Classical and Stochastic Slit Löwner Evolution

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

Dissertation date:

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TH2i2 ?QHQKQ`T?B+ p2+iQ` }2H/ B7 M/ QMHv B7 i?2`2 2tBbib TQBMiτ∈DM/

?QHQKQ`T?B+ 7mM+iBQMp:D→C- rBi? Rep≥0- bm+? i?i

V(z) = (z−τ)(¯τ z−1)p(z), z∈D. URXjV h?Bb `2T`2b2MiiBQM Bb mMB[m2 B7V(z)≡0X h?2 TQBMiτ BM i?Bb `2T`2b2Mi@

iBQM Bb T`2+Bb2Hv i?2 .2MDQv@qQHz TQBMi Q7 i?2 ;2M2`i2/ b2KB~Qr{g_t}t≥0X q2 +HH i?2 7mM+iBQM p(z) i?2 >2`;HQix 7mM+iBQM- M/ i?2 TQBMi τ i?2 i@

i`+iBM; TQBMi Q7 i?2 }2H/V(z)X h?2 i2`K ǳii`+iBM; TQBMiǴ Bb KQiBpi2/

#v i?2 7+i i?i g_t(z)→τ HQ+HHv mMB7Q`KHv BMD bt→+∞- 2t+2Ti 7Q`

i?2 +b2 r?2M {g_t}t≥0Bb b2KB~Qr +QMbBbiBM; Q7 2HHBTiB+ miQKQ`T?BbKbX Smb?7Q`r`/b Q7 ?QHQKQ`T?B+ p2+iQ` }2H/b

Ai Bb 2bv iQ 2ti2M/ i?2 /2}MBiBQM Q7 ?QHQKQ`T?B+ b2KB~Qrb iQ M `#Bi``v bBKTHv@+QMM2+i2/ ?vT2`#QHB+ /QKBMD- r?B+? KQmMib iQ `2TH+BM;D#v D BM .2}MBiBQM RX

G2if:D₁→D₂#2 +QM7Q`KH BbQKQ`T?BbK Q7 irQ bBKTHv +QMM2+i2/

?vT2`#QHB+ /QKBMbX A7{g_t¹}t≥0Bb b2KB~Qr BMD₁- i?2M{g_t²}t≥0={φ◦g_t¹◦ φ⁻¹}t≥0Bb b2KB~Qr BMD₂X G2iV₁M/V₂#2 i?2 BM}MBi2bBKH ;2M2`iQ`b Q7{g_t¹}t≥0M/{g_t²}t≥0- `2bT2+iBp2HvX h?2M r2 bv i?i i?2 p2+iQ` }2H/bV₁ M/V₂`2φ@`2Hi2/- M/ r2 +HHV₂i?2 Tmb?7Q`r`/ Q7V₁rBi? `2bT2+i iQ φX q2 r`Bi2V₂=φ_∗V₁X 1tTHB+BiHv-

φ_∗V₁(z) = 1

φ⁻¹(z)V₁(φ⁻¹(z)).

(14)

e AMi`Q/m+iBQM

GB2 /2`BpiBp2b Q7 +QKTH2t 7mM+iBQMb

G2i f #2 C¹@bKQQi? +QKTH2t@pHm2/ 7mM+iBQM BM /QKBM D- M/ H2i z=x+iyX h?2M Bi Bb +QKKQM iQ r`Bi2

∂f= ∂f

∂z = 1 2

∂

∂x−i ∂

∂y

f,

∂f= ∂f

∂z¯= 1 2

∂

∂x+i ∂

∂y

f.

G2i V #2 ?QHQKQ`T?B+ p2+iQ` }2H/ BM /QKBMDX h?2MV +ib QM +QKTH2t 7mM+iBQMf ∈C¹(D,C)++Q`/BM; iQ i?2 `mH2

f →V ∂+ ¯V∂¯f.

q2 /2MQi2 i?Bb QT2`iQ` #vLV- bQ i?i L_V =V ∂+ ¯V ∂.¯

Ai Bb bT2+BH +b2 Q7 i?2 GB2 /2`BpiBp2X :2QK2i`B+HHv-L_V `2T`2b2Mib i?2 /2`BpiBp2 Q7f BM i?2 /B`2+iBQM Q7 i?2 b2KB~Qr ;2M2`i2/ #vVX

aQK2 Q7 Bib #bB+ T`QT2`iB2b `2 HBbi2/ #2HQrX RX LVf Bb HBM2` BMf M/V;

kX LVf¯ = LVf , M/- +QMb2[m2MiHv- LV Ref = ReLVf- LV Imf = ImLVfc

jX LV(∂f) =∂LVf,LV( ¯∂f) = ¯∂LVfX

h?Bb /2}MBiBQM Q7 i?2 GB2 /2`BpiBp2 +M #2 2ti2M/2/ iQ i?2 +b2 Q7 b2p2`H +QKTH2t p`B#H2b- i?i Bb- r?2M V Bb p2+iQ` }2H/ BM /QKBM D⊂Cⁿ- M/f :Cⁿ →CX q2 r`Bi2z= (z₁, . . . , z_n)-∂ =_∂z^∂

1, . . . , ^∂

∂zn

-

∂¯=_∂¯^∂_z

1, . . . , _∂¯^∂

zn

- V(z) = (V₁(z), . . . , V_n(z))- M/ i?2M i?2 /2`BpiBp2 Q7 f HQM; i?2 ~Qr ;2M2`i2/ #vV Bb ;Bp2M #v i?2 bmK Q7 /Qi T`Q/m+ib

LVf(z) =V(z)·∂+V(z)·∂¯f(z). URX9V RXRXk hQTB+b BM biQ+?biB+ MHvbBb

lMH2bb bT2+B}2/ Qi?2`rBb2- i?2 /2iBH2/ 7Q`KmHiBQMb M/ T`QQ7b Q7 i?2

`2bmHib K2MiBQM2/ BM i?Bb b2+iBQM +M #2 7QmM/ BM (ǟFbyj) Q` (_uNN)X aiQ+?biB+ BMi2;`Hb Q7 AiƬ M/ ai`iQMQpB+?

h?2 irQ KQbi rB/2Hv mb2/ ivT2b Q7 biQ+?biB+ BMi2;`Hb `2 i?Qb2 Q7 AiƬ M/

ai`iQMQpB+?X h?2 +Q``2bTQM/BM; +QMbi`m+iBQMb `2 7B`Hv +QKTHB+i2/c Bi Bb TQbbB#H2 ?Qr2p2` iQ ;Bp2 BMimBiBp2Hv +H2` /2}MBiBQMb B7 i?2 BMi2;`M/b

`2 /Ti2/ +QMiBMmQmb T`Q+2bb2bX

G2iX_t#2 +QMiBMmQmb T`Q+2bb /Ti2/ iQ i?2 }Hi`iBQM{Ft}t≥0Q7 i?2 biM/`/ "`QrMBM KQiBQMB_t- M/ H2iT >0X h?2M i?2 AiƬ BMi2;`H Q7X_t

(15)

RXR S`2HBKBM`B2b d rBi? `2bT2+i iQ B_t QM [0, T] +M #2 /2}M2/ b i?2 HBKBi BM T`Q##BHBiv Q7 i?2 _B2KMM@aiB2HiD2b bmKb

