Estimates for conformal maps

For a proper simply connected domain D andw ∈ D, let CR(w;D)denote the conformal radius of Dwith respect tow, i.e., CR(w;D) ≡ f^′(0)for f the unique conformal mapD → Dwith f(0) = wand f^′(0) > 0. Let rad(w;D) ≡ inf{r : B_r(w) ⊇ D}denote the out-radius of Dwith respect tow. By the Schwarz lemma and the Koebe one-quarter theorem,

dist(w, ∂D)≤CR(w;D)≤ [4 dist(w, ∂D)] ∧rad(w;D). (2.15) Further (see e.g., [30, Theorem 1.3])

|ζ|

(1+ |ζ|)² ≤ |f(ζ )−w|

CR(w;D) ≤ |ζ|

(1− |ζ|)² (2.16) As a consequence,

|ζ|

4 ≤ |f(ζ )−w|

CR(w;D) ≤4|ζ| (2.17)

where the right-hand inequality above holds for|ζ| ≤1/2.

Finally, we state the Beurling estimate [16, Theorem 3.76] which we will frequently use in conjunction with the conformal invariance of Brownian motion.

Theorem 2.9 (Beurling estimate) Suppose that B is a Brownian motion inCand τD=inf{t≥0:B(t)∈∂D}. There exists a constant c<∞such that ifγ: [0,1] → Cis a curve withγ (0)=0and|γ (1)| =1, z∈D, andP^zis the law of B when started at z, then

P^z[B([0, τD])∩γ ([0,1])=∅] ≤c|z|^1/2. 3 The intersection of SLEκ(ρ)with the boundary 3.1 The upper bound

The main result of this section is the following theorem, which in turn implies Theo-rem1.8.

Theorem 3.1 Fixκ >0,ρ_1,R >−2, andρ_2,R∈Rsuch thatρ_1,R+ρ_2,R >^κ₂−4. Fix x_R ∈ [0⁺,1)and letηbe anSLEκ(ρ_1,R, ρ_2,R)process with force points(x_R,1). Let

α= 1

κ(ρ1,R+2)

ρ1,R+ρ2,R+4−κ 2

. (3.1)

For eachǫ > 0, letτ_ǫ = inf{t ≥ 0 : η(t) ∈ ∂B(1, ǫ)}and, for each r > 0, let σ_r =inf{t ≥0:η(t)∈∂(rD)}. For eachδ ∈ [0,1)and r≥2fixed, let

E_ǫ^δ,r = {τǫ< σr, Im(η(τǫ))≥δǫ}. (3.2) We have that

P[E_ǫ^δ,r] =ǫ^α+o(1) as ǫ→0. (3.3) Theo(1)in the exponent of (3.3) tends to 0 asǫ → 0 and depends only onκ, δ,x_R, and the weightsρ_1,R,ρ_2,R. Theo(1), however, is uniform inr ≥ 2. Taking ρ_1,R> (−2)∨(^κ₂−4)andρ_2,R=0, we have that

α= 1

κ(ρ+2)

ρ+4−κ 2

. (3.4)

Thus Theorem 3.1 leads to the upper bound of Theorem 1.6. We begin with the following lemma which contains the same statement as Theorem3.1except is restricted to the case thatδ∈(0,1)and, in particular, is not applicable forδ=0.

Lemma 3.2 Assume that we have the same setup and notation as in Theorem3.1.

Then for eachδ∈(0,1)and r≥2fixed, we have that P[E_ǫ^δ,r] ≍ǫ^α

where the constants in≍depend only onκ,δ, xR, and the weightsρ1,R,ρ2,R. Proof Forη, the SLEκ(ρ_1,R, ρ_2,R)process with force points(x_R,1), let(g_t)be the associated Loewner evolution and letV_t^Rdenote the evolution ofx_R. From (2.6) we know that

M_t =

gt(1)−V_t^R g_t^′(1)

−α

gt(1)−Wt

g_t(1)−V_t^R

₋²_κ(ρ_1,R+ρ_2,R+4−κ/2)

is a local martingale and the law ofηreweighted byMis that of an SLEκ(ρ_1,R,ρ_2,R) process whereρ_2,R= −2ρ1,R−ρ_2,R−8+κ. We writeK =K_τǫandK = {z:z∈ K}. LetGbe the extension ofg_τǫtoC\(K∪K)which is obtained by Schwarz reflection.

