A Dirichlet and a Thomson principle for non-selfadjoint elliptic operators, Metastability in non-reversible diffusion processes

0.97 MB·

by  C. Landim

A DIRICHLET AND A THOMSON PRINCIPLE FOR
NON-SELFADJOINT ELLIPTIC OPERATORS, METASTABILITY
IN NON-REVERSIBLE DIFFUSION PROCESSES.
7 C.LANDIM,M.MARIANI,I.SEO
1
0
Abstract. We present two variational formulae for the capacity in the con-
2
text of non-selfadjoint elliptic operators. The minimizers of these variational

n problems are expressed as solutions of boundary-value elliptic equations. We
a use these principles to provide a sharp estimate for the transition times be-
J tweentwodifferentwellsfornon-reversiblediffusionprocesses. Thisestimate

4 permitstodescribethemetastablebehaviorofthesystem.

]
R
P 1. Introduction
.
h
This article is divided in two parts. In the first one, we present two variational
t
a formulae which extend to non-selfadjoint elliptic operators the classical Dirichlet
m
and Thomson principle. In the second one, we use these formulae to describe the
[ metastable behavior of a non-reversible diffusion process in a double-well potential

1 field.
v Fix a smooth, bounded, domain (open and connected) Ω ⊂ Rd, d ≥ 2, and a
5 smooth function f : Rd → R. Denote by Ω the set of functions v : Ω → R such
f
8
thatv =f on∂Ω,theboundaryofΩ. TheclassicalDirichletprinciple[15,1]states
9
that the energy
0 (cid:90)
0 (cid:107)∇u(x)(cid:107)2dx
.
1 Ω
is minimized on Ω by the harmonic function on Ω which takes the value f at the
0 f
7 boundary, that is, by the solution of
1
∆u = 0 on Ω and u=f on ∂Ω. (1.1)
:
v
When Ω = D \B, where B ⊂ D are smooth domains, and f = 1, 0 on B,
i
X Dc, respectively, the minimal energy is called the capacity. In electrostatics, it
r corresponds to the total electric charge on the conductor ∂B held at unit potential
a and grounded at Dc. It is denoted by cap (B) and can be represented, by the
D
divergence theorem, as
(cid:90) ∂h
cap (B) = − dσ , (1.2)
D ∂n
∂B B
where h is the harmonic function which solves (1.1), n is the outward normal
B
vector to ∂B, and σ the surface measure at ∂B.
These results have long been established for self-adjoint operators of the form
(Lu)(x)=eV(x)∇·[e−V(x)S(x)∇u(x)], provided S(x) are smooth, positive-definite,
symmetric matrices, and V is a smooth potential. They have been extended, more
Key words and phrases. Non-reversible diffusions, Potential theory, Metastability, Dirichlet
principle,Thomsonprinciple,Eyring-Kramersformula.
1
2 C.LANDIM,M.MARIANI,I.SEO
recently, by Pinsky [19, 21] to the case in which the operator L is not self-adjoint.
In this situation, the minimization formula for the capacity, mentioned above, has
to be replaced by a minmax problem.
The first main result of this article provides two variational formulae for the
capacity (1.2) in terms of divergence-free flows. In contrast with the minmax for-
mulae, the first optimization problem is expressed as an infimum, while the second
one is expressed as a supremum, simplifying the task of obtaining lower and upper
bounds for the capacity.
Analogous Dirichlet and Thomson principle were obtained by Gaudilli`ere and
Landim [13] (the Dirichlet principle) and by Slowik [22] (the Thomson principle)
for continuous-time Markov chains.
Inthesecondpartofthearticle,weusetheformulaeforthecapacitytoexamine
the metastable behavior of a non-reversible diffusion in a double well potential.
LetU :Rd →Rbeasmooth,double-wellpotentialwhichdivergesatinfinity,and
let M be a non-symmetric, positive-definite matrix. We impose in the next section
furtherassumptionsonU. DenotebyM†thetransposeofMandbyS=(M+M†)/2
its symmetric part. Consider the diffusion X(cid:15), (cid:15)>0, described by the SDE
t
√
dX(cid:15) = −M(∇U)(X(cid:15))dt + 2(cid:15)KdW , (1.3)
t t t
where W is a standard d-dimensional Brownian motion, and K is the symmetric,
t
positive-definite square root of S, S=KK.
