Some limit theorems related to multi-dimensional diffusions in a random environment

J. Korean Math. Soc. 48 (2011), No. 1, pp. 147–158 DOI 10.4134/JKMS.2011.48.1.147

SOME LIMIT THEOREMS RELATED TO MULTI-DIMENSIONAL DIFFUSIONS IN

A RANDOM ENVIRONMENT

Daehong Kim

Abstract. In this paper, we consider a multi-dimensional diffusion pro- cess in a self-similar random environment and prove a limit theorem for the shape of the full trajectory of the diffusion by using the localization phenomenon.

1. Introduction

Let W be the space of continuous functions on R ⁿ vanishing at the origin and let Q be a probability measure on it. We call an element of W an environment.

For given an environment w, let P ^w _x be the probability measure on Ω, the canonical path space of real-valued continuous functions from [0, ∞) to R ⁿ , such that {X(t), P ^w x , x ∈ R ⁿ } is a diffusion process with generator

(1) L ^w = 1

2 e ^w

∑ n i=1

∂

∂x _i (

e ^−w ∂

∂x _i )

.

It is well known that such a process X(t) can be constructed from a Brownian motion by a drift transformation ([2]).

Let P x be the probability measure on W × Ω defined by P x (dwdω) = Q(dw)P ^w _x (dω). Then the diffusion X(t) can be regarded as a stochastic pro- cess defined on the probability space ( W × Ω, P x ) and it is called as a diffusion process in a random environment. This process has a close connection with the model of Sinai’s random walk and exhibits some interesting features ([1], [5], [8], [9], [10]). Among those, Brox [1] showed when n = 1 and w(x) is a Brown- ian environment that there exists a nontrivial measurable function b 1 : W 7→ R such that for any ε > 0,

(2) P x (α ⁻² X (e ^α ) − b 1 (w)  > ε )

−→ 0 as α → ∞

Received May 11, 2009; Revised October 13, 2009.

2010 Mathematics Subject Classification. Primary 60K37, 31C25, 60J60.

Key words and phrases. Dirichlet forms, random environment, set valued diffusion pro- cesses, subdiffusivity.

This research was partially supported by Grant-in-Aid for Scientific Research No.

17740062.

⃝2011 The Korean Mathematical Societyc

147

which is the so-called subdiffusivity. This result was a consequence of a local- ization phenomenon, the diffusion is trapped in some valleys of its potential w, and was extended to a large class of random environments ([5]). However, all their methods are heavily rely on the one dimensional situation.

The first analogue result of (2) in higher dimensional cases was obtained by Mathieu [8]. The novelty of his work was the introduction of some analytic approaches such as asymptotics of the first non-zero eigenvalue of the generator (1) with a small noise and the related Dirichlet form theory. Motivated by this work, Tanaka [13] also proved that {X(t), P x , x ∈ R ⁿ } is to be recurrent for all dimensions.

We are interested in the long time asymptotics for the shape of the full trajectory of the multi-dimensional diffusion X(t) in a random environment.

More precisely, let us consider the random set of trajectory of the diffusion X(t) up to time t:

X (t) := {X(s) : 0 ≤ s ≤ t},

where A stands for the closure of a set A. We may call the process X (t) on the probability space ( W × Ω, P x , x ∈ R ⁿ ) a set valued diffusion process in a random environment. Our purpose in this article is to extend the result (2) of Brox to the case of set valued diffusion X (t).

To state our main theorem precisely, let K be a family of non-empty compact sets of R ⁿ and D _r (w) the connected component containing the origin of the sub-level domain {x ∈ R ⁿ : w(x) < r }. Let d H be the Hausdorff distance on K, that is, for K 1 , K 2 ∈ K

d H (K 1 , K 2 ) = inf {ε > 0 : U ε (K 1 ) ⊃ K 2 , U ε (K 2 ) ⊃ K 1 } ,

where U ε (K) denotes the ε-neighborhood of K. Our result is based on some assumptions for the random characteristics of the environment. The main result of this article is stated as follows:

Theorem 1.1. Let f be a probability density on R ⁿ with compact support.

