apj1 SSGA* hapla P6 _1G hao1 1Lh_PSu AL..AhAo1 PJ"AL hp_aa

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1 n n 1/2 n (n 1) 0/1

2 l 2

3 E x X X x X E x X g(x) := 1 g(x). X f : X C L p f p := (E x X f(x) p ) 1/p f,g := E x X f(x)g(x) x X X X X := {f : X [0, ) : f 1 =1}. X µ A A X x X µ A (x) :=α 1 1 A (x) 1 A A α := A / X A = µ A f X f (f) :=E x X f(x) f(x). f = µ A A X (f) = (α 1 ) H(Y ) Y X f(x) = X P[Y = x]

4 H(Y ):= x X P[Y = x] 1 P[Y = x] = X (f). H(Y ) X (f) 0 (f) f f,g X f g (f) (g) D KL (f g) :=E x X f(x) f(x) g(x) D KL (f 1) = (f) f(x) = X P[Y = x] Y Z X D KL (Y Z) := x X P[Y = x] + (Y ) (Z) P[Y = x] P[Z = x], Z Y Z Y δ(y,z) Y,Z 1 δ(y,z) 2 D KL(Y Z). P[Z S] δ(y,z) P[Y S] P[Z S]+δ(Y,Z),

5 D KL (Y Z) Z S Y S X X 1/2 l 2 F L 2 (X) φ : X R φ 1 F h X h F := h, φ φ F d 1 R L 2 (X) F d R(x) = d (1 + ϵ i φ i (x)) i=1 ϵ 1,...,ϵ d { 1, 0, 1} φ 1,...,φ d F φ i X f X F F 0 <ϵ<e 3 f X g X f g F ϵ F F F 9ϵ 2 (f) g F ( ) ϵ d 18ϵ 1 1 (f)+o. ϵ 1 T>0 {g t : t [0,T]} X

6 g 0 := 1 g t := ( t φ 0 sds) E ( t φ 0 sds) t [0,T] s φ s s φ s [0,t 1 ), [t 1,t 2 ),...,[t i 1,t i ) i N 0 <t 1 <t 2 < <t i t i+1 φ s [t i,t i+1 ) φ F g ti,φ f,φ > 2ϵ/3, t i+1 := {t t i : g t,φ f,φ ϵ/3}, φ s := ( f g ti,φ )φ s [t i,t i+1 ) φ t i T := t i I := i g t φ (g) g X f g F η L(g, λ) = (g) φ F λ φ( f g, φ η) g (g) = φ F λ φφ g = ( φ F λ φφ) f g F g φ l 2 D KL (f g t ) g t f (f) f T I T 3ϵ 1 (f)

7 t [0,T) d dt D KL(f g t )= d dt E[f f g t ]= E(f/g t ) d dt g t, g t t g t (φ t g t,φ t ) d dt D KL(f g t )= φ t,g t f. t [t i,t i+1 ) φ t = ( f g ti,φ )φ φ f g t,φ ϵ/3 t [t i,t i+1 ) d dt D KL(f g t )= φ t,g t f = ( g t f,φ ) φ, g t f ϵ/3. D KL (f g 0 )= (f) D KL (f g t ) 0 t T 0 (f) ϵ/3 T I 9ϵ 2 (f) t i t i 1 ϵ/3 i I I I T /(ϵ/3) [t i 1,t i ) i I φ = φ ti 1 φ t t [t i 1,t i ) g t t [t i 1,t i ) d dt φ, g t = φ, g t (φ t g t,φ t ) = φ 2,g t + φ, g t 2 φ 2 g t 1 1. φ t i φ, g ti 1 φ, g ti = φ, f g ti 1 φ, f g ti 2ϵ/3 ϵ/3 = ϵ/3, t i t i 1 ϵ/3 g T X f g T F 2ϵ/3 g T ψ t := t (1 + φ 0 s)ds p d (x) := d x j j=0 j! d g := p d(ψ T ) Ep d (ψ T ) X

