2011 8 26

3 I 5 1 7 1.1 Markov................................ 7 2 Gau 13 2.1.................................. 13 2.2............................... 18 2.3............................ 23 3 Gau (Le vy ) 25 3.1......................... 25 3.2................................... 25 3.3................................... 25 3.4 Fokker-Planck....................... 25 II 27 4 29 4.1................................ 29 4.2 Fokker-Planck....................... 29 5 31 5.1.................................. 31 5.2................................ 31

I

7 1 1.1 Markov t 1 τ 1... t 1,..., t n x 1,..., x n p( x 1, t 1 ;... ; x n, t n ) (1.1) p( x 1, t 1 ;... y 1, τ 1 ;... ) p( x 1, t 1 ;... ; t 1, τ 1 ;... ) p( y 1, τ 1 ;... ) (1.2) Markov Markov p( x 1, t 1 ;... y 1, τ 1 ;... ) = p( x 1.t 1 ;... y 1, τ 1 ) (1.3) ( ) Chapman-Kolmogorov Markov p( x 1, t 1 x 3, t 3 ) = d x 2 p( x 1, t 1 x 2, t 2 )p( x 2, t 2 x 3, t 3 ) (1.4)

8 1 ϵ > 0 lim dt +0 1 d xp( x, t + dt z, t) = 0 (1.5) dt x z >ϵ (Lindeberg ) Chapman-Kolmogorov i)ii)iii) i) p( x, t + t z, t) lim = W ( x z, t) t +0 t (1.6) ( x z < ϵ x, z, t ) ii) iii) lim t +0 lim t +0 1 t 1 t x z <ϵ x z <ϵ d x(x i z i )p( x, t + t z, t) = A i ( z, t) + O(ϵ) (1.7) d x(x i z i )(x j z j )p( x, t + t z, t) = B ij ( z, t) + O(ϵ) (1.8) Chapman-Kolmogorov p( z, t y.t ) t = [A i ( z, t)p( z, t y, t )] + 1 2 [B ij ( z, t)p( z, t y, t )] z i i 2 z i,j i z j + d x[w ( z x, t)p( x, t y, t ) W ( x z, t)p( z, t y, t )] (1.9) ϕ( η) < e i η ˆ x >= d xp( x)e i η x (1.10)

1.1 Markov 9 1 (1) (1.6) W ( x z, t) = 0 (for x z) (1.11) 0 2 (2) (1) p(x, t + dt z, t) = 1 (x z)2 e 2σ 2 dt 2πσ2 dt (Wiener ) (2) p(x, t + dt z, t) = dt π[(x z) 2 + dt 2 ] (Cauchy ) 3 Chapman-Kolmogorov (1) (1.6),(1.7) 2 3 1 d x(x i z i )(x j z j )(x k z k )p( x, t + t z, t) = C ijk ( z, t) + O(ϵ) t lim t +0 x z <ϵ C( α, z, t) ijk (1.12) α i α j α k C ijk ( z, t) (1.13) C C( α, z, t) = O(ϵ) 4 Chapman-Kolmogorov (2) Chapman-Kolmogorov 2 f( z) d d xf( x)p( x, t y, t ) (1.14) dt

