Title Asymptotic behaviors on the small parameter exit problemsand the singularly perturbation problems

Author(s) Sugiura, Makoto

Citation Ryukyu mathematical journal, 14: 79-118

Issue Date 2001-12-30

URL http://hdl.handle.net/20.500.12000/15071

Rights

Ryukyu Math. J., 14(2001), 79-118

ASYMPTOTIC BEHAVIORS ON THE SMALL

PARAMETER EXIT PROBLEMS AND

THE SINGULARLY PERTURBATION PROBLEMS

MAKOTO SUGIURA

ABSTRACT. We consider the small parameter exit problems for dif­fusion processes and the associated singular perturbed Dirichletproblems. We investigate the asymptotic relations between themean exit time and the principal eigenvalue. Two problems areconsidered under the gradient condition for the corresponding dy­namical system. One is under the uniqueness of deepest valley,where we show that the product of the mean exit time and theprincipal eigenvalue converges to one exponentially fast. The otheris related to the sharp asymptotics of the mean exit times, the eigen­values and eigenfunctions, where we characterize the scaling limitsof them by the Markov chain which appears metastable behaviorof the corresponding diffusion process. To do this, we extend themethods used in our previous papers [10] and [11].

1. Introduction

Let M be a d-dimensional Riemannian manifold of class Coo withRiemannian metric g = (gij) and let Coo-vector fields bE: on M,E > 0, be given. Consider the second order differential operator .cE:defined by

(1.1 ) E> 0,

and the diffusion process (x~, Px ) generated by .cE:, where ~ is theLaplace-Beltrami operator on M. We assume that {bE:} convergesuniformly to a Coo-vector field b on M as E ..j.. °on each compact

Received November 30, 2001.

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subset of M. Then, the process (x~, Px ) can be thought of as a smallrandom perturbation of the dynamical system:

(1.2) Xo = x.

Let D be a connected domain in M with a non-empty smooth bound­ary aD and a compact closure D. Define the first exit time Th fromD by

(1.3) Th = inf{ t > 0; x~ ~ D}.

We denote by At and r.pt the principal eigenvalue and the associ­ated eigenfunction, respectively, for the singularly perturbed Dirich­let boundary value problem:

(1.4) [,tr.p = -Ar.p in D, r.p = 0 on aD.

Noting that r.pt is smooth and positive, we normalize r.pt as

(1.5) sup r.pt(x) = sup Ir.pt(x) I = 1.xED xED

If all the trajectories of (1.2) are attracted toward a single pointwithout leaving D, it is known that both of At Ex [Th] and r.pt con­verge to 1 uniformly on compact subsets of D. In this paper, weshall generalize these results to systems with more than one w-limitsets. More precisely, it will be shown that, for every valley V of thepotential U, each of At Ex [Th] and r.pt converges to some constantuniformly on compact subsets of V, where "valley" and U will beintroduced below. In particular, we shall find that each constantappearing as a limit is not necessarily equal to 1.

Let us suppose that the vector field b satisfies the following as­sumptions:

(AI) (gradient type condition) b = -~ grad U on D for some poten­tial function U E C= (D);

(A2 ) b =/; 0 on aD;(A3 ) the set K = {x E D; b(x) = O} is decomposed into the sum

of finite connected components {Kil, each of which is called

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compactum, in the sense that one can find an absolutely con­tinuous function ¢ E C~iY(Ki) satisfying Jolll~(t)112dt < 00 forall two points x, y belonging to the same compactum;

(A4 ) there exists at least one asymptotically stable compactum withrespect to the flow determined by b in D.

Here grad means the Riemannian gradient, 11·11 = Jg( . , . ) denotesthe Riemannian norm and

C~;f(F) = {¢ E C([O, T], F); ¢(O) = x, ¢(T) = y}, x, Y E F, T> 0,

for an open or closed set F. We remark that, under the assumptions(A I )-(A3 ), for every xED, it holds either that w(x) = (/) or thatw(x) C K i for some i = 1, ... ,L, where w(x) denotes the w-limit setof x E D for the system (1.2): see (2.1a) below (cf. Palis and deMelo [7]).

In order to describe exponential asymptotics of N' or Ex [TV], letus introduce a notation of "valley" relative to the potential functionU. To this end, set U(FI ) = minxEFl U(x) and

(1.6) UFl (F2 ) = max inf max {U(¢(t)) - U(x)},xEFl </JEcx,F2 tE[O,I]

for compact subsets FI ,F2 of D, where cx,F = UYEFC~iY(D). Forevery stable compactum Ki , we define a valley V(Ki) containingK i in D as the connected component of {x E D; U(x) < U(Ki ) +UK; (8D)} which contains K i . Denote by {h,"', VL} the set of allvalleys {V(Ki)} and write U/ = SUPXEVi U(x), Ui- = minxEv; U(x),1 :s i :s L. Then, Vi has depth Ui+ - Ui-. We number the valleys sothat Vo = Uf - Uj-, 1 :s j :s l, and Vo > Ui+ - Ui-, l + 1 :s i :s L,where Vo = maxI:Si:SL{Ui+ -Ui-}, and assume U1 = minl:Sj:s/ Uj- =°without loss of generality.

The theory of Wentzell and Freidlin [3] describes exponentialasymptotics of mean exit times, harmonic measures, etc. Underthe assumptions (Ad-(A4 ), one can see the assumption (A) in [3,p.169] is fulfilled and find the exponential rate of a mean exit time in[11]. Namely, lime:-J,.o c210g Ex [TV] = Vo for x belonging to a certainsubdomain nof D and the left hand side (LHS) is smaller than Vofor x E D\n. This subdomain is defined by n= D n U~=o Ok if one

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sets Do = U~=l Vj and constructs the sequence {D k }k=l inductivelyby the same formulae as (2.2b)-(2.2d) below. Moreover, it has thefollowing property: for an arbitrary fixed_ positive number 5, theprobability that xf starting from a point in D visits one of the deepestvalleys of U before leaving D is larger than e-8/

c2 for all sufficientlysmall E > O. Here xf denotes a sample path of the diffusion processgenerated by £c. On the other hand, in the globally attractive case,Friedman [4] shows that the principal eigenvalue )..C vanishes as E .J- 0with exponential rate - Va: limdO E2 log)..c = - Va.

The purpose of the present article is, as the next stage of thisproblem, to investigate asymptotic relations among the mean exittime Ex[Tb], the principal eigenvalue )..C and the associated eigen­function <pc. In Sect.3 we shall prove that both <pc and )..cEx [TV]converge to 1 exponentially fast on a certain subdomain D of Dassuming the uniqueness of the deepest valley and the local strictgradient condition (see (Bd and (B2 ) in Sect.3). This subdomainD will be defined precisely in Sect.2 and has the following property:the probability that xf starting from a point in D visits the deepestvalley before leaving D i~ larger than 1 - e-8/

c2 for some 5 > O. Inparticular we have D c D. Sect.4 will be devoted to the case wherethe potential U having more than one valleys with depth Va ill D.

The asymptotics of )..C Ex [TV] has been considered formally bySchuss [9] and more rigorously by Williams [12], while Devinatz andFriedman [2] have shown that, when <pc is normalized appropriately,<pc converges to a constant uniformly on each compact subset of D(cf. Corollary 2.3). Moreover Day [1] obtains a similar result fornon-gradient systems. But their arguments were restricted to theglobally attractive case.

In the next section, we shall consider exponential asymptotics ofthe first exit time and the principal eigenvalue, which will be appliedin Sect. 's 3 and 4.

The proofs of the results in Sect.3 are similar to those in [1]' buttheir key estimates are not valid in our case. We shall apply theWentzell-Freidlin large deviation theorem [3] in order to get our fun­damental estimates. More precisely, we consider exponential asymp­toties of the probability that xf stays in a small neighborhood ofsome trajectory "ljJ having sufficiently small energy, i.e., ST("ljJ) < 5for small 5 > 0, where ST is the action functional defined in Sect.2-

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2 (cf. Lemmas 3.4 and 3.5). There we shall also show that >,cTnconverges in distribution to an exponential random variable of meanone.

In Sect.4, assuming that bc = - ~ grad U, c > 0, and that Mis orientable, we shall calculate sharp asymptotics and asymptoticrelations of >.c, <pc and Ex [Tn]' Here the sharp asymptotics means,in case of the mean exit time for instance, that of the form

for some constants Il, Co and C1 . Indeed, if C1 =I- 0, it is sharper thanthe exponential estimates limc:.j..o c2 log Ex [Tn] = Co in the Wentzell­Freidlin theory. One can find sharp asymptotics of mean exit timesin case l = 1 in [10], whose main tools are the Rayleigh-Ritz formula,the Fermi coordinates and the Laplace methods. But the techniquesin [10] are not applicable directly to our problems. For a mean exittime, we shall calculate sharp asymptotics of the conditional meanexit time from a domain containing (essentially) only one deepestvalley, and then use the Wentzell-Freidlin {aD}-graph. The estimateof the principal eigenvalue problem is more complicated. If l = 1,the asymptotics of >.C follows only from limit values of <pc. On theother hand, in case l 2: 2, it may depends also on sharp asymptoticsof <pc on valleys where <pc converges to zero. In order to solve thisproblem, we shall find the rates of convergence on each of the deepestvalleys by using (4.14) below, and then investigate the rates on theother valleys.