Suppose that

(A.1) For any α > 0 and any fixed λ > 0, the environment {w α,λ (x) := α ⁻¹ w(α ^λ x), x ∈ R ⁿ }

has the same law as {w(x), x ∈ R ⁿ } under Q, namely (W, Q) is a λ ⁻¹ -self-similar random environment.

(A.2) For Q-a.s. D r (w) is bounded for any r > 0 and for Q-a.s. w(x) ≥ 0 for all x ∈ R ⁿ .

Then for any r > 0, ε > 0 and any fixed λ > 0, P f

( d H

(

α ^−λ X (e ^αr ) , D r (w α,λ ) )

> ε

) −→ 0 as α → ∞.

That is, the law of the random set α ^−λ X (e ^αr ) under P f converges to the law

of the compact set D r (w) under Q as α → ∞.

The natural and typical examples of the random environment satisfying the assumptions (A.1) and (A.2) can be provided by W = {w : w(x) = |B(x)|, x ∈ R ⁿ }, where B(x) is a L´evy’s Brownian motion with a n-dimensional time (cf. [8]) and W = {w : w(x) = ∑ n

i=1 |B(x i ) |, x = (x 1 , x 2 , . . . , x n ) ∈ R ⁿ }, where {B i (x) } 1 ≤i≤n is a family of one dimensional Brownian motions which are mutually independent (cf. [12]).

Our theorem tells us that the shape of the K-valued stochastic process r ^λ X (t) converges very slowly to that of the connected component containing the origin of {x ∈ R ⁿ : w(x) < r } by the effect of environment.

To prove our theorem, we address in Section 2 some notations and facts on a one parameter family of diffusion processes and their associated Dirichlet forms (for the general theory of Dirichlet forms, we refer readers to [2]). In Section 3, we obtain some asymptotic results on the first exit times of parameterized diffusions originally due to Mathieu [8] with some modifications (cf. [6], [7]).

The proof of the main theorem will be given in Section 4. A key point of the proof is to use a modified environment to show that the parameterized diffusions enter any ball before leaving the domain D r (w).

2. Preliminaries on parameterized diffusions

For a given w ∈ W, consider a symmetric closable form on L ² (R ⁿ ; e ^−αw dx) (energy form) parameterized by α > 0:

(3) E ^αw (u, v) = 1 2

∫

Rⁿ

∇u(x) · ∇v(x) e ^−αw(x) dx, u, v ∈ C 0 ¹ (R ⁿ ),

where C ₀ ¹ (R ⁿ ) is the space of continuously differentiable functions with compact support in R ⁿ . Then ( E ^αw , H ¹ (R ⁿ )) becomes a regular (local) Dirichlet form defined by the smallest closed extension of (3) (see [2]). Here H ¹ (R ⁿ ) = {u ∈ L ² (R ⁿ ) : ∂ i u ∈ L ² (R ⁿ ), 1 ≤ i ≤ n}. By a general theory of Dirichlet forms, there exists a diffusion process associated with ( E ^αw , H ¹ (R ⁿ )) and we denote it by {X(t), P ^αw x , x ∈ R ⁿ }. Let f be a probability density function on R ⁿ and set

P f ( ·) :=

∫

Rⁿ

P x ( ·)f(x)dx

for a probability distribution P _x . Then {X(t), P ^αw f } can be regarded as the diffusion process with an initial distribution f of the generator (1) replaced w by αw.

For a fixed λ > 0, we simply write as

(4) w α (x) := α ⁻¹ w(α ^λ x), f α (x) := α ^λn f (α ^λ x), x ∈ R ⁿ .

