Dissertation for the degree philosophiae doctor (PhD) at the University of Bergen

Dissertation for the degree philosophiae doctor (PhD) at the University of Bergen Dissertation date:

GF F GF F SLE GF F

D Ĉ = C { } Ĉ \ D D D = {z : z < 1} f : D D D D = D D, D = D D f f : D D D f P (D) D ˆD = D P (D) ˆD f : D D ˆf : D ˆD ˆD D D ˆD D Ĉ a D, b 1,b 2, b 3 P (D)

f : D D, ˆf : ˆD ˆD. f ˆf(0) = a ˆf (0) > 0, ˆf(0) = a ˆf(1) = b1, ˆf( i) =b1, ˆf(1) = b2, ˆf(i) =b 3 Hol(D, D) D Aut(D) Hol(D, D) D {g t } t 0 Hol(D, D) D D g 0 =id D, g t+s = g t g s, s, t 0 lim t 0 + g t (z) =z z D g t D z D t g t (z) t [0, + ) g t Aut(D) t 0, {g t } t 0 {g t } t 0 V : D C, {g t } t R g t := g 1 t t g t(z) =V (g t (z)), g 0 (z) =z, t 0, z D. V {g t } t 0

V (z) V (z) =V (0) zq(z) V (0) z 2, q(z) Re q(z) 0 V (z) q(z) =ib, b R φ Hol(D, D) φ id D τ D φ τ D lim z τ φ(z) =τ α = φ(z) τ lim z τ z τ 0 <α 1 0 <α<1 φ α =1 φ τ φ {g t } t 0 g 0 =id D τ τ {g t } t 0 V (z) τ D p : D C Re p 0 V (z) =(z τ)( τ z 1) p(z), z D. V (z) 0 τ {g t } t 0 p(z) τ V (z) g t (z) τ D t + {g t } t 0 D D D f : D 1 D 2 {g 1 t } t 0 D 1 {g 2 t } t 0 = {φ g 1 t φ 1 } t 0 D 2 V 1 V 2 {g 1 t } t 0 {g 2 t } t 0 V 1 V 2 φ V 2 V 1 φ V 2 = φ V 1 φ V 1 (z) = 1 φ 1 (z) V 1(φ 1 (z)).

f C 1 D z = x + iy f = f z = 1 ( 2 x i ) f, y f = f z = 1 ( 2 x + i ) f. y V D V f C 1 (D, C) f ( V + V ) f. L V L V = V + V. L V f V L V f f V ; L V f = LV f, L V Re f = ReL V f L V Im f = Im L V f L V ( f) = L V f, L V ( f) = L V f V D C n f : C n C z =(z 1,...,z n ) = ( ) z 1,..., z n = ( ) z 1,..., z n V (z) =(V1 (z),...,v n (z)) f V L V f(z) = ( V (z) + V (z) ) f(z). X t {F t } t 0 B t T>0 X t

B t [0,T] T 0 X t db t := lim n Δt 0 j=1 X tj 1 ( Btj B tj 1 ), 0=t 0 <t 1 <...<t n = T Δt := max j=1,..., n (t j t j 1 ) T 0 X t db t := lim n Δt 0 j=1 X t j 1 +t j 2 ( Btj B tj 1 ). T>0 T 0 X s db s t 0 X s db s Y t {F t } t 0 ( ) t P Y t = X s db s =1 t 0. 0 t 0 X s db s E ( T0 Xs 2 ds ) < T>0 t 0 X s db s P ( T0 Xs 2 ds < ) =1 Y t {F} t 0 M t M t = M 0 + t 0 X s db s F t X t X t Y t X T,Y T = X, Y T := lim n Δt 0 j=1 (X tj X tj 1 )(Y tj Y tj 1 ) 0 = t 0 < t 1 <... < t n = T Δt := max j=1,..., n (t j t j 1 ) X t Y t X t (ω) = Y t (ω) = t t b 1(ω, s) ds + σ 1(ω, s) db s (ω), 0 0 t t b 2(ω, s) ds + σ 2(ω, s) db s (ω), 0 0

