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Reversibility of Whole-Plane SLE

Dapeng Zhan

Michigan State University

Dapeng Zhan Reversibility of Whole-Plane SLE

Two most well-know SLE: chordal SLE and radial SLE. They have the following properties:

◮ A random curve in a simply connected domain; ◮ Starts from a boundary point; ◮ Domain Markov Property; ◮ The behavior of the curve depends on κ ∈ (0, 4], (4, 8), or

[8, ∞).

Dapeng Zhan Reversibility of Whole-Plane SLE

A chordal SLE curve ends at another boundary point.

Dapeng Zhan Reversibility of Whole-Plane SLE

A radial SLE curve ends at an interior point.

Dapeng Zhan Reversibility of Whole-Plane SLE

It is proven earlier, for κ ≤ 4, chordal SLE satisfies reversibility: The time-reversal of a chordal SLEκ from a to b is a chordal SLEκ from b to a in the same domain. A radial SLE can not satisfy reversibility because the initial point and the end point are topologically different. To study the time-reversal of a radial SLE, we consider the reversibility of whole-plane SLE.

Dapeng Zhan Reversibility of Whole-Plane SLE

A whole-plane SLE

◮ Domain:

C = C ∪ {∞};

◮ Initial: an interior point; ◮ End: another interior point.

Conditioned on a part of a whole-plane SLE curve, the rest of the curve is a radial SLE growing in the remaining domain.

Dapeng Zhan Reversibility of Whole-Plane SLE

In the proof of reversibility of chordal SLE, two SLEκ curves γ1 and γ2 are constructed in a simply connected domain D to grow towards each other. They satisfy: if Tj is a stopping time for one curve γj, then conditioned on γj up to Tj, the other curve γ3−j up to the time hitting γj([0, Tj]), is a chordal SLE in D \ γj([0, Tj]) aimed at γj(tj). So γ3−j does visit γj(Tj). So every point on one curve will be visited by the other, and they must overlap. We call this an

• verlap coupling.

Dapeng Zhan Reversibility of Whole-Plane SLE

To construct the above coupling, we use the following facts.

◮ Two chordal SLEκ curves which run towards each other

satisfy commutation relation. This enables us to construct local overlap couplings.

◮ Every local overlap coupling is absolutely continuous w.r.t. the

local independent coupling, and the RN derivative can be expressed in terms of Loewner maps.

◮ Using the stochastic coupling technique, we can construct a

global overlap coupling from these local couplings.

Dapeng Zhan Reversibility of Whole-Plane SLE

The left curve is SLEκ aiming at b. The right curve is SLEκ aiming at a. Conditioned on the left curve up to a′, the right curve is SLEκ in the remaining domain aiming at a′, and vice versa.

Dapeng Zhan Reversibility of Whole-Plane SLE

For the reversibility of whole-plane SLE, we also want to construct a pair of whole plane SLE curves that run towards each other such that every point on one curve will be visited by the other. The situation is different. The whole domain – Riemann sphere C minus a part of one curve is a simply connected domain, so the

• ther curve restricted in the remaining domain is no longer a

whole-plane SLE. And it is neither a radial nor a chordal SLE because it starts from an interior point.

Dapeng Zhan Reversibility of Whole-Plane SLE

So we need to define SLE in simply connected domains starting from one interior point and ending at a boundary point.

Dapeng Zhan Reversibility of Whole-Plane SLE

Note that after a positive initial segment, the remaining curve grows in a doubly connected domain. To define such SLE, we first define SLE in doubly connected domains.

Dapeng Zhan Reversibility of Whole-Plane SLE

The definition of SLE in doubly connected domains uses annulus Loewner equation. Let Ap = {e−p < |z| < 1}, T = {|z| = 1}, and Tp = {|z| = e−p}. Let S(p, z) = P. V.

∞

• k=−∞

e2kp + z e2kp − z = P. V.

• 2|n

enp + z enp − z . S(p, ·) is analytic in Ap, has a simple pole at 1 ∈ T; Re S(p, ·) ≡ 0

• n T \ {1}; Re S(p, ·) ≡ 1 on Tp.

Dapeng Zhan Reversibility of Whole-Plane SLE

The annulus Loewner equation of modulus p > 0 driven by ξ is ∂tgt(z) = gt(z)S(p − t, gt(z)/eiξ(t)), g0(z) = z. To define SLEκ curve, we let ξ(t) be a semi-martingale with dξ(t) = √κdB(t) + c(t)dt. The trace is defined by β(t) := g−1

t

(eiξ(t)), 0 ≤ t < p.

Dapeng Zhan Reversibility of Whole-Plane SLE

The trace β grows in Ap, starts from a point on T, and grows towards Tp. For each t, gt maps Ap \ β(0, t] conformally onto Ap−t, and maps Tp onto Tp−t.

