Conformal Restriction and Brownian Motion

We review some of the results related to conformal restriction: the chordal case and the radial case. We describe Brownian intersection exponents, conformal restriction property and SLE, and study their properties.

This shows in particular thatξ is encoded in U.
The function η is called a generalized disconnection exponent and it is a continuous increasing function.

From Brownian motion to Brownian excursion
Consider the simplest exponentξ (1) = 1. Suppose B is a planar BM started from i, then we have Suppose W is a planar BM started from εi, then since (W (ε 2 t)/ε,t ≥ 0) has the same law as B. Consider the law of W conditioned on the event [W [0,t] ⊂ H], we can see that the limit as t → ∞, ε → 0 exists. We call the limit as Brownian excursion and denote its law as µ H (0, ∞). There is another equivalent way to define µ H (0, ∞): Suppose W is a planar BM started from εi, consider the law of W conditioned on the event [W hits R + iR before R]. Let R → ∞, ε → 0, the limit is the same as µ H (0, ∞). (We will discuss Brownian excursion more in detail in Lecture 2). Suppose Z is a Brownian excursion, A is a bounded closed subset ofH such that H \ A is simply connected and 0 ∈ A. Let Φ A be the conformal map from H \ A onto H such that Consider the law of Φ A (Z) conditioned on [Z ∩ A = / 0]. We have that, for any function F, In other words, the Brownian excursion Z satisfies the following conformal restriction property: the law of Φ A (Z) conditioned on [Z ∩ A = / 0] is the same as Z itself. Conformal restriction property is closely related to the half-plane/whole-plane intersection exponents.

Chordal conformal restriction property
Definition 1.1. Let A c be the collection of all bounded closed subset A ⊂H such that 0 ∈ A, A = A ∩ H, and H \ A is simply connected.
Denote Φ A as the conformal map from H \ A onto H such that 3 Φ A (0) = 0, Φ A (∞) = ∞, Φ A (z)/z → 1 as z → ∞. 3 Riemann's Mapping Theorem asserts that, if we have three boundary points, there exists a unique conformal map from H \ A onto H that fixes the three points. More detail in Lecture 3.
We are interested in closed random subset K ofH such that 1. K ∩ R = {0}, K is unbounded, K is connected and H \ K has two connected components 2. ∀λ > 0, λ K has the same law as K 3. For any A ∈ A c , we have that the law of Φ A (K) conditioned on [K ∩ A = / 0] is the same as K.
The combination of the above properties is called chordal conformal restriction property, and the law of such a random set is called chordal restriction measure. Clearly, the "fill-in" of the Brownian excursion satisfies the chordal conformal restriction property. And for n ≥ 1, the "fill-in" of the union of n independent Brownian excursions also satisfies the chordal conformal restriction property. It turns out that there exists only a one-parameter family P(β ) of such probability measures for β ≥ 5/8. (More detail in Lecture 3 and Lecture 4). As expected, for n ≥ 1, the "fill-in" of the union of n independent Brownian excursions corresponds to P(n) and, for β ≥ 5/8, P(β ) can viewed as the law of a packet of β independent Brownian excursions. The chordal restriction measures are closely related to half-plane intersection exponent (more detail in Lecture 4): Suppose K 1 , ..., K p are p independent chordal restriction samples of parameters β 1 , ..., β p respectively. The "fill-in" of the union of these sets p j=1 K j conditioned on the event (viewed as a limit) has the same law as a chordal restriction sample of parameterξ (β 1 , ..., β p ).

Radial conformal restriction property
Definition 1.2. Let A r be the collection of all compact subset A ⊂Ū such that 0 ∈ A, 1 ∈ A, A = A ∩ U, and U \ A is simply connected.
We are interested in closed random subset K ofŪ such that The combination of the above properties is called radial conformal restriction property, and the law of such a random set is called radial restriction measure. It turns out there exists only a two-parameter family Q(α, β ) of such probability measures for (More detail in Lecture 5 and Lecture 6). And the radial restriction measures are closely related to whole-plane intersection exponent (more detail in Lecture 6): Suppose K 1 , ..., K p are p independent radial restriction samples of parameters (ξ (β 1 ), β 1 ), ..., (ξ (β p ), β p ) respectively. The "fill-in" of the union of these sets p j=1 K j conditioned on the event (viewed as a limit) has the same law as a radial restriction sample of parameter (ξ (β 1 , ..., β p ),ξ (β 1 , ..., β p )).

Notations
It is characterized by the following four facts: 3. W has independent increments, i.e. for any t ≥ s ≥ 0, 4. For any t ≥ s ≥ 0, (W t −W s ) is Gaussian with mean zero and variance (t − s).
We list several basic properties of 1-dimensional BM.
From the first basic property, we have that the following corollary.

2-dimensional BM/complex BM
Suppose W 1 ,W 2 are two independent 1-dimensional BMs, then Lemma 2.3. Suppose B is a complex BM and u is a harmonic function, then u(B) is a local martingale.
Proof. By Itô Formula, Proposition 2.4. Suppose D is a domain and f : D → C is a conformal map. Let B be a complex BM starting from z ∈ D, stopped at τ D := inf{t ≥ 0 : B t ∈ D}.
Then the time-changed process f (B) has the same law as a complex BM starting from f (z) stopped at τ f (D) . Precisely, define has the same law as BM starting from f (z) stopped at τ f (D) .
Proof. Write f = u + iv where u, v are harmonic and We have Thus the two coordinates of f (B) are local martingales and the quadratic variation is Thus the two coordinates of Y are independent local martingales with quadratic variation t which implies that Y is a complex BM.

