α
≤ exp ε ˜−(1/γ ) −
k (α ) − I (f ) + δ1 τ 1/γ
.
Now we take ε ˜1/γ = (LL(tn ))−1 and τ = (1 − ε) have Cr,p = Cr,p (LL(tn ))1−α w(tn ·) tn LL(tn ) −f
α
k(α) 1−I (f )
r
Dr,p ´Wiener•¼ Sobolev˜m, =
p, F
r,p
= (1 − L)r/2 F
∈ Lp , r
0, 1
p < ∞.
Lp ˜ m§ , L ´(B, H, µ) þ
Ornstein-Uhlenbeck Ž f.
0, p > 1, (r, p)−NݽÂXe Cr,p (O) = inf { F
n→∞
γ
w(tn ·) tn LL(tn )
−f
α
≥
k (α ) 1 − I (f )
γ
,
Cr,p − q.s.
(3.2)
Ún 2.5
For any f ∈ K with I (f ) < 1, we have w(t·) tLL(t) −f
α
lim inf (LL(t− I (f )
'…c1; '…c2; '…c3; '…c4 XXXX ; XXXX © Ù ? Ò: ¥ ã © a Ò: XXXX A 0255-7797(2007)07-0001-00
MR(2010)ÌK©aÒ: © z I £ è:
1 Úó
•Ä²; Wiener˜m(B, H, µ) , Dr,p = (1 − L)− 2 Lp , F Ù ¥Lp P •(B, µ)þ ér ¢Š¼ê
p r,p ; F
∈ Dr,p , F
1, µ − a.s. on O},
ém8 O ⊂ B,
…é?¿8ÜA ⊂ B k, Cr,p (A) = inf {Cr,p (O); A ⊂ O ⊂ B, O ´m8}.
d ë Y ¼ ê ˜ m,D ƒ þ( . ‰ ê f = sup0 t 1 |f (t)| . PC0 = {f ∈ t 1 2 d d d d 2 ˙ ˙ C ; f (0) = 0}, H = {f ∈ C0 ; f (t) = 0 f (s)ds, f Hd = 0 |f (t)| dt < ∞}, H ´ ˜ ½
k k n 1/q2
n ≤ c δ ˜i •Fi Ù¥F Ún 2.2
(i) Fε (w) = ε
1 + max Fi
1≤i≤n
k,kq1
µ
i=1
{ai − δ < Fi (z ) < bi + δ }
[7,ëY?
.
k, p, q1 , q2 XÚn2.1¥½Â. é?¿f ∈ K 9ε > 0, w(ti + ·hi ) − w(ti ) √ hi −f
α
?˜Ú, é?¿f ∈ K †γ = (1 − 2α)/2k
ε→ 0
lim ε1/γ log P
w− ©
1 2
f ε1/(2γ )
≤ rε
α
= −I (f ) −
k (α ) . r1/γ
(1.2)
Ú\PÒLL(t) = log log t, ½n 1.1
t→∞
̇(ØXeµ k (α ) 1 − I (f )
Similar to the proof of (5.3) in [2], we have f( Tn ·) − f (·) tn ≤2
α
Tn −1 tn
1 2 −α
≤2
tn+1 −1 tn
1 2 −α
.
(3.5)
By the inequatlity exp(−x) ≥ 1 − x, we have tn n+1 n ≥1− + . tn+1 (log(n + 1))a (log n)a
γ
0<α<
1 , 2
γ=
− α , …f ∈ K . ef ÷vI (f ) < 1, Kk −f
α
1−α
lim inf LL(t) Ù¥k (α) > 0 X(1.1)¤½Â.
w(t·) tLL(t)
=
,
Cr,p − q.s.
(1.3)
2 ½n1.1 y²
!0 Ú n 2.1 áµ
n 1/p
eZÚn§½n1.1 y²òd±eÚn ¤. [Ú n2.1 in [7] ]
lim ε1/γ log Cr,p lim ε1/γ log µ k (α ) − I (f ). τ 1/γ
=
ε→ 0
= − y².
k, p, q1 , q2 XÚn¥¤½Â. Cr,p (·) ≥ µ(·), K•Iy²eª=Œµ
ε→0
lim ε1/γ log Cr,p lim ε1/γ log µ
α
,
0 ≤ ti < ∞, hi > 0, i = 1, 2, · · · , n.