We shall obtain two properties of asymptotic relations for exitproblems and eigenvalue problems in Sect.4-3. One concerns a for­mula which the limits of >.C Ex [Tn] and <pc(x), x E Vi, 1 ::; i ::; l,should satisfy. This formula can be regarded as a generalization ofthe result derived in Sect.3. In fact, one can have limc:.j..o >.C Ex [Tn] = 1immediately by setting l = 1 in the formula. The other is relatedto metastable behaviors. Let (Xt, P) be a process appearing as ametastable behavior of the diffusion process (xiI\TE' Px ) developed

D

in [10]. Namely, (Xt, P) is a Markov jump process living in N(j),

1 ::; j ::; lo, where each N(j) is represented as a union of bottomsof several deepest valleys, and having an absorbing state aD. Hereeach of N(l), .. " N(lo) and aD should be understood as a point.

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(See Sect.4-3 for the precise definitions of (Xl, P) and N(j).) De­note the generator of (Xt , P) by 9. We shall show that, if (Xt , P)is irreducible, then for every valley Vi, 1 :::; i :::; L, 'PC: converges to aconstant, say TJi, uniformly on compact subsets of Vi. Furthermore,if one sets ep(N(j)) = TJj' for Vjl n N(j) =I 0 and ep(aD) = 0, then epis a principal eigenvector of the eigenvalue problem of 9 with condi­tion ep(aD) = 0 (cf. (4.22) below). And each TJi, t + 1 :::; i :::; L, isrepresented as L~~l iiijep(N(j)) for some iiij, where iiij is character­ized by the limit as E -!- 0 of the probability that xi starting from apoint in Vi goes close to N(j) before it either goes close to the otherN(j'), 1 :::; j' :::; to, jf =I j, or leaves D. There we shall treat the k-theigenvalue >'k' k = 2, ... , to, of the problem (1.4) as well as >.C: (= >'i).It will be shown that for k = 1,···, to limc:.j..o EI-£evo/c:

2>'k = >'k with

some constant J1 (defined in Sect.4-1) and that the eigenfunction as­sociated to >'k converges to an eigenvector associated to >'k in anappropriate sense, where >'I, ... , >'10 denote all the eigenvalues of 9such that 0 < >'1 :::; ... :::; >'10. Concerning this problem, Mathieu [6]considers the Neumann boundary value problem only under the con­tinuity of U in case that D is a valley in itself. He obtains exponentialasymptotics of >.~ (>'i = 0 in this case) by using methods of Dirichletforms, which is quite different from ours.

2. Preliminaries

The present section is devoted to the exponential asymptotics con­cerning some exit problems and that of the principal eigenvalue,which will be used in Sect.'s 3 and 4. The assumptions (A 1)-(A4 )

are supposed throughout this section.

2-1. Wentzell-Freidlin's results under gradient type condi­tion

Let G be a subdomain of D with a smooth boundary in such a way- I

that grad U =I 0 on aG, G c Uj =l Vj and G n Vj =I 0 for 1 :::; j :::; t.

Then, set Do = D\G. We state the following result without proof,since it is shown in the same manner as the Wentzell-Freidlin theorem[3, Chap.6, Theorem 5.1].

Theorem 2.1. Let 710

denote the first exit time from Do givenby (1.3) when D is replaced by Do. There exists a non-negative

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continuous function W Do (x) on Do such that

limE2logPx(x~E E aD) = -WDo(X)EtO Do

uniformly in x belonging to every compact subset of Do.

Let us consider WDo (x) in our model. We prepare several nota­tions. Let {Xt(x); t ~ 0, x E D} be the flow determined by b, i.e.,Xt = Xt(x) is a unique solution of the ordinary differential equation(1.2). We define the w-limit set w(x) of a point x E D and the do­main V(F) of the attraction of a connected open or closed set F inD with respect to this flow in the following manner: if Xt(x) E D forall t > 0,

(2.1a) w(x) = {y E D; Xt n (x) ----* y for some sequence tn ----* oo},

otherwise w(x) = 0, and

(2.1b) V(F) = {x E D;w(x) C F,w(x) =I- 0}.

The associated a-limit set a(x) of a point x E D is defined by

(2.1c) a(x) = {y E D; Xt n (x) ----* y for some sequence t n ---+ -oo},

in case that Xt(x) E D for all t < °and a(x) = 0 otherwise. Wewrite K s and Ku for the set of all stable compacta and that of allunstable ones respectively. Now we define subsets !1k , k = 0,1", "and !1~, 0%, k = 1,2"", of D as follows:

(2.2a)

(2.2b)

!1o = {x E D; w(x) = 0},

!1% = !1k-l u U K i ,

KiEKu:Kinnk-l#0

k = 1,2","

(2.2c) !1~ = !1% U U k = 1,2,""j=l+1,. ..,L:Vjnn~#0

(2.2d) !1k = !1~ U {x E D; w(x) C !1~}, k = 1,2,···.

Noting that the sequence {!1dk~O is not decreasing and !1koOko+1 = ... for a sufficiently large number ko, we define !1oo

Oko n D and!1 = D\!1oo .

Here one remarks !1 is open and notices Vj C !1 for 1 ::; j ::; l.By using a similar argument to Theorem 4.1 of [11], one can showProposition 2.2 as follows.

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Proposition 2.2. We have WDo(X) = 0 for x E Do\O and WDo(x)> 0 for x EOn Do.

Corollary 2.3. Let F be a compact subset of o. There is a positiveconstant l' so that

for all sufficiently small c > o. Here we note that Px (x~. E G) = 1DO

for x E G.

From the following proposition, one can find the constant l' inCorollary 2.3 more concretely. Since the proof is quite similar toProposition 4.3 in [11], we also omit it.

Proposition 2.4. We have W Do (x) ~ U{x} (BD) - U{x} (BG) for allx E Do. Recall (1.6) for the notation UF1 (F2 ).

The next proposition is a slight generalization of Theorem 4.2 in[3, ChapA].

Proposition 2.5. For Q > 0, we have

uniformly in x belonging to every compact subset of o.

Proof. Let us fix Q > 0 sufficiently small and a compact subset F ofO. From Theorem 4.1 in [11], we have

(2.3) limc2 10g sup Ex [rn] = Vo·E-!-O xED

By using Chebyshev's inequality, this implies

lim sup Px(rn ~ e(VO+O)/E2

) = o.E-!-O xED

Find subdomains r 1, r 2 of D with smooth boundaries satisfying thefollowing conditions:

- I -(1) r 2 nK = Ul{X E Vj;U(x) = Uj };

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I -- 1(2) r 1 ::) U1{X E Vj ;U(X) = Uj- } and rIC U1{x E Vj ;U(x) <

Uj- + a/4};1 -- 1

(3) r 2 ::) Ul{X E Vj;U(x) ::; Uj- + a/4} and f 2 C Ul{X E

Vj; U(x) < Uj- + a/2}.

We introduce an increasing sequence of stopping times ao, TO, a1, T1,

a2,"', below: ao = 0 and Tn = inf{t > an; xf E r 1 or xf 1:. D},an = inf{t > Tn-I; xf 1:. r 2 }. Then, Proposition 204 and Corollary2.3 respectively verify

inf Px(x~~ E aD) 2: e-(Vo-a./2)/£2,XE r 2 D\r 1

inf Px(X~~ E rd 2: 1 - e- r /£2,xEF D\r1

for some r > 0 and all sufficiently small E > O. Indeed, we note thatWD\r 1 (x) 2: Uf - U(x) > Vo - a/2 for all x E r 2 n Vj\r1 for thefirst inequality. Moreover one can find B > 0 so that

inLPx (Tl > B) 2: inLPx (al > B) 2: 1/2,xEr l xEr l

which follows from [11, Lemma 2.8] combined with [3, Chap.5, The­orem 3.2]. By using the above estimates, one can show this proposi­tion by the same procedure as [3, ChapA, Theorem 4.2]' where Tm

= (Tl - TO) + (T2 - Tl) +... + (Tm - Tm -l) is regarded as the number ofsuccesses in Tn Bernoulli trials by the strong Markov property. 0

2-2. Exponential asymptotics of the principal eigenvalue

Recall that >.£ is the principal eigenvalue of the Dirichlet boundaryvalue problem (104). We mainly use the following theorem by Fried­man [4, Chap.14, Theorem 10.1]. This gives a basic relationshipbetween the first exit time and the principal eigenvalue.