AssumethatU hastwolocalminima,denotedbym ,m ,separatedbyasingle
1 2
saddle point σ, and that U(m ) ≤ U(m ). The stationary state of X(cid:15), given by
2 1 t
µ (dx) ∼ exp{−U(x)/(cid:15)}dx, is concentrated in a neighborhood of m when the
(cid:15) 2
previous inequality is strict.
If X(cid:15) starts from a neighborhood of m , it remains there for a long time in the
t 1
small noise limit (cid:15) → 0 until it overcomes the potential barrier and jumps to a
neighborhood of m through the saddle point σ. Denote by τ the hitting time of
2 (cid:15)
a neighborhood of m . The asymptotic behavior of the mean value of τ as (cid:15)→0
2 (cid:15)
has been the object of many studies.
The Arrhenius’ law [3] asserts that the mean value is logarithmic equivalent
to the potential barrier: lim (cid:15)logE [τ ] = U(σ) − U(m ) =: ∆U, where
(cid:15)→0 m1 (cid:15) 1
E represents the expectation of the diffusion X(cid:15) starting from m . The sub-
m1 t 1
exponential corrections, known as the Eyring-Kramers formula [10, 16], have been
computed when the matrix M is symmetric and the potential non-degenerate at
the critical points. Assuming that the Hessian of the potential is positive definite
at m and that it has a unique negative eigenvalue at σ, denoted by −λ, while all
1
the others are strictly positive, the sub-exponential prefactor is given by
(cid:112)
2π −det[(Hess U)(σ)]
E [τ ] = [1+o (1)] e∆U/(cid:15) ,
m1 (cid:15) (cid:15) λ (cid:112)det[(Hess U)(m )]
1
where o (1)→0 as (cid:15) vanishes.
(cid:15)
Thisestimateappearsinarticlespublishedinthe60’s. Arigorousproofwasfirst
obtainedbyBovier,Eckhoff,Gayrard,andKlein[8]withargumentsfrompotential
theory, and right after by Helffer, Klein and Nier [14] through Witten Laplacian
analysis. We refer to Berglund [6] and Bouchet and Reygner [7] for an historical
overview and further references.
METASTABILITY IN NON-REVERSIBLE DIFFUSION PROCESSES 3
Recentlty,BouchetandReygner[7]extendedtheEyring-Kramersformulatothe
non-reversible setting. They showed that in this context the negative eigenvalue
−λ has to be replaced by the unique negative eigenvalue of (Hess U)(σ)M.
Wepresentbelowarigorousproofofthisresult,basedonthevariationalformulae
obtained for the capacity in the first part of the article, and on the approach
developed by Bovier, Eckhoff, Gayrard, and Klein [8] in the reversible case. This
estimatepermitstodescribethemetastablebehaviorofthediffusionX(cid:15)inthesmall
t
noise limit. Analogous results have been derived for random walks in a potential
field in [17, 18].
2. Notation and Results
We start by introducing the main assumptions. We frequently refer to [12] for
results on elliptic equations and to [11, 21] for results on diffusions.
2.1. A Dirichlet and a Thomson principle. Fix d≥2, and denote by Ck(Rd),
0≤k ≤∞, the space of real functions on Rd whose partial derivatives up to order
k are continuous. Let M , 1≤m,n≤d, be functions in C2(Rd) for which there
m,n
exists a finite constant C such that
0
d
(cid:88)
M (x)2 ≤ C for all x∈Rd. (2.1)
m,n 0
m,n=1
Denote by M(x) the matrix whose entries are M (x). Assume that the matrices
m,n
M(x), x∈Rd, are uniformly positive-definite: There exist 0<λ<Λ such that for
all x, ξ ∈Rd,
λ(cid:107)ξ(cid:107)2 ≤ ξ·M(x)ξ ≤Λ(cid:107)ξ(cid:107)2 , (2.2)
where η·ξ represents the scalar product in Rd, and (cid:107)x(cid:107) the Euclidean norm.