K•3˜~êc = c(k, p, q1 , f ) > 0, é?¿δ ∈ (0, 1], ε ∈ (0, 1], k
n
1 p
Ck,p
(i) (z ) < bi } {z : ai < Fε i=1 n
1 q2
≤ cδ
γ
. Then for n large enough, we
≤ (1 − ε)
k (α ) 1 − I (f )
γ
k (α) w(tn ·) 1 √ − (LL(tn ))1/2 f ≤ (LL(tn ))− 2 +α (1 − ε) 1 − I (f ) tn α η0 1 − I (f ) 1 − I (f ) + δ 1 . = ≤ exp LL(tn ) − 1 /γ (1 − ε) log tn By Borel-Cantelli’s Lemma lim inf (LL(tn ))1−α
=
sup
s, t∈[0,1]
s= t
|f (t)−f (s)| |t−s|α
< ∞},
C
∗
α, 0
= {f ∈ C ; lim
α
δ →0 s, t∈[0,1]
0<|t−s|<δ
sup
|f (t)−f (s)| |t−s|α
= 0},
ÂvFÏ: XXXX-XX-XX ÂFÏ: XXXX-XX-XX Ä7‘8µXXXXÄ7]Ï (12345678); XXXXÄ7]Ï (12345679). Š ö { 0: 1 ˜ Š ö(Ñ )ž m–), 5 O, ¬ x(Ç x Ž ), 7 0, … ¡, Ì ‡ ï Ä • •: xxxxxx. E-
w(t + h·) − w(t) f √ − 1/(2γ ) ≤ ετ ε h α f w(·) − 1/(2γ ) ≤ ε(τ + δ ) ε α
-δ → 0, q2 → p, Ún Ún 2.4
For f ∈ K with I (f ) < 1, we have w(tn ·) tn LL(tn ) −f
γ
,
Cr,p − q.s.
(3.3)
w(ts) Proof. Let ψt (s) = √ , s ∈ [0, 1]. Let tn be as in Lemma . For tn < t ≤ tn+1 , set tLL(t)
X (t) = (LL(t))1−α ψt (·) − f x=
utn ,y Tn
C d • l[0, 1]
Rd
ÂXeSÈ
d (C 0 , Hd , µ)
Hilbert ˜ m r1 , r2
Hd
=
1 (r ˙ 1 (s), r ˙2 (s))ds. 0
d µ ´C0 þ
Wienerÿ Ý, l
´˜²;Wiener˜m.
e¡•ÄXeü‡Banach˜m
d C α = {f ∈ C0 ; f (·) α
z
2 Hd
I (z ) = PK = {f ∈ Hd ; I (f ) 0, ¦
ε→0
2
, if z ∈ Hd , Ù¦. •3k (α) ≥ (1.1)
∞,
1}.
{ω (t) : t ≥ 0}´˜IOÙK$Ä.©[2]y² ≤ ε} = −k (α).
lim ε2/(1−2α) log P { w
−2k2 −k k
n µ
{z : ai − δ <
i=1
(i) Fε (z )
< bi + δ }
.
y². A^[3]¥½n4.4§Ó[4]¥Ún2.4aq y. Ún 2.3 ¿τ > 0 k
ε→ 0
,γ= 0<α< 1 2
1 2
− α, t ≥ 0§f ∈ K , k (α) > 0 X(1.1)¥¤½Â§Ké? f w(t + h·) − w(t) √ − 1/(2γ ) ≤ ετ ε h α w(t + h·) − w(t) f √ − 1/(2γ ) ≤ ετ ε h α
≤
sup
tn 0≤x<y ≤ T n
|x − y |α
α
≤ γn ψTn (·) − f (·) √ Tn LL(Tn ) where we use notation γn = √ .
tn LL(tn )
+ |γn − 1| · f (·)