Theorem 2.6. >.£ = sup{>' 2: O;suPxEDEx[e'XT.b] < +oo}, E > O.

Friedman also proved that >.£ vanishes exponentially fast as E tO. The next theorem gives the exponential rate of this convergenceunder the assumptions (Ad-(A4 ).

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Theorem 2.7. limc.j.o c: 2 log Ac = - Vo.

Proof. Let a > 0 be fixed arbitrarily. We claim the following twoinequalities:

(2.4a)

(2.4b)

limc:210g AC 2: -(Vo + 3a),dO

limc:210g AC :s: -(Vo - 3a).dO

In view of Theorem 2.6 it suffices for (2.4a) to show

(2.5)

from Theorem 2.6. However, since sUPxED Px(Tn> e(Vo+20)/c2) :s:

e- o /c2 by (2.3), the Markov property implies

m= 1,2,···.

Hence, by observing

00

+ L {e(m+I)e-(Vo +3<»/e2

e(Vo+2<»/e2

_ e me-(Vo+3<»/e2

e(Vo+2<»/e2

}

m=I

(2.5) is easily verified.We move to the proof of (2.4b). Recalling U1 - minxEvl U(x) =

0, we assume U has no critical points in VI \F30 without loss ofgenerality, where F/j = {x E VI : U(x) :s: Vo - <>}, <> > O. Denote byD I a subdomain of VI with a smooth boundary so that VI ~ D I ,

D I ~ Fa· Then, there is To > 0 satisfying U(ITo (x)) < Vo - 3a forall x E Fo / 2. The fundamental fact in the Wentzell-Freidlin theoryguarantees the existence of <> > 0 such that

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for all sufficiently small E > O. Here we assume that 6 > 0 is smallerthan both of the distance between aDI and Fa and that betweenD 1\F3a and {XTo(x); x E Fa} without loss of generality. FromProposition 2.5, it holds that infxEF3 o. Px ( T Pa > e(Vo-3a/2)/E

2) >

1 - 6 for all sufficiently small E > O. Then, we have

. f T) (E > (Vo-2a)/E2

E E F )In r x TD e ,x ('" -2a)/,2 axEF3a 1 e 0

> inf P (TE > e(Vo-2a)/E2

) > 1 - 6.- F x Fo. -xE 3a

On the other hand, the Markov property shows

sup Ix~ - Xt(x)1 < 6/2)O~t~To

> . f E [T) (E (Vo-2a)/E2

,." E F )._ In x rx~ TD 1 > e - .LO, X e(Vo-2a)/,2 -Too Eo,xEFo. \F3o. 0

2: inf Px ( TD 1xEF3a

2: inf Px (TPaxEF3a

sup Ix~ - Xt(x)1 < 6/2]09~To

(Vo -2a)/E2

,." E F) ( 5:)>e -.L 0 ,Xe (Vo-2a)/,2_To

E a X I-v

Hence, we obtain

. f T) (E > e(Vo-2a)/E2 x E E Fa) >_ (1 _ 5:)2In r x TD , e(~o-2a)/,2 v

XEFa 1

and also

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> . f E [P (C > (m _ 1)e(Vo-20:)/c2

),_ In x x e TVxEF<> e(VO-2<»/e2 1

T C > e(VO-20:)/c2

XCVI ' e(VO-2<»/e2

2: inf Px(TD > (m - 1)e(Vo-20:)/c2

) x (1 _ (5)2xEF<> 1

From the above estimates, we have

00 2 2 22: L {e(m+l)e-(VO-3o<l/e e(Vo-2<>J/e _ e me-(VO-3<>J/e2e(VO-2<>J/e }

m=1

00 2> L eme<>/e (1 _ (5)2m

m=l

= +00.

Since Px (T1h ::::; T1)) = 1 for all x E F0:, this implies that, for allsufficiently small E > 0,

By using Theorem 2.6 again, we obtain (2.4b). 0

3. The case under uniqueness of the deepest valley

In this section, we suppose uniqueness of the deepest valley (B1 ) andlocal strict gradient condition (B2 ) as well as (A 1)-(A4 ):

(BI ) there is exactly one deepest valley, i.e., 1 = 1;(B2 ) there exists a neighborhood Go of a stable compactum K i in

VI such that Go is orientable as a manifold and that bc =

-~ grad U on Go for all E > O.

Recall that <pc is the eigenfunction corresponding to the principaleigenvalue ,XC of the Dirichlet boundary value problem (1.4) and isnormalized as (1.5).

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Theorem 3.1. Let F be a compact subset of n, where n is definedin Sect.2-1. There exists a positive constant r so that

(3.1) sup 11 - CIl(x) I ~ e- r /£2xEF

for all sufficiently small c > o.The next lemma is shown similarly to [1, Lemma 7] by means of

the Schauder interior estimate.

Lemma 3.2. Let F be a compact subset of Go. There is C1 > 0 sothat

sup II gradcp£(x)11 ~ C1c- 2

xEF

for all c > o.

We denote by G"( the connected component of {x E VI; U(x) <U(Ki ) + 'Y} which contains K i for'Y > 0, where K i is a stable com­pactum in Go· Let 'Yo > 0 satisfy G"(o C Go. By combining (B2 )

with Theorem 2.7 and Lemma 3.2, one can obtain the next lemmain the same manner employed by Devinatz and Friedman [2, Lemma2.2].

Lemma 3.3. There is a positive constant r so that

sup Ilgradcp£(x)11 ~ e-r /e2

xEG-yo

for all sufficiently small c > o.

Proof of Theorem 3.1. Let F be a compact subset of n and setDo = D\G"(o· By (B1 ), Theorem 1 in [11] verifies

(3.2)

Then, from Corollary 2.3 and Lemma 3.3, there is rl > 0 such that

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for all sufficiently small E > O. On the other hand, (1.4) and Ito'sformula prove

(3.3)

for all x E Do. From the normalization (1.5), we obtain the followingestimates:

inf cpC(x) 2: { sup cpc(x) - e- rI/c2} . (1 _ e- rI/c

2),

xEF E-Gx /'0

sup cpC(x):S inf cpC(x) + e- r1 /c2

+)..c sup Ex [TDo]'xEF xEG/,o xEF

Together with Theorem 2.7 and (3.2), one can find r2 > 0 so that

sup Icpc (x) - cpc (y) I :S e- r2 / c2

x,yEF

for all sufficiently small E > O. The normalization (1.5) also impliesthat, for each E > 0, there is a point xC; E D such that cpc (xC;) = 1.By using (3.3) and (1.5) again, we obtain

1 :S inFf cpc (x) + e-r2 /c2 + )..cExc,[TDo].

xE

Therefore, combining with Theorem 2.7 and (3.2) again, we can finda constant r > 0 so that (3.1) holds. 0

Let us assume that V2 is the deepest valley of all the valleys{V2 ,' . " VL} with depth < Vo without loss of generality. Set V2 =

max2~i~L{Ui+ - Ui-}·

Lemma 3.4. For an arbitrary a > 0, there is a compact subset Fin n such that

for all sufficiently small E > O.

Proof. Recall Lemma 1.1 in [3, Chap.6]; there exist ).., C1 > 0 suchthat, for all x, y belonging to a small neighborhood D 1 of D with

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p(x, Y) ~ A, one can find a function 'l/J in DI satisfying 'l/J(O) = x,'l/J(T) = y and ST('l/J) ~ CIP(x, y), where T = p(x, y). Since aD issmooth, one can find a sufficiently small 0 < 8 < a/8CI such that,for each x E [aD](b) n D, there is a trajectory 'l/Jx : [0, Tx] -+ D Isatisfying 'l/Jx(O) = x, 'l/Jx(Tx) E [aD](b\D, Tx = l'l/Jx(O) - 'l/Jx(Tx)I <28, B b/ 2('l/Jx(Tx)) n D = 0 and STx('l/Jx) ~ a/4. Here ST denotesthe action functional defined on C([O, T]' M), T ~ 0, below: if

¢ E C([O, T], M) is absolutely continuous, ST(¢) = ~ J:{ 111>(t) ­b(¢(t))112dt; otherwise ST(¢) = +00; p( ., . ) is the Riemannian dis­tance and [aD](b) and Bb(Y) stand for the 8-neighborhood of aD andY respectively. From Theorem 3.2 in [3, Chap.5], we have

(3.4) inf Px (Tn < Tx )xE[8Dj(o)nD

~ inf Px( sup Ix~ - 'l/Jx(t) I < 8/4)xE[8Dj(o)nD 0:St:STx

~ e-a / 3 E: 2•

For some maXXEv1nK U(x) < " < Vo, set G = {x E VI; U(x) < ,'}and Do = D\G. Since F I = {x E Do; WDo(x) ~ a/4} is a subset ofn from Proposition 2.2, one can find compact sets F ' C Do, Fensatisfying D C F ' u F u [aD](b) and F ' n FI = 0; recall Theorem 2.1for the notations WDo(X). Theorem 2.1 shows

inf Px(x~£ E aD) ~ e-a/2E:2.

xEF' DO

We know limE:../.oE2 1ogsuPxEDo Ex [TDo ]= V2 from Theorem 1 in [11]and also

sup P (TE: > e(V2+2a)/E:2) < e-a/E:2x Do - -

xEF'by Chebyshev's inequality. Hence, by noting Tno = Tn for all pathssatisfying xg E Do and x~£ E aD, it holds that

Do

inf P (TE: < e(V2+2a )/E:2) > inf P (xE: £ E aD TE: < e(V2+ 2a )/E: 2

)xEF' x D - xEF' x T Do ' Do

> inf P (xE:£ E aD) - sup P (TE: > e(V2 +2a)/E:2

)- F' x T D x D o -xE 0 xEF'

~ e-a/2E:2(1 _ e-a/2E:2).