10 1 (1) (1.14) 1 (1.14) = lim t +0 t { d x d zf( x)p( x, t + t z, t)p( z, t y, t ) d zf( z)p( z, t y, t )} (1.15) (2) (1.15) 1 (1.15) = lim t +0 t { d xd z[ (x i z i ) f + 1 (x i z i )(x j z j ) 2 f ] x z <ϵ z i i 2 z i,j i z j + + + x z <ϵ x z ϵ x z <ϵ p( x, t + t z, t)p( z, t y, t ) (1.16) d xd z x z 2 R( x, z)p( x, t + t z, t)p( z, t y, t ) (1.17) d xd zf( x)p( x, t + t z, t)p( z, t y, t ) (1.18) d xd zf( z)p( x, t + t z, t)p( z, t y, t ) (1.19) d xd zf( z)p( x, t + t z, t)p( z, t y, t )} (1.20) R( x, z) Taylor f( x) = f( z) + i f( z) (x i z i ) + 1 2 f( z) (x i z i )(x j z j ) + x z 2 R( x, z) z i 2 z i,j i z j (3) (1.16) d z[ A i ( z) f + 1 B ij ( z) 2 f ]p( z, t y, t ) + O(ϵ) (1.21) z i i 2 z i,j i z j (4) (1.17) (1.17) = o(ϵ) (5) (1.18),(1.19),(1.20) d xd zf( z)[w ( z x, t)p( x, t y, t ) W ( x z, t)p( z, t y, t )] (1.22) x z ϵ (6) Chapman-Kolmogorov 5 Wiener p(x, t + dt z, t) = 1 (x z)2 e 2σ 2 dt 2πσ2 dt

1.1 Markov 11 Chapman-Kolmogorov p(x 0 ) = δ(x 0 ) 6 Wiener 5 7 Gau d ˆX dt = γ ˆX + ˆξ (1.23) ˆξ p(ξ) = 1 ξ2 e 2σ 2 dt (1.24) 2πσ2 dt A(x, t), B(x, t), W (x z, t) *1 Chapman- Kolmogorov 8 Cauchy p(x, t + dt z, t) = dt π[(x z) 2 + dt 2 ] Chapman-Kolmogorov 9 Poion p(x, t + dt z, t) = λδ(z + 1 x)dt (1.25) Chapman-Kolmogorov 10 Poion 9 11 *1 d ˆX dt = γ ˆX + ˆξ (1.26)

12 1 ˆξ ξ 4 p(ξ) = (4dt) 1 4 σγ( 1 4 ) e 2σ 4 dt (1.27) A(x, t), B(x, t), W (x z, t) Chapman- Kolmogorov dt +0 14 dt +0 12 Lévy Gau Gau Lévy 13 Lévy 12 Lévy 14 -Lévy Gau Poion ( : 13)

13 2 Gau 2.1 [, v] ( < v) n t 0 = < t 1 = 1 n (v ) < < t i = i n (v ) < < t n 1 = v  <  > dap (a)a n + dt v n 0 Gau Gau * 1 dt ξ p(ξ) = 2 2σ 2 dt (2.1) 2πσ 2 e Wiener 1 Gau ˆξ(t) Wiener Ŵ (t) t dt ˆξ(t ) = lim n + n i=1 t n ˆξ(t i ) (2.2) dŵ (t) = ˆξ(t)dt Wiener (1) < dŵ >= 0 (2) < dŵ 2 >= dt (3)dŴ 2 = dt (4)dŴ n = 0 (n > 2) (2.3) *1 σ = 1 Gau

14 2 Gau (3),(4) Mean Square Mean Square (M S) lim n + ˆX n = ˆX lim < ( ˆX n ˆX) 2 >= 0 (2.4) n + v (I) dŵ (t)f(ŵ (t)) lim n 1 n + i=0 [Ŵ (t i+1) Ŵ (t i)]f(ŵ (t i)) (2.5) df[ŵ ] = f W dŵ + 1 2 f dt (2.6) 2 W 2 Stratonovich v (S) dŵ (t)f(ŵ (t)) lim n 1 n + i=0 [Ŵ (t i+1) Ŵ (t i)]f(ŵ (t i) + Ŵ (t i+1) 2 Stratonovich ) (2.7) v v (S) dŵ (t)f(ŵ (t)) = (I) dŵ (t)f(ŵ (t)) + 1 v dt df (Ŵ (t)) 2 dw

