aa r X i v : . [ m a t h . P R ] M a y Two-curve Green’s function for 2-SLE: the boundary case
Dapeng ZhanMichigan State UniversityMay 5, 2020
Abstract
We prove that for κ ∈ (0 , η , η ) is a 2-SLE κ pair in a simply connected domain D with an analytic boundary point z , then lim r → + r − α P [dist( z , η j ) < r, j = 1 ,
2] convergesto a positive number for some α >
0, which is called the two-curve Green’s function. Theexponent α equals κ − κ −
1) depending on whether z is one of the endpoints of η and η . We also find the convergence rate and the exact formula of the Green’s function upto a multiplicative constant. To derive these results, we construct two-dimensional diffusionprocesses and use orthogonal polynomials to obtain their transition density. Contents H -hulls and chordal Loewner equation . . . . . . . . . . . . . . . . . . . . . . . . 62.2 Chordal SLE κ and 2-SLE κ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.3 SLE κ ( ρ ) processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.4 Hypergeometric SLE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.5 Two-parameter stochastic processes . . . . . . . . . . . . . . . . . . . . . . . . . 132.6 Jacobi polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 Commuting Pair of SLE κ (2 , ρ ) Curves 31 κ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475.2 Opposite pair of hSLE κ curves, the generic case . . . . . . . . . . . . . . . . . . . 515.3 Opposite pair of hSLE κ curves, a limit case . . . . . . . . . . . . . . . . . . . . . 53 This paper is the follow-up of [21], in which we proved the existence of two-curve Green’sfunction for 2-SLE κ at an interior point, and obtained the formula of the Green’s function upto a multiplicative constant. In the present paper, we will study the case when the interiorpoint is replaced by a boundary point.As a particular case of multiple SLE κ , a 2-SLE κ consists of two random curves in a simplyconnected domain connecting two pairs of boundary points (more precisely, prime ends), whichsatisfy the property that, when any one curve is given, the conditional law of the other curveis that of a chordal SLE κ in a complement domain of the first curve.The two-curve Green’s function of a 2-SLE κ is about the rescaled limit of the probabilitythat the two curves in the 2-SLE κ both approach a marked point in D . More specifically, itwas proved in [21] that, for any κ ∈ (0 , η , η ) is a 2-SLE κ in D , and z ∈ D , then thelimit G ( z ) := lim r → + r − α P [dist( η j , z ) < r, j = 1 ,
2] (1.1)converges to a positive number, where the exponent α is α := (12 − κ )( κ +4)8 κ . The limit G ( z )is called the (interior) two-curve Green’s function for ( η , η ). The paper [21] also derived theconvergence rate and the exact formula of G ( z ) up to an unknown constant.In this paper we study the limit in the case that z ∈ ∂D assuming that ∂D is analytic near z . Below is our main theorem. Theorem 1.1.
Let κ ∈ (0 , . Let ( η , η ) be a -SLE κ in a simply connected domain D . Let z ∈ ∂D . Suppose ∂D is analytic near z . We have the following results in two cases.(A) If z is not any endpoint of η or η , then the limit in (1.1) exists and lies in (0 , ∞ ) for α = α = α := 2( κ − . B) If z is one of the endpoints of η and η , then the limit in (1.1) exists and lies in (0 , ∞ ) for α = α := κ − .Moreover, in each case we may compute G D ( z ) up to some constant C > as follows. Let F denote the hypergeometric function F ( κ , − κ ; κ , · ) . Let f map D conformally onto H suchthat f ( z ) = ∞ . Let J denote the map z
7→ − /z .(A1) Suppose Case (A) happens and none of η and η separates z from the other curve. Welabel the f -images of the four endpoints of η and η by v − < w − < w + < v + . Then G D ( z ) = C | ( J ◦ f ) ′ ( z ) | α G ( w ; v ) , where C > is a constant depending only on κ , and G ( w ; v ) := Y σ ∈{ + , −} ( | w σ − v σ | κ − | w σ − v − σ | κ ) F (cid:16) ( w + − w − )( v + − v − )( w + − v − )( v + − w − ) (cid:17) − . (1.2) (A2) Suppose Case (A) happens and one of η and η separates z from the other curve. Welabel the f -images of the four endpoints of η and η by v − < w − < w + < v + . Then G D ( z ) = C | ( J ◦ f ) ′ ( z ) | α G ( w ; v ) where C > is a constant depending only on κ , and G ( w ; v ) := Y u ∈{ w,v } | u + − u − | κ − Y σ ∈{ + , −} | w σ − v − σ | κ F (cid:16) ( v + − w + )( w − − v − )( w + − v − )( v + − w − ) (cid:17) − . (1.3) (B) Suppose Case (B) happens. We label the f -images of the other three endpoints of η and η by w + , w − , v + , such that f − ( v + ) and z are endpoints of the same curve, and w + , v + lie on the same side of w − . Then G D ( z ) = C | ( J ◦ f ) ′ ( z ) | α G ( w ; v + ) , where C > is a constant depending only on κ , and G ( w ; v + ) = | w + − w − | κ − | v + − w − | κ F (cid:16) v + − w + v + − w − (cid:17) − . (1.4)Our long-term goal is to prove the existence of Minkowski content of double points of chordalSLE κ for κ ∈ (4 , κ . Following the approach in [6], we need to prove theexistence of two-curve two-point Green’s function for 2-SLE κ , where Theorem 1.1 is expectedto serve as the boundary estimate in the proof.3 .2 Strategy For the proof of the main theorem, we use a two-curve technique, which was introduced in [21],and recently used in [20] to study the Green’s function for the cut points of chordal SLE κ .By conformal invariance of 2-SLE κ , we may assume that D = H := { z ∈ C : Im z > } , and z = ∞ . It suffices to consider the limit lim L →∞ L α P [ η j ∩ {| z | > L } 6 = ∅ , j = 1 , η and η by v + > w + > w − > v − . There aretwo possible link patterns: ( w + ↔ v + ; w − ↔ v − ) and ( w + ↔ w − ; v + ↔ v − ), which respectivelycorrespond to Case (A1) and Case (A2) of Theorem 1.1.For the first link pattern, we label the two curves by η + and η − . By translation and dilation,we may assume that v ± = ±
1. We assume that 0 ∈ [ w − , w + ] and let v = 0. We then grow η + and η − simultaneously from w + and w − towards v + and v − , respectively, up to the timethat either curve reaches its target, or separates v + or v − from ∞ . For each t in the lifespan[0 , T u ), let H t denote the unbounded connected component of H \ ( η + [0 , t ] ∪ η − [0 , t ]). Duringthe lifespan [0 , T u ) of the process, the speeds of η + and η − are controlled by two factors:(F1) the harmonic measure of [ v − , v + ] ∪ η + [0 , t ] ∪ η − [0 , t ] in H t viewed from ∞ increases in t exponentially with factor 2, and(F2) [ v − , v ] ∪ η − [0 , t ] and [ v , v + ] ∪ η + [0 , t ] have the same harmonic measure viewed from ∞ .Suppose g t maps H t conformally onto H and satisfies g t ( z ) = z + o (1) as z → ∞ . Define V + ( t ) = lim x ↓ max([ v ,v + ] ∪ η + [0 ,t ] ∩ R ) g t ( x ) and V − ( t ) = lim x ↑ max([ v − ,v ] ∪ η − [0 ,t ] ∩ R ) g t ( x ). Then (F1)is equivalent to that V + ( t ) − V − ( t ) = e t ( v + − v − ). The inverse g − t extends continuously to H .We will see that there is a unique V ( t ) ∈ ( V − ( t ) , V + ( t )) such that g − t maps [ V ( t ) , V σ ( t )] into[ v , v σ ] ∪ η σ [0 , t ] for σ ∈ { + , −} . Then (F2) is equivalent to that V + ( t ) − V ( t ) = V ( t ) − V − ( t ).In the case κ ≤ V σ ( t ) is simply g t ( v σ ) for σ ∈ { + , − , } . We will also be able to deal withthe case κ ∈ (4 , T u , one of the two curves, say η + , separates v + or v − from ∞ . If η + separates v + , the rest of η + grows in a bounded connected component of H \ η + [0 , T u ); if η + separates v − , the whole η − is disconnected from ∞ by η + [0 , T u ). Thus, after T u , at least one curve cannot get closer to ∞ . So we may focus on the parts of η + and η − before T u . Using Koebe’s 1 / g t at ∞ ) and Beurling’s estimate (applied to a planar Brownian motionstarted near ∞ ), we find that for 0 ≤ t < T u , the diameter of both η + [0 , t ] and η − [0 , t ] arecomparable to e t .We define a two-dimensional diffusion process R ( t ) = ( R + ( t ) , R − ( t )) ∈ [0 , , 0 ≤ t < T u ,by R σ ( t ) = W σ ( t ) − V ( t ) V σ ( t ) − V ( t ) , σ ∈ { + , −} , where W σ ( t ) = g t ( η σ ( t )) ∈ [ V ( t ) , V σ ( t )]. Here η σ ( t ) isunderstood as a prime end of H t . We then use the knowledge of 2-SLE κ partition function anda technique of orthogonal polynomials to derive the transition density of ( R ), which will play acentral role in the proof of Case A1 of Theorem 1.1.For the link pattern ( w + ↔ w − ; v + ↔ v − ), we label the curves by η w and η v . We observethat η v disconnects η w from ∞ . Thus, for L > max {| v + | , | v − |} , η w intersects {| z | > L } impliesthat η v does the intersection as well. Then the two-curve Green’s function reduces to a single-curve Green’s function. But we will still use a two curve approach. We assume that v ± = ± ∈ ( w − , w + ), and let v = 0 as in the previous case. This time, we grow η + and η − simultaneously along the same curve η w such that η σ runs from w σ towards w − σ , σ ∈ { + , −} .The growth is stopped if η + and η − together exhaust the range of η w , or any of them disconnectsits target from ∞ . The speeds of the curves are also controlled by (F1) and (F2). Then we define V , V ± , W ± , R ± in the same way as before, and derive the transition density of R = ( R + , R − ),which also plays a central role in the proof.In Case (B), we may assume that v + = 1 and w + + w − = 0. Now we introduce two newpoints: v = 0 and v − = −
1. Unlike the previous cases, v − is not an end point of any curve.For this case, we grow η + and η − simultaneously from w + and w − along the same curve η w asin Case (A2). The rest of the proof almost follows the same approach as in Case (A2). Below is the outline of the paper. In Section 2, we recall definitions, notations, and some basicresults that will be needed in this paper. In Section 3 we develop a framework on a commutingpair of deterministic chordal Loewner curves, which do not cross but may touch each other.The work extends the disjoint ensemble of Loewner curves that appeared in [27, 26]. At theend of the section, we describe the way to grow the two curves simultaneously with properties(F1) and (F2). In Section 4, we use the results from the previous section to study a pairof multi-force-point SLE κ ( ρ ) curves, which commute with each other in the sense of [2]. Weobtain a two-dimensional diffusion process R ( t ) = ( R + ( t ) , R − ( t )), 0 ≤ t < ∞ , and derive itstransition density using orthogonal two-variable polynomials. In Section 5, we study threetypes of commuting pair of hSLE κ curves, which correspond to the three cases in Theorem1.1. We prove that each of them is locally absolutely continuous w.r.t. a commuting pair ofSLE κ ( ρ ) curves for certain force values, and also find the Radon-Nikodym derivative at differenttimes. For each commuting pair of hSLE κ curves, we obtain a two-dimensional diffusion process R ( t ) = ( R + ( t ) , R − ( t )) with random finite lifetime, and derive its transition density and quasi-invariant density. In the last section we finish the proof of Theorem 1.1. Acknowledgments
The author thanks Xin Sun for suggesting the problem on the (interior and boundary) two-curveGreen’s function for 2-SLE.
We first fix some notation. Let H = { z ∈ C : Im z > } . For z ∈ C and S ⊂ C , letrad z ( S ) = sup {| z − z | : z ∈ S ∪{ z }} . If a function f is absolutely continuous on a real interval I , and f ′ = g a.e. on I , then we write f ′ ae = g on I . This means that f ( x ) − f ( x ) = R x x g ( x ) dx for any x < x ∈ I . Here g may not be defined on a subset of I with Lebesgue measure zero.We will also use “ ae =” for PDE or SDE in some similar sense.5 .1 H -hulls and chordal Loewner equation A relatively closed subset K of H is called an H -hull if K is bounded and H \ K is a simplyconnected domain. For a set S ⊂ C , if there is an H -hull K such that H \ K is the unboundedconnected component of H \ S , then we say that K is the H -hull generated by S , and write K = Hull( S ). For an H -hull K , there is a unique conformal map g K from H \ K onto H suchthat g K ( z ) = z + cz + O (1 /z ) as z → ∞ for some c ≥
0. The constant c , denoted by hcap( K ),is called the H -capacity of K , which is zero iff K = ∅ . We write hcap ( K ) for hcap( K ) / ∂ ( H \ K ) is locally connected, then g − K extends continuously from H to H , and we use f K to denote the continuation. If K = Hull( S ), then we write g S , f S , hcap( S ) , hcap ( S ) for g K , f K , hcap( K ) , hcap ( K ), respectively.If K ⊂ K are two H -hulls, then we define K /K = g K ( K \ K ), which is also an H -hull.Note that g K = g K /K ◦ g K and hcap( K ) = hcap( K /K ) + hcap( K ), which imply thathcap( K ) , hcap( K /K ) ≤ hcap( K ). If K ⊂ K ⊂ K are H -hulls, then K /K ⊂ K /K and ( K /K ) / ( K /K ) = K /K . (2.1)Let K be a non-empty H -hull. Let K doub = K ∪ { z : z ∈ K } , where K is the closure of K , and z is the complex conjugate of z . By Schwarz reflection principle, there is a compactset S K ⊂ R such that g K extends to a conformal map from C \ K doub onto C \ S K . Let a K = min( K ∩ R ), b K = max( K ∩ R ), c K = min S K , d K = max S K . Then the extended g K maps C \ ( K doub ∪ [ a K , b K ]) conformally onto C \ [ c K , d K ]. Since g K ( z ) = z + o (1) as z → ∞ ,by Koebe’s 1 / K ) ≍ diam( K doub ∪ [ a K , b K ]) ≍ d K − c K . Example . Let x ∈ R , r >
0. Then H := { z ∈ H : | z − x | ≤ r } is an H -hull with g H ( z ) = z + r z − x , hcap( H ) = r , a H = x − r , b H = x + r , H doub = { z ∈ C : | z − x | ≤ r } , c H = x − r , d H = x + 2 r .The next proposition combines [28, Lemmas 5.2 and 5.3]. Proposition 2.1. If L ⊂ K are two non-empty H -hulls, then [ a K , b K ] ⊂ [ c K , d K ] , [ c L , d L ] ⊂ [ c K , d K ] , and [ c K/L , d
K/L ] ⊂ [ c K , d K ] . Proposition 2.2.
For any x ∈ R \ K doub , < g ′ K ( x ) ≤ . Moreover, g ′ K is decreasing on ( −∞ , a K ) and increasing on ( b K , ∞ ) .Proof. By [17, Lemma C.1], there is a measure µ K supported on S K with | µ K | = hcap( K ) suchthat g − K ( z ) − z = R S K − z − y dµ K ( y ) for any x ∈ R \ S K . Differentiating this formula and letting z = x ∈ R \ S K , we get ( g − K ) ′ ( x ) = 1 + R S K x − y ) dµ K ( y ) ≥
1. So 0 < g ′ K ≤ R \ K doub .Further differentiating the integral formula w.r.t. x , we find that ( g − K ) ′′ ( x ) = R S K − x − y ) dµ K ( y )is positive on ( −∞ , c K ) and negative on ( d K , ∞ ), which means that ( g − K ) ′ is increasing on( −∞ , c K ) and decreasing on ( d K , ∞ ). Since g K maps ( −∞ , a K ) and ( b K , ∞ ) onto ( −∞ , c K )and ( d K , ∞ ), respectively, we get the monotonicity of g ′ K .6 roposition 2.3. If K is an H -hull with rad x ( K ) ≤ r for some x ∈ R , then hcap( K ) ≤ r , rad x ( S K ) ≤ r , and | g K ( z ) − z | ≤ r for any z ∈ C \ K doub .Proof. We have K ⊂ H := { z ∈ H : | z − x | ≤ r } . By Proposition 2.1, hcap( K ) ≤ hcap( H ) = r , S K ⊂ [ c K , d K ] ⊂ [ c H , d H ] = [ x − r, x + 2 r ]. Since g K ( z ) − z is analytic on C \ K doub andtends to 0 as z → ∞ , by the maximum modulus principle,sup z ∈ C \ K doub | g K ( z ) − z | ≤ lim sup C \ K doub ∋ z → K doub | g K ( z ) − z | ≤ rad x ( K doub ) + rad x ( S K ) ≤ r, where the second inequality holds because z → K doub implies that g K ( z ) → S K . Proposition 2.4.
For two nonempty H -hulls K ⊂ K such that K /K ∩ [ c K , d K ] = ∅ , wehave | c K − c K | , | d K − d K | ≤ K /K ) .Proof. By symmetry it suffices to estimate | c K − c K | . Let c ′ = lim x ↑ a K g K ( x ) and ∆ K = K /K . Since g K maps H \ K onto H \ ∆ K , we have c ′ = min { c K , a ∆ K } . Since ∆ K ∩ [ c K , d K ] = ∅ , c ′ ≥ c K − diam(∆ K ). Thus, by Proposition 2.3, c K = lim x ↑ a K g ∆ K ◦ g K ( x ) = lim y ↑ c ′ g ∆ K ( y ) ≥ c ′ − K ) ≥ c K − K ) . By Proposition 2.1, c K ≤ c K . So we get | c K − c K | ≤ K ).The following proposition is [5, Proposition 3.42]. Proposition 2.5.
Suppose K , K , K are H -hulls such that K ⊂ K ∩ K . Then hcap( K ) + hcap( K ) ≥ hcap(Hull( K ∪ K )) + hcap( K ) . Let b w ∈ C ([0 , T ) , R ) for some T ∈ (0 , ∞ ]. The chordal Loewner equation driven by b w is ∂ t g t ( z ) = 2 g t ( z ) − b w ( t ) , ≤ t < T ; g ( z ) = z. For every z ∈ C , let τ z be the first time that the solution g · ( z ) blows up; if such time does notexist, then set τ z = ∞ . For t ∈ [0 , T ), let K t = { z ∈ H : τ z ≤ t } . It turns out that each K t is an H -hull with hcap ( K t ) = t , K doub t = { z ∈ C : τ z ≤ t } , which is connected, and each g t agreeswith g K t . We call g t and K t the chordal Loewner maps and hulls, respectively, driven by b w .If for every t ∈ [0 , T ), f K t is well defined, and η ( t ) := f K t ( b w ( t )), 0 ≤ t < T , is continuousin t , then we say that η is the chordal Loewner curve driven by b w . Such η may not exist ingeneral. When it exists, we have η (0) = b w (0) ∈ R , and K t = Hull( η [0 , t ]) for all t , and we saythat K t , 0 ≤ t < T , are generated by η .Let u be a continuous and strictly increasing function on [0 , T ). Let v be the inverse of u − u (0). Suppose that g ut and K ut , 0 ≤ t < T , satisfy that g uv ( t ) and K uv ( t ) , 0 ≤ t < u ( T ) − u (0),7re chordal Loewner maps and hulls, respectively, driven by b w ◦ v . Then we say that g ut and K ut ,0 ≤ t < T , are chordal Loewner maps and hulls, respectively, driven by b w with speed u , and call( K uv ( t ) ) the normalization of ( K ut ). If ( K ut ) are generated by a curve η u , i.e., K ut = Hull( η u [0 , t ])for all t , then η u is called a chordal Loewner curve driven by b w with speed u , and η u ◦ v iscalled the normalization of η u . If u is absolutely continuous with u ′ ae = q , then we also saythat the speed is q . In this case, the chordal Loewner maps satisfy the differential equation ∂ t g ut ( z ) ae = q ( t ) g ut − b w ( t ) . We omit the speed when it is constant 1.The following proposition is straightforward. Proposition 2.6.