Dapeng Zhan Reversibility of Whole-Plane SLE

For b ∈ ∂Ap, θ(t) := arg gt(b) satisfies θ′(t) = H(p − t, θ(t) − ξ(t)), if b ∈ T, θ′(t) = HI(p − t, θ(t) − ξ(t)), if b ∈ Tp, where H(t, z) = −iS(t, eiz) = P. V.

• 2|n

cot2(z − nt); HI(t, z) = H(t, z + it) + i = P. V.

• 2∤n

cot2(z − nt). Here we set cot2(z) = cot(z/2).

Dapeng Zhan Reversibility of Whole-Plane SLE

These H and HI are related to Jacobi theta functions. Let θk(z, q), k = 1, 2, 3, 4, be Jacobe theta functions. Let Θk(t, z) = θk(z, e−t). Then H = 2Θ′

1

Θ1 ; HI = 2Θ′

4

Θ4 . Here we use ′ to denote the derivative wrt the second variable.

Dapeng Zhan Reversibility of Whole-Plane SLE

If ξ(t) = √κB(t), we have annulus SLEκ with no marked point. Given a function Λ(t, x) on (0, ∞) × R with period 2π in the second variable, we may define the annulus SLE(κ; Λ) in Ap started from a = eix0 ∈ T with marked point b ∈ Tp by letting ξ(t) be the solution of the SDE: dξ(t) = √κdB(t) + Λ(p − t, ξ(t) − arg(gt(b)))dt, ξ(0) = x0.

Dapeng Zhan Reversibility of Whole-Plane SLE

Using conformal maps we can define annulus SLE(κ; Λ) in Ap started from a ∈ Tp with marked point b ∈ T. Let b be fixed, and let p → ∞. Then Ap → D = {|z| < 1} and Tp → {0}. The SLE curve tends to a random curve in D started from 0. The limit curve is called a disc SLE(κ; Λ) in D started from 0 with marked point b. This definition extends to any simply connected domain.

Dapeng Zhan Reversibility of Whole-Plane SLE

We want to find some drift function Λ, and construct a pair of whole-plane SLEκ curves γ1 and γ2 that grow towards each other, such that, conditioned on γj(0, Tj], the other curve γ3−j restricted in C \ γj(0, Tj] is a disc SLE(κ; Λ) process in this domain, and has γj(Tj) as its marked point.

Dapeng Zhan Reversibility of Whole-Plane SLE

We find that for such a coupling exists, the Λ must satisfy a PDE

• n (0, ∞) × R:

∂tΛ = κ 2Λ′′ +

• 3 − κ

2

• H′′

I + (ΛHI)′ + ΛΛ′.

(1) If in addition, a disc SLE(κ; Λ) curve eventually ends at the marked point then using the coupling technique, we obtain the desired

• verlap coupling.

To prove the reversibility of whole-plane SLE, we need to prove the existence of the solution to (1) with the above property.

Dapeng Zhan Reversibility of Whole-Plane SLE

For κ ∈ (0, 4], the Λ is proved to exist, so we have the reversibility

• f whole-plane SLEκ.

Such Λ is uniquely determined by κ. In fact, a byproduct we have is that the reversal of a radial SLEκ curve is a disc SLE(κ; Λ) curve. The reversibility is also satisfied by whole-plane SLE with a constant drift, which is the whole-plane Loewner process driven by √κB(t) + ρt.

Dapeng Zhan Reversibility of Whole-Plane SLE

We may define annulus SLE(κ, Λ) process such that the marked point and the initial point lie on the same boundary component. For the commutation relation to hold, the drift function λ must solve a similar PDE ∂tΛ = κ 2Λ′′ +

• 3 − κ

2

• H′′ + (ΛH)′ + ΛΛ′.

(2) This equation is similar to (1). If a lattice path in a doubly connected domain satisfies reversibility, and is expected to converge to some SLE. Then the SLE must satisfy reversibility, and so (1) or (2) should hold.

Dapeng Zhan Reversibility of Whole-Plane SLE

Equation (1) is solved using transformations and Feynman-Kac expression. First,may we transform (1) into a linear PDE using Λ = κ Γ′

Γ :

∂tΓ = κ 2Γ′′ + HIΓ′ + 3 κ − 1 2

• H′

IΓ.

(3) Let Ψ = ΓΘ

2 κ

4 , and σ = 4/κ − 1. Then (3) is equivalent to

∂tΨ = κ 2Ψ′′ + σH′

IΨ.

(4)

Dapeng Zhan Reversibility of Whole-Plane SLE

We rescale HI by

• HI(t, z) = π

t HI π2 t , π t z

• + z

t = P. V.

• 2|n

tanh2(z − nt), where tanh2(z) = tanh(z/2). We have HI(t, ·) → tanh2 as t → ∞. Define

• Ψ(t, x) = e

x2 2κt Ψ

π2 t , π t x

• .

