of Theorem 2.1. Let Q∗ be a limit point of the family {Qϵπϵ }ϵ∈(0, 1] . Subsequently, as demonstrated above, (M (t))t≥0 defined by (3.18) is a martingale under Q∗ ;furthermore, by Proposition 3.2, we obtain Q∗ [x(0) = i] = 1 and Q∗ (x(t) ̸= x(t−)) = 0
for all t > 0. The only probability measure on D([0, ∞), S) satisfying these properties simultaneously is Qi ; thus, we can conclude that Q∗ = Qi . This completes theproof.
§ 3.3.

Analysis of Poisson Equation

In view of the argument presented in the previous section, the entire analysis isdependent on the construction of ϕfϵ introduced in Proposition 3.1. This should be
proven for each model.
For the non-reversible diﬀusion processes in a 1-dimensional torus considered in[24], we found an explicit solution ψϵ of the equation (3.4). It is noteworthy that foreach constant cϵ , the function ψϵ + cϵ is also a solution of (3.4). Hence, in this case we
can select cϵ carefully such that the function ϕϵ = ψϵ + cϵ satisfies (3.5) as well. We
refer to [24, Section 9] for the details.
For the SRPDS (1.1) in Rd , we cannot expect such a closed form solution. We shallsketch our idea of the proof in the next paragraph. We believe that the idea presented
herein can be applied to a broad scope of examples as well after a suitable modification.
We refer to [26, Section 4] for the full details.
∫
Let Dϵ (ϕ) = Ω ϕ(−Lϵ ϕ)dµϵ be the Dirichlet form associated to the process y ϵ (t)defined in (1.1). Since Lϵ is self-adjoint with respect to dµϵ , we can find a solution of
the Poisson equation (3.4) by a minimizer of the functional defined by∫∑1aϵ (i) (Lf )(i)ϕ dµϵ .
Iϵ (ϕ) = θϵ Dϵ (ϕ) +
2
Ei
i∈S

Take a minimizer ψϵ of this functional so that ψϵ solves the equation (3.4). For thisminimizer, one can readily show that, for some λϵ > 0,∫∑θϵ Dϵ (ψϵ ) = λϵ andaϵ (i) (Lf )(i)ψϵ dµϵ = −λϵ .
i∈SEiThen, one can show that λϵ = Oϵ (1). Denote by m the Lebesgue measure on Rd andAnalysis of Metastable Behavior via Solutions of Poisson Equations

15

define q = (q(i) : i ∈ S) ∈ RS such a manner that∫1q(i) =ψϵ (x)dx ; i ∈ S .
m(Ei ) Ei
Subsequently, by the bound on Dϵ (ψϵ ) and Poincare’s inequality we can verify that∫
(3.19)∥ψϵ − q(i)∥L2 (Ei ) =(ψϵ − q(i))2 dx = oϵ (1) for all i ∈ S.
Ei

Then, a technique developed in [12, 28] based on the interior elliptic estimate in thepartial diﬀerential equations theory allows us to prove that
(3.20)∥ψϵ − q(i)∥L∞ (Ei ) = λϵ oϵ (1) for all i ∈ S.

To obtain this, we first prove (3.19) for a slightly larger set Eei ⋑ Ei and then use theinterior elliptic estimate [15, Theorem 8.17] to enhance the L2 -estimate to the interior
L∞ -estimate. For the detail we refer to [26, Proposition 4.8].
The final step is to find a constant cϵ such that q(i) + cϵ = f (i) for all i ∈ S. We
will not provide the detailed proof for this, but we strongly recommend the readers toread [26, Section 4.5], in which a novel method to prove the characterization of q has
been developed. The primary idea is to couple the function ψϵ with a test function thatis already popular in the study of metastability. We believe that the argument thereincan be applied to a broad scope of models.
§ 4.

Conclusion

At the time when this review paper was written, the approach via the Poisson
equation introduced herein has been applied to the reversible SRPDS (1.1) in [26], andthe non-reversible SRPDS in a 1-dimensional torus in [24]. We believe that this methodology can be applied to a wide range of models exhibiting metastability. In particular,
the approach based on the Poisson equation did not heavily use the reversibility of theunderlying metastable process; hence, we hope that our approach paves the way forthe quantitative analysis of non-reversible metastable processes, in which many open
problems still remain.
Acknowledgements
This article is written as a proceeding to the 17th International Symposium Stochastic Analysis on Large Scale Interacting Systems held in the Research Institute for Mathematical Sciences (RIMS), Kyoto University from November 5–8, 2018. The author
would like to thank the organizers for the invitation to the conference and the kind

16

Insuk Seo

support during the stay. The author also thanks Professor Kenkichi Tsunoda for theinvitation to Osaka University before the conference was held, during which most of thecurrent article was written.
The author is deeply indebted to Professor Claudio Landim, who introduced themetastability theory to the author and shared numerous valuable ideas, and to Professor
Fraydoun Rezakhanlou who first envisioned the approach explained herein and shared
it with the author.

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