The exit from a metastable state: concentration of the exit point distribution on the low energy saddle points

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The exit from a metastable state: concentration of the

exit point distribution on the low energy saddle points

Giacomo Di Gesù, Tony Lelièvre, Dorian Le Peutrec, Boris Nectoux

To cite this version:

The

exit

from

a

metastable

state:

Concentration

of

the

exit

point

distribution

on

the

low

energy

saddle

points,

part

1

Giacomo Di Gesùa,b, Tony Lelièvreb,∗, Dorian Le Peutrecc, Boris Nectouxa,b a

InstitutfürAnalysisundScientificComputing,E101-TUWien,WiednerHauptstr.8,1040Wien, Austria

b _{CERMICS,ÉcoledesPonts,UniversitéParis-Est,INRIA,77455Champs-sur-Marne,France}

c_{LaboratoiredeMathématiquesd’Orsay,Univ.Paris-Sud,CNRS,UniversitéParis-Saclay,91405Orsay,} France a b s t r a c t MSC: 60J60 58J65 35Q82 58C40 81Q20 Keywords: OverdampedLangevin Exitproblem

Smalltemperatureregime Semi-classicalanalysis

We consider the first exit point distribution from a bounded domain Ω of the stochastic process (Xt)t≥0solution to the overdamped Langevin dynamics

dXt=−∇f(Xt)dt +

√ h dBt

starting from the quasi-stationary distribution in Ω. In the small temperature regime (h → 0) and under rather general assumptions on f (in particular, f may have several critical points in Ω), it is proven that the support of the distribution of the first exit point concentrates on some points realizing the minimum of f on ∂Ω. Some estimates on the relative likelihood of these points are provided. The proof relies on tools from semi-classical analysis.

r és u m é

Dans ce travail, nous étudions la distribution du point de sortie d’un domaine borné Ω pour le processus stochastique (Xt)t≥0 solution de la dynamique de Langevin

suramortie

dXt=−∇f(Xt)dt +

√ h dBt

initialement distribué suivant la distribution quasi-stationnaire dans Ω. Dans la limite basse température h→ 0 et sous des hypothèses générales sur la fonction f

(f pouvant notamment avoir plusieurs points critiques dans Ω), nous montrons que la distribution du lieu de sortie se concentre sur certains points réalisant le minimum de f sur ∂Ω. Nous calculons aussi les probabilités relatives de sortir autour de chacun de ces points. Nos preuves reposent sur des outils issus de l’analyse semi-classique.

© 2019 Elsevier Masson SAS. All rights reserved.

* Correspondingauthor.

E-mailaddresses:giacomo.di.gesu@asc.tuwien.ac.at(G. Di Gesù),tony.lelievre@enpc.fr(T. Lelièvre),

1. Introduction andmainresults

1.1. Setting andmotivation

WeareinterestedintheoverdampedLangevindynamics

dXt=−∇f(Xt)dt + √

h dBt, (1)

whereXtisavectorinRd, f : Rd→ R isaC∞function,h isapositiveparameterand(Bt)t≥0isastandard d-dimensional Brownianmotion.Such adynamics isprototypical ofmodels used forexamplein computa-tionalstatistical physicsto simulatethe evolutionof amolecular systemat afixedtemperature, inwhich casef isthepotential energyfunctionandh isproportionalto thetemperature.It admitsas aninvariant measure theBoltzmann-Gibbsmeasure(canonicalensemble)Z−1e−h2f (x)dx whereZ =

Rde−

2

hf <∞. In the small temperature regime h → 0, the stochastic process (Xt)t≥0 is typically metastable: it staysfor avery long periodof time in aneighborhood of a local minimum of f (called ametastable state)before hoppingto another metastablestate. In thecontext of statistical physics, this behavior is expectedsince the molecular system typically jumps between various conformations, which are indeedthese metastable states.Formodeling purposesas wellasforbuildingefficient numericalmethods(seeforinstance [1–3]),it isthusinterestingtobeabletopreciselydescribetheexiteventfromametastablestate,namelythelawof thefirstexittime andthefirstexitpoint.

Themainobjectiveofthisworkistoaddressthefollowingquestion:givenametastabledomainΩ⊂ Rd, whataretheexitpointsinthesmalltemperatureregime h→ 0?Thisisformalizedmathematicallybythe notionof concentration,which isnow introduced. Foradomain Ω⊂ Rd _{andagiven initial conditionX}

0, letusconsidertheexitevent(τΩ,XτΩ) from Ω where

τΩ= inf{t ≥ 0|Xt∈ Ω}/ (2)

isthefirst exittime fromΩ. Wewillconsider thefamilyoflawsof XτΩ, ash goesto zero,andprovethat

thesedistributionsconcentrateonasubsetof∂Ω,asdefinedbelow.

Definition1.LetY ⊂ ∂Ω andletus considerafamilyof randomvariables (Yh)h≥0 whichadmits alimitin distributionwhen h→ 0.The law of Yh concentratesonY inthelimit h→ 0 if foreveryneighborhood VY of Y in ∂Ω,

lim

h→0P [Yh∈ VY] = 1, andif forallx∈ Y andforallneighborhoodsVxof x in ∂Ω,

lim

h→0P [Yh∈ Vx] > 0. In otherwords,Y isthesupportof thelaw ofYh inthelimit h→ 0.

Previousresultsonthe behavior ofthe lawof XτΩ whenh→ 0.Let us reportonpreviousresultson the

lawofXτΩ inthelimith→ 0 (seealso[4] foracomprehensivereviewoftheliterature).

Using partial differential equation techniques, some of the formal results above have been rigorously proven.Forexample,when

∂nf > 0 on ∂Ω, (3)

and

{x ∈ Ω, |∇f(x)| = 0} = {x0} with f(x0) = min Ω

f and det Hessf (x0) > 0, (4) the concentrationof thelaw of XτΩ inthelimit h→ 0 on arg min∂Ωf hasbeen obtained in [8–10], when X0= x∈ Ω, seealso [11,12] formorerecentresultswithsimilar techniques.

Finally, anotherrigorous approachto study theexitpoint distributionis torely onthetheory of large deviations. When (3)–(4) hold and f attains its minimum on ∂Ω at asingle point y0, [13, Theorem 2.1 in Chapter 4.2] implies that the law of XτΩ concentrates on y0 in the limit h → 0, when X0 = x ∈ Ω.

This result has then been generalized in [14,15] when only (3)–(4) are satisfied. In [13, Theorem 5.1 in Chapter 6.5],undermoregeneralassumptionsonf , forΣ⊂ ∂Ω,thelimitofhln P [XτΩ ∈ Σ] when h→ 0

is related to a minimizationproblem involving the quasipotential of the process (1). Let us mention two limitations when applying [13, Theorem 5.1 in Chapter 6.5] in order to obtain some information on the first exitpointdistribution.First,this theorem requiresto be abletocomputethe quasipotentialinorder to getuseful information:this istrivialundertheassumptions (3)–(4) butmorecomplicatedfor ageneral function f (inparticular whenf has severalcritical pointsinΩ).Second,even whenthequasipotential is analyticallyknown,thisresultonlygivesthesubsetof∂Ω throughwhichexitoccurswithanexponentially small probabilityinthelimith→ 0. Itdoesnotallowto excludepointsthroughwhich exitoccurs witha polynomiallysmallprobabilityinh forexample(thisindeedhappens,see Section 1.4).Besides,itdoesnot givetherelative probabilityoftheexitpointswhichhaveanon-zeroprobabilityinthelimith→ 0.

The quasi-stationarydistribution approachwhichisused inthis worktostudy theexiteventhas been introducedin [16,17,1].Noticethatcompared tothework [1], wehereonlyidentifythesupportofthefirst exitpointdistribution,andtherelativelikelihoodofthepointsinthissupport,whereasin [1],wealsostudy the exit throughpoints which occur with exponentiallysmall probability inthe limith → 0. Theresults herearethuslessprecisethanin [1],buttheassumptionsonf arealsomuchmoregeneral.

Toconcludethisshort reviewoftheliterature,letusmentionthat[8–10,14,15,13] also coverthecaseof non reversiblediffusions,whilewehereonlyconsiderthereversibledynamics (1).

Purpose ofthis work: ageneralgeometricsettingand aprecisedescriptionofthe exitpoint distribution. In this work,we study theconcentrationof thelaw of XτΩ on arg min∂Ωf in thelimith→ 0.Compared to results previouslyobtainedintheliterature, thenovelty istwofold: first, thegeometric settingis much moregeneral, andsecond,we obtainaprecisedescription ofthefirst exitpointdistribution,by providing therelative probabilitiesoftheexitpoints.