Let {R ^w _β } β>0 be the resolvent of {X(t), P ^w _f }. Then for any φ ∈ L ² (R ⁿ ; e ^−w dx) and v ∈ H ¹ (R ⁿ ),

(5) E β ^w

( R ^w _β φ, v )

=

∫

Rⁿ

φ(x)v(x)e ^−w(x) dx,

where E _β ^w (ϕ, ψ) := E ^w (ϕ, ψ) + β ∫

Rⁿ

ϕ(x)ψ(x)e ^−w(x) dx for ϕ, ψ ∈ H ¹ (R ⁿ ). By the change of variable with x = α ^λ y, the equation (5) yields that

E _α ^αw

β

( R ^w _β φ(α ^λ ·), v(α ^λ ·) )

=

∫

Rⁿ

α ^2λ φ(α ^λ y)v(α ^λ y)e ^−αw

^(y) dy

= E _α ^αw

β

(

R ^αw _α

_β (α ^2λ φ)(α ^λ ·), v(α ^λ ·) ) . So the fact that {α ^2λ R ^αw _α

_β } β>0 is the resolvent of {α ^λ X(α ^−2λ t), P ^αw _f

α

} implies the following lemma.

Lemma 2.1. For w ∈ W and α > 0,

(6) {

α ^−λ X(t), P ^w _f } d

= {

X(α ^−2λ t), P ^αw _f

α

} , where = means the equality in distribution. ^d

For any r > 0, let r(α) := r − (2λ/α) log α. Clearly r(α) → r as α → ∞.

Then (6) can be rewritten as

(7) {

α ^−λ X(e ^αr ), P ^w _f } d

= {

X(e ^αr(α) ), P ^αw _f

α

}

by taking t = e ^αr .

Remark 2.2. Under the assumption (A.1), P ^αw _·

and P ^αw _· are same for Q- a.s.. Therefore we see from (7) that a subdiffusivity problem of the diffusion X(t) is reformulated as an α-asymptotic problem of the parameterized diffusion {X(e ^αr(α) ), P _f ^α

}. Here P x ^α (dwdω) := Q(dw)P ^αw _x (dω).

Let D be an arbitrary bounded domain in R ⁿ . We introduce the so-called 3-depths of D relative to an environment (cf. [4], [8]): the depths of D,

d = d(D, w) = inf

∂D w − inf

D w, d ^′ = d ^′ (D, w) = sup

D

w − inf

D w and the critical depth (or elevation) of D,

c = c(D, w) = sup

x,y ∈D

{ inf

ϕ sup

t ∈[0,1]

w(ϕ(t)) − w(x) − w(y) }

+ inf

D w, where ϕ is a continuous path from [0, 1] to D such that ϕ(0) = x, ϕ(1) = y.

Note that the quantities d ^′ , d and c depend only on the landscape of w and do not depend on any boundary conditions (such as smoothness) of ∂D. Moreover, it is easy to check that if D ≡ D r (w) (r > 0), then d ^′ = d and d > c.

Let H ¹ (D) = {u ∈ L ² (D) : ∂ i u ∈ L ² (D), 1 ≤ i ≤ n} and denotes H 0 ¹ (D) by the closure of C ₀ ¹ (D) in H ¹ (D). It is well known that the absorbing process of the diffusion {X(t), P ^αw x , x ∈ R ⁿ } on D is by definition the diffusion pro- cess {X(t), P ^αw,D x , x ∈ D} whose regular Dirichlet form on L ² (D; e ^−αw dx) is coincided with ( E ^αw,D , H ₀ ¹ (D)), where

E ^αw,D (u, v) = 1 2

∫

D

∇u(x) · ∇v(x) e ^−αw(x) dx

([2]). On the other hand, by defining H _∗ ¹ (D) := (H ₀ ¹ (D) + constant) as a new Sobolev space, ( E ^αw,D , H _∗ ¹ (D)) becomes a regular Dirichlet form on L ² (D ^∗ ; e ^−αw dx) as well ([6]). In fact, this form is the smallest one in the family of Dirichlet forms M = {(E ^αw,D , F ^D ) : H ₀ ¹ (D) ⊂ F ^D ⊆ H ¹ (D), 1 ∈ F ^D }. Here D ^∗ denotes the one-point compactification of D. Let us denote by {X(t), P ^αw, x ^∗ , x ∈ D ^∗ } the diffusion process associated with (E ^αw,D , H _∗ ¹ (D)).