Suppose K t , ≤ t < T , are chordal Loewner hulls driven by b w ( t ) , ≤ t < T ,with speed u . Then for any t ∈ [0 , T ) , K t + t /K t , ≤ t < T − t , are chordal Loewner hullsdriven by b w ( t + t ) , ≤ t < T − t , with speed u ( t + · ) . One immediate consequence is that,for any t < t ∈ [0 , T ) , K t /K t is connected. The following proposition is a slight variation of [7, Theorem 2.6].
Proposition 2.7.
The H -hulls K t , ≤ t < T , are chordal Loewner hulls with some speed ifand only if for any fixed a ∈ [0 , T ) , lim δ ↓ sup ≤ t ≤ a diam( K t + δ /K t ) = 0 . Moreover, the drivingfunction b w satisfies that { b w ( t ) } = T δ> K t + δ /K t , ≤ t < T ; and the speed u could be chosento be u ( t ) = hcap ( K t ) , ≤ t < T . Proposition 2.8.
Suppose K t , ≤ t < T , are chordal Loewner hulls driven by b w with somespeed. Then for any t ∈ (0 , T ) , c K t ≤ b w ( t ) ≤ d K t for all t ∈ [0 , t ] .Proof. Let t ∈ (0 , T ). If 0 ≤ t < t , by Propositions 2.1 and 2.7, b w ( t ) ∈ [ a K t /K t , b K t /K t ] ⊂ [ c K t /K t , d K t /K t ] ⊂ [ c K t , d K t ]. By the continuity of b w , we also have b w ( t ) ∈ [ c K t , d K t ].The following proposition combines [11, Lemma 2.5] and [10, Lemma 3.3]. Proposition 2.9.
Suppose b w ∈ C ([0 , T ) , R ) generates a chordal Loewner curve η and chordalLoewner hulls K t , ≤ t < T . Then the set { t ∈ [0 , T ) : η ( t ) ∈ R } has Lebesgue measure zero.Moreover, if the Lebesgue measure of η [0 , T ) ∩ R is zero, then the functions c ( t ) and d ( t ) definedby c (0) = d (0) := b w (0) , and c ( t ) := c K t and d ( t ) := d K t , < t < T , are absolutely continuouswith c ′ ( t ) ae = c ( t ) − b w ( t ) and d ′ ( t ) ae = d ( t ) − b w ( t ) , and are respectively monotonically decreasing andincreasing. Moreover, c ( t ) and d ( t ) are continuously differentiable at the set of times t suchthat η ( t ) R , and in that case “ ae = ” can be replaced by “ = ”. Definition 2.10.
We define the following notation.(i) Modified real line. For w ∈ R , we define R w = ( R \ { w } ) ∪ { w − , w + } , which has a totalorder endowed from R and the relation x < w − < w + < y for any x, y ∈ R such that x < w and y > w . It is assigned the topology such that ( −∞ , w − ] := ( −∞ , w ) ∪ { w − } and [ w + , ∞ ) := { w + } ∪ ( w, ∞ ) are two connected components, and are respectivelyhomeomorphic to ( −∞ , w ] and [ w, ∞ ) through the map π w : R w → R with π w ( w ± ) = w and π w ( x ) = x for x ∈ R \ { w } . 8ii) Modified Loewner map. Let K be an H -hull and w ∈ R . Let a wK = min { w, a K } , b wK =max { w, b K } , c wK = lim x ↑ a wK g K ( x ), and d wK = lim x ↓ b wK g K ( x ). They are all equal to w if K = ∅ . Define g wK on R w ∪ { + ∞ , −∞} such that g wK ( ±∞ ) = ±∞ , g wK ( x ) = g K ( x ) if x ∈ R \ [ a wK , b wK ]; g wK ( x ) = c wK if x = w − or x ∈ [ a wK , b wK ] ∩ ( −∞ , w ); and g wK ( x ) = d wK if x = w + or x ∈ [ a wK , b wK ] ∩ ( w, ∞ ). Note that g wK is continuous and increasing. Proposition 2.11.
Let K ⊂ K be two H -hulls. Let w ∈ R and e w ∈ [ c wK , d wK ] . Revise g wK such that when g wK ( w ) = e w , we define g wK ( x ) = e w sign( x − w ) . Then g e wK /K ◦ g wK = g wK , on R w ∪ { + ∞ , −∞} . (2.2) Proof.
By symmetry, it suffices to show that (2.2) holds on [ w + , ∞ ]. Since for x ≥ w + , g wK ( x ) ≥ d wK ≥ e w , the revised g wK is a continuous map from [ w + , ∞ ] into [ e w + , ∞ ], and so both sidesof (2.2) are continuous on [ w + , ∞ ]. If x > b wK , then x > max { b wK , b K } , which implies that g wK ( x ) = g K ( x ) > max { d wK , b K /K } ≥ b e wK /K . Thus, g e wK /K ◦ g wK ( x ) = g K /K ◦ g K ( x ) = g K ( x ) = g wK ( x ) on ( b wK , ∞ ]. We know that g wK is constant on [ w + , b wK ]. To prove that (2.2)holds on [ w + , ∞ ], by continuity it suffices to show that the LHS of (2.2) is constant on [ w + , b wK ].This is obvious if b wK = b wK since g wK is constant on [ w + , b wK ]. Suppose b wK < b wK . Then wehave b K , w < b wK = b K . So [ w + , b wK ] is mapped by g wK onto [ d wK , b K /K ] (or [ e w + , b K /K ]),which is in turn mapped by g e wK /K to a constant. Proposition 2.12.
Let K t and η ( t ) , ≤ t < T , be chordal Loewner hulls and curve drivenby b w with speed q . Suppose the Lebesgue measure of η [0 , T ) ∩ R is . Let w = b w (0) , and x ∈ R w . Define X ( t ) = g wK t ( x ) , ≤ t < T . Then X is absolutely continuous and satisfies thedifferential equation X ′ ( t ) ae = q ( t ) X ( t ) − b w ( t ) on [0 , T ) ; if x > w (resp. x < w ), then X ( t ) ≥ b w ( t ) (resp. X ( t ) ≤ b w ( t ) ) on [0 , T ) , and so is increasing (resp. decreasing) on [0 , T ) . Moreover, forany ≤ t < t < T , | X ( t ) − X ( t ) | ≤ K t /K t ) .Proof. We may assume that the speed q is constant 1. By symmetry, we may assume that x ∈ ( −∞ , w − ]. If x = w − , then X ( t ) = c K t for t > X (0) = b w (0). Then the conclusionfollows from Propositions 2.4 and 2.9. Now suppose x ∈ ( −∞ , w ).Fix 0 ≤ t < t < T . We first prove the upper bound for | X ( t ) − X ( t ) | . There arethree cases. Case 1. x K t j , j = 1 ,
2. In this case, X ( t ) = g K t /K t ( X ( t )), and theupper bound for | X ( t ) − X ( t ) | follows from Proposition 2.3. Case 2. x ∈ K t ⊂ K t . Inthis case X ( t j ) = c K tj , j = 1 ,
2, and the conclusion follows from Proposition 2.4. Case 3. x K t and x ∈ K t . Then X ( t ) = g K t ( x ) < c K t and X ( t ) = c K t . Moreover, wehave τ x ∈ ( t , t ], lim t ↑ τ x X ( t ) = b w ( τ x ), and X ( t ) satisfies X ′ ( t ) = X ( t ) − b w ( t ) < t , τ x ).By Propositions 2.8 and 2.1, c K ( t ) > X ( t ) ≥ b w ( τ x ) ≥ c K τx ≥ c K t = X ( t ). So we have | X ( t ) − X ( t ) | ≤ | c K t − c K t | ≤ K t /K t ) by Propositions 2.4. By Proposition 2.7, X is continuous on [0 , T ). 9ince X ( t ) = g K t ( x ) satisfies the chordal Loewner equation driven by b w up to τ x , weknow that X ′ ( t ) = X ( t ) − b w ( t ) on [0 , τ x ). From Proposition 2.9 we know that X ′ ( t ) ae = X ( t ) − b w ( t ) on ( τ x , T ).The differential equation on [0 , T ) then follows from the continuity of X . Since X ( t ) ≤ c K ( t ) ≤ b w ( t ) by Proposition 2.8, X is decreasing on [0 , T ). κ and -SLE κ If b w ( t ) = √ κB ( t ), 0 ≤ t < ∞ , where κ > B ( t ) is a standard Brownian motion, then thechordal Loewner curve η driven by b w is known to exist (cf. [16]). We now call it a standardchordal SLE κ curve. It satisfies that η (0) = 0 and lim t →∞ η ( t ) = ∞ . The behavior of η depends on κ : if κ ∈ (0 , η is simple and intersects R only at 0; if κ ≥ η is space-filling,i.e., H = η ( R + ); if κ ∈ (4 , η is neither simple nor space-filling. If D is a simply connecteddomain with two distinct marked boundary points (or more precisely, prime ends) a and b , thechordal SLE κ curve in D from a to b is defined to be the conformal image of a standard chordalSLE κ curve under a conformal map from ( H ; 0 , ∞ ) onto ( D ; a, b ).Chordal SLE κ satisfies Domain Markov Property (DMP): if η is a chordal SLE κ curve in D from a to b , and T is a stopping time, then conditionally on the part of η before T and theevent that η does not reach b at the time T , the part of η after T is a chordal SLE κ curve from η ( T ) to b in a connected component of D \ η [0 , T ].We will focus on the range κ ∈ (0 ,
8) so that SLE κ is non-space-filling. One remarkableproperty of these chordal SLE κ is reversibility: the time-reversal of a chordal SLE κ curve in D from a to b is a chordal SLE κ curve in D from b to a , up to a time-change ([27, 9]). Anotherfact that is important to us is the existence of 2-SLE κ . Let D be a simply connected domainwith distinct boundary points a , b , a , b such that a and b together do not separate a from b on ∂D (and vice versa). A 2-SLE κ in D with link pattern ( a ↔ b ; a ↔ b ) is a pair ofrandom curves ( η , η ) in D such that η j connects a j with b j for j = 1 ,
2, and conditionallyon any one curve, the other is a chordal SLE κ curve a complement domain of the given curvein D . Because of reversibility, we do not need to specify the orientation of η and η . If wewant to emphasize the orientation, then we use an arrow like a → b in the link pattern. Theexistence of 2-SLE κ was proved in [3] for κ ∈ (0 ,
4] using Brownian loop measure and in [11, 9]for κ ∈ (4 ,
8) using imaginary geometry theory. The uniqueness of 2-SLE κ (for a fixed domainand link pattern) was proved in [10] (for κ ∈ (0 , κ ∈ (4 , κ ( ρ ) processes First introduced in [8], SLE κ ( ρ ) processes are natural variations of SLE κ , where one keeps trackof additional marked points, often called force points, which may lie on the boundary or interior.For the generality needed here, all force points will lie on the boundary. In this subsection, wereview the definition and properties of SLE κ ( ρ ) developed in [11].Let n ∈ N , κ > ρ = ( ρ , . . . , ρ n ) ∈ R n . Let w ∈ R and v = ( v , . . . , v n ) ∈ R nw . The chordalSLE κ ( ρ ) process in H started from w with force points v is the chordal Loewner process driven10y the function b w ( t ), which drives chordal Loewner hulls K t , and solves the SDE d b w ( t ) ae = √ κdB ( t ) + n X j =1 ρ j b w ( t ) − g wK t ( v j ) dt, b w (0) = w, where B ( t ) is a standard Brownian motion, and we used Definition 2.10. We require that for σ ∈ { + , −} , P j : v j = w σ ρ j > −
2. The solution exists uniquely up to the first time (called a con-tinuation threshold) that P j : b v j ( t )= c Kt ρ j ≤ − P j : b v j ( t )= d Kt ρ j ≤ −
2, whichever comes first.If a continuation threshold does not exist, then the lifetime is ∞ . For each j , b v j ( t ) := g wK ( t ) ( v j )is called the force point function started from v j , satisfies b v ′ j ae = b v j − b w , and is monotonicallyincreasing or decreasing depending on whether v j > w or v j < w .A chordal SLE κ ( ρ ) process generates a chordal Loewner curve η in H started from w up tothe continuation threshold. If no force point is swallowed by the process at any time, this factfollows from the existence of chordal SLE κ curve ([16]) and Girsanov Theorem. The existence ofthe curve in the general case was proved in [11]. By Proposition 2.11 and the Markov propertyof Brownian motion, a chordal SLE κ ( ρ ) curve η satisfies the following DMP. If τ is a stoppingtime for η , then conditionally on the process before τ and the event that τ is less than thelifetime T , b w ( τ + t ) and b v j ( τ + t ), 1 ≤ j ≤ n , 0 ≤ t < T − τ , are the driving function andforce point functions for a chordal SLE κ ( ρ ) curve η τ started from b w ( τ ) with force points at b v ( τ ) , . . . , b v n ( τ ), and η ( τ + · ) = f K τ ( η τ ), where K τ := Hull( η [0 , τ ]). Here if b v j ( τ ) = b w ( τ ), then b v j ( τ ) as a force point is treated as b w ( τ ) sign( v j − w ) .We now relabel the force points v , . . . , v n by v − n − ≤ · · · ≤ v − < w < v +1 ≤ · · · ≤ v + n + , where n − + n + = n ( n − or n + could be 0). Then for any t in the lifespan, b v − n − ( t ) ≤ · · · ≤ b v − ( t ) ≤ b w ( t ) ≤ b v +1 ( t ) ≤ · · · ≤ b v + n + ( t ). If for any σ ∈ {− , + } and 1 ≤ k ≤ n σ , P kj =1 ρ σj > −
2, thenthe process will never reach a continuation threshold, and so its lifetime is ∞ , in which caselim t →∞ η ( t ) = ∞ . If for some σ ∈ { + , −} and 1 ≤ k ≤ n σ , P kj =1 ρ σj ≥ κ −
2, then η does nothit v σk and the open interval between v σk and v σk +1 ( v σn σ +1 := σ · ∞ ). If κ ∈ (0 ,
8) and for any σ ∈ { + , −} and 1 ≤ k ≤ n σ , P kj =1 ρ σj > κ −
4, then for every x ∈ R \ { w } , a.s. η does not visit x , which implies by Fubini Theorem that a.s. η ∩ R has Lebesgue measure zero. For a, b, c ∈ C such that c
6∈ { , − , − , · · · } , the hypergeometric function F ( a, b ; c ; z ) (cf.[14]) is defined by the Gauss series on the disc {| z | < } : F ( a, b ; c ; z ) = ∞ X n =0 ( a ) n ( b ) n ( c ) n n ! z n , where ( x ) n is rising factorial: ( x ) = 1 and ( x ) n = x ( x + 1) · · · ( x + n −
1) if n ≥
1. It satisfiesthe ODE z (1 − z ) F ′′ ( z ) − [( a + b + 1) z − c ] F ′ ( z ) − abF ( z ) = 0 . (2.3)11or the purpose of this paper, we chose the parameters a, b, c by a = κ , b = 1 − κ , c = κ , anddefine F ( x ) = F (1 − κ , κ ; κ ; x ). It is known that such F extends to a continuous and positivefunction on [0 , e G ( x ) = κx F ′ ( x ) F ( x ) + 2. Definition 2.13.
Let κ ∈ (0 , v ≤ v ∈ [0 + , + ∞ ] or v ≥ v ∈ [ −∞ , − ]. Suppose b w ( t ),0 ≤ t < ∞ , solves the following SDE: d b w ( t ) ae = √ κdB ( t ) + (cid:16) b w ( t ) − b v ( t ) − b w ( t ) − b v ( t ) (cid:17) e G (cid:16) b w ( t ) − b v ( t ) b w ( t ) − b v ( t ) (cid:17) dt, b w (0) = 0 , where B ( t ) is a standard Brownian motion, b v j ( t ) = g K t ( v j ), j = 1 ,
2, and K t are chordalLoewner hulls driven by b w . The chordal Loewner curve driven by b w is called a hypergeometricSLE κ , or simply hSLE κ , curve in H from 0 to ∞ with force points v , v . We call v j ( t ) theforce point function started from v j , j = 1 ,
2. We then extend the definition of hSLE κ curvesto general simply connected domains using conformal maps.Hypergeometric SLE is important because if ( η , η ) is a 2-SLE κ in D with link pattern( a → b ; a → b ), then for j = 1 ,
2, the marginal law of η j is that of an hSLE κ curve in D from a j to b j with force points b − j and a − j (cf. [19, Proposition 6.10]).Using the standard argument in [18], we obtain the following proposition describing anhSLE κ curve in H in the chordal coordinate in the case that the target is not ∞ . Proposition 2.14.
Let w = w ∞ ∈ R . Let v ∈ R w ∪ {∞} \ { w ∞ } and v ∈ R w ∞ ∪ {∞} \ { w } be such that the cross ratio R := ( w − v )( w ∞ − v )( w − v )( w ∞ − v ) ∈ [0 + , . Let κ ∈ (0 , . Let b η be an hSLE κ curve in H from w to w ∞ with force points at v , v . Stop b η at the first time that it separates w ∞ from ∞ , and parametrize the stopped curve by H -capacity. Then the new curve, denoted by η , is the chordal Loewner curve driven by some function b w , which satisfies the following SDEwith initial value b w (0) = w : d b w ( t ) ae = √ κdB ( t ) + κ − b w ( t ) − b w ∞ ( t ) dt ++ (cid:16) b w ( t ) − b v ( t ) − b w ( t ) − b v ( t ) (cid:17) · e G (cid:16) ( b w ( t ) − b v ( t ))( b v ( t ) − b w ∞ ( t ))( b w ( t ) − b v ( t ))( b v ( t ) − b w ∞ ( t )) (cid:17) dt, where B ( t ) is a standard Brownian motion, b w ∞ ( t ) = g K t ( w ∞ ) and b v j ( t ) = g w K t ( v j ) , j = 1 , ,and K t are the chordal Loewner hulls driven by b w . Definition 2.15.
We call the η in Proposition 2.14 an hSLE κ curve in H from w to w ∞ withforce points at v , v , in the chordal coordinate; call b w the driving function; and call b w ∞ , b v and b v the force point functions respectively started from w ∞ , v and v . Proposition 2.16.
