( ) Loewner SLE 13 February

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1 ( ) Loewner SLE 3 February 00

2 G. F. Lawler, Conformally Invariant Processes in the Plane, (American Mathematical Society, 005)., Summer School 009 ( )

3 . d- (BES d ) d B t = (Bt, B t,, Bd t ) (d N) X t = B(t) = d (Bt j ) j= d- (BES d ). F (x, x,, x d ) = d x j j=, F (B(t)), X t (stochastic differential equation, SDE). dx t = db t + d dt () X t 3

4 F x k = x k F, d k= F x k F x k = F x k F 3 = F, Bt,, Bd t dx t = X t d F d k= d x k k= = d F db k t dbl t = δ kldt, k, l d B k t db k t + d d Bt k X dbk t = d t k= Xt t k=(b k ) (dbt k ) = d Xt (Bt k ) dt = dt k=, {B j t }d j= BM, B t dt X t 4

5 dx t = db t + d dt () X t d,, X t d ()., d ν. ν = d d = (ν + ) () ν, ν. 5

6 BES d, SDE () (Kolmogorov backward equation) t p(t; x, y) = d xp(t; x, y) + p(t; x, y) x x d <, p(t, y x) = t y ν+ x ν exp, I ν (z) I ν (z) = n=0 Γ(z) Γ(z) = ( x + y t Γ(n + )Γ(n + + ν) ) ( ) xy I ν t ( ) z n+ν 0 e u u z du, Rz > 0 6

7 . d- (BES d ) x > 0 BES d X x t dx x t = d dt X x t + db t, t 0, X x 0 = x > 0 (3) x > 0 x Xx x t (law) = X t (4) x > 0 BES d T x } T x = inf {t > 0 : Xt x = 0. (5) SDE (3) t T x well-defined 7

8 . (i) d = T x =, x > 0 (ii) d > = lim t X x t =, x > 0 (iii) d = = inf t>0 Xx t = 0, x > 0 (iv) d < = T x <, x > 0. (i) 3 < d < = x < y, P(T x = T y ) > 0. (ii) d 3 = x < y, T x < T y 8

9 SLE κ BES d Schramm, U t = κb t, κ > 0, B 0 = 0 t g t(z) = g t (z) κb t, g 0 (z) = z (6) ( t 0 ) {g t } t 0., (chordal) (Schramm- Loewner evolution). κ SLE κ U t = lim s t g s (γ(t)) γ = {γ(t) : t [0, )}. 9

10 γ (t) H t g t 0 K t 0 U t = g t (γ (t)) = /κ B t SLE κ γ H t = H \ γ[0, t] K t = H \ H t K t SLE κ γ[0, t] hull H SLE κ γ(0, t] hull K t g t H γ(t) H U t κb t 0

11 SLE κ BM BES d SLE κ SLE κ r > 0 r g (law) r t (rz) = g t (z) r γ(r t) (law) = γ

12 γ (t) H t g t 0 K t 0 U t = g t (γ (t)) = /κ B t { T z = sup t 0 : g t (z) well-defined g t (z) H } = inf {t 0 : z K t } (7) H t = { z H : T z > t }, K t = { } z H : T z t

13 ĝ t (z) = g t(z) κb t κ ĝ t (z) dĝ t (z) = /κ ĝ t (z) dt + dw t, ĝ 0 (z) = z κ, W t = B t. κ = 4 d dĝ t (z) = d, BES d. d = 4 κ + (8) ĝ t (z) dt + dw t (9) 3

14 R[ĝ t (z)] = R t, I[ĝ t (z)] = I t dr t = d R t R t + I t, dt + dw t, di t = d I t R t + I t dt. d(r t + I t r(t) 0 t R ) = (d ) Rt + I t dt (d )dt + R t dw t 4R t dt = t r(t), Y t = R r(t) +I r(t), R t = R r(t), W t = r(t) 0 R t dw t W t, dy t = d W t + d. dt (d )dt X t 4 R = d W t + dt d t X t ( dt X t dt R t ) 4

15 . (i) d = T x =, x > 0 (ii) d > = lim t X x t =, x > 0 (iii) d = = inf t>0 Xx t = 0, x > 0 (iv) d < = T x <, x > 0. (i) 3 < d < = x < y, P(T x = T y ) > 0. (ii) d 3 = x < y, T x < T y 5

16 . (i) 0 < κ 4 γ γ(0, ) H lim γ(t) =. t (ii) 4 < κ < 8 γ t>0 K t = H γ(t) γ[0, ) H H H. (iii) κ 8 γ H γ[0, ) = H. 6