This process is irreducible and recurrent on D ^∗ . More general profound prop- erties on this kind of process and its associated Dirichlet form can be found in [3].

For a notational convenience, we set u(α) ≻ a if lim inf

α →∞

1

α log u(α) ≥ a, u(α) ≺ a if lim sup

α →∞

1

α log u(α) ≤ a.

Let m α (dx) be the normalized underlying measure defined by m α (dx) = I D Z _α ⁻¹ e ^−αw(x) dx, Z α =

∫

D

e ^−αw(x) dx.

Note that (1/α) log Z α → − inf D w as α → ∞. Let γ(α) be the first non-zero eigenvalue of ( E ^αw,D , H _∗ ¹ (D)), the so-called spectral gap, defined by

γ(α) = inf

u ∈H

(D)

E ^αw,D (u, u)

∫

D (u − ∫

D u dm _α ) ² e ^−αw dx .

The following asymptotic lower bound of γ(α) for an arbitrary bounded domain D of R ⁿ was obtained by [6].

Lemma 2.3. γ(α) ≻ −c.

3. Lemmas on exit times

Let τ _D ^αw be the first exit time of the diffusion X(t) out of D, that is, τ _D ^αw :=

inf {t > 0 : X(t) /∈ D}. For any β > 0, set h ^αw β (x) := E ^αw, _x ^∗ (exp( −βτ D ^αw )) the Laplace transform of τ _D ^αw , where E ^αw, _x ^∗ denotes the expectation relative to {X(t), P ^αw, x ^∗ , x ∈ D ^∗ }.

In this section, we consider the asymptotics in α of the distribution of τ _D ^αw studied in [6], [8] under some modifications. We note that no addi- tional condition is imposed (smoothness as like in [8]) on a domain D ⊂ R ⁿ , and it makes no difference to replace the diffusion {X(t), P ^αw x , x ∈ R ⁿ } by {X(t), P ^αw, x ^∗ , x ∈ D ^∗ } (or {X(t), P ^αw,D x , x ∈ D}) as far as the exit time distri- bution from D is concerned.

Lemma 3.1. Let θ(α) be the function such that ∫

D h ^αw _θ(α) dm _α = 1/2.

(i) −d ^′ ≺ θ(α) ≺ −d.

(ii) Assume d > c. Then for any k > 0, ∫

D h ^αw _kθ(α) dm α → 1/(1 + k) as

α → ∞. In particular, set D ≡ D r (w) (r > 0). Then for any open ball

B such that B ⊂ D,

(8)

h ^{αw kθ(α)} − 1 1 + k

L

(B)

−→ 0

as α → ∞.

Proof. We may assume that d > 0 (otherwise the present lemma is trivial).

Note that it follows from the proof of Theorem 3.1 in [6] that for any ε > 0, (9)

∫

D

h ^αw _e

dm α −→ 0 as α → ∞.

On the other hand, for the absorbing diffusion process {X(t), P ^αw,D x , x ∈ D}, put u ^D _α (t, x) = P ^αw,D _x (t < τ _D ^αw ). Then by definition,

E ^αw,D (

u ^D _α (t, ·), u ^D α (t, ·) )

= lim

s →0

( u ^D _α (t, ·) − u ^D α (s + t, ·)

s , u ^D _α (t, ·) )

e

dx

= − 1 2

∫

D

∂

∂t u ^D _α (t, x) ² e ^−αw(x) dx (10)

= − 1 2

d dt H _α ^D (t),

where ( ·, ·) µ denotes the inner product on L ² (D; µ) and H _α ^D (t) =

∫

D

 u ^D _α (t, x)  ² e ^−αw(x) dx.

