Ç ?vT2`#QHB+- B72 (−Im ˜p(0)− |p(0)˜ |)< k <2 (−Im ˜p(0) +|p(0)˜ | )-Ç T`#QHB+- B72 (−Im ˜p(0)− |˜p(0)|)< k <2 (−Im ˜p(0) +|p(0)|)-˜ Ç 2HHBTiB+- B7k <2 (−Im ˜p(0)− |p(0)|)- Q`˜ k >2 (−Im ˜p(0) +|p(0)|)X˜
AM i?2 +b2 Q7 2HHBTiB+ miQKQ`T?BbKb- i?2 i`D2+iQ`B2bφt(z)`2 +m`p2b bBKBH` iQ i?2 #2miB7mH `Qb2 U`?Q/QM2V +m`p2bX h?2 i`D2+iQ`B2b `2 +HQb2/
B7 M/ QMHv B7 √−4A2+4k B k+k2 Bb `iBQMH MmK#2`X
h?2 b2KB;`QmT{ψt}t≥0 #2+QK2b 2HHBTiB+ 7Q` bm{+B2MiHv ?B;? `/BH p2@
HQ+BiB2b Q7 i?2 ii`+iBM; TQBMi HbQ BM i?2 ;2M2`H +b2 Q7 ?QHQKQ`T?B+
2M/QKQ`T?BbKb- MQ Kii2` r?i >2`;HQix 7mM+iBQMp(z)˜ r2 +?QQb2X AM i?2 biQ+?biB+ +b2- >2`;HQix 7mM+iBQMb Q7 i?2 7Q`Kp(z, t) = ˜p(z/τ(t)) HbQ im`M Qmi iQ THv M BKTQ`iMi `QH2X h?2 +?M;2 Q7 p`B#H2b wt = φt/ei k Bt MQr H2/b iQ iBK2@?QKQ;2M2Qmb AiƬ /BzmbBQM
⎧⎨
⎩
dwt=−k22wt+ (wt−1)2p(w˜ t)dt−i k wtdBt,
w0(z) =z. UkX9V
h?Bb ?b irQ BKTQ`iMi BKTHB+iBQMbX 6B`bi- i?2 J`FQp T`QT2`iv Q7 i?2 biQ+?biB+ ~Qr{wt}t≥0 H2/b iQ p2`bBQM Q7 i?2 /QKBM J`FQp T`QT2`iv 7Q` i?2 /QKBM 7KBHv{φt(D)}t≥0X a2+QM/- QM2 +M mb2 i?2 `B+? K+?BM2`v Q7 biQ+?biB+ +H+mHmb- bm+? b EQHKQ;Q`Qp 2[miBQMb- iQ bim/v i?2 T`Q+2bbX q2 ?p2 i?mb 7QmM/ M2r ivT2 Q7 i?2 ;2M2`H GƺrM2` 2pQHmiBQM UBM //BiBQM iQ i?2 H`2/v FMQrM `/BH- +?Q`/H M/ /BTQH` 2[miBQMbV- i?i H2/b iQ iBK2@?QKQ;2M2Qmb /BzmbBQM 2[miBQM BM i?2 biQ+?biB+ +b2X
*M r2 }M/ KQ`2 bm+? ivT2b Q7 i?2 ;2M2`H GƺrM2` 2pQHmiBQM\ *M r2 +?`+i2`Bx2 i?2K HH\ *M i?2v ?p2 bHBi ;2QK2i`v bBKBH` iQ i?2 +HbbB+H SLE +b2b\ A7 v2b- rBHH i?2 `M/QK bHBib #2 bBKBH` iQ i?2 +HbbB+H SLE +m`p2b\
h?2b2 [m2biBQMb `2 Mbr2`2/ BM i?2 M2ti TT2`X
kXj :2M2`H bHBi GƺrM2` +?BMb
>QHQKQ`T?B+ biQ+?biB+ b2KB~Qrb q2 bi`i #v +QMbB/2`BM; i?2 biQ+?biB+ ~Qr