Notations
Let K be the set of all parameterized continuous planar curves γ defined on a time interval [0,t γ ]. K can be viewed as a metric space where the inf is taken over all increasing homeomorphisms θ : If µ is any measure on K , let |µ| = µ(K ) denote the total mass. If 0 < |µ| < ∞, let µ = µ/|µ| be µ normalized to be a probability measure. Let M denote the set of finite Borel measures on K . This is a metric space under Prohorov metric [Bil99, Section 6]. To show that a sequence of finite measures µ n converges to a finite measure µ, it suffices to show that |µ n | → |µ|, µ n → µ .
If D is a domain, we say that γ is in D if γ(0,t γ ) ⊂ D, and let K (D) be the set of γ ∈ K that are in D. Note that, we do not require the endpoints of γ to be in D. Suppose f : D → D is a conformal map and γ ∈ K (D). Let If µ is a measure supported on the set of γ in K (D) such that f • γ is well-defined and in K (D ), then

From interior point to interior point
Let µ(z, ·;t) denote the law of complex BM (B s , 0 ≤ s ≤ t) starting from z. We can write where dA(w) denotes the area measure and µ(z, w;t) is a measure on continuous curve from z to w. The total mass of µ(z, w;t) is ). (2. 2) The normalized measure µ (z, w;t) = µ(z, w;t)/|µ(z, w;t)| is a probability measure, and it is called a Brownian bridge from z to w in time t.
Remark 2.5. |µ(z, w;t)| is also called heat kernel and Equation (2.2) can be obtained through The measure µ(z, w) is defined by This is a σ -finite infinite measure. If D is a domain and z, w ∈ D, define µ D (z, w) to be µ(z, w) restricted to curves stayed in D. If z = w, and D is a domain such a BM in D eventually exits D, then |µ D (z, w)| < ∞. Define Green's function G D (z, w) = π|µ D (z, w)|.

From interior point to boundary point
Suppose D is a connected domain. Let B be a BM starting from z ∈ D and stopped at Define µ D (z, ∂ D) to be the law of (B s , 0 ≤ s ≤ τ D ). If D has nice boundary (i.e. ∂ D is piecewise analytic), we can write where dw is the length measure and µ D (z, w) is a measure on continuous curves from z to w. Define Poisson's kernel In particular, H U (0, w) = 1/(2π). The normalized measure µ D (z, w) = µ D (z, w)/|µ D (z, w)| can also be viewed as the law of BM conditioned to exit D at w when w is a nice boundary point: Proposition 2.7. (Conformal Covariance) Suppose D is a connected domain with nice boundary, z ∈ D,w ∈ ∂ D is a nice boundary point. Let f : D → D be a conformal map. Then In particular, Relation between the two Proposition 2.8. Suppose D is a connected domain with nice boundary, z ∈ D, w ∈ ∂ D is a nice boundary point. Let n w denote the inward normal at w, then In particular,

Brownian excursion
Suppose D is a connected domain with nice boundary and z, w are two distinct nice boundary points. Define the measure on Brownian path from z to w in D: The normalized measure µ D (z, w) is called Brownian excursion measure in D with two end points z, w ∈ ∂ D. Note that Proposition 2.9. (Conformal Covariance) Suppose that f : D → D is a conformal map, and z, w ∈ ∂ D, f (z), f (w) ∈ ∂ f (D) are nice boundary points. Then In particular, The following proposition is an equivalent expression of the conformal restriction property of Brownian excursion we discussed in Subsection 1.2.
Proposition 2.10. Suppose A ∈ A c and Φ A is the conformal map defined in Definition 1.1. Let e be a Brownian excursion whose law is µ H (0, ∞). Then Proof. Although µ H (0, ∞) has zero total mass, the normalized measure can still be defined through the limit procedure: Corollary 2.11. Suppose e 1 , ..., e n are n independent Brownian excursion with law µ H (0, ∞), denote Σ = ∪ n j=1 e j , then for any A ∈ A c , Corollary 2.12. Let e be a Brownian excursion with law µ H (x, y) where x, y ∈ R, x = y. Then, for any closed subset A ⊂H such that x, y ∈ A and H \ A is simply connected, we have that where Φ is any conformal map from H \ A onto H that fixes x and y.
Homework: Prove this corollary and check that the quantity Φ (x)Φ (y) is unique although Φ is not unique.
Definition 2.13. Suppose D has nice boundary, then Brownian excursion measure is defined as Generally, if I is a subsect of ∂ D, define Theorem 2.15. Let (e j , j ∈ J) be a Poisson point process with intensity πβ µ exc H,R − for some β > 0. Set Σ = ∪ j e j . For any A ∈ A c such that A ∩ R ⊂ (0, ∞), we have that we need to show that This will be obtained by two steps: First, there exists a constant c such that For the first step, we need to introduce a set A · B: Suppose A, B ∈ A c such that A ∩ R ⊂ (0, ∞) and B ∩ R ⊂ (0, ∞). Define (see Figure 2.1) (2.5) For the Brownian excursion measure, we have Next, we will decide the constant. where Φ x,y is any conformal map from H \ A onto H that fixes x and y. Define the Mobius transformation It is not clear to see how this double integral would give c log(Φ A (a)Φ A (b)). However, we only need to decide the the constant c which is much easier. Suppose I = [−ε, 0], and set a 1 = Φ A (0) and a 2 = Φ A (0)/2, we have that Compare the two expansions, the constant c = −1/π.
We say that two loops γ, γ are equivalent if for some r, we have γ = θ r γ. DenoteK u as the set of unrooted loops, i.e. the equivalent classes. We will define Brownian loop measure on unrooted loops. Recall that µ(z, ·;t) denotes the law of complex BM (B s , 0 ≤ s ≤ t) and µ(z, ·;t) = µ(z, w;t)dA(w).
Now we are interested in loops, i.e. µ(z, z;t) where the path starts from z and returns back to z. We have that We define Brownian loop measure µ loop by (2.7) The term 1/t γ corresponds to averaging over the root and µ loop is defined on unrooted loops. If D is a domain, define µ loop D to be µ loop restricted to the curves totally contained in D.
Proof. We call a Borel measurable function T :K → [0, ∞) a unit weight if, for any γ ∈K , we have One example is T (γ) = 1/t γ . For any unit weight T , since µ loop is defined on unrooted loops, we have that (2.8) We will give a short proof of Equation (2.8). For any function G onK , it induces a function G u onK u : For any function F onK u , we define It is easy to see that G 1 u = G 2 u = F. Thus Compare the two, we get Equation (2.8). Now we are ready to prove the proposition. Define T f : for any γ ∈K , Recall the time change in Equation (2.1), we can see that T f is a unit weight: Thus, Denote µ loop U,0 as µ loop U restricted to the loops surrounding the origin.
Theorem 2.17. Let (l j , j ∈ J) be a Poisson point process with intensity α µ loop U,0 for some α > 0. Set Σ = ∪ j l j . For any closed subset A ⊂Ū such that 0 ∈ A, U \ A is simply connected, we have that Proof. Since The same as before, this can be obtained by two steps: First, there exists a constant c such that For the first step, it can be proved in the similar way as the proof of the first step of Theorem 2.15, and the precise proof can be found in [Wer08, Lemma 4]. But for the second step, it is more complicate. We omit this part and the interested readers can consult [Wer08, SW12, LW04].

