STOCHASTIC ANALYSIS AND THE KdV EQUATION Setsuo TANIGUCHI Faculty - - PowerPoint PPT Presentation

STOCHASTIC ANALYSIS AND THE KdV EQUATION Setsuo TANIGUCHI Faculty of Mathematics, Kyushu Univ. Joint work with N. Ikeda +α http://www.math.kyushu-u.ac.jp/~taniguch/

PDE and Stochastic Analysis Heat eq:

✓ ✒ ✏ ✑

u(x, t); ut = LV u, u(·, 0) = f

• LV = 1

2

• i,j aij∂xi∂xj +

i bi∂xi + V

• u(x, t) = E
• f(X(t, x))eΦ(x;V )
• X(t, x):L0-diff.pr.
• 1942 K.Itˆ
• ; stoch. integral, Itˆ
• ’s formula

1944 R.Cameron-W.Martin; 1

2 d dx 2 − x2 2

1947 M.Kac; the Feynman-Kac formula

⋄ Refrectionless potential,

generalized refrectionless potential,

n-solitons of the KdV eq. vt = 3 2vvx + 1 4vxxx.

1

v(x, t) = 2sech2(x + t − 2)

v(x,y)

• 10
• 5

5 10 x-axis

• 4
• 2

2 4 t-axis 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

2

v(x, t) = 12 3+4 cosh(2x+2t)+cosh(4x+16t) 3 cosh(x+7t)+cosh(3x+9t)

v(x,y)

• 15
• 10
• 5

5 10 15 x-axis

• 2
• 1

1 2 3 4 5 t-axis 1 2 3 4 5 6 7 8

3

• us (s ∈ S): reflectionless potential

✛ ✚ ✘ ✙

us(x) = −2 d2 dx2 log det(I + Gs(x)) , where S =

• {ηj, mj}1≤j≤n
• n ∈ N, ηj, mj > 0, ηi = ηj
• Gs(x) =

mimj e−(ηi+ηj)x ηi + ηj

• 1≤i,j≤n

.

• Schr¨
• op. − d2

dx2 + us → Scattering data s ∈ S Ξ0 = {us | s ∈ S} bijective ← → S,

• Ξ ∋ u ⇔ ∃µ > 0, un ∈ Ξ0 s.t.

Spec(− d2

dx2 + un) ⊂ [−µ, ∞), n = 1, 2, . . . un → u (unif on cpts)

(Marchenko)

4

• W ={w : [0, ∞)→R | conti, w(0)=0}; 1-dim W.sp
• X(x) : W → R: X(x, w) = w(x), w ∈ W
• Σ = {σ | finite meas on R with cpt supp}
• P σ: the prob meas on W under which

{X(x)} is the cent. Gaussian pr with cov fn

• W

X(x)X(y)dP =

• R

eζ(x+y) − eζ|x−y| 2ζ σ(dζ).

• G = {P σ | σ ∈ Σ} bijective

← → Σ ∵) d dx

• W

X(x)2dP σ =

• R

e2ζxσ(dζ)

5

ψ(P σ)(x)=4 d2 dx2 log

• W

exp

• −1

2 x X(y)2dy

• dP σ

, x≥0

The Plan of talk

✬ ✫ ✩ ✪ ✬ ✫ ✩ ✪ ✬ ✫ ✩ ✪ ✬ ✫ ✩ ✪ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅

• ψ

(3)

G:Cen. Gauss P σ, σ ∈ Σ G0 : P σ, σ = j c2 jδpj Ξ:gen. rl. pot u Ξ0: rl pot us, s ∈ S

❅ ❅ ❅ ❅ ❅ ❅

• (2)
• ❅

❅ ❅ ❅ ❅ ❅

(1)

❅ ❅ ❅ ❅ ❅ ❅

• ❅

❅ ❅ ❅ ❅ ❅

• unif conv.
• n cpts

Realization of P σ, Spelling out s ∈ S, Solitons

6

reflectionless potential and n-soliton

Σ0=

• σ =n

j=1 c2 jδpj

• n ∈ N, pj ∈ R, pj = pi, cj > 0
• σ ∈ Σ0. {b(x)}x≥0; an n-dim B.m.on (Ω, F, P )