Let V be a function in C3(Rd) such that (cid:82) exp{−V(x)}dx<∞, and assume,
Rd
(cid:82)
withoutlossofgenerality,that exp{−V(x)}dx=1. Denotebyµtheprobability
Rd
measure on Rd defined by µ(dx)=exp{−V(x)}dx.
Denote by L the differential operator which acts on functions in C2(Rd) as
(Lf)(x) = eV(x)∇·(cid:8)e−V(x)M(x)(∇f)(x)(cid:9).
In this formula, ∇g represents the gradient of a function g :Rd →R and ∇·Φ the
divergenceofavectorfieldΦ:Rd →Rd. Thepreviousformulacanberewrittenas
d d
(cid:88) (cid:88)
(Lf)(x) = b (x)∂ f(x) + S (x)∂2 f(x), (2.3)
j xj j,k xj,xk
j=1 j,k=1
where the drift b=(b ,...,b ) is given by
1 d
d
(cid:88)(cid:110) (cid:111)
b (x) = ∂ M (x) − (∂ V)(x)M (x) ,
j xk k,j xk k,j
k=1
andwhereS(x)representsthesymmetricpartofthematrixM(x),S(x)=(1/2)[M(x)+
M†(x)], M†(x) being the transpose of M(x).
Denote by B(r), r > 0, the open ball of radius r centered at the origin, and by
∂B(r) its boundary. We assume that
lim inf V(z) = ∞, (2.4)
n→∞z(cid:54)∈B(n)
4 C.LANDIM,M.MARIANI,I.SEO
and that there exist r >0, c >0 such that
1 1
(LV)(x) ≤ −c (2.5)
1
for all x such that (cid:107)x(cid:107)≥r . By (2.3), this last condition can be rewritten as
1
d d
(cid:88) (cid:88)
(∂ M )(x)(∂ V)(x) + S (x)∂2 V(x) + c
xj j,k xk j,k xj,xk 1
j,k=1 j,k=1
≤ (∇V)(x)·S(x)(∇V)(x)
for all x such that (cid:107)x(cid:107)≥r .
1
It follows from the first condition in (2.4) that V is bounded below by a finite
constant: there exists c ∈ R such that V(y) ≥ c for all y ∈ Rd. Of course,
2 2
M(x) = I, where I represents the identity matrix, and V(x) = (cid:107)x(cid:107)2+c satisfy all
previous hypotheses for an appropriate constant c.
TheregularityofMandV,andassumptions(2.1),(2.2)aresufficienttoguaran-
tee the existence of smooth solutions of some Dirichlet problems. Conditions (2.4),
(2.5)guaranteethattheprocesswhosegeneratorisgivenbyLispositiverecurrent.
Elliptic equations Fix a domain (open and connected set) Ω ⊆ Rd. Denote by
Ck(Ω), k ≥ 0, the space of functions on Ω whose partial derivatives up to order k
are continuous, and by Ck,α(Ω), 0 < α < 1, the space of function in Ck(Ω) whose
k-th order partial derivatives are H¨older continuous with exponent α.
Denote by Ω the closure of Ω and by ∂Ω its boundary. The domain Ω is said to
have a Ck,α-boundary, if for each point x∈∂Ω, there is a ball B centered at x and
a one-to-one map ψ from B onto C ⊂Rd such that ψ(B∩Ω)⊂{z ∈Rd :z >0},
d
ψ(B∩∂Ω)⊂{z ∈Rd :z =0}, ψ ∈Ck,α(B), ψ−1 ∈Ck,α(C).
d
Denote by L2(Ω) the space of functions f : Ω → R endowed with the scalar
product (cid:104)·, ·(cid:105) given by
µ
(cid:90)
(cid:104)f, g(cid:105) = fgdµ,
µ
Ω
and by W1,2(Ω) the Hilbert space of weakly differentiable functions endowed with
the scalar product (cid:104)f,g(cid:105) given by
1
(cid:90) (cid:90)
(cid:104)f,g(cid:105) = fgdµ + ∇f ·∇gdµ.