Introduction
Schramm Lowner Evolution (SLE for short) is introduced by Oded Schramm in 1999 [Sch00] as the candidates of the scaling limits of discrete statistical physics models. We will take percolation as an example. Suppose D is a domain and we have a discrete lattice of size ε inside D, say the triangular lattice εT ∩ D. The critical percolation on the discrete lattice is the following: At each vertex of the lattice, there is a random variable which is black or white with equal probability 1/2. All these random variables are independent. We can see that there are interfaces separating black vertices from white vertices. To be precise, let us fix two distinct boundary points a, b ∈ ∂ D. Denote ∂ L (resp. ∂ R ) as the part of the boundary from a to b clockwise (resp. counterclockwise). We fix all vertices on ∂ L (resp. ∂ R ) to be white (resp. black). And then sample independent black/white random variables at the vertices inside D. Then there exists a unique interface from a to b separating black vertices from white vertices (see Figure 3.1). We denote this interface as γ ε , and call it as the critical percolation interface in D from a to b. It is worthwhile to point out the domain Markov property in this discrete model: Starting from a, we move along γ ε and stopped at some point γ ε (n). Given L = (γ ε (1), ..., γ ε (n)), the future part of γ ε has the same law as the critical percolation interface in D \ L from γ ε (n) to b.
People believe that the discrete interface γ ε will converge to some continuous path in D from a to b as ε goes to zero. Assume this is true and suppose γ is the limit continuous curve in D from a to b. Then we would expect that the limit should satisfies the following two properties: Conformal Invariance and Domain Markov Property which is the continuous analog of discrete domain Markov property. SLE curves are introduced from this motivation: chordal SLE curves are random curves in simply connected domains connecting two boundary points such that they satisfy: (see Figure 3.2) • Conformal Invariance: γ is an SLE curve in D from a to b, ϕ is a conformal map, then ϕ(γ) has the same law as an SLE curve in ϕ(D) from ϕ(a) to ϕ(b).
• Domain Markov Property: γ is an SLE curve in D from a to b, given γ([0,t]), γ([t, ∞)) has the same law as an SLE curve in The following of the lecture is organized as follows: In Subsection 3.2, we introduce one time parameterization of continuous curves, called Loewner chain, that is suitable to describe the domain Markov property of the curves. In Subsection 3.3, we introduce the definition of chordal SLE and discuss its basic properties.
Without loss of generality, we choose to work in the upper half-plane H and suppose the two boundary points are 0 and ∞.

Half-plane capacity
We call a compact subset K ofH a hull if H = H \ K is simply connected. Riemann's mapping theorem asserts that there exists a conformal map Ψ from H onto H that Ψ(∞) = ∞. In fact, if Ψ is such a map, then cΨ + c for c > 0, c ∈ R is also a map from H onto ∞ fixing ∞. We choose to fix the two-degree freedom in the following way. Since Ψ can be expanded near ∞: Furthermore, since Ψ preserves the real axis near ∞, all coefficients b j are real. Hence, for each K, there exists a unique conformal map Ψ from H = H \ K onto H such that We call such a conformal map as the conformal map from H = H \ K onto H normalized at ∞, and denote it as Ψ K . In particular, there exists a real a = a(K) such that This number a(K) is a way to measure the size of K.
Lemma 3.1. a(K) is a non-negative increasing function of the set K.
Proof. We first show that a is non-negative. Suppose that Z = X + iY is a complex BM starting from Z 0 = iy for some y > 0 large (so that iy ∈ H = H \ K) and stopped at its first exit time τ of H. Let Ψ be the conformal map from H onto H normalized at the infinity, then ℑ(Ψ(z) − z) is a bounded harmonic function in H. The martingale stopping theorem therefore shows that Next we show that a is increasing. Suppose K, K are hulls and K ⊂ K . Let Ψ 1 = Ψ K , and let Ψ 2 be the conformal map from H \ Ψ K (K \ K) onto H normalized at infinity. Then Ψ K = Ψ 2 • Ψ 1 , and We call a(K) as the capacity of K in H seen from ∞ or half-plane capacity. Here are several simple facts Homework: check these two items.