17 0 0 0 (a) (b) (c) (a) 0 < κ 4 SLE (b) ( ) H H 4 < κ < 8 SLE (c) H κ 8 SLE 7

18 0 < x < x < x < { } σ = inf t > 0 : Xt x = x or Xt x = x φ(x) = φ(x; x, x ) = P(X x σ = x ) φ(x ) = 0, φ(x ) = t σ min{t, σ},. M t = φ(x x t σ ) ] M t = E [φ(x σ x ) F t E[M t F s ] = M s, 0 s t M t 8

19 (φ(x) ) BES d SDE (3) [ t σ M t = φ(x) + φ (Xs x ) db s + d ] ds t σ 0 Xs x + 0 φ (Xs x )(db s) t σ [ t σ = φ(x) + φ (Xs x )db s + φ (Xs x 0 0 ) + d ] Xs x φ (Xs x ) ds. M t ( ) φ (x) + d x φ (x) = 0, x < x < x (0) φ(x ) = 0, φ(x ) = x d x d x d x d d φ(x) = φ(x; x, x ) = () log x log x d = log x log x 9

20 (i) d > d < 0 () x = L > x φ(x; 0, L) lim x 0 φ(x; x, L) = lim x 0 x d x d L d x d =. x > 0 BES d L > 0 T x = inf{t > 0 : Xt x = 0} = d = () φ(x; 0, L) = lim x 0 T x = log x log x log L log x = 0

21 (ii) α > x k = α k x, k =,, 3,... d > d β < 0 () φ(x k ; x k, x k+ ) = xβ k xβ k x β k+ xβ k = αβ α β = α β + >. = αkβ α (k )β α (k+)β α (k )β Z {,,, 0,,, } n > 0, n Z p = /(α β +), p BES d X x t, x > 0

22 (iii) () d = x = /n < x < x = e n, n φ(x; /n, e n ) = log x + log n n + log n 0 n > 0 Xt x /n (iv) d <, lim x 0 x d = 0 () L T x < φ(x; 0, L) = x d 0. L d

23 0 < x < y {T x = T y } 0 X sup t<t x y t Xx t Xt x <. () 0 < x < y ( y X Z t = log t Xt x ) Xt x = f(xt x, Xy t ), t < T x. (3) f(x, y) = log{(y x)/x}. 3

24 f x (x, y) = y x x, f y (x, y) = y x f xx (x, y) = (y x) + x, f yy (x, y) = (y x), f xy (x, y) = f yx (x, y) = (y x) 4

25 [ dz t = f x (Xt x, Xy t ) db t + d ] [ dt Xt x + f y (X xt, Xyt ) db t + d ] dt Xt y + ] [f xx (Xt x, Xy t ) + f xy(xt x, Xy t ) + f yy(xt x, Xy t ) dt = [ (3 ) Xt x db t + d (Xt x + d Xt y ] Xx t ) (Xt x) Xt y dt (4) t r r(t) ds 0 (Xs x = t. ) dr(t)/(xx r(t) ) = dt. (4) r(t) dz r(t) = X r(t) x db r(t) + ( 3 d ) + d X y r(t) Xx r(t) X y r(t) dr(t) (X x r(t) ) 5

26 (d B t ) = B t = r(t) 0 db s X x s (X r(t) x )(db r(t) ) = dr(t) (X r(t) x = dt ) B t BM Z t = Z r(t) d Z t = d B t + SDE. ( 3 d ) + d X y r(t) Xx r(t) dt (5) X y r(t) 6

27 (i) 3 < d < d (3/, d) ε = (d d ) d y = ( + ε/)x { σ = inf t > 0 : X y r(t) Xx r(t) = εxy r(t) 0 t < T x σ (X y r(t) Xx r(t) )/Xy r(t) ε (5) } ( ) 3 d + d X y r(t) Xx r(t) X y r(t) ( ) 3 d + d (d d ) d = 3 d 7

28 Z t d Z t = d B t + ( ) 3 d dt, Z 0 = Z 0 = log ε Z t Z t, 0 t < T x σ d > 3/ Z t. Z t log(ε/) log ε Z t log ε ( y X log t Xt x ) Xt x < log ε Xy t Xx t X x t () P(T x = T y ) > 0 < ε 8

29 (ii) d 3 3/ d 0 X y r(t) Xx r(t) X y r(t) > 0, 0 t < T x sup Z t = sup e Z y t X = sup t Xx t t<t x t<t x t<t x P(T x = T y ) = 0 X x t = 9

30 G. F. Lawler, Conformally Invariant Processes in the Plane, (American Mathematical Society, 005)., Summer School 009 ( ) 30

Katori_SLE_Sep07_v2c.dvi

Katori_SLE_Sep07_v2c.dvi Schramm-Loewner Evolution (SLE) and Conformal Restriction 4 September 007 (version c) SLE [6] Lawler ( [7]), 3 Werner ( []), 4 Friedrich and Werner ( [3]) 3........................ 3. d-.................................