We adopt the notation in the last proposition. Let T be the first time that w ∞ or v is swallowed by the hulls. Note that | b w − b w ∞ | , | b v − b v | , b w − b v | , and | b w ∞ − b v | are all positive on [0 , T ) . We define M on [0 , T ) by M = G ( b w , b v ; b w ∞ , b v ) , where G is given y (1.2). Then M is a positive local martingale, and if we tilt the law of η by M , then we getthe law of a chordal SLE κ (2 , , curve in H started from w with force points w ∞ , v and v .More precisely, if τ < T is a stopping time such that M is uniformly bounded on [0 , τ ] , then ifwe weight the underlying probability measure by M ( τ ) /M (0) , then we get a probability measureunder which the law of η stopped at the time τ is that of a chordal SLE κ (2 , , curve in H started from w with force points w ∞ , v and v stopped at the time τ .Proof. This follows from straightforward applications of Itˆo’s formula and Girsanov Theorem,where we use (2.3), Propositions 2.12 and 2.14. Actually, the calculation could be simpler if wetilt the law of a chordal SLE κ (2 , ,
2) curve by M − to get an hSLE κ curve. In this subsection we briefly recall the framework used in [21, Section 2.3]. We assign a partialorder ≤ to R = [0 , ∞ ) such that t = ( t + , t − ) ≤ ( s + , s − ) = s iff t + ≤ s + and t − ≤ s − .It has a minimal element 0 = (0 , t < s if t + < s + and t − < s − . We define t ∧ s = ( t ∧ s , t ∧ s ). Given t, s ∈ R , we define [ t, s ] = { r ∈ R : t ≤ r ≤ s } . Let e + = (1 , e − = (0 , t + , t − ) = t + e + + t − e − . Definition 2.17. An R -indexed filtration F on a measurable space Ω is a family of σ -algebras F t , t ∈ R , on Ω such that F t ⊂ F s whenever t ≤ s . Define F by F t = T s>t F s , t ∈ R .Then we call F the right-continuous augmentation of F . We say that F is right-continuous if F = F . A process X = ( X ( t )) t ∈ R defined on Ω is called F -adapted if for any t ∈ R , X ( t ) is F t -measurable. It is called continuous if t X ( t ) is sample-wise continuous.For the rest of this subsection, let F be an R -indexed filtration with right-continuousaugmentation F , and let F ∞ = W t ∈ R F t . Definition 2.18.
A [0 , ∞ ] -valued random element T is called an F -stopping time if for anydeterministic t ∈ R , { T ≤ t } ∈ F t . It is called finite if T ∈ R , and is called bounded if thereis a deterministic t ∈ R such that T ≤ t . For an F -stopping time T , we define a new σ -algebra F T by F T = { A ∈ F ∞ : A ∩ { T ≤ t } ∈ F t , ∀ t ∈ R } .The following proposition follows from a standard argument. Proposition 2.19.
The right-continuous augmentation of F is itself, and so F is right-continuous. A [0 , ∞ ] -valued random map T is an F -stopping time if and only if { T < t } ∈ F t for any t ∈ R . For an F -stopping time T , A ∈ F T if and only if A ∩ { T < t } ∈ F t for any t ∈ R . If ( T n ) n ∈ N is a decreasing sequence of F -stopping times, then T := inf n T n is also an F -stopping time, and F T = T n F T n . Definition 2.20.
A relatively open subset R of R is called a history complete region, or simplyan HC region, if for any t ∈ R , we have [0 , t ] ⊂ R . Given an HC region R , for σ ∈ { + , −} ,define T R σ : R + → R + ∪ {∞} by T R σ ( t ) = sup { s ≥ se σ + te − σ ∈ R} , where we set sup ∅ = 0.13n HC region-valued random element D is called an F -stopping region if for any t ∈ R , { ω ∈ Ω : t ∈ D ( ω ) } ∈ F t . A random function X ( t ) with a random domain D is called an F -adapted HC process if D is an F -stopping region, and for every t ∈ R , X t restricted to { t ∈ D} is F t -measurable.The following propositions are [21, Lemmas 2.7 and 2.9]. Proposition 2.21.
Let T and S be two F -stopping times. Then (i) { T ≤ S } ∈ F S ; (ii) if S is a constant s ∈ R , then { T ≤ S } ∈ F T ; and (iii) if f is an F T -measurable function, then { T ≤ S } f is F S -measurable. In particular, if T ≤ S , then F T ⊂ F S . We will need the following proposition to do localization. The reader should note that foran F -stopping time T and a deterministic time t ∈ R , T ∧ t may not be an F -stopping time.This is the reason why we introduce a more complicated stopping time. Proposition 2.22.
Let T be an F -stopping time. Fix a deterministic time t ∈ R . Define T t such that if T ≤ t , then T t = T ; and if T t , then T t = t . Then T t is an F -stopping timebounded above by t , and F T t agrees with F T on { T ≤ t } , i.e., { T ≤ t } ∈ F T t ∩ F T , and for any A ⊂ { T ≤ t } , A ∈ F T t if and only if A ∈ F T .Proof. Clearly T t ≤ t . Let s ∈ R . If t ≤ s , then { T t ≤ s } is the whole space. If t s , then { T t ≤ s } = { T ≤ t } ∩ { T ≤ s } = { T ≤ t ∧ s } ∈ F t ∧ s ⊂ F s . So T t is an F -stopping time.By Proposition 2.21, { T ≤ t } ∈ F T . Suppose A ⊂ { T ≤ t } and A ∈ F T . Let s ∈ R . If t ≤ s , then A ∩ { T t ≤ s } = A = A ∩ { T ≤ t } ∈ F t ⊂ F s . If t s , then A ∩ { T t ≤ s } = A ∩ { T ≤ t ∧ s } ∈ F t ∧ s ⊂ F s . So A ∈ F T t . In particular, { T ≤ t } ∈ F T t . On the other hand, suppose A ⊂ { T ≤ t } and A ∈ F T t . Let s ∈ R . If t ≤ s , then A ∩ { T ≤ s } = A = A ∩ { T t ≤ t } ∈ F t ⊂F s . If t s , then A ∩ { T ≤ s } = A ∩ { T ≤ t } ∩ { T ≤ s } = A ∩ { T t ≤ s } ∈ F s . Thus, A ∈ F T .So for A ⊂ { T ≤ t } , A ∈ F T t if and only if A ∈ F T .Now we fix a probability measure P , and let E denote the corresponding expectation. Definition 2.23. An F -adapted process ( X t ) t ∈ R is called an F -martingale (w.r.t. P ) if forany s ≤ t ∈ R , a.s. E [ X t |F s ] = X s . If there is ζ ∈ L (Ω , F , P ) such that X t = E [ ζ |F t ] forevery t ∈ R , then we say that X is an F -martingale closed by ζ .The following proposition is [21, Lemma 2.11]. Proposition 2.24 (Optional Stopping Theorem) . Suppose X is a continuous F -martingale.The following are true. (i) If X is closed by ζ , then for any finite F -stopping time T , X T = E [ ζ |F T ] . (ii) If T ≤ S are two bounded F -stopping times, then E [ X S |F T ] = X T . For α, β > −
1, Jacobi polynomials ([14, Chapter 18]) P ( α,β ) n ( x ), n = 0 , , , , . . . , are a class ofclassical orthogonal polynomials with respect to the weight Ψ ( α,β ) ( x ) := ( − , (1 − x ) α (1 + x ) β .14his means that each P ( α,β ) n ( x ) is a polynomial of degree n , and for the inner product definedby h f, g i Ψ ( α,β ) := R − f ( x ) g ( x )Ψ ( α,β ) ( x ) dx , we have h P ( α,β ) n , P ( α,β ) m i Ψ ( α,β ) = 0 when n = m . Thenormalization is that P ( α,β ) n (1) = Γ( α + n +1) n !Γ( α +1) , P ( α,β ) n ( −
1) = ( − n Γ( β + n +1) n !Γ( β +1) , and k P ( α,β ) n k ( α,β ) = 2 α + β +1 n + α + β + 1 · Γ( n + α + 1)Γ( n + β + 1) n !Γ( n + α + β + 1) . (2.4)For each n ≥ P ( α,β ) n ( x ) is a solution of the second order differential equation:(1 − x ) y ′′ − [( α + β + 2) x + ( α − β )] y ′ + n ( n + α + β + 1) y = 0 . (2.5)When max { α, β } > − , we have an exact value of the supernorm of P ( α,β ) n over [ − , k P ( α,β ) n k ∞ = max {| P ( α,β ) n (1) | , | P ( α,β ) n ( − |} = Γ(max { α, β } + n + 1) n !Γ(max { α, β } + 1) . (2.6)For general α, β > −
1, we get an upper bound of k P ( α,β ) n k ∞ using (2.6), the exact value of P ( α,β ) n (1), and the derivative formula ddx P ( α,β ) n ( x ) = α + β + n +12 P ( α +1 ,β +1) n − ( x ) for n ≥ k P ( α,β ) n k ∞ ≤ Γ( α + n + 1) n !Γ( α + 1) + ( α + β + n + 1) · Γ(max { α, β } + n + 1)Γ( n )Γ(max { α, β } + 2) . (2.7) In this section, we develop a framework about commuting pairs of deterministic chordal Loewnercurves, which will be needed to study the commuting pairs of random chordal Loewner curvesin the next two sections. The major length of this section is caused by the fact that we allowthat the two Loewner curves have intersections. This is needed in order to handle the case κ ∈ (4 , Let w − < w + ∈ R . Suppose for σ ∈ { + , −} , η σ ( t ), 0 ≤ t < T σ , is a chordal Loewner curve(with speed 1) driven by b w σ started from w σ , such that η + does not hit ( −∞ , w − ], and η − doesnot hit [ w + , ∞ ). Let K σ ( t σ ) = Hull( η [0 , t σ ]), 0 ≤ t σ < T σ , σ ∈ { + , −} . Then K σ ( · ) are chordalLoewner hulls driven by b w σ , hcap ( K σ ( t σ )) = t σ , and by Proposition 2.7, { b w σ ( t σ ) } = \ δ> K σ ( t σ + δ ) /K σ ( t σ ) , ≤ t σ < T σ . (3.1)The corresponding chordal Loewner maps are g K σ ( t ) , 0 ≤ t < T σ , σ ∈ { + , −} . From theassumption on η + and η − we get a K − ( t − ) ≤ w − < a K + ( t + ) , b K − ( t − ) < w + ≤ b K + ( t + ) , for t σ ∈ (0 , T σ ) , σ ∈ { + , −} . (3.2)15ince each K σ ( t ) is generated by a curve, f K σ ( t ) is well defined. Let I σ = [0 , T σ ), σ ∈ { + , −} ,and for t = ( t + , t − ) ∈ I + × I − , define K ( t ) = Hull( η + [0 , t + ] ∪ η − [0 , t − ]) , m( t ) = hcap ( K ( t )) , H ( t ) = H \ K ( t ) . (3.3)It is obvious that K ( · , · ) and m( · , · ) are increasing (may not strictly) in both variables. Since ∂K ( t + , t − ) is locally connected, f K ( t + ,t − ) is well defined. For σ ∈ { + , −} , t − σ ∈ I − σ and t σ ∈ I σ , define K t − σ σ ( t σ ) = K ( t + , t − ) /K − σ ( t − σ ). Then we have g K ( t + ,t − ) = g K t − + ( t + ) ◦ g K − ( t − ) = g K t + − ( t − ) ◦ g K + ( t + ) . (3.4)By (3.2) and the assumption on η + , η − , we have a K ( t + ,t − ) = a K − ( t − ) if t − >
0, and b K ( t + ,t − ) = b K + ( t + ) if t + > Lemma 3.1.
For any t + ≤ t ′ + ∈ I + and t − ≤ t ′− ∈ I − , we have m( t ′ + , t ′− ) − m( t ′ + , t − ) − m( t + , t ′− ) + m( t + , t − ) ≤ . (3.5) Especially, m is Lipschitz continuous with constant in any variable, and so is continuous on I + × I − .Proof. Let t + ≤ t ′ + ∈ I + and t − ≤ t ′− ∈ I − . Since K ( t ′ + , t − ) and K ( t + , t ′− ) together generatethe H -hull K ( t ′ + , t ′− ), and they both contain K ( t + , t − ), we obtain (3.5) from Proposition 2.5.The rest of the statements follow easily from (3.5), the monotonicity of m, and the fact thatm( t σ e σ ) = t σ for any t σ ∈ I σ , σ ∈ { + , −} . Definition 3.2.
We use the above setting. Let
D ⊂ I + × I − be an HC region as in Definition2.20. Suppose that there are dense subsets I ∗ + and I ∗− of I + and I − , respectively, such that forany σ ∈ { + , −} and t − σ ∈ I ∗− σ , the following two conditions hold:(I) K t − σ σ ( t ), 0 ≤ t σ < T D σ ( t − σ ), are chordal Loewner hulls generated by a chordal Loewnercurve, denoted by η t − σ σ , with some speed.(II) η t − σ σ [0 , T D σ ( t − σ )) ∩ R has Lebesgue measure zero.Then we call ( η + , η − ; D ) a commuting pair of chordal Loewner curves, and call K ( · , · ) andm( · , · ) the hull function and the capacity function, respectively, for this pair. Remark 3.3.
Later in Lemma 3.9 we will show that for the commuting pair in Definition 3.2,Conditions (I) and (II) actually hold for all t − σ ∈ I − σ , σ ∈ { + , −} .From now on, let ( η + , η − ; D ) be a commuting pair of chordal Loewner curves, and let I ∗ + and I ∗− be as in Definition 3.2. Lemma 3.4. K ( · , · ) and m( · , · ) restricted to D are strictly increasing in both variables. roof. By Condition (I), for any σ ∈ { + , −} and t − σ ∈ I ∗− σ , t K ( t − σ e − σ + te σ ) and t m( t − σ e − σ + te σ ) are strictly increasing on [0 , T D σ ( t − σ )). By (3.5) and the denseness of I ∗− σ in I − σ , this property extends to any t − σ ∈ I − σ .In the rest of the section, when we talk about K ( t + , t − ), m( t + , t − ), K t − + ( t + ) and K t + − ( t − ),it is always implicitly assumed that ( t + , t − ) ∈ D . So we may now simply say that K ( · , · ) andm( · , · ) are strictly increasing in both variables. Lemma 3.5.
We have the following facts.(i) Let a = ( a + , a − ) ∈ D and L = max {| z | : z ∈ K ( a + , a − ) } . Let σ ∈ { + , −} . Suppose t σ < t ′ σ ∈ [0 , a σ ] satisfy that diam( η σ [ t σ , t ′ σ ]) < r for some r ∈ (0 , L ) . Then for any t − σ ∈ [0 , a − σ ] , diam( K t − σ σ ( t σ + δ ) /K t − σ σ ( t σ )) ≤ πL ∗ log( L/r ) − / .(ii) For any ( a + , a − ) ∈ D and σ ∈ { + , −} , lim δ ↓ sup ≤ t σ ≤ a σ sup t ′ σ ∈ ( t σ ,t σ + δ ) sup ≤ t − σ ≤ a − σ sup z ∈ C \ K t − σσ ( t ′ σ ) doub | g K t − σσ ( t ′ σ ) ( z ) − g K t − σσ ( t σ ) ( z ) | = 0 . (iii) The map ( t, z ) g K ( t ) ( z ) is continuous on { ( t, z ) : t ∈ D , z ∈ C \ K ( t ) doub } .Proof. (i) Suppose σ = + by symmetry. We may assume that a ± ∈ I ∗± . Let ∆ η + = η + [ t + , t ′ + ]and S = {| z − η + ( t + ) | = r } . By assumption, ∆ η + ⊂ {| z − η + ( t + ) | < r } . By Lemma 3.4, thereis z ∗ ∈ ∆ η + ∩ H ( t + , a − ) ⊂ H ( t + , t − ). Since z ∗ ∈ {| z − η + ( t + ) | < r } , the set S ∩ H ( t + , t − ) hasa connected component, denoted by J , which separates z ∗ from ∞ in H ( t + , t − ). Such J is acrosscut of H ( t + , t − ), which divides H ( t + , t − ) into two domains, where the bounded domain,denoted by D J , contains z ∗ .Now ∆ η + ∩ H ( t + , a − ) ⊂ H ( t + , a − ) \ J . We claim that ∆ η + ∩ H ( t + , a − ) is contained inone connected component of H ( t + , a − ) \ J . Note that J ∩ H ( t + , a − ) is a disjoint union ofcrosscuts, each of which divides H ( t + , a − ) into two domains. To prove the claim, it suffices toshow that, for each connected component J of J ∩ H ( t + , a − ), ∆ η + ∩ H ( t + , a − ) is containedin one connected component of H ( t + , a − ) \ J . Suppose that this is not true for some J . Let J e = g K ( t + ,a − ) ( J ). Then J e is a crosscut of H , which divides H into two domains, both ofwhich intersect ∆ b η + := g K ( t + ,a − ) (∆ η + ∩ H ( t + , a − )). Since ∆ η + has positive distance from S ⊃ J , and g − K ( t + ,a − ) | H extends continuously to H , ∆ b η + has positive distance from J e . Thus,there is another crosscut J i of H , which is disjoint from and surrounded by J e , such that thesubdomain H bounded by J i and J e is disjoint from ∆ b η + . Label the three connected componentsof H \ ( J e ∪ J i ) by D e , A, D i , respectively, from outside to inside. Then ∆ b η + intersects both D e and D i , but is disjoint from A . Let K i = D i ∪ J i and K e = K i ∪ A ∪ J e be two H -hulls.Let η ∗ + = η + ( t + + · ) and b η ∗ + = g K ( t + ,a − ) ◦ η ∗ + , whose domain is S := { s ∈ [0 , T + − t + ) : η ∗ + ( s ) ∈ H \ K ( t + , a − ) } . For each s ∈ [0 , δ := t ′ + − t + ], K ( t + + s, a − ) = Hull( K ( t + , a − ) ∪ ∆ η s + ), and so K ′ + ( s ) := K a − + ( t + + s ) /K a − + ( t + ) = K ( t + + s, a − ) /K ( t + , a − ) (by (2.1)) is the H -hull generated by b η ∗ + ([0 , s ] ∩ S ). For 0 ≤ s ≤ δ , since A is disjoint from b η ∗ + ([0 , δ ] ∩ S ) ⊂ ∆ b η + , it is either contained17n or disjoint from K ′ + ( s ). If K ′ + ( s ) ⊃ A , then K ′ + ( s ) ⊃ Hull( A ) = K i ; if K ′ + ( s ) ∩ A = ∅ , then bythe connectedness of K ′ + ( s ), K ′ + ( s ) is contained in either K i or H \ ( K i ∪ A ). Since K ′ + ( δ ) ⊃ ∆ b η + intersects both D e and D i , we get K ′ + ( δ ) ⊃ K e . Let s = inf { s ∈ [0 , δ ] : K e ⊂ K ′ + ( s ) } .By Proposition 2.7, we have s ∈ (0 , δ ] and K ′ + ( s ) ⊃ K e . By the increasingness of K ′ + ( · ), S ≤ s
2. By the definition of extremal length, there exists a curvein Ω with Euclidean length less than 10 πL ∗ (log( L/r )) − / , which separates g ( J ) from g ( J ′ } ).This implies that the diam( g ( J )) is bounded above by 10 πL ∗ (log( L/r )) − / , and so is that of K ′ + ( δ ) = K t − + ( t ′ + ) /K t − + ( t + ). This finishes the proof of (i).(ii) This follows from (i), Proposition 2.3, the continuity of η σ , the limit lim r ↓ πL ∗ log( L/r ) − / = 0, and the equality g K t ∓± ( t ′± ) = g K t ∓± ( t ′± ) /K t ∓± ( t ± ) ◦ g K t ∓± ( t ± ) .(iii) This follows from (ii), (3.4) and the fact that g K ( t ) is analytic on C \ K ( t ) doub .For a function X defined on D , σ ∈ { + , −} and t − σ ∈ R + , we use X | − σt − σ to denote thesingle-variable function t σ X ( t σ e σ + t − σ e − σ ), 0 ≤ t σ < T D σ ( t − σ ), and use ∂ σ X ( t + , t − ) todenote the derivative of this function at t σ . Lemma 3.6.