2.1 15 1 Gau (1) < ˆξ(t) > (2) < ˆξ(t) 2 > (2.8) p(ξ(t i )) = 1 ξ2 e 2σ 2 dt (2.9) 2πσ2 dt Gau δ(0) 1 dt < ˆξ(t )ˆξ(t ) > δ(t t ) 2 Wiener (3) Mean Square (1) < dŵ >= 0 (2) < dŵ 2 >= dt (3)dŴ 2 = dt (2.10) 3 (1) (1) (I) v dŵ (t )Ŵ (t ) (2) (I) v dŵ (t )[Ŵ (t )] 2 v (3) (I) dŵ (t )[Ŵ (t )] n d[ŵ (t)]2 = [Ŵ (t) + dŵ (t)]2 Ŵ (t)2 = 2Ŵ (t)dŵ (t) + [dŵ (t)]2 = 2Ŵ (t)dŵ (t) + dt d[ŵ (t)]n = nŵ (t)n 1 dŵ n(n 1) (t) + Ŵ (t) n 2 dt 2 x n x n+1 dŵ (t)n (n 2) (2.3)

16 2 Gau 4 Stratonovich (1) (1) (S) v dŵ (t )Ŵ (t ) (2) (S) v dŵ (t )[Ŵ (t )] 2 v (3) (S) dŵ (t )[Ŵ (t )] n Stratonovich 5 ( Taylor ) df[ŵ ] = f W dŵ + 1 2 f dt (2.11) 2 W 2 6 ( (1)(2) 3 ) (1) d[ŵ 2 ] (2) d[ŵ n ] (3)d[eŴ ] (4)d[in Ŵ ] (5) d[log Ŵ ] 7 f(x) F (x) dxf(x) Stratonovich (S) v dŵ (t )f(ŵ (t )) = F (Ŵ (v)) F (Ŵ ()) (2.12) Stratonovich 8 Stratonovich (2) 4 7 9 Stratonovich (S) v v dŵ (t)f(ŵ (t)) = (I) dŵ (t)f(ŵ (t)) + 1 2 v dt df (Ŵ (t)) dw

2.1 17 10 (1) Stratonovich (1) (S) v (t) dŵ (t)e Ŵ (2) (S) v dŵ (t) log Ŵ (t) (3) (S) v dŵ (t) co Ŵ (t) 11 (2) Stratonovich (1) (I) v (t) dŵ (t)e Ŵ (2) (I) v dŵ (t) log Ŵ (t) (3) (I) v dŵ (t) co Ŵ (t) 12 (2) 11 (1) (3) ( 7) Stratonovich Stratoinovich 13 Stratonovich lim n + n i=0 [Ŵi+1 Ŵi]f(Ŵi+1 + Ŵi 2 ) = lim n + n i=0 [Ŵi+1 Ŵi] f(ŵi+1) + f(ŵi) 2 Stratonovich Stratonovich * 2 *2 Stratonovich ( )

18 2 Gau 2.2 d ˆX dt = f( ˆX) + g( ˆX)ˆξ (2.13) ˆξ Gau ˆX(t) = ˆX(0) + t 0 df( ˆX()) + t 0 dŵ ()g( ˆX()) (2.14) 3 v (I) dŵ (t)g( ˆX(t)) n 1 lim n + i=0 [Ŵ (t i+1) Ŵ (t i)]g( ˆX(t i )) (2.15) t t ˆX(t) = ˆX(0) + df( ˆX()) + (I) dŵ ()g( ˆX()) (2.16) 0 0 (I)d ˆX = f( ˆX)dt + g( ˆX)dŴ (2.17) Stratonovich Stratonovich v (S) dŵ (t)g( ˆX(t)) n 1 lim n + i=0 [Ŵ (t i+1) Ŵ (t i)]g( ˆX(t i ) + ˆX(t i+1 ) 2 ) (2.18)