Moreprecisely, weexhibitassumptions onf whichensurethatwhen X0 isdistributedaccordingtothe quasi-stationarydistribution νhin Ω (see Definition2below),thelawofXτΩ concentratesinthelimith→ 0

on some global minima of f on ∂Ω: these global minima and the relative probabilityof these exitpoints aremadeexplicit.Thegeometricsettingismuchmoregeneralthan (3)–(4).Forinstance,itisnotassumed that∂nf > 0 on ∂Ω,there is norestrictiononthe numberofcritical pointsof f in Ω and f is allowed to have critical pointsin Ω with largerenergies than min∂Ωf (however we do notconsider the casewhen f hascriticalpointson ∂Ω,andweworkundersomeMorseassumptiononf ).

and [1–3]).Thecompanionpaper [18] buildsonandextendstheresultofthepresentworktogeneralinitial conditionsin Ω,as explainedinRemark8below.

Herearerepresentative examplesofoutputsofthiswork.First,if{y ∈ Ω,f (y)< min∂Ωf} isconnected andcontainsallthecriticalpointsof f in Ω,andif∂nf > 0 onarg min∂Ωf ,thenthelawof XτΩconcentrates

on arg min_∂Ωf when X0 ∼ νh. Besides, when some critical points of f in Ω are larger in energy than min∂Ωf ,thenthelawofXτΩ concentratesonasubsetof ∂Ω whichmaybestrictlyincludedinarg min∂Ωf .

Inparticular,thefollowingphenomena mayoccur:

(i) There may exist points z ∈ arg min_∂Ωf , C > 0 and c > 0, such that for every sufficiently small neighborhoodΣz of z in∂Ω,inthelimith→ 0, Pνh[XτΩ ∈ Σz]≤ C e−

c

h (see (24) in Theorem1and thediscussion afterthestatementofTheorem1).

(ii) Theremayexistpointsz∈ arg min_∂Ωf and C > 0 suchthatforeverysufficientlysmallneighborhood Σz ofz in∂Ω,Pνh[XτΩ ∈ Σz] = C

√

h (1+ o(1)). ThisisexplainedinSection1.4.

Letusfinally mentionthatwhileproving theseresults,wealso obtainasharpasymptotic estimateonthe principaleigenvalue(when h→ 0)and ontheprincipaleigenfunction ofthe infinitesimalgeneratorof the diffusion (1) with Dirichlet boundary conditions on ∂Ω, see Section 1.3.4. These results have their own interests.

OrganizationoftheendofSection1.InSection1.2,thequasi-stationarydistributionoftheprocess (1) inΩ isintroduced.InSection1.3,theassumptionsonf whichwillbeusedthroughoutthis paperarepresented and the main result of this work is stated (see Theorem 1). Finally, the necessity of the assumptions of Theorem1isdiscussedinSection1.4.

1.2. Metastability andthequasi-stationary distribution

The quasi-stationary distribution is the cornerstone of our analysis. Here, we assume that Ω ⊂ Rd is smooth,open, boundedandconnected(seeSection1.3forthegeneralgeometric setting).

Definition 2.A quasi-stationary distribution forthestochastic process (Xt)t≥0 inthe domain Ω⊂ Rd is a probabilitymeasureνh supportedin Ω such thatforallmeasurable setsA⊂ Ω andforallt≥ 0

νh(A) =  Ω Px[Xt∈ A, t < τΩ] νh(dx)  Ω Px[t < τΩ] νh(dx) . (5)

Here and in the following, the subscript x indicates that the stochastic process starts from x ∈ Rd (X0 = x). In words, (5) means that if X0 is distributed according to νh, then for all t > 0, Xt is still distributedaccordingtoνh conditionallyonXs∈ Ω foralls∈ [0,t].Thefollowingresultshavebeenproven in [16] (seealso [19] formuchmoregeneralresultsonquasi-stationarydistributions):

Proposition 3.LetΩ⊂ Rd be aboundeddomain and considerthe dynamics (1). Thenthere existsa prob-ability measure νh with support in Ω such that, whatever the law of the initial condition X0 with support

in Ω,itholds:

lim

Here, Law(Xt|t < τΩ) denotes the law of Xt conditional to the event {t < τΩ}. A corollary of this propositionisthatthequasi-stationarydistributionνh existsandisunique.Foragiveninitialdistribution oftheprocess (1),iftheconvergencein (6) ismuchquickerthantheexitfromΩ,theexitfromthedomain Ω issaidtobemetastable.WhentheexitfromΩ ismetastable,itisthusrelevanttostudytheexiteventfrom Ω assumingthattheprocess(1) isinitiallydistributedaccordingtothequasi-stationarydistribution νh.This will bethesettingofthis work.

Letusnowrelatethenotionofquasi-stationarydistributiontotheinfinitesimalgeneratorofthe dynam-ics (1)

L(0)_f,h=−∇f · ∇ +h

2 Δ. (7)

InthenotationL(0)_f,h,thesuperscript(0) indicatesthatweconsideranoperatoronfunctions,namely0-forms. The basic observation to define our functional framework is thatthe operator L(0)_f,h is self-adjoint on the weightedL2 _space L2_w(Ω) = ⎧ ⎨ ⎩u : Ω→ R,  Ω u2e−2hf <∞ ⎫ ⎬ ⎭

(theweightedSobolevspaces Hwk(Ω) aredefinedsimilarly).Indeed,for anysmoothtest functionsu andv with compactsupportsin Ω, onehas

 Ω (L(0)_f,hu)v e−h2f =  Ω (L(0)_f,hv)u e−h2f =−h 2  Ω ∇u · ∇v e−2 hf.

This givesa proper framework to introduce the Dirichletrealization LD,(0)_f,h onΩ of the operator L(0)_f,h as follows:

Proposition4.TheFriedrichsextensionassociatedwiththequadraticformφ∈ C_c∞(Ω)→ h₂_Ω|∇φ|2e−h2f is

denoted by−LD,(0)_f,h .Itisanonnegativeunboundedself-adjointoperatoronL2

w(Ω) with domainD

LD,(0)_f,h 

= H_w,01 (Ω)∩ H_w2(Ω), whereH_w,01 (Ω)={u∈ H_w1(Ω),u= 0 on ∂Ω}. ThecompactinjectionH1

w(Ω)⊂ L2w(Ω) impliesthattheoperatorL D,(0)

f,h hasacompactresolventandits spectrumisconsequentlypurelydiscrete.Letusintroduceλh> 0 thesmallesteigenvalueof−LD,(0)f,h (a.k.a. theprincipaleigenvalue):

λh= inf σ  − LD,(0) f,h . (8)

From standardresultsonellipticoperators (seee.g. [20,21]), λh isnondegenerateanditsassociated eigen-function uh has a sign on Ω. Moreover, uh ∈ C∞(Ω). Without loss of generality, one can then assume that:

uh> 0 on Ω and  Ω

u2_he−h2f _{= 1. (9)}

Theeigenvalue-eigenfunctionpair(λh,uh) satisfies: 

−L(0)

f,huh= λhuh on Ω, uh= 0 on ∂Ω.

Thelinkbetweenthequasi-stationary distributionνh andthefunctionuh isgivenbythefollowing propo-sition(seeforexample[16]):

Proposition 5.The unique quasi-stationary distribution νh associated with the dynamics (1) and the do-main Ω is givenby:

νh(dx) = uh(x)e− 2 hf (x)  Ω uh(y)e− 2 hf (y)dy dx. (11)

Letusrecall that ∂n = n· ∇ standsforthenormalderivativeand n istheunitoutward normalon ∂Ω. Thenextproposition(seeagain[16])characterizesthelawoftheexiteventfromΩ.

Proposition 6.Let us consider the dynamics (1) and the quasi-stationary distribution νh associated with the domain Ω. If X0 is distributed according to νh, the random variables τΩ and XτΩ are independent. FurthermoreτΩisexponentiallydistributedwithparameterλh andthelaw ofXτΩ hasadensitywithrespect totheLebesguemeasureon ∂Ω givenby

z∈ ∂Ω → − h 2λh ∂nuh(z)e− 2 hf (z)  Ω uh(y)e− 2 hf (y)dy . (12)

1.3. Hypothesesandmain results 1.3.1. Hypothesesandnotation

Inthefollowing,we consider asettingthatis actuallymoregeneral thantheoneof Section1.2:Ω is a

C∞ orientedcompactandconnectedRiemannianmanifoldof dimensiond withboundary∂Ω.

Thefollowingnotationwillbeused:fora∈ R,{f < a}={x∈ Ω, f (x)< a},{f ≤ a}={x∈ Ω, f (x)≤ a} and {f = a} = {x ∈ Ω, f (x) = a}. Let us now introduce the basic assumption on f which is used throughoutthis work:

The function f : Ω→ R is C∞, and for all x∈ ∂Ω, |∇f(x)| = 0. The functions f : Ω→ R and f : {x ∈ ∂Ω, ∂nf (x) > 0} → R are Morse. Moreover, f has at least one local minimum in Ω.