b j (ω, t) σ j (ω, t) j =1, 2 F t ( ) ( T ) T P b j(ω, s) ds < =1, P σ j(ω, s) 2 ds =1, 0 0 T > 0 j =1, 2 X, Y t (ω) = t 0 σ 1(ω, s) σ 2 (ω, s) ds. T 0 X t db t = T 0 X t db t + 1 2 X, B T. w t = X 1 t + ix 2 t X 1 t X 2 t f : C C w t t f(w t )=f(w 0 )+ f(w s) dw s + 0 t + 1 2 0 2 f(w s ) d w s + 1 2 t + f(w s ) d w, w s, 0 t 0 t 0 f(w s ) d w s 2 f(w s ) d w s df (w t )= f(w t ) dw t + f(w t ) d w t + 1 2 2 f(w t ) d w t + 1 2 2 f(w t ) d w t + f(w t ) d w, w t. w t t n t w t = w 0 + b(w s) ds + σ k(w s ) dbs k, 0 0 b σ n B 1 t,...,b n t k=1 dw t = b(w t ) dt + n k=1 σ k (w t ) db k t.

dw t = b(w t )+ 1 2 n k=1 σ k (w t ) σ k(w t ) dt + w t = w 0 + t 0 b(w s )+ 1 2 n k=1 n k=1 σ k (w s ) σ k(w s ) ds + σ k (w t ) db k t, n t k=1 0 σ k(w s ) db k s. f : C C f(w t ) df (w t )=L b f(w t ) dt + = L b + 1 2 n k=1 n L 2 σ k k=1 L σk f(w t ) db k t f(w t ) dt + n k=1 L σk f(w t ) db k t. w t C n f C 2 n b : C n C σ k : C n C {g t } t 0 t g t(z) =V (g t (z)), g 0 (z) =z, t g t ( ) D V D g t Hol(D, D) t 0 z D w t (z) w 0 = z {w t } t 0 dw t (z) =b(w t (z),t) dt + n k=1 σ k (w t (z),t) dbt k, w 0 (z) =z, z D. σ k (w) w σ k(w) f t(z) z f t(z)

b(z,t), σ 1 (z,t),...σ n (z,t) C 2 z C 1 t T (z) w t (z) w t (z) =z T (z) D t t>0 D t = {z D : T (z) >t} D s D t s t R t D t w t R t := w t (D t ) w t : D t R t t 0 1 b(z,t) C 1 t>0 C d z D σ 1 (z,t),...,σ n (z,t) C 1 t C d+1 z w t : D t R t C d t 0 1 b(z,t), σ 1 (z,t),...σ n (z,t) C 1 t>0 C z D w t (z) :D t R t C t 0 1 b(z,t), σ 1 (z,t),...σ n (z,t) C 1 t>0 z D w t (z) :D t R t t 0 1 w t (z) dw t (z) =b(w t (z)) dt + n k=1 σ k (w t (z)) dbt k, w 0 (z) =z, z D, b(z) σ 1 (z),...,σ n (z) C w t (z) D t 0 g w t (z) dw t (z) =b(w t (z),t) dt + n k=1 σ k (w t (z),t) dbt k, w 0 (z) =z, z D,

d w t (z) = b( w t (z),t) dt + n k=1 σ k ( w t (z),t) dbt k, w 0 (z) =z, z D, T (z), T (z) {ζ t } t 0 ζ t := w t w t U(z) := min[inf{t >0:w t (z) D t },T(z)] D t = {z D : T (z) >t} dζ t (z) = [ b(ζt (z),t)+ w t b(ζ t (z),t) ] dt + n k=1 [ σ k (ζ t (z),t)+ w t σ k (ζ t (z),t)] db k t, {η t } t 0,η t = wt 1 dη t (z) = (η t b)(η t (z),t) dt = η t (z) b(z,t) dt n k=1 n k=0 (η t σ k )(η t (z),t) db k t η t (z) σ k (z,t) db k t, SLE κ SLE(κ, ρ)