By (2.15), we have

G^′(x)dist(x,K)≍dist(G(x),G(K∪K)). (3.5) Observe that G(K ∪K) = [O_τ^L

ǫ,O_τ^R

ǫ] where O_t^L (resp. O_t^R) is the image of the leftmost (resp. rightmost) point ofK_t∩Runderg_t. Note that (3.5) implies

ǫg_τ^′

ǫ(1)≍g_τǫ(1)−O_τ^R

ǫ.

It is clear thatg_t(1)−W_t ≥ g_t(1)−O_t^R ≥ g_t(1)−V_t^R. On the eventE_ǫ^δ,r, we run a Brownian motion started from the midpoint of the line segment[1, η(τǫ)]. Then

0 xR 1 η

B(1 ) ϕ

−=ϕ(∞) ϕ(xR) ϕ(η)

B(− 1) =ϕ(B(1 ))

0

Fig. 9 The image of an SLEκ(ρ1,R, ρ2,R)process inHfrom 0 to∞with force points(x_R,1)under ϕ(z) =ǫz/(1−z)has the same law as an SLEκ(ρ_L;ρ_R)process inHfrom 0 to∞with force points (−ǫ;ǫx_R/(1−x_R))whereρ_R=ρ_1,Randρ_L=κ−6−(ρ_1,R+ρ_2,R)

this Brownian motion has uniformly positive (thoughδ-dependent) probability to exit H\K through each of the left side ofK, the right side ofK, the interval[xR,1], and the interval(1,∞). Consequently, by the conformal invariance of Brownian motion,

g_τǫ(1)−W_τǫ ≍g_τǫ(1)−O_τ^R

ǫ ≍g_τǫ(1)−V_τ^R

ǫ on E_ǫ^δ,r.

These facts imply that Mτ_ǫ ≍ǫ^−α onE_ǫ^δ,r where the constants in≍depend only onκ,δ,x_R, and the weightsρ1,R,ρ2,R. Thus

P[E_ǫ^δ,r] ≍ǫ^αE[M_τǫ1

E^δ,rǫ ] =ǫ^αP^⋆[E^δ,r_ǫ ]

whereP^⋆is the law ofηweighted by the martingale M. As we remarked earlier,P^⋆ is the law of an SLEκ(ρ_1,R,ρ_2,R)with force points(x_R,1).

We now perform a coordinate change using the Möbius transformationϕ(z) = ǫz/(1−z). Then the law of the image of a path distributed according toP^⋆underϕis equal to that of an SLEκ(2+ρ_1,R+ρ_2,R;ρ_1,R)process inHfrom 0 to∞with force points(−ǫ;ǫx_R/(1−x_R))(see Fig.9). Note that 2+ρ1,R+ρ2,R ≥ ^κ₂ −2 by the hypotheses of the lemma. Letη^⋆be an SLEκ(2+ρ1,R+ρ2,R;ρ1,R)process inHfrom 0 to∞with force points(−ǫ;ǫxR/(1−xR)). In particular, by Lemma2.1,η^⋆almost surely does not hit(−∞,−ǫ). Under the coordinate change, the eventE_ǫ^δ,r becomes {σ_1,ǫ^⋆ < ξ_ǫ,r^⋆ , Im(η^⋆(σ_1,ǫ^⋆ )) ≥ δ}whereσ_1,ǫ^⋆ is the first time thatη^⋆ hits∂B(−ǫ,1), ξ_ǫ,r^⋆ is the first time thatη^⋆hits∂B(−ǫr²/(r²−1), ǫr/(r²−1)). By Lemma2.4, the probability of the event{σ_1,ǫ^⋆ < ξ_ǫ,r^⋆ , Im(η^⋆(σ_1,ǫ^⋆ ))≥δ}is bounded from below by a positive constant depending only onκ,δ,ρ_1,R, andρ_2,R. ThusP^⋆[E_ǫ^δ,r] ≍1 which impliesP[E_ǫ^δ,r] ≍ǫ^αand the constants in≍depend only onκ,δ,x_R, and the weights