⎧⎨
⎩
dwt(z) =b(wt(z))dt+σ(wt(z))◦dBt,
w0(z) =z, z∈D, UkX8V
BM bBKTHv +QMM2+i2/ ?vT2`#QHB+ /QKBM D- M/ bF r?i +QM/BiBQMb b?QmH/ #2 BKTQb2/ QM i?2 7mM+iBQMb b M/σ- bQ i?iwt ∈ Hol(D, D)7Q`
HH t≥0X
Ai Bb FMQrM 7`QK i?2 rQ`F Q7 EmMBi [mQi2/ BM a2+iBQM RXRXk i?i B7 i?2 +Q2{+B2MibbM/σ`2 +QKTH2i2 ?QHQKQ`T?B+ p2+iQ` }2H/b- i?2Mwt∈ Aut(D)7Q` HHt≥0X
jk JBM _2bmHib q2 mb2 i?2 ;2M2`H GƺrM2` i?2Q`v iQ bim/v i?2 +b2 r?2Mb(z)Bb b2KB@
+QKTH2i2X 6B`bi r2 +QMbB/2` i?2 mtBHB`v ~Qr
⎧⎨
⎩
dHt(z) =σ(Ht(z))◦dBt,
H0(z) =z, z∈D, UkXeV
Q7 miQKQ`T?BbKb Q7 DX AM 7+i- {Ht}t≥0 Bb `2Hi2/ iQ i?2 /2i2`KBMBbiB+
~Qr ⎧
⎨
⎩
∂t∂ht(z) =σ(ht(z)),
h0(z) =z, z∈D UkXdV
#v i?2 7Q`KmHHt=hBtX
"v URXR3V M/ URXRNV- i?2 +QKTQbBi2 ~Qr{gt}t≥0- r?2`2gt:=Ht−1◦wt= h−1B
t ◦wt- biBb}2b
dgt(z) =h−1B
t∗b (gt(z))dt, g0(z) =z, z∈D. UkX3V Ai Bb 2bv iQ +?2+F i?i h−1Bt∗b Bb >2`;HQix p2+iQ` }2H/ Q7 Q`/2` ∞-BKTHvBM; i?i {gt}t≥0 ⊂ Hol(D, D)- r?B+? Bb 2[mBpH2Mi iQ {wt}t≥0 ⊂ Hol(D, D)X
h?mb- B7 bBb b2KB+QKTH2i2 ?QHQKQ`T?B+ p2+iQ` }2H/ M/ σ Bb +QK@
TH2i2 ?QHQKQ`T?B+ p2+iQ` }2H/- i?2M i?2 ~Qr UkX8V +QMbBbib Q7 ?QHQKQ`T?B+
2M/QKQ`T?BbKb Q7DX
AM i?2 T`2+2/BM; TT2` r2 mb2/ KQbiHv i?2 #+Fr`/ UBM+`2bBM;V 7`K2@
rQ`FX AM i?Bb TT2`- i?MFb iQ i?2 TT2`M+2 Q7 i?2 rQ`F (*.J:R9)-r2 (*.J:R9)-r2`2 #H2 iQ /B`2+i Qm` ii2MiBQM iQr`/b i?2 7Q`r`/ U/2+`2bBM;V 7`K2rQ`F- i?2 KBM iQQH Q7 i?2SLE i?2Q`vX
lbBM; i?2 7Q`r`/ GƺrM2` i?2Q`v- r2 bim/v i?2 bQHmiBQMb iQ
⎧⎨
⎩
dwt(z) =−b(wt(z))dt+σ(wt(z))◦dBtk,
w0(z) =z, z∈D, UkXNV
r?2`2 b Bb b2KB+QKTH2i2 ?QHQKQ`T?B+ p2+iQ` }2H/ M/ σ Bb +QKTH2i2
?QHQKQ`T?B+ }2H/X q2 /2MQi2
Dt={z ∈D:wt(z)Bb /2}M2/ i H2bi mT iQ iBK2t},