There are two functions W + , W − ∈ C ( D , R ) such that for any σ ∈ { + , −} and t − σ ∈ I − σ , K t − σ σ ( t σ ) , ≤ t σ < T D σ ( t − σ ) , are chordal Loewner hulls driven by W σ | − σt − σ with speed m | − σt − σ , and for any t = ( t + , t − ) ∈ D , η σ ( t σ ) = f K ( t ) ( W σ ( t )) . roof. By symmetry, we only need to prove the case that σ = +. Sincehcap ( K t − + ( t + + δ )) − hcap ( K t − + ( t + )) = m( t + + δ, t − ) − m( t + , t − ) , by Lemma 3.5 (i), the continuity of η σ and Proposition 2.7, for every t − ∈ I − , K t − + ( t + ),0 ≤ t + < T D + ( t − ), are chordal Loewner hulls with speed m | − t − , and the driving function,denoted by W + ( · , t − ), satisfies that { W + ( t + , t − ) } = \ δ> K t − + ( t + + δ ) /K t − + ( t + ) = \ δ> K ( t + + δ, t − ) /K ( t + , t − ) . (3.6)Fix t = ( t + , t − ) ∈ D . We now show that f K ( t ) ( W + ( t )) = η + ( t + ). By Lemma 3.4, there existsa sequence t n + ↓ t + such that η + ( t n + ) ∈ K ( t n + , t − ) \ K ( t + , t − ) for all n . Then g K ( t ) ( η + ( t n + )) ∈ K ( t n + , t − ) /K ( t ) = K t − + ( t n + ) /K t − + ( t + ). So we have g K ( t ) ( η + ( t n + )) → W + ( t ) by (3.6). From thecontinuity of f K ( t ) and η + , we then get η + ( t + ) = lim n →∞ η + ( t n + ) = lim n →∞ f K ( t ) ( g K ( t ) ( η + ( t n + ))) = f K ( t ) ( W + ( t )) . It remains to show that W + is continuous on D . Let t + , t − , t − ∈ R + be such that t − < t − and( t + , t − ) ∈ D . By Lemma 3.4, there is a sequence δ n ↓ z n := η + ( t + + δ n ) ∈ H ( t + , t − ).Then g K ( t + ,t j − ) ( z n ) ∈ K ( t + + δ n , t j − ) /K ( t + , t j − ) = K t j − + ( t + + δ n ) /K t j − + ( t + ), j = 1 ,
2. From (3.6)we get | W + ( t + , t j − ) − g K ( t + ,t j − ) ( z n ) | ≤ diam( K t j − + ( t + + δ n ) /K t j − + ( t + )) , j = 1 , . Since g K ( t + ,t − ) ( z n ) = g K ( t + ,t − ) /K ( t + ,t − ) ◦ g K ( t + ,t − ) ( z n ), by Proposition 2.3 we get | g K ( t + ,t − ) ( z n ) − g K ( t + ,t − ) ( z n ) | ≤ K ( t + , t − ) /K ( t + , t − )) = 3 diam( K t + − ( t − ) /K t + − ( t − )) . Combining the above displayed formulas and letting n → ∞ , we get | W + ( t + , t − ) − W + ( t + , t − ) | ≤ K t + − ( t − ) /K t + − ( t − )) , (3.7)which together with Lemma 3.5 (i) implies that, for any ( a + , a − ) ∈ D , the family of functions[0 , a − ] ∋ t − W + ( t + , t − ), 0 ≤ t + ≤ a + , are equicontinuous. Since W + is continuous in t + asa driving function, we conclude that W + is continuous on D . Definition 3.7.
We call W + and W − the driving functions for the commuting pair ( η + , η − ; D ).It is obvious that W σ | − σ = b w σ , σ ∈ { + , −} . Remark 3.8.
By (3.6) and Propositions 2.6 and 2.8, for t < t ∈ I + and t − ∈ I − such that( t , t − ) ∈ D , | W + ( t , t − ) − W + ( t , t − ) | ≤ K t − + ( t ) /K t − + ( t )). This combined with (3.7)and Lemma 3.5 (i) implies that, if η σ extends continuously to [0 , T σ ] for σ ∈ { + , −} , then W + and W − are uniformly continuous on D , and so extend continuously to D .19 emma 3.9. For any σ ∈ { + , −} and t − σ ∈ I − σ , the chordal Loewner hulls K t − σ σ ( t σ ) = K ( t + , t − ) /K − σ ( t − σ ) , ≤ t σ < T D σ ( t − σ ) , are generated by a chordal Loewner curve, denotedby η t − σ σ , which intersects R at a set with Lebesgue measure zero such that η σ | [0 ,T D σ ( t − σ )) = f K − σ ( t − σ ) ◦ η t − σ σ . Moreover, for σ ∈ { + , −} , ( t + , t − ) η t − σ σ ( t σ ) is continuous on D .Proof. It suffices to work on the case that σ = +. First we show that there exists a continuousfunction ( t + , t − ) η t − + ( t + ) from D into H such that η + ( t + ) = f K − ( t − ) ( η t − + ( t + )) , ∀ ( t + , t − ) ∈ D . (3.8)Let ( t + , t − ) ∈ D . By Lemma 3.4, there is a sequence t n + ↓ t + such that for all n , ( t n + , t − ) ∈ D and η + ( t n + ) ∈ H \ K ( t + , t − ). Then we get g K − ( t − ) ( η + ( t n + )) ∈ g K − ( t − ) ( K ( t n + , t − ) \ K ( t + , t − )) = K t − + ( t n + ) /K t − + ( t + ). If t − ∈ I ∗− , then by Condition (I), T n K t − + ( t n + ) /K t − + ( t + ) = { η t − + ( t + ) } , whichimplies that g K − ( t − ) ( η + ( t n + )) → η t − + ( t + ). From the continuity of f K − ( t − ) and η + , we find that(3.8) holds if t − ∈ I ∗− . Thus, η t − + ( t + ) = g K − ( t − ) ( η + ( t + )) , if ( t + , t − ) ∈ D , t − ∈ I ∗− and η + ( t + ) ∈ H \ K − ( t − ) . (3.9)Fix a − ∈ I ∗− . Let R = { t + ∈ I + : ( t + , a − ) ∈ D , η + ( t + ) ∈ H \ K − ( a − ) } , which by Lemma 3.4 isdense in [0 , T D + ( a − )). By Propositions 2.3 and 2.7,lim δ ↓ sup t − ∈ [0 ,a − ] sup t ′− ∈ [0 ,a − ] ∩ ( t − − δ,t − + δ ) sup t + ∈R | g K − ( t − ) ( η + ( t + )) − g K − ( t ′− ) ( η + ( t + )) | = 0 . (3.10)This combined with (3.9) implies thatlim δ ↓ sup t − ∈ [0 ,a − ] ∩I ∗− sup t ′− ∈ [0 ,a − ] ∩I ∗− ∩ ( t − − δ,t − + δ ) sup t + ∈R | η t − + ( t + ) − η t ′− + ( t + ) | = 0 . (3.11)By the denseness of R in [0 , T D + ( a − )) and the continuity of each η t − + , t − ∈ I ∗− , we knowthat (3.11) still holds if sup t + ∈R is replaced by sup t + ∈ [0 ,T D + ( a − )) . Since I ∗− is dense in I − , thecontinuity of each η t − + , t − ∈ I ∗− , together with (3.11) implies that there exists a continuousfunction [0 , T D + ( a − )) × [0 , a − ] ∋ ( t + , t − ) η t − + ( t + ) ∈ H , which extends those η t − + ( t + ) for t − ∈ I ∗− ∩ [0 , a − ] and t + ∈ [0 , T D + ( a − )). Running a − from 0 to T − , we get a continuous function D ∋ ( t + , t − ) η t − + ( t + ) ∈ H , which extends those η t − + ( t + ) for ( t + , t − ) ∈ D and t − ∈ I ∗− . Since η t − + ( t + ) = g K − ( t − ) ( η + ( t + )) for all t + ∈ R and t − ∈ [0 , a − ] ∩ I ∗− , from (3.9,3.10) we know that itis also true for any t − ∈ [0 , a − ]. Thus, η + ( t + ) = f K − ( t − ) ( η t − + ( t + )) for all t + ∈ R and t − ∈ [0 , a ].By the denseness of R in [0 , T D + ( a − )) and the continuity of η + , f K − ( t − ) and η t − + , we get (3.8)for all t − ∈ [0 , a − ] and t + ∈ [0 , T D + ( a − )). So (3.8) holds for all ( t + , t − ) ∈ D .For ( t + , t − ) ∈ D , since K ( t + , t − ) = Hull( K − ( t − ) ∪ ( η + [0 , t + ] ∩ ( H \ K − ( t − ))), we see that K t − + ( t + ) = g K − ( t − ) ( K ( t + , t − ) \ K − ( t − )) is the H -hull generated by g K − ( t − ) ( η + [0 , t + ] ∩ ( H \ K − ( t − ))) = η t − + [0 , t + ] ∩ H . So K t − + ( t + ) = Hull( η t − + [0 , t + ]). By Lemma 3.6, for any t − ∈ , T − ), η t − + ( t + ), 0 ≤ t + < T D + ( t − ), is the chordal Loewner curve driven by W + ( · , t − ) withspeed m( · , t − ). So we have η t − + ( t + ) = f K t − + ( t + ) ( W + ( t + , t − )), which together with η + ( t + ) = f K ( t + ,t − ) ( W + ( t + , t − )) implies that η + ( t + ) = f K − ( t − ) ( η t − + ( t + )).Finally, we show that η t − + ∩ R has Lebesgue measure zero for all t − ∈ I − . Fix t − ∈ I − and b t + ∈ I + such that ( b t + , t − ) ∈ D . It suffices to show that η t − + [0 , b t + ] ∩ R has Lebesguemeasure zero. There exists a sequence I ∗− ∋ t n − ↓ t − such that ( b t + , t n − ) ∈ D for all n . Let K n = K − ( t n − ) /K − ( t − ), g n = g K n , and f n = g − n . Then f K − ( t − ) = f K − ( t n − ) ◦ g n on H \ K n . Let t + ∈ [0 , b t + ]. From f K − ( t − ) ( η t − + ( t + )) = η + ( t + ) = f K − ( t n − ) ( η t n − + ( t + )) we get η t n − + ( t + ) = g n ( η t − + ( t + ))if η t − + ( t + ) ∈ H \ K n . By continuity we get η t − + ( t + ) = f n ( η t − + ( t n + )) if η t n − + ( t + ) ∈ R \ [ c K n , d K n ],0 ≤ t + ≤ b t + . Thus, η t − + [0 , b t + ] ∩ ( R \ [ a K n , b K n ]) ⊂ f n ( η t n − + [0 , b t + ] ∩ ( R \ [ c K n , d K n ])). Since t n − ∈ I ∗− , by Condition (II) and the analyticity of f n on R \ [ c K n , d K n ] we know that η t − + [0 , b t + ] ∩ ( R \ [ a K n , b K n ]) has Lebesgue measure zero for each n . Sending n → ∞ and using the fact that[ a K n , b K n ] ↓ { b w − ( t − ) } , we see that η t − + [0 , b t + ] ∩ R also has Lebesgue measure zero. Lemma 3.10.
For any σ ∈ { + , −} and ( t + , t − ) ∈ D , b w σ ( t σ ) = f K tσ − σ ( t − σ ) ( W σ ( t + , t − )) ∈ ∂ ( H \ K t σ − σ ( t − σ )) .Proof. By symmetry, it suffices to work on the case σ = +. For any ( t + , t − ) ∈ D , by Lemma3.4 there is a sequence t n + ↓ t + such that η + ( t n + ) ∈ K ( t n + , t − ) \ K ( t + , t − ) for all n . From (3.1)and Lemma 3.6 we get g K + ( t + ) ( η + ( t n + )) → b w + ( t + ) and g K ( t + ,t − ) ( η + ( t n + )) → W + ( t + , t − ). From(3.4) we get g K + ( t + ) = f K t + − ( t − ) ◦ g K ( t + ,t − ) . From the continuity of f K t + − ( t − ) on H , we then get b w + ( t + ) = f K t + − ( t − ) ( W + ( t + , t − )). Finally, b w + ( t + ) ∈ ∂ ( H \ K t + − ( t − )) because W + ( t + , t − ) ∈ ∂ H and f K t + − ( t − ) maps H conformally onto H \ K t + − ( t − ). For σ ∈ { + , −} , define C σ and D σ on D such that if t σ > C σ ( t + , t − ) = c K t − σσ ( t σ ) and D σ ( t + , t − ) = d K t − σσ ( t σ ) ; and if t σ = 0, then C σ = D σ = W σ at t − σ e − σ . Since K t − σ σ ( · ) arechordal Loewner hulls driven by W σ | − σt − σ with some speed, by Proposition 2.8 we get C σ ≤ W σ ≤ D σ on D , σ ∈ { + , −} . (3.12)Since K t − σ σ ( t σ ) is the H -hull generated by η t − σ σ [0 , t σ ], we get f K t − σσ ( t σ ) [ C σ ( t + , t − ) , D σ ( t + , t − )] ⊂ η t − σ σ [0 , t σ ] . (3.13)Recall that w − < w + ∈ R . We write w for ( w + , w − ). Define R w = ( R \ { w + , w − } ) ∪{ w ++ , w − + , w + − , w −− } with the obvious order endowed from R . Assign the topology to R w suchthat I − := ( −∞ , w −− ] , I := [ w + − , w − + ] , I + := [ w ++ , ∞ ) are three connected components of R w ,21hich are respectively homeomorphic to ( −∞ , w − ] , [ w − , w + ] , [ w + , ∞ ). Recall that for σ ∈{ + , −} and t ∈ I σ , g w σ K σ ( t ) (Definition 2.10) is defined on R w σ , and agrees with g K σ ( t ) on R \ ([ a K σ ( t ) , b K σ ( t ) ] ∪ { w σ } ). By Lemma 3.10 and the fact that w − σ [ a K σ ( t ) , b K σ ( t ) ] ∪ { w σ } ,we then know that g w σ K σ ( t ) ( w − σ ) = W − σ ( te σ ). So we define g w σ K σ ( t ) ( w ±− σ ) = W − σ ( te σ ) ± , andunderstand g w σ K σ ( t ) as a continuous function from R w to R W − σ ( te σ ) . Lemma 3.11.
For any t = ( t + , t − ) ∈ D , g W + (0 ,t − ) K t − + ( t + ) ◦ g w − K − ( t − ) and g W − ( t + , K t + − ( t − ) ◦ g w + K + ( t + ) agree on R w ,and the common function in the equality, denoted by g wK ( t ) , satisfies the following properties.(i) g wK ( t ) is increasing and continuous on R w , and agrees with g K ( t ) on R \ ( K ( t ) .(ii) g wK ( t ) maps I + ∩ ( K ( t ) ∪ { w ++ } ) and I − ∩ ( K ( t ) ∪ { w −− } ) to D + ( t ) and C − ( t ) , respectively.(iii) If K + ( t + ) ∩ K − ( t − ) = ∅ , g wK ( t ) maps I ∩ ( K + ( t + ) ∪ { w − + } ) and I ∩ ( K − ( t − ) ∪ { w + − } ) to C + ( t ) and D − ( t ) , respectively.(iv) If K + ( t + ) ∩ K − ( t − ) = ∅ , g wK ( t ) maps I to C + ( t ) = D − ( t ) .(v) The map ( t, v ) g wK ( t ) ( v ) from D × R w to R is jointly continuous.Proof. Fix t = ( t + , t − ) ∈ D . For σ ∈ { + , −} , we write K for K ( t ), K σ for K σ ( t σ ), e K σ for K t − σ σ ( t σ ), e w σ for W σ ( t − σ e − σ ), C σ for C σ ( t ), and D σ for D σ ( t ). The equality now reads g e w + e K + ◦ g w − K − = g e w − e K − ◦ g w + K + . Before proving the equality, we first show that both sides are welldefined and satisfy (i-iii) and a weaker version of (iv) (see below). First consider g e w − e K − ◦ g w + K + .Since g w + K + : R w → R e w − , the composition is well defined on R w . We denote it by g w, + K ( t ) .(i) The continuity and monotonicity of the composition follows from the continuity andmonotonicity of both g e w − e K − and g w + K + . Let v ∈ R \ K . Then v K + , and g w + K + ( v ) = g K + ( v ). Since e K − = K/K + , K \ K + = f K + ( e K − ). From v = f K + ( g K + ( v )) K \ K + and the continuity of f K + on H , we know that g K + ( v ) e K − , which implies that g w, + K ( t ) ( v ) = g e K − ◦ g K + ( v ) = g K ( v ).In the proof of (ii,iii) below, we write η σ for η σ [0 , t σ ] and e η σ for η t − σ σ [0 , t σ ]; when t σ = 0, i.e., K σ = e K σ = ∅ , we understand a K σ = b K σ = c K σ = d K σ = w σ , and a e K σ = b e K σ = c e K σ = d e K σ = e w σ . Then it is always true that a K σ = min { η σ ∩ R } , b K σ = max { η σ ∩ R } , a e K σ = min { e η σ ∩ R } , b e K σ = max { e η σ ∩ R } , c e K σ = C σ , and d e K σ = D σ . Since η ± = f K ∓ ( e η ± ), we get b e K + = g K − ( b K + ), a e K − = g K + ( a K − ). If K + ∩ K − = ∅ , then a e K + = g K − ( a K + ), b e K − = g K + ( b K − ).(ii) Since I + ∩ ( K ∪ { w ++ } ) = { w ++ } ∪ ( w + , b K ] = { w ++ } ∪ ( w ++ , b K + ] is mapped by g w + K + to asingle point, it is also mapped by g w, + K ( t ) to a single point, which by (i) is equal tolim x ↓ b K g K ( x ) = lim x ↓ b K + g e K + ◦ g K − ( x ) = lim y ↓ b e K + g e K + ( y ) = d e K + = D + .
