REFLECTIONLESS POTENTIALS AND STOCHASTIC ANALYSIS. Setsuo TANIGUCHI. Faculty of Mathematics, Kyushu Univ.

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1 REFLECTIONLESS POTENTIALS AND STOCHASTIC ANALYSIS Setsuo TANIGUCHI Faculty of Mathematics, Kyushu Univ. 0

2 PDE and Stochastic Analysis 194 K.Itô; stoch. integral, Itô s formula R.Feynman; Feynman path integral ( d ) x dx 1944 R.Cameron-W.Martin; M.Kac; the Feynman-Kac formula Heat eq: u = u(x, t) ( L = 1 u t = Lxu, u(, 0) = f i,j aij x i x j + ) i bi x i + V u(x, t) = E [ f(x(t, x))e Φ(x;V )] ( X(t, x):l-diff.pr. ) Reflectionless potentials, generalized... n-solitons of the KdV eq. 1

3 Reflectionless potential us (s S); us(x) = (d/dx) log det(i + Gs(x)), where S = {{η j, m j } 1 j n n N, η j, m j > 0, η i η j } Gs(x) = ( m i m j e (η i+η j )x η i + η j ) 1 i,j n. Schrö Op. (d/dx) + u; Scattering data Ξ 0 = {us s S}, Ξ u µ > 0, un Ξ 0 s.t. Spec( (d/dx) + un) [ µ, ), n = 1,,... un u (unif on cpts)

4 Σ = {σ finite meas on R with cpt supp} W = {w : [0, ) R conti, w(0) = 0} X(x) : W R: X(x, w) = w(x), w W P σ : the prob meas on W under which {X(x)} is the centered Gaussian pr with cov fn e ζ(x+y) e ζ x y X(x)X(y)dP = σ(dζ). W R ζ a 1 X(x 1 ) +... a k X(x k ) N(0, ), (x j 0) G = {P σ σ Σ} (bij) Σ ) X(x) dp σ = W R e ζx 1 σ(dζ) ζ 3

5 ψ(p σ )(x) = 4 d dx W ( exp 1 x ) 0 X(y) dy dp, x 0 The Plan of talk G:Cen. Gauss P σ, σ Σ () G 0 : P σ, σ = j c j δ p j ψ (1) Ξ:gen. rl. pot u unif conv. on cpts Ξ 0 : rl pot us, s S Realization of P σ, Spelling out s S, Solitons 4

6 reflectionless potential and n-soliton Σ 0 = { σ = n j=1 c } j δ p j n N, p j R, p j p i, c j > 0 σ Σ 0. {b(x)} x 0 ; an n-dim B.m.on (Ω, F, P ) ξσ(x) = e xd σ x 0 e yd σdb(y) (Dσ =diag[p j ]) ξ i σ (x) = exp i x 0 e yp idb i (y), i = 1,..., n Xσ(x) = c, ξσ(x) (c = (c j )) P σ (A) = P (Xσ 1 (A)) Xσ : Ω ω Xσ(, ω) W σ Σ 0 ; m < n, 1 j(1) < < j(m) n s.t. p j p j+1, p j(l) > 0, p j(l)+1 = p j(l) #{ p 1,..., pn } = n m. 5

7 ψ : Σ 0 σ {η j, m j } S p 1 p n 1 {η 1 < < ηn} = {p j(1),..., p j(m), r 1,..., r n m } m i = (0 < r 1 < < r n m : n j=1 c j /(p j η i c j(l)+1 c j(l) η i k i k i η k + η i η k η i η k + η i η k η i n k=1 k j(l),j(l)+1 p k + η i p k η i, r) = 1) p k + η i p k η i, if i = j(l), otherwise. 6

8 Thm 1. Let P σ G 0 = {P σ σ Σ 0 }. Then ( ( 4 log exp 1 x W 0 X(y) dy )dp σ) ( ) = log det I + G (x) ψ(σ) ( ) + log det I + G (0) ψ(σ) x n (p i + η i ). i=1 In particular, ψ : G 0 Ξ 0 and ψ(p σ ) = u ψ(σ). Moreover, ψ : G 0 Ξ 0 is bijective. ψ(p σ ) = u ψ(σ) on [0, ); The real analyticity does the rest of job u ψ(σ) (x) = ψ(p µ )( x), x (, 0] (µ(a) = σ( A)) 7