t t H = {z :Imz>0} S = {z :0< Im z<1} d [1, + ] {φ s,t } 0 s t<+ φ s,s = id D φ s,t = φ u,t φ s,u 0 s u t<+ z D T > 0 k z,t L d ([0,T], R), φ s,u (z) φ s,t (z) 0 s u t T. t u k z,t(ξ)dξ

z D φ s,t (z) t [s, + ) d [1, + ] V : D [0, + ) C [0, + ) t V (z,t) z D z V (z,t) t [0, + ) K D T > 0 k K,T L d ([0,T], R), V (z,t) k K,T (z) z K t [0,T] t [0, + ) V (,t) {φ s,t } d 1 V (z,t) d, z D t [0, + ) t φ s,t(z) =V (φ s,t (z),t). V (z,t) d 1 {φ s,t } 0 s t<+ d H(z,t) H(z,t) =V (z,t) z D t [0, + ) {f t } 0 t< f t : D C d [1, + ] f t f s (D) f t (D) 0 s<t<+, K D T>0 k K,T L d ([0,T], R) f s (z) f t (z) k K,T (ξ)dξ s z K 0 s t T. t

f t (D) t {f t } t 0 d {φ s,t } d φ s,t = f 1 t f s. {φ s,t } 0 s t<+ d {f t } t 0 d, φ s,t = ft 1 f s 0 s t f(0) = 0 f (0) = 1 Ω:= t 0 f t (D) {z : z < R} R (0, + ]. {g t } t 0 = {F f t } t 0, F :Ω C R 1/β 0, φ β 0 = lim 0,t(0) t + 1 φ 0,t (z) 2. {f t } t 0 d s f s(z) = V (z,s)f s(z) ( s 0), V (z,s) {φ s,t } 0 s t<+. V (,t) t 0 p t d [1, + ) p : D [0, + ) C, t p(z,t) L d loc([0, + ), C) z D z p(z,t) D t [0, + )

Re p(z,t) 0 z D t [0, + ) V (z,t) d 1 t V (,t) 0 τ :[0, + ) D p(z,t) d, z D t [0, + ) V (z,t) =(z τ(t)) (τ(t) z 1) p(z,t). τ :[0, + ) D p(z,t) d 1, d {φ s,t } V (z,t) (p, τ) V (z,t) {φ s,t } 0 s t<+ τ(t) V (z,t) {φ s,t } 0 s t<+ d [1, + ] {φ s,t } 0 s t<+ Hol(D, D) d φ s,s =id D φ s,t = φ s,u φ u,t 0 s u t< z D T > 0 k z,t L 2 ([0,T], R) φ s,u (z) φ s,t (z) k z,t(ξ) dξ, u s, t, u [0,T] s u t d [1, + ] {f t } t 0 Hol(D, D) d t

f t : D D f 0 =id D, f s (D) f t (D) 0 s<t<+ K D T > 0 k K,T L d ([0,T], R) f s (z) f t (z) k K,T (ξ)dξ s z K 0 s t T {f t } t 0 t φ s,t (z) =f 1 s f t, 0 s t<. {φ s,t } 0 s t<+ f t (z) :=φ 0,t V d [1, + ] z D, g t (z) D t g t(z) = V (g t (z),t), g 0 (z) =z. t 0, D t z D, g t (z) t, g t (z) z D t D t D f t := gt 1 d t f t(z) =f t(z) V (z,t), f 0 =id D. D t = g 1 t (D) =f t (D) t K t = D \ D t t {D t } t 0 {K t } t 0 0 s t< D s D t K s K t

τ(t) τ 0 τ 0 D τ 0 =0 p(0,t) 1 p(z,t) V (z,t) = zp(z,t), p(z,t) p(0,t) 1. p(0,t) 1 φ s,t (z) =e s t z +..., z D, f t (z) =e t z +..., z D. f t (z) =e t z +..., z D. {φ s,t } 0 s t<+ {f t } t 0 lim t et φ 0,t (z) =f 0 (z) V (z,t) = z eiu(t) + z e iu(t), u(t). z

