Equivalence of annulus and radial SLE

2.4 Annulus SLE

2.4.2 Equivalence of annulus and radial SLE

Suppose Ω is a simply connected domain, ais a prime end, and b is an interior point.

Suppose F %{b} is an interior hull in Ω. Then Ω\F is a doubly connected domain with two boundary components ∂Ω and ∂F. Let p:=M(Ω\F). For a fixed κ >0, let (K_t) has the law of radial SLE_κ(Ω;a → b), and (L_s) has the law of annulus SLE_κ(Ω\F;a→∂F). Let T_F be the first time that K_t intersects F. Then we have Theorem 2.4.1 (i)If κ = 6, the law of (K_t)_0≤t<TF, is equal to that of (L_s)_0≤s<p, up to a time-change.

(ii)Ifκ >0andκ6= 6, there exist two sequences of stopping times{T_n}and{S_n}such

that T =∨nTn, p=∨nSn, and for each n ∈N, the law of (Kt)0≤t≤Tn is equivalent to that of (L_s)_0≤s≤Sn, up to a time-change.

By conformal invariance, we may assume that Ω = D, a = 1 and b = 0. Then (Kt,0 ≤ t < ∞) has the law of the standard radial SLEκ. Suppose ϕt and ξ(t) =

√κB(t), 0 ≤ t < ∞, are the corresponding standard radial LE maps and driving function, respectively. SupposeW mapsD\F conformally ontoAp so thatW(1) = 1.

Sincet7→K_t, 0 ≤t <∞, is a Loewner chain inD, it is clear thatt 7→K_t, 0≤t < T_F, is a Loewner chain inD\F onC0. By conformal invariance,t 7→W(Kt), 0≤t < TF, is a Loewner chain in A_p on C₀. Let u(t) :=C_D,∂F(K) = C_Ap,Cp(W(K)). Then u is a continuous increasing function and maps [0, TF) onto [0, p). Letv be the inverse of u. By Proposition 2.4.1, W(K_v(s)), 0 ≤ s < p, are the standard annulus modulus p LE hulls driven by some continuous function ζ : [0, p) → R. Let ψs, 0≤ s < p, be the corresponding standard annulus LE maps.

Now ϕt maps D\F \Kt conformally onto D\ϕt(F). Let ft :=ψ_u(t)◦W ◦ϕ⁻¹_t . Then f_t maps D\ϕ_t(F) conformally onto A_p−u(t), and f_t(C₀) = C₀. By Schwarz reflection, we may extend ft conformally to Σt, which is the union of D\ ϕt(F), C₀, and the reflection of D\ϕ_t(F) w.r.t. C₀. Note that f_t maps ϕ_t(K_t+a\K_t) to ψ_u(t)(W(Kt+a)\W(Kt)) for a > 0. From Proposition 2.4.1, we see that {e^iζ(t)} =

∩_a>0ψ_u(t)(W(K_t+a)\W(K_t)). And from formula (2.2.2), we know that {e^iξ(t)} =

∩a>0ϕt(Kt+a\Kt). Thus e^iζ(t) =ft(e^iξ(t)).

Recall that for a hullK in a dimply connected domainDandz ∈D\K,C_D;z(K) is the capacity of K in D w.r.t. z. Similarly as Lemma 2.8 in [4], using the integral formulas for capacities of hulls in D and A_p, it is not hard to derive the following Lemma:

Lemma 2.4.1 Supposex, y ∈C₀, andG is a conformal map from a neighborhoodU of x onto a neighborhood V of y such that G(U ∩D) = V ∩D. Fix any p > 0. For every ε >0, there is r=r(ε)>0such that if K is a non-empty hull in D on C₀ and K ⊂ B(x;r), which is the open ball of radius r about x, then G(K) is a hull in Ap

on C0, and ¯

¯¯

¯C_Ap,Cp(G(K))

CD,0(K) − |G⁰(x)|²

¯¯

¯¯< ε.

Now ϕ_t(K_t+a\K_t) is a hull in D, ϕ_t+a◦ϕ⁻¹_t maps D\ϕ_t(K_t+a\K_t) conformally ontoD, fixes 0, and (ϕt+a◦ϕ⁻¹_t )⁰(0) =e^a. So the capacity ofϕt(Kt+a\Kt) inDw.r.t.