Let denote ( ¹ ₂ D, H ₀ ¹ (D)) by the Dirichlet form ( E ^αw,D , H ₀ ¹ (D)) when α = 0.

By the boundedness of D, it is transient (Example 1.5.3 in [2]) and thus for any u ∈ H 0 ¹ (D), it holds that

∫

D

u(x) ² dx ≤ 2∥R ^D 1 ∥ _∞ D(u, u),

where R ^D is the Green operator of ( ¹ ₂ D, H ₀ ¹ (D)) ([11]). Using this, we have (11) H _α ^D (t) ≤ 2∥R ^D 1 ∥ ∞ e ^αd

E ^αw,D (

u ^D _α (t, ·), u ^D α (t, ·) ) . By combining (10) and (11),

e ^−αd

∥R ^D 1 ∥ ⁻¹ _∞ ≤ − d

dt log H _α ^D (t) that implies

(12) H _α ^D (t) ≤ H α ^D (0) exp

( −∥R ^D 1 ∥ ⁻¹ _∞ e ^−αd

t )

.

Take r ^′ > 0 such that r ^′ ∈ (d ^′ , d ^′ + ε) for any ε > 0. Applying t = e ^αr

to (12), we see then

P ^αw, _m

^∗ (

e ^αr

< τ _D ^αw )

= P ^αw,D _m

(

e ^αr

< τ _D ^αw )

= Z _α ⁻¹

∫

D

u ^D _α (

e ^αr

, x )

e ^−αw(x) dx

≤ e ^α(inf

^w+ε) H _α ^D (

e ^αr

) 1/2

−→ 0 as α → ∞, which implies that

(13)

∫

D

h ^αw _e

dm _α −→ 1 as α → ∞.

Now, (i) of the lemma is a consequence of (9) and (13).

The first assertion of (ii) was proved in Theorem [6] (also, in Theorem II.1 in [8] related to the reflected Dirichlet form ( E ^αw,D , H ¹ (D)) under the smoothness of D). Finally, we prove the last assertion of (ii). The idea of the proof is originally due to [8], but we need to improve some technical tools in our settings (D is not smooth). Let B be an open ball such that B ⊂ D ≡ D r (w) (r > 0) and r B := max _B w ∈ (0, d). Let B 1 be an open ball such that B 1 ⊂ B and r B

= max _B

1

w ∈ (0, d − c). Then for any ε ∈ (0, d − c − r B

), h ^{αw kθ(α)} −

∫

D

h ^αw _kθ(α) dm α

²

L

(B

)

(14)

≤ e ^αr

h ^{αw kθ(α)} −

∫

D

h ^αw _kθ(α) dm α

²

L

(B

;e

dx)

≤ γ(α) ⁻¹ e ^αr

E _kθ(α) ^αw,D (

h ^αw _kθ(α) , h ^αw _kθ(α) )

≤ ≺ −(d − c − r B

− ε)

by virtue of Lemma 2.3, Lemma 3.1(i) and the fact that for β > 0 E β ^αw,D (h ^αw _β , h ^αw _β ) = E β ^αw,D (h ^αw _β , 1) = β

∫

D

h ^αw _β e ^−αw dx ≤ β.

Therefore we see that (8) holds for the open ball B ₁ by combining (14) and the first assertion of (ii). On the other hand, by Poincar´ e inequality related to H ¹ (B) for the open ball B, there exists a constant γ _B ⁻¹ such that for any ε ∈ (0, d − r B ),

h ^αw kθ(α) − ⟨ h ^αw _kθ(α)

⟩

B

²

L

(B) ≤ γ B ⁻¹

∫

B

 ∇h ^αw kθ(α)   ² dx

≤ 2γ B ⁻¹ e ^αr

E _kθ(α) ^αw,D (

h ^αw _kθ(α) , h ^αw _kθ(α) ) (15)

≺ −(d − r B − ε),

where ⟨h⟩ B := ∫

B h dx/ ∫

B dx. Hence, for the open balls B ₁ and B, h ^{αw kθ(α)} − 1

1 + k

L

(B

)

−→ 0, h ^αw kθ(α) − ⟨ h ^αw _kθ(α)

⟩

B

L

(B

) −→ 0 as α → ∞ and thus

(16)

⟨ h ^αw _kθ(α)

⟩

B −→ 1

1 + k as α → ∞.