Loewner chain
Suppose that (W t ,t ≥ 0) is a continuous real function with W 0 = 0. For each z ∈H, define the function g t (z) as the solution to the ODE This is well-defined as long as g t (z) −W t does not hit 0. Define This is the largest time up to which g t (z) is well-defined. Set We can check that • g t is a conformal map from H t onto H normalized at ∞.
• For each t, In other words, a(K t ) = t.
The family (K t ,t ≥ 0) is called the Loewner chain driven by (W t ,t ≥ 0). One example:

Definition
Chordal SLE κ for κ ≥ 0 is the Loewner chain driven by W t = √ κB t where B is a 1-dimensional BM starting from 0.
3. The law of SLE κ is symmetric with respect to the imaginary axis.
Proof. Proof of scale-invariance: Since W is scale-invariant, i.e. for any λ > 0, the process Thus (K λt / √ λ ,t ≥ 0) has the same law as K.
Proof of domain Markov property: Since BM is a stong Markov process with independent increments. Thus for any stopping time T , the process and has the same law as K.
Proof of symmetry: W and −W has the same law. The proof of this proposition is difficult, we will omit it in the lecture. The interested readers could consult [RS05].
Restriction property of SLE 8/3 In this part, we will compute the probability of SLE 8/3 γ to avoid a set A ∈ A c . To this end, we need to analyze the behavior of the imageγ = Φ A (γ): Proposition 3.4. When κ = 8/3, the process .
A time-change shows that Plugin Equation (3.1), we have that . (3.2) We can first decide ∂ t a: multiply h t (z) − h t (W t ) to both sides of Equation (3.2), and then let z → W t , we have Then Equation (3.2) becomes Proof. We may assume A has smooth boundary. Set Roughly speaking, when T = ∞, g t (A) will be far away from W t as t → ∞ and thus the probability for e to avoid g t (A) converges to 1; whereas, when T < ∞, g t (A) will be very close to W t as t → ∞. (See [LSW03] for details.) Since M converges in L 1 and a.s. when t → T , we have that

Setup for chordal restriction sample
Let Ω be the collection of closed sets K ofH such that K ∩ R = {0}, K is unbounded, K is connected and H \ K has two connected components.
And recall A c in Definition 1.1. We endow Ω with the σ -field generated by the events [K ∈ Ω : K ∩ A = / 0] where A ∈ A c . This family of events is closed under finite intersection, so that a probability measure on Ω is characterized by the values of P[K ∩ A = / 0] for A ∈ A c : Let P, P are two probability measures on Definition 4.1. A probability measure P on Ω is said to satisfy chordal conformal restriction property, if the following is true: 1. For any λ > 0, λ K has the same law as K; 2. For any A ∈ A c , Φ A (K) conditioned on [K ∩ A = / 0] has the same law as K.
1. (Characterization) A chordal restriction measure is fully characterized by a positive real β > 0 such that, for every A ∈ A c , We denote the corresponding chordal restriction measure as P(β ).
Homework: Suppose that K is scale-invariant and satisfies Equation (4.1) for every A ∈ A c , then check that K satisfies chordal conformal restriction property.
Proof of Theorem 4.2. Characterization. We will omit the details related to regularities and only keep the details that are related to the key idea. Fix x ∈ R \ {0} and let ε > 0. We claim that the probability decays like ε 2 as ε goes to zero. And the limit exists which we denote as λ (x). Furthermore, λ (x) ∈ (0, ∞). Since K is scale-invariant, we have that, for any y > 0, Since λ is an even function, we have that, there exists c > 0 such that Since there is only one-degree of freedom, when K satisfies chordal restriction property, we must have that Equation (4.1) holds for some β > 0.
Denote f x,ε = ΦŪ (x,ε) . In fact, Note that, and that In the following of the lecture, we will first show that P(β ) does not exist for β < 5/8 and then construct all P(β ) for β > 5/8.

Chordal SLE κ (ρ) process
Definition Suppose κ > 0, ρ > −2. Chordal SLE κ (ρ) process is the Loewner chain driven by W which is the solution to the following SDE: The evolution is well-defined at times when W t > O t , but a bit delicate when W t = O t . We first show the existence of the solution to this SDE. Define Z t as the solution to the Bessel equation In other words, Z is √ κ times a Bessel process of dimension And this process is well-defined for all ρ > −2. Note also that, for all t ≥ 0, Then Clearly, (W t , O t ) is a solution to Equation (4.2). And when ρ = 0, we get the ordinary SLE κ .
Second, we explain the geometric meaning of the process (O t ,W t ). Recall Suppose (K t ,t ≥ 0) is the Loewner chain generated by W , then g t is the conformal map from H \ K t onto H normalized at ∞. W t is the image of the tip, and O t is the image of the leftmost point of R ∩ K t . See • It is scale-invariant: for any λ > 0, (λ −1 K λ 2 t ,t ≥ 0) has the same law as K.
• If ρ ≥ κ/2 − 2, the dimension of the Bessel process Z t = W t − O t is greater than 2 and Z does not hit zero, thus almost surely  Theorem 4.4. Fix ρ > −2. Let (K t ,t ≥ 0) be the hulls of chordal SLE 8/3 (ρ) and K = ∪ t≥0 K t . Then K satisfies the right-sided restriction property with exponent In other words, for every A ∈ A c such that A ∩ R ⊂ (0, ∞), we have Proof. The definitions of g t ,g t , h t are recalled in Figure 4.2. Set T = inf{t : K t ∩ A = / 0}, and define, for t < T , . 6 When ρ > 0, W t gets a push away from O t , the curve is repelled from R − . When ρ < 0, the curve is attracted to R − . When ρ < κ/2 − 2, the attraction is big enough that the curve touches R − . Then (M t ,t < T ) is a local martingale [LSW03, Lemma 8.9]: Combine these, M is a local martingale. Since In fact, there exists δ > 0 such that M t ≤ h t (W t ) δ . (We omit the proof of this point, details could be found in [LSW03, Lemma 8.10]). In particular, M t ≤ 1 and (M t ,t < T ) is a bounded martingale.