ρ_1,R,ρ_2,R. ⊓⊔

Corollary 3.3 Fixκ >0,ρ_L >−2, ρ1,R >−2andρ_2,R ∈Rsuch thatρ_1,R+ρ_2,R>

κ

2 −4. Fix xL ≤ 0,x_R ∈ [0⁺,1)and let η be an SLEκ(ρ_L;ρ_1,R, ρ_2,R)process with force points(x_L;x_R,1). Let E^δ,r_ǫ be the event as in Theorem3.1, then for each δ∈(0,1)and r ≥2fixed, we have that

P[E_ǫ^δ,r] ≍ǫ^α

where the constants in≍depend only onκ,δ, r , x_L, x_R, and the weightsρ_L, ρ_1,R,ρ_2,R. Proof Let(g_t)be the Loewner evolution associated withηand letV_t^L,V_t^Rdenote the evolution ofx_L,x_R, respectively, underg_t. From (2.6) we know that

M_t =

g_t(1)−V_t^R g^′_t(1)

−α

×

g_t(1)−W_t g_t(1)−V_t^R

₋²_κ(ρ1,R+ρ2,R+4−κ/2)

×(g_t(1)−V_t^L)^{−ρκL(ρ1,R+ρ2,R+4−κ/2)}

is a local martingale which yields that the law of ηreweighted by M is that of an SLEκ(ρL;ρ1,R,ρ2,R)process whereρ2,R = −2ρ1,R−ρ2,R−8+κ. Note that, by similar analysis in Lemma3.4, the termgτ_ǫ(1)−V_τ^L_ǫ is bounded both from below and above by positive finite constants depending only onr on the eventE_ǫ^δ,r. The rest of the analysis in the proof of Lemma3.2applies similarly in this setting. ⊓⊔

Throughout the rest of this subsection, we let:

T=R×(0,1). (3.6)

Lemma 3.4 Letηbe a continuous curve inHstarting from0with continuous Loewner driving function W and let(gt)be the corresponding family of conformal maps. For each t ≥0, let O_t^L(resp. O_t^R) be the leftmost (resp. rightmost) point of gt(η([0,t]))in R. There exists a universal constant C ≥1such that the following is true. Fixϑ >0 and letσ be the first time thatηexitsϑT. Then

|W_σ −O_σ^q| ≥ ϑ

C for q∈ {L,R}. (3.7)

Letζ be the first time thatηexitsD∩ϑT. Then

|Wt−O_t^q| ≤Cϑ for q∈ {L,R} and all t ∈ [0, ζ]. (3.8) Finally, ifηexitsD∩ϑTthrough the right side of∂D∩ϑT, then

|Wζ−O_ζ^L| ≥ 1

C. (3.9)

Proof Forz∈C, we letP^z denote the law of a Brownian motionBinCstarted atz.