⎫ ⎪ ⎬ ⎪

⎭ (A0)

Letusrecallthatafunctionφ: Ω→ R isaMorsefunctionifallitscriticalpointsarenondegenerate(which impliesinparticularthatφ hasafinite numberofcriticalpointssinceΩ iscompactand anon degenerate criticalpointis isolatedfromtheother criticalpoints).Acritical pointz∈ Ω of φ isnon degenerateifthe Hessianmatrixof φ atz,denotedbyHess φ(z),isinvertible.Wereferfor exampleto [22,Definition4.3.5] foradefinitionof theHessian matrixon amanifold.A non degeneratecriticalpoint z∈ Ω of φ is saidto haveindex p∈ {0,. . . ,d} if Hess φ(z) has precisely p negativeeigenvalues (counted with multiplicity). In thecasep= 1, z iscalledasaddlepoint.

Foranylocal minimumx of f in Ω,theheightoftheenergybarrierto leaveΩ fromx is

Hf(x) := inf γ∈C0_([0,1],Ω) γ(0)=x, γ(1)∈∂Ω max t∈[0,1]f  γ(t) , (13)

• (A0) holdsand

∃!Cmax∈ C such that max C∈C  max C f− min C f= max Cmax f− min Cmax f (A1) where C :=C(x), x is a local minimum of f in Ω, (14) with,forany localminimumx off inΩ,

C(x) is the connected component of{f < Hf(x)} containing x. (15) • (A1) holdsand

∂Cmax∩ ∂Ω = ∅. (A2)

• (A1) holdsand

∂Cmax∩ ∂Ω ⊂ arg min

∂Ω

f. (A3)

It will be shown that the assumptions (A0), (A1), (A2), and (A3) ensure that when X0 ∼ νh, the law of XτΩ concentrates on the set ∂Cmax ∩ ∂Ω, see items 1 and 2 in Theorem 1. Finally, let us introduce

assumption (A4): • (A1) holdsand

∂Cmax∩ Ω contains no separating saddle point of f. (A4) Roughly speaking,asaddlepoint z off is separatingifforany sufficiently smallconnected neighborhood

Vz of z, Vz ∩ {f < f(z)} has two connected components included in two different connected compo-nents of {f < f(z)}. We refer to Definition 13 below for more details. The assumption (A4) together with (A0), (A1), (A2), and (A3), ensures thatthe probability thatthe process (1) leaves Ω through any sufficiently small neighborhoodof z∈ ∂Ω\ ∂Cmax in ∂Ω isexponentiallysmall when h→ 0,see item3in Theorem 1.

Fig. 1 givesa one-dimensional example where (A1), (A2), (A3) and (A4) aresatisfied. In Section 1.4, thenecessityofassumptions (A1), (A2), (A3),and (A4) isdiscussed.Wewillactuallyworkwithequivalent formulationsoftheassumptions (A1), (A2), (A3),and (A4) whichwillbe giveninSection2.4.

1.3.2. Notationforthelocal minimaandsaddlepoints ofthefunction f

The main purpose of this section is to introduce the local minima and the generalized saddle points of f .TheseelementsofΩ areusedextensivelythroughoutthisworkandplayacrucialroleinouranalysis. Roughlyspeaking,thegeneralizedsaddlepointsof f arethesaddlepointsz∈ Ω ofthefunctionf extended

by−∞ outsideΩ (whichisindeedconsistentwiththehomogeneousDirichletboundaryconditionsin (10)). Thus, when the function f satisfies the assumption (A0), a generalized saddle point of f (as introduced in [23])iseitherasaddlepointz∈ Ω of f or alocalminimum z∈ ∂Ω off|∂Ω suchthat∂nf (z)> 0.

Letus assumethatthefunctionf satisfiestheassumption (A0).Letus denoteby UΩ₀ ={x1, . . . , xmΩ

Fig. 1. Aone-dimensionalcasewhere (A1), (A2), (A3) and (A4) aresatisfied.Onthefigure,f (x1)= f (x5),Hf(x1)= Hf(x4)=

Hf(x5),C = {Cmax,C2,C3},∂C2∩ ∂Cmax=∅ and∂C3∩ ∂Cmax=∅.

thesetoflocalminimaof f in Ω wheremΩ

0 ∈ N isthenumberoflocal minimaof f in Ω.Noticethatsince

f satisfies(A0), mΩ

0 ≥ 1.Theset ofsaddlepointsof f ofindex1 in Ω isdenotedby UΩ1 and itscardinality by mΩ

1.Letusdefine

U∂Ω₁ :={z ∈ ∂Ω, z is a local minimum of f|∂Ω but not a local minimum of f in Ω}. Noticethatanequivalent definitionof U∂Ω

1 is

U∂Ω1 ={z ∈ ∂Ω, z is a local minimum of f|∂Ω and ∂nf (z) > 0}, (17) whichfollowsfrom thefactthat∇f(x)= 0 forallx∈ ∂Ω.Letusintroduce

m∂Ω₁ := Card(U∂Ω₁ ). (18)

Inaddition,onedefines:

UΩ₁ := U∂Ω₁ ∪ UΩ₁ and m₁Ω:= Card(UΩ₁) = m∂Ω₁ + mΩ₁. (19) As explained above, the set UΩ₁ is the set of the generalized saddle points of f . If U∂Ω₁ is not empty,its elementsaredenotedby:

U∂Ω₁ ={z1, . . . , zm∂Ω

1 } ⊂ ∂Ω, (20)

andifUΩ

1 isnotempty,itselementsarelabeled asfollows:

UΩ₁ ={z_m∂Ω

1 +1, . . . , zmΩ1} ⊂ Ω. (21)

Thus, one has:UΩ

1 ={z1,. . . ,zm∂Ω

1 ,zm∂Ω1 +1,. . . ,zmΩ1}.Moreover, we assumethat the elements of U ∂Ω

1 are orderedsuchthat:

Fig. 2. Schematic representationofC (see (14))andf|∂Ω whentheassumptions (A0),(A1), (A2) and (A3) aresatisfied.In this representation,x1∈ Ω istheglobalminimumoff inΩ andtheotherlocalminimaoff inΩ arex2andx3(thusUΩ0 ={x1,x2,x3}

and mΩ

0 = 3).Moreover, min∂Ωf = f (z1) = f (z2) = f (z3) = Hf(x1) = Hf(x2) < Hf(x3) = f (z4), {f < Hf(x1)} hastwo

connectedcomponents:Cmax(see (A1))whichcontainsx1andC2whichcontainsx2.Thus,onehasC = {Cmax,C2,C3}.Inaddition,

U∂Ω1 ={z1,z2,z3,z4} (m∂Ω1 = 4),{z1,z2,z3}= arg min∂Ωf (k ∂Ω 1 = 3 andk ∂Cmax 1 = 2),U Ω 1 ={z5,z6,z7} where{z5}= Cmax∩ C2

(mΩ1 = 3 and (A4) isnotsatisfied)andmin(f (z6),f (z7))> f (z4),∂Cmax∩ ∂Ω={z1,z2} (k∂C1 max= 2).Finally,onehasm Ω

1 = 7.The

pointym∈ Ω isalocalmaximumoff withf (ym)> f (zi) foralli∈ {1,. . . ,7}.

Letusassumethat (A1), (A2),and (A3) aresatisfied.LetusrecallthatCmaxisdefinedby (A1).Moreover, inthiscase,onehask∂Ω₁ ≥ 1 and

∂Cmax∩ ∂Ω ⊂ {z1, . . . , zk∂Ω

1 }.

Indeed, by assumption ∂Cmax ∩ ∂Ω⊂ {f = min∂Ωf} (see (A3))and there is no local minima of f in Ω on ∂Cmax (since Cmax is a connected component of a sublevel set of f ). We assume lastly that the set

{z1,. . . ,zk∂Ω

1 } isorderedsuchthat:

{z1, . . . , z_k∂Cmax

1 } = {z1, . . . , zk

∂Ω

1 } ∩ ∂Cmax. (23)

Noticethatk∂Cmax

1 ∈ {0,. . . ,k∂Ω1 }.WeprovideanexampleinFig.2toillustrate thenotationsintroducedin this section.

As introducedin [23,Section5.2], UΩ0 istheset ofgeneralizedcriticalpointsof f of index 0,associated witheigenformsoftheWittenLaplacianΔD,(0)_f,h andUΩ₁ isthesetofgeneralizedcriticalpointsof f ofindex 1,associated witheigenformsof theWitten LaplacianΔD,(1)_f,h , seeSection3.1.2formoredetails.

1.3.3. Mainresultson theexit pointdistribution

Themain resultofthisworkisthefollowing.