T

0 X_tdB_t:= lim

Δt→0 n j=1

X_t_j−1B_t_j−B_t_j−1, URX8V r?2`2 0 =t₀< t₁< . . . < t_n=T- M/Δt:= max_{j=1,..., n}(t_j−t_j−1)X

aBKBH`Hv- i?2 ai`iQMQpB+? BMi2;`H +M #2 /2}M2/ b i?2 HBKBi BM T`Q#@

#BHBiv

T

0 X_t◦dB_t:= lim

Δt→0 n j=1

X_tj−1+tj

2

B_t_j−B_t_j−1. URXeV

h?2 /2}MBiBQMb Q7 i?2 AiƬ M/ ai`iQMQpB+? biQ+?biB+ BMi2;`Hb +M #2 2ti2M/2/ iQ KQ`2 +QKTHB+i2/ BMi2;`M/b M/ BMi2;`iQ`b- #mi i?2 BMimBiBp2 7Q`KmHb URX8V M/ URXeV rBHH MQi ?QH/ BM ;2M2`HX

6Q` }t2/T >0- i?2 AiƬ BMi2;`H₀^TX_sdB_sBb `M/QK p`B#H2- M/

Bi Bb /2}M2/ mT iQ b2i Q7 K2bm`2 x2`QX A7 r2 HHQr i?2 mTT2` BMi2;`H HBKBi p`v- i?2M ₀^tX_sdB_s Kv #2 `2;`/2/ b biQ+?biB+ T`Q+2bbX JQ`2 T`2+Bb2Hv- Bi +M #2 T`Qp2/ i?i i?2`2 2tBbib +QMiBMmQmb T`Q+2bbY_t/Ti2/

iQ{Ft}t≥0bm+? i?i P

Y_t= ^t

0 X_sdB_s

= 1 7Q` HHt≥0.

aBKBH`Hv- i?2 ai`iQMQpB+? BMi2;`H₀^tX_s◦dB_srBi? p`B#H2 mTT2` HBKBi HbQ /2}M2b +QMiBMmQmb biQ+?biB+ T`Q+2bbX

A7E₀^TX_s²ds<∞7Q` HHT >0- i?2M₀^tX_sdB_sBb K`iBM;H2X A7 r2

`2Ht i?Bb +QM/BiBQM iQP₀^TX_s²ds <∞= 1- i?2M r2 +M QMHv ;m`Mi22 i?i Y_tBb HQ+H KìBM;H2X JQ`2Qp2`- #v i?2 KìBM;H2 `2T`2b2MiiBQM i?2Q`2K- 2p2`v +QMiBMmQmb {F}t≥0@/Ti2/ HQ+H KìBM;H2 M_t +M #2 r`Bii2M bM_t=M₀+₀^tX_sdB_s 7Q` bQK2F_t@/Ti2/ T`Q+2bb X_tX

Zm/`iB+ +Qp`BiBQM

_2+HH i?i i?2 [m/`iB+ +Qp`BiBQM Q7 irQ T`Q+2bb2bX_tM/Y_tBb /2}M2/

b i?2 HBKBi BM T`Q##BHBiv

X_T, Y_T=X, YT := lim

Δt→0 n j=1

(X_t_j−X_t_j−1) (Y_t_j−Y_t_j−1)

UB7 Bi 2tBbibVX b #27Q`2- 0 = t₀ < t₁ < . . . < t_n = T- M/ Δt :=

max_{j=1,..., n}(t_j−t_j−1)X

amTTQb2 i?i i?2 T`Q+2bb2bX_t M/Y_t+M #2 `2T`2b2Mi2/ #v X_t(ω) = ^t

0 b₁(ω, s)ds+ ^t

0 σ₁(ω, s)dB_s(ω), URXdV M/

Y_t(ω) = ^t

0 b₂(ω, s)ds+ ^t

0 σ₂(ω, s)dB_s(ω),

(16)

3 AMi`Q/m+iBQM

r?2`2b_j(ω, t)- σ_j(ω, t)-j = 1,2`2Ft@/Ti2/ T`Q+2bb2b biBb7vBM;

P ^T

0 |b_j(ω, s)|ds <∞

= 1, P ^T

0 |σ_j(ω, s)|²ds

= 1,

7Q` HH T > 0 M/ j = 1,2X h?2M i?2 [m/`iB+ p`BiBQM +M #2 +QMp2@

MB2MiHv r`Bii2M b i?2 7QHHQrBM; BMi2;`H X, Yt(ω) = ^t

0 σ₁(ω, s)σ₂(ω, s)ds.

h?2 MQiBQM Q7 [m/`iB+ +Qp`BiBQM T`QpB/2b bBKTH2 7Q`KmH `2HiBM;

i?2 irQ ivT2b Q7 i?2 biQ+?biB+ BMi2;`H K2MiBQM2/ #Qp2,

T

0 X_t◦dB_t = ^T

0 X_tdB_t+1

2X, BT. URX3V

*QKTH2t AiQ 7Q`KmH

G2i w_t = X_t¹ +i X_t² #2 +QKTH2t@pHm2/ T`Q+2bb bm+? i?i i?2 `2H@

pHm2/ T`Q+2bb2b X_t¹- X_t² `2 Q7 i?2 7Q`K URXdVX G2i f : C → C #2 irB+2 /Bz2`2MiB#H2 +QKTH2t 7mM+iBQMX h?2 +QKTH2t AiƬ 7Q`KmH (l#Ǡ3d) /2b+`B#2b ?Qr w_ti`Mb7Q`Kb mM/2` bm+? 7mM+iBQMb,

URXNV f(w_t) =f(w₀) + ^t

0 ∂f(w_s)dw_s+ ^t

0

∂f(w¯ _s)dw¯_s +1

2

t

0 ∂²f(w_s)dws+1 2

t

0 ∂¯²f(w_s)dw¯s

+ ^t

0 ∂∂f(w¯ _s)dw,w¯s,

r?B+? Bb mbmHHv r`Bii2M BM i?2 b?Q`i2` /Bz2`2MiBH 7Q`K

URXRyV df(w_t) =∂f(w_t)dw_t+ ¯∂f(w_t)dw¯_t

+1

2∂²f(w_t)dwt+1

2∂¯²f(w_t)dw¯t+∂∂f(w¯ _t)dw,w¯t. amTTQb2 w_tbiBb}2b i?2 7QHHQrBM; ai`iQMQpB+? BMi2;`H 2[miBQM

w_t=w₀+ ^t

0 b(w_s)ds+ⁿ

k=1 t

0 σ_k(w_s)◦dB^k_s, URXRRV rBi? ?QHQKQ`T?B+ +Q2{+B2Mib bM/σ- M/n BM/2T2M/2Mi "`QrMBM KQ@

iBQMb B¹_t, . . . , B_tⁿX KQ`2 mbmH- #mi 7Q`KH- rv iQ r`Bi2 URXRRV Bb i?2 ai`iQMQpB+? biQ+?biB+ /Bz2`2MiBH 2[miBQM Ua.1V

dw_t=b(w_t)dt+

n k=1

σ_k(w_t)◦dB_t^k. URXRkV

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RXR S`2HBKBM`B2b N Ai +M #2 /2/m+2/ 7`QK URX3V M/ URXRyV i?i URXRkV Bb 2[mBpH2Mi iQ i?2 7QHHQrBM; AiƬ a.1^R

dw_t=

⎛

⎝b(w_t) +1 2

n k=1

σ_k(w_t)σ_k(w_t)

⎞

⎠dt+

n k=1

σ_k(w_t)dB^k_t, r?B+? Bb b?Q`i?M/ 7Q` i?2 AiƬ BMi2;`H 2[miBQM

w_t=w₀+ ^t

0

⎛

⎝b(w_s) +1 2

n k=1

σ_k(w_s)σ_k(w_s)

⎞

⎠ds+

n k=1

t

0 σ_k(w_s)dB_s^k. S`Q+2bb2b biBb7vBM; URXRkV `2 bT2+BH +b2 Q7 /BzmbBQM T`Q+2bb2bX 6Q`