Setup for right-sided restriction property
Let Ω + be the collection of closed sets K ofH such that K ∩ R = (−∞, 0], K is connected and H \ K is connected. And recall A c in Definition 1.1. And let A + c denote the set of A ∈ A c such that A ∩ R ⊂ (0, ∞). We endow Ω + with the σ -field generated by the events [K ∈ Ω + : Definition 4.5. A probability measure P on Ω + is said to satisfy right-sided restriction property, if the following is true 1. For any λ > 0, λ K has the same law as K; 2. For any A ∈ A + c , Φ A (K) conditioned on [K ∩ A = / 0] has the same law as K.
Similar to the proof of Theorem 4.2, we know that, if P satisfies the right-sided restriction property, then there exists β > 0 such that Remark 4.6. Theorem 4.4 states that SLE 8/3 (ρ) has the same law as the right boundary of the right-sided restriction sample with exponent β which is related to ρ through Equation (4.3). Note that when ρ spans (−2, ∞), β spans (0, ∞). In particular, Theorem 4.4 also states the existence of right-sided restriction measure for all β > 0.
On the one hand, K is symmetric with respect to the imaginary axis, thus the probability of i staying to the right of γ is less than 1/2.
On the other hand, since ρ < 0, the probability of i staying to the right of γ is strictly larger than the probability of i staying to the right of SLE 8/3 which equals 1/2. Contradiction.

Construction of P(β ) for β > 5/8
In the previous definition of SLE κ (ρ) process, there is a repulsion (when ρ > 0) or attraction (when ρ < 0) from R − . We will denote this process as SLE L κ (ρ). And symmetrically, we denote SLE R κ (ρ) as the same process only except that the repulsion or attraction is from R + . Precisely, SLE R κ (ρ) is the Loewner chain driven by W which is the solution to the following SDE: can also be viewed as the image of SLE L κ (ρ) under the reflection with respective to the imaginary axis.
From Theorem 4.4, we know that SLE L 8/3 (ρ) satisfies right-sided restriction property and SLE R 8/3 (ρ) satisfies left-sided restriction property. The idea to construct K whose law is P(β ) for β > 5/8 is the following: we first run an SLE L 8/3 (ρ) as the right-boundary of K, and then given the right boundary, we run the left boundary according to the conditional law.
Proposition 4.9. Fix β > 5/8, and ρ = ρ(β ) > 0 where ρ(β ) is given by Equation (4.4). Suppose γ R is a chordal SLE L 8/3 (ρ) process inH from 0 to ∞. Given γ R , in the left-connected component of H \ γ R , sample an SLE R 8/3 (ρ − 2) from 0 to ∞ which is denoted as γ L . Let K be the closure of the union of the domains between γ L and γ R . Then K has the law of P(β ).

Proof. We only need to check, for all
From the construction, we know that this is true for A ∈ A + c . We only need to prove it for A ∈ A c such that A ∩ R ⊂ (−∞, 0). Let (g t ,t ≥ 0) be the solution of the Loewner chain for the process γ R and (O t ,W t ,t ≥ 0) be the solution of the SDE (4.2). Set T = inf{t : γ R (t) ∈ A}. And,for t < T , let h t be the conformal map from H \ g t (A) onto H normalized at ∞. See Figure 4.3. Recall that is a local martingale. And that Since ρ > 0, we have that M t ≤ h t (W t ) β and thus M is a bounded martingale. If T < ∞, then h t (W t ) → 0, and M t → 0 as t → T.
In the following theorem, we will consider the law of K 1 , ..., K p conditioned on "non-intersection". Since the event of "non-intersection" has zero probability, we need to explain the precise meaning: the conditioned law would be obtained through a limiting procedure: first consider the law of K 1 , ..., K p conditioned on and then let R → ∞ and ε → 0.
For Proposition 4.10 and Theorem 4.11, we only need to show the results for p = 2 and other p can be proved by induction. Proposition 4.10 for p = 2 is a direct consequence of the following lemma.
Lemma 4.12. Suppose K is a right-sided restriction sample with exponent β > 0. Let γ be an independent chordal SLE R 8/3 (ρ) process for some ρ > −2. Fix t > 0 and let ε > 0 be small, we have Note that, if β 1 = β , β 2 = (3ρ 2 + 16ρ + 20)/32, we have Proof. Let (g t ,t ≥ 0) be the Loewner chain for γ and (O t ,W t ) be the solution to the SDE. Precisely, Given γ[0,t], since K satisfies right-sided restriction property, we have that One can check that M is a local martingale. Thus Proof of Theorem 4.11. Assume p = 2. For any A ∈ A c , we need to estimate the following probability for ε > 0 small: Since K i satisfies chordal conformal restriction property, conditioned on [K i ∩ A = / 0], the conditional law of Φ A (K i ) has the same law as K i , for i = 1, 2. Thus lim R→∞,ε→0 Fig. 4.4: K L is a right-sided restriction sample with exponent β L > 0 and K R is a left-sided restriction sample with exponent β R > 0. γ is an SLE κ (ρ L ; ρ R ) with force points (x L ; x R ) with x L < 0 < x R .