8 g g T 1 ϵ/3 g g T F ϵ/3 g T = ( T φ 0 sds) E ( T φ 0 sds) = ( T (1 + φ 0 s)ds) E ( T (1 + φ 0 s)ds) = (ψ T ) E (ψ T ) ( ) ϵ 1 d := 6T + O ϵ 1 d (x) ψ T 2T (x) p d (x) x [0,2T ] (x) (2T )d+1 (d +1)! ϵ/6. p d (ψ T ) (ψ T ) 1 (ϵ/6)e (ψ T ) g g T 1 = p d (ψ T ) Ep d (ψ T ) (ψ T ) E (ψ T ) p d (ψ T ) 1 E (ψ T ) (ψ T ) E (ψ T ) + p d (ψ T ) 1 Ep d (ψ T ) p d(ψ T ) E (ψ T ). 1 ϵ/6 Ep d (ψ T ) 1 Ep d (ψ T ) 1 E (ψ T ) = 1 Ep d(ψ T ) E (ψ T ), Ep d (ψ T ) (1 ϵ/6)e (ψ T ) ϵ/6 g g T 1 ϵ/3 φ F φ 1 g g T F φ g g T 1 ϵ/3 g T f g F f g T F + g T g F 2ϵ/3 + ϵ/3 = ϵ g F φ s F := {φ s : s [0,T)} F F = I 9ϵ 2 (f) ψ T φ F λ φ (1+φ) λ φ p d (ψ T ) g F d

9 G f : Ĝ C f : G C γ Ĝ f(γ) :=E x G f(x)γ(x). f f(x) = γ Ĝ f(γ)γ(x). f,g = f,ĝ, f,g := E x f(x)g(x) f,ĝ := γ f(γ)ĝ(γ). f g L p p (G) l (Ĝ) ρ>0 f : G C ρ f ρ (f) :={γ Ĝ : f(γ) ρ f 1 }. >0 (f) :={γ Ĝ : f(γ) > 0}. ρ f 2 2 = f 2 2 f(γ) 2 ρ 2 f 2 1 ρ (f), γ ρ (f) ρ (f) ρ 2 f 2 2/ f 2 1 f = µ A A G α := A / G ρ (µ A ) ρ 2 α 1

10 A F n 2 s s Γ Ĝ s S Ĝ S s { } Γ ϵ γ γ : ϵ γ { 1, 0, 1}. γ S l 1 A G α ρ>0 ρ (µ A ) s s 18ρ 2 (α 1 ). G = F n 2 (s, d) s, d Γ Ĝ (s, d) S Ĝ S s { Γ ϵ γ γ : ϵ γ Z, } ϵ γ d. γ S γ S

11 Γ Ĝ s s N (s, s) G = F n 2 A G α ρ>0 ρ (µ A ) (s, d) s 2 30 ρ 2 (α 1 ) d 8 (α 1 ). A G = Z/pZ p 2A 2A := {a 1 +a 2 a 3 a 4 : a 1,a 2,a 3,a 4 A} B(K, δ) :={x G : xt/p δ t K} K Ĝ K 233 α 1 (α 1 ) δ =(2 8 (α 1 )) 1 δ = α(2 8 (α 1 )) 1 F = {Rγ,Iγ : γ Ĝ} F F F F F= {Rγ,Iγ : γ Ĝ} δ>0 R : G R F d r : G R F d r G δ (R) d δ (r) (2 F,d) F R = d i=1 (1+ϵ iφ i ) d φ 1,...,φ d F ϵ 1,...,ϵ d { 1, 0, 1} j = 1,...,d φ j Rγ j = 1 2 (γ j + γ j ) Iγ j = 1 2 (γ j γ j ) γ j Ĝ R(γ) 0 γ d S := {γ 1,...,γ d, γ 1,...,γ d } Ĝ >0 (R) (2d, d) γ k

12 S {γ,γ 2,...,γ k, γ, γ 2,...,γ k } S δ >0 (R) d δ (R) >0 (R) δ (R) r = i c ir i F d r G 1= r 1 = i c i R i 1 γ δ (r) i c i Ri (γ) = r(γ) δ r 1 = δ i c i R i 1, i R i (γ) δ R i 1, γ δ (R i ) δ (r) (2 F,d) f = µ A A G (f) (α 1 ) ϵ = ρ/4 g G f g F ρ/4 F F= {Rγ,Iγ : γ Ĝ} F 144ρ 2 (f) g F d 72ρ 1 (f) + O( ρ 1 / ρ 1 ) f g F ρ/4 ρ (f) ρ/2 (g) ρ/2 (g) (2 F,d) ρ (f) δ = α 1/2 (2 8 (α 1 )) 1