22o show that I − ∩ ( K ∪ { w −− } ) = [ a K , w −− ) ∪ { w −− } is mapped by g w, + K ( t ) to C − , by (i) it sufficesto show that lim x ↑ a K g K ( x ) = g e w − e K − ◦ g w + K + ( w −− ) = c e K − . This holds because g e w − e K − ◦ g w + K + ( w −− ) = g e w − e K − ( e w −− ) = c e K − = lim x ↑ a e K − g e K − ( x ) = lim x ↑ a K − g e K − ◦ g K + ( x ) = lim x ↑ a K g K ( x ) . (iii) Suppose K + ∩ K − = ∅ . Then I ∩ ( K + ∪ { w − + } ) = [ a K + , w − + ) ∪ { w − + } is mapped by g w + K + to a single point, so is also mapped by g w, + K ( t ) to a single point. By (i) the latter point islim x ↑ a K + g K ( x ) = lim x ↑ a K + g e K + ◦ g K − ( x ) = lim y ↑ a e K + g e K + ( y ) = c e K + = C + . Since I ∩ ( K − ∪ { w + − } ) = { w + − } ∪ ( w + − , b K − ] is mapped by g w + K + to { e w + − } ∪ ( e w + − , b e K − ], whichis further mapped by g e w − e K − to d e K − = D − , we see that g w, + K ( t ) maps I ∩ ( K − ∪ { w + − } ) to D − .(iv) Suppose K + ∩ K − = ∅ . For now, we only show that I is mapped by g w, + K ( t ) to D − . Bythe assumption we have t + , t − > c K + , d K + ] ∩ e K − = ∅ , which implies that c K + ≤ b e K − .Thus, g w + K + ( I ) = [ e w + − , c K + ] ⊂ [ e w + − , b e K − ], from which follows that g w, + K ( t ) ( I ) = { d e K − } = { D − } .Now g e w − e K − ◦ g w + K + satisfies (i-iii) and a weaker version of (iv). By symmetry, this is also truefor g e w + e K + ◦ g w − K − , where for (iv), I is mapped to { C + } . We now show that the two functions agreeon R w . By (i), g e w + e K + ◦ g w − K − and g e w − e K − ◦ g w + K + agree on R \ K . By (ii), the two functions also agreeon I + ∩ ( K ( t ) ∪ { w ++ } ) and I − ∩ ( K ( t ) ∪ { w −− } ). Thus they agree on both I + and I − . By (i,iii)they agree on I when K + ∩ K − = ∅ . To prove that they agree on I when K + ∩ K − = ∅ , bythe weaker versions of (iv) we only need to show that c e K + = d e K − in that case.First, we show that d e K − ≤ c e K + . Suppose d e K − > c e K + . Then J := ( c e K + , d e K − ) ⊂ [ c e K − , d e K − ] ∩ [ c e K + , d e K + ]. So f e K + ( J ) ⊂ ∂ ( H \ e K + ). If f e K + ( J ) ⊂ R , then it is disjoint from e K + , and so is disjointfrom [ a e K + , b e K + ] since e K + is generated by e η + , which does not spend any nonempty interval oftime on R . That f e K + ( J ) ∩ [ a e K + , b e K + ] = ∅ then implies that J ∩ [ c e K + , d e K + ] = ∅ , a contradiction.So there is x ∈ J such that f e K + ( x ) ⊂ H , which implies that f K ( x ) = f K − ◦ f e K + ( x ) ∈ H \ K − .On the other hand, since x ∈ [ c e K − , d e K − ], f K ( x ) = f K + ◦ f e K − ( x ) ⊂ f K + ( e η − ) = η − , whichcontradicts that f K ( x ) ∈ H \ K − . So d e K − ≤ c e K + .Second, we show that d e K − ≥ c e K + . Suppose d e K − < c e K + . Let J = ( d e K − , c e K + ). Then f e K + ( J ) = ( f e K + ( d e K − ) , a e K + ). From K + ∩ K − = ∅ we know a e K + ≤ d K − . From a e K − = g K + ( a K − )we get d e K − ≥ c e K − = lim x ↑ a e K − g e K − ( x ) = lim y ↑ a K − g e K − ◦ g K + ( y ) = lim y ↑ a K − g e K + ◦ g K − ( y ).Thus, f e K + ( d e K − ) ≥ lim y ↑ a K − g K − ( y ) = c K − . So we get f e K + ( J ) ⊂ [ c K − , d K − ], which is mappedinto η − by f K − . Thus, f K ( J ) ⊂ η − . Symmetrically, f K ( J ) ⊂ η + . Since η − = f K + ( e η − ) and f K ( J ) ⊂ ∂ ( H \ K ), for every x ∈ J , there is z − ∈ e η − ∩ ∂ ( H \ e K − ) such that f K ( x ) = f K + ( z − ).Then there is y − ∈ [ c e K − , d e K − ] such that z − = f e K − ( y − ). So f K ( x ) = f K ( y − ). Similarly,23or every x ∈ J , there is y + ∈ [ c e K − , d e K − ] such that f K ( x ) = f K ( y + ). Pick x < x ∈ J such that f K ( x ) = f K ( x ). This is possible because f K ( J ) has positive harmonic measure in H \ K . Then there exist y ∈ [ c e K + , d e K + ] and y − ∈ [ c e K − , d e K − ] such that f K ( x ) = f K ( y +1 ) and f K ( x ) = f K ( y − ). This contradicts that y > x > x > y − . So d e K − ≥ c e K + .Combining the last two paragraphs, we get c e K + = d e K − . So g W + (0 ,t − ) K t − + ( t + ) ◦ g w − K − ( t − ) and g W − ( t + , K t + − ( t − ) ◦ g w + K + ( t + ) agree on I + ∪ I − ∪ I = R w , and the original (iv) holds for both functions.(v) By (i), the composition g wK ( t ) is continuous on R w for any t ∈ D . It suffices to showthat, for any ( a + , a − ) ∈ D and σ ∈ { + , −} , the family of maps [0 , a σ ] ∋ t σ g wK ( t ) ( v ),( t − σ , v ) ∈ [0 , a − σ ] × R w , are equicontinuous. This statement follows from the expression g wK ( t ) = g W σ ( t − σ e − σ ) K t − σσ ( t σ ) ◦ g w − σ K − σ ( t − σ ) , Proposition 2.12 and Lemma 3.5 (i). Lemma 3.12.
For any ( t + , t − ) ∈ D and σ ∈ { + , −} , W σ ( t + , t − ) = g W − σ ( t σ e σ ) K tσ − σ ( t − σ ) ( b w σ ( t σ )) .Proof. Fix t = ( t + , t − ) ∈ D . By symmetry, we may assume that σ = +. If t − = 0, it is obvioussince W + ( · ,
0) = b w + and K t + − (0) = ∅ . Suppose t − >
0. From (3.12) and Lemma 3.11 (i,iii,iv)we know that W + ( t ) ≥ C + ( t ) ≥ D − ( t ) = d K t + − ( t − ) . Since b w + ( t + ) = f K t + − ( t − ) ( W + ( t )) by Lemma3.10, we find that either W + ( t ) = d K t + − ( t − ) and b w + ( t + ) = b K t + − ( t − ) , or W + ( t ) > d K t + − ( t − ) and W + ( t ) = g K t + − ( t − ) ( b w + ( t + )). In either case, we get W + ( t ) = g W − ( t + , K t + − ( t − ) ( b w + ( t + )). Definition 3.13.
For v ∈ R w , we call V ( t ) := g wK ( t ) ( v ), t ∈ D , the force point function (for thecommuting pair ( η + , η − ; D )) started from v , which is continuous by Lemma 3.11 (v). Definition 3.14.
Let ( η + , η − ; D ) be a commuting pair of chordal Loewner curves started from( w + , w − ) with hull function K ( · , · ).(i) For σ ∈ { + , −} , let φ σ be a continuous and strictly increasing function defined on thelifespan of η σ with φ σ (0) = 0, and let φ ⊕ ( t + , t − ) = ( φ + ( t + ) , φ − ( t − )). Let e η σ = η ◦ φ − σ , σ ∈ { + , −} , and e D = φ ⊕ ( D ). Then we call ( e η + , e η − ; e D ) a commuting pair of chordalLoewner curves with speeds ( φ + , φ − ), and call ( η + , η − ; D ) its normalization.(ii) Let τ ∈ D . Suppose there is a commuting pair of chordal Loewner curves ( e η + , e η − ; e D ) withsome speeds such that e D = { t ∈ R : τ + t ∈ D} , and η σ ( τ σ + · ) = f K ( τ ) ◦ η σ , σ ∈ { + , −} .Then we call ( e η + , e η − ; e D ) the part of ( η + , η − ; D ) after τ up to a conformal map.For a commuting pair ( e η + , e η − ; e D ) with some speeds, we still define the hull function e K ( · , · )and the capacity function e m( · , · ) using (3.3), define the driving functions f W + and f W − usingLemma 3.6, and define the force point functions by e V ( t ) = g w e K ( t ) ( v ) started from v for any v ∈ R w . Most lemmas in this section still hold (except that m may not be Lipschitz continuous).24 emma 3.15. (i) For the ( η + , η − ; D ) , ( e η + , e η − ; e D ) and φ ⊕ in Definition 3.14 (i), we have e X = X ◦ φ − ⊕ for X ∈ { K, m , W ± , V } , where V and e V are force point functions respectivelyfor ( η + , η − ; D ) and ( e η + , e η − ; e D ) started from the same v ∈ R w .(ii) For the ( η + , η − ; D ) , ( e η + , e η − ; e D ) and τ in Definition 3.14 (ii), we have e K = K ( τ + · ) /K ( τ ) , e m = m( τ + · ) − m( τ ) , and f W σ = W σ ( τ + · ) , σ ∈ { + , −} . Let v ∈ R ( w + ,w − ) , and let V be the force point function for ( η + , η − ; D ) started from v . Let e w ± = f W ± (0) = W ± ( τ ) .Define e v ∈ R ( e w + , e w − ) such that if V ( τ )
6∈ { e w + , e w − } , then e v = V ( τ ) ; and if V ( τ ) = e w σ ,then e v = e w sign( v − w σ ) σ , σ ∈ { + , −} . Let e V be the force point function for ( e η + , e η − ; e D ) startedfrom e v . Then e V = V ( τ + · ) on e D .Proof. Part (i) is obvious. We now work on (ii). Let t = ( t + , t − ) ∈ e D . From K ( τ + t ) =Hull( S σ η σ [0 , τ σ + t σ ]), we get K ( τ + t ) = Hull( K ( τ ) ∪ [ σ η σ [ τ σ , τ σ + t σ ]) = Hull( K ( τ ) ∪ f K ( τ ) ( [ σ e η σ [0 , t σ ])) . This implies that e K ( t ) = Hull( S σ e η σ [0 , t σ ]) = K ( τ + t ) /K ( τ ), which then implies that e m( t ) =m( τ + t ) − m( τ ). It together with (2.1,3.6) implies that f W σ ( t ) = W σ ( τ + t ).By (i), Proposition 2.11 and Lemma 3.11, if V ( τ )
6∈ { e w + , e w − } , e V ( t ) = g f W + (0 ,t − ) e K ( t ) / e K − ( t − ) ◦ g e w − e K − ( t − ) ( e v ) = g W + ( τ + ,τ − + t − ) K ( τ + t ) /K ( τ + ,τ − + t − ) ◦ g W − ( τ ) K ( τ + ,τ − + t − ) /K ( τ ) ( e v )= g W + ( τ + ,τ − + t − ) K ( τ + t ) /K ( τ + ,τ − + t − ) ◦ g W − ( τ ) K ( τ + ,τ − + t − ) /K ( τ ) ◦ g W − ( τ + , K ( τ ) /K ( τ + , ◦ g w + K ( τ + , ( v )= g W + ( τ + ,τ − + t − ) K ( τ + t ) /K ( τ + ,τ − + t − ) ◦ g W − ( τ + , K ( τ + ,τ − + t − ) /K ( τ + , ◦ g w + K ( τ + , ( v )= g W + ( τ + ,τ − + t − ) K ( τ + t ) /K ( τ + ,τ − + t − ) ◦ g W + (0 ,τ − + t − ) K ( τ + ,τ − + t − ) /K (0 ,τ − + t − ) ◦ g w − K (0 ,τ − + t − ) ( v )= g W + (0 ,τ − + t − ) K ( τ + t ) ◦ g w − K (0 ,τ − + t − ) ( v ) = g ( w + ,w − ) K ( τ + t ) ( v ) = V ( τ + t ) . Here the “=” in the 1 st line follow from the definition of e V ( t ) and that e K = K ( τ + · ) /K ( τ ), the“=” in the 2 nd line follows from the definition of e v , the “=” in the 3 rd line and the first “=” inthe 5 th line follow from Proposition 2.11, and the “=” in the 4 th line follows from Lemma 3.11.Now suppose V ( τ ) ∈ { e w + , e w − } . By symmetry, assume that V ( τ ) = e w − = W − ( τ ). If v ≥ w + − , we understand g W − ( τ + , K ( τ ) /K ( τ + , as a map from [ W − ( τ + , + , ∞ ) into [ e w + − , ∞ ), i.e., when g W − ( τ + , K ( τ ) /K ( τ + , takes value e w − at some point in [ W − ( τ + , + , ∞ ), we redefine the value as e w + − .Then the above displayed formula still holds. The case that v ≤ w −− is similar, in which weunderstand g W − ( τ + , K ( τ ) /K ( τ + , as a map from ( −∞ , W − ( τ + , − ] into ( −∞ , e w −− ].25rom now on, we fix v ∈ I = [ w + − , w − + ], v + ∈ I + = [ w ++ , ∞ ), and v − ∈ I − = ( −∞ , w −− ],and let V ν ( t ), t ∈ D , be the force point function started from v ν , ν ∈ { , + , −} . By Lemma3.11, V − ≤ C − ≤ D − ≤ V ≤ C + ≤ D + ≤ V + , which combined with (3.12) implies V − ≤ C − ≤ W − ≤ D − ≤ V ≤ C + ≤ W + ≤ D + ≤ V + . (3.14) Lemma 3.16.
For any t = ( t + , t − ) ∈ D , we have | V + ( t ) − V − ( t ) | / ≤ diam( K ( t ) ∪ [ v − , v + ]) ≤ | V + ( t ) − V − ( t ) | . (3.15) f K ( t ) [ V ( t ) , V ν ( t )] ⊂ η ν [0 , t ν ] ∪ [ v , v ν ] , ν ∈ { + , −} (3.16) Here for x, y ∈ R , the [ x, y ] in (3.16) is the line segment connecting x with y , which is the sameas [ y, x ] ; and if any v ν , ν ∈ { , + , −} , takes value w ± σ for some σ ∈ { + , −} , then its appearancein (3.15,3.16) is understood as w σ .Proof. Fix t = ( t + , t − ) ∈ D . We write K for K ( t ), K ± for K ± ( t ± ), e K ± for K t ∓ ± ( t ± ), η ± for η ± [0 , t ± ], e η ± for η t ∓ ± [0 , t ± ], and X for X ( t ), X ∈ { V , V + , V − , C + , C − , D + , D − } .Since g K maps C \ ( K doub ∪ [ v − , v + ]) conformally onto C \ [ V − , V + ], fixes ∞ , and has derivative1 at ∞ , by Koebe’s 1 / ν = +. By (3.14), V ≤ C + ≤ D + ≤ V + . By (3.13), f K [ C + , D + ] ⊂ f K − ( e η + ) = η + .It remains to show that f K ( D + , V + ] ⊂ [ w , v + ] and f K [ V , C + ) ⊂ [ v , w ]. If V + = D + , then( D + , V + ] = ∅ , and f K ( D + , V + ] = ∅ ⊂ [ w + , v + ]. Suppose V + > D + . By Lemma 3.11, D + =lim x ↓ max(( K ∩ R ) ∪{ w + } ) g K ( x ), and V + = g K ( v + ). So f K ( D + , V + ] = (max(( K ∩ R ) ∪ { w + } ) , v + ] ⊂ [ w + , v + ]. If V = C + , then [ V , C + ) = ∅ , and f K [ V , C + ) = ∅ ⊂ [ v , w + ]. If V < C + , byLemma 3.11 (iii,iv), K + ∩ K − = ∅ , v = K + , and C + = lim x ↑ min(( K + ∩ R ) ∪{ w + } ) g K ( x ). Noweither v K ∪ { w + − } and V = g K ( v ), or v ∈ K − ∪ { w + − } and V = D − . In the formercase, f K [ V , C + ) ⊂ [ v , min(( K + ∩ R ) ∪ { w + } )) ⊂ [ v , w + ]. In the latter case, f K [ V , C + ) =[max(( K − ∩ R ) ∪ { w − } ) , min(( K + ∩ R ) ∪ { w + } )) ⊂ [ v , w + ]. Lemma 3.17.