Theorem1.Letusassumethat(A0), (A1), (A2),and (A3) aresatisfied.Recallthatνhisthequasi-stationary distributionoftheprocess (1) inΩ (seeDefinition2).LetF ∈ L∞(∂Ω,R) and(Σi)i∈{1,...,k∂Ω

1 } beafamilyof disjointopensubsetsof ∂Ω suchthatfor all i∈1,. . . ,k∂Ω

1 

, zi∈ Σi,wherewerecallthat  z1,. . . ,zk∂Ω 1  = U∂Ω

1. There existsc> 0 suchthat inthelimit h→ 0: Eνh[F (XτΩ)] = k∂Ω1  i=1 Eνh[1ΣiF (XτΩ)] + O  e−ch (24) and k∂Ω 1  i=k1∂Cmax+1 Eνh[1ΣiF (XτΩ)] = O  h14 , (25)

wherewerecall that z1,. . . ,z_k∂Cmax

1



= ∂Cmax∩ ∂Ω (see (23)). 2. When forsomei∈1,. . . ,k∂Cmax

1 

thefunctionF isC∞ inaneighborhood ofzi,onehas whenh→ 0: Eνh[1ΣiF (XτΩ)] = F (zi) ai+ O(h 1 4), (26) where ai= ∂nf (zi)  det Hessf|∂Ω(zi) ⎛ ⎝k ∂Cmax 1  j=1 ∂nf (zj)  det Hessf|∂Ω(zj) ⎞ ⎠ −1 . (27)

3. When (A4) is satisfied, theremainder term O(h14) in (25) isof theorder Oe−

c

h forsome c> 0 and

theremainderterm Oh14 in(26) isoftheorderO(h) andadmitsafullasymptoticexpansionin h (as definedin Remark7below).

Remark7.Letusrecall thatforα > 0, (r(h))h>0 admits afull asymptoticexpansionin hαif thereexistsa sequence(ak)k≥0∈ RN suchthat foranyN ∈ N,itholds inthelimit h→ 0:

r(h) = N  k=0 akhαk+ O  hα(N +1) .

Theorem1impliesthatinthelimith→ 0,whenX0∼ νh,thelawofXτΩ admitsalimitindistribution

andconcentratesontheset{z1,. . . ,z_kCmax

1 }= ∂Ω∩ ∂Cmax withexplicitformulasfortheprobabilitiestoexit

througheachofthezi’s.

Asasimplecorollary,noticethatwhenthefunctionF belongstoC∞(∂Ω,R),onehasinthelimith→ 0:

Eνh[F (XτΩ)] = k∂Cmax1 i=1 aiF (zi) + O(h 1 4) = k∂Cmax1 i=1  Σi F ∂nf e− 2 hf k∂Cmax1 i=1  Σi ∂nf e− 2 hf + oh(1),

where theorderinh oftheremaindertermoh(1) dependsonthesupportofF and onwhetheror notthe assumption (A4) issatisfied.

AnotherconsequenceofTheorem1isthefollowing.Theprobabilitytoexitthroughaglobalminimumz of f|∂Ωwhichsatisfies∂nf (z)< 0 isexponentiallysmallinthelimith→ 0 (see (24))andwhenassuming (A4), theprobabilitytoexitthroughthepointsz_kCmax

1 +1,. . . ,zk

∂Ω

1 isalsoexponentiallysmalleventhoughallthese

Remark8. In [18],weshow thattheresultsofTheorem1stillholdwhenX0= x∈ Cmax.Moreover,wealso

prove that when X0 = x∈ C, forC∈ C such that ∂C∩ ∂Ω = ∅, thelaw of XτΩ concentrates on ∂C∩ ∂Ω whenh→ 0,withexplicit exitprobabilities.Wealso refertothepreprint [24] whichconcatenatestheresults of this manuscriptandof [18], andto [25] whichpresentsasimplified versionoftheresultsof theseworks.

TheproofofTheorem1reliesonacrucialresultontheconcentrationofthequasi-stationarydistribution onneighborhoods oftheglobal minimaoff inCmax.

Proposition 9.Assumethat (A0) and(A1) are satisfied. Furthermore,letus assumethat

min Cmax

f = min

Ω

f,

wherewerecallthatCmax isintroducedin (A1).LetO beanopensubsetofΩ.Then,ifO∩arg minCmaxf = ∅, one has inthelimit h→ 0:

νh 

O = 

x∈O∩arg minCmaxf

det Hessf (x) −

1 2



x∈arg minCmaxf

det Hessf (x) −12

1 + O(h) .

When O∩ arg min_Cmaxf =∅, thereexistsc> 0 such thatwhen h→ 0: νh

O = Oe−ch .

Proposition9isadirectconsequenceof (11) andProposition58below.Noticethatmin_C

maxf = minΩf

is satisfiedwhen(A1),(A2),and(A3) hold,seeLemma22.

1.3.4. Intermediateresultsonthespectrumof −LD,(0)_f,h

Letus recallthatfrom (12),onehas:

Eνh[F (XτΩ)] =− h 2λh  ∂Ω F ∂nuhe− 2 hf  Ω uhe− 2 hf .

Therefore, to obtain the asymptotic estimates on Eνh[F (XτΩ)] stated in Theorem 1 when h → 0, it is

sufficient tostudy theasymptotic behaviorof thequantitiesλh, 

Ωuhe−

2

hf and∂_nu_h.Letus pointto the resultswhichwillbe provenbelowonthesequantities, andwhichmayhavetheirowninterest:

1. In Theorem 4, one gives for h → 0 small enough, alower and an upper bound for all the mΩ0 small eigenvaluesof−LD,(0)_f,h when (A0) issatisfied.

2. In Theorems 2 and 3, one gives a sharp asymptotic equivalent in the limit h → 0 of the smallest eigenvalueλh of−LD,(0)f,h when (A0) and (A1) aresatisfied.

3. InProposition58,when (A0), (A1) andmin_Cmaxf = min_Ωf hold,oneshowsthatuhe− 2

hf concentrates intheL1_{(Ω)-normontheglobalminimaof f in C}

max inthelimith→ 0.

1.4. Discussion ofthehypotheses

In this section, we discuss the necessity of the assumptions (A1), (A2), (A3) and (A4) to obtain the resultsstatedinTheorem 1.

On the assumption (A1). To study the concentration of the law of XτΩ when h → 0 when X0 ∼ νh,

oneneeds inparticular to have access to the repartitionof νh inneighborhoods of the local minima of f inΩ when h→ 0. When (A1) is notsatisfied,theanalysis oftherepartition ofνh istricky.When (A1) is not satisfied,one hasfrom Theorem 4below (see Section4.2.2), lim_h→0hln λh = limh→0hln λ2,h, where

λ2,histhesecondsmallesteigenvalueof−Lf,hD,(0).Thefirsttwoeigenvaluesthushavethesameexponential scalinginh.Asaconsequence,itisdifficulttomeasurethequalityoftheapproximationofuhbyanansatz projectedonSpan(uh),sincetheerrorisrelatedto theratioofλhoverλ2,h(seeLemma25).Forexample, when (A1) isnotsatisfied,itisdifficultto predictinwhichwellνh concentrates,as explainedin [26].

Ontheassumptions (A2) and (A3). When (A1) issatisfiedbutnot (A2),orwhen(A1),(A2) aresatisfied butnot (A3),itispossibletoexhibitfunctionsf : [z1,z2]→ R (withf (z1)< f (z2))suchthatPνh[Xτ(z1,z2) = z2]= 1+ O(e−

c

h) forsome positiveconstantc (see [24, Section 1.4.3,Section 1.4.4] fordetails). Inthose cases,theprocess (1) thusleavesΩ= (z1,z2) throughthepoint z2 whichisnotaglobalminimumoff|∂Ω. Ontheassumption (A4). In[24,Section1.4.5],wegiveaone-dimensionalexampletoshowthatwhen (A4) isnotsatisfied,theremaindertermO(h14) in(25) isnotoftheorderO(e−

c

h) forsomec> 0,butactuallyof theorder O(√h).This canbe generalizedto higher-dimensionalsettings, see forexample [27, Proposition C.40, item3]. Wethus expectthattheremainder termsOh14 in (25) and (26) are actuallyof theorder O(√h),butproving thisfactwouldrequiresomesubstantiallyfineranalysis.

1.5. Organization ofthepaperand outlineof theproof

Theaimofthis sectionisto giveanoverviewofthestrategyoftheproofof Theorem1.From (12) and inorder toobtainanasymptotic estimateofEνh[F (XτΩ)],we studytheasymptoticbehaviorwhenh→ 0

ofthequantitiesλh, 

Ωuhe−

2

hf and ∂_nu_h,whereλ_h isdefinedby (8) andu_h by (10).