1
Ω Ω
Fix 0 < α < 1, a function g in L2(Ω)∩Cα(Ω) and a function b in W1,2(Ω)∩
C2,α(Ω). Assume that Ω has a C2,α-boundary. It follows from assumptions (2.1),
(2.2) and Theorems 8.3, 8.8 and 9.19 in [12] that the Dirichlet boundary-value
problem
(cid:40)
(Lu)(x) = −g(x) x∈Ω,
(2.6)
u(x) = b(x) x∈∂Ω.
has a unique solution in W1,2(Ω)∩C2,α(Ω). Moreover, by the maximum principle,
[12, Theorem 8.1], if g=0,
inf b(x) ≤ inf u(y) ≤ supu(y) ≤ sup b(x). (2.7)
x∈∂Ω y∈Ω y∈Ω x∈∂Ω
The proofs of Theorems 8.3 and 8.8 in [12] require simple modifications since it is
easier to work with µ(dx) as reference measure than the Lebesgue measure.
Dirichlet and Thomson principle We will frequently assume that a pair of sets
A, B with C1-boundaries fulfill the following conditions. Denote by d(A,B) the
METASTABILITY IN NON-REVERSIBLE DIFFUSION PROCESSES 5
distance between the sets A, B, d(A,B) = inf{(cid:107)x−y(cid:107) : x ∈ A, y ∈ B}, and by
σ(∂A) the measure of the boundary of A.
AssumptionS.ThesetsA,BaretwoboundeddomainsofRdwithC2,α-boundaries,
for some 0 < α < 1, and finite perimeter, σ(A) < ∞, σ(B) < ∞. Moreover,
d(A,B)>0, and the set Ω=(A∪B)c is a domain.
Denote by h the unique solution of the Dirichlet problem (2.6) with Ω =
A,B
(A∪B)c, g = 0, and b such that b(x) = 1, 0 if x ∈ ∂A, ∂B, respectively. The
function h is called the equilibrium potential between A and B. Similarly,
A,B
denote by h∗ the solution to (2.6) with M replaced by its transpose M† and the
A,B
same functions g and b.
The capacity between A and B, denoted by cap(A,B), is defined as
(cid:90)
cap(A,B) = (cid:2)M(x)∇h (x)(cid:3)·n (x)e−V(x)σ(dx), (2.8)
A,B Ω
∂A
whereσ(dx)representsthesurfacemeasureontheboundary∂Ωandn represents
Ω
the outward normal vector to ∂Ω (and, therefore, the inward normal vector to
∂A∪∂B). Since ∂A is the 1-level set of the equilibrium potential h which, by
A,B
the maximum principle, is bounded by 1, ∇h =cn (x) for some c≥0 so that
A,B Ω
M(x)(∇h )(x)·n (x)=S(x)(∇h )(x)·n (x)≥0. The capacity can also be
A,B Ω A,B Ω
expressed as
(cid:90) ∂h
cap(A,B) = − A,B e−V dσ ,
∂v
∂A
wherev(x)=M†(x)n (x). WepresentinSection3somepropertiesofthecapacity.
A
Let F = F be the Hilbert space of vector fields ϕ : Ω → Rd endowed with
A,B
the scalar product given by
(cid:90)
(cid:104)ϕ, ψ(cid:105) = ϕ(x)·S(x)−1ψ(x)eV(x)dx.
Ω
By assumption (2.2), (cid:104)ϕ, ϕ(cid:105)≤λ−1(cid:82) (cid:107)ϕ(x)(cid:107)2eV(x)dx. By Schwarz inequality, for
Ω
every ϕ, ψ ∈F,
(cid:104)ϕ, ψ(cid:105)2 ≤ (cid:104)ϕ, ϕ(cid:105)(cid:104)ψ, ψ(cid:105). (2.9)
Denote by F(c), c∈R, the space of vector fields ϕ∈F of class C1(Ω) such that
(∇·ϕ)(x) = 0 for x∈Ω,
(cid:90) (cid:90)
− ϕ(x)·n (x)σ(dx) = c = ϕ(x)·n (x)σ(dx),
Ω Ω
∂A ∂B
The reason for the minus sign is due to the convention that n (x) is the inward
Ω
normal to ∂A. The integrals over ∂A, ∂B are well defined because ϕ is continuous,
and A, B have finite perimeter. On the other hand, the integral over ∂B must be
equal to minus the one over ∂A because ϕ is divergence free on Ω.