Suppose for some t = ( t + , t − ) ∈ D and σ ∈ { + , −} , η σ ( t σ ) ∈ η − σ [0 , t − σ ] ∪ [ v − σ , v ] . Then W σ ( t ) = V ( t ) .Proof. We assume σ = + by symmetry. By Lemma 3.12, W + ( t ) = g W − ( t + , K t + − ( t − ) ( b w + ( t + )). ByLemma 3.11, V ( t ) = g W − ( t + , K t + − ( t − ) ◦ g w + K + ( t + ) ( v ). If η + ( t + ) ∈ [ v − , v ], then b w + ( t + ) = c K + ( t + ) = g w + K + ( t + ) ( v ), and so we get W σ ( t ) = V ( t ). Now suppose η + ( t + ) ∈ η − [0 , t − ]. Then b w + ( t + ) ∈ K t + − ( t − ), which together with b w + ( t + ) = W + ( t + , ≥ V ( t + ,
0) = g w + K + ( t + ) ( v ) ≥ W − ( t + , g W − ( t + , K t + − ( t − ) ◦ g w + K + ( t + ) ( v ) = g W − ( t + , K t + − ( t − ) ( b w + ( t + )), as desired.26 .3 Ensemble without intersections We say that ( η + , η − ; D ) is disjoint if K + ( t + ) ∩ K − ( t − ) = ∅ for any ( t + , t − ) ∈ D . Given acommuting pair ( η + , η − ; D ), we get a disjoint commuting par ( η + , η − ; D disj ) by defining D disj = { ( t + , t − ) ∈ D : K + ( t + ) ∩ K − ( t − ) = ∅} . (3.17)In this subsection, we assume that ( η + , η − ; D ) is disjoint. Most results appeared earlierin [27, 26]. Here we also need to deal with the cases that some force point may equal to w νσ , σ, ν ∈ { + , −} . We now write g t − σ σ ( t σ , · ) for g K t − σσ ( t σ ) , σ ∈ { + , −} . For ( t + , t − ) ∈ D and σ ∈ { + , −} , from K + ( t + ) ∩ K − ( t − ) = ∅ we know [ c K σ ( t σ ) , d K σ ( t σ ) ] has positive distancefrom K t σ − σ ( t σ )). So g t σ − σ ( t − σ , · ) is analytic at b w σ ( t σ ) ∈ [ c K σ ( t σ ) , d K σ ( t σ ) ]. By Lemma 3.12, W σ ( t + , t − ) = g t σ − σ ( t − σ , b w σ ( t σ )). We further define W σ,j , j = 1 , ,
3, and W σ,S on D by W σ,j ( t + , t − ) = ( g t σ − σ ) ( j ) ( t − σ , b w σ ( t σ )) , W σ,S = W σ, W σ, − (cid:16) W σ, W σ, (cid:17) , σ ∈ { + , −} . (3.18)Here the superscript ( j ) means the j -th complex derivative w.r.t. the space variable. Thefunctions are all continuous on D because ( t + , t − , z ) ( g t σ − σ ) ( j ) ( t − σ , z ) is continuous by Lemma3.5. Note that W σ,S ( t + , t − ) is the Schwarzian derivative of g t σ − σ ( t − σ , · ) at b w σ ( t σ ). Lemma 3.18. m is continuously differentiable with ∂ σ m = W σ, , σ ∈ { + , −} .Proof. This follows from a standard argument, which first appeared in [7, Lemma 2.8]. Thestatement for ensemble of chordal Loewner curves appeared in [27, Formula (3.7)].So for any σ ∈ { + , −} and t − σ ∈ I − σ , K t − σ σ ( t σ ), 0 ≤ t σ < T D σ ( t − σ ), are chordal Loewnerhulls driven by W σ | − σt − σ with speed ( W σ, | − σt − σ ) , and we get the differential equation: ∂ t σ g t − σ σ ( t σ , z ) = 2( W σ, ( t + , t − ) g t − σ σ ( t σ , z ) − W σ ( t + , t − ) , (3.19)which together with Lemmas 3.12 and 3.11 implies the differential equations for V , V + , V − : ∂ σ V ν ae = 2 W σ, V ν − W σ , ν ∈ { , + , −} , (3.20)and the differential equations for W − σ , W − σ, and W − σ,S : ∂ σ W − σ = 2 W σ, W − σ − W σ , ∂ σ W − σ, W − σ, = − W σ, ( W + − W − ) , ∂ σ W − σ,S = − W , W − , ( W + − W − ) . (3.21)Define Q on D by Q ( t ) = exp (cid:16) Z [0 ,t ] − W + , ( s ) W − , ( s ) ( W + ( s ) − W − ( s )) d s (cid:17) . (3.22)27hen Q is continuous and positive with Q ( t + , t − ) = 1 when t + · t − = 0. From (3.21) we get ∂ σ QQ = W σ,S , σ ∈ { + , −} . (3.23)By (3.14), V + ≥ W + ≥ C + ≥ V ≥ D − ≥ W − ≥ V − on D . Because of disjointness, wefurther have C + > D − by Lemma 3.11. By the same lemma, V σ ( t + , t − ) = g t σ − σ ( t − σ , V σ ( t σ e σ )); (3.24) V ( t + , t − ) = g t σ − σ ( t − σ , V ( t σ e σ )) , if v K − σ ( t − σ ) . (3.25)Let t = ( t + , t − ) ∈ D . For σ ∈ { + , −} , differentiating (3.4) w.r.t. t σ , letting b z = g K σ ( t σ ) ( z ),and using Lemma 3.12 and (3.19,3.18) we get ∂ t σ g t σ − σ ( t − σ , b z ) = 2( g t σ − σ ) ′ ( t − σ , b w σ ( t σ )) g t σ − σ ( t − σ , b z ) − g t σ − σ ( t − σ , b w σ ( t σ )) − g t σ − σ ) ′ ( t − σ , b z ) b z − b w σ ( t σ ) . (3.26)Letting H \ K t σ − σ ( t − σ ) ∋ b z → b w σ ( t σ ) and using the power series expansion of g t σ − σ ( t − σ , · ) at b w σ ( t σ ), we get ∂ t σ g t σ − σ ( t − σ , b z ) | b z = b w σ ( t σ ) = − W σ, ( t ) , σ ∈ { + , −} . (3.27)Differentiating (3.26) w.r.t. b z and letting b z → b w σ ( t σ ), we get ∂ t σ ( g t σ − σ ) ′ ( t − σ , b z )( g t σ − σ ) ′ ( t − σ , b z ) (cid:12)(cid:12)(cid:12)(cid:12) b z = b w σ ( t σ ) = 12 (cid:16) W σ, ( t ) W σ, ( t ) (cid:17) − W σ, ( t ) W σ, ( t ) , σ ∈ { + , −} . (3.28)For σ ∈ { + , −} , define W σ,N on D by W σ,N = W σ, W σ, | σ . Since W σ, | − σ ≡
1, we get W σ,N ( t + , t − ) = 1 when t + t − = 0. From (3.21) we get ∂ σ W − σ,N W − σ,N = − W σ, ( W − σ − W σ ) ∂t σ − − W σ, ( W − σ − W σ ) (cid:12)(cid:12)(cid:12)(cid:12) − σ ∂t σ , σ ∈ { + , −} . (3.29)We now define V ,N , V + ,N , V − ,N on D by V ν,N ( t ) = ( g t ν − ν ) ′ ( t − ν , V ν ( t ν e ν )) / ( g − ν ) ′ ( t − ν , v ν ) , ν ∈ { + , −} ; V ,N ( t ) = ( g t σ − σ ) ′ ( t − σ , V ( t σ e σ )) / ( g t σ − σ ) ′ ( t − σ , v ) , if v K − σ ( t − σ ) , σ ∈ { + , −} . (3.30)The functions are well defined because either v K + ( t + ) or v K − ( t − ), and when they bothhold, the RHS of (3.30) equals g ′ K ( t + ,t − ) ( v ) / ( g ′ K + ( t + ) ( v ) g ′ K − ( t − ) ( v )) by (3.4).Note that V ν,N ( t + , t − ) = 1 if t + t − = 0 for ν ∈ { , + , −} . From (3.24-3.25) and (3.4,3.19)we find that these functions satisfy the following differential equations on D : ∂ σ V ν,N V ν,N = − W σ, ( V ν − W σ ) ∂t σ − − W σ, ( V ν − W σ ) (cid:12)(cid:12)(cid:12)(cid:12) − σ ∂t σ , σ ∈ { + , −} , ν ∈ { , − σ } , if v ν K σ ( t σ ) . (3.31)28e now define E X,Y on D for X = Y ∈ { W + , W − , V , V + , V − } as follows. First, let E X,Y ( t + , t − ) = ( X ( t + , t − ) − Y ( t + , t − ))( X (0 , − Y (0 , X ( t + , − Y ( t + , X (0 , t − ) − Y (0 , t − )) , (3.32)if the denominator is not 0. If the denominator is 0, then since V + ≥ W + ≥ V ≥ W − ≥ V − and W + > W − , there is σ ∈ { + , −} such that { X, Y } ⊂ { W σ , V σ , V } . By symmetry, we willonly describe the definition of E X,Y in the case that σ = +. If X ( t + ,
0) = Y ( t + , X ( t + , · ) ≡ Y ( t + , · ). If X (0 , t − ) = Y (0 , t − ), then we must have X (0) = Y (0),and so X (0 , · ) ≡ Y (0 , · ). For the definition of E X,Y , we modify (3.32) by writing the RHS as X ( t + ,t − ) − Y ( t + ,t − ) X ( t + , − Y ( t + , : X (0 ,t − ) − Y (0 ,t − ) X (0 , − Y (0 , , replacing the first factor (before “:”) by ( g t + − ) ′ ( t − , X ( t + , X ( t + ,
0) = Y ( t + , g ′ K − ( t − ) ( X (0 , X (0 , t − ) = Y (0 , t − ); and do both replacements when two equalities both hold. Then in allcases, E X,Y is continuous and positive on D , and E X,Y ( t + , t − ) = 1 if t + · t − = 0. By (3.20,3.21),for σ ∈ { + , −} , if X, Y = W σ , then ∂ σ E X,Y E X,Y ae = − W σ, ( X − W σ )( Y − W σ ) ∂t σ − − W σ, ( X − W σ )( Y − W σ ) (cid:12)(cid:12)(cid:12) − σ ∂t σ . (3.33) In this subsection we do not assume that ( η + , η − ; D ) is disjoint. Let v ν and V ν , ν ∈ { , + , −} ,be as before. We assume in this subsection that v + − v = v − v − >
0. Define Λ and Υ on D by Λ = log V + − V V − V − and Υ = log V + − V − v + − v − . By assumption, Λ(0) = Υ(0) = 0. Lemma 3.19.
There exists a unique continuous and strictly increasing function u : [0 , T u ) →D , for some T u ∈ (0 , ∞ ] , with u (0) = 0 , such that for any ≤ t < T u and σ ∈ { + , −} , | V σ ( u ( t )) − V ( u ( t )) | = e t | v σ − v | ; and u can not be extended beyond T u with such property.Sketch of the proof. We use an argument that is similar to that of [21, Section 4]. Since V + ≥ W + ≥ V ≥ W − ≥ V − , by the definition of V ν , Proposition 2.12 and Lemma 3.11, for σ ∈ { + , −} , | V σ − V | and | V σ − V − σ | are strictly increasing in t σ , and | V − V − σ | is strictly decreasing in t σ .Thus, Λ is strictly increasing in t + and strictly decreasing in t − , and Υ is strictly increasingin both t + and t − . These monotone properties guarantee the existence and uniqueness of u : [0 , T u ) → D with Λ( u ( t )) = 0 and Υ( u ( t )) = t for all t . Lemma 3.20.
For any t ∈ [0 , T u ) and σ ∈ { + , −} , e t | v + − v − | / ≤ rad v ( η σ [0 , u σ ( t )] ∪ [ v , v σ ]) ≤ e t | v + − v − | . (3.34) If T u < ∞ , then lim t ↑ T u u ( t ) converges to a point in ∂ D ∩ (0 , ∞ ) . If D = R , then T u = ∞ .If T u = ∞ , then diam( η + ) = diam( η − ) = ∞ . roof. Fix t ∈ [0 , T u ). For σ ∈ { + , . −} , let S σ = [ v , v σ ] ∪ η σ [0 , u σ ( t )] ∪ η σ [0 , u σ ( t )], where thebar stands for complex conjugation, and L σ = rad v ( S σ ). From (3.15) and that | V + ( u ( t )) − V − ( u ( t )) | = e t | v + − v − | , we get e t | v + − v − | / ≤ L + ∨ L − ≤ e t | v + − v − | . Since V + ( u ( t )) − V ( u ( t )) = V ( u ( t )) − V − ( u ( t )), by Lemma 3.16, S + and S − have the same harmonic measureviewed from ∞ . By Beurling’s estimate, L + ∨ L − ≤ L + ∧ L − ). So we get (3.34). Forany σ ∈ { + , −} , u σ ( t ) = hcap ( η σ [0 , u σ ( t )]) ≤ L σ ≤ e t | v + − v − | . Suppose T u < ∞ . Then u + and u − are bounded on [0 , T u ). Since u is increasing, lim t ↑ T u u ( t ) converges to a point in(0 , ∞ ) , which must lie on ∂ D because u cannot be extended beyond T u . If D = R , then ∂ D ∩ (0 , ∞ ) = ∅ , and so T u = ∞ . If T u = ∞ , then by (3.34), diam( η ± ) = ∞ .Define X u = X ◦ u . Let I = | v + − v | = | v − − v | . From the definition of u , we have | V u ± ( t ) − V u ( t ) | = e t I for any t ∈ [0 , T u ). Let R σ = W uσ − V u V uσ − V u ∈ [0 , σ ∈ { + , −} , and R = ( R + , R − ). Let e c · denote the function t e ct for c ∈ R . Lemma 3.21.
Let D disj be defined by (3.17). Let T u disj ∈ (0 , T u ] be such that u ( t ) ∈ D disj for ≤ t < T u disj . Then u is continuously differentiable on [0 , T u disj ) , and ( W uσ, ) u ′ σ = R σ (1 − R σ ) R + + R − e · I on [0 , T u disj ) , σ ∈ { + , −} . (3.35) Proof.
By (3.20), Λ and Υ satisfy the following differential equations on D disj : ∂ σ Λ ae = ( V + − V − ) W σ, Q ν ∈{ , + , −} ( V uν − W uσ ) and ∂ σ Υ ae = − W σ, Q ν ∈{ + , −} ( V uν − W uσ ) , σ ∈ { + , −} . From Λ u ( t ) = 0 and Υ u ( t ) = t , we get X σ ∈{ + , −} ( W uσ, ) u ′ σ Q ν ∈{ , + , −} ( V uν − W uσ ) ae = 0 and X σ ∈{ + , −} − ( W uσ, ) u ′ σ Q ν ∈{ + , −} ( V uν − W uσ ) ae = 1 . Solving the system of equations, we get ( W uσ, ) u ′ σ ae = ( Q ν ∈{ , + , −} ( V uν − W uσ )) / ( W σ − W − σ ), σ ∈ { + , −} . Using V uσ − V u = σe · I and W uσ − V u = R σ ( V uσ − V u ), we find that (3.35) holdswith “ ae =” in place of “=”. Since W + > W − on D disj , we get R + + R − > , T u disj ). So theoriginal (3.35) holds by the continuity of its RHS.Now suppose that η + and η − are random curves, and D is a random region. Then u and T u are also random. Suppose that there is an R -indexed filtration F such that D is an F -stoppingregion, and V , V + , V − are all F -adapted. Now we extend u to R + such that if T u < ∞ , then u ( s ) = lim t ↑ T u u ( t ) for s ∈ [ T u , ∞ ). The following proposition is similar to [21, Lemma 4.1]. Proposition 3.22.
For every t ∈ R + , u ( t ) is an F -stopping time. Since u is non-decreasing, we get an R + -indexed filtration F u : F ut = F u ( t ) , t ≥
0, byPropositions 2.21 and 3.22. 30
Commuting Pair of SLE κ (2 , ρ ) Curves
In this section, we apply the results from the previous section to study a pair of commutingSLE κ (2 , ρ ) curves, which arise as flow lines of a GFF with piecewise constant boundary data(cf. [11]). The results of this section will be used in the next section to study commuting pairof hSLE κ curves that we are mostly interested in. Throughout this section, we fix κ, ρ , ρ + , ρ − such that κ ∈ (0 , ρ + , ρ − > max {− , κ − } , ρ ≥ κ −
2, and ρ + ρ σ ≥ κ − σ ∈ { + , −} . Let w − < w + ∈ R . Let v + ∈ [ w ++ , ∞ ), v − ∈ ( −∞ , w −− ], and v ∈ [ w + − , w − + ]. Write ρ for ( ρ , ρ + , ρ − ). From ([11]) we know that thereis a coupling of two chordal Loewner curves η + ( t + ), 0 ≤ t + < ∞ , and η − ( t − ), 0 ≤ t − < ∞ ,driven by b w + and b w − (with speed 1), respectively, with the following properties.(A) For σ ∈ { + , −} , η σ is a chordal SLE κ (2 , ρ ) curve in H started from w σ with force pointsat w − σ and v ν , ν ∈ { , + , −} . Here if any v ν equals w ±− σ , then as a force point for η σ , it istreated as w − σ . Let b w σ − σ , b v σν denote the force point functions for η σ started from w − σ , v ν , ν ∈ { , + , −} , respectively.(B) Let σ ∈ { + , −} . If τ − σ is a finite stopping time w.r.t. the filtration F − σ generated by η − σ , then a.s. there is a chordal Loewner curve η t − σ σ ( t ), 0 ≤ t < ∞ , with some speed suchthat η σ = f K − σ ( τ − σ ) ◦ η τ − σ σ . Moreover, the conditional law of the normalization of η τ − σ σ given F − στ − σ is that of a chordal SLE κ (2 , ρ ) curve in H started from b w − σσ ( τ − σ ) with forcepoints at b w − σ ( τ − σ ) , b v − σν ( τ − σ ), ν ∈ { , + , −} .In fact, one may construct η + and η − as two flow lines of the same GFF on H with some piecewiseboundary conditions (cf. [11]). The conditions that κ ∈ (0 , ρ , ρ + , ρ − > max {− , κ − } and ρ + ρ σ ≥ κ − σ ∈ { + , −} , ensure that (i) there is no continuation threshold for either η + or η − , and so η + and η − both have lifetime ∞ and η ± ( t ) → ∞ as t → ∞ ; (ii) η + does nothit ( −∞ , w − ], and η − does not hit [ w + , ∞ ); and (iii) η ± ∩ R has Lebesgue measure zero. Thestronger condition that ρ ≥ κ − ρ > max {− , κ − } ) will be neededlater (see Remark 4.11). We call the above ( η + , η − ) a commuting pair of chordal SLE κ (2 , ρ )curves in H started from ( w + , w − ; v , v + , v − ). If ρ = 0, which satisfies ρ > κ − κ < v does not play a role, and we omit ρ and v in the name.We may take τ − σ in (B) to be a deterministic time. So for each t − σ ∈ Q + , a.s. there is anSLE κ -type curve η t − σ σ defined on R + such that η σ = f K − σ ( t − σ ) ◦ η t − σ σ . The conditions on κ and ρ implies that a.s. the Lebesgue measure of η t − σ σ ∩ R is 0. This implies that a.s. η + and η − satisfythe conditions in Definition 3.2 with I + = I − = R + , I ∗ + = I ∗− = Q + , and D = R . So ( η + , η − )is a.s. a commuting pair of chordal Loewner curves. Here we omit D when it is R . Let K and m be the hull function and the capacity function, W + , W − be the driving functions, and V , V + , V − be the force point functions started from v , v + , v − , respectively. Then b w σ = W σ | − σ , b w σ − σ = W − σ | − σ , and b v σν = V ν | − σ , ν ∈ { , + , −} . For each F − σ -stopping time τ − σ , η τ − σ σ is the31hordal Loewner curve driven by W σ | − στ − σ with speed m | − στ − σ , and the force point functions are W − σ | − στ − σ and V ν | − στ − σ , ν ∈ { , + , −} .Let F ± be the R + -indexed filtration as in (B). Let F be the separable R -indexed filtrationgenerated by F + and F − . From (A) we know that, for σ ∈ { + , −} , there exists a standard F σ -Brownian motions B σ such that the driving functions b w σ satisfies the SDE d b w σ ae = √ κdB σ + h b w σ − b w σ − σ + X ν ∈{ , + , −} ρ ν b w σ − b v σν i dt σ . (4.1) Lemma 4.1 (Two-curve DMP) . Let G be a σ -algebra. Let D = R . Suppose that, con-ditionally on G , ( η + , η − ; D ) is a commuting pair of chordal SLE κ (2 , ρ ) curves started from ( w + , w − ; v , v + , v − ) , which are G -measurable random points. Let K, W σ , V ν , σ ∈ { + , −} , ν ∈{ , + , −} , be respectively the hull function, driving functions, and force point functions. Let F σ be the R + -indexed filtration defined by F σt = σ ( G , η σ [0 , t ]) , t ≥ , σ ∈ { + , −} . Let F be the right-continuous augmentation of the separable R -indexed filtration generated by F + and F − . Let τ be an F -stopping time. Then on the event E τ := { τ ∈ R , η σ ( τ σ ) η − σ [0 , τ − σ ] , σ ∈ { + , −}} ,there is a random commuting pair of chordal Loewner curves ( e η , e η ; e D ) with some speeds,which is the part of ( η + , η − ; D ) after τ up to a conformal map (Definition 3.14), and whosenormalization conditionally on F τ ∩ E τ has the law of a commuting pair of chordal SLE κ (2 , ρ ) curves started from ( W + , W − ; V , V + , V − ) | τ . Here if V ν ( τ ) = W σ ( τ ) for some σ ∈ { + , −} and ν ∈ { , + , −} , then as a force point V ν ( τ ) is treated as W σ ( τ ) sign( v ν − w σ ) .Proof. This lemma is similar to [20, Lemma A.5], which is about the two-directional DMP ofchordal SLE κ for κ ≤