ξσ(x) = exDσ x e−yDσdb(y) (Dσ =diag[pj]) Xσ(x) = c, ξσ(x) = n i=1 ciξi σ(x) (c=(ci)) P σ = P ◦ X−1 σ (Xσ : Ω → W)

• σ ∈ Σ0; ∃m < n, 1 ≤ j(1) < · · · < j(m) ≤ n s.t.

|pj| ≤ |pj+1|, pj(ℓ) > 0, pj(ℓ)+1 = −pj(ℓ) #{|p1|, . . . , |pn|} = n − m.

7

ψ : Σ0 ∋ σ → {ηj, mj} ∈ S

r r r r

p2

1

p2

n

−1

✉ ✉ ✉ ✉

{η1 < · · · < ηn}={pj(1),. . ., pj(m), √r1,. . ., √rn−m}

(0 < r1 < · · · < rn−m: n

j=1 c2 j/(p2 j − r) = −1) mi =                        2ηi c2 j(ℓ)+1 c2 j(ℓ)

• k=i

ηk + ηi ηk − ηi

• k=j(ℓ),j(ℓ)+1

pk + ηi pk − ηi ,

if i = j(ℓ),

− 2ηi

• k=i

ηk + ηi ηk − ηi n

• k=1

pk + ηi pk − ηi ,

• therwise.

8

Thm 1. Let P σ ∈ G0 = {P σ|σ ∈ Σ0}. Then

4 log

• W

exp

• −1

2 x X(y)2dy

• dP σ

= −2 log det

• I + Gψ(σ)(x)
• + 2 log det
• I + Gψ(σ)(0)
• − 2x

n

• i=1

(pi + ηi).

Moreover, ψ : G0 → Ξ0 and ψ(P σ) = uψ(σ). Finally, ψ : G0 → Ξ0 is bijective.

• ψ(P σ)(x) = 4 d2

dx2 log

• W

exp(· · · )dP σ

• ψ(P σ) = uψ(σ) on [0, ∞); “ψ(G0) ⊂ Ξ0”

The real analyticity does the rest of job

9

Cor. (i) If µ(A) = σ(−A), then

uψ(σ)(x) = ψ(P µ)(−x) for x ∈ (−∞, 0].

(ii) For y ≤ 0, let b(y) = b(−y), and

ξσ(y) = −eyDσ y e−zDσdb(z) Xσ(y) = c, ξσ(y).

Then u = ψ(P σ) is represented as

u(x) = 4 d2 dx2 log

• Ω

exp

• −1

2 0∨x 0∧x Xσ(y)2dy

• dP

for every x ∈ R.

10

• The τ-function of the KdV hierarchy is

τ(x, t ) = det(I + A(x, t ))

where x ∈ R,

t = (tj) ∈ RN with #{tj = 0} < ∞, A(x, t ) = mimj ηi + ηj e−{ζi(x, t )+ζj(x, t )}

• 1≤i,j≤n

, {ηj, mj} ∈ S, ζi = ζi(x, t ) = xηi + ∞

• α=1

tαη2α+1 i

.

• If

t = (t, 0, . . . ), then v(x, t) = 2∂2 x log τ(x, t ) solves

the KdV eq;

vt = 3 2vvx + 1 4vxxx.

11

• For σ ∈ Σ0, let ψ(σ) = {ηj, mj} ∈ Ξ0. Define

β t = −

• (∂xφ)φ−1

(0, t ), ζ = diag[ζj], φ(x, t ) = U

• cosh(ζ) − sinh(ζ)R−1U−1DσU
• U−1,

U ∈ O(n); D2 σ+c ⊗ c=UR2U−1 (R=diag[ηj]), Iσ(x, t ) =

• Ω

exp

• −1

2 x Xσ(y)2dy +1 2(β t − Dσ)ξσ(x), ξσ(x)

• dP.

