Asymptotic expansion of stochastic oscillatory integrals with rotation invariance

Ann. Inst. Henri Poincaré, Vol. 35, n◦ 4, 1999, p. 417-457.

Probabilités et Statistiques

Asymptotic expansion of stochastic oscillatory integrals with rotation invariance by

Naomasa UEKI 1 Department of Mathematics, Faculty of Science, Himeji Institute of Technology, Shosha 2167, Himeji 671-2201, Japan Manuscript received on 19th March 1997, revised 17th December 1998

A BSTRACT. – Asymptotics of oscillatory integrals on the classical 2-dimensional Wiener space, whose phase functional is the stochastic line integral of a 1-form, is considered. Under the assumption that the exterior derivative of the 1-form is rotation invariant, an asymptotic expansion is obtained. This result is extended to the process associated to a general rotation invariant metric.  Elsevier, Paris Key words: Wiener integrals, Asymptotic expansions, Oscillatory integrals, Diffusion processes

R ÉSUMÉ. – Nous étudions le comportement asymptotique d’intégrales oscillantes sur l’espace de Wiener classique en dimension 2, lorsque la phase est l’intégrale stochastique d’une 1-forme. Sous l’hypothèse que la dérivée extérieure de la 1-forme est invariante par rotation, nous obtenons ce développement asymptotique. Nous étendons ce résultat à la diffusion associée à une métrique invariante par rotation.  Elsevier, Paris

1 Present address: Graduate School of Human and Environmental Studies, Kyoto

University, Kyoto 606-8501, Japan. E-mail: [email protected]. Annales de l’Institut Henri Poincaré - Probabilités et Statistiques - 0246-0203 Vol. 35/99/04/ Elsevier, Paris

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1. INTRODUCTION Let (W, P ) be the 2-dimensional Wiener space: W = {w ∈ C([0, ∞) → R ): w(0) = 0} and P is the Wiener measure on W . Let b be a C 1 differential 1-form on R2 satisfying the following rotation invariance condition: 2

db = f (|x|) dx 1 ∧ dx 2 .

(1.1)

In this paper we investigate the asymptotic behavior of "

I (ξ ; a) = E exp iξ

Z1

#
b w(t) ◦ dw(t) a w(·) w(1) = 0 




(1.2)

0

and "



I(ξ ; a) = E exp iξ

Z1






#


b w(t) ◦ dw(t) a w(·)

(1.3)

0

√ as ξ → ∞, where i = −1 and ◦ dw(t) is the Stratonovitch stochastic integral. Under some conditions on the function f in (1.1) and the amplitude functional a, we give an asymptotic expansion as I (ξ ; a) ∼ exp − ξf (0)/2 −

p


2

(

ξ µ10 − µ0 ξ f (0)a(0) +

∞ X

)

ξ −n/2 In

n=1

(1.4) and a similar expansion for I(ξ ; a). Moreover we extend these results to the case, where the Wiener process w is replaced by the process associated to a general rotation invariant metric and the 1-form b may degenerate finitely at the origin. The integrals (1.2) and (1.3) are called stochastic oscillatory integrals. These integrals appear, for example, in the study of the Schrödinger operator with a magnetic field and the 2-form db in (1.1) corresponds to the magnetic field (cf. [21]). For these integrals the asymptotic behavior is estimated from above in more general setting [3,4,12–14,29]. For the exact leading term, Ikeda–Manabe studied the following two cases: one is the case where the phase functional is a quadratic functional, and the other is the above rotation invariant case [9]. For the quadratic case, we have a general exact formula and many related results are obtained. For this aspect, see [7,9,14,23,24,26,27] and references therein. However, as Annales de l’Institut Henri Poincaré - Probabilités et Statistiques

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opposed to finite-dimensional oscillatory integrals, it is difficult to obtain the asymptotic behavior for more general phase functionals from these results. On the other hand, the rotation invariant case is the simplest nonquadratic case for which the asymptotic behavior can be investigated in detail. In this paper we will extend the results in [9] on the rotation invariant case. As for the asymptotic expansion, Ben Arous gave√for a function corresponding to (1.3), where w(s) is replaced by w(s)/ ξ in more general setting without rotation invariance conditions [1]. Under the rotation invariance condition (1.1), by taking the expectation in the spherical direction, our problem is reduced to that on Laplace type asymptotic behavior. For example, I (ξ ; 1) is written as "

ξ2 I (ξ ; 1) = E exp − 2

Z1


2

F |w(t)|

! # dt w(1) = 0 ,

(1.5)

0

where F (r) =

Z1

f (ur)ur du. 0

For more general cases, see Lemmas 2.1 and 3.1 below. By the Feynman– Kac formula the right hand side is related to the integral kernel of the heat semigroup of the Schrödinger operator −1 + ξ 2 F 2 . Then the problem on the asymptotic behavior corresponds to a problem on the semiclassical approximation. We now use the technique of Simon [22] for the asymptotic expansion of the eigenvalues and eigenfunctions in the semiclassical limit. For this Laplace type asymptotic behavior, more general results are obtained for the case where w(s) is replaced by w(s)/ξ . For these results, see [2] and references therein. The aim of Ikeda–Manabe [9] was to show an infinite dimensional analogue of the principle of the stationary phase, in the setting of Wiener functional integrals. In fact, we can formally regard our results as examples of this principle. For this aspect, see Remark 3.2(i) below. For a general rotation invariant metric dr ⊗ dr + g(r)2 dθ ⊗ dθ in terms of the polar coordinate (r, θ), r > 0, θ ∈ S 1 , we reduce the problem to that on the radial process and use techniques of the theory of the 1-dimensional diffusion processes and ordinary differential equations. Vol. 35, n◦ 4-1999.

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Since the asymptotic behavior of our oscillatory integrals are determined by the behavior of the metric g(r) and the 1-form b at the boundary r = 0, we need precise analysis at this boundary. Under the condition that limr→0 g(r)/r α = 1 for some α ∈ R, r = 0 is an entrance boundary point for the radial process if α > 1, a regular boundary point if α ∈ (−1, 1), and an exit boundary point if α 6 −1, in the sense of Feller [5]. Accordingly we pose the Neumann boundary condition at r = 0 for α > −1 and the Dirichlet boundary condition for α 6 1. If we moreover assume that g(r)/r α = 1 + o(r 2 ) as r ↓ 0, then the generator is regarded as a simple perturbation of that for the case g(r) = r α . Then we use the explicit representation of the transition density of the Bessel process with index α + 1 for α > −1 and the corresponding representation for the Dirichlet case with α 6 1 (see Lemma 5.1 below). When this condition is not satisfied, the existence of the function such as I (ξ ; a) in (1.3) is not trivial. However, by a work of Hille [6] (see also Matsumoto [15]) on the continuity of the transition density at the boundary, we obtain a sharp result for the case of the Neumann condition with −1 < α < 3, which corresponds to the limit circle case (see Theorem 6 below). The organization of this paper is as follows: In Section 2, we treat a fundamental case for the function I (ξ ; a) in (1.2) with a simple amplitude functional a. In Section 3, we consider more general amplitude functionals a for both functions I (ξ ; a) and I(ξ ; a). In this section, we also give a fundamental remark for our problem (see Remark 3.2 below). In Sections 4 and 5, we consider the case of a general rotation invariant metric. Section 4 is devoted to the case of Neumann condition and Section 5 is devoted to the case of Dirichlet condition. 2. A FUNDAMENTAL CASE On R2 with the standard metric, we take a C 1 differential 1-form b satisfying the following condition: (A.1) (i) db = f (r) dx 1 ∧ dx 2 , r = |x|; (ii) f (0) > 0; (iii) f (r) > 0; (iv) limr→∞ r 1−ε f (r) > 0 for some ε > 0; (v) f is smooth on the open interval (0, ∞) and each derivative of f is dominated by a polynomial. We take a functional a on the Wiener space W . For this functional, we introduce the following conditions: Annales de l’Institut Henri Poincaré - Probabilités et Statistiques

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(A.2) a(w) = A(r(t1 ), r(t2 ), . . . , r(tK )), where r(·) = |w(·)|, 0 < t1 < t2 < · · · < tK < 1 and A is a smooth function on [0, ∞)K whose derivatives are dominated by polynomials. Let µ0 < µ1 6 µ2 6 · · · be the eigenvalues of the harmonic oscillator 

H=

−1 1 f (0)r + 2 2 2

2

on L2 (R2 ), ϕ0 , ϕ1 , ϕ2 , . . . be the corresponding normalized eigenfunctions, and P⊥ be the orthogonal projection to the orthogonal compleP ment to the ground states, where 1 = 2j =1 ∂ 2 /∂(x j )2 . In this case, µ0 = f (0)/2 and ϕ0 can be taken as s





f (0) 2 f (0) exp − r . 2π 4

ϕ0 (x) =

We denote the norm of L2 (R2 ) by k · k and the inner product by (·, ·). For the function I (ξ ; a), ξ ∈ R, defined in (1.2), we prove the following in this section: T HEOREM 1. – Under the conditions (A.1)–(A.2), we have the following asymptotic expansion as ξ → ∞: I (ξ ; a) ∼ exp −ξ µ0 −

∞
X

p

ξ µ10 − µ20 ξ

ξ −n/2 In ,

n=0

in the sense that I (ξ ; a) exp ξ µ0 +

p

ξ µ10 + µ20


1

ξ

−

N−1 X

ξ −n/2 In = O ξ −N/2




n=0

for any N > 1, where f (0) , 2 s π f 0 (0) 1 µ0 = , 2 2f (0) µ0 =



µ20

f 00 (0) 2f 0 (0) + = 2f (0) 3f (0)

I0 = f (0)a(0), Vol. 35, n◦ 4-1999.

