2次Wiener汎関数について

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1 .. 2 Wiener Wiener / 18

2 Plan Plan Wiener F-L. 3 2 Wiener. 4 2 Wiener. 5 Wiener 2 Wiener / 18

3 1979B4 198M1 Dym-McKean: Fourier series and inegrals Sroock-Varadhan: Mulidimensional diffusion processes Varadarajan: Lie groups, Lie algebras, and heir represenaion Iô-McKean: Diffusion processes and hier sample pahs 1985 he Malliavin calculus and long ime asympoics of cerain Wiener inegrals 1992 Dirichle forms on separable meric spaces A Kähler meric on a based loop group and a covarian differeniaion 2 Wiener / 18

4 まずは思い出話からお写真なども谷口説男九州大学基幹教育院 2 次 Wiener 汎関数について 213 年 9 月 21 日 4 / 18

5 2 Wiener F-L 2 Wiener W, H, µ: Wiener W Banach W coni dense e ilw µdw e 1 2 l 2 H H Hilber l W H H. q L 2 µ:2 Wiener D 3 q q L 2 µ:2 Wiener iff q 1 2 D 2 A + D h + c E[F] W Fdµ, A D2 q, h E[Dq], c E[q]. HEOREM 1.. E[e ζq ] e ζc de2 I ζ Ae I ζ A 1 h,h H ζ C Reζ 1 2 Wiener / 18

6 2 Wiener h W W 1, h θ 2 d W d {w : [, ] R d coni. w }, θ w w h D 2 A + 2 Ah hsds 2 2 A n n π h 2 n h n E[e 1 2 ζh 1 ] cosh ζ E[e 1 2 ζh δ θ ] ζ 1 2π sinh ζ h n 2 n+ 1 2 π sin n+ 1 2 π 2 Wiener / 18

7 2 Wiener v EJP2 w.h.masumoo W W 1, v v D 2 A A n1 θ θ 2 d nπ 2 h n h n ζ 1 E[e 1 2 ζv 2 ] sinh ζ 1 E[e 1 2 ζv 2 ζ δ θ ] sinh 1 2 ζ Ah h n θ 1 2 nπ θ d hs hds { cos nπ 1 } 2 Wiener / 18

8 2 Wiener Lévy s W W 2, s 1 1 Jθ 2, dθ J 1 s 1 2 D 2 A Ah J h 1 2 h A 2n + 1π {h n h n + ĥ n ĥ n } n Z h n 2n+1π cos 1 2n+1π sin 2n+1π, ĥ n Jh n E[e ζs 1 ] cos 1 ζ 2 E[e ζs δ θ ] 1 2π 1 2 ζ sin 1 2 ζ 2 Wiener / 18

9 2 Wiener D- ECP6 2 W W 1, 1 Dh 4 2 θ H s ds d h h D 2 A A n + 1 π 2 n Ah s ds du dvhv u 4 h n h n h n h A a n g n g n D 2 A a n {D g n 2 1} DD 2 A 2 a n D g n g n DD 2 A 2 H 4 a 2 n D g n 2 E[e ζ 2 h ] {coshζ 1/4 cosζ 1/4 } 1/2 E[e ζ 2 h δ θ ] ζ 1/8 π{sinζ 1/4 coshζ 1/4 + sinhζ 1/4 cosζ 1/4 } 1/2 2 Wiener / 18

10 2 Wiener KdV n- JFA4 W W n, q KdV 1 c, ξ p 2 2 d + 1 βξ p, ξ p 2 c, p R n, β R n n, ξ p θ + D p ξ p s ds, D p diag[p] q KdV D 2 A + ra Ah e D p e sd p c cξs; hds + e D p βξ; h ξ; h e D p e sd p ḣsds λ σa \ {} iff de[c; B 1 λ { 1 λ β + D p}s; B 1 λ ] E.vec: ḣ {c; B 1 λ D ps; B 1 λ }u c; M j 2 j M j, s; M 2 j+1 M j 2 j! j 2 j+1! Bz D 2 p + zc c 2 Wiener / 18

11 2 Wiener KdV n- E[e ζq KdV ] 1 ζλ 1/2 λ σa { e rd p de [ c; Bζ ζβ + D p s; Bζ ]} 1/2 2 Wiener / 18

12 2 Wiener COSA8 W {W : [, ] 2 R coni.w, W, } H 1 Ws, 2 dsd 2 [,] 2 H 1 2 D 2 A + 4 Ahs, 8 [s,] [,] 2 2 A m,n m + 1 π n + 1 π h m,n h m,n 2 2 h m,n s, 2 sin 1 m+ 2 πs n+ 1 m+ 1 2 n+ 1 2 π2 sin { E[e ζh ] m 2 ζ } 1/2 cos 2m + 1π hu, vdudv 2 πs Deheuvels, P., Peccai, G., and Yor, M.: On quadraic funcionals of he Brownian shee and relaed processes, SPA Wiener / 18