Now, applying (16) to (15), we conclude that (8) also holds for any open ball

B such that B ⊂ D. □

Lemma 3.2. Let f be a probability density function on D ≡ D r (w) (r > 0) and let f _α be the scaled function of f defined in (4). Then for any k > 0, (17)

∫

D

h ^αw _kθ(α) df _α −→ 1

1 + k as α → ∞.

Proof. Let B 0 be an open ball centered at 0 satisfying B 0 ⊂ D. In view of the proof of the last assertion of Lemma 3.1(ii), we see that (8) holds for B 0 and its speed of decay is exponential. Note that the support of f α is contained in B 0 for large enough α. So by the similar argument in the proof of Theorem II.3 in [8], we have

h ^{αw kθ(α)} − 1 1 + k

L

(D;f

dx)

≤ 4

∫

{f≥N}

f (x) dx + α ^λn N

h ^{αw kθ(α)} − 1 1 + k

L

(B

)

−→ 0

by letting α → ∞ and N → ∞. This ends the proof of the lemma. □ 4. Proof of Theorem 1.1

In what follows, let d _H be the Hausdorff distance on a family of non-empty compact sets K of R ⁿ . Clearly, D r (w) (for fixed w) is a non-decreasing set valued function and is an element of K for any r > 0.

Lemma 4.1. For Q-a.s., D _· (w) is continuous on (0, ∞) with respect to d H . Proof. First, we prove that (for fixed w) D _· (w) is left continuous on (0, ∞) with respect to d H . To do this, it suffices to show that for any ε > 0, there exists s ∈ (0, r) such that U ε (D s (w)) ⊃ D r (w). Set

ℓ s (x) := inf

y ∈D

(w)

|x − y|, x ∈ D r (w).

Then ℓ s ( ·) is a continuous function on D r (w). Moreover, lim s ↑r ℓ s (x) = 0

for any x ∈ D r (w). Indeed, by the connectedness of D r (w), there exists a

continuous path ϕ : [0, 1] → D r (w) such that ϕ(0) = 0 and ϕ(1) = x for

x ∈ D r (w). For this ϕ, we can choose s > 0 such that s ∈ (sup t ∈[0,1] w(ϕ(t)), r)

and x ∈ D s (w). Therefore we see that ℓ _s (x) converges uniformly to 0 on D _r (w) as s ↑ r and consequently, there exists s ∈ (0, r) such that ℓ s (x) < ε (ε > 0) for all x ∈ D r (w). Now, we prove the assertion of the lemma. Let J (w) be the set of discontinuous points of D r (w). By the left continuity of D _· (w), J (w) is a denumerable set. Therefore ∫

J (w) dr = 0 and (18)

∫ _∞

0

Q (r ∈ J(w)) dr = E ^Q (∫

J (w)

dr )

= 0,

where E ^Q denotes the expectation related to ( W, Q). Since for any α > 0 and r > 0

D _r (w _α ) = α ^−λ D _αr (w), the λ ⁻¹ -self-similarity of the environment implies that

Q(r ∈ J(w)) = Q(r ∈ J(w α )) = Q(αr ∈ J(w)) = 0.

Hence we see that Q(r ∈ J(w)) = 0 does not depend on r > 0 and we obtain

the desired result. □

Lemma 4.2. Let f , f α and D be the same as in Lemma 3.2. Let r(α) be a function such that r(α) → r (r > 0) as α → ∞. Then

 P ^αw, f

^∗

(

X(e ^αr(α) ) ∈ B )

− m α (B)   −→ 0 as α → ∞ for any open ball B such that B ⊂ D.