0 is a. Similarly,ψ_u(t)(W(K_t+a\W(K_t)) is a hull in A_p−u(t) on C₀, and the capacity is u(t+a)−u(t). From Lemma 2.4.1 we conclude that u⁰₊(t) =|f_t⁰(e^iξ(t))|².

Let H := {(t, z) : 0 ≤ t < T_F, z ∈ Σ_t} and G(ξ) = {(t, e^iξ(t)) : 0 ≤ t < T_F}.

By the definition of ft, we see that (t, z) 7→ f_t⁰(z) is continuous in H \G(χ). Note that f_t⁰ is analytic in Σ_t for each t ∈ [0, T_F). The maximum principle implies that (t, z) 7→ f_t⁰(z) is continuous in H. In particular, t 7→ f_t⁰(e^iξ(t)) is continuous. So we have

Lemma 2.4.2 u(t) is C¹ continuous, and u⁰(t) = |f_t⁰(e^iξ(t))|².

The fact W(1) = 1 implies thate^iζ(0) = 1. We may chooseζ(0) = 0, and lift ft to the covering space. Leteⁱ denote the functionz 7→e^iz. LetΣe_t:= (eⁱ)⁻¹(Σ_t). There is a unique family of conformal maps fe_t onΣe_t such that eⁱ◦fe_t=f_t◦eⁱ,fe_t(ξ(t)) =ζ(t), and fe_t takes real values on R. Then we have u⁰(t) =fe_t⁰(ξ(t))².

Lemma 2.4.3 (t, x) 7→ fet(x) is C^1,∞ continuous on [0, TF)×R. And for all t ∈ [0, T_F), ∂_tfe_t(ξ(t)) =−3fe_t⁰⁰(ξ(t)).

Proof. For any t ∈ [0, T_F), and z ∈ D\F \K_t, we have f_t◦ϕ_t(z) = ψ_u(t)◦W(z).

Taking the derivative w.r.t. t, we compute

∂_tf_t(ϕ_t(z)) +f_t⁰(ϕ_t(z))ϕ_t(z)e^iξ(t)+ϕ_t(z) e^iξ(t)−ϕt(z)

=u⁰(t)ψ_u(t)(W(z))S_p−u(t)(ψ_u(t)(W(z))/e^iζ(t)).

By Lemma 2.4.2, u⁰(t) = |f_t⁰(e^iξ(t))|². Thus for any t ∈[0, T_F) and z ∈D\F \K_t,

∂_tf_t(ϕ_t(z)) =|f_t⁰(e^iξ(t))|²f_t(ϕ_t(z))S_p−u(t)(f_t(ϕ_t(z))/f_t(e^iξ(t)))

−f_t⁰(ϕ_t(z))ϕ_t(z)e^iξ(t)+ϕ_t(z) e^iξ(t)−ϕt(z).

For any t∈[0, T_F), and w∈D\ϕ_t(F), we have ϕ⁻¹_t (w)∈D\F \K_t. Thus

∂_tf_t(w) =|f_t⁰(e^iξ(t))|²f_t(w)S_p−u(t)(f_t(w)/f_t(e^iξ(t)))−f_t⁰(w)we^iξ(t)+w e^iξ(t)−w.

Let g_t(w) be the right-hand side of the above formula for t ∈ [0, T_F) and w ∈ Σ_t\ {e^iξ(t)}. Then for eacht∈[0, T_F),g_t(w) is analytic in Σ_t\{e^iξ(t)}. And (t, w)7→g_t(w) is C^0,∞ continuous on H\G(ξ).

Now fix t₀ ∈[0, T_F). Let us compute the limit of g_t0(w) whenw→e^iξ(t0). Since

S_p−u(t0)(f_t0(w)/f_t0(e^iξ(t0)))− f_t0(e^iξ(t0)) +f_t0(w)

f_t0(e^iξ(t0))−f_t0(w) →0, as w→χ_t0, so the limit of gt0(w) is equal to the limit of the following function:

|f_t⁰₀(e^iξ(t0))|²f_t0(w)f_t0(e^iξ(t0)) +f_t0(w)

f_t0(e^iξ(t0))−f_t0(w)−f_t⁰₀(w)we^iξ(t0)+w e^iξ(t0)−w.