By [16, Remark 3.50] we have that

|W_σ −O_σ^L| = lim

y→∞yP^yi[BexitsH\η[0, σ]on the left side ofη([0, σ])].

|W_σ −O_σ^L| ≥ lim

y→∞yP^yi[Bτ ∈I]

×P^yi[BexitsH\η([0, σ])on the left side ofη([0, σ])|B_τ ∈I]. (3.10) We have,

ylim→∞yP^yi[Bτ ∈ I]= lim

y→∞

I−ϑi

1 π

y(y−ϑ ) w²+(y−ϑ )²dw

=

I−ϑi

1

πdw= ϑ

π (3.11)

(recall the form of the Poisson kernel onH, see e.g., [16, Exercise 2.23]). It is easy to see that there exists a universal constant p0>0 such that for anyz∈I,

P^z[BexitsH\η[0, σ]on the left side ofη([0, σ])]≥ p₀. (3.12) Combining (3.10) with (3.11) and (3.12) gives (3.7). The bounds (3.8) and (3.9) are

proved similarly. ⊓⊔

Lemma 3.5 Fixκ >0,ρ_L ∈(^κ₂−4,^κ₂−2), andρ_R>−2. Letηbe anSLEκ(ρ_L;ρ_R) process with force points(−ǫ;x_R)for x_R ≥ 0⁺andǫ > 0. Letσ1 =inf{t ≥ 0 : η(t)∈∂D}. Define, for u≥0, T_u^L =inf{t ≥0:Wt−V_t^L =u}, where V_t^Ldenotes the evolution of x^L. Let p2 = p2(¹₂)be the constant from Lemma2.6. There exists constantsǫ0>0,ϑ0>0, and C>0such that for allǫ∈(0, ǫ0)andϑ∈(0, ϑ0)we have

P[σ₁<T₀^L∧T_ϑ^L] ≤ p₂^{1/(Cϑ )}. Proof LetE_ϑ = {σ₁<T₀^L∧T_ϑ^L}. By definition, we have that

|Wt−V_t^L|< ϑ for all t ∈ [0, σ1] on Eϑ. (3.13) By (3.7) of Lemma3.4there exists a constantC₁>0 such thatη([0, σ1])⊆C₁ϑT.

Moreover,ηexitsD∩ C₁ϑT

on its left side for allϑ >0 small enough because a Brownian motion argument [(analogous to (3.9)] implies there exists a constant C2 >0 such that|Wσ₁−V_σ^L

1| ≥C2on the event thatηexits through the right side, contradicting (3.13).

SupposeC>0; we will set its value later in the proof. For each 1≤k≤ Cϑ¹ , we let

L_k= {z∈H:Re(z)= −kCϑ} and ζ_k =inf{t≥0:η(t)∈L_k}.

g_ζk(·)−W_ζk W_ζk−V^L

ζk

−kCϑ

−(k+1)Cϑ

η z_k^ϑ=η(ζk)−ϑ

Lk

Lk+1

z^ϑ_k ϑ

0

−1 L_k+1

Fig. 10 Illustration of the justification of (3.14) in the proof of Lemma3.5

On Eϑ, we have thatζ1 < ζ2 <· · · < σ1 <T₀^L. For eachk, let Fk = {ζk <T_ϑ^L} and letF_kbe theσ-algebra generated byη|[0,ζ_k]. To complete the proof, we will show that

P[ζk+1<T₀^L|F_k]1F_k ≤ p21F_k for each 1≤k≤ 1 Cϑ

where p₂=p₂(¹₂)is the constant from Lemma2.6. To see this, we just need to show thatg_ζk(η|[ζk,ζk+1])satisfies the hypotheses of Lemma2.6and that with

Lk+1= g_ζk(L_k+1)−W_ζk W_ζk−V_ζ^L

k

we have thatL_k+1∩2D=∅onF_k. Therefore it suffices to prove

dist(Wζ_k,gζ_k(Lk+1)) W_ζk−V_ζ^L

k

→ ∞ on F_k as C→ ∞. (3.14)

LetBbe a Brownian motion starting fromz^ϑ_k =η(ζk)−ϑand letHk+1= {z∈H: Re(z)≥ −(k+1)Cϑ}be the subset ofHwhich is to the right ofLk+1(see Fig.10).