2

2

−

f (0)f 0 (0) k(H − µ0 )−1/2 P⊥ r 3 ϕ0 k 6

,

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and In , n ∈ N, are finite linear combinations of {(∂ α A)(0): α ∈ ZK +} whose coefficients are polynomials of 




f (m) (0), ϕ0 , r `(0)(µ0 − H )−1 P⊥ · · · r `(m−1) (µ0 − H )−1 P⊥ r `(m) ϕ0 :

m, `(0), `(1), . . . , `(m) ∈ Z+ . We first show the following: L EMMA 2.1. – "

Zξ

1 I (ξ ; a) = E exp − 2

F r(t), ξ


2

0

! 

#



r(ξ ·) dt A √ w(ξ ) = 0 ξ

(2.1)

where F (r, ξ ) =

Z1 0





ur f √ ur du. ξ

Proof. – By the stochastic version of Stokes’ theorem (cf. [8,25]), we have Z1




b w(t) ◦ dw(t) =

0

Z1




f1 r(t) ◦ dS(t) +

0

Z1




b uw(1) · w(1) du, (2.2)

0

where f1 (r) =

Z1

f (ur)2u du 0

and S(t) is Lévy’s stochastic area defined by 1 S(t) = 2

Zt





w 1 (s) dw 2 (s) − w 2 (s) dw 1 (s) .

0

As is explained in Section VI-6 of [10], the stochastic area is represented as 1 S(t) = B 4

!

Zt 2

r(s) ds ,

(2.3)

0

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where B(·) is a 1-dimensional Brownian motion independent of the process {r(·)}. By taking the expectation in the Brownian motion B(·), we have "

ξ2 I (ξ ; a) = E exp − 2

Z1


2

F r(t), 1

!

#
dt A r(·) w(1) = 0 .

(2.4)

0

By using the scaling property of the Brownian motion 



w(ξ ·) {w(·)} ∼ √ , ξ L

2

we obtain (2.1).

We introduce an operator H (ξ ) =


1 −1 + F (r, ξ )2 , 2

where F (r, ξ ) is the function defined in Lemma 2.1. This operator is essentially self adjoint on C0∞ (R2 ) by Theorem X.28 in [18]. We denote the self adjoint extension by the same symbol. The integral kernel e−t H (ξ ) (x, x 0 ), t > 0, x, x 0 ∈ R2 of the heat semigroup e−t H (ξ ) , which we call the heat kernel of H (ξ ) in the following, has the following representation: −t H (ξ )

e

"




1 x, x = E exp − 2 0

Zt

F r(s), ξ


2

! # 0 ds x + w(t) = x

0

exp(−|x − x 0 |2 /(2t)) 2π t (cf. [21] Theorem 6.2). Thus (2.4) is rewritten as follows: ×

I (ξ ; a) = 2π ξ

Z

e−ξ t1 H (ξ ) (0, x1 )e−ξ(t2 −t1 )H (ξ )(x1 , x2 )

R2K

× · · · × e−ξ(tK −tK−1 )H (ξ ) (xK−1 , xK )e−ξ(1−tK )H (ξ ) (xK , 0)   rK r1 r2 × A √ , √ , . . . , √ dx1 dx2 · · · dxK . ξ ξ ξ On the other hand, by the condition (A.1) (iv), the operator H (ξ ) has purely discrete spectrum (cf. [19] Theorem XIII.67). We denote the Vol. 35, n◦ 4-1999.

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eigenvalues by µ0 (ξ ) 6 µ1 (ξ ) 6 µ2 (ξ ) 6 · · · and the corresponding normalized eigenfunctions by ϕ0 (x, ξ ), ϕ1 (x, ξ ), ϕ2 (x, ξ ), . . . . Since the heat semigroup e−t H (ξ ) generated by H (ξ ) is positivity improving, we see that µ0 (ξ ) < µ1 (ξ ) and we can take ϕ0 (x, ξ ) as a positive function (see [19] Theorem XIII.44). By Mercer’s expansion theorem, the heat kernel e−t H (ξ ) (x, y) is represented as the following convergent series: −t H (ξ )

e

(x, y) =

∞ X

e−t µn (ξ ) ϕn (x, ξ )ϕn (y, ξ ).

(2.5)

n=0

For the asymptotic behavior as ξ → ∞, we follow the argument of Simon [22]. We first show the following: P ROPOSITION 2.1. – For each n > 0, limξ →∞ µn (ξ ) = µn . Proof. – For the upper bound, we use the Rayleigh–Ritz principle:


µn (ξ ) 6 the largest eigenvalue of (ϕj , H (ξ )ϕk )

16j,k6n .

Since each ϕj is a polynomial times a Gaussian function, we easily see that ϕj , H (ξ )ϕk








1 = µj δj k + O √ . ξ

Therefore we have lim µn (ξ ) 6 µn .

ξ →∞

We next show the lower bound. We take smooth functions {h, k} on R2 so that h2 + k 2 ≡ 1, supp h ⊂ {|x| < 2} and supp k ⊂ {|x| > 1}. For any ξ > 0, we set hξ (x) = h(x/ξ 1/10) and kξ (x) = k(x/ξ 1/10 ). By the Ismagilov–Morgan–Sigal–Simon localization ([22] Lemma 3.1), we have








ϕ, H (ξ )ϕ > hξ ϕ, H (ξ )hξ ϕ + kξ ϕ, H (ξ )kξ ϕ −

C1 kϕk2 ξ 1/5

for some C1 > 0 and any ϕ ∈ C0∞ (R2 ). For the first term, we easily see that


hξ ϕ, H (ξ )hξ ϕ > (hξ ϕ, H hξ ϕ) −

C2 kϕk2 . ξ 1/5

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For the second term, we estimate as follows: 2

2 1 
1 kξ ϕ, H (ξ )kξ ϕ > F (r, ξ )kξ ϕ > inf F (r, ξ ) kkξ ϕk2 . 2 2 r>ξ 1/10

By the conditions (A.1) (ii)–(iv), we can estimate as inf F (r, ξ ) > C3 ξ 1/10

r>ξ 1/10

for some C3 > 0. We now take µ ∈ (µn , µn+1 ) and set G = hξ (H − µ)P [µ]hξ , where P [µ] is the orthogonal projection onto the eigenspace of H corresponding to the eigenvalues less than µ and hξ is regarded as a multiplication operator. Then, by the above estimates, we have


ϕ, H (ξ )ϕ > µkϕk2 + (ϕ, Gϕ) −

C4 kϕk2 ξ 1/5

for large enough ξ . Since the rank of G is at most n, we have µn+1 (ξ ) > µ −

C4 ξ 1/5

by the min-max principle. From this we obtain lim µn (ξ ) > µn .

(2.6)

ξ →∞

2 Moreover, by the same procedure as in Simon [22], we have the following: P ROPOSITION 2.2. – (i) As ξ → ∞, we have µ0 (ξ ) ∼ µ0 +

∞ X µn0 n=1

√ n, ξ

where µ10 , µ20 are those of Theorem 1. (ii) As ξ → ∞, we have ϕ0 (x, ξ ) ∼ ϕ0 (x) +

∞ X ϕ0n (x) n=1

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√ n , ξ

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N. UEKI

where ϕ0n , n ∈ N, are continuous functions such that r m ϕ0n ∈ L2 for any m > 0, in the following sense: N−1 X ϕ n (x)
0 sup ϕ0 (x, ξ ) − ϕ0 (x) − √ n = O ξ −N/2 ξ x∈K

(2.7)

n=1

for any compact set K in R2 , and

( ) N−1

X ϕ0n (x)


m

ϕ0 (x, ξ ) − ϕ0 (x) − √ n = O ξ −N/2

r

ξ

(2.8)

n=1

for any m > 0. Proof. – The proof of (i) and (2.8) with m = 0 is identical to that of Simon [22] Theorem 4.1. The rest is also proven by modifying slightly the proof of Simon [22] Theorem 4.1. In fact, if we write as ϕ0 (x, ξ ) = ϕ0 (x) +

N−1 X n=1

ϕ0n (x) ϕ0[N] (x, ξ ) √ n + √ N , ξ ξ

then ϕ0n are linear combinations of Z |z−µ0 |=ε

1 (z − H )−1 r k(1) (z − H )−1 r k(2) · · · (z − H )−1 r k(m) ϕ0 dz, (z − µ0 )

m, k(1), k(2), . . . , k(m) ∈ Z+ and ϕ0[N] is also some polynomial of the functions dominated by functions of this type and Z |z−µ0 |=ε

 N
−1 1 (z − H )−1 V (r, ξ ) z − H (ξ ) ϕ0 dz, (z − µ0 )

where V (r, ξ ) = F (r, ξ )2 − (f (0)r/2)2 and ε > 0 is taken small enough. Then to prove (2.8), as in the proof of Theorem 4.1 of [22], it is enough to show the subsequent Lemma 2.2. To prove (2.7), by the Sobolev lemma, it is enough to show that

( ) N−1

X ϕ0n (x)


√ n = O ξ −N/2

(1 − 1)ζ ϕ0 (x, ξ ) − ϕ0 (x) −

ξ n=1

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for any ζ ∈ C0∞ (R2 ). This is proven by the subsequent Lemmas 2.2 and 2.3. 2 L EMMA 2.2 (Lemma 4.2 of [22]). – For each fixed m ∈ Z+ , we have

sup 1 + r 2


m/2

|z−µ0 |=ε

(z − H )−1 1 + r 2


−m/2

op

<∞

and

sup 1 + r 2


m/2

|z−µ0 |=ε


−1

z − H (ξ )

1 + r2


−m/2

op

< ∞,

where k · kop is the operator norm and ε > 0 is taken small enough. L EMMA 2.3. – For any ζ ∈ C0∞ (R2 ), we have



sup (1 − 1)ζ(z − H )−1 op < ∞

|z−µ0 |=ε

and


−1

sup (1 − 1)ζ z − H (ξ )

|z−µ0 |=ε

op

< ∞.