/BzmbBQM T`Q+2bb2b rBi? ?QHQKQ`T?B+ +Q2{+B2Mib- b BM URXRkV- i?2 +QKTH2t AiƬ 7Q`KmH +M #2 2H2;MiHv 7Q`KmHi2/ mbBM; i?2 GB2 /2`BpiBp2b MQiiBQMX :Bp2M irB+2 /Bz2`2MiB#H2 7mM+iBQMf :C→C- i?2 T`Q+2bbf(w_t)biBb}2b i?2 7QHHQrBM; a.1

URXRjV df(w_t) =Lbf(w_t)dt+ⁿ

k=1Lσkf(w_t)◦dB_t^k

=

⎛

⎝Lb+1 2

n k=1L²_σ_k

⎞

⎠f(w_t)dt+ⁿ

k=1Lσkf(w_t)dB_t^k.

h?2 7Q`KmH `2KBMb pHB/ B7 i?2 T`Q+2bb w_t iF2b pHm2b BMCⁿ-f Bb C² +QKTH2t 7mM+iBQM Q7 n+QKTH2t p`B#H2b- M/ i?2 +Q2{+B2Mibb:Cⁿ→C- σ_k:Cⁿ→C `2 ?QHQKQ`T?B+ BM 2+? p`B#H2X

aiQ+?biB+ ~Qrb BM i?2 +QKTH2t THM2

AM i?2 /2i2`KBMBbiB+ +b2- i?2 ~Qr Ub2KB~QrV {g_t}t≥0 Q7 M Q`/BM`v m@

iQMQKQmb /Bz2`2MiBH 2[miBQM

∂

∂tg_t(z) =V(g_t(z)), g₀(z) =z,

/2b+`B#2b ?Qr i?2 P.1Ƕb bQHmiBQM /2T2M/b QM i?2 BMBiBH +QM/BiBQM- bQ i?i 7Q` }t2/ pHm2 Q7t- i?2 KTg_t(·)Bb +2`iBM i`Mb7Q`KiBQM Q7 i?2 bT+2 QM r?B+? i?2 P.1 Bb +QMbB/2`2/X 6Q` BMbiM+2- B7 i?2 2[miBQM Bb +QMbB/2`2/ BM i?2 mMBi /BbFDM/V Bb b2KB+QKTH2i2 ?QHQKQ`T?B+ p2+iQ`

}2H/ BM D- i?2Mg_t∈Hol(D,D)7Q` 2+?t≥0- b r2 ?p2 b22M #Qp2X AM bBKBH` p2BM- QM2 +M bim/v ~Qrb Q7 biQ+?biB+ /Bz2`2MiBH 2[m@

iBQMbX G2i z ∈ D- M/ H2i w_t(z) /2MQi2 i?2 bQHmiBQM Q7 URXRkV rBi? i?2 BMBiBH +QM/BiBQMw₀=zX h?2 7KBHv{w_t}t≥0bQHpBM; i?2 T`Q#H2K

⎧⎨

⎩

dw_t(z) =b(w_t(z), t)dt+ⁿ_k=1σ_k(w_t(z), t)◦dB^k_t,

w₀(z) =z, z∈D. URXR9V

RAM i?Bb i?2bBb- i?2 T`BK2 K`F Hrvb /2MQi2b i?2 /2`BpiBp2 Q7 ?QHQKQ`T?B+ 7mM+iBQM rBi? `2bT2+i iQ i?2 +QKTH2t p`B#H2- i?mb- 7Q` 2tKTH2-σ_k(w)/2MQi2b _∂w^∂ σk(w)- M/f_t(z)

?b i?2 bK2 K2MBM; b _∂z^∂ft(z)X

(18)

Ry AMi`Q/m+iBQM Bb +HH2/ i?2 biQ+?biB+ ~Qr Q7 URXR9VX *QKT`2?2MbBp2 Qp2`pB2rb Q7 i?2 i?2Q`v Q7 biQ+?biB+ ~Qrb +M #2 7QmM/ BM (EmM39)- (EmMNd) M/ (Aq3N)X

6Q` i?2 2tBbi2M+2 Q7 mMB[m2 KtBKH bQHmiBQM Bib Bb bm{+B2Mi i?i i?2 p2+iQ` }2H/b b(z, t), σ₁(z, t), . . . σ_n(z, t) `2 C² BM z- M/ C¹ BM t(EmM39- h?2Q`2K AAX3Xj)X

MHviB+ T`QT2`iB2b Q7 biQ+?biB+ ~Qrb

JMv MHviB+ T`QT2`iB2b Q7 biQ+?biB+ ~Qr +M #2 /2/m+2/ 7`QK i?2 MHviB+ T`QT2`iB2b Q7 i?2 p2+iQ` }2H/b /2i2`KBMBM; i?2 ~QrX

G2i T(z) /2MQi2 i?2 `M/QK iBK2 r?2M w_t(z)- i?2 bQHmiBQM iQ URXR9V rBi? i?2 BMBiBH +QM/BiBQMw_t(z) =z- +2b2b iQ 2tBbiX T(z)Bb mbmHHv +HH2/

i?2 2tTHQbBQM iBK2 Q` i?2 2b+T2 iBK2X

G2iD_t/2MQi2 i?2 `M/QK b2i Q7 BMBiBH +QM/BiBQMb 7Q` r?B+? i?2 bQHmiBQM 2tBbib i H2bi mT iQ iBK2t >0- BX2X-D_t ={z ∈ D:T(z)> t}X Ai Bb 2bv iQ b22 i?iD_s⊇D_t 7Q` s≤tX

G2i R_t /2MQi2 i?2 BK;2 Q7D_t rBi? `2bT2+i iQ w_t- BX2X- R_t := w_t(D_t)X h?2M w_t :D_t →R_t Bb ?QK2QKQ`T?BbK 7Q` HH t≥ 0rBi? T`Q##BHBiv 1 (EmM39- h?2Q`2K AAXNXR)X

amTTQb2 i?i i?2 p2+iQ` }2H/ b(z, t) Bb C¹ BMt > 0 M/C^d BMz ∈ D- M/ i?2 p2+iQ` }2H/bσ₁(z, t), . . . , σ_n(z, t)`2C¹BMtM/C^d+1 BMzX h?2M w_t : D_t → R_t Bb C^d@/Bz2QKQ`T?BbK 7Q` HH t ≥ 0 rBi? T`Q##BHBiv 1 (EmM39- h?2Q`2K AAXNXk)X

AM T`iB+mH`- B7 i?2 }2H/b b(z, t), σ₁(z, t), . . . σ_n(z, t) `2 C¹ BM t > 0 M/C^∞ BMz ∈D- i?2M i?2w_t(z) :D_t →R_t Bb C^∞@/Bz2QKQ`T?BbK 7Q`

HHt≥0rBi? T`Q##BHBiv1X

A7 i?2 p2+iQ` }2H/b b(z, t), σ₁(z, t), . . . σ_n(z, t)`2C¹BMt >0M/ ?QHQ@

KQ`T?B+ BMz ∈D- i?2M i?2w_t(z) :D_t→R_t Bb +QM7Q`KH BbQKQ`T?BbK 7Q` HHt≥0rBi? T`Q##BHBiv 1(EmM39- h?2Q`2K AAAX8Xd)X

*QMbB/2` i?2 ~Qrw_t(z) Q7 i?2 miQMQKQmb a.1

⎧⎨

⎩

dw_t(z) =b(w_t(z))dt+ⁿ_k=1σ_k(w_t(z))◦dB^k_t,

w₀(z) =z, z ∈D, URXR8V

r?2`2 b(z)- σ₁(z), . . . , σ_n(z) `2 C^∞- +QKTH2i2- iBK2@BM/2T2M/2Mi p2+iQ`

}2H/bX h?2Mw_t(z)Bb /Bz2QKQ`T?BbK Q7D7Q` 2+?t≥0rBi? T`Q##BHBiv RX A7 i?2 GB2 H;2#`g;2M2`i2/ #v i?2b2 p2+iQ` }2H/b Bb }MBi2@/BK2MbBQMH- i?2M w_t(z) iF2b pHm2b BM i?2 bbQ+Bi2/ GB2 ;`QmT Q7 /Bz2QKQ`T?BbKb (EmM39- h?2Q`2K AAAX8XR)X