D C D C \ C D {f t } t 0 t f t(z) =z eiu(t) + z e iu(t) z f t(z) u(t) f t(0) = e t ɛ>0 δ>0 s, t 0 0 t s δ f t (D) ɛ 0 f t (D)\f s (D) {f t } t 0 f t(0) = e t Γ:[0, + ) C t 0 f t (D) C \ Γ[t, + ) 0 Γ {f t } t 0 u(t) u(t) f(0) = 0 f (0) = 1 S S f S f(z) =z + a 2 z 2 + a 3 z 3 +..., z D. a n n S n =2 a 2 =2 f(z) =e iθ k(e iθ z) θ [0, 2π) k(z) k(z) = z (1 z) 2 = z +2z2 +3z 3 +..., z D. n =3

n =4, 5 6 n a n = n f S n N f(z) =e iθ k(e iθ z) θ [0, 2π) f S f(d) =C \ Γ Γ Γ=(, 1/4] S f {f t } t 0 f(z) =f 0 (z) {f t } t 0 {φ s,t } 0 s t<+ lim t et φ 0,t (z) =f 0 (z). F : S C S S S n S n S u(t) S S τ(t) τ 0 τ 0 D

D τ 0 = 1 V (z,t) = (z +1) 2 p(z,t). H τ 0 D H z 2 i 1 z 1+z, ( ) 2i z V H (z,t) =4ip 2i + z,t = i p(z,t), p(z,t) :=4p ( 2i z 2i+z,t) z H t 0 Re p(z,t) 0 z H V H 1 (z,t) = u(t) z, u(t) t i p(z,t) = z u(t), 2 V H 2 (z,t) = u(t) z. t f 2 t(z) = f, t(z) u(t) f 0 (z) =z, z H, t f 1 t(z) = f 0 (z) =z, tanh[(f t(z) u(t))/2], z S,

u(t) :[0, + ) R S {z :0< Im z<π} { f t } t 0 f t = φ f t φ 1 φ : S D φ(z) :=i ez i e z + i. φ 0 1 i + i { f t } t 0 f t t (z) = V ( f t (z),t), f 0 (z) =z, z D V (z,t) = 1 2 (1 + z2 ) 1 iz+ eu(t) (z i) i + z e u(t) (1 + iz) < 1 V (z,t) = tanh[(z u(t))/2] φ V (i, t) = V ( i, t) = 0 t 0 ±i { f t } t 0 τ(t) = sech u(t)+i tanh u(t). u(t) u(t) {φ s,t } 0 s t<+ φ s,t (D) 0 s t<+ [0, ) C u(t) Lip(1/2) 1/2

u 1/2loc < 4 u 1/2loc := inf sup u(t) u(s). ɛ>0 t s <ɛ t s u(t) u(s) lim t s t s SLE κ u(t) = κb t B t κ>0 SLE κ t f t(z) =f t (z) ei κbt + f t (z) e i κb t ft (z), f 0(z) =z, z D, SLE κ t f 2 t(z) = f t (z), f 0 (z) =z, z H, κb t SLE κ t f 2 t(z) = tanh[(f t (z) κb t )/2], f 0(z) =z, z S. γ γ SLE κ SLE κ SLE κ SLE κ>0 SLE κ γ κ [0, 4] κ (4, 8) κ [8, ) SLE κ min (2, 1+κ/8)

SLE SLE SLE κ 1 0 1 0 SLE κ (D, 1, 0) D a D b D φ : D D φ(0) = b ˆφ(1) = a SLE κ (D, a, b) =SLE κ (φ(d), ˆφ(1),φ(0)) D a b γ SLE κ (D, 1, 0) φ γ SLE κ (φ(d), ˆφ(1),φ(0)) γ[0,t] SLE κ (D t,γ(t),b) D t D \ γ[0,t] b SLE κ (D t,γ(t),b) γ SLE SLE κ w t (z) =f t (z)/e i κb t dw t (z) =w t (z) 1+w t(z) 1 w t (z) dt i κw t db t, w 0 (z) =z, z D. D C 0 (D) D {p n } n=1 C 0 (D) p C 0 (D) K supp(p n p) K n =1, 2,...