Proof of Theorem 1. – We decompose as I (ξ ; a) = I0 (ξ ; a) + IR (ξ ; a), where −ξ µ0 (ξ )

I0 (ξ ; a) = 2π ξ e

Z 2

ϕ0 (0, ξ )

R2K



ϕ0 (x1 , ξ )2 ϕ0 (x2 , ξ )2 · · · ϕ0 (xK , ξ )2 

rK r1 r2 × A √ , √ , . . . , √ dx1 dx2 · · · dxK ξ ξ ξ and IR (ξ ; a) = I (ξ ; a)−I0 (ξ ; a). I0 (ξ ; a) is expanded by Proposition 2.2 and IR (ξ ; a) is negligible by the subsequent Lemma 2.4. 2 L EMMA 2.4. – lim

ξ →∞

1 log |IR (ξ ; a)| < −µ0 . ξ

Proof. – Since A is of polynomial growth, it is enough to show that 1 log |I (ξ ; α1 , α2 , . . . , αK )| < −µ0 , ξ →∞ ξ lim

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(2.9)

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N. UEKI

for any m ∈ {0, 1, . . . , K}, α1 , α2 , . . . , αK ∈ Z+ , where I (ξ ; α1 , α2 , . . . , αK ) = e−ξ tm µ0 (ξ ) ϕ0 (0, ξ )

Z

ϕ0 (x1 , ξ )2 ϕ0 (x2 , ξ )2 · · · ϕ0 (xm−1 , ξ )2 ϕ0 (xm , ξ )

R2K −ξ(tm+1 −tm )H (ξ )

× e

(xm , xm+1 )

−ξ(tm+1 −tm )µ0 (ξ )

−e




ϕ0 (xm , ξ )ϕ0 (xm+1 , ξ )

× e−ξ(tm+2 −tm+1 )H (ξ )(xm+1 , xm+2 ) · · · e−ξ(1−tK )H (ξ ) (xK , 0) × r1α1 r2α2 · · · rKαK dx1 dx2 · · · dxK .

(2.10)

We take δ > 0 small enough. By using (2.5) and Schwarz’s inequality we estimate as follows: e−ξ t H (ξ ) (x, y)

v v u∞ u∞ uX uX −ξ t µ (ξ ) 2 t 0 e ϕ0 (x, ξ ) t e−ξ t µ0 (ξ ) ϕ0 (y, ξ )2 6 n=0

n=0

q

q

6 e−(ξ t −δ)µ0 (ξ ) e−δH (ξ ) (x, x) e−δH (ξ ) (y, y) and −ξ t H (ξ ) e (x, y) − e−ξ t µ0 (ξ ) ϕ0 (x, ξ )ϕ0 (y, ξ ) v v u∞ u∞ uX uX t −ξ t µ (ξ ) 2 0 6 e ϕ0 (x, ξ ) t e−ξ t µ0 (ξ ) ϕ0 (y, ξ )2 n=1 −(ξ t −δ)µ1 (ξ )

6e

n=1

q

e−δH (ξ ) (x, x)

q

e−δH (ξ ) (y, y).

Moreover we use the following: for each k ∈ N,



sup ϕ0 (x, ξ )r k < ∞

(2.11)

ξ

and there are C > 0 and N ∈ N such that Z




e−δH (ξ ) (x, x)r k dx 6 C 1 + ξ N .

(2.12)

R2

(2.11) is deduced from Proposition 2.2 and (2.12) can be shown as follows: by using the Feynman–Kac formula, Jensen’s inequality, the Annales de l’Institut Henri Poincaré - Probabilités et Statistiques

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ASYMPTOTIC EXPANSION

translation invariance of the Lebesgue measure and the assumption (A.1) (iv), we estimate as follows: Z

e−δH (ξ ) (x, x)r k dx

R2

6

Z R2

=E

rk dx E 2π δ 2 " Zδ

Z

dt 0

(

6 C1 1 +

R2 Z∞

" Zδ 0



#




2 δ w(δ) = 0 dt exp − F |x + w(t)|, ξ 2 



#

|x − w(t)|k δ 2 dx exp − F (r, ξ ) w(δ) = 0 2π δ 2 2 

dr · r

0
N

k+1

δ exp − F (r, ξ )2 2

)

6 C2 1 + ξ

for some N > 0. Now we can show (2.9) by using Schwarz’s inequality and these estimates in the right hand side of (2.10). 2 3. OTHER AMPLITUDES In this section we consider more general amplitude functionals. We first give a fundamental theorem for the function I(ξ ; a), ξ ∈ R, defined in (1.3). For this we introduce the following conditions: (A.3) b(0) = 0, b is smooth, and each derivative of b is dominated by a polynomial. (A.4) a(w) = A(r(t1 ), r(t2 ), . . . , r(tK ))C(w(1)), where A(r(t1 ), r(t2 ), . . . , r(tK )) is same as in (A.2) and C is a smooth function on R2 whose derivatives are dominated by polynomials. Then we have the following: T HEOREM 2. – Under the conditions (A.1), (A.3) and (A.4), we have the following asymptotic expansion as ξ → ∞: I(ξ ; a) ∼ exp − ξ µ0 − µ20

∞
X

ξ −n In ,

n=0

in the same sense as √ in Theorem 1, where µ0 , µ20 are same as in Theorem 1, I0 = a(0)/ det{E2 − i(∇b(0) +t ∇b(0))/f (0)}, E2 is the Vol. 35, n◦ 4-1999.

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N. UEKI

2 × 2 identity matrix and In , n ∈ N, are finite linear combinations 2 of {(∂ α A)(0) × (∂ β C)(0): α ∈ ZK + , β ∈ Z+ } whose coefficients are polynomials of 




f (2m) (0), ϕ0 , r `(0) (µ0 − H )−1 P⊥ · · · r `(m−1) (µ0 − H )−1 P⊥ r `(m) ϕ0 :

m, `(0), `(1), . . . , `(m) ∈ Z+ . Remark 3.1. – Under the condition (A.3), f 0 (0) = µ10 = 0. To consider more general amplitude functionals, we introduce the following conditions: (A.5) limr→∞ f (r) > 0. (A.6) a = A(r(·)) ∈ L1 (P (·|w(1) = 0)), limkr(·)k2 →0 A(r(·)) = A(0) and |A(0)| < ∞, where k · k2 is the norm on the L2 space on the interval [0, 1]. (A.7) a = A(r(·))C(w(1)) where A ∈ L1 (P ), |A(0)| < ∞, lim

kr(·)k2 →0




A r(·) = A(0),

C is C 1 and C, ∇C are bounded. Then we have the following: T HEOREM 3. – Under the conditions (A.1), (A.5) and (A.6), we have I (ξ ; a) = exp − ξ µ0 −

p







ξ µ10 − µ20 ξ I0 + o(1) ,

as ξ → ∞, where µ0 , µ10 , µ20 and I0 are given in Theorem 1. T HEOREM 4. – Under the conditions (A.1), (A.3), (A.5) and (A.7), we have





I(ξ ; a) = exp − ξ µ0 − µ20 I0 + o(1) , as ξ → ∞, where µ0 , µ10 , µ20 , I0 are given in Theorem 2. Moreover, we introduce the following condition: (A.8) a = A(r(·)), A > 0, sup A < ∞ and inf{A(r(·)): sup r(t) 6 R} > 0 for some R > 0. [0,1]

Then, for the function I (ξ ; a), we have the following: T HEOREM 5. – Under the conditions (A.1) and (A.8), we have 1 f (0) log I (ξ ; a) = − . ξ →∞ ξ 2 lim

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ASYMPTOTIC EXPANSION

We here give general remarks: Remark 3.2. – (i) As is explained in Ikeda–Manabe [9], our results are regarded formally as an infinite-dimensional example of the principle of the stationary phase. We here discuss this aspect. We assume b(0) = 0. On the Cameron–Martin subspace H of W , we consider a functional β(h), h ∈ H , defined by β(h) =

Z1




b h(s) · dh(s).

0

This functional is sometimes called the skeleton of our phase functional. For k ∈ H such that k(1) = 0, the derivative in the direction k is Dβ(h)[k] =

Z1 Z t 0






f h(s) dh2 (s) dk 1 (t) − dh1 (s) dk 2 (t) .

0

Therefore, h ≡ 0 is the only stationary point. The Hessian at h ≡ 0 is


D 2 β (0)[h, k] = h∇b(0), h(1) ⊗ k(1)i + f (0)

Z1





h1 (t) dk 2 (t) − h2 (t) dk 1 (t)

0

for h, k ∈ H . Another expression of this is


D 2 β (0)[h, k] = (Bh, k)H , where B is a linear map on H determined by 



 d 0 −1 h(t). (Bh)(t) = ∇ h(1) · b (0) + f (0) 1 0 dt

This operator B is one to one. Thus h ≡ 0 satisfies the condition of the nondegenerate stationary point except that the operator B may not be onto (cf. [17]). On the other hand, our results state that the asymptotic behavior of the oscillatory integrals I (ξ ; a) and I(ξ ; a) as ξ → ∞ are determined mainly by the behavior of the functionals β(h) and a at the point h ≡ 0 on H . More precisely, if we assume b is smooth at 0 and put β2 (w) = f (0)S(1) and I2 (ξ ) = E[exp(iξβ2 (w)) | w(1) = 0], then we Vol. 35, n◦ 4-1999.

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have




lim I (ξ, 1)/I2 (ξ ) = exp − µ20 .