*QKTQbBiBQM M/ BMp2`bBQM Q7 biQ+?biB+ ~Qrb

aiQ+?biB+ /Bz2`2MiBH 2[miBQMb r`Bii2M BM i?2 ai`iQMQpB+? 7Q`K Q#2v i?2 i`Mb7Q`KiBQM `mH2b Q7 +HbbB+H +H+mHmbX h?2 +?BM `mH2 M/ `2Hi2/ 7Q`@

KmHb +M #2 r`Bii2M BM M 2bT2+BHHv M2i 7Q`K B7 r2 mb2 i?2 Tmb?7Q`r`/

MQiiBQM 7`QK a2+iBQM RXRXRX

*QMbB/2` i?2 ~Qrb_⎧

⎨

⎩

dw_t(z) =b(w_t(z), t)dt+ⁿ_k=1σ_k(w_t(z), t)◦dB_t^k,

w₀(z) =z, z ∈D, URXReV

(19)

RXk GƺrM2` i?2Q`v RR

M/ ⎧

⎨

⎩

dw˜_t(z) = ˜b( ˜w_t(z), t)dt+ⁿ_k=1σ˜_k( ˜w_t(z), t)◦dB^k_t,

˜

w₀(z) =z, z∈D, URXRdV

M/ H2iT(z), T˜(z)/2MQi2 i?2 +Q``2bTQM/BM; 2tTHQbBQM iBK2bX

*QMbB/2` i?2 +QKTQbBi2 ~Qr{ζ_t}t≥0-ζ_t:= ˜w_t◦w_tX h?2 2tTHQbBQM iBK2 7Q` i?2 +QKTQbBi2 ~Qr Bb i?2M ;Bp2M #v U(z) := min[inf{t > 0 :w_t(z)∈ D˜_t}, T(z)]- r?2`2D˜_t={z∈D: ˜T(z)> t}X

h?2 +QKTQbBi2 ~Qr biBb}2b

URXR3V dζ_t(z) =˜b(ζ_t(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, b22 (EmM39- h?2Q`2K AAAXjXR)X

h?2 BMp2`b2 ~Qr{η_t}t≥0, η_t=w⁻¹_t - biBb}2b

URXRNV dη_t(z) =−(η_t_∗b) (η_t(z), t)dt−ⁿ

k=1

(η_t_∗σ_k) (η_t(z), t)◦dB_t^k

=−η_t(z)b(z, t)dt−ⁿ

k=0

η_t(z)σ_k(z, t)◦dB_t^k, b22 (EmM39- *Q`QHH`v AAAXjX9)X

RXk GƺrM2` i?2Q`v

LBM2iv v2`b ?p2 Tbb2/ bBM+2 E`H GƺrM2` U*?`H2b GQ2rM2`V BMi`Q/m+2/

?Bb Q`B;BMH 2[miBQM BM (Gƺrkj)- M/ iQ/v i?2`2 Bb i H2bi /Qx2M Q7 Q#@

D2+ib `272``2/ iQ b i?2 ǳGƺrM2` 2[miBQMǴX h?2`2 `2 +?Q`/H- `/BH- /BTQ@

H` GƺrM2` 2[miBQMb- GƺrM2`Ƕb Q`/BM`v M/ T`iBH /Bz2`2MiBH 2[miBQMb- 7Q`r`/ M/ #+Fr`/ U`2p2`b2V 2[miBQMb- i?2 GƺrM2`@Em7`2p 2[miBQMb- i?2 ;2M2`H GƺrM2` 2[miBQMb 2i+X

JQ`2Qp2`- i?2`2 Bb `B+? 7KBHv Q7 biQ+?biB+ Gƺ2rM2` 2[miBQMb, +?Q`@

/H- `/BH M/ /BTQH` SLE_κ- b r2HH b SLE(κ, ρ)X aiQ+?biB+ GƺrM2`

2[miBQMb `2- BM 7+i- i?2 KBM `2bQM r?v GƺrM2` i?2Q`v ii`+ib bQ Km+? BMi2`2bi MQr/vbX

hQ /2b+`B#2 i?Bb TH2i?Q` Q7 2[miBQMb BM +QM+Bb2 rv- r2 iF2 i?2 7QHHQrBM; TT`Q+?X 6B`bi- r2 BMi`Q/m+2 i?2 KBM MQiBQMb Q7 i?2 bQ@+HH2/

;2M2`H GƺrM2` i?2Q`v- `i?2` `2+2Mi 7`K2rQ`F /2p2HQT2/ BM ("*.JyN)- (*.J:Ry)- ("*.JRk) M/ (*.J:R9)X h?2M r2 /2}M2 HH i?2 Qi?2`

2[miBQMb K2MiBQM2/ #Qp2 b bT2+BH +b2b Q7 i?2 ;2M2`H 7Q`KmHiBQMX AM i?2 bBKTHv +QMM2+i2/ +b2- i?2 ;2M2`H GƺrM2` i?2Q`v ?b #22M /2@

p2HQT2/ 7Q` QM2 Ui?2 mMBi /BbFV M/ b2p2`H +QKTH2t /BK2MbBQMb U+QKTH2i2

?vT2`#QHB+ KMB7QH/bVX AM (*.J:Rj) M/ (*.J:RR) i?2 ;2M2`H GƺrM2`

i?2Q`v Bb +QMbi`m+i2/ 7Q` i?2 /Qm#Hv +QMM2+i2/ QM2@/BK2MbBQMH +b2 Ui?2 MMmHmbVX

(20)

Rk AMi`Q/m+iBQM PMHv i?2 QM2@/BK2MbBQMH bBKTHv +QMM2+i2/ i?2Q`v Bb `2H2pMi iQ i?2 T`Q#H2Kb +QMbB/2`2/ BM i?Bb i?2bBbX

h?2 ;2M2`H GƺrM2` i?2Q`v ?b #22M 7Q`KmHi2/ BM irQ p2`bBQMb- r?B+?

`2~2+ib i?2 7+i i?i i?2 i?2 MmK2`Qmb GƺrM2` 2[miBQMb 7HH BMiQ irQ +Hbb2b- bQ@+HH2/ ǳ7Q`r`/Ǵ M/ ǳ#+Fr`/Ǵ 2[miBQMbX h?2 `2bQM 7Q` i?Bb i2`KBMQHQ;v Bb i?i i?2 2[miBQMb Q7 QM2 +Hbb +M #2 Q#iBM2/ b iBK2@

`2p2`b2/ p2`bBQMb Q7 i?2 2[miBQMb 7`QK i?2 Qi?2` +Hbb- BX2X- #v `2TH+BM;

i?2 iBK2 p`B#H2t#v−tX

h?2`2 b22Kb iQ #2 +2`iBM /2;`22 Q7 /Bb;`22K2Mi BM i?2 i2`KBMQHQ;vX 6Q` BMbiM+2- i?2 rQ`/b ǳ#+Fr`/Ǵ M/ ǳiBK2@`2p2`b2/Ǵ ?p2 QTTQbBi2 K2MBM;b B7 r2 +QKT`2- 7Q` 2tKTH2- (*.J:R9) rBi? (GryN)- (.aRR)- (_wRj)X AM i?Bb i?2bBb r2 i`v iQ 7QHHQr i?2 Hii2` +QMp2MiBQMX