The probability thatBexitsHk+1\η([0, ζk])through the right side ofη([0, ζk])(blue) is1, through(−(k+1)Cϑ,−kCϑ )(green) is1, and throughLk+1(orange) is 1/C (since this probability is less than the probability that the Brownian motion exits{z ∈ C : −(k+1)Cϑ < Re(z) < −kCϑ}through L_k+1which is less than 1/C). Let

z^ϑ_k ≡x_k^ϑ+y_k^ϑi ≡ g_ζk(z_k^ϑ)−W_ζk W_ζk−V_ζ^L

k

.

By the conformal invariance of Brownian motion, we have that dist(z^ϑ_k,L_k+1)

y_k^ϑ C. (3.15)

(gζ_k(Hk+1)−Wζ_k)/(Wζ_k −V_ζ^L

k)throughLk+1is bounded from below by a positive universal constant times the probability that a Brownian motion starting fromz^ϑ_k exits B(z^ϑ_k,d)∩H,d =dist(z^ϑ_k,Lk+1), through∂B(z^ϑ_k,d)∩H. This latter probability is bounded from below by a positive universal constant timesy_k^ϑ/d. Thus 1/Cy_k^ϑ/d, as desired.

The conformal invariance of Brownian motion and the estimates above also imply that sin(arg(z^ϑ_k))≍1, hence|z^ϑ_k| ≍ |y_k^ϑ|. Combining this with (3.15) implies that

dist(z^ϑ_k,Lk+1)

|z^ϑ_k| C.

Thus, by the triangle inequality,

dist(Lk+1,0)C|z^ϑ_k|

(providedCis large enough). Since|z^ϑ_k| ≍1, this proves (3.14), hence the lemma.⊓⊔ Proof of Theorem3.1 Lemma3.2implies the lower bound in (3.3) because we can take, e.g.,δ =¹₂. In order to prove the upper bound, it is sufficient to show

P[τǫ <∞] ≤ǫ^α+o(1) as ǫ→0.

We are first going to perform a change of coordinates. Letϕ: H → Hbe the Möbius transformationz → ϕ(z):=ǫz/(1−z). Fixx^R ∈ [0⁺,1)and letηbe an SLEκ(ρ_1,R, ρ_2,R)process with force points located at(x^R,1)as in Theorem3.1. Then the law ofη=ϕ(η)is that of an SLEκ(ρ_L;ρ_R)process with force points(−ǫ;x_R) wherex_R=ǫx^R/(1−x^R)and

ρ_L =κ−6−

ρ_1,R+ρ_2,R

and ρ_R=ρ_1,R. (3.16) Letσ1be the first time thatηhits∂Dand letV_t^L,V_t^Rdenote the evoltuion ofxL,xR

undergt, respectively. Foru ≥ 0, defineT_u^L = inf{t ≥ 0 : Wt −V_t^L = u}(as in the statement of Lemma3.5). Then it is sufficient to proveP[σ1 <T₀^L] ≤ ǫ^α+o(1). Note that the exponentαcomes from the sum of the exponent of|V_t^L−V_t^R|and the exponent of|W_t−V_t^L|in the left martingaleM^Lfrom (2.7) with these weights. For u ≥0, defineτ_u^L =inf{t ≥0: M_t^L =u}. Note thatτ₀^L =T₀^L. Fixβ∈(0,1)and set ϑ=ǫ^β. Foru >0, we have the bound

P[σ₁< τ₀^L] ≤P[τ_u^L < τ₀^L] +P[σ₁< τ₀^L < τ_u^L]. (3.17) We claim that exists constantsC₁>0 andγ >0 depending only onρ_L,ρ_R, and κsuch that

|W_t−V_t^L|^γ ≤C1M_t^L for all t ∈ [0, σ1]. (3.18)