ξ →∞

The right hand side becomes 1 if we replace I2 (ξ ) by 






I4 (ξ ) = E exp iξβ4 (w) |w(1) = 0 , where f 00 (0) β4 (w) = β2 (w) + 8

Z1 Z t 0




|w(s)|2 dw 2 (s) dw 1 (t) − dw 1 (s) dw 2 (t) .

0

The skeleton of β2 (w) and β4 (w) are the 1st and 2nd approximating functionals appearing in the Taylor expansion of β(h) around the critical point h = 0, respectively. (ii) In√ Ikeda–Manabe [9], Theorem 3 is obtained in the case that f (r) = 1 + r β for some β > 0 and a is represented as a(w) = A

Z1

Z1 2

r(s) h1 (s) ds, 0

!

Z1 2

2

r(s) h2 (s) ds, . . . , 0

r(s) hm (s) ds , 0

where A is a bounded uniformly continuous function and h1 , h2 , . . . , hm are smooth functions on the interval [0.1]. (iii) The estimate in [28] asserts that 1 f (r) log |I (ξ, 1)| 6 − inf ξ →∞ ξ r>0 2 lim

(3.1)

for the present situation. Since infr>0 f (r) may be less than f (0), the result in this paper is an example that (3.1) is not best possible. Theorem 2 is proven as in the last section by using the following: L EMMA 3.1. – "

1 I(ξ ; a) = E exp − 2 p

+i ξ

Z1 0



Zξ

F r(t), ξ 0




2

dt !

uw(ξ ) b √ · w(ξ ) du ξ

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r(ξ ·) w(ξ ) ×A √ C √ ξ ξ

#

.

(3.2)

The following proof is essentially that of Theorem 3.1 of [9]: Proof of Theorem 3. – Taking K > 0 large, we decompose the right hand side of (2.4) as I (ξ ; a) = I (ξ ; 1)a(0) + I1 (ξ ; a) + I2 (ξ ; a), where "

ξ2 I1 (ξ ; a) = E exp − 2 

Z1


2

!

F r(t), 1 dt 0




s



× A r(·) − A(0) : kr(·)k2 >



#



#

K w(1) = 0 ξ

and "

ξ2 I2 (ξ ; a) = E exp − 2 




Z1


2

!

F r(t), 1 dt 0

s



× A r(·) − A(0) : kr(·)k2 6

K w(1) = 0 . ξ

By the conditions (A.1) (ii), (iii) and (A.5), there exists C1 > 0 such that F (r, 1) > C1 r. Then, by Hölder’s inequality, we have |I1 (ξ ; a)| 6 C2 exp(−C3 ξ K). This is negligible if K is taken large enough. For any ε > 0, by the condition √ (A.6), there exists ξε > 0 such that |A(r(·)) − A(0)| < ε if kr(·)k2 6 K/ξε . Therefore we have |I2 (ξ ; a)| 6 ε|I (ξ ; 1)| for any ξ > ξε . By all these, we can complete the proof.

2

The proof of Theorem 4 is almost identical with that of Theorem 3. Vol. 35, n◦ 4-1999.

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N. UEKI

Proof of Theorem 5. – For large enough ξ > 0, we have "

ξ2 I (ξ ; a) > CE exp − 2 "

1 = CE exp − 2

Z1 0

!



#

1 F r(t), 1 dt : sup r(t) < w(1) = 0 ξ [0,1]
2

Zξ

F r(t), ξ


2

0

!

# dt : sup r(t) < 1 w(ξ ) = 0 , [0,ξ ]

where C = inf{A(r(·)): sup[0,1] r(t) < R}. From this we have the lower bound. 2 Remark 3.3. – (i) In this proof of Theorem 5, it is essential that the integrand of I (ξ ; a) is real nonnegative. Such an expression is not obtained for I(ξ ; a) (cf. Lemma 3.1). (ii) By using Hörder’s inequality, the condition (A.8) is weakend as follows:


0 6 a = A r(·) ∈

\




Lp P (· | w(1) = 0) ,

p>1

a −1 χR ∈

\




Lp P (· | w(1) = 0)

p>1

for some R > 0, where χR is the indicator function of the set {w: sup[0,1] r(t) < R} on W . 4. A GENERAL ROTATION INVARIANT METRIC In this section we extend our results in last sections to the case where the Wiener process w is replaced by a process associated to a general rotation invariant metric and the 1-form b may degenerate finitely at the origin. For the formulation, we refer to a work of Sheu [20]. Using the polar coordinate (r, θ), we assume that a Riemannian metric g on R2 is given by g = dr ⊗ dr + g(r)2 dθ ⊗ dθ, where g(r) is a positive smooth function on the interval (0, ∞) satisfying the following: (g-i) As r → 0, g(r) ∼ r α

X

gn r α(n)

n>0

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ASYMPTOTIC EXPANSION

where α ∈ R, g0 = 1 and 0 = α(0) < α(1) < α(2) < · · · % ∞; 0 (g-ii) infr>1 g(r)/r α > 0 for some α 0 ∈ R; 00 0 (g-iii) supr>1 |g 0 (r)|/r α < ∞ for some α 00 < α 0 and supr>1 |g 00 (r)|/r α < ∞; (g-iv) −1 < α < 3 and α(1) > (1 − |α|)/2. The corresponding Laplace–Beltrami operator is represented as 1=

g0 ∂ 1 ∂2 ∂2 + (r) + ∂r 2 g ∂r g(r)2 ∂θ 2

on {r > 0}. Its radial part is a Sturm–Liouville operator 1r =

d d d d2 g0 (r) = g(r) + dr 2 g dr g(r) dr dr

on (0, ∞). The boundary 0 is entrance if α > 1 and regular if −1 < α < 1 in the sense of Feller [5]. The boundary ∞ is always natural. Let r(t, r), t > 0, r > 0 on [0, ∞) be the diffusion process generated by 1r /2 with the domain 






D = ϕ(r) = Φ r 2 : Φ ∈ C0∞ ([0, ∞)) .

(4.1)

Then we can construct a diffusion process X(t, x), t > 0, x 6= 0, generated by 1/2 as the skew product of the process r(t, r) and an independent spherical Brownian motion BM(S 1 ) run with the clock R t −2 0 g (r(s, r)) ds (cf. [11]). We consider a differential 1-form b given by b = k(r) dθ, where k(r) is a positive smooth function on (0, ∞). The fibre norm of this form with respect to the above metric is F (r) := k(r)/g(r). We assume that this function F satisfies the following: (b-i) As r → 0, F (r) ∼ r ρ

X

fn r ρ(n)

n>0

where ρ > 0, f0 > 0, 0 = ρ(0) < ρ(1) < ρ(2) < · · · % ∞; (b-ii) There are 0 < ρ 0 < ρ 00 and c0 , c00 > 0 such that 0

c0 r ρ 6 F (r) 6 c00 r ρ Vol. 35, n◦ 4-1999.

00

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N. UEKI

for any r > 1. We consider the asymptotic behavior, as ξ → ∞, of a function defined by I (ξ ) := lim lim 0

"

Z

R →0 R→0 B(R)

E exp iξ

X(1, x) ∈ B R ×

( Z

Z1

#
0

!




b X(t, x) ◦ dX(t, x) :

0

d vol(x)

P X(1, x) ∈ B R



0

)−1

d vol(x)

,

(4.2)

B(R)

where vol is the volume with respect to the above Riemannian metric and B(R) := {x ∈ R2 : |x| < R} for R > 0. By taking the expectation in BM(S 1 ), we can rewrite this as "

ξ2 I (ξ ) = lim E exp − R 0 →0 2

Z1


2

F r(t)

! # dt r(1) < R 0 ,

(4.3)

0

where r(t) := r(t, 0). To state the theorem, we introduce an operator H=


1 − 1α + F0 (r)2 2

with the domain D defined in (4.1), where 1α =

d2 α d + 2 dr r dr

and F0 (r) := f0 r ρ . By Lemma 4.1 below, this operator is essentially self adjoint as an operator on L2 ((0, ∞), mα ), where mα (dr) := r α dr. We denote the self adjoint extension by the same symbol. By Lemma 4.3 below, this operator has purely discrete spectrum. We denote the eigenvalues by µ0 < µ1 < µ2 < · · ·, the corresponding normalized eigenfunctions by ϕ0 , ϕ1 , ϕ2 , . . . , and the orthogonal projection to the orthogonal complement to the ground states by P⊥ . We easily see that µ0 > 0 and that ϕ0 can be taken as a positive continuous function on the closed interval [0, ∞). Let et 1r /2 (r, r 0 ), t > 0, r ∈ [0, ∞), r 0 ∈ (0, ∞), be the transition density of the process r(t, r) with respect to the speed measure Annales de l’Institut Henri Poincaré - Probabilités et Statistiques

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ASYMPTOTIC EXPANSION

m(dr) := g(r) dr. By the proof of Lemma 4.7 below, we see that t 1r /2

e

(0, 0) = lim

ZR

et 1r /2 (0, r)m(dr)/m((0, R))

R→0 0

exists. This value is positive (see, e.g., McKean [16]). Let 

A = A = (k1 , k2 , . . . , km ; `1 , `2 , . . . , `n ): m, k1 , k2 , . . . , km ,



`1 , `2 , . . . , `n ∈ N, n ∈ Z+ . For A = (k1 , k2 , . . . , km ; `1 , `2 , . . . , `n ) ∈ A, we set ι(A) = ρ(k1 ) + ρ(k2 ) + · · · + ρ(km ) + α(`1) + α(`2 ) + · · · + α(`n). Let



B = B = (A0 , A1 , . . . , Am ): m ∈ Z+ , A0 , A1 , . . . ,



Am ∈ A, ι(A1 ), ι(A2 ), . . . , ι(Am ) > 2 . For B = (A0 , A1 , . . . , Am ) ∈ B, we set κ(B) = ι(A0 ) + ι(A1 ) + · · · + ι(Am ) − 2m. The main theorem is the following: T HEOREM 6. – Under the conditions (g-i)–(g-iv) and (b-i)–(b-ii), we have the following asymptotic expansion as ξ → ∞: (

I (ξ ) ∼ exp

− ξ µ0 −

(

× I0 +

2γ

X

)

ξ

γ (2−ι(A))

µA 0

ξ (α+1)γ

A∈A: ι(A)62

X

)

IB ξ −γ κ(B) ,

(4.4)

B∈B

where γ = 1/(ρ + 1), {µA 0 : A ∈ A} are polynomials of 

gn , fn , r k(0) r −ε(0) d/dr


`(0)

ϕ0 , (µ0 − H )−1 P⊥ r k(1) r −ε(1) d/dr

(µ0 − H )−1 P⊥ r k(n) r −ε(n) d/dr


`(n)





`(1)

ϕ0 : n ∈ Z+ , k(0), k(1), . . . , k(n)

···

> 0, 0 6 ε(0), ε(1), . . . , ε(n) 6 1 − α(1), `(0), `(1), . . . , `(n) ∈ {0, 1} , I0 = Vol. 35, n◦ 4-1999.