M/ mbBM; i?2 7Q`r`/ GƺrM2` i?2Q`v T`Qp2 i?iwt :Dt→DBb +QM7Q`@
KH BbQKQ`T?BbK 7Q` HHt≥0rBi? T`Q##BHBiv RX aBKBH`Hv iQ i?2 +b2 Q7 UkX8V- i?2 +`m+BH bi2T ?2`2 Bb iQ MQi2 i?i i?2 +QKTQbBiBQM gt=Ht−1◦wt biBb}2b
dgt(z) =−h−1Bt∗b(gt(z))dt, g0(z) =z, z ∈D, UkXRyV M/ ?2M+2 {gt}t≥0 Bb 7Q`r`/ GƺrM2` +?BM Q7 Q`/2`∞X
kXj :2M2`H bHBi GƺrM2` +?BMb jj
aHBi p2+iQ` }2H/b M/ oB`bQ`Q ;2M2`iQ`b
h?2 M2ti [m2biBQM Bb r?i +QM/BiBQMb b?QmH/ #2 BKTQb2/ QM i?2 p2+iQ`
}2H/ biQ 2Mbm`2 i?i i?2 ;2M2`i2/ GƺrM2` +?BMb ?p2 bHBi ;2QK2i`v- b BM i?2 +HbbB+H SLEsX 6`QK i?2 ;2QK2i`B+H TQBMi Q7 pB2r- i?2 Mim`H M2+2bb`v +QM/BiBQM Bb iQ `2[mB`2 i?i i?2 }2H/bBb iM;2Mi iQ i?2 #QmM/`v Q7 i?2 +MQMB+H /QKBM i HH #QmM/`v TQBMib #mi QM2X
T`2+Bb2 7Q`KmHiBQM Q7 i?Bb +QM/BiBQM BMDBb i?i
limr→1Reb(reiθ)re−iθ = 0 UkXRRV 7Q` HH eiθ ∈ ∂D 2t+2Ti- T2`?Tb- 7Q` QM2 TQBMi eiθ0X qBi?Qmi HQbb Q7
;2M2`HBiv r2 Kv bbmK2 eiθ0 = 1X q2 +HH bm+? }2H/b bHBi p2+iQ` }2H/b BM DX AM i?2 miQMQKQmb /2i2`KBMBbiB+ +b2 bm+? p2+iQ` }2H/b BM/22/ ;2M2`i2 b2KB~Qrb ?pBM; bHBi ;2QK2i`vX
q2 T`Qp2 i?ibBb bHBi p2+iQ` }2H/ BMDB7 M/ QMHv B7 b(z) =α−z
i β+γ1 +z 1−z
−α z2, z∈D, UkXRkV 7Q` bQK2α∈C, β∈RM/γ≥0X
h?2 p2+iQ` }2H/b
Hn(z) :=−zn+1, n∈Z, z∈H,
`2 +HH2/ i?2 oB`bQ`Q ;2M2`iQ`b 7Q` i?2 mTT2` ?H7@THM2X lbBM; i?2 Tmb?7Q`r`/ QT2`iBQM- i?2 ;2M2`iQ`b +M #2 /2}M2/ 7Q` Mv bBKTHv +QM@
M2+i2/ ?vT2`#QHB+ /QKBMD#v i?2 7Q`KmHDn :=φ∗Hn- r?2`2φ:H→D Bb +QM7Q`KH BbQKQ`T?BbKX h?Bb /2}MBiBQM Q7Dn /2T2M/b QM i?2 +?QB+2 Q7 φX
6Q` i?2 mMBi /BbF- r2 +?QQb2φ(z) =−z−2iz+2i- M/ ;2i Dn(z) =−2n−1(−i)n(z−1)n+1(z+ 1)−n+1.
6Q` i?2 BM}MBi2 bi`BTS={z: 0<Imz < π}rBi?ψ(z) = Log2+z2−z r2 ;2i Sn(z) =ψ∗Hn(z) =−2n sinh(z) tanhn
z 2
.
M BMi2`2biBM; 7+i Bb i?i i?2 bT+2 Q7 +QKTH2i2 ?QHQKQ`T?B+ p2+iQ`
}2H/b BM DBb ;Bp2M #v spanR{D−1, D0, D1}X q?i Bb 2p2M KQ`2 BMi2`2biBM;
7Q` mb- Bb i?i r2 +M mb2 Dn, n=−2, . . . ,1iQ `2T`2b2Mi bHBi p2+iQ` }2H/b BM DX p2+iQ` }2H/ b Bb bHBi p2+iQ` }2H/ BM D B7 M/ QMHv B7 Bi +M #2
`2T`2b2Mi2/ BM i?2 7Q`K
b(z) =b−2D−2(z) +b−1D−1(z) +b0D0(z) +b1D1(z), UkXRjV rBi?b−2≥0M/b−1, b0, b1∈RX
q2 +M i?2M /2}M2 bHBi p2+iQ` }2H/b BM M `#Bi``v bBKTHv +QMM2+i2/
?vT2`#QHB+ /QKBMD #v `2TH+BM;Dn →Dn BM UkXRjVX
j9 JBM _2bmHib
.2}MBiBQM Q7 bHBi GƺrM2` +?BMb
G2i D#2 bBKTHv +QMM2+i2/ ?vT2`#QHB+ /QKBMDX G2iσ #2 +QKTH2i2 p2+iQ` }2H/ BMD- M/b#2 bHBi p2+iQ` }2H/ BMD- bQ i?i
b(z) =b−2D−2(z)+b−1D−1(z)+b0D0(z)+b1D1(z), b−2≥0, b−1, b0, b1∈R, M/
σ(z) =σ−1D−1(z) +σ0D0(z) +σ11, σ−1, σ0, σ1∈R, σ−1= 0.
G2i ut : [0,+∞) →R #2 +QMiBMmQmb 7mM+iBQMX h?2 bHBi GƺrM2` +?BM /`Bp2M #v b, σ M/ ut Bb #v /2}MBiBQM i?2 7Q`r`/ GƺrM2` +?BM {ft}t≥0
+Q``2bTQM/BM; iQ i?2 >2`;HQix p2+iQ` }2H/
V(t, z) =h−1ut∗b(z),
r?2`2{ht}t≥0Bb i?2 ~Qr Q7 miQKQ`T?BbKb Q7D;2M2`i2/ #vσX
qBi?Qmi HQbb Q7 ;2M2`HBiv r2 Kv BKTQb2 i?2 MQ`KHBxiBQM +QM/BiBQMb b−2= 2, σ−1= 1,
M/ r2 +HH GƺrM2` +?BMb biBb7vBM; i?2b2 +QM/BiBQMb MQ`KHBx2/ bHBi GƺrM2`
+?BMbX
Pm` ;2M2`H /2}MBiBQM Q7 MQ`KHBx2/ bHBi GƺrM2` +?BMb BM+Hm/2b i?2 +HbbB+H `/BH- +?Q`/H M/ /BTQH` GƺrM2` +?BMb b bT2+BH +b2b,
GƺrM2` +?BM ivT2 b σ
*?Q`/H 2−2 −1
_/BH 2−2+120 −1+141 .BTQH` 2−2−120 −1−141 A7 ut = √
κBt- κ ≥ 0- i?2M- b r2 /Bb+mbb2/ 2`HB2`- i?2 +?M;2 Q7 p`B#H2bwt=h√κBt ◦gt H2/b iQ i?2 /BzmbBQM
dwt(z) =−b(wt(z))dt+√
κ σ(wt(z))◦dBt, w0(z) =z, z∈D.
q2 +HH {wt}t≥0i?2 bHBi ?QHQKQ`T?B+ biQ+?biB+ ~Qr /`Bp2M #vbM/ σX S`QT2`iB2b Q7 i?2 ?mHHb
i H2bi 7Q` bKHH iBK2b- Bi Bb TQbbB#H2 iQ /2b+`B#2 i?2 ?mHHb ;2M2`i2/ #v M
`#Bi``v bHBi GƺrM2` +?BM BM i2`Kb Q7 `/BH GƺrM2` +?BMX h?2 `/BH /`BpBM; 7mM+iBQM Bb `2Hi2/ iQ i?2 /`BpBM; 7mM+iBQM Q7 i?2 Q`B;BMH +?BM #v K2Mb Q7 /Bz2`2MiBH 2[miBQMX