8. The argument also works here. See [20, Remark A.4].
Write w and v respectively for ( w + , w − ) and ( v , v + , v − ). Let P ρw ; v denote the joint law of thedriving functions ( b w + , b w − ) of a commuting pair of chordal SLE κ (2 , ρ ) curves in H started from( w ; v ). Now we fix w and v , and omit the subscript in the joint law.The P ρ is a probability measure on Σ , where Σ := S
For any ξ ∈ Ξ and R > , there is a constant C > depending only on κ, ρ, ξ, R ,such that if | v + − v − | ≤ R , then | log M | ≤ C on [0 , τ ξ ] .Proof. Fix ξ = ( ξ + , ξ − ) ∈ Ξ and
R >
0. Suppose | v + − v − | ≤ R . Throughout the proof, aconstant is a number that depends only on ξ, R ; and a function defined on [0 , τ ξ ] is said tobe uniformly bounded if its absolute value on [0 , τ ξ ] is bounded above by a constant. By thedefinition of M , it suffices to prove that | log Q | , | log E Y ,Y | , Y = Y ∈ { W + , W − , V , V + , V − } , | log W σ,N | , σ ∈ { + , −} , and | log V ν,N | , ν ∈ { , + , −} , are all uniformly bounded.Let K ξ σ = Hull( ξ σ ), σ ∈ { + , −} and K ξ = K ξ + ∪ K ξ − . Let I = (max( ξ − ∩ R ) , min( ξ + ∩ R )).Then | g K ξ ( I ) | is a positive constant. By symmetry we assume that either v ∈ K ξ − or v ∈ I and g K ξ ( v ) is no more than the middle of g K ξ ( I ). So we may pick v < v ∈ I with v ≤ v such that | g K ξ ( v ) − g K ξ ( v ) | ≥ | g K ξ ( I ) | /
3. Let V j be the force point function started from v j , j = 1 ,
2. By (3.14), V + ≥ W + ≥ V > V ≥ V ≥ W − ≥ V − on [0 , τ ξ ].By Proposition 2.2, W + , , W − , are uniformly bounded by 1. For σ ∈ { + , −} , the function( t + , t − ) t σ is bounded on [0 , τ ξ ] by hcap ( K ξ ). For any t ∈ [0 , τ ξ ], since g K ξ = g K ξ /K ( t ) ◦ g K ( t ) ,by Proposition 2.2 we get 0 < g ′ K ξ ≤ g ′ K ( t ) ≤ v , v ]. Since V j ( t ) = g K ( t ) ( v j ), j = 1 , | V ( t ) − V ( t ) | ≥ | g K ξ ( v ) − g K ξ ( v ) | ≥ | g K ξ ( I ) | /
3. So V − V is uniformly bounded,which then implies that | W σ − W − σ | and | W σ − V − σ | are uniformly bounded, σ ∈ { + , −} . From(3.22) we see that | log Q | is uniformly bounded. From (3.29,3.31) and the fact that W − σ,N | σ = V − σ,N | σ = 1, we see that | log W − σ,N | and | log V − σ,N | , σ ∈ { + , −} , are uniformly bounded. Wealso know that | W + − V | ≤ | V − V | is uniformly bounded. From (3.31) with ν = 0 and σ = +and the fact that V ,N | +0 ≡ | log V ,N | is uniformly bounded.Now we estimate | log E Y ,Y | . By (3.15), | V + − V − | is uniformly bounded. Thus, for any Y = Y ∈ { W + , W − , V , V + , V − } , | Y − Y | ≤ | V + − V − | is uniformly bounded. If Y ∈ { W + , V + } and Y ∈ { W − , V − } , then | Y − Y | ≤ | V − V | is uniformly bounded. From (3.32) we see that | log E Y ,Y | is uniformly bounded. If Y , Y ∈ { W − σ , V − σ } for some σ ∈ { + , −} , then | Y j − W σ | , j = 1 ,
2, are uniformly bounded, and then the uniformly boundedness of | log E Y ,Y | followsfrom (3.33) and the fact that E Y ,Y | σ ≡
1. Finally, we consider the case that Y = V . If Y ∈ { W + , V + } , then | Y − V | ≤ | V − V | , which is uniformly bounded. We can again use (3.32)to get the uniformly boundedness of | log E V ,Y | . If Y ∈ { W − , V − } , then | V − W + | and | Y − W + | are uniformly bounded. The uniformly boundedness of | log E V ,Y | then follows from (3.33)with σ = +, X = V , Y = Y , and the fact that E V ,Y | +0 ≡ Corollary 4.3.
For any ξ ∈ Ξ , M ( · ∧ τ ξ ) is an F -martingale closed by M ( τ ξ ) w.r.t. P ρi . roof. This follows from (4.7), Lemma 4.2, and the same argument for [21, Corollary 3.2].
Lemma 4.4.
For any ξ = ( ξ + , ξ − ) ∈ Ξ , P ρ is absolutely continuous w.r.t. P ρi on F τ ξ ∨ F τ ξ ,and the RN derivative is M ( τ ξ ) .Proof. Let ξ = ( ξ + , ξ − ) ∈ Ξ. The above corollary implies that E ρi [ M ( τ ξ )] = M (0) = 1. So wemay define a new probability measure P ρξ by d P ρξ = M ( τ ξ ) d P ρi .Since M ( t + , t − ) = 1 when t + t − = 0, from the above corollary we know that the marginallaws of P ρξ agree with that of P ρi , which are P ρ + and P ρ − . Suppose ( b w + , b w − ) follows the law P ρξ . Then they satisfy Condition (A) in Section 4.1. Now we write τ ± for τ ± ξ ± , and τ for τ ξ .Let σ − ≤ τ − be an F − -stopping time. From Lemma 2.24 and Corollary 4.3, d P ρξ |F ( t + ,σ − ) d P ρi |F ( t + ,σ − ) = M ( t + ∧ τ + , σ − ), 0 ≤ t + < ∞ . From Girsanov Theorem and (4.1,4.7), we see that, under P ρξ , b w + satisfies the following SDE up to τ + : d b w + = √ κdB τ − + + κ b W + , W + , (cid:12)(cid:12)(cid:12) − τ − dt + + 2 W + , W + − W − (cid:12)(cid:12)(cid:12) − τ − dt + + X ν ∈{ , + , −} ρ ν W + , W + − V ν (cid:12)(cid:12)(cid:12) − τ − dt + , where B τ − + is a standard ( F ( t + ,σ − ) ) t + ≥ -Brownian motion under P ρξ . Using Lemma 3.12 and(3.27) we find that W + ( · , σ − ) under P ρξ satisfies the following SDE up to τ + : dW + | − σ − ae = √ κW + , | − σ − dB σ − + + 2 W , W + − W − (cid:12)(cid:12)(cid:12)(cid:12) − σ − dt + + X ν ∈{ , + , −} ρ ν W , W + − V ν (cid:12)(cid:12)(cid:12)(cid:12) − σ − dt + . (4.8)Note that the SDE (4.8) agrees with the SDE for W + ( · , σ − ) if ( η + , η − ) is a commuting pairof chordal SLE κ (2 , ρ ) curves started from ( w ; v ), where the speed is W + , ( · , σ − ) . There is asimilar SDE for W − ( σ + , · ) if σ + ≤ τ + is an F + -stopping time. Thus, P ρξ agrees with P ρ on F τ ξ ∨ F τ ξ , which implies the conclusion of the lemma. Corollary 4.5. If T is an F -stopping time, then P ρ is absolutely continuous w.r.t. P ρi on F T ∩ { T ∈ D disj } , and the RN derivative is M ( T ) . In other words, if A ∈ F T and A ⊂ { T ∈D disj } , then P ρ [ A ] = E ρi [ A M ( T )] .Proof. Since { T ∈ D disj } = S ξ ∈ Ξ ∗ { T < τ ξ } and Ξ ∗ is countable, it suffices to prove the state-ment with { T < τ ξ } in place of { T ∈ D disj } for every ξ ∈ Ξ ∗ . Fix ξ = ( ξ + , ξ − ) ∈ Ξ ∗ . We write F ξ for F + τ + ξ + ∨ F − τ − ξ − . Let A ∈ F T ∩ { T < τ ξ } . Fix t = ( t + , t − ) ∈ Q . Let A t = A ∩ { T ≤ t < τ ξ } .For every B + ∈ F + t + and B − ∈ F − t − , B + ∩ B − ∩ { t < τ ξ } ∈ F + τ + ξ + ∨ F − τ − ξ − = F ξ . Using a monotoneclass argument, we conclude that F t ∩ { t < τ ξ } ∈ F ξ . Thus, A t ∈ F t ∩ { t < τ ξ } ⊂ F ξ . Since A = S t ∈ Q A t , we get A ∈ F ξ . By Lemma 4.4, Proposition 2.24, and the martingale propertyof M ( · ∧ τ ξ ), we get E ρ [ A ] = E ρi [ A M ( τ ξ )] = E ρi [ A M ( T ∧ τ ξ )] = E ρi [ A M ( T )].35 .3 SDE along a time curve up to intersection Now assume that v + − v = v − v − . Let u = ( u + , u − ) : [0 , T u ) → R be as in Section 3.4. ByLemma 3.20, a.s. T u = ∞ . By Proposition 3.22, u ( t ) is an ( F t )-stopping time for each t ≥ R + -indexed filtration F u by F ut := F u ( t ) , t ≥
0. For ξ = ( ξ + , ξ − ) ∈ Ξ, let τ uξ denote the first t ≥ u ( t ) = τ ξ or u ( t ) = τ ξ , whichever comes first. Note thatsuch time exists and is finite because ( τ ξ , τ ξ ) ∈ D . The following proposition has the sameform as [21, Lemma 4.2]. Proposition 4.6.
For every ξ ∈ Ξ , τ uξ is an F u -stopping time, and u ( τ uξ ) and u ( t ∧ τ uξ ) , t ≥ ,are all F -stopping times. Assume that ( b w + , b w − ) follows the law P ρi . Let η ± be the chordal Loewner curve driven by b w ± . Let D disj be as before. Let b w σ − σ ( t ) and b v σν ( t ) be the force point functions for η σ startedfrom w − σ and v ν respectively, ν ∈ { , + , −} , σ ∈ { + , −} . Define b B σ , σ ∈ { + , −} , by √ κ b B σ ( t ) = b w σ ( t ) − w σ − Z t ds b w σ ( s ) − b w σ − σ ( s ) − X ν ∈{ , + , −} Z t ρ ν ds b w σ ( s ) − b v σν ( s ) . (4.9)Then b B + and b B − are independent standard Brownian motions. So we get five F -martingaleson D disj : b B + ( t + ), b B − ( t − ), b B + ( t + ) − t + , b B − ( t − ) − t − , and b B + ( t + ) b B − ( t − ). Fix ξ ∈ Ξ. UsingPropositions 2.24 and 3.22 and the facts that u ± is uniformly bounded above on [0 , τ ξ ], weconclude that b B uσ ( t ∧ τ uξ ), b B uσ ( t ∧ τ uξ ) − u σ ( t ∧ τ uξ ), σ ∈ { + , −} , and b B u + ( t ∧ τ uξ ) b B u − ( t ∧ τ uξ ) areall F u -martingales under P ρi . Recall that for a function X defined on D , we use X u to denotethe function X ◦ u defined on [0 , T u ). This rule applies even if X depends only on t + or t − (for example, b B uσ ( t ) = b B σ ( u σ ( t ))); but does not apply to F u , T u , T u disj , τ uξ . Thus, the quadraticvariation and covariation of b B u + and b B u − satisfy h b B u + i t ae = u + ( t ) , h b B u − i t ae = u − ( t ) , h b B u + , b B u − i t = 0 , (4.10)up to τ uξ . By Corollary 4.3 and Proposition 2.24, M u ( · ∧ τ uξ ) is an F u -martingale. Let T u disj denote the first t such that u ( t )
6∈ D disj . Since T u disj = sup ξ ∈ Ξ τ uξ = sup ξ ∈ Ξ ∗ τ uξ , and Ξ ∗ iscountable, we see that, T u disj is an F u -stopping time. We now compute the SDE for M u upto T u disj in terms of b B u + and b B u − . Using (4.6) we may express M u as a product of severalfactors, among which E uW + ,W − , ( W uσ,N ) b , ( E uW σ ,V ν ) ρ ν /κ , σ ∈ { + , −} , ν ∈ { , + , −} , contributethe local martingale part, and other factors are differentiable in t . For σ ∈ { + , −} , since W σ ( t + , t − ) = g K tσ − σ ( t − σ ) ( b w σ ( t σ )), using (3.27) we get the F u -adapted SDEs: dW uσ = W uσ, d b w uσ + (cid:16) κ − (cid:17) W uσ, u ′ σ dt + 2( W u − σ, ) u ′− σ W uσ − W u − σ dt, (4.11)36ince W σ,N = W σ, W σ, | σ , W σ ( t + , t − ) = g ′ K tσ − σ ( t − σ ) ( b w σ ( t σ )), W σ | σ is differentiable in t − σ , and g ′ K tσ − σ ( t − σ ) is differentiable in both t σ and t − σ , we get the SDE for ( W uσ,N ) b : d ( W uσ,N ) b ( W uσ,N ) b ae = b W uσ, W uσ, √ κd b B uσ + drift terms . For the SDE for ( E uW + ,W − ) κ , note that when X = W + and Y = W − , the numerators anddenominators in (3.32) never vanish. So using (4.11) we get d ( E uW + ,W − ) κ ( E uW + ,W − ) κ = 2 κ X σ ∈{ + , −} h W uσ, W uσ − W u − σ − b w uσ − ( b w σ − σ ) u i √ κd b B uσ + drift terms . Note that E uW σ ,V ν ( t ) equals f ( u ( t ) , b w uσ ( t ) , ( b v σν ) u ( t )) times a differential function in u − σ ( t ), where f ( · , · , · ) is given by (4.4). Using (4.11) we get the SDE for ( E uW σ ,V ν ) ρνκ : d ( E uW σ ,V ν ) ρνκ ( E uW σ ,V ν ) ρνκ ae = ρ ν κ h W uσ, W uσ − V uν − b w uσ − ( b v σν ) u i √ κd b B uσ + drift terms . Here if at any time t , ( b v σν ) u ( t ) = b w uσ ( t ), then the function inside the square brackets is understoodas W uσ, ( t ) W uσ, ( t ) , which is the limit of the function as ( b v σν ) u ( t ) → b w uσ ( t ).Combining the above displayed formulas and using the fact that M u and b B u ± are all F u -localmartingales under P ρi , we get dM u M u ae = X σ ∈{ + , −} (cid:20) κ b W uσ, W uσ, + 2 h W uσ, W uσ − W u − σ − b w uσ − ( b w σ − σ ) u i ++ X ν ∈{ , + , −} ρ ν h W uσ, W uσ − V uν − b w uσ − ( b v σν ) u i(cid:21) d b B uσ √ κ . (4.12)From Corollary 4.5 and Proposition 4.6 we know that, for any ξ ∈ Ξ and t ≥ d P ρ |F u ( t ∧ τ uξ ) d P ρi |F u ( t ∧ τ uξ ) = M u ( t ∧ τ uξ ) . (4.13)We will use a Girsanov argument to derive the SDEs for b w u + and b w u − up to T u disj under P ρ .For σ ∈ { + , −} , define a process e B uσ ( t ) such that e B u ( t ) = 0 and d e B uσ = d b B uσ − (cid:20) κ b W uσ, W uσ, + h W uσ, W uσ − W u − σ − b w uσ − ( b w σ − σ ) u i + X ν ∈{ , + , −} h ρ ν W uσ, W uσ − V uν − ρ ν b w uσ − ( b v σν ) u i(cid:21) u ′ σ ( t ) √ κ dt. (4.14)37 emma 4.7. For any σ ∈ { + , −} and ξ ∈ Ξ , | e B uσ | is bounded on [0 , τ uξ ] by a constant dependingonly on κ, ρ, w, v, ξ .Proof. Throughout the proof, a positive number that depends only on κ, ρ, w, v, ξ is called aconstant. By Proposition 2.3, V + and V − are bounded in absolute value by a constant on[0 , τ ξ ], and so are W + , V , W − because V + ≥ W + ≥ V ≥ W − ≥ V − . It is clear that b B uσ ( t ) = U ( u σ ( t ) e σ ) − U (0), σ ∈ { + , −} , where U := W + + W − + P ν ∈{ , + , −} ρ ν V ν . Thus, b B uσ , σ ∈ { + , −} ,are bounded in absolute value by a constant on [0 , τ uξ ]. By (3.15) and that V u + ( t ) − V u − ( t ) = e t ( v + − v − ) for 0 ≤ t < T u , we know that e τ uξ ≤ ξ + ∪ ξ − ∪ [ v − , v + ]) / | v + − v − | . Thismeans that τ uξ is bounded above by a constant. Since u [0 , τ uξ ] ⊂ [0 , τ ξ ], it remains to show that,for σ ∈ { + , −} , W σ, W σ, , W σ, W σ − W − σ − b w σ − b w σ − σ , W σ, W σ − V ν − b w σ − b v σν , ν ∈ { , + , −} , are all bounded in absolute value on [0 , τ ξ ] by a constant.Because b w σ − b w σ − σ = W σ, W σ − W − σ (cid:12)(cid:12)(cid:12) − σ , the boundedness of W σ, W σ − W − σ − b w σ − b w σ − σ on [0 , τ ξ ] simplyfollows from the boundedness of W σ, W σ − W − σ , which in turn follows from 0 ≤ W σ, ≤ | W σ − W − σ | is bounded from below on [0 , τ ξ ] by a positive constant, where the latter boundwas given in the proof of Lemma 4.2.For the boundedness of W σ, W σ, on [0 , τ ξ ], we assume σ = + by symmetry. Since W + ,j ( t + , t − ) = g ( j ) K t + − ( t − ) ( b w + ( t + )), j = 1 ,
2, and K t + − ( · ) are chordal Loewner hulls driven by W − ( t + , · ) with speed W − , ( t + , · ) , by differentiating g ′′ K t + − ( t − ) /g ′ K t + − ( t − ) at b w + ( t + ) w.r.t. t − , we get W + , ( t + , t − ) W + , ( t + , t − ) = Z t − W − , W + , ( W + − W − ) (cid:12)(cid:12)(cid:12)(cid:12) ( t + ,s − ) ds. From the facts that 0 ≤ W + , , W − , ≤ | W + − W − | is bounded from below by aconstant on [0 , τ ξ ], we get the boundedness of W + , W + , .For the boundedness of W σ, W σ − V ν − b w σ − b v σν , we assume by symmetry that σ = +. By differen-tiating w.r.t. t − and using (3.20,3.21), we get W + , ( t + , t − ) W + ( t + , t − ) − V ν ( t + , t − ) − b w + ( t + ) − b v + ν ( t + ) = Z t − W − , W + , ( W + − W − ) ( V ν − W − ) (cid:12)(cid:12)(cid:12)(cid:12) ( t + ,s − ) ds. Since 0 ≤ W + , ≤ | W + − W − | is bounded from below by a constant on [0 , τ ξ ], and V ν − W − does not change sign (but could be 0), it suffices to show that (cid:12)(cid:12) R t − W − , V ν − W − | ( t + ,s − ) ds (cid:12)(cid:12) is boundedby a constant on [0 , τ ξ ]. This holds because the integral equals V ν ( t + , t − ) − V ν ( t + ,
0) by (3.20),and | V ν | is bounded by a constant on [0 , τ ξ ]. 38 emma 4.8. Under P ρ , there is a stopped planar Brownian motion B ( t ) = ( B + ( t ) , B − ( t )) , ≤ t < T u disj , such that, for σ ∈ { + , −} , b w uσ satisfies the SDE d b w uσ ae = p κu ′ σ dB σ + h κ b W uσ, W uσ, + 2 W uσ, W uσ − W u − σ + X ν ∈{ , + , −} ρ ν W uσ, W uσ − V uν i u ′ σ dt, ≤ t < T u disj . Here by saying that ( B + ( t ) , B − ( t )) , ≤ t < T u disj , is a stopped planar Brownian motion, wemean that B + ( t ) and B − ( t ) , ≤ t < T u disj , are local martingales with d h B σ i t = t , σ ∈ { + , −} , d h B + , B − i t = 0 , ≤ t < T u disj .Proof. For σ ∈ { + , −} , define e B uσ using (4.14). By (4.12), e B uσ ( t ) M u ( t ), 0 ≤ t < T u disj , is an F u -local martingale under P ρi . By Lemmas 4.2 and 4.7, for any ξ ∈ Ξ, e B uσ ( ·∧ τ uξ ) M u ( ·∧ τ uξ ) is an F u -martingale under P ρi . Since this process is ( F u ( ·∧ τ uξ ) )-adapted, and F u ( t ∧ τ uξ ) ⊂ F u ( t ) = F ut , itis also an ( F u ( ·∧ τ uξ ) )-martingale. From (4.13) we see that e B uσ ( · ∧ τ uξ ), is an ( F u ( ·∧ τ uξ ) )-martingaleunder P ρ . Then we see that e B uσ ( · ∧ τ uξ ) is an F u -martingale under P ρ since for any t ≥ t ≥ A ∈ F ut = F u ( t ) , A ∩ { t ≤ τ uξ } ⊂ F u ( t ∧ τ uξ ) , and on the event { t > τ uξ } , e B uσ ( t ∧ τ uξ ) = e B uσ ( t ∧ τ uξ ). Since T u disj = sup ξ ∈ Ξ ∗ τ uξ , we see that, for σ ∈ { + , −} , e B uσ ( t ), 0 ≤ t < T u disj , is an F u -local martingale under P ρ .From (4.10) and that for any ξ ∈ Ξ ∗ and t ≥ P ρ ≪ P ρi on F u ( t ∧ τ uξ ) , we know that, under P ρ , (4.10) holds up to τ uξ . Since T u disj = sup ξ ∈ Ξ ∗ τ uξ , and e B u ± − b B u ± are differentiable, (4.10) holdsfor e B u + and e B u − under P ρ up to T u disj . Since e B u + and e B u − up to T u disj are local martingales under P ρ ,the (4.10) for e B u ± implies that there exists a stopped planar Brownian motion ( B + ( t ) , B − ( t )),0 ≤ t < T u disj , under P ρ , such that d e B uσ ( t ) = p u ′ σ ( t ) dB σ ( t ), σ ∈ { + , −} . Combining this factwith (4.9) and (4.14), we then complete the proof.From now on, we work under the probability measure P ρ . Combining Lemma 4.8 with (4.11)and (3.20), we get an SDE for W uσ − V u up to T u disj : d ( W uσ − V u ) ae = W uσ, p κu ′ σ dB σ + X ν ∈{ , + , −} ρ ν ( W uσ, ) u ′ σ W uσ − V uν dt + 2( W uσ, ) u ′ σ W uσ − W u − σ dt + 2( W u − σ, ) u ′− σ W uσ − W u − σ dt + 2( W uσ, ) u ′ σ W uσ − V u dt + 2( W u − σ, ) u ′− σ W u − σ − V u dt. Recall that R σ = W uσ − V u V uσ − V u ∈ [0 , σ ∈ { + , −} , and V uσ − V u = σe · I . Combining the aboveSDE with (3.35) and using the continuity of R σ and the positiveness of R + + R − (because W u + > W u − ), we find that R σ , σ ∈ { + , −} , satisfy the following SDE up to T u disj : dR σ = σ s κR σ (1 − R σ ) R + + R − dB σ + (2 + ρ ) − ( ρ σ − ρ − σ ) R σ − ( ρ + + ρ − + ρ + 6) R σ R + + R − dt. (4.15)39 .4 SDE in the whole lifespan We are going to prove the following theorem in this section.