ϕ0 (0)2 , e1r /2 (0, 0)

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N. UEKI

and {IB : B ∈ B} are polynomials of 

gn , fn , r k(0) r −ε(0) d/dr


`(0)

ϕ0 , (µ0 − H )−1 P⊥ r k(1) r −ε(1) d/dr

(µ0 − H )−1 P⊥ r k(n) r −ε(n) d/dr


`(n)

(µ0 − H )−1 P⊥ r k(0) r −ε(0) d/dr (µ0 − H )−1 P⊥ r k(n) r −ε(n) d/dr




···

ϕ0 ,


`(0)


`(n)


`(1)

···


ϕ0 (0): n ∈ Z+ , k(0), k(1), . . . , k(n)

> 0, 0 6 ε(0), ε(1), . . . , ε(n) 6 1 − α(1), `(0), `(1), . . . , `(n) ∈ {0, 1} . The condition (g-iv) α < 3 ensures that the boundary 0 is of limit circle type for the operator 1r (see, e.g., [18] Appendix to X.1). When this condition is not satisfied, we assume the following: (g-v) α > −1 and α(1) > 2. Then we have the following: T HEOREM 7. – Under the conditions (g-i)–(g-iii), (g-v) and (b-i)– (b-ii), we have the following asymptotic expansion as ξ → ∞: (

I (ξ ) ∼ exp

X

− ξ µ0 − 2γ

)

ξ

γ (2−ι(A))

µA 0




ξ (α+1)γ I0 + o(1) ,

A∈A: ι(A)62

(4.5) where γ , µA 0 and I0 are those in Theorem 6. Remark 4.1. – (i) Under the condition α(1) > 2, {µA 0 : A ∈ A, ι(A) 6 2} are polynomials of only 




fn , r k(0) ϕ0 , (µ0 − H )−1 P⊥ r k(1) · · · (µ0 − H )−1 P⊥ r k(n) ϕ0 : n ∈ Z+ ,

k(0), k(1), . . . , k(n) > 0 . (ii) We can treat also general amplitudes as in Sections 2 and 3. However, since the discussion is almost identical, we omit it. In the following we always assume only (g-i)–(g-iii), α > −1 and (b-i)–(b-ii), unless otherwise stated. To analyze the function I (ξ ), we first represent this as I (ξ ) = e−H(ξ ) (0, 0)/e1r /2 (0, 0),

(4.6)

where e−t H(ξ ) (r, r 0 ) and et 1r /2 (r, r 0 ), t > 0, r, r 0 > 0 are the heat kernels of the operators H(ξ ) =


1 − 1r + ξ 2 F (r)2 2

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ASYMPTOTIC EXPANSION

and −1r /2, respectively, with respect to the measure m(dr). To define these operators, we use the following lemma, which is proven by the same technique of Sheu [20]: L EMMA 4.1. – The operators H(ξ ) and 1r are essentially self adjoint on the domain D defined in (4.1) as operators on L2 ((0, ∞), m). Moreover, by this lemma, we can represent the heat kernel as follows: e−t H(ξ ) r, r 0




"

ξ2 = E exp − 2

Zt


2

F r(s, r)

! #
0 ds r(t, r) = r et 1r /2 r, r 0

(4.7)

0

(see, e.g., [21] Theorem 1.1). By a general theory (see, e.g., [11] Section 4.11), the heat kernel e−t H(ξ ) (r, r 0 ) is a positive smooth function of (r, r 0 ) ∈ (0, ∞)2 . For the representation (4.6), we use the following: L EMMA 4.2. – e−t H(ξ ) (r, r 0 ) is a positive continuous function of (r, r 0 ) ∈ [0, ∞)2 . Proof. – When α < 1, this fact is well known since the boundary 0 is regular (see, e.g., McKean [16]). When α > 1, we use a result of Hille [6] as follows (see also Matsumoto [15]). We assume ξ = 1 and write H := H(1). We use a canonical scale x = T (r), where T (r) =

Zr 1

ds . g(s)

(4.8)

Then the operator H is represented as 




2 d d 1 + F T −1 (x) , H= − 2 dM(x) dx

where dM(x) = g(T −1 (x))2 dx. Since the boundary 0 is entrance, we have limr↓0 T (r) = −∞ and Z0 −∞

Vol. 35, n◦ 4-1999.

(−x) dM(x) =

Z1 Z1 0

r

ds g(r) dr < ∞. g(s)

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N. UEKI

Moreover, if we set (

ρ(x) = sup −y x>y

)

Zy

dM(z) ,

−∞

then we have Z0 −∞

ρ(x) dx = −x

Z1

Z1

ds g(s)

g(r) r

0

!−1

Z1

sup r>s

0

dt g(t)

Zs

!

g(t) dt dr < ∞.

0

Therefore, by the same method of the proof of Theorem 4.1 of Matsumoto [15] (cf. Hille [6]), we have 1 log sup |Φn (x)| 6 0 n→∞ mn x6x0 lim

(4.9)

for any x0 ∈ R, where m1 < m2 < · · · are the eigenvalues of H and Φ1 , Φ2 , . . . are the corresponding normalized eigenfunctions. The discreteness of the spectrum of H is ensured by the subsequent Lemma 4.3. Moreover by the same lemma and Mercer’s expansion theorem, we see that e−t H (r, r 0 ) is a bounded continuous function of (r, r 0 ) ∈ [0, ∞)2 . Since the boundary 0 is entrance, we have lim Φn (x) 6= 0.

x→−∞

Thus we have


e−t H r, r 0 > 0 for any (r, r 0 ) ∈ [0, ∞)2 .

2

L EMMA 4.3. – The heat semigroup generated by H(ξ ) consists of trace class operators. Proof. – As in the last proof, we consider only H := H(1). We set 

µn (H) :=

sup

inf

ϕ1 ,ϕ2 ,...,ϕn−1 ∈L2 (gdr)


1 kψ 0 k2g + kF ψk2g : ψ ∈ D, 2



kψkg = 1, (ψ, ϕj )g = 0 for j = 1, 2, . . . , n − 1 , Annales de l’Institut Henri Poincaré - Probabilités et Statistiques

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ASYMPTOTIC EXPANSION

where k · kg and (·, ·)g are the norm and the inner product of L2 (g dr), respectively. By the condition (g-i), we can take positive smooth functions g1 (r) and g2 (r) on the interval (0, ∞) such that   g1 (r) 6 g(r) 6 g2 (r),

on (0, ∞), g (r) = C1 r , g2 (r) = C2 r , on (0, 1),  1 g1 (r) = g(r) = g2 (r), on (2, ∞), α

α

for some 0 < C1 < C2 . Then we have b µn (H) > µn (H),

where b := µn (H)



sup

inf

ϕ1 ,ϕ2 ,...,ϕn−1 ∈L2 (g1 dr)


1 k(J ψ)0 k2g1 + kF J ψk2g1 : ψ ∈ D, 2



kψkg1 = 1, (ψ, ϕj )g1 = 0 for j = 1, 2, . . . , n − 1 , √ and J (r) := g1 (r)/g2 (r). By the min-max principle, it is enough to show that the self adjoint operator 

J d d g1 J + (F J )2 − 2 g1 dr dr

b=1 H



on L2 (g1 dr) generates a heat semigroup consisting of trace class operators. We use a scale σ = S(r) defined by S(r) =

Zr 0

ds . J (s)

b is regarded as a self adjoint operator Then H  1 1 b H= −

2



d d G(σ ) + V(σ ) , G(σ ) dσ dσ

on L2 ((0, ∞), G(σ )dσ ), where


G(σ ) := (g1 J ) S −1 (σ ) and








V(σ ) := (F J )2 − J J 0 g10 /g1 − J J 00 S −1 (σ ) . Vol. 35, n◦ 4-1999.

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N. UEKI

√ By the unitary operator G(σ )/σ α from L2 (G(σ ) dσ ) to L2 (σ α dσ ), the b is transformed to a self adjoint operator operator H e := H


1 e ) + V(σ ) − 1α − G(σ 2

on L2 (σ α dσ ), where e )= G(σ

α(α − 2) G 02 − 2GG 00 + . 4σ 2 4G 2

e ) is a bounded continuous By the conditions (g-ii) and (g-iii), G(σ e has the function on [0, ∞). Therefore the heat kernel of the operator H following representation: e

e−t H σ, σ 0




"

1 = E exp 2

! #
0 Ge − V rα (s, σ ) ds rα (t, σ ) = σ et 1α /2 σ, σ 0

Zt







0

0

for (σ, σ ) ∈ [0, ∞)2 , where rα (t, σ ), t > 0, σ > 0, is the Bessel diffusion process with index α + 1 and et 1α /2 (σ, σ 0 ) is its transition density function:


et 1α /2 σ, σ 0 =

∞ exp(−(σ 2 + σ 02 )/(2t)) X (σ σ 0 /(2t))2n (4.10) 2(α−1)/2t (α+1)/2 n!0(n + (α + 1)/2) n=0

(cf., e.g., [10] IV-(8.20) and [21] Theorem 6.1). By this explicit representation, we have





sup et 1α /2 σ, σ 0 / 1 + (σ σ 0 )2 < ∞ σ,σ 0

(4.11)

for any t > 0. Moreover if we take large R > 0, then we have e

0

e−t H (σ, σ ) < C3 exp − C4 σ 2(1∧ρ )




(4.12)

for any σ > R, where C3 and C4 are positive constants depending only on R. In fact, if we divide as e −H

e

"

1 (σ, σ ) = E exp 2

Z1

!
e G − V (rα (t, σ )) dt :

0

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443

ASYMPTOTIC EXPANSION

#



σ inf rα (t, σ ) 6 rα (1, σ ) = σ e1α /2 (σ, σ ) 06t 61 2 "

1 + E exp 2

Z1




!




Ge − V rα (t, σ ) dt :

0

#



σ inf rα (t, σ ) > rα (1, σ ) = σ e1α /2 (σ, σ ), 06t 61 2 then the first term is estimated by the Chapman–Kolmogorov equality, (4.11) and an estimate of stopping times (cf. [10] Lemma V-10.5) as follows: "

1 E exp 2

Z1

0 1α /2

×e

(σ, σ )



σ inf rα (t, σ ) 6 rα (1, σ ) = σ 06t 61 2

#

!



σ inf rα (t, σ ) 6 rα (1, σ ) = σ e1α /2 (σ, σ ) 06t 61 2

6 C5 P = 2C5

!

e G − V rα (t, σ ) dt :



Z∞ 

σ 1 P inf rα (t, σ ) 6 rα , σ 06t 61/2 2 2

0



=σ

0




2

× e1α /4 σ, σ 0 σ 0α dσ 0 

6 C6 E



1 + rα 



1 ,σ 2

1 ,σ 2
6 C7 exp − C8 σ 2 , 6 C6 E

2



σ2 : 2

1 + rα

inf

06t 61/2

rα (t, σ ) 6

2 1/2 

σ2

P

inf

06t 61/2

σ 2



rα (t, σ ) 6

σ 2

1/2

where C7 and C8 are positive constants depending only on R. The second term is estimated as "

1 E exp 2 ×e

t 1α /2

Z1



e G − V rα (t, σ ) dt :

0

(σ, σ )

6 C9 exp − C10 σ 2ρ Vol. 35, n◦ 4-1999.

0






σ inf rα (t, σ ) > rα (t, σ ) = σ 06t 61 2

#

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N. UEKI

by the condition (b-ii). From (4.11) and (4.12), we obtain Z∞

e

e−H (σ, σ )σ α dσ < ∞,

0

which completes the proof.

2

As in Section 2, we rewrite (4.6) as I (ξ ) = ξ (α+1)γ e−ξ

2γ H (ξ )

.

(0, 0) e1r /2 (0, 0),

(4.13)

where e−t H (ξ ) (r, r 0 ), t > 0, r, r 0 > 0 is the heat kernel of the self adjoint operator
1 − 1ξ + F (r, ξ )2 2 2 on L (mξ ) with respect to the speed measure mξ (dr) = gξ (r) dr,

H (ξ ) =

1ξ =

gξ0 d2 d + (r) , 2 dr gξ dr 



k r F (r, ξ ) = ξ g ξγ and gξ (r) = ξ αγ g(r/ξ γ ). (4.13) is obtained from the scaling property 1−γ





L

r(·) ∼

(

)

rξ (ξ 2γ ·) , ξγ

(4.14)

where rξ (·) is the diffusion process generated by 1ξ /2. Moreover we rewrite (4.13) as I (ξ ) = ξ (α+1)γ

∞ X

e−ξ

2γ µ (ξ ) n

ϕn (0, ξ )2 /e1r /2 (0, 0),

n=0

where µ0 (ξ ) < µ1 (ξ ) < µ2 (ξ ) < · · · are the eigenvalues of H (ξ ) and ϕ0 (r, ξ ), ϕ1 (r, ξ ), ϕ2 (r, ξ ), . . . are the corresponding normalized eigenfunctions. The aymptotic behavior of µn (ξ ) is as follows: P ROPOSITION 4.1. – limξ →∞ µn (ξ ) = µn for each n > 0. Annales de l’Institut Henri Poincaré - Probabilités et Statistiques

445

ASYMPTOTIC EXPANSION

P ROPOSITION 4.2. – We have the following asymptotic expansion as ξ → ∞: µ0 (ξ ) ∼ µ0 +

X µA 0 A∈A

ξ γ ι(A)

,

where A, µA 0 , γ and ι(A) are those in Theorem 6. To prove these propositions we use the following: L EMMA 4.4. – For each ϕn , we have the following: (i) For some c1 , c2 , c3 > 0 depending only on n, we have


|ϕn (r)| 6 c1 exp − c2 r c3 .

(4.15)

(ii) limr↓0 r α∧0 ϕn0 (r) = 0. (iii) For any N = 0, 1, 2, . . . , we have Z∞

0 2 N α ϕ r m (dr) < ∞.

(4.16)

n

0

(iv) ϕn ∈ Dom(H (ξ )). L EMMA 4.5. – For each fixed m ∈ Z+ and a ∈ [0, 1 ∧ {(|α| + 1)/2}), we have

sup 1 + r 2

|z−µ0 |=ε


m/2

(z − H )−1 1 + r 2


−m/2

op

<∞

(4.17)

and

sup 1 + r 2


m/2 1 d

|z−µ0 |=ε

r a dr

(z − H )−1 1 + r 2


−m/2

op

< ∞,

(4.18)

where k · kop is the operator norm on L2 (mα ). Proof of Lemma 4.4. – (i) By estimating the probabilistic representation e−t H r, r 0




"

1 = E exp − 2

Zt


2

F0 rα (s, r)

! #
0 ds rα (t, r) = r et 1α /2 r, r 0 (4.19)

0

as in the proof of (4.12), we obtain (4.15) with c3 = 2(1 ∧ ρ 0 ). Vol. 35, n◦ 4-1999.

446

N. UEKI

(ii) Let ψ be a smooth function on (0, ∞) such that ψ(r) = r −δ on (0, R1 ] and ψ(r) ≡ 0 on [R2 , ∞) for some small δ > 0 and some 0 < R1 < R2 . For ε ∈ (0, R1 ], we have Z∞

ϕn ψm (dr) =

Z∞

α

µn ε

(H ϕn )ψmα (dr) ε

1 = 2

Z∞



1 ϕn0 ψ 0 + F0 (r)2 ϕn ψ mα (dr) + ε α−δ ϕn0 (ε). 2

ε

Since the left hand side and the first term of the right hand side have limits as ε ↓ 0, the limit limr↓0 r α−δ ϕn0 (r) should exist. Thus we have limr↓0 r α ϕn0 (r) = 0. On the other hand, we have α 0
0 r ϕ 6 C2 r α n

for 0 < r 6 C3

by the characteristic equation and (i) of this lemma. Thus we have α 0 r ϕ 6

Zr

n

α 0
0 r ϕ dr 6 C4 r α+1 , n

0

from which we obtain limr↓0 ϕn0 (r) = 0. (iii) (4.16) with N = 0 follows from ϕn ∈ Dom(H ). For N > 1, we use the integration by parts and the characteristic equation for ϕn as follows: Z

0 2 α+N ϕ r dr = − n

Z

ϕn (1α ϕn )r

α+N

dr − N

Z

ϕn ϕn0 r α+N−1 dr.

The first term equals to Z




2µn − F0 (r)2 ϕn2 r α+N dr,

which is finite by (4.15). The second term is dominated by sZ

sZ

ϕn2 r α+2N−2 dr

2 ϕ 0 r α dr, n

which is finite by (4.15) and (4.16) with N = 0. (iv) We define a domain by Annales de l’Institut Henri Poincaré - Probabilités et Statistiques

447

ASYMPTOTIC EXPANSION

( e := ϕ ∈ C 2 ((0, ∞)): ϕ satisfies (i)–(iii) of this lemma, and D Z∞

)

|1α ϕ| r m (dr) < ∞ for any N = 0, 1, 2, . . . . . 2 N

α

0

e and that H (ξ ) is defined as a symmetric We easily see that D ⊂ D e by the condition (g-iv). Therefore, by Lemma 4.1, we have operator on D e e by (i)–(iii) of this D ⊂ Dom(H (ξ )). On the other hand, we have ϕn ∈ D lemma. Thus we have ϕn ∈ Dom(H (ξ )). 2

Lemma 4.5 is proven by the technique in McKean [16]. The asymptotic behavior of ϕ0 (0, ξ ) is as follows: P ROPOSITION 4.3. – Under the conditions of Theorem 6, we have the following asymptotic expansion as ξ → ∞: ϕ0 (0, ξ ) ∼ ϕ0 (0) +

X ϕ A (0) 0 A∈A

ξ γ ι(A)

,

where A, γ , ι(A) are those in Theorem 6 and {ϕ0A (0): A ∈ A} are polynomials of 

gn , fn , r k(0) r −ε(0) d/dr


`(0)

ϕ0 , (µ0 − H )−1 P⊥ r k(1) r −ε(1) d/dr

(µ0 − H )−1 P⊥ r k(n) r −ε(n) d/dr


`(n)

(µ0 − H )−1 P⊥ r k(0) r −ε(0) d/dr (µ0 − H )−1 P⊥ r k(n) r −ε(n) d/dr




···

ϕ0 ,


`(0)


`(n)


`(1)

···


ϕ0 (0): n ∈ Z+ , k(0), k(1), . . . , k(n)

> 0, 0 6 ε(0), ε(1), . . . , ε(n) 6 1 − α(1), `(0), `(1), . . . , `(n) ∈ {0, 1} . To prove this proposition, we use the following: L EMMA 4.6. – Under the conditions of Theorem 6, we have sup







sup (z − H )−1 ϕ (0) < ∞

(4.20)

|z−µ0 |=ε kϕk=1

and sup sup



ξ >1 |z−µ0 |=ε kϕk=1

for large enough m > 0. Vol. 35, n◦ 4-1999.


−1

sup z − H (ξ )

1 + r2


−m/2



ϕ (0) < ∞

(4.21)

448

N. UEKI

The proof of this lemma is reduced to the following: L EMMA 4.7. – Under the conditions (g-i)–(g-iii) and (b-i)–(b-ii), we have sup sup e−H (ξ,δ) (0, 0) < ∞,

(4.22)

ξ >1 0<δ61

where
1 − 1ξ + δF (r, ξ )2 2 2 is a self adjoint operator on L (mξ ).

H (ξ, δ) =

Proof. – We note that e−H (ξ,δ) (0, 0) "

δ = lim E exp − ε→0 2

Z1

F rξ (t), ξ


2

!

#

dt : rξ (1) < ε




mξ (0, ε)

0

and that Rξ (t) := rξ (t) , t > 1 is regarded as the pathwise unique solution of the stochastic integral equation 2

Rξ (t) =

Zt q

2 Rξ (s) dw(s) +

0

Zt




Gξ Rξ (s) ds, 0

where w(t) is the 1-dimensional standard Brownian motion with w(0) = 0 and √ gξ0 √
Gξ (R) = R R +1 gξ (cf. [10] Examples IV-8.2 and 3). By the conditions (g-i)–(g-iii), we have Gξ (R) > G0 (R) for any ξ > 1, where G0 (R) =

√ g00 √
R R + 1, g0

g0 (r) is a smooth function on (0, ∞) such that (

g0 (r) =

r α exp(C1 r α(1) ), for 0 6 r 6 1, C2 r −2 exp(−r 3 ), for r > C3 , Annales de l’Institut Henri Poincaré - Probabilités et Statistiques

449

ASYMPTOTIC EXPANSION

and C1 , C2 , C3 are some positive constants. Let R0 (t, R), t > 0, R > 0, be the solution of R0 (t, R) = R +

Zt p

Zt

0

0

2 R0 (s, R) dw(s) +




G0 R0 (s, R) ds

p

and let r0 (t, r) := R0 (t, r 2 ) for r > 0 and r0 (t) := r0 (t, 0). Then, by the comparison theorem (cf. Ikeda–Watanabe [10] Theorem VI-1.1), we have rξ (t) > r0 (t) and "

δ E exp − 2

Z1

F rξ (t), ξ


2

!

#




dt : rξ (1) < ε 6 P r0 (1) < ε .

0

The generator of the process {r0 (t, r)} is 10 /2, where 10 =

d d g0 (r) . g0 (r) dr dr

For this operator, the ∞ is entrance and the boundary 0 is same as for the operator 1r or 1α . Thus the resolvent (µ − 10 )−1 for each µ > 0 is a trace class operator (see, e.g., McKean [16]). Therefore, as in Lemma 4.2, we can show the existence of P (r0 (1) < ε) , ε→0 m0 ((0, ε))

e10 /2 (0, 0) = lim

where m0 (dr) = g0 (r) dr (cf. Hille [6], Matsumoto [15]). On the other hand, by the condition (g-i), we easily see that sup lim

ξ >1 ε→0

Therefore we obtain (4.22).

m0 ((0, ε)) < ∞. mξ ((0, ε))

2

Now Theorem 6 is proven similarly as in Section 2 by using Propositions 4.1–4.3. For the proof of Theorem 7, we use the following: Vol. 35, n◦ 4-1999.

450

N. UEKI

L EMMA 4.8. – Under the conditions of Theorem 7, we have lim ϕ0 (0, ξ )2 = ϕ0 (0)2 .

ξ →∞

p

Proof. – By the unitary operator gξ /r α from L2 (mξ ) to L2 (mα ), the operator H (ξ ) is transformed to a self adjoint operator He (ξ ) =


1 − 1α − gfξ (r) + F (r, ξ )2 2

on L2 (mα ), where fξ (r) = g

α(α − 2) gξ02 − 2gξ gξ00 + . 4r 2 4gξ2

fξ is a bounded Under the conditions of Theorem 7, we can show that g continuous function on the closed interval [0, ∞). Therefore the heat kernel has the following representation: s −t H (ξ )

e

0

(r, r ) =

"

rα 1 E exp gξ (r) 2

Zt






fξ rα (s, r) g

0

s ! #
2
r 0α 0 t 1 /2 0 − F rα (s, r), ξ ds rα (t, r) = r e α r, r g (r 0 ) ξ

for (r, r 0 ) ∈ [0, ∞)2 . By the Lebesgue convergence theorem, we have lim e−t H (ξ ) (0, 0) = e−t H (0, 0).

ξ →∞

By regarding −t H (ξ )

e

(0, 0) =

∞ X

e−t µn (ξ ) ϕn (0, ξ )2

n=0

as a Laplace transform, we have ∞ X n=0

ϕn (0, ξ )2 δµn (ξ ) (λ) dλ →

∞ X

ϕn (0)2 δµn (λ) dλ

(4.23)

n=0

vaguely, as ξ → ∞, from which we can obtain the result.

2

Annales de l’Institut Henri Poincaré - Probabilités et Statistiques

451

ASYMPTOTIC EXPANSION

5. AN ABSORBING CASE In this section, we assume the conditions (g-i)–(g-iii) and (g-vi) α < 1 and α(1) > 2. In this case, the boundary 0 is exit if α 6 −1 and regular if −1 < α < 1 for the operator 1r . Thus, to obtain a radial process on (0, ∞), we should take the boundary 0 as its absorbing barrier especially if α 6 −1. We assume this absorbing condition also for −1 < α < 1. We denote the corresponding process by r 0 (t, r), t > 0, r > 0. As in the last section, we construct a diffusion process X0 (t, x), t > 0, x 6= 0, as the skew product of the process r 0 (t, r) and an R independent spherical Brownian motion BM(S 1 ) run with the clock 0t g −2 (r 0 (s, r)) ds (cf. [11]). Then X0 (t, x) is a diffusion process generated by the half of the Laplace–Beltrami operator 1/2 with the point 0 as its terminal point. Let b be a differential 1-form satisfying the conditions (b-i)–(b-ii) in the last section. As in the last section, we consider the asymptotic behavior of a function defined by I (ξ ) :=

"

Z

0

lim

(R,R 0 )→(0,0) B(R)

Z1

E exp iξ

b X (t, x) ◦ dX (t, x) :

0

X (1, x) ∈ B R 0

×

( Z

P X (1, x) ∈ B R 0

0

!




0





0

0

#


d vol(x) )−1

d vol(x)

.

(5.1)

B(R)

By taking the expectation in BM(S 1 ), we can rewrite this as I (ξ ) =

ZR

0

lim

(R,R 0 )→(0,0) 0 0

"

ξ2 E exp − 2 #

0

( ZR

Z1 0

0

We define a self adjoint operator

Vol. 35, n◦ 4-1999.

0




)−1

P r (1, r) < R m(dr) 0

H0 =

!

F r (t, r) dt :

0

r (1, r) < R m(dr)


2


1 − 1α + F0 (r)2 2

.

(5.2)

452

N. UEKI

on L2 ((0, ∞), mα ) as the Friedrichs extension of the corresponding operator on C0∞ ((0, ∞)). This operator has purely discrete spectrum. We denote the eigenvalues by µ00 < µ01 < µ02 < · · ·, the corresponding normalized eigenfunctions by ϕ00 , ϕ10 , ϕ20 , . . . , and the orthogonal projection to the orthogonal complement to the ground states by P⊥0 . We easily see that µ00 > 0 and the nonzero existence of the derivative 0 lim r α ϕ000 (r) =: ϕ0,+ (0)

r→0

with respect to the canonical scale (see, e.g., McKean [16]). Let 0 et 1r /2 (r, r 0 ), t, r, r 0 > 0, be the transition density of the process r 0 (t, r) with respect to the speed measure m(dr) := g(r) dr. The nonzero existence of lim 0

(r,r )→(0,0)

g(r)


∂ 10 /2
∂ 10r /2 g r0 e r r, r 0 =: e++ (0, 0) 0 ∂r ∂r

is proven by using Lemma 5.2 below. Let A, ι(A), B and K(B) be as in the last section. The main theorem in this section is the following: T HEOREM 8. – We assume the conditions (g-i)–(g-iii), (g-vi) and (b-i)–(b-ii). (i) When α > −1, we have the following asymptotic expansion as ξ → ∞: (

I (ξ ) ∼ exp −ξ 0

2γ

×

)

X

−

ξ

γ (2−ι(A))

µ0,A 0

ξ (3−α)γ

A∈A: ι(A)62

(

I00

µ00

+

X

)

IB0 ξ −γ κ(B)

,

(5.3)

B∈B

where γ = 1/(ρ + 1), {µ0,A 0 : A ∈ A} are polynomials of 

gn , fn , r k(0) ϕ00 , µ00 − H 0


−1

P⊥0 r k(1) · · · µ00 − H 0


−1




P⊥0 r k(n) ϕ00 :

n ∈ Z+ , k(0), k(1), . . . , k(n) > 0 , I00 =

0 (0)2 ϕ0,+ 10 /2

,

r e++ (0, 0)

and {IB0 : B ∈ B} are polynomials of Annales de l’Institut Henri Poincaré - Probabilités et Statistiques

453

ASYMPTOTIC EXPANSION



gn , fn , r k(0) ϕ00 , µ00 − H 0 lim r α−1 µ00 − H 0


−1

r→0


−1

P⊥0 r k(1) · · · µ00 − H 0

P⊥0 r k(0) · · · µ00 − H 0


−1




−1




P⊥0 r k(n) ϕ00 ,


P⊥0 r k(n) ϕ00 (r):

n ∈ Z+ , k(0), k(1), . . . , k(n) > 0 . (ii) For any α < 1, we have the following asymptotic expansion as ξ → ∞: (

)

X

I 0 (ξ ) ∼ exp −ξ 2γ µ00 −





ξ γ (2−ι(A))µ0,A ξ (3−α)γ I00 + o(1) . 0

A∈A: ι(A)62

(5.4) A ∈ A, ι(A) 6 2} are independent of {gn : n > As in Theorem 7, 1}. The fundamental tool is the following representation of the transition 0 density et 1α /2 (r, r 0 ), t, r, r 0 > 0, with respect to the speed measure mα (dr), of the process rα0 (t, r) generated by the operator 1α /2 with the absorbing condition at the boundary 0: {µ0,A 0 :

L EMMA 5.1. – 




et 1α /2 r, r 0 = exp − 0

 

r 2 + r 02 1 rr 0 √ 2t t 2t

1−α X ∞

(rr 0 /(2t))2n . n!0(n + (3 − α)/2) n=0 (5.5)

We can prove this samely as in Example IV-8.3 of [10]. Remark 5.1. – In terms of the modified Bessel function Iν (x) =

 ν X ∞ x

2

(x/2)2n , n!0(ν + n + 1) n=0

(5.5) is rewritten as




et 1α /2 r, r 0 = exp − 0







r 2 + r 02 (rr 0 )(1−α)/2 rr 0 . I(1−α)/2 2t t t

This expression is similar to that for the usual Bessel process with index α + 1 for α > −1: t 1α /2

e

Vol. 35, n◦ 4-1999.









rr 0 r 2 + r 02 (rr 0 )(1−α)/2 r, r = exp − . I(α−1)/2 2t t t 0




454

N. UEKI

By this lemma and the subsequent Lemma 5.2, we can represent as H I 0 (ξ ) = e− αα

0 (ξ )

0

1r /2 (0, 0)/eαα (0, 0),

(5.6)

where −t H eαα

0 (ξ )

(0, 0) =

0

1r /2 (0, 0) = etαα




lim

r α−1 r 0α−1 e−t H

lim

r α−1 r 0α−1 et 1r /2 r, r 0

0 (ξ )

(r,r 0 )→(0,0)

r, r 0 ,

0

(r,r 0 )→(0,0)




and H0 (ξ ) =


1 −1r + ξ 2 F (r)2 2

on L2 ((0, ∞), m) is the Friedrichs extension of the corresponding operator on C0∞ ((0, ∞)). L EMMA 5.2. – The heat kernel e−t H (ξ ) (r, r 0 ) and the transition 0 density et 1r /2 (r, r 0 ) have the representations 0

−t H0 (ξ )

e

0

s




r, r =

"

rα 1 E exp g(r) 2 #

rα0 (t, r) = r 0

Zt

ge − ξ F 2

0

t 10α /2

e

s

r, r

0


2







rα0 (s, r)

r 0α g(r 0 )

! ds

(5.7)

and t 10r /2

e

0


r, r =

s

"

rα 1 E exp g(r) 2 0

× et 1α /2 r, r

Zt

! # ge rα0 (s, r) ds rα0 (t, r) = r 0


0

s
r 0α 0

g(r 0 )

,

(5.8)

where ge(r) =

α(α − 2) g 02 − 2gg 00 + . 4r 2 4g 2

As in the last section, we use the scaling property to rewrite (5.6) as −ξ I 0 (ξ ) = ξ (3−α)γ eαα

2γ H 0 (ξ )

0

1r /2 (0, 0)/eαα (0, 0),

Annales de l’Institut Henri Poincaré - Probabilités et Statistiques

455

ASYMPTOTIC EXPANSION

where
1 − 1ξ + F (r, ξ )2 . 2

H 0 (ξ ) = Moreover we rewrite this as I (ξ ) = ξ 0

(3−α)γ

∞ X

e−ξ

2γ µ0 (ξ ) n

0

0 ϕn,α (0, ξ )2 /e1r /2 (0, 0),

n=0

where 0 (0, ξ ) = lim r α−1 ϕn0 (r, ξ ), ϕn,α r→0

µ00 (ξ ) < µ01 (ξ ) < µ02 (ξ ) < · · · are the eigenvalues of H 0 (ξ ) and ϕ00 (r, ξ ), ϕ10 (r, ξ ), ϕ20 (r, ξ ), . . . are the corresponding normalized eigenfunctions. As in the last section, we have the following: P ROPOSITION 5.1. – (i) limξ →∞ µ0n (ξ ) = µ0n for each n > 0. (ii) We have the following asymptotic expansion as ξ → ∞: X µ0,A 0

µ00 (ξ ) ∼ µ00 +

A∈A

ξ γ ι(A)

,

where A, µA 0 , γ and ι(A) are those in Theorem 8. P ROPOSITION 5.2. – Under the conditions of Theorem 8(i), we have the following asymptotic expansion as ξ → ∞: X ϕ00,A (0)

0 0 (0, ξ ) ∼ ϕ0,α (0) + ϕ0,α

A∈A

ξ γ ι(A)

,

where A, γ , ι(A) are those in Theorem 8, 0 (0) = lim r α−1 ϕ00 (r) ϕ0,α r→0

and {ϕ00,A (0): A ∈ A} are polynomials of 

gn , fn , r k(0) ϕ00 , µ00 − H 0 lim r α−1 µ00 − H 0

r→0


−1


−1

P⊥0 r k(1) · · · µ00 − H 0

P⊥0 r k(0) · · · µ00 − H 0

n ∈ Z+ , k(0), k(1), . . . , k(n) > 0 . Vol. 35, n◦ 4-1999.


−1


−1




P⊥0 r k(n) ϕ00 ,


P⊥0 r k(n) ϕ00 (r):

456

N. UEKI

By these propositions, Theorem 8 is proven similarly as in the last 0 0 (0) = ϕ0,+ (0)/(1 − α) and section, since ϕ0,α 0

1r /2 etαα (0, 0) =

1 t 10 /2 e++r (0, 0). 2 (1 − α)

ACKNOWLEDGEMENTS The author would like to express his gratitude to Professor H. Matsumoto for fruitful discussions. He would also like to express his gratitude to the referee for valuable comments. REFERENCES [1] G. B EN A ROUS , Methods de Laplace et de la phase stationnaire sur l’espace de Wiener, Stochasics 25 (1988) 125–153. [2] I.M. DAVIES and A. T RUMAN , Laplace asymptotic expansions of conditional Wiener integrals and generalised Mehler kernel formulae for Hamiltonians on L2 (Rn ), J. Phys. A: Math. Gen. 17 (1984) 2773–2789. [3] L. E RDÖS , Estimates on stochastic oscillatory integrals and on the heat kernel of the magnetic Schrödinger operator, Duke Math. J. 76 (1994) 541–566. [4] L. E RDÖS , Magnetic Lieb–Thirring inequalities, Comm. Math. Phys. 170 (1995) 629–668. [5] W. F ELLER , The parabolic differential equations and the associated semi-groups of transformations, Ann. of Math. 55 (1952) 468–519. [6] E. H ILLE , On a class of orthonormal functions, Rend. Sem. Math. Univ. Padova 25 (1956) 214–249. [7] N. I KEDA , S. K USUOKA and S. M ANABE , Lévy’s stochastic area formula and related problem, in: M.C. Cranston and M.A. Pinsky, eds., Stochastic Analysis, Proc. Symp. Pure Math. 57, Amer. Math. Soc., 1995, pp. 281–305. [8] N. I KEDA and S. M ANABE , Integral of differential forms along the path of diffusion processes, Publ. RIMS, Kyoto Univ. 15 (1979) 827–852. [9] N. I KEDA and S. M ANABE , Asymptotic formulas for stochastic oscillatory integral, in: K.D. Elworthy and N. Ikeda, eds., Asymptotic Problems in Probability Theory: Wiener Functionals and Asymptotics, Longman, New York, 1993, pp. 136–155. [10] N. I KEDA and S. WATANABE , Stochastic Differential Equations and Diffusion Processes, 2nd. ed., Kodansha/North-Holland, Tokyo/Amsterdam, 1989. [11] K. I TÔ and H.P. M C K EAN , Diffusion Processes and Their Sample Paths, Springer, Berlin, 1974. [12] P. M ALLIAVIN , Analyticité transverse d’opérateurs hypoelliptiques C 3 sur des fibrés principaux. Spectre équivariant et courbure, C. R. Acad. Sci. 301 (1985) 767– 770. [13] P. M ALLIAVIN , Analyticité réelle des lois conditionnelles de fonctionnelles additives, C. R. Acad. Sci. 302 (1986) 73–78. Annales de l’Institut Henri Poincaré - Probabilités et Statistiques

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Vol. 35, n◦ 4-1999.