m+p m x p y p n m+p m x p yp n K m, p = 1, 2,... C0 (D) T C0 (D) T (p n ) T (p) p n p C0 (D) D D(D) D (D) A D (, ) L 2 (D, A) (p, q) := D p(z) q(z) da(z), p q L 1 loc(d, A) D h L 1 loc(d, A) h L 1 (U) U U D L 1 loc(d) L p (D, A) L 1 loc(d, A) p 1 h L 1 loc(d, A) h L 2 (D, A) (h, p) p D(D) h D p (h, p), p D(D), L 1 loc(d) D (D) D (h, p) h p D w D f w : D D f w (w) =0, f w(w) > 0 D G D (z,w) = log f w (z). G D (z,w) = log 1 wz z w G H (z,w) = log z w z w p D(D) 1 2π G D(z,w)Δp(w) da(w) =p(z). D Δ w G D (z,w) =2πδ(z w), z D, G D (z,w) =0, z D.

Δ D(D) ker Δ = {0} D(D) Δ Δ 1 p(z) = 1 2π G D(z,w) p(w) da(w). D D(D) L 2 (D, A) (, ) (, ) L 2 (D,A) p, q D(D) (p, q) := D p(z) q(z) da(z). p, q D(D) (p, q) := D p(z) q(z) da(z). D(D) (p, q) E(D) := 2 G D(z 1,z 2 ) p(z 1 ) q(z 2 ) da(z 1 ) da(z 2 ). D D p, q D(D) (p, q) = 1 4π ( Δp, q) E(D), (p, q) =( Δp, q), (p, q) = 1 4π (Δp, Δq) E(D). D(D) p := (p, p), p := (p, p), p E(D) := (p, p) E(D). p p p E(D) p (Ω, F, P) D Φ:Ω D (D) Φ GF F D p D(D) (Φ, p) p 2 E(D)

Cov ((Φ, p), (Φ, q)) = (p, q) E(D). H(D) D(D) (, ) H(D) H(D) L 2 (D, A) L 1 loc(d) D (D) {e n } n=1 H(D) {α n } n=1 2 π α n e n. n=1 p D(D) 2 π n=1 α n (p,e n ) L 2 (Ω, P) 2 π n=1 α n (p,e n ) 2 L 2 (Ω,P)= 4π n=1 (p,e n ) 2, 4π n=1 (p,e n ) 2 =4π n=1 ( Δ 1 p,e n ) 2 =4π Δ 1 p 2 = p 2 E(D)<, p D(D) Φ D Φ=2π n=1 α n e n {e n } n=1 H(D) {α n } n=1 B D D H(B) H(D) Harm(B) H(D) H(D) =H(B) Harm(B). P H(B) P Harm(B)

f Harm(B) (f,g) =0 g H(B) (f,δg) L 2 (B,A) =0 g H(B). B Harm(B) H(D) B Φ B B Φ B := 2 π n=1 α n f n {f n } n=1 H(B) {α n } n=1 Φ B D p D(D) (Φ B, p) p 2 E(B) p D(D) (Φ B, p) =2 π =2 π =2 π n=1 n=1 n=1 α n (f n, p) α n (f n, Δ 1 p) α n (f n,p H(B) ( Δ 1 p)), (Φ B, p) 4π P H(B) ( Δ 1 p) 2 = ΔP H(B) ( Δ 1 p) 2 E(B) = ΔP H(B) ( Δ 1 p)+δp Harm(B) ( Δ 1 p) 2 E(B)= p 2 E(B). GF F h D h D (D) GF F h ˆΦ D =Φ D +h GF F h D ˆΦ D h h μ h D B D A(D\B) ˆΦ B =Φ B +h GF F B p D(D) Φ B D h B (h, p) L 2 (D,A) = D h(z) p(z) da(z)