13 0 <ρ<e 3 f G S ρ (f) d S ρ 2 ρ(f) d 36 ( ) ρ 2ρ 1 1 (f)+o. ρ 1 ϵ = ρ/(2 2) g G f g F ρ/(2 2) F F F 72ρ 2 (f) g = l c i R i, i=1 l c i R i F d 36 ( ) ρ 2ρ 1 1 (f)+o. ρ 1 R i d i >0 (R i ) ρ (f) {1, 2,...,l} j c j ER j g G l i=1 c ier i = E l i=1 c ir i = Eg =1 γ ρ (f) ρ/2 (g) f g F ρ/(2 2) E j γ,r j /(ER j ) = l c i ER i γ,r i /(ER i ) γ, i=1 l c i R i = γ,g ρ/2, j {1, 2,...,l} γ ρ (f) i=1 P j [ R j (γ) 0]=P j [ γ,r j /(ER j ) > 0] ρ/2, γ ρ (f) ρ/2 >0 (R i ) i E j >0 (R j ) (ρ/2) ρ (f) j >0 (R j ) (ρ/2) ρ (f) S := >0 (R i ) ρ (f)

14 G G C U k k 2 U k f : G C f 2k U k := E x,h1,...,h k G h1 h2... hk f(x), h f(x) :=f(x)f(x + h) f U 2 = f 4, U 2 f U k k 2 f U 2 f U 3 f U k f. U k+1 k k

15 k +2 L k f : G C f 1 m E x1,...,x d f(l i (x 1,...,x d )) f U k+1. i=1 3 k =1 A G U 2 A k =2 U 3 A G α G α 4 G 2 U 3 A p =2 p >2 U 3 δ>0 f : F n p C f 1 f U 3 δ q F n p E x f(x)ω q(x) c(δ), ω := (2πi/p) c(δ) δ U 3

16 f : F n p [ 1, 1] δ>0 (B 1,B 2 ) O δ (1) f = E(f (B 1,B 2 )) + g, g U 3 δ (B 1,B 2 ) O δ (1) O δ (1) E(f (B 1,B 2 )) f f l 2 f : F p n C f 2 1 ϵ>0 η>0 M = (O( (ηϵ) O(1) )) f f = i λ i ω q i + h + l, q i F n p h U 3 ϵ l 1 η i λ i M h M

17 F φ : X [ 1, 1] ϵ, δ > 0 B>1 A f : X [ B,B] f ϵ 1 δ φ F f,φ η η = η(ϵ, B) f : X [ 1, 1] 1 η 2 δ f = k c i φ i + h + l i=1 k η 2 h ϵ l 1 (2B) 1 φ i F i =1,...,k k A F U 3 A f : F n p R + f <ϵ<e 3 g : F n p R + Ef = Eg Q Q 9ϵ 2 (f/ f 1 ) f g = i λ iω q i f = g + h, λ i h Q ϵ q i q i = q q, q Q i q Q i Q i,q i Q ( ) (ϵ Q i + Q i 18ϵ 1 1 ) (f/ f 1 )+O. (ϵ 1 ) F := {Rω q(x), Iω q(x) : q F n p} f/ f 1 F F Q

18 g : F n p R + g G g Q g := g f 1 h U 3 l 2 F {φ : X [0, 1]} k 1 F k k F ϵ > 0 k = ϵ O(1) η = ( ϵ O(1) ) F = {φ : X [0, 1]} ν : X R + η F k ν 1 X F k η f : X R + 0 f ν Ef 1 g : X [0, 1] Eg = Ef f g F ϵ F ν

19 (F) F k (F) k F t R t : R {0, 1} t (x) =1 x t t (x) =0 x<t

20 ϵ, δ > 0 F = {φ : X [0, 1]} ν : X R + f : X R + 0 f ν Ef = δ g : X [0, 1] Eg = δ f g F ϵδ ν k = O(ϵ 2 (ϵ 1 δ 1 )) t R ψ = t (φ) φ k (F) ψ, ν 1 X =Ω(ϵδ). k = (ϵ 1,δ 1 ) φ F k φ, ν 1 X ( (ϵ 1,δ 1 )). F c V X W Ef(V,W) f : X W [0, 1] Ef(V,W) = W (W) V V V (V) Ef(V,W). W W V (V) W (W) c V V Ef(V,W )= Ef(V,W)=c. W W H g : X [0, 1] Eg = δ h H