Together with (3.4), we obtain the desired estimate. 0

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Lemma 3.5. For an arbitrary a > 0, there exists a compact subsetF in [2 so that

(3.5) Px(xf ~ F,7n > t) Ot/€2sup sup 2 < et>O xED Px(t < 7n ~ t + e(V2+2Ot )/€ ) -

for all sufficiently small E > O.

PT'Oof. By virtue of the Markov property combined with the estimatein Lemma 3.4, we have

P (t < 7€ < t+e(V2+ 2Ot )/€2) = E [P € (7€ < e(V2+2Ot)/€2) 7€ > t]x D - x X t D - , D

> E [P € (7€ < e(V2+2Ot)/€2) x€ rf F 7€ > t]- X X t D - , t 'F- 'D

> e- Ot /€2 P (x€ d F 7€ > t)- x t 'F- 'D

for all t > 0 and xED, and (3.5) is obtained. 0

Theorem 3.6. Let F be a compact subset of n. There exists apositive constant J so that

(3.6) sup I'\€Ex [7n] - 11 :::; e- 8/€2xEF

for all sufficiently small E > O.

Proof. Set v€(x, t) = e->'€trp€(x) and u€(x, t) = Px(Tn > t). From(1.4) and Ito's formula, we know v€(x, t) = Ex [rp€(xf) , Tn > tJ. Letus take 0 < a < (Vo - V2 ) /4 and a compact subset F of [2 so that(3.5) holds. From Theorem 3.1, there is r > 0 such that (3.1) holds.Then, we have

Iv€(x, t) - u€(x, t)1

~ Ex [ll - rp€(xDI, x~ E F, Tn > t] + Px(X~ ~ F,7n > t)

< e-r /€2 P (7€ > t) + eOt /€2 P (t < 7€ < t + e(V2+2Ot )/€2)- x D x D- ,

Iv€(x, t) - e->.€tl ~ e- r /€2 e->!t,

for all x E F. Since ,\€ Jooo e->.€t dt = 1, the above estimates imply

(3.7) I'\€Ex [Tn] - 11 ~ ,\€100

lu€(x, t) - e->.€tl dt

< e- r /€2 ,\€ E [7€] + ,\€e(V2+3Ot)/€2 + e- r /€2- x D

for all x E F. Hence, by using (2.3) and Theorem 2.7, we obtain(3.6). 0

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Theorem 3.7. We have limc-J-o PxCAcTb > t) = e- t for all x E 0and t 2 o.Proof. It suffices to prove

(3.8)

for each x E 0 and z E ]RI. But since

Ex[eiz>!rb ] = 1 + iz)..C 1= eiz>.EtuC(x, t) dt,

1/(1 - iz) = 1 + iz)..C 1= eiz>.Ete->.Et dt,

one notices

Hence, by using a similar estimate to (3.7), we obtain (3.8). 0

Remark 3.8. By making some extra efforts, one can obtain a uniformversion of Theorem 3.7; namely,

for every compact subset F in O. (See, e.g., [10, Proposition 5.1].)

Remark 3.9. Theorems 3.1 and 3.6 seem to provide no informationson the limit values of <pc(x) and )..C Ex [Tb] respectively for x E D\O.In fact, their limits are not equal to 1 generally, unless D = O. Onecan find the precise limits of them for some potential function in [10]for instance. (See Theorems 4.12 and 4.19 also.)

4. Cases of more than one deepest valleys

As mentioned in Remark 3.9, one can find in [10] sharp asymptoticsof the principal eigenvalue )..C, the associated eigenfunction <pc andthe mean exit time Ex[TbJ, etc., under several conditions in additionto (Ad-(A4 ). But the results are restricted to the case that thepotential U has a unique deepest valley in D. In the present section

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we calculate sharp asymptotics of them for the potential with morethan one deepest valleys in D and then consider asymptotic relationsamong them.

Let us state our precise assumptions after making some prepa­rations. Set N(i) = {x E Vi; U(x) = Ui-}, 1 ::; i ::; L, and M =U7=1 UK

pCK:K

pnav;;i:0 K p , where one notes that N(i), 1 ::; i ::; L,

and M consist of critical points of U. The Hessian H = H(x) of U isa symmetric tensor field of type (0,2). Especially for a critical pointx of U, it is defined by

l::;i,j::;d,

in terms of an arbitrary local coordinate x = (xi). A tensor H* =H*(x) of type (1,1) is naturally associated with H by means of themetric g:

g(H* X, Y) = H(X, Y), X,YETxM,

where TxM stands for the tangent space to M at x E M.We shall suppose the following assumptions (Cd-(Cs) with (A 2 )

-(A4 ) throughout this section, where we note that (C2 ) implies (AI):

(C1) M is orientable;(C2 ) (strict gradient condition) there exists a potential function U E

COO(D) such that bE: = -~ grad U on D for all E > 0;(C3 ) Vi c D for alII::; i::; L;(C4 ) each N(i), 1 ::; i ::; L, consists of finite connected n~)-di­

mensional compact submanifolds N~i), 1 ::; Q ::; li, of M,

N(i) = U~=l N~i), and if aN~i) I: (/) there exists a connected

n~)-dimensional submanifold N~i) of M such that the interiorof N~i) contains N~i);

(C5 ) each N~i), 1 ::; Q ::; li, 1 ::; i ::; L, is non-degenerate in the

sense that the Hessian H* (x) has rank d - n~);(C6 ) each compactum M a , 1 ::; Q ::; K, in M is an rna-dimensional

compact submanifold of M, M = U~=l M a , and if 8M<:. I: (/)there exists a connected rna-_dimensional submanifold Ma ofM such that the interior of M a contains M a ;

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(C7 ) each Men 1 ::; a ::; K, is non-degenerate and index 1, namely,for every x E Men rankH*(x) = d-mo. and H*(x) has exactlyone negative eigenvalue;

(Cs) for each Mo., 1 ::; a ::; K, one can find a neighborhood G ofMo. satisfying one of the next conditions: (i) w(x) of. 0 for allx E G and there exist two valleys ViI' Vi2 , 1 ::; i 1 < i 2 ::; L,such that w(x) eVil UVi2 UMo. for all x E G, (ii) there existsone valley Vi, 1 ::; i ::; L, such that each x E Gsatisfies eitherw(x) = 0 or w(x) C Vi U Mo..

Moreover, in Sect. 's 4-2 and 4-3, we shall also suppose the nextassumption (Cg ):

(Cg ) the minimum of U in D is attained by all the points in N(j)

(1 ::; j ::; l) and only by them, i.e., U1 = ... = Ul- and

U(x) > U1 for all x E D\ U~=l N(j).

Remark 4.1. The assumptions (C4 ) and (C5 ) are utilized for simpli­fying the proof of sharp asymptotics of Ex [Tn]' In fact, it suffices tosuppose them only on the bottoms N(j), 1 ::; j ::; l, of the deepestvalleys in order to obtain sharp asymptotics of '\£, <p£ and Ex[Tn]'

4-1. Sharp asymptotics of the mean exit time

For an integer a, let us introduce an equivalence relation "'-'a on theset of valleys V ={VI, .. " VL} in the following manner:

(1) Vi "'-'a Vi;(2) Vi "'-'a Vi' if Vi of. Vi' and there are ViI'···' Vip E V so that

maxM cTn-v- mo. > a, 0 ::; q ::; p, where we write Vio = Vi0' tq 'lq+l

and Vip+ I = Vi' simply.