9 The τ-function of the KdV hierarchy is τ (x, t) = 1 + p j=1 m ij η ij n p=1 1 i 1 < <i p n (η ij η ik ) [ p exp ζ η 1 j<k p ij +η ij ], ik j=1 where x R, t = (t j ) R N with #{t j 0} <, {η j, m j } Ξ 0, ζ j = ζ j (x, t) = xη j + α=1 tαη j α+1. If t = (t, 0,... ), then v(x, t) = x log τ (x, t) solves the KdV eq; v t = 3 vv x v xxx. 8

10 For σ Σ 0, let ψ(σ) = {η j, m j } Ξ 0 and Iσ(x, t) = Ω [ exp 1 x 0 X σ(y) dy + 1 ] (β t D σ)ξσ(x), ξσ(x) dp, where β t = ( ( xφ)φ 1) (0, t), ζ = diag[ζ j ], φ(x, t) = U { cosh(ζ) sinh(ζ)r 1 U 1 DσU } U 1, U O(n); D σ +c c = UR U 1 (R =diag[η j ]) Thm (i) log ( Iσ(x, t) ) = (1/) log τ (x, t) +(1/) log τ (0, t) (x/) n i=1 (p i + η i ) (ii) If t = (t, 0,... ), then vσ(x, t) = 4 x log( Iσ ) is an n-soliton of the KdV eq. 9

11 Change of variables formulae on W; Prop 1. Let φ(y) R n n be a sol of φ Eσφ = 0, where Eσ = D σ + c c. Let x > 0 and assume (A.1) det φ(y) 0, (A.) β(y) = (φ φ 1 )(y) is symm (0 y x). Then ( Ω exp 1 x 0 X σ(y) dy + 1 ) (β(0) D σ)ξσ(x), ξσ(x) dp = ( det φ(0) ) 1/ ( e xtrdσ det φ(x) ) 1/. 10

12 Pf of Thm 1: φ Eσφ = 0, φ(0) = I, φ (0) = Dσ; φ(y) = cosh(yeσ 1/ ) E 1/ sinh(yeσ 1/ )Dσ (Case1) p i < p i+1, i = 1,..., n 1. φ(y) = (1/)UV R 1 B { I + G ψ(σ) (y) } e yr B 1 XC, V = diag [ (D σ r j I) 1 c 1], R = diag[η j ], a(i) = sgn [ nβ=1 (p β η i ) ], { α i b(i) = a(i) η (η α η i ) } 1/, i nβ=1 (p β η i ) B = diag[b(j)], X ij = ( p j + η i ) 1. (Case) p ε j = p j ε m i=1 δ j,l(i)+1, ε 0. 11

13 Pf of Thm : φ(y) = φ(y, t); (A.1),(A.) are fulfilled (Case1) p i < p i+1, i = 1,..., n 1. φ(y, t) = 1 UR 1 V B{I + A(y, t)}e ζ(y,t) B 1 XC, where A(y, t) = ( mi m j e {ζ ) i(y,t)+ζ j (y,t)}. η i + η j 1 i,j n (Case) p ε j = p j ε m i=1 δ j,l(i)+1, ε 0. 1

14 ch of var formulae Dw; the Lebesgue meas on W n 1 (W n = {w : [0, ) R n conti, w(0) = 0}) ( fdp = f(w) exp 1 x W n W n 0 w ) Dw ( = f w ) ( Dw exp 1 x W n 0 w Dw ) Dw = e trd/ W n f (ch var w w Dw; Volterra) ( w ) Dw Itô s formula: Dw(x), w(x) = exp( 1 Dw(x), w(x) 1 x ) 0 Dw dp (w) x 0 Dw, w + trd 13

15 ξσ(x, w) = e xd σ x 0 e yd σdw(y) = w(y) + e xd σ x 0 e yd σdσw(y)dy ξσ(, w Dσw) = w Apply 1 with f(w) = g(ξσ( ; w)) ( exp 1 x W n 0 X σ(y) dy + 1 ) (γ(x) D σ)ξσ(x), ξσ(x) dp = e trd σ/ ( exp 1 x W n 0 E σw, w dy + 1 )dp γ(x)w(x), w(x) 14

16 ( 3 1 = exp 1 x W n 0 w ) Dw γ : [0, x] R n n ( = exp 1 x W n 0 w γw ) Dw. x x x γ(x)w(x), w(x) = 0 γ w, w + 0 γw, w + trγ. ( 0 1 = exp 1 x W n 0 (γ + γ )w, w + 1 ) γ(x)w(x), w(x) ( exp 1 x 0 w ) ( Dw exp 1 x ) 0 trγ ( x ) ( exp 1 0 trγ = exp 1 x W n 0 (γ + γ )w, w + 1 )dp γ(x)w(x), w(x) 4 γ + γ = Eσ, γ(x) = CH tr: γ(y) = (φ φ 1 )(x y), φ Eσφ = 0 15