ˆΦ B D GF F φ : D 1 D 2 Ψ D (D 2 ) Ψ φ Ψ φ D 1 (Ψ φ, p) =(Ψ, (φ 1 ) 2 p(φ 1 )), p C 0 (D 1 ). Ψ L 1 loc(d 2 ) Ψ(φ(z)) p(z) da(z) = Ψ(w) (φ 1 (w)) 2 p(φ 1 (w)) da(w). D 1 D 2 Φ D2 φ D 2 GF F D 1 SLE GF F SLE 4 GF F SLE κ κ SLE GF F SLE 4 Φ H Φ H ˆΦ H =Φ H 2 arg z. ˆΦ H ˆΦ H =0 ˆΦ H = π B t ˆΦ H {w t } t 0 SLE 4 dw t (z) = 2 w t (z) dt 2 db t, w 0 (z) =z, z H. ˆΦ H SLE 4 T > 0 ˆΦ H ˆΦ H w T

{φ s,t } 0 s t<+ t φ s,t(z) = φ s,t (z) eiu(t) +φ s,t(z) e iu(t) φ, s,t(z) φ s,s (z) =z, z D, {f t } t 0 t f t(z) =zf t(z) eiu(t) + z e iu(t) z, f 0(z) = lim t e t φ 0,t (z). {f t } t 0 φ s,t = ft 1 f s {f t } t 0 [0, + ) φ s,t (D) D \ φ s,t (D) γ D \ γ 0 u(t) D \ γ u(t) φ 0,t0 (D) =D \ γ t 0 > 0 u(t) Ĉ φ s,t (D)

u(t) Lip(1/2) u(t) 1/2loc (4, ) Lip(1/2) u(t) 1/2loc =0 V (t, z) =(z τ(t))(τ(t) z 1) p(z,t), τ(t) =e ikbt,k 0 τ(t) =e ikbt k R t φ t(z) = (τ(t) φt(z))2 τ(t) p(φ t (z),t), φ 0 (z) =z, z D. p(z,t) = p(z/τ(t)) p(z) :D C ψ t (z) = φt(z) τ(t) t ψ t(z) =(ψ t (z) 1) 2 p(ψ t (z)) ikψ t (z), ψ 0 (z) =z. {ψ t } t 0 {ψ t } t 0 {φ t } t 0 {φ t } t 0 Aut(D) p(z,t) = p(z/τ(t)) p(z) =A 1+z + Bi, A,B R. 1 z ψ t k τ(t)

2( Im p(0) p(0) ) <k<2( Im p(0) + p(0) ) 2( Im p(0) p(0) ) <k<2( Im p(0) + p(0) ) k<2( Im p(0) p(0) ) k>2( Im p(0) + p(0) ) φ t (z) k 4 A 2 +4 Bk+k 2 {ψ t } t 0 p(z) p(z,t) = p(z/τ(t)) w t = φ t /e ikbt dw t = ( k2 2 w t +(w t 1) 2 p(w t ) ) dt ikw t db t, w 0 (z) =z. {w t } t 0 {φ t (D)} t 0 SLE SLE dw t (z) =b(w t (z)) dt + σ(w t (z)) db t, w 0 (z) =z, z D, D b σ w t Hol(D, D) t 0 b σ w t Aut(D) t 0

b(z) dh t (z) =σ(h t (z)) db t, H 0 (z) =z, z D, D {H t } t 0 t h t(z) =σ(h t (z)), h 0 (z) =z, z D H t = h Bt {g t } t 0 g t := Ht 1 w t = w t h 1 B t dg t (z) = ( h 1 B t b ) (g t (z)) dt, g 0 (z) =z, z D. ( h 1 B t b ) {g t } t 0 Hol(D, D) {w t } t 0 Hol(D, D) b σ D SLE dw t (z) = b(w t (z)) dt + σ(w t (z)) dbt k, w 0 (z) =z, z D, b σ D t = {z D : w t (z) t}, w t : D t D t 0 g t = Ht 1 w t dg t (z) = ( h 1 B t b ) (g t (z)) dt, g 0 (z) =z, z D, {g t } t 0