Forafunctionf :Ω→RinC2(Ω)∩W1,2(Ω),denotebyΦ ,Φ∗,Ψ theelements
f f f
of F given by
Ψ = e−VS∇f , Φ = e−VM†∇f , Φ∗ = e−VM∇f . (2.10)
f f f
Denote by Ca,b , a, b∈R the set of bounded functions f in C2(Ω)∩W1,2(Ω) such
A,B
that f(x)=a, x∈∂A, f(y)=b, y ∈∂B.
6 C.LANDIM,M.MARIANI,I.SEO
Proposition 2.1 (Dirichlet Principle). Let A, B be two open subsets satisfying
Assumption S. Then,
cap(A,B) = inf inf (cid:104)Φ −ϕ, Φ −ϕ(cid:105). (2.11)
f f
f∈C1,0 ϕ∈F(0)
A,B
The minimum is attained at f =(1/2)(h +h∗ ), and ϕ=Φ −Ψ .
A,B A,B f hA,B
Proposition 2.2 (Thomson principle). Let A, B be two open subsets satisfying
Assumptions S. Then,
1
cap(A,B) = sup sup · (2.12)
(cid:104)Φ −ϕ, Φ −ϕ(cid:105)
f∈C0,0 ϕ∈F(1) f f
A,B
The maximum is attained at f =(h −h∗ )/2cap(A,B), and ϕ=Φ −Ψ ,
A,B A,B f gA,B
where g =h /cap(A,B).
A,B A,B
In section 4, we present a Dirichlet and a Thomson principle in compact mani-
folds.
2.2. Diffusions in double-well potential field. In this subsection, we state the
Eyring-Kramers formula for a non-reversible diffusion in a double-well potential
field.
Consider a potential U : Rd → R. Denote by H the height of the saddle
x,y
points between x and y ∈Rd:
H = infH(γ) := inf sup U(z), (2.13)
x,y
γ γ z∈γ
wheretheinfimumiscarriedoverthesetΓ ofallcontinuouspathsγ :[0,1]→Rd
x,y
such that γ(0)=x, γ(1)=y. Let G bethe smallestsubsetof {z ∈Rd :U(z)=
x,y
H } with the property that any path γ ∈ Γ such that H(γ) = H contains
x,y x,y x,y
a point in G . The set G is called the set of gates between x and y.
x,y x,y
The Potential. We assume that the potential field U is such that
(P1) U ∈C3(Rd) and lim inf U(x)=∞.
n→∞ x:(cid:107)x(cid:107)≥n
(P2) ThefunctionU hasfinitelymanycriticalpoints. Onlytwoofthem,denoted
bym andm ,arelocalminima. TheHessianofU ateachoftheseminima
1 2
has d strictly positive eigenvalues.
(P3) The set of gates between m and m is formed by (cid:96) ≥ 1 saddle points,
1 2
denotedbyσ ,...,σ . TheHessianofU ateachsaddlepointσ hasexactly
1 (cid:96) i
one strictly negative eigenvalue and (d−1) strictly positive eigenvalues.
(P4) The function U satisfies
x (cid:110) (cid:111)
lim ·∇U(x) = lim (cid:107)∇U(x)(cid:107)−2∆U(x) = ∞, (2.14)
(cid:107)x(cid:107)→∞(cid:107)x(cid:107) (cid:107)x(cid:107)→∞
and
(cid:90)
Z := exp{−U(x)/(cid:15)}dx < ∞
(cid:15)
Rd
for all (cid:15)>0.
It is not difficult to show that the conditions (2.14) imply that
(cid:90)
e−U(x)/(cid:15)dx ≤ C(a)e−a/(cid:15) (2.15)
x:U(x)≥a
where the constant C(a) is uniform in (cid:15)≤1.