AM i?2 /2i2`KBMBbiB+ +b2 i?Bb `2HiBQMb?BT HHQrb mb iQ i`MbHi2 i?2 r2HH@FMQrM `2bmHib #Qmi i?2 T`QT2`iB2b Q7 i?2 /`BpBM; 7mM+iBQM M/ i?2 +Q``2bTQM/BM; ?mHHb iQ i?2 +b2 Q7 ;2M2`H bHBi +?BMbX
AM i?2 biQ+?biB+ +b2 i?Bb HHQrb mb iQ T`Qp2 i?i i?2 Hr Q7 i?2 ?mHHb Q7 ;2M2`H bHBi ?QHQKQ`T?B+ biQ+?biB+ ~Qr Bb HQ+HHv #bQHmi2Hv +QMiBMmQmb rBi? `2bT2+i iQ i?2 `/BHSLE ?mHHb rBi? i?2 bK2 pHm2 Q7κX
AM T`iB+mH`- r2 T`Qp2 i?i i?2 ?mHHb Q7 ;2M2`H bHBi ?QHQKQ`T?B+
biQ+?biB+ ~Qr `2 HKQbi bm`2Hv ;2M2`i2/ #v +m`p2X MHQ;QmbHv iQ i?2 +HbbB+H +b2b- i?2 +m`p2 Bb bBKTH2 7Q`κ∈[0,4]- ?b b2H7@BMi2`b2+iBQMb 7Q` κ∈(4,8)M/ Bb bT+2@}HHBM; 7Q` κ∈[8,∞)X
kX9 aHBi ?QHQKQ`T?B+ biQ+?biB+ ~Qrb M/ :mbbBM 7`22 }2H/ j8
LQM@+HbbB+H 2tKTH2, ABP SLE
A7- +QK#BMBM; i?2 +?Q`/H M/ i?2 `/BH +b2b- r2 +?QQb2 b= 2−2 M/
σ=−1+141- i?2 2[miBQM 7Q` i?2 7Q`r`/ GƺrM2` +?BM BM i?2 mMBi /BbF Bb ;Bp2M #v
∂
∂tft(z) = 1 4eiu(t)
(eiu(t)+ft(z))3 eiu(t)−ft(z) .
h?Bb Bb M 2tKTH2 Q7 GƺrM2` +?BM rBi? M ii`+iBM; #QmM/`v TQBMi Ub22 ST2` "VX AM i?2 "2`FbQM@SQ`i `2T`2b2MiiBQM 7Q` i?2 +Q``2bTQM/BM;
>2`;HQix p2+iQ` }2H/- τ(t) =−eiu(t)-p(z) = 141+z1−zX h?2 T`Q+2bb Q#iBM2/ #v TmiiBM;u(t) =√
κ Bt- κ≥ 0- +M #2 +HH2/
T`QpBbBQMHHvABP SLEUABP biM/BM; 7Q` dzii`+iBM; #QmM/`v TQBMiǴVX AM +QMi`bi iQ +?Q`/H- `/BH M/ /BTQH` GƺrM2` +?BMb- i?2 7KBHv Q7 +QM7Q`KH KTb{ft}t≥0/Q2b MQi ?p2 +QKKQM }t2/ TQBMib M2Bi?2` BMbB/2 i?2 +MQMB+H /QKBM- MQ` QM Bib #QmM/`vX h?Bb 7+ib +QKTHB+i2b i?2 bim/v Q7 i?2 T`Q+2bb [mBi2 #BiX aBKmHiBQMb b?Qr i?iSLE ABP +m`p2b i2`KBMi2 i `M/QK TQBMib BM i?2 mMBi /BbFX LQi Km+? 2Hb2- ?Qr2p2`- Bb FMQrM #Qmi i?2B` HQM;@i2`K #2?pBQ` bQ 7`X
kX9 aHBi ?QHQKQ`T?B+ biQ+?biB+ ~Qrb M/
:mbbBM 7`22 }2H/
G2i {wt}t≥0 #2 MQ`KHBx2/ bHBi ?QHQKQ`T?B+ biQ+?biB+ ~Qr /`Bp2M #v b M/ σ- rBi? κ = 4X 6Q` i2+?MB+H `2bQMb- r2 T`272` iQ rQ`F BM i?2
?H7@THM2X h?2 ~Qr{wt}t≥0i?2M biBb}2b
dwt(z) =−b(wt(z))dt+ 2σ(wt(z))◦dBt, w0(z) =z, z ∈H. G2iΦH/2MQi2 :mbbBM 7`22 }2H/ BMHBM/2T2M/2Mi Q7BtX q2 //`2bb i?2 7QHHQrBM; T`Q#H2KX 6Q` r?ibM/σBb Bi TQbbB#H2 iQ }M/ ?`KQMB+