b SLEs b D lim Re r 1 b(reiθ ) re iθ =0 e iθ D e iθ0 e iθ0 =1 D b D ( b(z) =α z iβ+ γ 1+z ) αz 2, 1 z z D, α C, β R γ 0 l H n (z) := z n+1, n Z, z H, D l D n := φ l H n φ : H D l D n φ φ(z) = z 2i z+2i l D n(z) = 2 n 1 ( i) n (z 1) n+1 (z +1) n+1. S = {z :0< Im z<π} ψ(z) = Log 2+z 2 z ( ) l S n(z) =ψ l H n (z) = 2 n sinh(z) tanh n z. 2 D span R {l D 1,l D 0,l D 1 } l D n,n= 2,...,1 D b D b(z) =b 2 l D 2(z)+b 1 l D 1(z)+b 0 l D 0 (z)+b 1 l D 1 (z), b 2 0 b 1,b 0,b 1 R D l D n l D n

D D σ D b D b(z) =b 2 l D 2(z)+b 1 l D 1(z)+b 0 l D 0 (z)+b 1 l D 1 (z), b 2 0,b 1,b 0,b 1 R, σ(z) =σ 1 l D 1(z)+σ 0 l D 0 (z)+σ 1 l 1, σ 1,σ 0,σ 1 R, σ 1 0. u t :[0, + ) R b, σ u t {f t } t 0 V (t, z) = ( h 1 u t b ) (z), {h t } t 0 D σ b 2 =2, σ 1 =1, b σ 2l 2 l 1 2l 2 + 1 2 l 0 l 1 + 1 4 l 1 2l 2 1 2 l 0 l 1 1 4 l 1 u t = κb t κ 0 w t = h κb t g t dw t (z) = b(w t (z)) dt + κσ(w t (z)) db t, w 0 (z) =z, z D. {w t } t 0 b σ SLE κ κ [0, 4] κ (4, 8) κ [8, )

ABP SLE b =2l 2 σ = l 1 + 1 4 l 1 t f t(z) = 1 (e iu(t) + f t (z)) 3 4 e iu(t) e iu(t) f t (z). τ(t) = e iu(t) p(z) = 1 1+z 4 1 z u(t) = κb t κ 0 ABP SLE ABP {f t } t 0 SLE ABP {w t } t 0 b σ κ =4 {w t } t 0 dw t (z) = b(w t (z)) dt +2σ(w t (z)) db t, w 0 (z) =z, z H. Φ H H B t b σ h H ˆΦ H := Φ H + h(z), ˆΦ H w T ˆΦ H T>0 GF F SLEs dg H (w t (z 1 ),w t (z 2 )) = 1 2 h(w t(z 1 )),h(w t (z 2 )), G H H, z 1,z 2 H z 1 z 2 h

h(w t (z)) t h SLE 4 2 B t αt, α R αt b(z) = 2 z α, σ = 1, h(z) = α 2 Im z 2 arg z. b(z) = 2 z βz,β R, σ = 1, h(z) = 2 arg z. SLE 4 2 B t αt, α R b(z) = 2 z α + z 2 + α 4 z2, h(z) = 1 α arg(2 z) 2 arg z + 1+α arg(2 + z). 2 2 b(z) = 2 ( z +1 β 1 ) ( 1 z 2 4 β ) z 2,β R, 2 h(z) = 2 arg(2 z) 2 arg z, b(z) = 2 ( z 1 β 1 ) ( β z 2 2 1 ) z 2,β R, 4 h(z) = 2 arg(2 + z) 2 arg z, SLE 4 2 B t αt α R b(z) = 2 z α z 2 α z2 4, α R, h(z) = 2 α Im arctan 2 z 2 arg z + 1 arg(4 + z 2 ). 2 GF F α = 0 SLE 4 h GF F

b σ SLEs ABP SLE ABP SLE SLEs SLE SLE(κ; ρ) SLEs SLE(κ; ρ) SLE(κ; ρ)

SLE(4)

SLE 4