Sinceρ_1,R+ρ_2,R > ^κ₂ −4 it follows thatρ_L < ^κ₂ −2. Therefore the sign of the exponent of|V_t^L−V_t^R|in the definition ofM_t^Lis the same as the sign ofρR. IfρR≥0, then the exponent has a positive sign. In this case,M_t^L ≥ |Wt−V_t^L|^αso that we can takeγ =α. Now suppose thatρR <0. By (3.8) of Lemma3.4we know that there exists a constantC2>0 such that

|V_t^L−V_t^R| ≤C₂ for all t∈ [0, σ1]. (3.19) Thus, in this case, there exists a constant C₃ > 0 such that M_t^L ≥ C₃|W_t − V_t^L|^{(κ−4−2ρL)/κ}. Therefore we can takeγ =(κ−4−2ρL)/κ. This proves the claimed bound in (3.18).

Setu=ϑ^γ/C₁. To bound the second term on the right side of (3.17), we first note by (3.18) that

P[σ₁< τ₀^L < τ_u^L] ≤P[σ₁<T₀^L∧T_ϑ^L]. (3.20) By Lemma3.5, we know that

P[σ1<T₀^L∧T_ϑ^L] ≤ p^{1/(Cϑ )}₂ . (3.21) We will now bound the first term on the right side of (3.17). Since τ₀^L, τ_u^L are stopping times for the martingaleM^LandM_τ0∧τu =uP[τ_u^L < τ₀^L], we have that

P[τ_u^L < τ₀^L] = 1 uE[M_τ^L

0∧τ_u] = M₀^L

u = ǫ^α

u(1−x^R)^{(κ−4−2ρL)ρR/(2κ)}. (3.22) Combining (3.17) with (3.21) and (3.22) we get that P[σ₁ < T₀^L] ≤ ǫ^α+o(1), as

desired. ⊓⊔

Recall that (see for example [24, Section 4]) the β-Hausdorff measure of a set A⊆Ris defined as

H^β(A)= lim

ǫ→0+

H^β

ǫ(A)

where H^β

ǫ(A):=inf

⎧⎨

⎩ _j |I_j|^β : A⊆ ∪jI_j and |I_j| ≤ǫ for all j

⎫⎬

⎭.

Proof of Theorem1.6forκ ∈(0,4),upper bound Fixκ∈(0,4), ρ ∈(−2,^κ₂−2).

Letηbe an SLEκ(ρ)process with a single force point located at 0⁺. Letα∈(0,1) be as in (3.4). Fix 0<x <y. We are going to prove the result by showing that

dimH(η∩ [x,y])≤1−α almost surely. (3.23) For eachk∈Zandn∈Nwe letI_k,n= [k2⁻ⁿ, (k+1)2⁻ⁿ]and letz_k,nbe the center ofI_k,n. LetI_nbe the set ofksuch thatI_k,n⊆ [x/2,2y]and letE_k,nbe the event that

everyn ≥n0we have that{I_k,n :k∈I_n, E_k,noccurs}is a cover ofη∩ [x,y]. Fixζ >0. Theorem3.1implies that there exists a constantC1>0 (independent ofn) andn1=n1(ζ )such that

P[E_k,n] ≤C₁2^{−(α−ζ )n} for each n≥n₁ and k∈I_n. Consequently, there exists a constantC₂>0 such that

E

H^β

2⁻ⁿ(η∩ [x,y])

≤E

⎡

⎣

k∈I_n

2^−βn1E_k,n

⎤

⎦≤C22^−βn×2ⁿ×2^{−(α−ζ )n}.

By Fatou’s lemma, E

H^1−α+2ζ(η∩ [x,y])

≤lim inf

n E

H^1−α+2ζ

2⁻ⁿ (η∩ [x,y])

≤lim inf

n C₂2^−nζ =0.

This implies thatH¹−α+2ζ(η∩ [x,y])=0 almost surely. This proves (3.23) which

completes the proof of the upper bound. ⊓⊔

Dans le document Intersections of SLE Paths: the double and cut point dimension of SLE (Page 21-29)