We denote the equivalence class of Vi E V by Ca(Vi ) = {Vi' E

V; Vi' (Va Vd. Note that, if Vi (Va Vi' for some a ~ -1, thenU/ = Ui:. Set W(i) = C- 1 (Vi) and

vcii) = {x E D; U(X) < Ui+}\ U Vi',Vi,EW(i)

1 ::; i ::; L,

VJO) = {x E D; w(x) = 0,0 of. a(x) C Mo., 1 ::; a ::; K};

recall (2.1) for the notation a(x). We will sometimes regard VJl), ... ,

VJL) or VJO) as a valley. Next, we define a {Vi, Vcii)}-route through

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W(i), 1 :::; i :::; L, by a finite sequence of steps ViI -+ Vi2 (ViI' Vi2 E

W(i) U {VJi)}, ViI i= Vi2 ) satisfying the following conditions:

(1) the valleys of each step Vi I -+ Vi2 satisfy M 0. C Vi I n Vi2 for

some Mo., 1 :::; a :::; K, where we write ViI = VJi) simply in casethat i l = 0;

(2) the first step starts from Vi and the last one ends at VJi);(3) the end point of each step becomes the starting point of the next

step except the last step;(4) there are no closed cycles in each route.

Write the set of {Vi, VJi)}-routes through W(i) by ryt(i).Set for 1 :::; i :::; L

m(i) = max min max mO.,rE9l(i) (Vil-+Vi2)Er M<>c~nvi2

Let H+ (x) be the product of all the positive eigenvalues of the Hes­sian H*(x) for x E Uf=l N(i) U M and -H_(x) denotes the nega­tive eigenvalue of H* (x) for x E M. For a 2': 0, 1 :::; i :::; Lando:::; iI, i2 :::; L, i l i= i2, we define

Vi(a) = (21f)(d-a)/2 L L(i) H+(y)-t dy,

o.:n~t)=a <>

and, if maxM E{V- -+v- } mo. < a,Q tl 'l2 -

and, otherwise, H((va

) V-) 00, where the summation over an11.-......t 12

empty set is equal to 0 and the maximum of an empty set is equal to

-00. Here dy stands for the volume element of Nii) or Mo. induced

from 9 on M; if dim Nii) = 0 or dim Mo. = 0, then dy should be

understood as the 6-mass. Moreover we used a notation:

(4.1) {ViI -+ Vi2} = {Mo.; Mo. n aVi l i= 0,a(x) c Mo.

for some x E Vi2 },

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where we write Vi = V6°) simply in case of i = O.Let us define {Cdl~i~Lo = {Cm(i,)(Vi')h~i'~L so that {VI,"',

Vt} C U~o=1 Cj and {VI,"', Vt} n Cj :f 0 for 1 ~ j ~ lo, and set

Co = {V6°)}· Here we notice that Ci1 n Ci2 = 0 if i l :f i 2 . Then, setW/ = maxvi , EC i U/;, Wi- = minVi' EC i Ui-:, Wi = W/ - Wi- and

n(i) = max n(i')Vi,ECi '

Note W/ = Ui; and m(i) = m(i') for all Vi' E Ci. Choose regularvalues Ui-: < U[, < Ui;, 1 ~ if ~ L, of U and denote

(4.2a)

(4.2b)

Bi, = {x E Vi,;U(x) < U[,},

B(i) = UBi" 1 ~ i ~ L o,Vi,ECi

1 < if < L- - ,

(4.2c)

(4.2d)

D(id = D\ U B(i2), 1 ~ i l ~ L o,l~i2~Loh#il

D(O) = D\ U B(i).

l:<=;i:<=;Lo

We construct a subdomain n(i) of D(i) in like manner used in Sect.2­1 and have the same estimates of Corollary 2.3, in which nand Gshould be replaced respectively by n(i) and B(i), for 1 ~ i ~ L o.

Let 1 ~ i ~ L o be fixed arbitrarily for a while. Find a suf­ficiently small connected domain Di containing Uvi,EC

iVi' with a

smooth boundary, and construct a subdomain nDi of D i analogouslyto a subdomain n(i) of D(i). We denote by Xb

iand CPD

ithe prin­

cipal eigenvalue and the corresponding eigenfunction respectively ofthe Dirichlet boundary value problem (1.4), in which D should bereplaced with D i , and normalize CPDi in the same manner as (1.5).

The next lemma can be shown similarly to Theorem 3.1 in [10]and we omit the proof. (See also the proof of Theorem 4.12.)

Lemma 4.2. It holds that

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where we write

(4.3)

Moreover we have lime.j..o infxEF 'Ph (x) = 1 for every compact subsetF of nDi , and lime.j..o sUPXEFo 'PDi (x) = 0 for every compact subset

- -. +Fo of Di n D({x E Di,U(x) < Wi }\ UVi,ECi Vi').

Lemma 4.3. For compact subsets F of nDi , we have

lim sup IADiEx[TvJ - 11 = o.e.j..O xEF

Moreover we have limdO Px(ADi TVi > t) = e- t for all x E nDi andt 2: o.Remark 4.4. One can obtain a uniform version of the second state­ment:

lim sup sup Ipx(ADiTVi > t) - e- t I= 0e.j..O t2:0 xEF

for every compact subset F in nDi . (cf. Remark 3.8.)

Proof of Lemma 4.3. It can be shown by the same way as Theorems3.6 and 3.7 if one finds the estimates in Lemma 4.5 below. 0

Lemma 4.5. limdo ADi sUPxEDi Ex[T1J < +00.

Proof. Let n~i) = -1 and

nii) = min{n(i'); Vi' E Ci , Ui-: = Wi-, n(i') > nii~l}' k = 1,2,· .. ,

and set Di,k = Di\UV,ECoU-=W-n(i'l>n(ilBil, k = 0,1,···. Itt t· i' t' k

suffices to show

(4.4)

for each k = 0,1,···. We shall use induction on k. Theorem 1 in [11]and Theorem 2.7 implies (4.4) for k = o. From the formula (4.20)below, we have

ADi1 Ex [TvJ'Ph (x)e- U(x)/e2

dx = 1 'Ph (x)e- U(x)/e2

dx,Di Di

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where dx stands for the Riemannian volume element. Since Ex [TV ]"k

:S Ex[TvJ,

Ah j Ex [TDi,k]'Ph (x)e- U(x)/t:2

dx :S j 'PDi (x)e- U(x)/t:2

dx.Di,k Di

In view of Lemma 2.2 in [2]' if Bi, C Di,k, there exists r > 0 suchthat

su~IEx[TDi,J - Ey[TDi,JI :S e-r

/t:2

• sup Ex[TDi,J, k = 1,2"",x,yEBil xEDi,k

sup I'PDi (x) - 'PDi (y)1 :S e- r /t:2

,

x,yEBi,

for all sufficiently small E > O. Indeed, for the former estimate, usethe equation (4.19) below, in which vt: and D should be replaced byE.[TVi.J and Di,k respectively, and limdO E

2 log sUPXEBil Ex[TDi,J= Wi, which is shown from Theorem 1 in [11]. The latter is alreadyknown in Lemma 3.3. Hence, together with Lemma 4.2, the Laplacemethods imply

limE-(d-n~i))eWi-/t: 2 At: . j Ex[Tt:. ]'Pt:. (x)e- U(x)/t:2

dx£.1.0 D, D D"k D,

i,k

= C1 ·limADi sup sup Ex [TDi,J,t:.j.0 V" ECi :u-; =W.- ,nU') =n(ki) xEBil, , '

limE-(d-n(i))eWi- /t: 2 j 'PDi(x)e- U(x)/t: 2 dx = C2 ,

dO Di

(n(i)) (n(i))where C1 = "2:V /ECi:U-;=W- Vii k and C2 = "2:V"ECi:U-;=W- Vii .

t t t t t t

On the other hand, from the Markov property,

Ex [TDi,J

E U Bil]

Vii ECi:Ui-;=Wi- ,n(i/)=n~i)

+ Ex[TDi,k_J

By combining the above estimates with the assumption of induction,we obtain (4.4). 0

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Lemma 4.6. We have limf:-!-o SUPt~O sUPxEF IPx('\Di71(i) > t)­

e-tl = 0 for every compact subset F in n(i).

Proof. This is shown immediately from Lemma 4.3 and Remark 4.4if one notices the formula:

Px (& :::; ADi71(i) :::; t)

= ( Py(ADi71(i)

J(0,t-8] x aD i

for 0 < & < t. 0

Next we shall consider the distribution of exit positions. Let usfix 1 :::; i 1 :::; L o and 0 :::; i 2 :::; L o, i 1 I- i 2 , arbitrarily for a while.The following proposition is derived in the same manner as Theorem4.2 in [10] if one refines Theorem 3.1 in [10]. Since this can be donequite similarly, we omit the proof.

Proposition 4.7. Let us suppose that there exist Vi~ E Ci 1 andVi; E Ci2 so that #{Vi~ ----* Vi;} 2:: 1; recall (4.1) for {Vi~ ----* Vi;}'Then, we have

uniformly in x belonging to every compact subset of n(il), where weset

a 2:: 0,

and write aB(O) = aD simply.

Corollary 4.8. Let us suppose Wi~ = Wi~' Then, for every com­

pact subset F of n(id, there exists a constant C such that

P ( f: aB(i2)) < C (m(i] )-m(i2»)vOsup x xTo E _ ExEF D(i])

for all sufficiently small E > O.

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uniformly in x belonging to every compact subset of n(i l ), where we. _ (m('I»)

wllte P~I ~2 - Pi l ~2

Proof. From Lemma 4.2, it suffices to show (4.5) in case of Pi1i2 > O.By using Lemma 4.6 and Proposition 4.7, one can show(4.6)

lim sup sup Ipx(,ADir~(il) > t, x~.. E 8B(i2)) - Pi1i2e-t! = 0

dO t~O xEF 1 D('Il

in the same manner as Theorem 5.1 in [10]. If one sets r(t) =sUPxED(i l ) Px(ADil r1(ill > t), then the Markov property verifies

r(t + s) :s; r(t)r(s), t, s > O. But, since

Px(Ahlr1(ill > 1)

= Px(AD r1(i l ) > 1, x~. E 8D(id )'I D(O)

+ Px(AD r1(ill > l,x~. E B(id)'I D(O)

:S;PX(xE• E8D(id)+Px(A'°. rIO 0 >e-'Y/E2

)T D(O) D'l D()

+ Ex [Px' • (A Di r1(il) > 1 - e-'Y/E2

), x~. E B(id]T D(O) 1 D(O)

for some small, :> 0, one has limE.!.o r(l) :s; e- 1 by using Corol­lary 2.3, Theorem 1 in [11] with Chebyshev's inequality and (4.6)respectively for the three terms of the right hand side (RHS) of theabove inequality. Hence, SUPO<E<E r(t) belongs to L1([0,+oo),dt)_ 0

for some EO > 0, and from (4.6) we get (4.5) for each x E n(id owingto the Lebesgue dominated convergence theorem.

For uniform estimates, we set

and use the fact that they satisfy the equation:

(4.7)

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Let us fix Vii E Ci1 and put h€"(x) = g€(x)/[l", where g€" =

sUPXEV~ g€(x) and VJ = {x EVil; U(x) < UJ - -y}, -y > 0. Then,,"/ 2one can show sup v-y / llgradh€"(x)lI:::;e-' € forsome-y',-y">O

xE _/

in a similar manner 'to Lemma 2.2 in [2]. In order to obtain this,use (4.7) together with lim€-!.o E

2 log g€" = Wi1 , which is derived fromProposition 2.5 and Pil i2 > 0. Therefore, by combining Corollary 2.3with the Markov property, the above estimates immediately verifyour assertion. 0

We introduce the Wentzell and Freidlin W-graph. Let J be afinite set and let W be a subset of J. A graph consisting of arrows0: -t (3 (0: E J\W, (3 E J, 0: =1= (3) is called W-graph on J ifit satisfiesthe following conditions:

(1) every point 0: E J\W is the initial point of exactly one arrow;(2) there are no closed cycles in the graph.

We denote by (5J(W) the set of W-graphs on J. For 0: E J\W and(3 E W, (5~I3(W) stands for the set of W-graphs on J in which thesequence of arrows leading from 0: to (3. (See Wentzell and Freidlin[3,pp.177-182].)

The following lemma is a slight modification of Lemma 3.3 in [3,Chap.6]. One can prove it in a quite parallel manner.

Lemma 4.10. Let (Zt, P x) be a strong Markov diffusion process on

a phase space X = Ut~-I Xi, Xi n Xii = 0 (i =1= i'), with stoppingtimes T i = inf{t > 0; Zt fJ. X-I U Xd, 1 :::; i :::; jo, and T =

inf{t > 0; Zt fJ. X-I U UiEW Xd, where W is an arbitrary subsetof J = {O,·· ·,jo}. Assume Px(Ti < +(0) = 1, x E X-I U Xi,1 :::; i :::; jo, and

a-I. Pi1i2 :::;PX(ZTil E XiJ :::; a· Pid2'

a-I. Si1Phi2 :::;Ex[Ti1 , ZTi1

E X i2 ] :::; a· Si 1 Pi1 i 2'

x E X i1l 1:::; i I :::; jo,O:::; i 2 :::; jo,i I =1= i2 ,

for some a > 1. Then, we have Px(T < +(0) = 1, x E X o U

UiEW Xi, and

k (l) (W). k(l) (W)a- r • q- - <P (Z E X- ) < a r • q- -~l ~2 - x T ~2 - tl t2 '

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a-k~2) . t(W) <E [T Z EX] < ak~2) . t(W)~l ~2 - X ,T ~2 - ~l ~2 '

X E X ip i 1 E J\W,i2 E W,

if LgE<BJ (W) 7[(g) is positive, where we write

(4.8a)

(4.8b)

(W) _ L 9E<B!1;2 (W) 7[(g)qi1 i2 - L9E<B J (W) 7[(g) ,

t(W) = s. . LgE<B J (Wu{ ill) 7[(g) . L 9E<B!1;2 (W) 7[(g)~l ~2 ~l LgE<B J (W) 7[(9) LgE<B J (W) 7[(g)

( )- n d k(l) - 3 2r - 1 2 k(2) - 2r - 1 (3 2)7[g - (i~i')EgPii' an r -' -, r - r-,

r = #[J\W].

We have the main theorem below immediately if one uses theestimates in Propositions 4.8 and 4.9 together with Lemma 4.10.Here one should consider io := L o and X-I := D(O), X o := M\D,

Xi := B(i), 1 ::; i ::; Lo.

Theorem 4.11. Let us suppose the assumptions (A2 )-(A4 ) and(C1 )-(C8 ). Set f.1 = minl:Sj9o[m(j) - n(j)]. Then, for 1 ::; i 1 ::; L o,we have

uniformly in x belonging to every compact subset of n(id, where we

write i~!g}) for (4.8b) given by replacing i 2 , Si and W respectivelywith 0, Si and {O}, and set

{s-- ~

Si = 0

recall (4.3) for Si.

1 ::; i ::; lo satisfying m(i) - n(i) = f.1,

otherwise;

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4-2. Singularly perturbed Dirichlet problems

We shall suppose the assumption (C9 ) as well as (A 2 )-(A4 ) and(C1)-(C8 ) throughout the rest of this section.

Let us introduce index sets N = {1 :S i :S L; U/ = Ui} andNo = {1 :S j :S l; m(j) - n(j) = J1}, where J1 is defined in Theorem4.11. We set Vo = {x E D;U(x) < Ui}\UiEN Vi. Because of (C9 ),

N ~ {1,···, l}. Set

and denote

i 1 ,i2ENm(i!l=m(i2)

for 0 :S (i :S 1, i E N, where we set m(O) = +00 simply. Then, wedefine -X by

(4.9)

Remark 0 < -X < +00. Indeed, this is shown similarly to Lemma 2.5in [10].

Remembering that <pc is normalized as (1.5), we claim the fol­lowing theorem.

Theorem 4.12. We have

limeJLevo/c2 -Xc = -X/2.dO

Moreover let us suppose that m(l) = ... = m(lo) or n(1) = ... =n(lo) holds and that the minimum in (4.9) is attained uniquely by( = ((i)iEN. Then, for each i EN, we have

(4.10) lim sup I<pC(x) - (il = 0c.j.O xEF

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for every compact subset F in n(io), where we choose the index i o sothat Vi E Cio .

In order to prove this, we need some preparations.From the Stokes' formula with the assumptions (Cd and (C2 ),

the operator (I/, Co(D)) on L2 (D,e- u/c2dx) is semi-bounded and

symmetric. If £c denotes its Friedrichs extension [8, vol. II, p.177],,\c becomes the minimum eigenvalue of _£c and one arrives at theRayleigh-Ritz formula [8, vol.IV, p.82]

(4.11)

where one writes

,\C = E2

inf JC(<p)2 cpECO' (D) 11<p11~ ,

JC(<p) = lvll grad <p112 e-U/c2dx,

11<pllc = 11<p11£2(D,e-u/e2 dx) = {lv l<p12e-U/c2dx } 1/2.

On the other hand, we notice that (4.11) can be rewritten into

by using the principal eigenfunction <pC.In order to show the upper bound estimate:

(4.12)

it suffices to construct a sequence of functions {1pc} C Co (D) satis­fying

(4.13)

from (4.11). Let ( = ((i)iEN attain the minimum in (4.9) andwe set (0 = O. Find rno = min{rn(j);j E No,(j > O}. First,we set 1pc(X) = E{(m(i)-mo )VO}/2 . (i on x E Vi \UMa (c5), c5 > 0,

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where Mer(b) stands for the b-tubular neighborhood of Mer; namely,Mer (b) = {x EM; there exists a geodesic less than b from x meet­ing Mer orthogonally}. Next, define 'IV on Mer(b) and on the otherregion similarly to Sect.3-2 in [10]. Then, one can use the sametechniques as Sect.'s 3-1 and 3-2 in [10] to get

~fc}e-d+(mO-Jj)eU;-/g2 11 "pgll; = v(((j)jENo)'

lime-d+mo+2eu; /g2 Jg("pg) = H(((i)iEN).dO

Hence we obtain (4.13) and so (4.12).

We move to the lower bound estimate. The next lemma is easilyderived from Proposition 2.2 and Lemma 3.3.

Lemma 4.13. (i) Let F be a compact subset of Vo. Then, thereexists l' > 0 such that sUPxEF lepg(x)1 ~ e-r /

g2for all sufficiently

small e > O.(ii) Let F be a compact subset of Vi, i = 1,' ", L. Then, there existsr > 0 such that SUPx,yEF lepg(x) - epg(y)1 ~ e-r /

g2for all sufficiently

small e > 0

Since we know the sharp asymptotics of Ex [T~\ F J (cf.jENo J

Theorem 4.11) and the upper bound estimate (4.12) for )/, we obtainLemma 4.14 below in the same manner as Theorem 3.1; recall (4.2)for the notation Bi .

Lemma 4.14. limg.J-omaXjENo infXEBj epg(x) = 1.

Let us fix bo E Vo and bi E Vi, i EN. One can find the proof ofthe next lemma in [10, Lemma 3.7J.

Lemma 4.15. Let a compactum Mer, 1 ~ a ~ K, and two valleysViI' Vi2 , ill i2 E N U {O}, i1 =J i2, satisfy Mer C Vi! n Vi2 . Suppose

limn-+oo ::: ~~:~~ = ( for some subsequence {en} of {E}. Then, wehave

lim {epgn (bi!)}-2e~d+mo+2eU~/g~ r II grad epgn (x) 112e-U(x)/g~dxn-+oo 1Mo(b)

~ (1- ()2(27r)(d-m o-2)/2 r {H_(Y)}t dy,1M", H+(y)

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for some b > O.

Proof of Theorem 4.12. Let {en} be an arbitrary subsequence of{e}. By replacing the indices of b.'s if necessary, one can find asubsequence {En'} of {en} and 0 ~ 6"",6 ~ 1 such that

(4.14) j = 1" .. , I - 1.

Set (1 = 1, (2 = 6, (3 = 6 . ~2, , (l = n~:,; ~j. For j. = inf{j 2:1; ~j = O}, we have (1 2: (2 2: 2: (j. > 0 = (j.+l = ... = (l. LetI-t. = minl::;j::;j. [m(j) - n(j)] and N. = {j = 1"", Ii m(j) - n(j) =

(m(l) -m(j»)/2I-t.}. Note that Lemma 4.14 implies cpcn' (b 1 ) 2: en' , wherej is chosen so that limn'_Hxlcpcn'(bj ) = 1. From (Cg ),

l l j

eU;- /c2

1Icpc ll; = L L 1. (j) !cpC(x)1 2e-{U(X)-U1-}/C2

dxj=I/3=1 N 13 (0)

+ O(e-r /c2

)

for some b > 0 and r > 0, where N~j) is a compact subset of N~j)

such that the open kernel of Ny) contains Ny). Then, the Laplacemethods imply

(4.15)

l -If one sets Do = D\ Uj =1 Bj , from Lemma 4.13 one has

(4.16)l T<

cpc (b i ) = ~ E bi [cpc (x~< ), x~< E B j ] + ,xcE bi [ f Do cpC (xn dt]L...J DO DO Joj=1

l

= L{cpC(bj ) + O(e-r /c2

)}. Pbi(X~Do E B j ) + O(e- r /c2

)j=1

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for some r > O. By using Proposition 4.7 and Corollary 4.8 withLemma 4.10, for 1 ::; i ::; Land 1 ::; j ::; l, i i= j, we obtain thefollowing asymptotics: if Vi E Cjo for some jo = 1" . " lo,

_ {Pi~j + 0(1), if Vj E Cjo 'Pb(XEE EBJo)= (0) (0)

i T DO (m JO -m J )VO+l{ 0 0 (I)} h .C PZ~J + 0 ,at erW1se;

for some Pi~j 2: 0, where 0(1) means that the remainder term con­verges to 0 as n' ---+ 00. Together with (4.14) and (4.16), one has

where we write

2:= (jPi~j,jEN. :Vj ECjo

2:= (jPi~j,jEN. :m(j)=m(i)

if Vi E Cjo , jo = 1"" ,lo,

Lo

if Vi f- U Cjo ,

jo=l

and note 0 ::; (i ::; 1 for each i. Owing to Lemma 4.15, we obtainthe estimate:

2: lim {<pEn' (bd} -2c~,d+m(1)+2n/~oo

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where mi I i2 = maxM c~n~ma and [ViI' Vi2 ] = {ViI ~ Vi2 } u0: tl t2

{Vi2 ~ ViJ. Combining with (4.15), we conclude

(4.17)

But the RHS of (4.17) is positive. Hence, by comparing (4.17) with(4.12), we obtain that /1* = /1 and that the RHS of (4.17) should beequal to >../2.

Next, let us suppose the assumption of the second assertion. Incase that m(1) = ... = m(lo) , we have (i = (i for i EN, which imme­diately verifies (4.10) by using Lemma 4.13 (ii). If n(1) = ... = n(lo),we can prove this theorem by quite parallel methods to Theorem 3.1in [10]. And so we omit the proof. 0

4-3. Asymptotic relations

In this subsection we exhibit two properties of the asymptotic rela­tions among exit problems and eigenvalue problems: one is a formulaamong the limits of ).cEx [T1] and 'Pc, x E nU), 1 :s: j :s: 1o, and theother appears through the metastable behaviors of (x~I\T' ,Px ). We

D

shall suppose the assumptions (A 2 )-(A4 ), (Cd-(Cg ) and employ thesame notations as those in Sect. 's 4-1 and 4-2.

For the former problem, we suppose that n(1) = ... = n(lo) andthat {'Pc (b j )} converges to some (j E [0,1] for every 1 :s: j :s: 1o.Then, one has the formulae:

(4.18a)

(4.18b)

uniformly in x belonging to every compact subset of n(j). (See The­orems 4.11 and 4.12.) On the other hand, if one sets vc(x) = Ex[Tb],then it satisfies the equation

(4.19) VC = 0 on aD.

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(4.20)

Together with (1.4), we have

>..e l V e cpee-U/e2

dx = -l ve . (£ecpe)e- U/e2

dx

= -l (£eve) . cpee- U/e2

dx = l cpee-U/e2

dx,

where we use the Stokes' formula. However, owing to (4.18), theLaplace methods verify

lo

1· -(d-o(l)) J e -{U-U-}je2 d _ ~ 1. ~ (0(1))ImE cp e 1 x - L.J'-,J L.J Vj' .

e-!-O Dj=1 V,EC:1<J·'<l

J J - -

Hence, together with (4.20), we obtain the formula:

(4.21 ) =1.

Remark 4.16. If l = 1, the above formula may be rewritten into>..ii~O})/2 = 1. Indeed, this is already known as Theorem 3.6. Inthis sense, the formula (4.21) can be regarded as an extension ofthe above theorem, although the model is restricted to the strictgradient type and the convergence of >..e, cpe and Ex [Tn] may not beexponentially fast.

We move to the latter problem. We suppose simply that Jl =m(j) - o(j) for all 1 ::; j ::; lo. By using the notations in Lemma 4.10,set J = {O, 1,· .. , Lo}, W = {O, 1,· .. , lo}, W j = W\ {j}, 1 ::; j ::; lo,and

1 ::; i ::; lo, 0::; j ::; lo, i i= j,

1 ::; i = j ::; lo,

lo + 1 ::; i ::; Lo, 0::; j ::; lo,

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1 ~ JI ~ lo, 0 ~ 12 ~ lo, jl # 12,1 ~ JI = 12 ~ lo,

and, for 1 ~ j ~ lo,

L:gE~J (W) 1r(g)

L:gE~J (Wj) 1r(g) .

Then, from Propositions 4.5, 4.7 and Lemma 4.10, we have for lo +1 ~ i ~ Lo and 0 ~ j ~ lo

uniformly in x belonging to every compact subset of n(i), and for1 ~ jl ~ lo and 0 ~ 12 ~ lo, jl # 12,

uniformly in x belonging to every compact subset of nUI), where

Do = D\ U B(j),l:::;j:::;lo

Dh = D\ U B(h).l:::;h:::;lo,jd=jl

Set N(O) - aD N(j) - U NU') 1 < J' < 1- , - V/EC:n(j')=n(j) 1<)"<1 ,- - 0,J J ,- -

and JR = {N(O) N(I) ... N(lo)} Furthermore noting t· > 0 we, ". ,),define

10

Qf(N(j)) = f;1 L qjh{f(N(h)) - f(N(j))},h=O

f E B(JR),

for N(j) E JR, where we set to1 = 0 simply and B(JR) stands forthe space of all bounded functions on lR. We write (Xt, p(i)), 1 ~

i ~ L o, for the Markov jump process realized on some probabilityspace (n,F,p) generated by Q satisfying p(i)(XO = N(j)) = Qij,o~ j ~ lo. Then, one can find the following theorem in [10, Theo­rem 3], although they have considered the case without a boundarycondition. Here we do not need the assumption (C9 ) for the proof.