Tostudy λh and∂nuh,thefirst keypointis tonotice that∇uh isasolutionto aneigenvalueproblem. Indeed,bydifferentiatingtheeigenvalueproblem (10) satisfied byuh,onegets:

⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ −L(1) f,h∇uh= λh∇uh on Ω, ∇Tuh= 0 on ∂Ω,  h 2div− ∇f·  ∇uh= 0 on ∂Ω, (28) where L(1)_f,h= h 2Δ− ∇f · ∇ − Hess f (29)

isanoperatoractingon1-forms(namelyonvectorfields).Therefore,thevectorfield∇uhisaneigen-1-form oftheoperator−LD,(1)_f,h whichistheoperator−L(1)_f,hwithtangentialDirichletboundaryconditions(see (28)), associatedwith theeigenvalue λh.

Thesecondkeypoint(seeforexample [23])isthat,when (A0) holds,−LD,(0)_f,h admitsexactlymΩ

0 eigenval-uessmallerthan √h

2 (where m Ω

0 isthenumberoflocalminimaof f in Ω,seeSection1.3.2)andthat−L

D,(1) f,h admitsexactlymΩ

1 eigenvaluessmallerthan

√ h

belowbyaconstantinthis regime.Thisimpliesinparticularthatλh isanexponentiallysmalleigenvalue of −LD,(1)_f,h . Let us denote by π_h(0) (resp.π(1)_h ) the orthogonal projector onto thevector space spanned by the eigenfunctions(resp. eigen-1-forms)associated with themΩ

0 (resp.mΩ1)smallest eigenvalueof−L

D,(0) f,h (resp.of−LD,(1)_f,h ).

To obtain an asymptotic estimate on λh when h → 0, the strategy consists in studying the singular valuesofthe(finite-dimensional)operator∇ acting from Ran π_h(0) to Ran π(1)_h ,both spacesbeing equipped with the scalar product of L2w(Ω). Indeed, from Proposition4, the squares of the singular values of this matrixarethesmallesteigenvaluesof−_h2L_f,hD,(0).Tothis end,oneconstructsanappropriatebasis(withso called quasi-modes) of Ran π_h(0) and Ran π(1)_h .Besides, from (28), ∇uh ∈ Ran πh(1) and thus, to study the asymptotic behaviorof∂nuh on∂Ω whenh→ 0,onedecomposes ∇uh alongabasisof Ran πh(1).

Intermsofmathematicaltools,theproofsheavilyrelyonconstructionsmadein [28,23].Themainnovelty is to combine those techniques to get precise estimates of ∂nuh: this requires to go beyond the standard semiclassicalestimateswhichfocusontheeigenvalues.Thecrucialstepsintheproofsare:(i)Proposition47 which givestheinteraction termsbetweenthequasi-modesforL_f,hD,(0)and forLD,(1)_f,h ,(ii)Lemma50which usesamatrixrepresentationoftheoperator∇: Ran π_h(0)→ Ran π(1)_h ,thanksto anappropriatelabeling of theconnectedcomponentsattachedtothelocalminimaoff and(iii)Lemma63whichexplainshowtoget

H1

w-estimatesoftheerrorbetweenuh anditsapproximationbyquasi-modes.

Thepaperisorganizedasfollows.InSection2,oneconstructstwomapsj and j whichwillbeextensively used inSection3.These mapsare usefulinorderto understandthedifferent timescalesoftheprocess (1) in Ω.Section3isdedicatedto theconstructionof quasi-modesfor −LD,(0)_f,h and−LD,(1)_f,h . InSection4, we study the asymptotic behaviors of the smallest eigenvalues of −LD,(0)_f,h (see Theorem 4) and we give an asymptotic estimate of λh when h → 0, see Theorem 2. In Section 5, we give asymptotic estimates for 

Ωuhe−

2

hf and for∂_nu_h on∂Ω whenh→ 0 (seeProposition58and Theorem 5),and wefinally conclude theproofofTheorem1.Fortheeaseofthereader,alistofthemain notationusedinthisworkisprovided at theend.

2. Couplinglocalminimaof f withsaddlepoints of f

Thissectionisdedicatedtotheconstructionoftwomaps:themapj whichassociateseachlocalminimum of f withanensembleofsaddlepointsof f andthemapj whichassociateseachlocalminimumof f witha connected componentofasublevelsetof f .These mapsareusefultodefine thequasi-modes inSection3.

This sectionisorganizedas follows.InSection2.1,oneintroducesasetof connectedcomponentswhich play acrucial role inour analysis.The constructionsof the maps j and j require two preliminaryresults (Propositions 15 and 18) which are introduced in Section 2.2. Then, the maps j and j are defined in Section2.3.Finallytheassumptions(A1)-(A4) arereformulatedintermsofthemap j inSection2.4.

2.1. Connectedcomponentsassociatedwith theelements ofUΩ 0

TheaimofthissectionistogiveanequivalentdefinitionoftheelementsinC = {C(x),x∈ UΩ

0} (see (14) and (16))which will be easierto handle inthefollowing.For thatpurpose,letus introducethe following definitions.

Definition 10.Letus assumethat (A0) holds.Forallx∈ UΩ

0 andλ> f (x),one defines

C(λ, x) as the connected component of{f < λ} in Ω containing x (30)

C+(λ, x) as the connected component of {f ≤ λ} in Ω containing x. (31)

Moreover, forallx∈ UΩ

0,onedefines

λ(x) := sup{λ > f(x) s.t. C(λ, x) ∩ ∂Ω = ∅} and C(x) := C(λ(x), x). (32) AdirectconsequenceofLemma11belowisthatforallx∈ UΩ

0,C(x) definedin (32) coincideswithC(x) introducedin (15).Noticethatunder(A0),forallx∈ UΩ0 ⊂ Ω,λ(x) iswelldefined.Indeed,forallx∈ UΩ0,

{λ> f (x)s.t. C(λ,x)∩ ∂Ω=∅} is boundedby sup_Ωf + 1 andnonempty becausefor β > 0 smallenough C(f (x)+ β,x) isincludedin Ω (sincex∈ Ω andf isMorse).Onehasthefollowingresultwhichpermitsto giveanotherdefinitionofHf (comparewith (13))whichwillbeeasiertohandle inthesequel.

Lemma11. Letusassume that (A0) holds.Then, forallx∈ UΩ 0

Hf(x) = λ(x), (33)

whereHf(x) isdefined by (13) and λ(x) isdefined by (32). TheproofofLemma11ismadein [24, Lemma15].

Definition12. Letusassume that(A0) holds.The integer N1 isdefined by: N1:= Card(C) = Card  {C(x), x ∈ UΩ 0} ∈ {1, . . . , mΩ 0}, (34)

wherewerecallthat mΩ

0 = Card (UΩ0) (see (16)),C(x) isdefinedby (32) andC = 

C(x),x∈ UΩ 0



(see (14)).

The elementsof C are denoted by C1,. . . ,CN1.Finally,forall∈ {1,. . . ,N1}, Ck isalso denoted by

E1,:= C. (35)

Forexample,onFig.1,onehasmΩ

0 = 4 andN1= 3.Thenotation (35) willbe usefulwhenconstructing themapsj andj inSection2.3below.

2.2. Topologicalresultsandseparating saddlepoints

Theconstructionsofthemaps j andj madeinSection2.3arebasedonthenotionsofseparatingsaddle points and of critical components as introduced in [28, Section 4.1] for acase without boundary. Let us define and slightly adapt these two notions to our setting. To this end, let us first recall that according to [23,Section5.2],foranynoncriticalpointz∈ Ω,forr > 0 smallenough

{f < f(z)} ∩ B(z, r) is connected, (36) and for any critical point z ∈ Ω of index p of the Morsefunction f , forr > 0 small enough, onehas the threepossible cases:

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

either p = 0 (z is a local minimum of f ) and {f < f(z)} ∩ B(z, r) = ∅, or p = 1 and{f < f(z)} ∩ B(z, r) has exactly two connected components, or p≥ 2 and {f < f(z)} ∩ B(z, r) is connected,

(37)

Fig. 3. Anexampleindimension2 ofasaddlepointwhichisnot separating.Thepointsx1and x2aretwolocalminimaof f ,and

thepointsy1andy2aretwolocalmaximaof f .Thetwoconnectedcomponentsof{f < f(z)}∩ B(z,r) arecontainedinthesame

connectedcomponentsof{f < f(z)} (seethearrowedpathonthefigure).

Definition13. Assume(A0).LetC = {C1,. . . ,CN1} bethesetofconnectedsetsasintroducedinDefinition12.

1. A point z∈ UΩ

1 is a separatingsaddle point if either z ∈ U∂Ω1 ∩ ∪ N1

i=1∂Ci or z ∈ UΩ1 ∩ ∪ N1

i=1Ci andfor r > 0 small enough, the two connectedcomponents of {f < f(z)}∩ B(z,r) are contained in different connected components of {f < f(z)}. Notice that in the former case z ∈ ∂Ω while in the latter case z∈ Ω.The setof separatingsaddlepointsisdenoted by Ussp₁ .