METASTABILITY IN NON-REVERSIBLE DIFFUSION PROCESSES 7
Diffusion model. Let M be a d×d (generally non-symmetric) positive-definite
matrix: v·Mv >0 for all v (cid:54)=0. Denote by {X(cid:15) :t∈[0,∞)}, (cid:15)>0, the diffusion
t
process associated to the generator L given by
(cid:15)
(cid:88)
(L f)(x) = −∇U(x)·M(∇f)(x) + (cid:15) M (∂2 f)(x).
(cid:15) ij xi,xj
1≤i,j≤d
Note that we can rewrite the generator L as
(cid:15)
(cid:104) (cid:105)
(L f)(x) = (cid:15)eU(x)/(cid:15)∇ · e−U(x)/(cid:15)M(∇f)(x) .
(cid:15)
In particular, the probability measure
µ (dx) := Z−1exp{−U(x)/(cid:15)}dx
(cid:15) (cid:15)
is the stationary state of the process X(cid:15).
t
The process X(cid:15) can also be written as the solution of a stochastic differential
t
equation. Recall that K represents the symmetric, positive-definite square root of
the symmetric matrix S=(M+M†)/2: S=KK. It is easy to check that X(cid:15) is the
t
solution of the stochastic differential equation (1.3).
Let A, B⊂Rd be two open subsets of Rd satisfying the assumptions S, and let
Ω=(A∪B)c. In the present context, the capacity, defined in (2.8), is given by
cap(A,B) = Z(cid:15) (cid:90) (cid:2)M(x)∇hA,B(x)(cid:3)·nΩ(x)e−U(x)/(cid:15)σ(dx). (2.16)
(cid:15) ∂A
Structure of valleys. Let h = U(m ), i = 1, 2, and assume without loss of
i i
generality that h ≥ h , so that m is the global minimum of the potential U.
1 2 2
Denote by H the height of the saddle points S:={σ , ..., σ }:
1 (cid:96)
H := U(σ ) = ··· = U(σ ).
1 (cid:96)
Let Ω be the level set defined by saddle points which separate m from m :
1 2
Ω := {x∈Rd :U(x)<H}.
Denote by W and W the two connected components of Ω such that m ∈ W ,
1 2 i i
i=1, 2, respectively. Note that W ∩W =S.
1 2
Denote by V and V two metastable sets containing m and m , respectively.
1 2 1 2
More precisely, V , i = 1, 2, is a open subset of W which satisfies assumptions S
i i
and such that
B (m ) ⊂ V ⊂ {x∈Rd :U(x)<U(σ)−κ}
(cid:15) i i
for some κ > 0, where B (m ) represents the ball of radius (cid:15) centered at m :
(cid:15) i i
B (m )={x:|x−m |<(cid:15)}.
(cid:15) i i
Metastabilityresults. FixasaddlepointσofthepotentialU. Denoteby−λσ <
1
0 <λσ < ··· <λσ the eigenvalues of (Hess U)(σ):= Lσ. By Lemma 10.1 of [17],
2 d
both LσM and LσM† have a unique negative eigenvalue. The negative eigenvalues
of LσM and LσM† coincide because LσM† = Lσ(LσM)†(Lσ)−1. Denote by −µσ
this common negative eigenvalue, and let
µσ
ω(σ) := ; σ ∈S. (2.17)
(cid:112)
−det[(Hess U)(σ)]
WeproveinSection5thefollowingsharpestimateforcapacitybetweenthevalleys
V and V .
1 2
8 C.LANDIM,M.MARIANI,I.SEO
Figure 1. The structure of metastable wells and valleys.
Theorem 2.3. We have the following estimate on the capacity.
1 (2π(cid:15))d/2 (cid:88)(cid:96)
cap(V , V ) = [1+o (1)] e−H/(cid:15) ω(σ ). (2.18)
1 2 (cid:15) Z 2π i
(cid:15)
i=1
ThemetastablebehaviorofX(cid:15) followsfromthisresult. InSection6,wederivea
t
sharpestimateforthetransitionstimebetweenthetwodifferentwellsstatedbelow.