RXkXR "+Fr`/ UBM+`2bBM;V ;2M2`H GƺrM2` i?2Q`v h?2 2[miBQM +QMbB/2`2/ #v GƺrM2` BM ?Bb b2KBMH TT2` (Gƺrkj) #2HQM;b iQ i?2 `2HK Q7 i?2 #+Fr`/ GƺrM2` i?2Q`v UQ` i?2 BM+`2bBM; GƺrM2`

i?2Q`v- B7 r2 mb2 i?2 i2`KBMQHQ;v Q7 (*.J:R9)VX "+Fr`/ i?2Q`v Bb +HQb2Hv `2Hi2/ iQ i?2 i?2Q`v Q7 b2KB;`QmTb Q7 MHviB+ 7mM+iBQMb- M/ iQ +2`iBM 2ti2Mi +M #2 `2;`/2/ b Bib MQM@miQMQKQmb ;2M2`HBxiBQMX AM KMv +b2b- #+Fr`/ 2[miBQMb `2 2bB2` iQ rQ`F rBi?- +QKT`2/ iQ i?2 7Q`r`/ 2[miBQMb i?i r2 +QMbB/2` BM i?2 M2ti b2+iBQMX

h?2 i?`22 +2Mi`H Q#D2+ib Q7 i?2 ;2M2`H GƺrM2` i?2Q`v `2 2pQHmiBQM 7KBHB2b- >2`;HQix p2+iQ` }2H/b M/ ;2M2`H GƺrM2` +?BMbX

h?`Qm;?Qmi i?Bb b2+iBQM r2 bbmK2 i?i i?2 2pQHmiBQM 7KBHB2b- >2`@

;HQix p2+iQ` }2H/b M/ GƺrM2` +?BMb `2 /2}M2/ BM i?2 mMBi /BbFX h?Bb Bb K2`2Hv +QMp2MiBQM- M/ HH i?2 `2bmHib Q7 i?2 ;2M2`H GƺrM2` i?2Q`v U2t+2Ti 7Q` bQK2 `2T`2b2MiiBQM 7Q`KmHbV i`MbHi2 iQ i?2 +b2 Q7 M `@

#Bi``v bBKTHv@+QMM2+i2/ ?vT2`#QHB+ /QKBM- 7Q` BMbiM+2- i?2 mTT2` ?H7@

THM2H={z: Imz >0}Q` i?2 BM}MBi2 bi`BT S={z: 0<Imz <1}X q2 +HH i?2 /QKBM BM r?B+? i?2 2pQHmiBQM 7KBHB2b `2 /2}M2/ i?2 +MQMB+H /QKBMX

1pQHmiBQM 7KBHB2b M/ >2`;HQix p2+iQ` }2H/b

.2}MBiBQM kX M 2pQHmiBQM 7KBHv Q7 Q`/2`d∈[1,+∞]Bb irQ@T`K2i2`

7KBHv{φ_s,t}0≤s≤t<+∞Q7 ?QHQKQ`T?B+ b2H7@KTb Q7 i?2 mMBi /BbF- bm+? i?i i?2 7QHHQrBM; i?`22 +QM/BiBQMb `2 biBb}2/X

Ç φ_s,s=id_Dc

Ç φ_s,t=φ_u,t◦φ_s,u7Q` HH0≤s≤u≤t <+∞c

Ç 7Q` Mv z ∈ D M/ T > 0 i?2`2 Bb MQM@M2;iBp2 7mM+iBQM k_z,T ∈ L^d([0, T],R),bm+? i?i

|φ_s,u(z)−φ_s,t(z)|≤ ^t

u k_z,T(ξ)dξ URXkyV 7Q` HH0≤s≤u≤t≤T.

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RXk GƺrM2` i?2Q`v Rj h?2 +QM/BiBQM URXkyV BKTHB2b i?i 7Q` 2p2`v }t2/ z ∈ D- i?2 7mM+iBQM φ_s,t(z)Bb /Bz2`2MiB#H2 rBi? `2bT2+i iQtHKQbi 2p2`vr?2`2 QM[s,+∞)U#v

#bQHmi2 +QMiBMmBivVX M BM}MBi2bBKH /2b+`BTiBQM Q7 M 2pQHmiBQM 7KBHv Bb ;Bp2M BM i2`Kb Q7 >2`;HQix p2+iQ` }2H/X

.2}MBiBQM jX U;2M2`HBx2/V >2`;HQix p2+iQ` }2H/ Q7 Q`/2` d∈ [1,+∞]

Bb 7mM+iBQM V :D×[0,+∞)→CbiBb7vBM; i?2 7QHHQrBM; +QM/BiBQMb, Ç i?2 7mM+iBQM[0,+∞)t →V(z, t)Bb K2bm`#H2 7Q` 2p2`vz∈Dc Ç i?2 7mM+iBQM z → V(z, t) Bb ?QHQKQ`T?B+ BM i?2 mMBi /BbF 7Q` t ∈

[0,+∞)c

Ç 7Q` Mv +QKT+i b2i K ⊂ D M/ 7Q` 2p2`v T > 0 i?2`2 2tBbib MQM@M2;iBp2 7mM+iBQMk_K,T ∈L^d([0, T],R),bm+? i?i

|V(z, t)|≤k_K,T(z) 7Q` HHz∈K M/ HKQbi 2p2`vt∈[0, T]c

Ç 7Q` HKQbi 2p2`vt∈[0,+∞)i?2 p2+iQ` }2H/V(·, t)Bb b2KB+QKTH2i2X h?2Q`2K R U("*.JRk- h?2Q`2K RXR)VX 6Q` Mv 2pQHmiBQM 7KBHv {φ_s,t} Q7 Q`/2` d≥1 BM i?2 mMBi /BbF i?2`2 2tBbib M 2bb2MiBHHv mMB[m2 >2`;HQix p2+iQ` }2H/ V(z, t) Q7 Q`/2` d, bm+? i?i 7Q` HH z ∈D M/ 7Q` HKQbi HH t∈[0,+∞)

∂

∂tφ_s,t(z) =V(φ_s,t(z), t). URXkRV

*QMp2`b2Hv- 7Q` Mv >2`;HQix p2+iQ` }2H/ V(z, t)Q7 Q`/2`d≥1BM i?2 mMBi /BbF i?2`2 2tBbib mMB[m2 2pQHmiBQM 7KBHv{φ_s,t}0≤s≤t<+∞ Q7 Q`/2`d- bm+?

i?i i?2 2[miBQM #Qp2 Bb biBb}2/X

1bb2MiBH mMB[m2M2bb BM i?2 i?2Q`2K #Qp2 K2Mb i?i 7Q` Mv Qi?2`

>2`;HQix p2+iQ` }2H/H(z, t)biBb7vBM; URXkRV- i?2 2[mHBivH(z, t) =V(z, t)

?QH/b 7Q` HHz ∈D M/ HKQbi HHt∈[0,+∞)X GƺrM2` +?BMb

_2+HH i?i i?2 i2`K ǳmMBpH2MiǴ Bb b?Q`i?M/ 7Q` ǳBMD2+iBp2 M/ ?QHQKQ`@

T?B+ǴX

.2}MBiBQM 9X 7KBHv {f_t}0≤t<∞ Q7 KTbf_t :D→CBb +HH2/ U#+F@

r`/V GƺrM2` +?BM Q7 Q`/2`d∈[1,+∞]B7 RX 2+? 7mM+iBQMf_t Bb mMBpH2Mi-

kX f_s(D)⊂f_t(D)7Q` 0≤s < t <+∞,

jX 7Q` Mv +QKT+i b2iK ⊂DM/ HHT >0i?2`2 2tBbib MQM@M2;iBp2 7mM+iBQMk_K,T ∈L^d([0, T],R)bm+? i?i

|f_s(z)−f_t(z)|≤ ^t

s k_K,T(ξ)dξ 7Q` HHz∈K M/ HH0≤s≤t≤T.