Proof. Note that it makes no difference to replace an environment w by w − inf D w as far as the critical depth c, diffusion {X(t), P ^αw, x ^∗ , x ∈ D ^∗ } and the associated spectral gap γ(α) are considered. So we may assume inf D w = 0 without loss of generality. Let {p ^αw, t ^∗ } t>0 be the L ² (D; m α )-semigroup of {X(t), P ^αw, x ^∗ , x ∈ D ^∗ } and {F γ ^{αw, ∗} } the associated spectral family of {p ^αw, t ^∗ } t>0 , that is, p ^αw, _t ^∗ = ∫ _∞

0 e ^−γt dF _γ ^{αw, ∗} . Take a large enough α > 0 satisfying c < r(α). For an open ball B such that B ⊂ D, define the function g on D by g(x) = I _B (x) − m α (B). Then

 P ^αw, f

^∗ (X(e ^αr(α) ) ∈ B) − m α (B)   (19)

=

∫

D

 E ^αw, x ^∗

(

g(X(e ^αr(α) )) )  f α (x) dx

≤

∫

{f≥N}

f (x) dx + α ^λn N

∫

D

p ^{αw, ∗}

e

g(x) dx

≤

∫

{f≥N}

f (x) dx + α ^λn N Z α e ^αr p ^αw, _e

^∗ g

L

(D;m

) .

On the other hand, by Lemma 2.3 and the fact that ∥g∥ L

(D;m

) = 0, p ^αw, _e

^∗ g ²

L

(D;m

) =

∫ _∞

γ(α)

exp

( −2γe ^αr(α) ) d (

F _γ ^{αw, ∗} g, g )

L

(D;m

)

≤ exp(−2γ(α)e ^αr(α) ∥g∥ ² L

(D;m

)

≤ exp (

−2e ^{α(r(α) −c−ε)} )

for any ε ∈ (0, r(α) − c). Applying this relation to (19), we obtain the lemma

by letting α → ∞ and N → ∞. □

Now, we are prepared to prove our main theorem.

Proof of Theorem 1.1. In view of (7) and its related remark mentioned right after, it holds that

{ α ⁻² X (e ^αr ), P f

} d

=

{ X (e ^αr(α) ), P f ^α

} ,

where r(α) = r − (2λ/α) log α, P x ^α (dwdω) = Q(dw)P ^αw _x (dω) and f _α is the scaled function of f defined in (4). Using this, we shall prove that for any ε > 0

P f ^α

( d H

( X (e ^αr(α) ), D r (w) )

> ε

) −→ 0 as α → ∞.

First, take ε ^′ such that ε ^′ ∈ (0, r − r(α)) and apply k = e ^αε

to (17). Then by Lemma 3.1(i), we have

P ^αw _f

α

( τ _D ^αw

r

(w) < e ^αr(α) )

= P ^αw, _f ^∗

α

( τ _D ^αw

r

(w) < e ^αr(α) )

≤ e

∫

D

(w)

h ^αw _e

_e

f _α dx

≤ e

∫

D

(w)

h ^αw _e

_θ(α) f α dx −→ 0 as α → ∞ which implies that for any ε > 0, X (e ^αr(α) ) ⊂ U ε (D _r (w)), P f ^α

-a.s. as α → ∞.

Now, it remains to prove that under P f ^α

,

(20) D r (w) ⊂ U ε ( X (e ^αr(α) )) as α → ∞.

To this end, let B(x) be an open ball of D r (w) centered at x with radius ε ^′′ /2, where x ∈ D r −ε

/2 (w) and ε ^′′ ∈ (0, ε). By the same reason in the proof of Lemma 4.2, we may assume that inf _D

_(w) w = 0. Consider a modified environment w(x) of w(x) on D e r (w) relative to B(x) defined as follows:

e

w(x) = w(x) on B(x) ^c , w(x) e ≤ w(x) on B(x) and

inf

x ∈B(x) w(x) = e −δ (δ > 0).