2. Forany σ∈ R,aconnectedcomponent E ofthesublevelset{f < σ} in Ω is called acriticalconnected componentif ∂E∩ Ussp₁ = ∅.The familyof criticalconnected componentsisdenotedby Ccrit.

Remark14.Itisnaturaltodefine generallyaseparatingsaddlepointofaMorsefunctionf asfollows: z is a separatingsaddlepointif forany sufficiently smallconnectedneighborhood Vz of z,Vz∩ {f < f(z)} has two connected components included in two different connected components of {f < f(z)}. Our definition of separating saddle point is equivalent to this general definition when the function f is extended by −∞ outsideΩ.Inparticular,withthisextendeddefinitionoff ,therecannotbeaseparatingsaddlepointoutside ∪N1

i=1Ci.Werefer to [24,Remark19] for moredetails.

Intherestofthissection,wegiveaseriesofresultson{C1,. . . ,CN1} whichwillbeusedthroughoutthis

paper.Theseresultsareratherintuitiveand,forthesakeofconciseness,wereferto [24] fordetailedproofs. Proposition15. Letusassumethat(A0) holds.LetC = {C1,. . . ,CN1} bethesetofconnectedsetsintroduced in Definition 12andlet (k,)∈ {1,. . . ,N1}2 with k= .Then,

Ck is an open subset of Ω and Ck∩ C=∅. (38) In addition,one has

∂Ck∩ ∂Ω ⊂ Ussp1 ∩ ∂Ω and ∂Ck∩ ∂C⊂ Ussp1 ∩ Ω, (39)

where thesetUssp₁ isintroduced initem1inDefinition 13.Finally, ∂Ck∩ Ussp1 = ∅.

Lemma 16. Letus assume that thefunctionf : Ω→ R is aC∞ function. Letx∈ UΩ

0.For allμ> f (x),

itholds C(μ,x)=_λ<μC(λ,x) and C+_(μ,x)=

λ>μC+(λ,x) whereC(μ,x) and C+(μ,x) arerespectively defined in (30) and (31).

Thefollowingtechnicalresultwillbe neededinthesequel.

Lemma17. Letusassumethat (A0) issatisfied.LetC = {C1,. . . ,CN1} bethesetofconnectedsetsintroduced

in Definition 12.Let us consider {j1,. . . ,jk}⊂ {1,. . . ,N1} withk ∈ {1,. . . ,N1} and j1 < . . . < jk such that∪k

=1Cj isconnectedandsuchthatforallq∈ {1,. . . ,N1}\ {j1,. . . ,jk},Cq∩ ∪

k

=1Cj =∅.Then,there

existz∈ Ussp₁ and0∈ {1,. . . ,k} suchthat

z∈ ∂Cj_0\ ∪k =1, =0∂Cj  .

TheproofofLemma17ismadein [24,Lemma21].Letusendthissectionwiththefollowingproposition. Proposition 18.Let us assume that (A0) is satisfied. Let us consider Cq for q ∈ {1,. . . ,N1} (see

Defini-tion12).From (30) and (32),thereexistsxq∈ UΩ0∩CqsuchthatCq= C(xq,λ(xq)).Letλ∈ (minCqf,λ(xq)] andC be aconnectedcomponentof Cq∩ {f < λ}.Then,

C∩ Ussp₁ = ∅ iff C∩ UΩ₀ contains more than one point. (40)

Moreover, letusdefine

σ := max y∈C∩Ussp

1 f (y)

with theconventionσ = min_Cf whenC∩ Ussp₁ =∅.Then, thefollowingassertionshold.

1. For allμ∈ (σ,λ],theset C∩ {f < μ} is aconnectedcomponentof {f < μ}.

2. IfC∩ Ussp₁ = ∅,onehas C∩ UΩ

0 ⊂ {f < σ} andtheconnectedcomponentsofC∩ {f < σ} belongtoCcrit. TheproofofProposition18ismadein [24,Proposition22].Itreliesonpropertiesofthesublevelsetsof aMorsefunctioninRd.

2.3. Constructions ofthemapsj andj

In this section we construct, under (A0), two maps j andj, using an association between the local minima UΩ

0 and the(generalized) saddle points UΩ1. Such maps havebeen introduced in[29–31,28] in the boundaryless case. This hasbeen generalized in[23] to theboundary case(where the authors introduced thenotionofgeneralizedsaddlepointsforΔD,(0)_f,h ).

Let us recall (see Lemma 23 below), that LD,(0)_f,h has exactly mΩ

0 eigenvalues smaller than

√ h/2 for

sufficiently small h. Actually, from [32,23], it can be shown that these mΩ

0 eigenvalues are exponentially small. The goalof the map j is to associate each local minimum x of f with a set of generalized saddle pointsj(x)⊂ UΩ₁ suchthatf isconstantoverj(x) andforsufficientlysmallh,

∃λ ∈ σ− LD,(0) f,h , lim h→0h ln λ =−2  f (j(x))− f(x) .

general analysis of the sublevelsets of aMorse function on amanifold withoutboundary of [28, Section 4.1] whichgeneralizestheproceduredescribed in [31].Theideaistoconsidertheconnectedcomponentsof

{f < λ}∩ Ussp

1 appearingasλ decreasesfrom max_∪N1

k=1Ckf to −∞.Each timeanewconnectedcomponent appears in ∪N1

k=1Ck, one picks arbitrarily a global minimum of f in it and then, one associates this local minimum withtheseparatingsaddlepointsontheboundaryofthisnewconnectedcomponent.

Letusassumethattheassumption(A0) holds.Theconstructionsofthemaps j and j aremaderecursively as follows:

1. Initialization (q = 1). We consider E1, = C for  ∈ {1,. . . ,N1} (see (35)). For each ∈ {1,. . . ,N1},

x1,denotesonepointin arg minE1,f = arg minE1,f .Thenwedefine,forall k∈ {1,. . . ,N1}, σ1,:= max

E1,

f, j(x1,) := E1,and j(x1,) := ∂E1,∩ Ussp1 . (41) NoticethataccordingtoProposition15anditem2inDefinition13,itholds

j(x1,)= ∅, ∂E1,⊂ {f = σ1,}, j(x1,)∈ Ccritand

N1

=1

j(x1,)∩ ∂Ω ⊂ U∂Ω1 .

Moreover,onehasfrom Proposition15(andmorepreciselythesecond inclusionin (39)),

∀ = q ∈ {1, . . . , N1}, ∂E1,∩ ∂E1,q⊂ Ussp1 ∩ Ω. (42) 2. First step (q = 2). From Proposition 18, for each  ∈ {1,. . . ,N1}, E1,∩ UΩ0 = {x1,} if and only if

Ussp₁ ∩ E1,= ∅. Asaconsequence,onehas: Ussp₁ ! ∪N1 =1E1,  = ∅ iff {x1,1, . . . , x1,N1} = U Ω 0. IfUssp₁  ∪N1 =1E1, 

=∅ (orequivalentlyN1= mΩ0),theconstructionsofthemapsj andj arefinished andonegoestoitem4below.IfUssp₁  ∪N1

=1E1, 

= ∅ (orequivalentlyN1< mΩ0),onedefines

σ2:= max x∈Ussp 1   ∪N1 =1E1, f (x) ∈ min ∪N1 =1E1, f, max ∈{1,...,N1} σ1,  . Theset N1 =1 E1,∩ {f < σ2} 

is then the union of finitely many connected components. We denote by E2,1,. . . ,E2,N2 (with N2 ≥

1) the connected components of N1 =1

E1, ∩ {f < σ2}

which do not contain any of the min-ima {x1,1,. . . ,x1,N1}. From items 1 and 2 in Proposition 18 (applied to C = E1,∩ {f < σ2} for

each∈ {1,. . . ,N1}),

∀ ∈ {1, . . . , N2}, E2,∈ Ccrit.

Notice thattheother connectedcomponents (i.e.those containing the x1,’s)maybe notcritical. Let usassociatewitheach E2,, 1≤ ≤ N2,onepoint x2,arbitrarilychosenin arg minE2,f = arg minE2,f

j(x2,) := E2, and j(x2,) := ∂E2,∩ Ussp1 ⊂ {f = σ2}.