Denote by P , x ∈ Rd, the probability measure on C(R ,Rd) induced by the
x +
Markov process X(cid:15) starting from x. Expectation with respect to X is represented
t t
by E .
x
Denote by H , C an open subset of Rd, the hitting time of the set C:
C
H = inf{t≥0:X ∈C}. (2.19)
C t
Theorem 2.4. Under the notations above,
2πe(H−h1)/(cid:15) (cid:16)(cid:88)(cid:96) (cid:17)−1
E [H ] = [1+o (1)] ω(σ ) . (2.20)
m1 V2 (cid:15) (cid:112)det[(Hess U)(m )] i
1 i=1
Remark 2.5. Let Ξ ⊂ Rd be a bounded domain with a boundary in C2,α for
some 0 < α < 1. Assume that the potential has no critical points at ∂Ξ and that
n ·∇U > 0 at ∂Ξ. A similar result can be proven for a diffusion evolving on Ξ
Ξ
with Neumann boundary conditions.
Remark2.6. Theorems2.3,2.4togetherwiththetheorydeveloppedin[4,5]permit
to describe the metastable behavior of the diffusion X(cid:15). Denote by θ the expression
t (cid:15)
multiplied by [1+o (1)] in (2.20). If U(m ) < U(m ), on the time time-scale θ ,
(cid:15) 2 1 (cid:15)
starting from m , the diffusion process X(cid:15) remains a mean 1 exponential time in a
1 t
neighborhoodofm ,afterwhichitjumpstoaneighborhoodofm andthereremains
1 2
for ever. If U(m )=U(m ), on the time time-scale θ , X(cid:15) behaves as a two-state
2 1 (cid:15) t
Markov chain which jumps from one state to the other at mean 1 exponential times.
Remark 2.7. The arguments presented in the next sections to prove Theorems 2.3
and 2.4 apply to the case in which the entries of the matrix M(x) belong to C2(Rd)
and satisfy conditions (2.1), (2.2).
METASTABILITY IN NON-REVERSIBLE DIFFUSION PROCESSES 9
The article is organized as follows. Sections 3 and 4 are devoted to Propositions
2.1 and 2.2 and to the extension of the Dirichlet and the Thomson principle for
elliptic operators in compact manifolds without boundaries. In section 5, we prove
Theorem2.3byconstructingvectorfieldswhichapproximatetheoptimalones. The
propertiesofthesevectorfieldsarederivedinSection7,basedongeneralestimates
presented in Section 6. The last section is dedicated to Theorem 2.4.
3. The Dirichlet and the Thomson Principles
Denote by C([0,∞),Rd) the space of continuous functions ω from R to Rd
+
endowed with the topology of uniform convergence on bounded intervals. Let X ,
t
t ≥ 0 be the one-dimensional projections: X (ω) = ω(t), ω ∈ C([0,∞),Rd). We
t
sometimes represent X as X(t).
t
It follows from the assumptions on M and V, and from Theorems 1.10.4 and
1.10.6 in [21], that there exists a unique solution, denoted hereafter by {P : x ∈
x
Rd},tothemartingaleproblemassociatedtothegeneratorL. Moreover,thefamily
{P :x∈Rd} possesses the strong Markov and the Feller properties. Expectation
x
with respect to P is expressed as E .
x x
The process X can be represented in terms of a stochastic differential equation.
t
Denote by Σ : Rd → Rd ⊗Rd the square root of S(x), in the sense that Σ(x) is
a positive-definite, symmetric matrix such that Σ(x)Σ(x) = S(x). By [11, Lemma
6.1.1],theentriesofΣinherittheregularitypropertiesofS: Σ belongstoC3(Rd)
m,n
for1≤m,n≤d. TheprocessX istheuniquesolutionofthestochasticdifferential
t
equation
√
dX = b(X )dt + 2Σ(X )dB ,
t t t t
where B stands for a d-dimensional Brownian motion.
t
In view of condition (2.5), by [21, Theorem 6.1.3], the process X is positive
t
recurrent. Furthermore, by [21, Theorem 4.9.6], E [H ] < ∞ for all open sets
x C
C and all x (cid:54)∈ C. Finally, an elementary computation shows that the probability
measure µ is stationary.