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R9 AMi`Q/m+iBQM h?2 /QKBM f_t(D)Bb +HH2/ i?2 2pQHmiBQM /QKBM i i?2 iBK2 tX 1p2`v GƺrM2` +?BM {f_t}t≥0 Q7 Q`/2` d ;2M2`i2b M 2pQHmiBQM 7KBHv {φ_s,t}Q7 i?2 bK2 Q`/2` d/2}M2/ #v

φ_s,t=f_t⁻¹◦f_s. URXkkV h?Bb +Q``2bTQM/2M+2 Bb- ?Qr2p2`- MQi QM2@iQ QM2 Ĝ i?2`2 Kv #2 KMv /Bz2`2Mi GƺrM2` +?BMb bbQ+Bi2/ iQ i?2 ;Bp2M 2pQHmiBQM 7KBHvX 6Q`@

imMi2Hv- i?2v `2 mMB[m2 mT iQ MQ`KHBxiBQM M/ +QKTQbBiBQM rBi?

mMBpH2Mi 7mM+iBQM- b i?2 7QHHQrBM; i?2Q`2K bii2bX

h?2Q`2K k U(*.J:Ry- h?2Q`2Kb RXe@RXd)VX 6Q` Mv 2pQHmiBQM 7KBHv {φ_s,t}0≤s≤t<+∞ Q7 Q`/2` d- i?2`2 2tBbib mMB[m2 GƺrM2` +?BM {f_t}t≥0 Q7 i?2 bK2 Q`/2`d, bm+? i?i

RX φ_s,t=f_t⁻¹◦f_s 7Q` Mv 0≤s≤tc kX f(0) = 0M/ f(0) = 1c

jX Ω := ∪t≥0f_t(D) Bb /BbF {z : |z|< R} Q7 TQbbB#Hv BM}MBi2 `/Bmb R∈(0,+∞].

Mv Qi?2` GƺrM2` +?BM biBb7vBM; *QM/BiBQM R Bb Q7 i?2 7Q`K {g_t}t≥0= {F◦f_t}t≥0, r?2`2F : Ω→C Bb mMBpH2MiX

h?2 `/Bmb R+M #2 +H+mHi2/ 2tTHB+BiHv b 1/β₀, r?2`2 β₀= lim

t→+∞

|φ_0,t(0)|

1− |φ_0,t(z)|².

Ai rb HbQ b?QrM (*.J:Ry- h?2Q`2K 9XR) i?i 2p2`v GƺrM2` +?BM {f_t}t≥0Q7 Q`/2`dbiBb}2b i?2 ;2M2`HBx2/ GƺrM2` S.1

∂

∂sf_s(z) =−V(z, s)f_s(z) (7Q` HKQbi HHs≥0), URXkjV r?2`2V(z, s)Bb i?2 >2`;HQix p2+iQ` }2H/ ;2M2`iBM; i?2 bbQ+Bi2/ 2pQHm@

iBQM 7KBHv {φ_s,t}0≤s≤t<+∞.

"2`FbQM@SQ`i `2T`2b2MiiBQM

.m2 iQ Hbi +QM/BiBQM BM .2}MBiBQM j- QM2 b?QmH/ #2 #H2 iQ `2T`2b2MiV(·, t) BM i?2 7Q`K URXjV 7Q` HKQbi 2p2`v }t2/t≥0X AM Q`/2` iQ /2b+`B#2 ?Qr i?2 7mM+iBQM p BM i?2 `2T`2b2MiiBQM /2T2M/b QM t- i?2 MQiBQM Q7 ;2M2`HBx2/

>2`;HQix 7mM+iBQMb rb BMi`Q/m+2/X

.2}MBiBQM 8X >2`;HQix 7mM+iBQM Q7 Q`/2` d ∈ [1,+∞) Bb 7mM+iBQM p:D×[0,+∞)→C,bm+? i?i

Ç i?2 7mM+iBQMt →p(z, t)#2HQM;b iQL^d_loc([0,+∞),C)7Q` HHz∈Dc Ç i?2 7mM+iBQM z → p(z, t) Bb ?QHQKQ`T?B+ BM D 7Q` 2p2`v }t2/ t ∈

[0,+∞)c

(23)

RXk GƺrM2` i?2Q`v R8

Ç Rep(z, t)≥07Q` HHz∈D M/ 7Q` HHt∈[0,+∞)X

h?Bb /2}MBiBQM H2ib mb 2ti2M/ i?2 "2`FbQM@SQ`i `2T`2b2MiiBQM 7Q`KmH 7`QK i?2 +b2 Q7 miQMQKQmb b2KB+QKTH2i2 }2H/b iQ ;2M2`H >2`;HQix p2+iQ`

}2H/bX

h?2Q`2K j U("*.JRk- h?2Q`2K RXk)VX :Bp2M >2`;HQix p2+iQ` }2H/

V(z, t) Q7 Q`/2` d≥ 1BM i?2 mMBi /BbF- i?2`2 2tBbib M 2bb2MiBHHv mMB[m2 UBX 2X- /2}M2/ mMB[m2Hv 7Q` HKQbi HH t7Q` r?B+? V(·, t)= 0V K2bm`#H2 7mM+iBQM τ : [0,+∞)→D M/ >2`;HQix 7mM+iBQMp(z, t) Q7 Q`/2`d, bm+?

i?i 7Q` HH z∈D M/ HKQbi HHt∈[0,+∞)

V(z, t) = (z−τ(t)) (τ(t)z−1)p(z, t). URXk9V

*QMp2`b2Hv- ;Bp2M K2bm`#H2 7mM+iBQM τ : [0,+∞)→D M/ >2`@

;HQix 7mM+iBQMp(z, t)Q7 Q`/2`d≥1,i?2 p2+iQ` }2H/ /2}M2/ #v i?2 7Q`KmH

#Qp2 Bb >2`;HQix p2+iQ` }2H/ Q7 Q`/2` dX

++Q`/BM; iQ h?2Q`2K R- iQ 2p2`v 2pQHmiBQM 7KBHv{φ_s,t} QM2 +M b@

bQ+Bi2 M 2bb2MiBHHv mMB[m2 >2`;HQix p2+iQ` }2H/ V(z, t)X h?2 TB` Q7 7mM+iBQMb(p, τ)`2T`2b2MiBM; i?2 p2+iQ` }2H/V(z, t)Bb +HH2/ i?2 "2`FbQM@

SQ`i /i Q7 i?2 2pQHmiBQM 7KBHv {φ_s,t}0≤s≤t<+∞X

PM2 Q7 i?2 /pMi;2b Q7 `2T`2b2MiiBQM URXk9V Bb i?i Bi T`QpB/2b ;2Q@

K2i`B+ BMimBiBQM #Qmi i?2 #2?pBQ` Q7 i?2 +Q``2bTQM/BM; 2pQHmiBQM 7KBHvX

"v MHQ;v rBi? i?2 iBK2@BM/2T2M/2Mi +b2- r2 +HH i?2 7mM+iBQMτ(t)i?2 ii`+iBM; TQBMi Q7 i?2 >2`;HQix p2+iQ` }2H/ V(z, t) M/ Q7 i?2 2pQHmiBQM 7KBHv{φ_s,t}0≤s≤t<+∞X