21 φ F f h, φ >ϵδ. ν 1 X ϵδ F h, φ h(x) =1 x (φ) f ν h ν 1 X,φ = ν, φ 1 X,φ f,φ 1 X,φ = f,φ h, φ = f h, φ, ν 1 X ϵδ F h F F := F (1 F) φ (F) h H, f h, φ >ϵδ g V:= δ 1 H X φ W:= F φ, δ 1 f g φ, δ 1 f g c g (δ 1 H) φ (F) φ F φ, δ 1 f g c, g δ 1 H φ,δ 1 f g c g δ 1 H δ 1 H φ F φ, δ 1 f g >ϵ c>ϵ φ,δ 1 f g c>ϵ g δ 1 H φ,f g >ϵδ g H φ φ F φ F S δ X X

22 φ S δ X 1 S H f 1 S, φ >ϵδ f 1 S t [ϵ/2, 1] f, t ( φ) 1 S, t ϵ/2 ( φ) >ϵδ/2. t t ( φ)(x)dt = φ(x), f, t ( φ) dt = f, φ > 1 S, φ + ϵδ = S, t ( φ) dt + ϵδ. t [ϵ/2, 1] f, t ( φ) 1 S, t ϵ/2 ( φ) ϵδ/2. 1 f, t ( φ) dt = ϵ/2 f, t ( φ) dt ϵ/2 ϵ/2 f,1 X dt ϵ/2 ϵ/2 δ S, t ( φ) dt + ϵδ/2 0 0 f, t ( φ) dt ( 1 S, t ϵ/2 ( φ) + ϵδ/2)dt 1 1 S, t ( φ) dt + ϵδ, φ f 1 S φ ν 1 X t := t ϵ/2 1 S ( t ( φ)) 1 S (x) < 1 x ( t ( φ)) S δ X X φ 1 S (y) =0 y / ( t ( φ)) 1 S, t ( φ) = 1 S, 1 X = δ = f,1 X f, t ( φ),

23 δ X 1 S t ( φ) 1 S f 1 S ν 1 X, t ( φ) = ν, t ( φ) 1 X, t ( φ) f, t ( φ) 1 S, t ( φ), f, t ( φ) 1 S, t ( φ) ϵδ/2 t ( φ) ν 1 X t ( φ) k F (F) t φ k (F) t 2ϵ/5 (φ),ν 1 X =Ω(ϵδ). φ F φ k φ 1,φ 2,...,φ k F φ k = O(ϵ 2 (ϵ 1 δ 1 )) y P φ1,φ 2,...,φ k [ φ(y) φ 1 (y)+φ 2 (y)+ + φ k (y) k ] >ϵ/10 ϵδ/100. Y [ [ ]] φ 1 (y)+φ 2 (y)+ + φ k (y) E φ1,φ 2,...,φ k P y Y φ(y) k >ϵ/10 ϵδ/100, [ [ ] ] φ 1 (y)+φ 2 (y)+ + φ k (y) P φ1,φ 2,...,φ k P y Y φ(y) k >ϵ/10 >ϵδ/10 1/10 y Y φ 1,φ 2,...,φ k φ := 1 k k i=1 φ i P y Y [ φ(y) ] φ(y) >ϵ/10 ϵδ/10

24 y ν 1 X t ϵ/10 (φ),ν t ( φ),ν ϵδ/5 t ( φ),f ϵδ/5 t 2ϵ/5 (φ), 1 X t ϵ/2 ( φ), 1 X + ϵδ/10 = t ϵ/2 ( φ), 1 S + ϵδ/10. t 2ϵ/5 (φ),ν 1 X t ϵ/10 (φ),ν t 2ϵ/5 (φ), 1 X t ( φ),f ϵδ/5 ( t ϵ/2 ( φ), 1 S + ϵδ/10), Ω(ϵδ) t 2ϵ/5 (φ) ν 1 X k O(ϵ 2 (δ 1 )) k δ ϵ m R + V X W Ef(V,W) f : X W [ m, m] 0 <ϵ<1 S V (1),V (2),...,V (S) (V) W (1),W (2),...,W (S) W W (W) Ef(V,W ) 1 i S Ef(V (i),w (i) ) O(mϵ).