Related calculation
Chordal SLE κ (ρ L ; ρ R ) process is the Loewner chain driven by W which is the solution to the following SDE: There exists piecewise unique solution to the above SDE. And there exists almost surely a continuous curve γ inH from 0 to ∞ associated to the SLE κ (ρ L ; ρ R ) process with force points (x L ; x R ). Note that, for small time t when x L , x R are not swallowed by K t , x L (resp. x R ) is the preimage of O L t (resp. O R t ) under g t .
It is worthwhile to point out the relation between SLE κ (ρ L ; ρ R ) processes with different ρ's. Fix Then M is a local martingale under the measure of SLE κ (ρ L ; ρ R ) process (see [SW05,Theorem 6]). Moreover, the measure weighted by M/M 0 (with an appropriate stopping time) is the same as the law of SLE κ (ρ L ;ρ R ) process.

General calculation related to Lemma 4.12
In fact, the estimate in Lemma 4.12 is a special case of the following estimate. Suppose K L is a right-sided restriction sample with exponent β L > 0, and K R is a left-sided restriction sample with exponent β R > 0. Let γ be a chordal SLE κ (ρ L ; ρ R ) in H from 0 to ∞ with force points (x L ; x R ) where κ > 0, ρ L > −2, ρ R > −2, and x L < 0 < x R . See Figure 4.4. Suppose K L , K R , γ are independent and let ρ L > ρ L ,ρ R > ρ R be the solutions to the equations Then, for fixed time t > 0, Proof. Let (g t ,t ≥ 0) be the Loewner chain for γ. Given γ[0,t], we have that Note that is a local martingale. Thus,

Radial Loewner chain
Capacity Consider a compact subset K ofŪ such that 0 ∈ U \ K and U \ K is simply connected. Then there exists a unique conformal map g K from U \ K onto U normalized at the origin, i.e. g K (0) = 0, g K (0) > 0. We call a(K) := log g K (0) as the capacity of K in U seen from the origin.
Lemma 5.1. a is non-negative increasing function.
Proof. a is non-negative: Denote U = U \ K. log g K (z)/z is an analytic function on U \ {0} and the origin is removable: we can define the function equals log g K (0) at the origin. Then h(z) = log |g K (z)/z| is a harmonic function on U. Thus it attains its min on ∂U. For z ∈ ∂ D, h(z) ≥ 0. Therefore h(z) ≥ 0 for all z ∈ U. In particular, h(0) ≥ 0.
a is increasing: Suppose K ⊂ K . Define g 1 = g K and let g 2 be the conformal map from U \ g K (K \ K) onto U normalized at the origin. Then g K = g 2 • g 1 . Thus a(K ) = log g 2 (0) + log g 1 (0) ≥ log g 1 (0) = a(K).
Remark 5.2. If we denote d(0, K) as Euclidean distance from the origin to K, by Koebe 1/4-Theorem, we have that

Loewner chain
Suppose (W t ,t ≥ 0) is a continuous real function with W 0 = 0. Define for z ∈Ū, the function g t (z) as the solution to the ODE ∂ t g t (z) = g t (z) e iW t + g t (z) e iW t − g t (z) , g 0 (z) = z.
The solution is well-defined as long as e iW t − g t (z) does not hit zero. Define Define This is the largest time up to which g t (z) is well-defined. Set We can check that • g t is a conformal map from U t onto U normalized at the origin.
• For each t, g t (0) = e t . In other words, a(K t ) = t.
The family (K t ,t ≥ 0) is called the radial Loewner chain driven by (W t ,t ≥ 0).

Definition
Radial SLE κ for κ ≥ 0 is the radial Loewner chain driven by W t = √ κB t where B is a 1-dimensional BM starting from B 0 = 0.
Lemma 5.3. Radial SLE satisfies domain Markov property: For any stopping time T , the process (g T (K t+T \ K T )e −iW T ,t ≥ 0) is independent of (K s , 0 ≤ s ≤ T ) and has the same law as K.  Restriction property of radial SLE 8/3 Recall Definition 1.2. Suppose A ∈ A r , and Φ A is the conformal map from U \ A onto U such that Φ A (0) = 0, Φ A (1) = 1. Let γ be a radial SLE 8/3 , we will compute the probability Similar as the chordal case, define .
onto U normalized at the origin. Defineg t as the conformal map from U \γ[0,t] onto U normalized at the origin and h t as the conformal map from U \ g t (A) onto U such that Equation (5.1) holds. See Figure 5.
where log denotes the branch of the logarithm such that −i log h t (e iW t ) = W t . Then Define A simple time change shows that We can first decide ∂ t a: multiply e iW t − h t (z) to both sides of Equation (5.2) and then let z → e iW t . We have Then Equation (5.2) becomes Plugin the relation h t (e iz ) = e iφ t (z) , we have that Thus dh t (e iW t ) = dφ t (W t ) = 8 3 X 2 dB t + ( For the term h t (0), we have that Combine Equations (5.4) and (5.5), we have that onto U normalized at the origin. Defineg t as the conformal map from U \γ[0,t] onto U normalized at the origin and h t as the conformal map from U \ g t (A) onto U such that h t • g t =g t • Φ A .
Theorem 5.6. Suppose γ is a radial SLE 8/3 in U from 1 to 0. Then for any A ∈ A r , we have Proof. Suppose M is the local martingale defined in Proposition 5.5. Note that

Radial SLE κ (ρ) process
Fix κ > 0, ρ > −2. Radial SLE κ (ρ) process is the radial Loewner chain driven by W which is the solution to the following SDE: When κ > 0, ρ > −2, there exists a piecewise unique solution to the SDE (5.6). There exists almost surely a continuous curve γ inŪ from 1 to 0 so that (K t ,t ≥ 0) is generated by γ. When κ ∈ [0, 4] and ρ ≥ κ/2 − 2, γ is a simple curve and K t = γ[0,t]. When κ ∈ [0, 4], ρ ∈ (−2, κ/2 − 2), γ almost surely hits the boundary. The tip γ(t) is the preimage of e iW t under g t . And e ix (when it is not swallowed by K t ) is the preimage of e iO t under g t . When e ix is swallowed by K t , then the preimage of e iO t under g t is the last point (before time t) on the curve that is on the boundary. See Figure 5.3.   Let x → 0+ (resp. x → 2π−), the process has a limit, and we call this limit as radial SLE R κ (ρ) (resp. SLE L κ (ρ)) inŪ from 1 to 0. Suppose γ is an SLE L 8/3 (ρ) process for some ρ > −2. For any A ∈ A r , we want to analyze the image of γ under Φ A . Define T = inf{t : γ(t) ∈ A}. For t < T , Φ A is the conformal map from U \ A onto U with Φ A (0) = 0, Φ A (1) = 1, g t is the conformal map from U \ γ[0,t] onto U normalized at the origin. Defineg t as the conformal map from U \γ[0,t] onto U normalized at the origin and h t as the conformal map from Proposition 5.7. Define Then M is a local martingale. Note that, if we set onto U normalized at the origin. Defineg t as the conformal map from U \γ[0,t] onto U normalized at the origin and h t as the conformal map from Proof. Define φ t (z) = −i log h t (e iz ) where log denotes the branch of the logarithm such that −i log h t (e iW t ) = W t . Then To simplify the notations, we set . By Itô formula, we have that Combine these, M is a local martingale.

Relation between radial SLE and chordal SLE
Roughly speaking, chordal SLE is the limit of radial SLE when we let the interior target point go towards a boundary target point. Precisely, for z ∈ H, suppose ϕ z is the Mobius transformation from U onto H that sends 0 to z and 1 to 0. We define radial SLE in H from 0 to z as the image of radial SLE in U from 1 to 0 under ϕ z . Then, as y → ∞, radial SLE κ in H from 0 to iy will converge to chordal SLE κ (under an appropriate topology).
Proof. Fix R > 0, suppose y > 0 large. Let γ iy be a radial SLE κ in H from 0 to iy and let γ be a chordal SLE κ in H from 0 to ∞. Let τ R be the first time that the curve exits U(0, R). Set ρ = 6 − κ, and define One can check that M is a local martingale under the law of γ (see [SW05,Theorem 6]). Moreover, the measure weighted by M(iy)/M 0 (iy) is the same as the law of γ iy (after time-change). In particular, the Radon-Nikodym between the law of γ iy [0, τ R ] and the law of γ[0, τ R ] is given by which converges to 1 as y → ∞.

Setup for radial restriction sample
Let Ω be the collection of compact subset K ofŪ such that Recall A r in Definition 1.2. Endow Ω with the σ -field generated by the events [K ∈ Ω : K ∩ A = / 0] where A ∈ A r . Clearly, a probability measure P on Ω is characterized by the values of P[K ∩ A = / 0] for A ∈ A r .
Definition 6.1. A probability measure P on Ω is said to satisfy radial restriction property if the following is true: ] has the same law as K.
Theorem 6.2. 1. (Characterization) A radial restriction measure is characterized by a pair of real numbers (α, β ) such that, for every A ∈ A r , We denote the corresponding radial restriction measure as Q(α, β ).

(Existence) The measure Q(α, β ) exists if and only if
Homework: Suppose that K satisfies Equation (6.1) for any A ∈ A r , then K satisfies radial restriction property.
Remark 6.4. In Equation (6.1), we have that |Φ A (0)| ≥ 1 and Φ A (1) ≤ 1. Since β is positive, we have Φ A (1) β ≤ 1. But α can be negative or positive, so that |Φ A (0)| α can be greater than 1. The product |Φ A (0)| α Φ A (1) β is always less than 1 which is guaranteed by the condition that α ≤ ξ (β ). (In fact, we always have |Φ A (0)|Φ A (1) 2 ≤ 1.) Remark 6.5. While the class of chordal restriction measures is characterized by one single parameter β ≥ 5/8, the class of radial restriction measures involves the additional parameter α. This is due to the fact that the radial restriction property is in a sense weaker than the chordal one: the chordal restriction samples in H are scale-invariant, while the radial ones are not.

Proof of Theorem 6.2. Characterization.
It is easier to carry out the calculation in the upper half-plane H instead of U. Suppose that K satisfies radial restriction property in H with interior point i and the boundary point 0. In other words, K is the image of radial restriction sample in U under the conformal map ϕ(z) = i(1 − z)/(1 + z). The proof consists of six steps. We will omit the proof of regularities and keep the proof that is related to the key idea.
Step 3. Fix x, y ∈ R \ {0}. We estimate the probability of Clearly, it decays like ε 2 δ 2 as ε, δ go to zero. Denote f x,ε as the conformal map from H \ U(x, ε) onto H that fixes 0 and i. In fact, we can write out the exact expression of f x,ε : suppose 0 < ε < |x|. Then where a = ℜ(g x,ε (i)), b = ℑ(g x,ε (i)), c = g x,ε (0). Then f x,ε is the conformal map from H \ U(x, ε) onto H that preserves 0 and i.
Proof. By radial restriction property, we have that Step 4. We are allowed to exchange the order in taking the limits in Lemma 6.6. In other words, we have that λ (y)F(x, y) + 2λ (y)G(x, y) = λ (x)F(y, x) + 2λ (x)G(y, x). (6.2) We call this equation as Commutation Relation.
Step 5. Decide the expression of the function λ .
Lemma 6.7. There exists two constants c 0 ≥ 0, c 2 ≥ 0 such that Proof. In Commutation Relation (6.2), let y → x, we obtain a differential equation for the function λ . To write it in a better way, define then the differential equation becomes P (x) = 0.
And we know that λ is an even function. Thus, there exist three constants c 0 , c 1 , c 2 such that We plugin the expression of λ in Commutation Relation (6.2) and take x > 0 > y, then we get c 1 = 0. Since λ is positive, we have c 0 ≥ 0, c 2 ≥ 0.

Several basic observations of radial restriction property
Recall a result for Brownian loop: Theorem 2.17. Let (l j , j ∈ J) be a Poisson point process with intensity cµ loop U,0 for some c > 0. Set Σ = ∪ j l j . Then we have that, for any A ∈ A r , Suppose K 0 is a radial restriction sample whose law is Q(α 0 , β 0 ). Take K as the "fill-in" of the union of Σ and K 0 , then clearly, K has the law of Q(α 0 − c, β 0 ). Thus we derived the following lemma.
Another basic observation is that Q(α, β ) does not exist when β < 5/8. Suppose K 0 is a radial restriction sample with law Q(α, β ). For any interior point z ∈ H, we define K z as the image of K 0 under the Mobius transformation from U onto H such that sends 1 to 0 and 0 to z. Similar as the relation between radial SLE and chordal SLE in Subsection 5.4, if we let z → ∞, K z converges weakly toward some probability measure, and the limit measure satisfies chordal restriction property with exponent β , thus β ≥ 5/8. 6.4 Construction of radial restriction measure Q(ξ (β ), β ) for β > 5/8 The construction of Q(ξ (β ), β ) is very similar to the construction of P(β ). Proposition 6.9. Fix β > 5/8 and let Let γ R be a radial SLE L 8/3 (ρ) inŪ from 1 to 0. Given γ R , let γ L be an independent chordal SLE R 8/3 (ρ − 2) in U \ γ R from 1 − to 0. Let K be the closure of the union of the domains between γ L and γ R . See Figure  6.1. Then the law of K is Q(ξ (β ), β ). In particular, the origin is almost surely on the boundary of K.  Fig. 6.1: γ R is a radial SLE L 8/3 (ρ) in U from 1 to 0. Conditioned on γ R , γ L is a chordal SLE R 8/3 (ρ − 2) in U \ γ R from 1 − to 0. K is the closure of the union of domains between the two curves.
In the following theorem, we will consider the law of K 1 , ..., K p conditioned on "non-intersection". Since the event of "non-intersection" has zero probability, we need to explain the precise meaning: the conditioned law would be obtained through a limiting procedure: first consider the law of K 1 , ..., K p conditioned on [(e ix j 1 K j 1 ∩ A r ) ∩ (e ix j 2 K j 2 ∩ A r ) = / 0, 1 ≤ j 1 < j 2 ≤ p] and then let r → 0 and ε → 0.
We only need to show the results for p = 2 and other p can be proved by induction. When p = 2, Proposition 6.10 is a direct consequence of the following lemma.
Proof. Let (g t ,t ≥ 0) be the Loewner chain for γ and (O t ,W t ) be the solution to the SDE. Precisely, ∂ t g t (z) = g t (z) e iW t + g t (z) e iW t − g t (z) , g 0 (z) = z; Define M t = e t(q−q) g t (e iε ) β |g t (e iε ) − e iW t | 3ρ/8 |g t (e iε ) − e iO t | 3ρρ/16 . Proof of Theorem 6.11. Assume p = 2. For any A ∈ A r , we need to estimate the following probability

Related calculation
Radial SLE κ (ρ L ; ρ R ) process Suppose κ > 0, ρ L > −2, ρ R > −2 and 0 < x R < x L < 2π. Radial SLE κ (ρ L ; ρ R ) process with force points (e ix L ; e ix R ) is the Loewner chain driven by W which is the solution to the following SDE: There exists piecewise unique solution to the above SDE. And there exists almost surely a continuous curve γ inŪ from 1 to 0 associated to the radial SLE κ (ρ L ; ρ R ) process with force points (e ix L ; e ix R ). Note that, for small time t when e ix L , e ix R are not swallowed by K t , e ix L (resp. e ix R ) is the preimage of e iO L t (resp. e iO R t ) under g t . It is worthwhile to point out the relation between radial SLE κ (ρ L ; ρ R ) processes with different ρ's. Fix κ > 0, 0 < x R < x L < 2π. Let ρ L > −2, ρ R > −2,ρ L > −2,ρ R > −2. Set Then M is a local martingale under the measure of radial SLE κ (ρ L ; ρ R ) process (see [SW05,Equation (9)]). Moreover, the measure weighted by M/M 0 (with an appropriate stopping time) is the same as the law of radial SLE κ (ρ L ;ρ R ) process.
Note that is a local martingale. Thus,