3. Recurrence (q≥ 3).Ifallthelocalminimaoff in Ω havebeen labeledattheendofthepreviousstep above(q = 2),i.e.if∪2_j=1{xj,1,. . . ,xj,Nj}= U

Ω

0 (orequivalently ifN1+ N2= mΩ0),theconstructionsof the mapsj and j arefinishedand onegoesto item4below.If itis notthecase,from Proposition18, there existsm∈ N∗ suchthat

∀q ∈ {2, . . . , m + 1}, Ussp 1 ! N1 =1 E1,∩ {f < σq}  = ∅ (43)

where onedefines recursivelythedecreasingsequence(σq)q=3,...,m+2by

σq:= max  f (x), x∈ Ussp₁ ! N1 =1  E1,∩ {f < σq−1} " . Noticethatσq∈ min_∪N1 =1E1,f, σq−1 

.Letusnowconsiderthelargestintegerm∗∈ N∗suchthat (43) holds. Noticethatm∗ iswelldefinedsincethecardinalofUΩ

0 isfinite.Bydefinitionofm∗, onehas:

Ussp₁ ! N1 =1 E1,∩ {f < σm∗+2}  = ∅. (44)

Then, one repeats recursively m∗ times the procedure described in the first step above. For q ∈ {2,. . . ,m∗+1},onedefines(Eq+1,)∈{1,...,Nq+1}asthesetofconnectedcomponentsof

N1 =1 E1,∩{f < σq+1} 

whichdoesnotcontainanyofthelocalminima∪_j=1q {xj,1,. . . ,xj,Nj} of f in Ω whichhavebeen previously chosen. From items1 and 2in Proposition18, ∀∈ {1,. . . ,Nq+1}, Eq+1, ∈ Ccrit. For ∈ {1,. . . ,Nq+1},weassociatewith each Eq+1,,onepoint xq+1,arbitrarilychosenin arg minEq+1,f .For

∈ {1,. . . ,Nq+1},letusdefine:

j(xq+1,) := Eq+1, and j(xq+1,) := ∂Eq+1,∩ Ussp1 ⊂ {f = σq+1}. From (44) andProposition18,UΩ

0 =∪m ∗₊₂

j=1 {xj,1,. . . ,xj,Nj} andthus,allthelocalminimaoff inΩ are labeled. Thisconcludestheconstructionsof themapsj andj.

4. Properties ofthemapsj andj. The twomaps j : UΩ

0 −→ Ccrit and j : UΩ0 −→ P(UΩ1) (45)

are clearly injective.Notice thatthej(x), x∈ UΩ0, arenot disjoint ingeneral.For allx∈ UΩ0, f (j(x)) containsexactlyonevalue,whichwillbedenotedby f (j(x)).Moreover,since∪N1

=1E1,⊂ Ω (seethefirst statementin (38)),onehasforallx∈ UΩ

0,j(x)⊂ Ω.Moreover,itholds

∀x ∈ UΩ

0 \ {x1,1, . . . , x1,N1}, j(x) ⊂ Ω ∩ U

ssp

1 . (46)

Finally,forallx∈ UΩ

0,f (j(x))− f(x)> 0 andforallx∈ UΩ0 ∩j(x1,)\ {x1,}

f (j(x))− f(x) < min =1,...,N1

Fig. 4. Themaps j andj onaone-dimensionalexampleforwhichthemapsareuniquelydefinedandtheconstructionrequiresthree steps.

Theconstructionsofthemapsj andj areillustratedinFig.4onaone-dimensionalexample.Sinceonecan pick aminimumor another inacriticalconnected componentat eachstep oftheconstruction ofj and j, the maps arenot uniquelydefinedif over oneofthe connectedcomponents Ek, (k≥ 1, ∈ {1,. . . ,Nk}), arg min f contains morethan one point (see [24, Fig. 9] for a one-dimensional example). As will become clearbelow,thisnon-uniquenesshasnoinfluenceontheresultsprovenhereafter.

Remark19.Inthecasewhenforalllocalminimax of f , j(x) isasinglepoint, j(x)∩ j(y)=∅ forallx= y andwhenalltheheights(f (j(x))− f(x))_x∈UΩ

1 aredistinct,themapj isexactlytheoneconstructed in [23].

Thenextdefinitionwillbeused inSection3.2toconstructthequasi-modes. Definition 20.Letus assumethat (A0) issatisfied. Letε be suchthat

0≤ ε < min k≥1, ∈{1,...,Nk} max Ek, f− max Ussp1 ∩Ek, f, (48)

where the family (Ek,)k≥1, ∈{1,...,Nk} is defined in the construction of the map j above. For k ≥ 1 and

∈ {1,. . . ,Nk}, onedefines Ek,(ε) = Ek,∩  f < max Ek, f − ε, (49)

which isaconnectedcomponent off < max_Ek,f − εaccordingtoitem1inProposition 18. 2.4. Rewritingtheassumptions(A1)-(A4) in termsof themap j

Lemma 21.Let us assume that (A0) is satisfied. Then, theassumption (A1) isequivalent to thefact that there exists∈ {1,. . . ,N1} suchthat forallk∈ {1,. . . ,N1}\ {},

f (j(x1,k))− f(x1,k) < f (j(x1,))− f(x1,).

Thus, when (A1) holds, theelements ofC = {C1,. . . ,CN1} (see Definition 12) areordered suchthat = 1, i.e. for allk∈ {2,. . . ,N1}:

f (j(x1,k))− f(x1,k) < f (j(x1,1))− f(x1,1). (50)

Proof. Assumethatthehypothesis(A0) issatisfied.LetusrecallthatC = {C(x), x∈ UΩ

0}={C1,. . . ,CN1}.

Let C∈ C and letk ∈ {1,. . . ,N1},such thatC= Ck. Then, from (33) and thefirst stepof the construc-tion of j in Section 2.3, one has for all q ∈ {1,. . . ,N1}: Hf(x1,q) = λ(x1,q) = f (j(x1,q)) = supCqf and

f (x1,q)= minCqf .Thus,itholds supCf − minCf = f (j(x1,k))− f(x1,k).Thisimpliestheresultsstatedin Lemma21. 2

InviewofLemma21andbyconstructionofthemapj (seethefirststepinSection2.3),onecanrewrite theassumptions(A1),(A2),(A3),and (A4) withthemapj asfollows:

• Theassumption (A1) isequivalenttothefactthat,uptoreorderingtheelementsofC = {C1,. . . ,CN1}

suchthat (50) issatisfied,itholds:

∀x ∈ {x1,2, . . . , x1,N1}, f(j(x)) − f(x) < f(j(x1,1))− f(x1,1). (A1j) Furthermore, inthiscase, C1= Cmax,whereCmax isdefinedby (A1).

• Theassumption (A2) rewriteswhen(A1j) holds,

∂C1∩ ∂Ω = ∅. (A2j)

• Theassumption (A3) rewriteswhen(A1j) holds,

∂C1∩ ∂Ω ⊂ arg min

∂Ω

f. (A3j)

• When (A1j) holds, theassumption (A4) isequivalent to

j(x1,1)⊂ ∂Ω. (A4j)

Thisequivalencebetween(A4) and (A4j) followsfrom (A1j) togetherwiththefactthatj(x1,1)= ∂C1∩ Ussp1 (see (41)) and by definition of a separating saddle point. From now on, we work with the formula-tions (A1j), (A2j), (A3j),and (A4j) oftheassumptions(A1),(A2), (A3),and (A4).

For one-dimensional illustrationsof the assumptions, we refer to [24, Figures 6 to 9]. Notice that un-der (A1j),itholdsfrom (47):

∀x ∈ UΩ

0 \ {x1,1}, f(j(x)) − f(x) < f(j(x1,1))− f(x1,1). (51) When(A1j) and(A2j) are satisfied, from Definition13 and Proposition15 (seethe firstinclusion in (39) and (41)),onehas

∂Ω∩ j(x1,1) = ∂Ω∩ ∂C1= U∂Ω1 ∩ ∂C1. (52) Inthatcase,weassumethattheelements{z1,. . . ,zm∂Ω

1 } ofU ∂Ω

1 (see (20))areorderedsuchthat

∂Ω∩ ∂C1={z1, . . . , z_k∂C1

1 } (53)

where k∂C1

1 ∈ N∗ satisfies k1∂C1 ≤ m∂Ω1 (see (20)).Notice thatfrom Lemma21, this labeling implieswhen (A3j) issatisfied: k∂C1 1 = k ∂Cmax 1 , (54) wherek∂Cmax

Lemma 22. Let us assume that (A0), (A1j), (A2j), and (A3j) hold. Then, one has minΩf = minΩf < min∂Ωf and arg min C1 f = arg min Ω f. (55)

Proof. The fact that min_Ωf < min∂Ωf is obvious. Let us prove (55). Let k ∈ {1,. . . ,N1} and let us recall that from Definition 12, there exists x ∈ UΩ₀ ∩ Ck such that Ck = C(λ(x),x).Let us assume that x∈ arg min_Ωf .Then,bydefinitionofthemapj andbydefinitionofλ(x) (see (32))togetherwiththefact that (A1j), (A2j), and (A3j) hold, one has f (j(x1,k))= λ(x) ≥ min∂Ωf = f (j(x1,1)). Thus, if f (x1,k) =

f (x)≤ f(x1,1),itholdsλ(x)− f(x)= f (j(x1,k))− f(x1,k)≥ f(j(x1,1))− f(x1,1).ThisimpliesCk = C1from theassumption (A1j).Thisconcludestheproofof (55). 2

3. Constructionsofthe quasi-modes

Thissectionisdedicatedtotheconstructionsoftwofamiliesofquasi-modesapproximatingtheeigenforms of −LD,(0)_f,h and−LD,(1)_f,h associatedwithexponentiallysmalleigenvalues.Theseconstructionsusethemaps j and j introducedintheprevioussection.

Thissectionisorganizedasfollows.InSection3.1,weintroducethenotationsusedthroughoutthispaper for operators,andthepropertiesofWitten Laplaciansand oftheoperators LD,(p)_f,h needed inouranalysis. Thequasi-modesare thenbuiltinSection3.2.

3.1. NotationsandWitten Laplacian 3.1.1. NotationforSobolevspaces

Forp∈ {0,. . . ,d},onedenotesbyΛp_C∞_{(Ω) the spaceofC}∞_{p-formsonΩ.Moreover, Λ}p_C∞

T (Ω) isthe set ofC∞ p-formsv suchthattv = 0 on∂Ω,where t denotesthetangentialtraceonforms.Theweighted space ΛpL2_w(Ω) isthecompletionofΛpC∞(Ω) forthenorm

w∈ ΛpC∞(Ω)→ # $ $ %  Ω |w|2_e−2hf. Likewise, for p ∈ {0,. . . ,d} and q ∈ N, Λp_Hq

w(Ω) is the weighted Sobolev spaces of p-forms on Ω with regularity index q: v ∈ Λp_Hq

w(Ω) if and only if forevery multi-index α with |α|≤ q, ∂αv is in ΛpL2w(Ω). Seeforexample [33] foranintroductiontoSobolevspacesonmanifoldswithboundaries.Forp∈ {0,. . . ,d}

and q > 1₂, theset ΛpH_w,Tq (Ω) isdefinedby

ΛpH_w,Tq (Ω) :={v ∈ ΛpH_wq(Ω)| tv = 0 on ∂Ω} . Notice thatΛp_L2

w(Ω) is thespaceΛpHw0(Ω),and thatΛ0Hw,T1 (Ω) isthespaceHw,01 (Ω) alreadyintroduced in Proposition 4. We will denote by .Hwq the norm on the weightedspace Λ

p_Hq

w(Ω). Moreover ·,·L2

w denotes thescalarproductin Λp_L2

w(Ω).Finally,wewill alsousethesamenotationwithouttheindexw to denote thestandardSobolevspacesdefinedwithrespectto theLebesguemeasureonΩ.

3.1.2. The WittenLaplacianandtheinfinitesimal generatorof thediffusion (1)

For p ∈ {0,. . . ,n}, one defines the distorted exterior derivative à la Witten d(p)_f,h : Λp_C∞_{(Ω) →} Λp+1_C∞_{(Ω) anditsformaladjoint:d}(p)∗

f,h : Λp+1C∞(Ω)→ ΛpC∞(Ω) by

d(p)_f,h:= e−h1f_{h d}(p)_eh1f _{and d}(p)∗

f,h := e

1

hf_{h d}(p)∗_e−h1f_.

TheWitten Laplacian,firstlyintroduced in[34], isthen definedsimilarlyas theHodgeLaplacianΔ(p)_H := (d+ d∗)2_by

Δ(p)_f,h:= (df,h+ d∗f,h)2= df,hd∗f,h+ d∗f,hdf,h : ΛpC∞(Ω)→ ΛpC∞(Ω). TheDirichletrealizationofΔ(p)_f,honΛp_L2_{(Ω) isdenotedby Δ}D,(p)

f,h anditsdomain is

DΔD,(p)_f,h =w∈ ΛpH2(Ω)| tw = 0, td∗_f,hw = 0.

TheoperatorΔD,(p)_f,h isself-adjoint,nonnegative,anditsassociatedquadraticform isgivenby

φ∈ ΛpH_T1(Ω) → d(p)_f,hφ2_L2+d (p)∗ f,h φ 2 L2, where ΛpH_T1(Ω) =w∈ ΛpH1(Ω)| tw = 0.

Wereferinparticularto[23,Section2.4] foracomprehensivedefinitionofWittenLaplacianswithDirichlet tangentialboundaryconditionsandstatementsontheirproperties.ThelinkbetweentheWittenLaplacian andtheinfinitesimalgeneratorL(0)_f,h ofthediffusion (1) isthefollowing:since

L(0)_f,h=−∇f · ∇ − h 2 Δ (0) H and Δ (0) f,h= h 2_Δ(0) H +|∇f| 2_{+ hΔ}(0) H f, (56) onehas: ΔD,(0)_f,h =−2 h U LD,(0)_f,h U−1

whereU istheunitaryoperator

U :



ΛpL2w(Ω) → ΛpL2(Ω)

φ→ e−h1fφ. (57)

Inparticular,theoperatorLD,(0)_f,h hasanaturalextension top-formsdefinedbytherelation

LD,(p)_f,h =−1 2hU

−1_ΔD,(p)

f,h U. (58)

Forp= 1,onerecoverstheoperatorL(1)_f,hwithtangentialDirichletboundaryconditionsdefinedby (28)–(29). TheoperatorLD,(p)_f,h withdomain

DLD,(p)_f,h = U−1DΔD,(p)_f,h =w∈ ΛpH_w2(Ω)| tw = 0, td∗2f

h,1

is thenself-adjointonΛp_L2

w(Ω),nonpositiveanditsassociatedquadraticform is

ΛpHT1(Ω)  φ → − h 2 &'_'d(p) φ''2_L2 w+ ''_d(p)∗ 2f h,1 φ''2_L2 w ( .

Letus alsorecallthat−LD,(p)_f,h (and equivalentlyΔD,(p)_f,h )hasacompactresolvent.From generalresultson elliptic operators when p= 0, −LD,(0)_f,h (and ΔD,(0)_f,h )admits anon degenerate smallesteigenvaluewith an associated eigenfunction which has asign on Ω. Denoting moreover by πE(LD,(p)f,h ) the spectral projector associated withLD,(p)_f,h andsomeBorelsetE⊂ R, thefollowingcommutationrelationshold onΛp_H1

T(Ω): d(p)πE(LD,(p)f,h ) = πE(LD,(p+1)f,h (Ω)) d(p)and d (p)∗ 2f h,1 πE(LD,(p)f,h ) = πE(LD,(p−1)f,h ) d (p)∗ 2f h,1 . (59)

LetusrecallthatfromtheellipticregularityofLD,(p)_f,h ,foranyboundedBorelsetE⊂ R, Ran πE(LD,(d)_f,h )⊂ Λp_C∞

T (Ω),therelation (59) thenleadstothefollowingcomplexstructure:

{0} −→ Ran πE(LD,(0)f,h ) df,h −−−−→ Ran πE(LD,(1)f,h ) df,h −−−−→ · · · df,h −−−−→ Ran πE(LD,(d)f,h ) df,h −−−−→ {0} and {0} d ∗ 2f h,1 ←−−−− Ran πE(LD,(0)_f,h ) d∗2f h,1 ←−−−− Ran πE(LD,(1)_f,h ) d∗2f h,1 ←−−−− · · · d ∗ 2f h,1 ←−−−− Ran πE(LD,(d)_f,h )←− {0}. Foreaseofnotation,onedefines:

∀p ∈ {0, . . . , d} , π(p)

h := π[0,√h

2 )(−L D,(p)

f,h ). (60)

The following result, instrumental in our investigation of the smallest eigenvalue λh of −LD,(0)_f,h , is an immediate consequenceof[23,Theorem3.2.3] together with (58).

Lemma 23.Under assumption(A0), thereexistsh0> 0 suchthat forallh∈ (0,h0), dim Ran π_h(0)= mΩ₀ and dim Ran π_h(1)= mΩ₁,

where mΩ0 = Card(UΩ0) andm1Ω= Card(UΩ1) aredefined inSection 1.3.2.

Remark 24.In [23, Theorem 3.2.3] it isassumed that f : ∂Ω→ R is aMorse functionwhile in (A0), we only assume that f : {x ∈ ∂Ω,∂nf (x) > 0}→ R is aMorse function. As mentioned in [35, Section 7.1], the statement of Lemma 23 still holds under this weaker assumption. This is explained in details in [24, Appendix A].

Inthesequel,withaslightabuseof notation,onedenotes theexteriordifferentiald actingonfunctions by∇.NotethatitfollowsfromtheaboveconsiderationsandLemma23thatunder (A0),itholds