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Theorem 4.17. Let us suppose (A2 )-(A4 ) and (Cd-(Cs) and as­

sume that m(j) - n(j) = J.L for all ) = 1,···, lo. If one sets (Xc =EJ.LeVO /c

2

and Yt = x(ta€ )l\To' t ~ 0, then for every 0 < t 1 < ... < tN,

1 .s io .s La, 1 .s )1,·· ·,)N-1 .s lo and 0 .s )N .s lo it holds that

limPx(Y: E BJ"J'· .. , y: E BJ"N)c~O J N

= p(io)(XtJ = NUd, ... ,XtN = NUN))

uniformly in x belonging to every compact subset of n(io).

Remark 4.18. We have i~JO}) = E(i)[inf{t > O;Xt = N(O)}] for1 .s i .s La, where E(i) stands for the expectation with respect top(i) .

Theorem 4.19. Let us suppose (A 2 )-(A4 ) and (Cd-(Cg ) and as­sume that (Xt , p(i)), 1 .s i .s La, is irreducible; namely, for every1 .s )0 .s lo and 0 .s )N .s la, there exist 1 .s )1, .. ·,)k .s lo such thatqjojl qjd2·· ·qjk-IjkqjdN > o. Then, the principal eigenvalue of theproblem (4.22) below is >../2:

(4.22) <p(N(O)) = 0,

and the eigenspace associated to >../2 is one-dimensional. In partic­ular, if ({; denotes the principal eigenvector, then we have

(4.23)

uniformly in x belonging to every compact subset of n(i), where we

normalize ({; as max1:Sj:s/o ((;(N(j)) = 1.

Proof. For the sake of irreducibility, we have mel) = ... = m(lo)

and so n(1) = ... = n(lo). Let us define a measure a on lB by- (") (0(1»). - (0)

a(N J ) = LVjIECj:l:Sjl:S1 Vjl ,1 .s J .s la, and a(N ) = O. Then,

we have a(N(j)) > 0 for 1 .s) .s lo and

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for <fJ E B(lB) with <fJ(N(O)) = O. Here we write

- ~ 2 1H((I,···, (1 0 ) = L ((h - (h) . (InCh))

0-::;12 <jl-::;lo LVi' ECh ,vi" itCh H(Vi,-tVi")

LgE~J . (WJ·1

) 7f(9)X J1J2

LgE~J(W) 7f(9) ,

in which we set (0 = 0 simply. For 0 :S (1,· .. , (10 :S 1, one can easilyshow that

and that the minimum is attained uniquely by (i = L~~o Qij(j,

Lo + 1 :S i :S L o, where one writes (I, = (i for Vi' E Ci . Hence, itholds that

(4.24)A2

mm<pEB(lBJ:<p(NCO} )=0

i.e., the minimum eigenvalue of -9 is A/2. Let r.j; denote an associ­ated eigenvector satisfying maxI-::;j-::;lo r.j;(N(j)) = 1. Owing to (4.24),one can suppose that r.j; is non-negative. Note that the irreducibilityimplies p(i!)(Xt = N(12)) > 0 for all t > 0 and 1 :S jI,h :S LooThen, if one takes h so that r.j;(N(12)) = 1, one has

r.j;(NUd) =EUI) [r.j;(Xt )] + ~E(jI)[ t r.j;(Xs ) ds]2 Jo

2r.j;(N(12))pUl)(Xt = N(12)) > O.

Therefore, since it is impossible that two of them are orthogonal withrespect to a, the eigenspace of A/2 is one-dimensional.

Because of the above arguments, (4.23) immediately follows fromTheorem 4.12. 0

Finally, we consider the asymptotics of AI, ... , Alo . Here {AUdenotes the sequence of the eigenvalues of the Dirichlet boundaryproblem (1.4) such that 0 < Al < A~ :S A3 :S ... , and <fJk is theeigenfunction associated to Ak, k = 1,2,···. We normalize <fJk assUPxED l<fJk(X) I = 1. We also write 0 < Al :S A2 :S ... :S Alo for allthe eigenvalues of the problem (4.22), where we remark Al = A/2.

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Theorem 4.20. Let us suppose (A 2 )-(A4 ) and (C1)-(C9 ). We alsoassume m(1) = ... = m(lo) and n(1) = ... = n(lo) (instead ofirreducibility) .(i) We have lime-!-o cJLevo/e2 Ak = Ak for k = 1, ... ,10 .

(ii) For every subsequence {cn} of {E}, there exist a subsequence{En /} of {En} and eigenvectors 'PI, ... , 'Plo associated to AI,···, Alo'respectively, such that fllJ 'Pk'Pk' da = 0 for 1 S; k < k' S; 10 and that

uniformly in x belonging to every compact subset ofO,(i), 1 S; i S; Lo.

Proof. We shall use induction on k = 1, ... ,10 . From Theorem 4.12and its proof, we know the statement for k = 1.

Let us suppose that it holds for 1, ... , k - 1. Then, by applyingthe Rayleigh-Ritz formula [8, vol.IV, p.82], it is derived that

(4.25) lim c~,evo/e~, A~n' S; Ak.n/ -too

Indeed, for an eigenvector ¢k associated to Ak such that fllJ ¢k'Pk' da= 0, k' = 1,···, k - 1, it suffices to construct a function which ap­proximates L~~oQij¢k (N(j)) on each Uv

i, EC

iVi'·

By using (4.25), we have similar estimates in Lemma 4.13 for lpkas well as lpe. From Proposition 4.7, for compact subsets F of O,(i) ,

i E N, one can also show that SUPx,yEF Ilpk(x) - lpk(y)1 S; CE for

some C > O. Let us choose points hi E UVi , ECi Vi" ho E Vo and fixthem. By replacing the subsequence {En'} if necessary, there exist-1 S; "Ij S; 1,1 S; j S; 10 , such that limn'-toolp~n/(bj) = "Ij andthat maxl~j90 l"Ijl = 1. We set 'Pk(N(j)) = "Ij, 1 S; j S; 10 , and- (N- (0)) - 0 Th h 1· en' (b

A

) - ~Io - dlpk -. en we ave Imn'-too lpk i - L.Jj=o qij"lj an

(4.26)

On the other hand, since lpk is orthogonal to lpI, ... , lpk-l with re­

spect to e-U/e2

dx, 'Pk is also orthogonal to 'PI, ... , 'Pk-l with respect

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to da. Hence, together with (4.25), the RHS of (4.26) should be equalto A.k and we conclude that (/Jk is an eigenvector associated to A.k.o

Acknowledgement. The author would like to express his to Pro­fessor T. Funaki for valuable suggestions and kind encouragements.

References

[1] M. V. Day, On the exponential exit law in the small parameterexit problem, Stochastics 8 (1983), 297-323.

[2] A. Devinatz and A. Friedman, Asymptotic behavior of the prin­cipal eigenfunction for a singularly perturbed Dirichlet problem,Indiana Univ. Math. J. 27 (1978), 143-157.

[3] M. I. Freidlin and A. D. Wentzell, "Random perturbations ofdynamical systems," Springer-Verlag, Berlin Heidelberg NewYork, 1984.

[4] A. Friedman, "Stochastic differential equations and applica­tions, Volume 2," Academic Press, New York, 1976.

[5] D. Gilbarg and N. S. Trudinger, "Elliptic partial differentialequations of second order, 2nd ed.," Springer-Verlag, BerlinHeidelberg New York, 1983.

[6] P. Mathieu, Zero white noise limit through Dirichlet forms.Application to diffusions in a random medium, Probab. The­ory Relat. Fields 99 (1994), 549-580.

[7] J. Palis, Jr. and W. de Melo, "Geometric theory of dynamicalsystems, An introduction," Springer-Verlag, Berlin HeidelbergNew York, 1982.

[8] M. Reed and B. Simon, "Methods of modern mathematicalphysics, Volumes II, IV," Academic Press, New York, 1975,1978.

[9] Z. Schuss, "Theory and applications of stochastic differentialequations," John Wiley & Sons, New York, 1980.

[10] M. Sugiura, Metastable behaviors of diffusion processes withsmall parameter, J. Math. Soc. Japan. 47 (1995), 755-788.

[11] M. Sugiura, Exponential asymptotics in the small parameterexit problem, Nagoya. Math. J. 144 (1996), 137-154.

[12] M. Williams, Asymptotic exit time distributions, SIAM J.Appl.Math. 42 (1982), 149-154.

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Department of Mathematical SciencesFaculty of ScienceUniversity of the RyukyusNishihara-cho, Okinawa 903-0213JAPAN

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