7mM+iBQM h BM H- bm+? i?i 7Q` i?2 KQ/B}2/ :mbbBM 7`22 }2H/ ΦˆH :=
ΦH+h(z),
i?2 }2H/ΦˆH◦wT ?b i?2 bK2 Hr bΦˆH7Q` MvT >0\ UkXR9V h?Bb Bb Mim`H [m2biBQM BM pB2r Q7 i?2 rQ`Fb (.m#yN- a?2Ry- aaRj) r?2`2 bBKBH` `2HiBQMb?BTb Q7GF F rBi? +HbbB+HSLEsr2`2 /Bb+Qp2`2/X AM T`iB+mH`- i?Bb Bb `2bQM#H2 }`bi bi2T BM +QMbi`m+iBM; +QM7Q`KH }2H/ i?2Q`v 7Q` ;2M2`H bHBi ?QHQKQ`T?B+ biQ+?biB+ ~QrbX
M BKTQ`iMi `QH2 BM T`QpBM; UkXR9V Bb THv2/ #v i?2 2[miBQM dGH(wt(z1), wt(z2)) =−1
2h(wt(z1)), h(wt(z2)), UkXR8V r?2`2 GH /2MQi2b i?2 :`22MǶb 7mM+iBQM Q7 H, M/ z1, z2 ∈H- z1=z2X A7 UkXR8V ?QH/b- i?2M r2 +M /Ti i?2 i2+?MB[m2b mb2/ BM (aaRj) iQ T`Qp2 i?i UkXR9V HbQ ?QH/bX h?2 QMHv i?2Q`2iB+H /B{+mHiv r2 7+2 Bb i?i BM (aaRj) i?2 mMB7Q`K #QmM/2/M2bb bbmKTiBQM Q7 h rb +`m+BH- r?B+?
Bb-je JBM _2bmHib mM7Q`imMi2Hv- iQQ `2bi`B+iBp2 7Q` mbX q2 `2Ht i?Bb bbmKTiBQM M/ `2[mB`2 QMHv i?ih(wt(z))Bb #QmM/2/ #v /2i2`KBMBbiB+ +QMiBMmQmb 7mM+iBQM MQi /2T2M/BM; QMtX
7i2` B/2MiB7vBM; ~Qrb i?i `2 +HQb2Hv `2Hi2/ iQ 2+? Qi?2`- r2 ``Bp2 i bBt QM2@T`K2i2` 7KBHB2bX "2HQr r2 HBbi i?Qb2 7KBHB2b iQ;2i?2` rBi?
i?2 +Q``2bTQM/BM; ?`KQMB+ 7mM+iBQMbhX 6Q` bBKTHB+Biv- r2 Hrvb bbmK2 i?i i?2 +MQMB+H /QKBM Bb i?2 mTT2` ?H7@THM2X
RX *?Q`/H SLE4 /`Bp2M #v 2Bt−αt, α ∈ R UBX2X- "`QrMBM KQiBQM
UBX2X-`/BH- /BTQH` M/ +?Q`/H SLE4 rBi?Qmi /`B7iV r2`2 FMQrM T`2pBQmbHvX AM i?2 +b2 e- i?2 7mM+iBQM h Bb KmHiBpHm2/- r?B+? H2/b iQ KmHiBpHm2/
KQ/B}+iBQMb Q7GF F //`2bb2/ BM (EJRk)X
kX8 6m`i?2` rQ`F jd
kX8 6m`i?2` rQ`F
"2HQr r2 7Q`KmHi2 b2p2`H [m2biBQMb i?i `Qb2 /m`BM; i?2 rQ`F QM i?Bb i?2bBbX h?2v `2 [mBi2 BMi2`2biBM; M/ Kv T`QpB/2 #bBb 7Q` 7m`i?2`
`2b2`+?X
Ç >Qr +M r2 TTHv +QMi`QH i?2Q`v i2+?MB[m2b iQ ;2M2`H bHBi GƺrM2`
+?BMb\ q?i `2 i?2 `2+?#H2 b2ib Q7 mMBpH2Mi 7mM+iBQMb +Q``2@
bTQM/BM; iQ p`BQmb +QK#BMiBQMb Q7b M/σ\ *M i?Bb ?p2 TTHB@
+iBQMb BM i?2 i?2Q`v Q7 mMBpH2Mi 7mM+iBQMb\
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