Thm 2 (i) log

• Iσ(x,

t )

• = −1

2 log τ(x, t ) +1 2 log τ(0, t ) − x 2 n i=1(pi + ηi)

(ii) If

t = (t, 0, . . . ), then vσ(x, t )=−4∂2 x log

• Iσ
• is an n-soliton of the KdV eq. (Super pos)

12

Change of variables formulae on W;

• Prop. Let φ(y) ∈ Rn×n be a sol of

φ′′ − Eσφ = 0,

where Eσ = D2

σ + c ⊗ c.

Let x > 0 and assume (A.1) det φ(y) = 0, (A.2) β(y) = −(φ′φ−1)(y) is symm (0 ≤ y ≤ x). Then

• Ω

exp

• −1

2 x Xσ(y)2dy +1 2(β(0) − Dσ)ξσ(x), ξσ(x)

• dP

=

• det φ(0)

1/2 extrDσ det φ(x) −1/2.

13

• OU pr H.Osc;

γ : [0, x] → Rn×n

• Ω

exp

• −1

2 x Xσ(y)2dy +1 2(γ(x)−Dσ)ξσ(x), ξσ(x)

• dP

= e−trDσ/2

• Ω

exp

• −1

2 x Eσb(y), b(y)dy +1 2γ(x)b(x), b(x)

• dP
• (C-M) Itˆ
• ⊕ Girsanov;
• Ω

exp

• −1

2 x (γ2 + γ′)b(y), b(y)dy +1 2γ(x)b(x), b(x)

• dP = exp

1 2 x 0 trγ

• γ2 + γ′ = Eσ,

γ(x) = β(0) − Dσ

Cole-Hopf; γ(y)=−(φ′φ−1)(x−y) ⇒ φ′′ − Eσφ = 0

14

Pf of Thm 1: φ′′−Eσφ=0, φ(0)=I, φ′(0)=−Dσ;

φ(y) = cosh(yE1/2 σ ) − E−1/2 σ sinh(yE1/2 σ )Dσ

(Case1) |pi| < |pi+1|, i = 1, . . . , n − 1.

φ(y)=−1 2UV R−1B

• I + Gψ(σ)(y)
• eyRB−1XC

V = diag

• |(D2

σ − rjI)−1c|−1, R = diag[ηj], a(i) = sgn n β=1(pβ − ηi) , b(i) = a(i)

• −2ηi
• α=i(η2

α−η2 i )

n

β=1(p2 β−η2 i )

1/2, B = diag[b(j)], Xij =

• pj + ηi

−1.

(Case2) pε

j = pj − ε m i=1 δj,ℓ(i)+1, ε → 0.

15

Pf of Thm 2:

φ(y) = φ(y, t ); (A.1),(A.2) are fulfilled

(Case1) |pi| < |pi+1|, i = 1, . . . , n − 1.

φ(y, t ) = −1 2UR−1V B{I + A(y, t )}eζ(y, t )B−1XC

(Case2) pε

j = pj − ε m i=1 δj,ℓ(i)+1, ε → 0.

16

Bijectivity & Cor

• Let u = us ∈ Ξ0 (s ∈ S), and

e+(x; ζ) be the right Jost sol of L=−(d/dx)2 + us; Le+(∗; ζ) = ζ2e+(∗; ζ), e+(x; ζ) ∼ eiζx (x → ∞)

Then ∃λj ∈ C∞(R; R), 1 ≤ j ≤ n, s.t.

e+(x; ζ)=e √−1ζx j ζ−√−1λj(x) ζ+√−1ηj .

Define κ : Ξ0 → Σ0 by κ(s) =

j(−λ′ j(0))δλj(0)

Then

ψ(κ(s)) = s, κ(ψ(σ)) = σ.

• Let

u(x) = us(−x). Then u = u s (κ( s) = µ)

17

generalized reflectionless potentials

• u ∈ Ξ ⇔ ∃µ > 0, un ∈ Ξ0 s.t.