13 2 Wiener Anderson-Darling JMI11w.Nisshin Fire W W 1 1 {θ 1 }, a a D 2 A + 1 Ah 1 θ 2 1 d hs 1 u s1 s ds + du 1 A nn + 1 h n h n n1 h n 41 P n 2 1, 1 2xy + y2 1/2 { 2ζ } 1/2 2πζ E[e ζa ] 1 nn + 1 n1 cos π ζ hs ds s1 s P n xy n n 2 Wiener / 18

14 2 Wiener OU SPA1w.N.Ikeda W W 2, sp 1 2 Girsanov Jξ p, dξ p p >, ξ p e p e ps dθ R 2 E[e iαsp C ξ p 2 δ x ξ p ] 1 2π { 1 exp 2 E[e iαsp C ξ p 2 ] m 1 /2 sinhm 1 /2 m 1 /2 } anhm 1 /2 + C p x 2 p 2 m 1 e p m 1 coshm 1 /2 + 4C 2p sinhm 1 /2 m1 α 2 + 4p 2 1/2 2 Wiener / 18

15 Wiener W W 2 ϕ C 2 [,,, ϕ > +α {Θ ϕ Θ ϕ,1, Θ ϕ,2 } ; 1 Θ ϕ,α ϕsdθ α s for >, ϕ for, α 1, 2. Θ ϕ µ-a.s. 1/2 law ϕsdθ s θ ϕs2 ds ϕs 2 ds ϕ 1/2 s ϕ 1 JΘ ϕ s, dθ ϕ s 2 1. p R, ϕ e p ; Θ ϕ ξ p O-U 2. p >, ϕ p ; Z.Jurek, M.Yor e al. Sa.Probab.Le.23 2 Wiener / 18

16 Wiener s ϕ 1 2 D 2 A ker A {} λ σa λ σa Ah J {Θ ϕ [h] ϕ Θ ϕ [h] h ϕ s } ϕs 2 Θϕ s [h]ds, ϕ s s ϕuḣudu ϕs 2 ds h kera λi Jh kera λi, Kh kera + λi K 1 1 σa, {λ 1, λ 1,..., λ n, λ n,... } { } 1 E[e ζsϕ ] 1 + ζ 2 λ 2 n n1 2 Wiener / 18

17 Wiener λ σa iff λḣ J {Θ ϕ [h] ϕ k ϕθ ϕ [h] ϕ λ k J {k s } ϕ 2 ϕs ksds 3 σa σl { K, s 1 [,] s + ϕ2 1 2 ϕ 1 2 ϕ s L : L 2 [, ]; R 2 L 2 [, ]; R 2 L f ϕ s } ϕs 2 Θϕ s [h]ds } J, K, s fsds 2 Wiener / 18

18 Wiener Girsanov dθ ϕ dθ + ψ ψ Θϕd ϕ >, ψ 1 ϕ E[e ζsϕ δ y Θ ϕ ] E [ exp ζ 2 1 2π Jθ, dθ e 1 2 C 1 ζ de Aζ de Cζ A ζ J ζ A ψ ζ E[e ζsϕ ] ψ θ ψ 2 d δ y θ ] e 1 ψ 2 ψ y 2 ψ ψ Iy,y PRF1999 A ψ ζ, A ζ I, A ζ J, ζ 2 C ζ A ζ 1 A ζ 1 d 1 de A ζ dei ψ ψ C ζ 2 Wiener / 18

#A A A F, F d F P + F P = d P F, F y P F F x A.1 ( α, 0), (α, 0) α > 0) (x, y) (x + α) 2 + y 2, (x α) 2 + y 2 d (x + α)2 + y 2 + (x α) 2 + y 2 =

#A A A F, F d F P + F P = d P F, F y P F F x A.1 ( α, 0), (α, 0) α > 0) (x, y) (x + α) 2 + y 2, (x α) 2 + y 2 d (x + α)2 + y 2 + (x α) 2 + y 2 = #A A A. F, F d F P + F P = d P F, F P F F A. α, 0, α, 0 α > 0, + α +, α + d + α + + α + = d d F, F 0 < α < d + α + = d α + + α + = d d α + + α + d α + = d 4 4d α + = d 4 8d + 6 http://mth.cs.kitmi-it.c.jp/