2.2 19 Stratonovich Stratonovich ˆX(t) = ˆX(0) + t 0 df( ˆX()) + (S) t 0 dŵ ()g( ˆX()) (2.19) (S)d ˆX = f( ˆX)dt + g( ˆX)dŴ (2.20) Stratonovich (I)d ˆX = a( ˆX)dt + b( ˆX)dŴ (2.21) (I)df[ ˆX] = f x [a( ˆX)dt + b( ˆX)dŴ ] + b2 ( ˆX) 2 f dt (2.22) 2 x2 Stratonovich (S)d ˆX = α( ˆX)dt + β( ˆX)dŴ (2.23) (S)df[ ˆX] = f x [α( ˆX)dt + β( ˆX)dŴ ] (2.24) (1) (I)d ˆX = adt + bdŵ (2.25) (S)d ˆX = (a 1 2 b xb)dt + bdŵ (2.26) (2) (S)d ˆX = αdt + βdŵ (2.27) (I)d ˆX = (α + 1 2 β xβ)dt + βdŵ (2.28) Stratonovich Stratonovich Stratonovich

20 2 Gau 1 (I)d ˆX = a( ˆX)dt + b( ˆX)dŴ (2.29) 2 f(x) (I)df[ ˆX] = f x [a( ˆX)dt + b( ˆX)dŴ ] + b2 ( ˆX) 2 f dt (2.30) 2 x2 2 (1) (I)d ˆX = ˆXdt + dŵ (2.31) (1) d[ ˆX 2 ] (2) d[ ˆX n ] (3)d[e ˆX] (4)d[in ˆX] (5) d[log ˆX] 3 (2) (I)d ˆX = ˆXdt + ˆXdŴ (2.32) (1) d[ ˆX 2 ] (2) d[ ˆX n ] (3)d[e ˆX] (4)d[in ˆX] (5) d[log ˆX] 4 (3) ˆX(0) = x 0 (I)d ˆX = ˆXdt + σ ˆXdŴ (2.33) σ2 { (1+ ˆX(t) = x 0 e 2 )t+ŵ (t) Ŵ (0)} (2.34)

2.2 21 5 Stratonovich Stratonovich (S)d ˆX = α( ˆX)dt + β( ˆX)dŴ (2.35) 2 f(x) (S)df[ ˆX] = f x [α( ˆX)dt + β( ˆX)dŴ ] (2.36) 6 Stratonovich (1) Stratonovich (S)d ˆX = ˆXdt + dŵ (2.37) (1) d[ ˆX 2 ] (2) d[ ˆX n ] (3)d[e ˆX] (4)d[in ˆX] (5) d[log ˆX] 7 Stratonovich (2) Stratonovich (S)d ˆX = ˆXdt + ˆXdŴ (2.38) (1) d[ ˆX 2 ] (2) d[ ˆX n ] (3)d[e ˆX] (4)d[in ˆX] (5) d[log ˆX] 8 ˆX = ˆX = t o t 0 t ad + (I) bdŵ (2.39) 0 (a 1 t 2 b xb)dt + (S) bdŵ (2.40) 0 (I)d ˆX = adt + bdŵ (S)d ˆX = (a 1 2 b xb)dt + bdŵ (2.41)

22 2 Gau 9 (1) ˆX(0) = x 0 (1) (S)d ˆX = γ ˆXdt + dŵ (2.42) (2) (S)d ˆX = ( γdt + σdŵ ) ˆX (2.43) (3) (S)d ˆX = ˆX 2 dŵ (2.44) 10 (2) ˆX(0) = x 0 (1) (I)d ˆX = γ ˆXdt + dŵ (2.45) (2) (I)d ˆX = ( γdt + σdŵ ) ˆX (2.46) (3) (I)d ˆX = ˆX 3 dt + ˆX 2 dŵ (2.47)

2.3 23 2.3 Fokker-Planck 1 Fokker-Planck (1) 2 Fokker-Planck (2)

25 3 Gau (Le vy ) 3.1 3.2 3.3 3.4 Fokker-Planck

II

29 4 4.1 4.2 Fokker-Planck

31 5 5.1 5.2