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1 Markov Markov / 25

2 1 (GA) 2 GA 3 4 Markov / 25

3 (GA) (GA) L ( 1) I := {0, 1} L f : I (0, ) M( 2) S := I M GA (GA) f (i) i I Markov / 25

4 (GA) ρ(i, j), i, j I 2 i, j (L ) d(x, y), x, y S 2 x, y (LM ) 0 < µ < 1 M µ LM µ x, y S x M µ y : M µ [x, y] := µ d(x,y) (1 µ) LM d(x,y) > 0 Markov / 25

5 (GA) C (i 1,, i L ), (i 1,, i L ) I, π {1,, L}: i k, i k, k π x, y S x C y : C[x, y] ( ) π = {1,, k}(1 k L) = ( ) k {1,, L} 1 Markov / 25

6 (GA) x = (x 1,, x M ), y = (y 1,, y M ) S x k, y k I, k = 1,, M x y {x 1,, x M } {y 1,, y M } β > 0 S β y k g β (y k ) := f (y k ) β / M f (x k ) β (β: ) k=1 x, y S x S β y : { M S β [x, y] := k=1 g β(y k ), x y 0, x y Markov / 25

7 (GA) µ, β GA µ, β GA x, u, v, y S x Mµ u C v S β y : Q µ,β [x, y] := M µ [x, u]c[u, v]s β [v, y] u,v S f (i) i I Markov / 25

8 GA GA Q µ,β = (Q µ,β [x, y]) x,y S : Markov (X n ) S ( Markov ) 0 < µ < 1 = Q µ,β [x, y] > 0 = k s.t. Qµ,β k [x, y] > 0 ( ) x S gcd{n P(X n = x X 0 = x) > 0} 1 ( ) Markov Markov = GA Markov (0 < µ < 1) = (q µ,β (x)) x S [ qµ,β (1) q µ,β (σ) ] = [ q µ,β (1),, q µ,β (σ) ] Q µ,β [1, 1] Q µ,β [1, σ]... Q µ,β [σ, 1] Q µ,β [σ, σ] (S = {1,, σ}) Markov / 25

9 GA (q µ,β (x)) x S β µ GA µ, β Markov / 25

10 GA U := {x = (x 1,, x M ) S x 1 = = x M } U := {x = (x 1,, x M ) U f (x k ) = max f (i), k = 1,, M} i I Davis-Princepe, 1991 β > 0 Suzuki, 1998 lim q µ,β(x) > 0 = x U µ 0 lim lim q µ,β(x) > 0 = x U β µ 0 ( ) Markov / 25

11 Markov (Q α [x, y]) x,y S : α > 0 (q α (x)) x S : Q α q (x) := lim α q α (x), x S x S y S V (x y) := lim α 1 α log Q α[x, y] [0, ] S = {1,, σ} ( ) G(y): y S ( ) V (γ) := (x y) γ W (x) := V (x y), γ G(x) min V (γ) γ G(x) W min := min x S W (x) Markov / 25

12 Freidlin-Wentzell, 1984 Freidlin-Wentzell, 1984 Q α (x) := γ G(x) (u,z) γ 1 (Q α [x, y]) x,y S q α (x) = Q α [u, z] Q α (x) y S Q α(y), x S 2 lim α 1 α log q α(x) = W (x) W min x S q (x) > 0 = W (x) = W min Markov / 25

13 Cerf, 1998 Cert, 1998 S U α = α(µ, β) α µ 0, β S U L, M, f, µ, β W (x), x S\U > W (x), x U = S U Markov / 25

14 Albuquerque-Mazza, 2001 := max log f (i) log f (j) i,j I µ(β) = ϵ exp( λβ) 0 < ϵ < 1, λ > 0 f : I (0, ) 1 1 Albuquerque-Mazza, S U 2 λ > M = U S Markov / 25

15 := max log f (i) log f (j) i,j I µ(β) = ϵ exp( λβ) 0 < ϵ < 1, λ > 0 f : I (0, ) λ > M M 1 = U S λ > 2 M 2 GA Markov / 25

16 : 1 ( ) k {1,, L} 1 ( 1 M 0 ) x y def C[x, y] > 0 Markov / 25

17 V (x y) x = (x 1,, x M ) S V (x) := M k=1 log f (x k) V (x y) = lim β 1 β log Q β[x, y] = min {λd(x, u) + V (y) min V (r)} u v,v y v r Markov / 25

18 V (x y) ( ) : M β [x, y] := µ(β) d(x,y) (1 µ(β)) LM d(x,y) > 0 g β (y k ) := f (y k ) β M k=1 f (x k) β C[x, y] > 0 S β [x, y] := { M k=1 g β(y k ), x y 0, x y Markov / 25

19 o(β) lim β β = 0 Q β [x, y] := V (x y) ( ) u,v S 1 β log Q β[x, y] := 1 β log{ M β [x, u]c[u, v]s β [v, y] u,v S = 1 β log u v,v y M β [x, u]c[u, v]s β [v, y]} exp{δ(u, v)β + o(β)} max δ(u, v) u v,v y = min {λd(x, u) + V (y) min V (r)} u v,v y v r Markov / 25

20 S U S S S + := S\S 2 1 x S + V (x y) = 0 y S 2 x S +, y S V (y x) > 0 = q (x) > 0 = x S S := U S + := S\U V (x y) = min {λd(x, u) + V (y) min V (r)} u v,v y v r 2 q (x) > 0 = x U Markov / 25

21 S U φ : S [0, ) def V (x y) [0, ) (x, y) S 2 V (x y) V (y x) = φ(y) φ(x) 1 φ : S (, ) x S q (x) = 0 = φ(x) > min y S φ(y) W V (x) = M log f (i), x = (i) U, i I S := U Markov / 25

22 S U ( ) V ((i) (j)) = f (i) > f (j) min {λd((i), u) + V (y) min V (r)} u v,v (j) v r 1 j j M 2 j M j M 3 j 1 f (i), f (j) M 1 j M min{λρ(i, j) + V ((j)) V ((i)), Mλρ(i, j), λ[ρ(i, j) + M 1]} V ((i) (j)) min{λρ(i, j) + V ((j)) V ((i)), Mλρ(i, j)} λ > M V ((i) (j)) = λρ(i, j) + V ((j)) V ((i)) M 1 Markov / 25

23 S U ( ) V ((i) (j)) V ((j) (i)) = M(log f (i) log f (j)) = V (j) V (i) f (i) f (j) V ((i) (j)) = λρ(i, j) f (i) > f (j),f (i) f (j) V ((i) (j)) V ((j) (i)) = V (j) V (i) ( ) Markov / 25

24 : M = 2 β f 1 1 i 1 i L i 1 i L M β i 1 i L i 1 i L C j 1 j L j 1 j L S β j 1 j L j 1 j L {i h, i h } = {j h, j h }, h = 1,, L Markov / 25

25 GA Markov 1 µ 0, β T 0 2 µ β 1 Albuquerque-Mazza, f : I (0, ) Cerf, 1998 Markov / 25

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