The solutions of the elliptic equation (2.6) can be represented in terms of the
process X . Fix 0<α<1, a bounded function g in L2(Ω)∩Cα(Ω) and a bounded
t
function b in W1,2(Ω)∩C2,α(Ω). Assume that Ω has a C2,α-boundary. It follows
from the proof of [11, Theorem 6.5.1] and from the positive recurrence that the
unique solution u of (2.6) can be represented as
u(x) = E (cid:2)b(X(H ))(cid:3) + E (cid:104)(cid:90) HΩc g(X )dt(cid:105). (3.1)
x Ωc x t
0
In particular if A, B represent two open sets satisfying the assumptions S, the
equilibrium potential between A and B, introduced just above (2.8), is given by
h (x) = P [H <H ]. (3.2)
A,B x A B
3.1. Properties of the capacity. Wepresentinthissubsectionsomeelementary
properties of the capacity. We begin with an alternative formula for the capacity.
Unlessotherwisestated, untiltheendofthissection, theopensubsetsA, B satisfy
the assumptions S.
Lemma 3.1. We have that
(cid:90)
cap(A,B) = ∇h (x)·S(x)∇h (x)µ(dx).
A,B A,B
Rd
10 C.LANDIM,M.MARIANI,I.SEO
Proof. Since the function h is harmonic on Ω=(A∪B)c, and since it is equal
A,B
to 1 on the set ∂A and 0 on the set ∂B, the capacity cap(A,B) can be written as
(cid:90)
h (x)(cid:2)M(x)∇h (x)(cid:3)·n (x)e−V(x)σ(dx)
A,B A,B Ω
∂A
(cid:90)
+ h (x)(cid:2)M(x)∇h (x)(cid:3)·n (x)e−V(x)σ(dx)
A,B A,B Ω
∂B
(cid:90)
− h (x)∇·(cid:2)e−V(x)M(x)∇h (x)(cid:3)dx.
A,B A,B
Ω
Since h belongs to C2+α(Ω)∩W1,2(Ω) and since S represents the symmetric
A,B
part of the matrix M, by the divergence theorem, the previous expression is equal
to
(cid:90)
∇h (x)S(x)∇h (x)µ(dx).
A,B A,B
Ω
As the equilibrium potential is constant in A∪B, we may replace in the previous
integral the domain Ω by Rd, which completes the proof of the lemma. (cid:3)
Since h = 1−h , it follows from the previous lemma that the capacity is
B,A A,B
symmetric: for every disjoint subsets A, B of Rd,
cap(A,B) = cap(B,A). (3.3)
Adjoint generator. Denote by L∗ the L2(µ) adjoint of the generator L, which
acts on functions in C2(Rd) as
(L∗f)(x) = eV(x)∇·(cid:8)e−V(x)M†(x)(∇f)(x)(cid:9).
Let S be the symmetric part of the generator L, defined as S =(1/2)(L+L∗) and
acting on C2(Rd) as Sf =eV∇·(e−VS∇f).
Denote by cap∗(A,B) the capacity between the open sets A, B with respect to
the adjoint generator L∗. In view of (2.8), this capacity cap∗(A,B) is defined as
(cid:90)
cap∗(A,B) = (cid:2)M†(x)∇h∗ (x)(cid:3)·n (x)e−V(x)σ(dx), (3.4)
A,B Ω
∂A
where h∗ :Rd →[0,1], called the equilibrium potential between A and B for the
A,B
adjointgenerator,istheuniquesolutioninC2(Ω)∩W1,2(Ω)oftheellipticequation
(cid:40)
(L∗u)(x) = 0 x∈Ω,
u(x) = χ (x) x∈∂Ω.
A
The next lemma states that the capacity between two disjoint subsets A, B of
Rd coincides with the capacity with respect to the adjoint process. Recall that we
are assuming that A and B fulfill the conditions S.
Lemma 3.2. For every open subsets A, B of Rd,
cap(A,B) = cap∗(A,B).