For this w, let consider the Dirichlet form ( e E ^{α w e} , H _∗ ¹ (D _r (w))), the associated diffusion {X(t), P ^α x ^{w, e ∗} , x ∈ D r (w) ^∗ }, the normalized underlying measure e m _α on D _r (w), the depth e d and the critical depth ec of D r (w):

d = d(D e r (w), w), e ec = c(D r (w), w) = e sup

x,y ∈D

(w)

c x,y ( w) e with

c x,y ( w) = inf e

ϕ sup

t ∈[0,1] w(ϕ(t)) e − e w(x) − e w(y) − δ,

in a similar way of Section 2. Then since D _r (w) is also a sub-level domain of e

w, it is easy to check that m e _α (B(x)) → 1 as α → ∞ and

(21) ec < r + δ.

In particular, we claim that (21) can be regarded as ec < r by choosing suffi- ciently small δ > 0. Indeed, let e E(r 0 ) be a connected component of the level set {x ∈ D r (w) : w(x) < r e 0 , sup _B(x) w < r 0 < r } containing B(x). Then,

ec 1 := sup

x ∈D

(w) \ e E(r

), y ∈ e E(r

)

c _x,y ( w) e and ec 2 := sup

x,y ∈D

(w) \ e E(r

)

c _x,y ( w) e are strictly less than r by the definition of the critical depth. On the other hand, since e E(r 0 ) is also a sub-level domain of w, e

ec 3 := sup

x,y ∈ e E(r

)

c _x,y ( w) = c( e e E(r ₀ ), w) < d( e e E(r ₀ ), w) = r e ₀ + δ

and thus, ec 3 is also strictly less than r by choosing the sufficiently small δ > 0.

Noting ec = max{ec 1 , ec 2 , ec 3 } we conclude that the claim is true. Therefore we see that ec < r(α) for sufficiently large α > 0 and

P ^α _f ^{w, e ∗}

α

(

σ ^α _B(x) ^{w e} < e ^αr(α)

) ≥ P ^α _f

^{w, e ∗} ( X

( e ^αr(α)

) ∈ B(x) )

−→ 1 as α → ∞ by virtue of Lemma 4.2. Here σ ^α _B ^{w e} denotes the first hitting time of the diffusion X(t) to B. Since the processes {X(t), P ^αw x } and {X(t), P ^α x ^{w e} } have the same law on B(x) ^c ,

P ^αw _f

(

σ ^αw _B(x) < e ^αr(α) )

= P ^α _f ^{w, e ∗}

α

(

σ _B(x) ^{α w e} < e ^αr(α)

) −→ 1 as α → ∞

which implies that B(x) ⊂ U ε ( X (e ^αr(α) )), P ^αw _f

α

-a.s.. Hence we have D r −ε

(w) ⊂ U ε ( X (e ^αr(α) )) as α → ∞

under P f ^α

. Letting ε ^′′ → 0, we can derive (20) from Lemma 4.1. □

Remark 4.3. (i) Excepting the one dimensional case, we do not know how to

determine the compact set K(w, r) ⊂ R ⁿ such that under P f , α ^−λ X (e ^αr )

converges in probability to K(w, r) as α → ∞ without the second assumption

of (A.2).

(ii) It is possible to apply our result to an arbitrary initial distribution (that is, a point). To deduce this, one may use a priori Gaussian bounds on the transition probabilities.

Acknowledgement. A part of this paper is base on my talk in “International Workshop on Probability Theory and its Applications” held at Seoul National University, Korea. I am grateful to Professor Panki Kim for his kind invitation.

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Department of Mathematics and Engineering Faculty of Engineering

Kumamoto University Kumamoto 860-8555, Japan

E-mail address: [email protected]