25 V V S D KL (V V (1) ) Sϵ 2 V V W (i) V (i) Ef(V (i),w (i) )= W W Ef(V (i),w). W (W) V V Ef(V,W ) 1 i S Ef(V (i),w (i) ) O(mϵ) = 1 i S Ef(V (i),w) O(mϵ). W W W W Ef(V (i),w) V (V) W W Ef(V,W) i W (W) Ef(V,W) V V V V Ef(V,W ) V (V) W W Ef(V,W) O(mϵ). ϵ 0 V V Z X V X Ŷ V Z V Ŷ = Y V D KL(Y Z). V X Ŷ V Z V y V D KL (Y Ŷ )+D KL(Ŷ Z) D KL(Y Z). 0 <η<1 Z X V X Ỹ

26 η Z V Ỹ V Y V D KL (Y Ỹ ) D KL(Y Z)+η. D KL (Y Ỹ ) D KL(Y Ŷ )+η Y V Ŷ Z V Ỹ η Z V V (1) (V) i =1, 2,...,S W (i) W V (i) V (i) P[V (i) = x] ( ϵf(x, W (i) )/2m) P[V (i) = x]; V (i+1) ϵ 2 V (i) (V) W := 1 S S W (i). A, B X h : X [0, 1] A P[A = x] (ϵh(x))p[a = x] 0 ϵ 1 i=1 D KL (B A ) D KL (B A) ( e)ϵ(e[h(b)] E[h(A)] ϵ). D KL (B A) D KL (B A )= x ( P[B = x] P[A = x] P[A = x] = x P[B = x] ) e ϵh(x) y eϵh(y) P[A = y], ( e) ϵeh(b) y e ϵh(y) P[A = y].

27 e z 1+z + z 2 1+z e z D KL (B A) D KL (B A ) ( e) ( ϵe[h(b)] (1 + ϵe[h(a)] + ϵ 2 ) ) ( e)ϵ(e[h(b)] E[h(A)] ϵ) S V V D KL (V V (1) ) Sϵ 2 A = V (i) A = V (i) B = V h(x) = f(x, W (i) )/2m ( Ef(V D KL (V V (i) (i),w (i) ) Ef(V,W (i) ) ) D KL (V V (i) ) ( e)ϵ 2m ) ϵ. V (i+1) ϵ 2 V (i) (V) D KL (V V (i+1) ) D KL (V V (i) )+ϵ 2 ( Ef(V D KL (V V (i) ) D KL (V V (i+1) (i),w (i) ) Ef(V,W (i) ) ) ( e)ϵ 2m 1 S D KL (V V (1) ) D KL (V V (S+1) ) ( e)ϵ S ( Ef(V (i),w (i) ) Ef(V,W (i) ) ( ) 1 i S Ef(V (i),w (i) ) Ef(V,W ) ( e)sϵ ϵ Sϵ 2. 2m D KL (V V (S+1) ) 0 i=1 1 i S Ef(V (i),w (i) ) Ef(V,W ) 2m 2m D KL(V V (1) )+Sϵ 2 ( e)sϵ + ϵ = O(ϵ), ) ϵ ϵ 2. ) ϵ Sϵ 2, o(1)

28 ϵ, δ > 0 F = {φ : X [0, 1]} ν : X R + f : X R + 0 f ν Ef = δ g : X [0, 1] Eg = δ f g F ϵδ ν k = O(ϵ 2 (δ 1 )) t R ψ = t (φ) φ k (F) ψ, ν 1 X =Ω(ϵδ). c =: V V Ef(V,W ) W W Ef(V,W), c = V V Ef(V,W ) 1 i S Ef(V (i),w (i) ) O(ϵ) m =1 V = δ 1 H X W = F V X V (1) V S V V D KL (V V (1) )=D KL (V U X )= (V ) Sϵ 2 S O(ϵ 2 (δ 1 )) V V W W Ef(V,W) ϵ V (i) 1 i S Ef(V (i),w (i) )= 1 i S W W Ef(V (i),w) ϵ, c ϵ φ g H φ,f g ϵδ, φ S F

29 F n p

30 U 3 (G)

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