Spec

• − d2

dx2 + un

• ⊂ [−µ, ∞), n = 1, 2, . . .

un → u (unif on cpts)

• Φσ(x) =
• W

exp

• −1

2 x X(y)2dy

• dP σ

ψ(P σ) = 4 d2 dx2 log Φσ G ⊃ G0 ∋ P σ → ψ(P σ) ∈ Ξ0 ⊂ Ξ : bijective

Question: “P σn → P σ” “un → u”,

ψ(G) Ξ

18

Thm 3 (i) Let σn ∈ Σ0, σ ∈ Σ. Assume

σn → σ (vaguely),

• n∈N suppσn ⊂ [−β, β] (∃β > 0),

Then

ψ(P σn) → ψ(P σ) (unif on cpts).

Moreover, for ∀ε > 0, ∃n0 ∈ N s.t.

• n≥n0

Spec(− d2

dx2 + ψ(P σn)) ⊂ [− β2−σ(R)−ε, ∞)

(iia) ∀P σ ∈ G, ∃u ∈ Ξ s.t. ψ(P σ) = u on [0, ∞). (iib) ∀u ∈ Ξ, ∃P σ ∈ G s.t. u = ψ(P σ) on [0, ∞).

19

• For σ ∈ Σ, to have the above σn, let

suppσ ⊂ [−β, β], aj = βj/n, Ij = [aj, aj+1), and

σn = n j=−n{σ(Ij) + 1/n}δaj.

• If we set µ(A)=σ(−A), and

u(x) =    ψ(P σ)(x), x ≥ 0, ψ(P µ)(−x), x ≤ 0

, then u ∈ Ξ. Conversely ∀u ∈ Ξ is of the above form. “G bijictive

← → Ξ”

20

Brownian sheet

{W (p, x)}(p,x)∈[0,∞)2;

• Ω

W (p, x)W (q, y)dP = min{p, q} min{x, y}

• Wiener integral

L2([0, ∞)2) ∋ h →

• [0, ∞)2 hdW ∈ L2(P ) (isom)

s.t.

• [0, ∞)2 1[a1,a2)×[b1,b2)dW =

2

• i,j=1

(−1)i+jW (ai, bj)

• For σ =

j c2 jδpj and a > 0, −a ≤ b < p1, set q0 = b + a, qk = q0 + k j=1 |pj − pj−1|, p0 = b, Xσ Xa,b,σ(y) =

• [0, ∞)2 h(∗; y)dW, where

h(q, z; y) = n

• j=1

e(y−z)pjcj qj−qj−1 1[qj−1,qj)×[0,y)(q, z).

21

quadratic Wiener functional

• Wn; n-dim Wie sp, P; Wie meas
• q : Wn → R; quadratic

def

⇐ ⇒ ∇3q = 0

iff

⇐ ⇒ q = 1 2(∇∗)2A + ∇∗b + c,

where A = ∇2q, b = E[∇q], c = E[q]

• stoch osc int;

✗ ✖ ✔ ✕

• Wn exp(ζq)dP, ζ ∈ C

Exact exp; C-M, L´ evy, det2 (∞ prod)

• For σ ∈ Σ0 and symm β ∈ Rn×n, define

qσ,x = −1 2 x Xσ(y)2dy + 1 2

• βξσ(x), ξσ(x)
• ∗ rl pot; β ≡ 0, KdV; β = β

t − Dσ.

22

Let A = ∇2qσ,x and Bσ(ζ) = D2

σ + ζc ⊗ c.

• ker A =
• h′=g−Dσ

g

• g ⊥c a.e.,

x g ∈ ker(β)

• Wneζqσ,xdP =

∞ j=1(1 − ζaj) −1

2

aj;ev’s of A

• = exp
• R
• eζs − 1
• fA(s)ds
• fA(s) =

1 2|s|

• j;ajs>0

exp(−s/aj)

• =
• extrDσ det
• cosh(xBσ(ζ)

1 2)

−(ζβ + Dσ)Bσ(ζ)−1

2 sinh(xBσ(ζ) 1 2)

−1

2.

23