17 Bijectivity Let u = us Ξ 0 (s S), and e + (x; ζ) be the right Jost sol of L = (d/dx) + us; Le + ( ; ζ) = ζ e + ( ; ζ), e + (x; ζ) e iζx (x ) Then λ j C (R; R), 1 j n, s.t. e + (x; ζ) = e 1ζx j Define κ : Ξ 0 Σ 0 by Then κ(s) = j ( λ j (0))δ λ j (0). ψ(κ(s)) = s, κ(ψ(σ)) = σ. ζ 1λ j (x) ζ+ 1η j. 16

18 generalized reflectionless potentials u Ξ µ > 0, un Ξ 0 s.t. Φσ(x) = Spec ( ( d dx ) + u n ) [ µ, ), n = 1,,... un u (unif on cpts) W ( exp 1 x ) 0 X(y) dy dp σ ψ(p σ ) = 4 ( d dx ) Φ σ G G 0 P σ ψ(p σ ) Ξ 0 Ξ : bijective Question: P σ n P σ un u, ψ(g) Ξ 17

19 Thm 3 (i) Φσ C ([0, )), σ Σ (ii) Let σn Σ 0, σ Σ. Suppose n N suppσ n [ β, β] ( β > 0), σn σ (vag). Then Φ (j) σ n Φ (j) σ (unif on cpts) (j = 0, 1, ). (iii) Let σ Σ and suppσ [ β, β]. Define σn(dξ) = n j= n { σ ([ jβn, (j+1)β n )) + n } δ jβ/n. Then n 0 N, λ 0 > 0 st Spec ( ( d dx ) + ψ(p σ n )) [ λ 0, ), n n 0. (iv) P σ G, u Ξ st ψ(p σ ) = u on [0, ). Conversely, u Ξ, P σ G st... u(x) = ψ(p µ )( x), x (, 0] (µ(a) = σ( A)) 18

20 Brownian sheet {W (p, x)} (p,x) [0, ) ; a 1 W (p 1, x 1 ) + + a k W (p k, x k ) N(0, ) W (p, x)w (q, y)dp = min{p, q} min{x, y} Ω Wiener integral s.t. L ([0, ) ) h [0, ) hdw L (P ) (isom) χ [0, ) [a,b) [c,d) dw = W (b, d) W (a, d) W (b, c) + W (a, c) d + c + a b 19

21 For σ = j c j δ p j and a > 0, a b < p 1, set q 0 = b + a, q k = q 0 + k j=1 p j p j 1, p 0 = b, ( W (qj,y) W (q Wq 0...q n (y) = j 1,y) ) {q j q j 1 } 1/ 1 j n. ξ a,b,σ (y) = e yd σ y 0 e zd σdwq 0...q n (z) X a,b,σ (y) = c, ξ a,b,σ (y) Xσ It holds that where X a,b,σ (y) = h(q, z; y) = n j=1 [0, ) h( ; y)dw, e (y z)pj c j χ [qj 1,q qj q j ) [0,y)(q, z). j 1 0

22 Gaussian filtering theory Let u Ξ and take P σ so that u = ψ(p σ ). {b(x)} x 0 : a 1-dim Bm indep of {Xσ(x)} x 0 Put Y (x) = x 0 Xσ(y)dy + b(x), Gx = σ{y (y) y x} ˆXσ(x) = E[Xσ(x) Gx] γ(x) = (X σ(x) ˆXσ(x)) dp Ω exp( 1 x Ω 0 X ) ( σ dp = exp 1 x 0 γ). K(x, y); K + 0 K(, z)r σ(z, )dz = Rσ γ(x) = K(x, x). u(x) = 4(d/dx) log ( exp( 1 x Ω 0 X ) ) σ dp = dx d ( K(x, x)). Marchenko s formula 1

23 quadratic Wiener functional For σ Σ 0 and symm β R n n, define qσ,x = 1 x 0 X σ(y) dy + 1 rl pot; β 0 KdV; β = β t Dσ. βξ σ(x), ξσ(x). Let A = qσ,x and Bσ(a) = Dσ + ac c. eζqσ,x dp = { j=1 (1 ζa Ω j ) } 1/ ( aj ;ev of A ) = exp ( R ( e ζx 1 ) fa (x)dx) ( f A (x) = 1 x = { e xtrd σ det [ cosh(xbσ(ζ) 1/ ) ) n;a n x>0 exp( x/a n) (ζβ + Dσ)Bσ(ζ) 1/ sinh(xbσ(ζ) 1/ ) ]} 1/.

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