Theorem 4.9.
Under P ρ , R + and R − satisfy (4.15) throughout R + for a pair of independentBrownian motions B + and B − . Lemma 4.10.
Suppose that R + and R − are [0 , -valued semimartingales satisfying (4.15) fora stopped planar Brownian motion ( B + , B − ) up to some stopping time τ . Then on the event { τ < ∞} , a.s. lim t ↑ τ R ± ( t ) converges and does not equal .Proof. Let X = R + − R − and Y = 1 − R + R − . Then | X | ≤ Y ≤ Y ± X =(1 ± R + )(1 ∓ R − ) ≥
0. By (4.15), X and Y satisfy the following SDEs up to τ : dX = dM X − [( ρ + + ρ − + ρ + 6) X + ( ρ + − ρ − )] dt, (4.16) dY = dM Y − [( ρ + + ρ − + ρ + 6) Y − ( ρ + + ρ − + 4)] dt, (4.17)where M X and M Y are local martingales whose quadratic variation and covariation satisfy thefollowing equations up to τ : d h M X i = κ ( Y − X ) dt, d h M X , M Y i = κ ( X − XY ) dt, d h M Y i = κ ( Y − Y ) dt. (4.18)By (4.18), h M X i τ , h M Y i τ ≤ κτ , which implies that lim t ↑ τ M X ( t ) and lim t ↑ τ M Y ( t ) a.s.converge on { τ < ∞} . By (4.16,4.17), lim t ↑ τ ( X ( t ) − M X ( t )) and lim t ↑ τ ( Y ( t ) − M Y ( t )) a.s.converge on { τ < ∞} . Combining these results, we see that, on the event { τ < ∞} , lim t ↑ τ X ( t )and lim t ↑ τ Y ( t ) a.s. converge, which implies the a.s. convergence of lim t ↑ τ R ± ( t ) .Since ( R + ( t ) , R − ( t )) → (0 ,
0) iff ( X ( t ) , Y ( t )) → (0 , X ( t ) , Y ( t ))does not converge to (0 ,
1) as t ↑ τ . Since ( X, Y ) is Markov, it suffices to show that, if Y (0) = 0,and if T = τ ∧ inf { t : Y ( t ) = 0 } , then ( X ( t ) , Y ( t )) does not converge to (0 ,
1) as t ↑ T . Since Y = 0 on [0 , T ), we may define a process Z = X/Y ∈ [ − ,
1] on [0 , T ). Now it suffices to showthat ( Z ( t ) , Y ( t )) does not converge to (0 ,
1) as t ↑ T .From (4.16-4.18) and Itˆo’s calculation, we see that there is a stopped planar Brownianmotion ( B Z ( t ) , B Y ( t )), 0 ≤ t < T , such that Z and Y satisfy the following SDEs on [0 , T ): dZ = r κ (1 − Z ) Y dB Z − ( ρ + + ρ − + 4) Z + ( ρ + − ρ − ) Y dt ; dY = p κY (1 − Y ) dB Y − [( ρ + + ρ − + ρ + 6) Y − ( ρ + + ρ − + 4)] dt. Let v ( t ) = R t κ/Y ( s ) ds , 0 ≤ t < T , and e T = sup v [0 , T ). Let e Z ( t ) = Z ( v − ( t )) and e Y ( t ) = Y ( v − ( t )), 0 ≤ t < e T . Then there is a stopped planar Brownian motion ( e B Z ( t ) , e B Y ( t )),0 ≤ t < e T , such that e Z and e Y satisfy the following SDEs on [0 , e T ): d e Z = q − e Z d e B Z − ( a Z e Z + b Z ) dt, e Y = e Y q − e Y d e B Y − e Y ( a Y ( e Y −
1) + b Y ) dt, where a Z = ( ρ + + ρ − + 4) /κ , b Z = ( ρ + − ρ − ) /κ , b Y = ( ρ + 2) /κ , a Y = a Z + b Y .Let Θ = arcsin( e Z ) ∈ [ − π/ , π/
2] and Φ = log( √ − e Y − √ − e Y ) ∈ R + . Then ( e Z ( t ) , e Y ( t )) → (0 , t ) + Φ( t ) →
0, and Θ and Φ satisfy the following SDEs on [0 , e T ): d Θ = d e B Z − ( a Z −
12 ) tan Θ dt − b Z sec Θ dt ; d Φ = − d e B Y + ( b Y −
14 ) coth (Φ) dt + ( 34 − a Y ) tanh (Φ) dt. Here tanh := tanh( · /
2) and coth := coth( · / B Θ , Φ , Θ + Φ satisfies the SDE d (Θ + Φ ) = 2 p Θ + Φ dB Θ , Φ + 2 dt + (2 b Y −
12 )Φ coth (Φ) dt +( 32 − a Y )Θ tanh (Φ) dt − (2 a Z − dt − b Z sec Θ dt. From the power series expansions of coth , tanh , tan , sec at 0, we see that when Θ + Φ isclose to 0, it behaves like a squared Bessel process of dimension 4 b Y + 1 = κ ( ρ + 2) + 1 ≥ ρ ≥ κ −
2. Thus, a.s. lim t ↑ e T Θ( t ) + Φ( t ) = 0, as desired. Remark 4.11.
The assumption ρ ≥ κ − Lemma 4.12.
For every
N > and L ≥ , there is C > depending only on κ, ρ, N, L suchthat for any v ∈ [( − + , − ] , v + ∈ [1 + , ∞ ) and v − ∈ ( −∞ , ( − − ] with | v + − v − | ≤ L , if ( η + , η − ) is a commuting pair of chordal SLE κ (2 , ρ ) curves started from (1 , − v , v + , v − ) , thenfor any y ∈ (0 , N ] , P [ E + ( y ) ∩ E − ( y )] ≥ C , where for σ ∈ { + , −} , E σ ( y ) is the event that η σ reaches { Im z = y } before { Re z = σ } ∪ { Re z = σ } .Proof. In this proof, a constant depends only on κ, ρ, N, L . Since E ± ( y ) is decreasing in y , itsuffices to prove that P [ E + ( N ) ∩ E − ( N )] is bounded from below by a positive constant. By[13, Lemma 2.4], there is a constant e C > P [ E σ ( N )] ≥ e C for σ ∈ { + , −} . Thus, if( η ′ + , η ′− ) is an independent coupling of η + and η − , then the events E ′± ( N ) for ( η ′ + , η ′− ) satisfythat P [ E ′ + ( N ) ∩ E ′− ( N )] ≥ e C . Let ξ σ = H ∩ ∂ { x + iy : | x − σ | ≤ , ≤ y ≤ N } , σ ∈ { + , −} .Since the law of ( η + , η − ) restricted to F + τ + ξ + ∨F − τ − ξ − is absolutely continuous w.r.t. that of ( η ′ + , η ′− )(Lemma 4.4), and the logarithm of the Radon-Nikodym derivative is bounded in absolute valueby a constant (Lemma 4.2), we get the desired lower bound for P [ E + ( N ) ∩ E − ( N )]. Corollary 4.13.
For any r ∈ (0 , , there is δ > depending only on κ, ρ, r such that thefollowing holds. Suppose w + > w − ∈ R , v ∈ [ w + − , w − + ] , v + ∈ [ w ++ , ∞ ) , v − ∈ ( −∞ , w −− ] satisfy | v + − v | = | v − − v | and | w + − w − | ≥ r | v + − v − | . Let ( η + , η − ) be a commuting air of chordal SLE κ (2 , ρ ) curves started from ( w + , w − ; v , v + , v − ) . Let ξ = ( ξ + , ξ − ) , where ξ σ = H ∩ ∂ { x + iy : | x − w σ | ≤ | w + − w − | / , ≤ y ≤ e | v + − v − |} , σ ∈ { + , −} . Let τ uξ be asdefined in Section 4.3. Then P [ τ uξ ≥ ≥ δ .Proof. Let E denote the event that for both σ ∈ { + , −} , η σ hits ξ σ at its top for the first time.Suppose that E happens. By the definition of τ uξ , for one of σ ∈ { + , −} , the imaginary part of η σ ( u σ ( τ uξ )) is e | v + − v − | . So rad v ( η σ [0 , u σ ( τ uξ )]) ≥ e | v + − v − | . By (3.34) we then get τ uξ ≥ P [ τ uξ ≥ ≥ P [ E ], which by Lemma 4.12 and scaling is bounded from below by a positiveconstant depending only on κ, ρ, r whenever | v + − v − | ≤ | w + − w − | /r . Proof of Theorem 4.9.
We have known that (4.15) holds up to T u disj . We will combine it withthe DMP of commuting pair of chordal SLE κ (2 , ρ ) curves (Lemma 4.1).Let η ± = η ± . Let G be the trivial σ -algebra. We will inductively define the followingrandom objects. Let n = 1. We have the σ -algebra G n − and the pair ( η n − , η n − − ), whoselaw conditionally on G n − is that of a commuting pair of chordal SLE κ (2 , ρ ) curves. Let K n − , m n − , W n − ± , V n − , V n − ± , be respectively the hull function, capacity function, drivingfunctions and force point functions. Let D n − and Ξ n − be respectively the D disj and Ξ definedfor the pair. Let F n − be the R -indexed filtration defined by F n − t + ,t − ) = σ ( G n − , η n − σ | [0 ,t σ ] , σ ∈{ + , −} ). Let u n − be the time curve for ( η n − , η n − − ) as defined in Section 3.4, which exits for n = 1 because we assume that | v + − v | = | v − v − | .Let ξ n − be the ξ obtained from applying Corollary 4.13 to w ± = W n − ± (0) and v ± = V n − ± (0). Then it is a G n − -measurable random element in Ξ n − . Let τ n − ξ n − be the random time τ uξ introduced in Section 4.3 for the ( η n − , η n − − ) and ξ n − here. Let τ n − = ( τ n − , τ n − − ) = u n − ( τ n − ,uξ n − ). Then τ n − is a finite F n − -stopping time that lies in D n − . Let G n = F n − τ n − .We then obtain by Lemma 4.1 a random commuting pair of chordal Loewner curves ( e η n + , e η n − )with some speeds, which is the part of ( η n − , η n − − ) after τ n − up to a conformal map, andthe normalization of ( e η n + , e η n − ), denoted by ( η n + , η n − ), conditionally on G n , is a commuting pairof chordal SLE κ (2 , ρ ) curve started from ( W n − , W n − − ; V n − , V n − , V n − − ) | τ n − . If for some σ ∈ { + , −} and ν ∈ { , + , −} , V n − ν ( τ n − ) = W n − σ ( τ n − ), then as a force point, V n − ν ( τ n − )is treated as ( W n − σ ( τ n − )) sign( v ν − w σ ) . By the assumption of u n − , we have | V n − − V n − | = | V n − − − V n − | at τ n − . So we may increase n by 1 and repeat the above construction.Iterating the above procedure, we obtain two sequences of pairs ( η n + , η n − ), n ≥
0, and( e η n + , e η n − ), n ≥
1. They satisfy that for any n ∈ N , ( η n + , η n − ) is the normalization of ( e η n + , e η n − ),and ( e η n + , e η n − ) is the part of ( η n − , η n − − ) after τ n − up to a conformal map. Let φ n ± be thespeed of e η n ± , and φ n ⊕ ( t + , t − ) = ( φ n + ( t + ) , φ n − ( t − )). By Lemma 3.15, for any n ∈ N and Z ∈{ W + , W − , V , V + , V − } , e Z n = Z n ◦ φ n ⊕ and e Z n = Z n − ( τ n − + · ).Recall that, for n ≥ u n is characterized by the property that | V n ± ( u n ( t )) − V n ( u n ( t )) | = e t | V n ± (0) − V n (0) | , t ≥
0. So we get u n = φ n ⊕ ( u n − ( τ n − ξ n − + · ) − u n − ( τ n − ξ n − )), which then impliesthat Z n − ◦ u n − ( τ n − ξ n − + · ) = Z n ◦ u n , Z ∈ { W + , W − , V , V + , V − } . Let R n ± be the R ± defined in42ection 4.3 for ( η n + , η n − ). Then we have R n − ± ( τ n − ξ n − + · ) = R n ± . Let T n = P n − j =0 τ jξ j , n ≥
0. Since R ± = R ± , we get R ± ( T n + · ) = R n ± . For n ≥
0, since conditionally on G n , ( η n + , η n − ) has thelaw of a commuting pair of chordal SLE κ (2 , ρ ) curves started from ( W n + , W n − ; V n , V n + , V n − ) | , bythe previous subsection, there is a stopped two-dimensional Brownian motion ( B n + , B n − ) w.r.t. F nu n ( · ) such that R n + and R n − satisfy the F nu n ( · ) -adapted SDE (4.15) with ( B n + , B n − ) in place of( B + , B − ) up to τ nξ n . Let T ∞ = lim n →∞ T n = P ∞ j =0 τ jξ j , and define a continuous processes B ± on [0 , T ∞ ) such that B ± ( t ) − B ± ( T n ) = B n ( t − T n ) for each t ∈ [ T n , T n +1 ] and n ≥
0. Then( B + , B − ) is a stopped two-dimensional Brownian motion, and R + and R − satisfy (4.15) up to T ∞ . It remains to show that a.s. T ∞ = ∞ .Suppose it does not hold that a.s. T ∞ = ∞ . By Lemma 4.10, there is an event E withpositive probability and a number r ∈ (0 ,
1] such that on the event E , R + + R − ≥ r on [0 , T ∞ ).For n ≥
0, let E n = {| W n + (0) − W n − (0) | ≥ r | V n + (0) − V n − (0) |} = { R n + (0) + R n − (0) ≥ r } , which is G n -measurable. Since R n ± = R ± ( T n + · ), we get E ⊂ T E n . Let A n = { τ nξ n ≥ } . By Corollary4.13, there is δ > κ, ρ, r such that for n ≥ P [ A n |G n , E n ] ≥ δ . Since E ⊂ { P n τ nξ n < ∞} , we get E ⊂ lim inf( E n ∩ A cn ). For any m ≥ n ∈ N , P [ m \ k = n ( E k ∩ A ck )] = E [ P [ m \ k = n ( E k ∩ A ck ) |G m ]] ≤ (1 − δ ) P [ m − \ k = n ( E k ∩ A ck )] . So we get P [ T mk = n ( E k ∩ A ck )] ≤ (1 − δ ) n − m +1 , which implies that P [ T ∞ k = n ( E k ