Reversibility of Whole-Plane SLE Dapeng Zhan Michigan State - PowerPoint PPT Presentation

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 SLE κ from a to b is a chordal SLE κ 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 SLE κ curves γ 1 and γ 2 are constructed in a simply connected domain D to grow towards each other. They satisfy: if T j is a stopping time for one curve γ j , then conditioned on γ j up to T j , the other curve γ 3 − j up to the time hitting γ j ([0 , T j ]), is a chordal SLE in D \ γ j ([0 , T j ]) aimed at γ j ( t j ). So γ 3 − j does visit γ j ( T j ). So every point on one curve will be visited by the other, and they must overlap. We call this an overlap coupling. Dapeng Zhan Reversibility of Whole-Plane SLE

To construct the above coupling, we use the following facts. ◮ Two chordal SLE κ 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 SLE κ aiming at b . The right curve is SLE κ aiming at a . Conditioned on the left curve up to a ′ , the right curve is SLE κ 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 other 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 A p = { e − p < | z | < 1 } , T = {| z | = 1 } , and T p = {| z | = e − p } . Let ∞ e 2 kp + z e np + z � � S ( p , z ) = P . V . e 2 kp − z = P . V . e np − z . k = −∞ 2 | n S ( p , · ) is analytic in A p , has a simple pole at 1 ∈ T ; Re S ( p , · ) ≡ 0 on T \ { 1 } ; Re S ( p , · ) ≡ 1 on T p . Dapeng Zhan Reversibility of Whole-Plane SLE

The annulus Loewner equation of modulus p > 0 driven by ξ is ∂ t g t ( z ) = g t ( z ) S ( p − t , g t ( z ) / e i ξ ( t ) ) , g 0 ( z ) = z . To define SLE κ curve, we let ξ ( t ) be a semi-martingale with d ξ ( t ) = √ κ dB ( t ) + c ( t ) dt . The trace is defined by β ( t ) := g − 1 ( e i ξ ( t ) ) , 0 ≤ t < p . t Dapeng Zhan Reversibility of Whole-Plane SLE

The trace β grows in A p , starts from a point on T , and grows towards T p . For each t , g t maps A p \ β (0 , t ] conformally onto A p − t , and maps T p onto T p − t . Dapeng Zhan Reversibility of Whole-Plane SLE

For b ∈ ∂ A p , θ ( t ) := arg g t ( b ) satisfies θ ′ ( t ) = H ( p − t , θ ( t ) − ξ ( t )) , if b ∈ T , θ ′ ( t ) = H I ( p − t , θ ( t ) − ξ ( t )) , if b ∈ T p , where � H ( t , z ) = − i S ( t , e iz ) = P . V . cot 2 ( z − nt ); 2 | n � H I ( t , z ) = H ( t , z + it ) + i = P . V . cot 2 ( z − nt ) . 2 ∤ n Here we set cot 2 ( z ) = cot( z / 2). Dapeng Zhan Reversibility of Whole-Plane SLE

These H and H I are related to Jacobi theta functions. Let θ k ( z , q ), k = 1 , 2 , 3 , 4, be Jacobe theta functions. Let Θ k ( t , z ) = θ k ( z , e − t ) . Then H = 2Θ ′ 1 ; Θ 1 H I = 2Θ ′ 4 . Θ 4 Here we use ′ to denote the derivative wrt the second variable. Dapeng Zhan Reversibility of Whole-Plane SLE

If ξ ( t ) = √ κ B ( t ), we have annulus SLE κ with no marked point. Given a function Λ( t , x ) on (0 , ∞ ) × R with period 2 π in the second variable, we may define the annulus SLE( κ ; Λ) in A p started from a = e ix 0 ∈ T with marked point b ∈ T p by letting ξ ( t ) be the solution of the SDE: d ξ ( t ) = √ κ dB ( t ) + Λ( p − t , ξ ( t ) − arg( g t ( b ))) dt , ξ (0) = x 0 . Dapeng Zhan Reversibility of Whole-Plane SLE

Using conformal maps we can define annulus SLE( κ ; Λ) in A p started from a ∈ T p with marked point b ∈ T . Let b be fixed, and let p → ∞ . Then A p → D = {| z | < 1 } and T p → { 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 SLE κ curves γ 1 and γ 2 that grow towards each other, such that, conditioned on γ j (0 , T j ], the other curve γ 3 − j restricted in � C \ γ j (0 , T j ] is a disc SLE( κ ; Λ) process in this domain, and has γ j ( T j ) 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 on (0 , ∞ ) × R : � � ∂ t Λ = κ 3 − κ 2Λ ′′ + I + (Λ H I ) ′ + ΛΛ ′ . H ′′ (1) 2 If in addition, a disc SLE( κ ; Λ) curve eventually ends at the marked point then using the coupling technique, we obtain the desired overlap 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 of whole-plane SLE κ . Such Λ is uniquely determined by κ . In fact, a byproduct we have is that the reversal of a radial SLE κ 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 √ κ B ( t ) + ρ t . 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 � � ∂ t Λ = κ 3 − κ 2Λ ′′ + H ′′ + (Λ H ) ′ + ΛΛ ′ . (2) 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 : � 3 � ∂ t Γ = κ κ − 1 2Γ ′′ + H I Γ ′ + H ′ I Γ . (3) 2 2 Let Ψ = ΓΘ 4 , and σ = 4 /κ − 1. Then (3) is equivalent to κ ∂ t Ψ = κ 2Ψ ′′ + σ H ′ I Ψ . (4) Dapeng Zhan Reversibility of Whole-Plane SLE

We rescale H I by � π 2 � � H I ( t , z ) = π t , π + z � t H I t z t = P . V . tanh 2 ( z − nt ) , 2 | n where tanh 2 ( z ) = tanh( z / 2). We have H I ( t , · ) → tanh 2 as t → ∞ . Define � π 2 � t , π x 2 � 2 κ t Ψ Ψ( t , x ) = e t x . Then (4) is equivalent to Ψ = κ ′ Ψ ′′ + σ � − ∂ t � � I � Ψ . (5) H 2 Dapeng Zhan Reversibility of Whole-Plane SLE

As t → ∞ , H I ( t , · ) → tanh 2 , so (5) tends to Ψ = κ Ψ ′′ + σ tanh ′ − ∂ t � � 2 � Ψ . (6) 2 Using separation of variables, we find a solution to (6): Ψ ∞ ( t , x ) = e − τ 2 t 2 κ τ � 2 κ cosh 2 ( x ) , where τ ≤ 0 is a root of τ 2 − κ 2 τ = κσ . Dapeng Zhan Reversibility of Whole-Plane SLE