RXkXk 6Q`r`/ U/2+`2bBM;V GƺrM2` i?2Q`v

h?2 7Q`r`/ GƺrM2` i?2Q`v Bb i?2 T`272``2/ iQQH BM i?2 bim/v Q7 a+?`KK@

GƺrM2` 2pQHmiBQM Ub22 a2+iBQM RXkXdVX

.2}MBiBQM e U(*.J:R9- .2}MBiBQM RXN)VX G2i d ∈ [1,+∞]X 7K@

BHv {φ_s,t}0≤s≤t<+∞ ⊂Hol(D,D) Bb +HH2/ `2p2`b2 UQ` 7Q`r`/V 2pQHmiBQM 7KBHv Q7 Q`/2` dB7 Bi biBb}2b i?2 7QHHQrBM; +QM/BiBQMb

RX φ_s,s= id_D-

kX φ_s,t=φ_s,u◦φ_u,t 7Q` HH0≤s≤u≤t <∞-

jX 7Q` Mv z ∈ D M/ Mv T > 0 i?2`2 2tBbib MQM@M2;iBp2 7mM+iBQM k_z,T ∈L²([0, T],R)bm+? i?i

|φ_s,u(z)−φ_s,t(z)|≤ ^t

u k_z,T(ξ)dξ, 7Q` HHs, t, u∈[0, T] biBb7vBM; BM2[mHBivs≤u≤tX

.2}MBiBQM d U(*.J:R9- .2}MBiBQM RXe)VX G2i d ∈ [1,+∞]X 7KBHv {f_t}t≥0 ⊂ Hol(D,D) Bb +HH2/ 7Q`r`/ UQ` /2+`2bBM;V GƺrM2` +?BM Q7 Q`/2` dB7 Bi biBb}2b i?2 7QHHQrBM; +QM/BiBQMb,

(24)

Re AMi`Q/m+iBQM

RX 2+? 7mM+iBQMf_t:D→D Bb mMBpH2Mi-

kX f₀= id_D, M/f_s(D)⊃f_t(D)7Q`0≤s < t <+∞-

jX 7Q` Mv +QKT+i b2i K ⊂ D M/ 7Q` HH T > 0 i?2`2 2tBbib MQM@

M2;iBp2 7mM+iBQMk_K,T ∈L^d([0, T],R)bm+? i?i

|f_s(z)−f_t(z)|≤ ^t

s k_K,T(ξ)dξ 7Q` HHz ∈K M/ 7Q` HH0≤s≤t≤TX

aBKBH`Hv iQ i?2 #+Fr`/ +b2- i?2`2 Bb QM2@iQ@QM2 +Q``2bTQM/2M+2

#2ir22M 7Q`r`/ 2pQHmiBQM 7KBHB2b M/ 7Q`r`/ GƺrM2` +?BMX :Bp2M GƺrM2` +?BM {f_t}t≥0- r2 +M Q#iBM i?2 +Q``2bTQM/BM; 2pQHmiBQM 7KBHv mbBM; i?2 7QHHQrBM; MHQ; Q7 URXkkV

φ_s,t(z) =f_s⁻¹◦f_t, 0≤s≤t <∞. URXk8V M/ +QMp2`b2Hv- ;Bp2M M 2pQHmiBQM 7KBHv{φ_s,t}0≤s≤t<+∞- r2 +M `2+Qp2`

i?2 GƺrM2` +?BM mbBM; i?2 bBKTH2 7Q`KmH f_t(z) :=φ_0,tX

GƺrM2` +?BMb `2 `2Hi2/ iQ >2`;HQix p2+iQ` }2H/b #v K2Mb Q7 i?2 7Q`r`/ GƺrM2` P.1X h?2 T`2+Bb2 bii2K2Mi Bb ;Bp2M BM i?2 7QHHQrBM;

i?2Q`2KX

h?2Q`2K 9 U(*.J:R9- h?2Q`2K RXRR)VX G2iV #2 >2`;HQix p2+iQ` }2H/

Q7 Q`/2` d∈[1,+∞]X h?2M-

RX 6Q` 2p2`vz∈D,i?2`2 2tBbib mMB[m2 KtBKH bQHmiBQMg_t(z)∈DiQ i?2 7QHHQrBM; BMBiBH pHm2 T`Q#H2K

⎧⎨

⎩

∂t∂g_t(z) =−V(g_t(z), t),

g₀(z) =z. URXkeV

kX 6Q` 2p2`v t≥ 0, i?2 b2i D_t Q7 HH z ∈ D, 7Q` r?B+? g_t(z) Bb /2}M2/

i i?2 TQBMit, Bb bBKTHv +QMM2+i2/ /QKBM- M/ i?2 7mM+iBQM g_t(z) /2}M2/ 7Q` HHz∈D_t KTbD_t +QM7Q`KHHv QMiQDX

jX h?2 7mM+iBQMb f_t :=g⁻¹_t 7Q`K /2+`2bBM; GƺrM2` +?BM Q7 Q`/2` d- r?B+? Bb i?2 mMB[m2 bQHmiBQM iQ i?2 7QHHQrBM; BMBiBH pHm2 T`Q#H2K 7Q`

S.1 _⎧

⎨

⎩

∂

∂tf_t(z) =f_t(z)V(z, t), f₀= id_D.

q2 +HH i?2 b2iD_t=g⁻¹_t (D) =f_t(D)i?2 2pQHmiBQM /QKBM i i?2 iBK2 t- M/ i?2 +QKTH2K2Mi K_t =D\D_t i?2 GƺrM2` ?mHH i i?2 iBK2 tX h?2 7KBHv{D_t}t≥0Bb /2+`2bBM;- M/ i?2 7KBHv{K_t}t≥0Bb BM+`2bBM;- BX2X- 7Q`

0≤s≤t <∞-D_s⊇D_t M/K_s⊆K_tX

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RXk GƺrM2` i?2Q`v Rd

RXkXj _/BH GƺrM2` i?2Q`v

h?2 `/BH- +?Q`/H M/ /BTQH` GƺrM2` 2[miBQMb- b r2HH b i?2B` T`Q#@

#BHBbiB+ +QmMi2`T`ib- +M #2 Q#iBM2/ 7`QK i?2 ;2M2`H GƺrM2` i?2Q`v

#v +?QQbBM; T`iB+mH` >2`;HQix p2+iQ` }2H/X q2 +HH i?2b2 i?`22 +b2b +HbbB+H BM +QMi`bi iQ Qi?2` 2pQHmiBQM 7KBHB2b M/ GƺrM2` +?BMb i?i +M #2 /2b+`B#2/ #v i?2 ;2M2`H GƺrM2` i?2Q`vX

"v i?2 `/BH GƺrM2` i?2Q`v r2 mM/2`biM/ i?2 +b2 r?2M τ(t) ≡ τ₀ BM i?2 `2T`2b2MiiBQM URXk9V 7Q` bQK2 τ₀ ∈ D- BX2X- r?2M i?2 ii`+iBM;

TQBMi Q7 i?2 >2`;HQix p2+iQ` }2H/ Bb }t2/ BMi2`BQ` TQBMi Q7 i?2 +MQMB+H /QKBMX 6Q` bBKTHB+Biv QM2 mbmHHv iF2b τ₀ = 0X Ai Bb HbQ +QKKQM iQ BKTQb2 i?2 MQ`KHBxiBQM +QM/BiBQM p(0, t) ≡ 1QM i?2 >2`;HQix 7mM+iBQM p(z, t)X

h?mb- i?2 `/BH GƺrM2` i?2Q`v Bb i?2 bim/v Q7 2pQHmiBQM 7KBHB2b M/

GƺrM2` +?BMb +Q``2bTQM/BM; iQ >2`;HQix p2+iQ` }2H/b Q7 i?2 7Q`K V(z, t) =−z p(z, t), p(z, t) Bb >2`;HQix 7mM+iBQM-p(0, t)≡1. URXkdV

h?2 `/BH i?2Q`v rb /2p2HQT2/ BM (Gƺrkj)- (:QHjN)- (Em79j)- (SQKe8) M/ (SQKee)X .m`BM; i?2 kyi? +2Mim`v QMHv i?2 #+Fr`/ i?2Q`v rb bim/B2/- ?Qr2p2`- 7i2` i?2 BMi`Q/m+iBQM Q7 i?2 a+?`KK@GƺrM2` 2pQHmiBQM BM (a+?yy)- i?2 7Q+mb b?B7i2/ iQr`/b 7Q`r`/ GƺrM2` 2[miBQMbX

h?2 MQ`KHBxiBQM +QM/BiBQM p(0, t)≡1 2Mbm`2b i?i #+Fr`/ `/BH 2pQHmiBQM 7KBHB2b ?p2 i?2 hvHQ` 2tTMbBQM

φ_s,t(z) =e^s−tz+. . . , z ∈D,

M/ MQ`KHBx2/ #+Fr`/ `/BH GƺrM2` +?BMb ?p2 i?2 2tTMbBQM f_t(z) =e^tz+. . . , z ∈D.