Then (4) is equivalent to −∂t Ψ = κ 2

• Ψ′′ + σ

H

′ I

Ψ. (5)

Dapeng Zhan Reversibility of Whole-Plane SLE

As t → ∞, HI(t, ·) → tanh2, so (5) tends to −∂t Ψ = κ 2

• Ψ′′ + σ tanh′

2

Ψ. (6) Using separation of variables, we find a solution to (6):

• Ψ∞(t, x) = e− τ2t

2κ cosh 2 κτ

2 (x),

where τ ≤ 0 is a root of τ 2 − κ

2τ = κσ.

Dapeng Zhan Reversibility of Whole-Plane SLE

If Ψ = Ψ∞ Ψq, and HI,q = HI − tanh2, then (5) is equivalent to −∂t Ψq = κ 2

• Ψ′′

q + τ tanh2

Ψ′

q + σ

H

′ I,q

Ψq. (7) Suppose Xx0(t) is a diffusion process that satisfies dXx0(t) = √κdB(t) + τ tanh2(Xx0(t))dt, Xx0(0) = x0. If Ψq solves (7), then we have a local martingale M(t) := Ψq(t0 + t, Xx0(t)) exp

• σ

t

• H

′ I,q(t0 + s, Xx0(s))ds

• .

Dapeng Zhan Reversibility of Whole-Plane SLE

If M(t) is a bounded martingale, and Ψq → 1 as t → ∞, then we have

• Ψq(t0, x0) = M(0) = E
• exp
• σ

∞

• H

′ I,q(t0 +s, Xx0(s))ds

• . (8)

Now we define Ψq using (8). Then we can prove that

◮

Ψq is well defined;

◮

Ψq is smooth;

◮

Ψq solves (7).

Dapeng Zhan Reversibility of Whole-Plane SLE

Using this Ψq and the above transformations, we find a solution of (3). We call it Γ0. Then Λ0 := κΓ′

0/Γ0 solves (1). But such Λ0

does not have period 2π. Let Γm(t, z) = Γ0(t, z − 2mπ). Since HI has period 2π, so Γm, m ∈ Z, all solve (3). Since (3) is a linear equation, so Γ :=

• m∈Z

Γm still solves (3). Here we use the fact that, for h = 0, 1, 2, Γ(h)

0 (t, x) ∼ e−x2,

as x → ±∞. Such Γ has period 2π, so is Λ := κΓ′/Γ. This is the Λ we are looking for.

Dapeng Zhan Reversibility of Whole-Plane SLE

To prove the reversibility of skew whole-plane SLE (driven by √κB(t) + ρt), we use Γρ =

• m∈Z

e

2π κ mρΓm.

Such Γρ does not have period 2π, but Λρ := κΓ′

ρ/Γρ does.

We may construct a pair of whole-plane SLE(κ, ρ) process such that, conditioned on a part of one curve, the other curve becomes a disc SLE(κ, Λρ) curve. So we can conclude that the reversal of a radial SLE(κ, ρ) curve is a disc SLE(κ, Λρ) curve.

Dapeng Zhan Reversibility of Whole-Plane SLE

Now we consider the annulus SLE(κ, Λρ) process. An annulus SLE(κ, Λρ) curve lies in a doubly connected domain, connects two boundary points on different boundary components, and satisfies reversibility. If we lift the curve to the covering space, we have a natural decomposition according to the end point in the covering space P ρ =

• m∈Z

C ρ

m P m.

Dapeng Zhan Reversibility of Whole-Plane SLE

We have two interesting facts.

◮ The distribution P m does not depend on ρ. ◮ The coefficients C ρ m

satisfies C ρ

m

= e

2π κ mρΓm(p, arg(b) − arg(a))

Γρ(p, arg(b) − arg(a)) , where p is the modulus, a and b are the initial and end points

• f the curve.

Dapeng Zhan Reversibility of Whole-Plane SLE

For some particular values of κ, we find solutions to (1) and (2), which can be expressed in terms of H and HI. Some particular solutions to (1):

◮ κ = 2: Λ(t, x) = 2tH′′ I (t, x)/(tH′ I(t, x) + 1). ◮ κ = 3: Λ(t, x) = 3 2HI(t, x) − 3 2HI(t, x − π). ◮ κ = 4: Λ(t, x) = 2HI(2t, x − π) − HI(t, x).

Dapeng Zhan Reversibility of Whole-Plane SLE

Some particular solutions to (2):

◮ κ = 0: Λ(t, x) = H(t, x) − 2H(t, x 2). ◮ κ = 2: Λ(t, x) = 2H′′(t, x)/H′(t, x). ◮ κ = 3: Λ(t, x) = 3 2H(t, x) − 3 2HI(t, x − π). ◮ κ = 4: Λ(t, x) = 2HI(2t, x) − H(t, x). ◮ κ = 16 3 : Λ(t, x) = 2 3H(t, x 2) − 1 3H(t, x).

Dapeng Zhan Reversibility of Whole-Plane SLE