Letw=e^ix, we may express the above formula in term ofx,ξ(t₀) and fe_t0, as follows:

fe_t⁰₀(ξ(t₀))²e^ifet0(x)e^{ifet0(ξ(t0))}+e^ifet0(x)

e^{ifet0(ξ(t0))}−e^ifet0(x) −fe_t⁰₀(x)e^ifet0(x)e^iξ(t0)+e^ix e^iξ(t0)−e^ix

=−ie^ifet0(x)[fe_t⁰₀(ξ(t₀))²cot(fet0(x)−fet0(ξ(t0))

2 )−fe_t⁰₀(x) cot(x−ξ(t0) 2 )].

By expanding the Laurent series of cot(z/2) near 0, we see that the limit of the above formula is 3ie^{ifet0(ξ(t0))}fe_t⁰⁰₀(ξ(t₀)) = 3if_t0(e^iξ(t0))fe_t⁰⁰₀(ξ(t₀)). Therefore g_t has an analytic extension to Σ_t for each t ∈ [0, T_F). The maximum principle also implies that g_t(w) is C^0,∞continuous in H, and∂_tf_t(w) =g_t(w) holds in the wholeH. Thus f_t(w) isC^1,∞ continuous on [0, T_F)×C₀, and fe_t(w) isC^1,∞continuous on [0, T_F)×R.

Finally,

∂_tfe_t(ξ(t)) = i∂_tf_t(e^iξ(t))

f_t(e^iξ(t)) = ig_t(e^iξ(t))

f_t(e^iξ(t)) = −3f_t(e^iξ(t))fe_t⁰⁰(ξ(t))

f_t(e^iξ(t)) =−3fe_t⁰⁰(ξ(t)). 2

Proof of Theorem 2.4.1. Note that ζ(t) =fet(ξ(t)), ξ(t) = √

κB(t), and from the last lemma, ∂_tfe_t(ξ(t)) =−3fe_t⁰⁰(ξ(t)). By Itˆo’s formula, we have

dζ(u(t)) =fe_t⁰(ξ(t))dξ(t) + (κ

2 −3)fe_t⁰⁰(ξ(t))dt.

Since u⁰(t) = fe_t⁰(ξ(t))², so

dζ(t) = dξ¹(t) + (κ

2 −3)fe_v(s)⁰⁰ (ξ(t))/fe_v(s)⁰ (ξ(t))²dt, where ξ¹(t) =√

κB¹(t), 0≤ t < p, and B¹(t) is another standard Brownian motion.

Note that ζ₀ = 0. If κ = 6, then ζ(t) = ξ¹(t), 0 ≤t < p. Thus (W(K_v(s)))_0≤s<p has the law of the standard moduluspannulus SLE_κ=6. So (K_v(s))_0≤s<p has the same law as (L_s)_0≤s<p.

If κ > 0 and κ 6= 6, then dζ(s) = dξ¹(s) + a drift term. The remaining part follows from Girsanov’s Theorem. 2

From this proof, it is clear that if a family of standard moduluspannulus LE hulls (K_t) is equivalent to radial SLE_κ in the sense of the above theorem, then the driving function is√

κB(t) plus some C¹ continuous function. If the law of (K_t) also satisfies the properties of symmetry and the conformally equivalent time-homogeneity, then the driving function must be √

κB(t). So the standard annulus SLE_κ is determined by the three properties.

This equivalence theorem implies the a.s. existence of annulus SLE trace. Suppose (K_t) has the law of the standard moduluspannulus SLE_κ. Then there is a.s. a random curve β : [0, p)→A_p∪C₀ with β(0) = 1 such that for each t, K_t is the complement in A_p of the component of A_p \β[0, t] whose boundary contains C_p. If κ ≤ 4, then β is simple and K_t = β(0, t]. Such β is called a standard modulus p annulus SLE_κ trace. Through a conformal map, we could define annulus SLE_κ trace in an arbitrary doubly connected domain.

From the strong equivalence forκ= 6, ifβ is a standard modulus pannulus SLE₆ trace, then lim_t→pβ(t) exists on C_p a.s.. The same is true for κ = 2 thanks to the

convergence of LERW to SLE2( see Chapter 4). It is not known now whether this is true for other κ.