AM i?2 7Q`r`/ +b2 i?2 }`bi +Q2{+B2Mi BM i?2 2tTMbBQM Q7 MQ`KHBx2/

GƺrM2` +?BMb Bb /Bz2`2Mi,

f_t(z) =e^−tz+. . . , z∈D.

h?2`2 Bb M BKTQ`iMi `2HiBQMb?BT #2ir22M #+Fr`/ 2pQHmiBQM 7KBHB2b {φ_s,t}0≤s≤t<+∞ M/ #+Fr`/ GƺrM2` +?BMb{f_t}t≥0- MK2Hv

t→∞lime^tφ_0,t(z) =f₀(z) HQ+HHv mMB7Q`KHvX _/BH GƺrM2` i?2Q`v M/ bHBi ;2QK2i`v

AM i?2 Q`B;BMH TT2` (Gƺrkj)- GƺrM2` rb T`iB+mH`Hv BMi2`2bi2/ BM bT2+BH +b2 Q7 URXkdV- MK2Hv

V(z, t) =−ze^iu(t)+z

e^iu(t)−z, u(t)Bb +QMiBMmQmb M/ `2H@pHm2/. URXk3V hQ /BbiBM;mBb? #2ir22M URXkdV M/ URXk3V- i?2 KQ`2 ;2M2`H +b2 URXkdV Bb bQK2iBK2b `272``2/ iQ b i?2 `/BH GƺrM2`@Em7`2p i?2Q`v- #2+mb2 Bi rb 2ti2MbBp2Hv bim/B2/ BM Em7`2pǶb rQ`F (Em79j)X

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R3 AMi`Q/m+iBQM _2+HH i?i +`Qbb+mi Q7 /QKBM D Bb M QT2M CQ`/M `+ C BM D bm+? i?iC¯\C +QMbBbib Q7 QM2 Q` irQ TQBMib QM∂DX

SQKK2`2MF2 (SQKee) ;p2 i?2 7QHHQrBM; ;2QK2i`B+ +?`+i2`BxiBQM Q7

#+Fr`/ GƺrM2` +?BMb biBb7vBM; URXkjV rBi? >2`;HQix p2+iQ` }2H/ Q7 i?2 7Q`K URXk3VX

MQ`KHBx2/ #+Fr`/ GƺrM2` +?BM{f_t}t≥0 biBb}2b

∂

∂tf_t(z) =ze^iu(t)+z

e^iu(t)−zf_t(z) URXkNV

7Q` bQK2 `2H@pHm2/ +QMiBMmQmbu(t)B7 M/ QMHv B7f_t(0) =e^tM/ 7Q` 2p2`v >0i?2`2 2tBbib δ >0bm+? i?i r?2M2p2`s, t≥0M/0≤t−s≤δ- bQK2 +`Qbb+mi Q7f_t(D)rBi? bT?2`B+H /BK2i2` H2bb i?Mb2T`i2b07`QK f_t(D)\f_s(D)X q2 +M `272` iQ i?Bb +QM/BiBQM b i?2 SQKK2`2MF2 +QM/BiBQMX H`;2 M/ BKTQ`iMi +Hbb Q7 #+Fr`/ `/BH GƺrM2` +?BMb biBb7vBM;

i?Bb +QM/BiBQM `2 i?2 bQ@+HH2/ +?BMb ;2M2`i2/ #v +m`p2X #+Fr`/

`/BH GƺrM2` +?BM{f_t}t≥0 rBi? f_t(0) =e^t Bb bB/ iQ #2 ;2M2`i2/ #v +m`p2Γ : [0,+∞)→C B7 7Q` 2p2`vt≥0i?2 BK;2f_t(D)Bb i?2 +QMM2+i2/

+QKTQM2Mi Q7C\Γ[t,+∞)+QMiBMBM;0X q2 HbQ bv i?i i?2 +m`p2ΓBb

;2M2`i2/ #v i?2 +?BM{f_t}t≥0X

PM2 +M +?2+F i?i BM i?Bb +b2 i?2 SQKK2`2MF2 +QM/BiBQM Bb 7mH}HH2/

M/- +QMb2[m2MiHv- HH MQ`KHBx2/ `/BH +?BMb ;2M2`i2/ #v +m`p2b bi@

Bb7v URXkNV 7Q` bQK2 +QMiBMmQmb /`BpBM; 7mM+iBQMu(t)X h?2`2 `2- ?Qr2p2`- +QmMi2`2tKTH2b- BX2X- MQ`KHBx2/ `/BH GƺrM2` +?BMb i?i `2 MQi ;2M@

2`i2/ #v +m`p2- #mi M2p2`i?2H2bb biBb7v SQKK2`2MF2Ƕb +`Bi2`BQM M/

URXkNV 7Q` bQK2 /`BpBM; 7mM+iBQM u(t)X a22- 7Q` BMbiM+2- i?2 +?BM b?QrM BM (SQKee- 6B;m`2 j) Q` i?2 2tKTH2 BM (J_y8)X

GƺrM2`Ƕb T`K2i`B+ K2i?Q/

h?2 +Hbb Q7 mMBpH2Mi 7mM+iBQMb BM i?2 mMBi /BbF MQ`KHBx2/ #v i?2 +QM/B@

iBQMb

f(0) = 0 M/ f(0) = 1

Bb /2MQi2/ #vSX h?2 +HbbS Bb +QKT+i rBi? `2bT2+i iQ i?2 HQ+H mMB7Q`K iQTQHQ;vX

.m2 iQ i?2 MQ`KHBxiBQM +QM/BiBQMb #Qp2- 2+? f ∈ S ?b hvHQ`

2tTMbBQM

f(z) =z+a₂z²+a₃z³+. . . , z∈D.

"B2#2`#+? ("B2Re) +QMD2+im`2/ i?i |a_n|≤ n 7Q` 7mM+iBQMb BM SX >2 T`Qp2 Bi 7Q`n = 2X >2 HbQ b?Qr2/ i?i i?2 2[mHBiv |a₂|= 2 Bb iiBM2/

B7 M/ QMHv B7f(z) =e^−iθk(e^iθz)- r?2`2θ∈[0,2π)M/k(z)Bb i?2 EQ2#2 7mM+iBQM

k(z) = z

(1−z)² =z+ 2z²+ 3z³+. . . , z ∈D.

GƺrM2` BMi`Q/m+2/ ?Bb MK2bF2 2[miBQM BM M ii2KTi iQ T`Qp2 i?2

"B2#2`#+? +QMD2+im`2- #mi QMHv bm++22/2/ 7Q`n= 3X 7i2` i?2 +QMD2+im`2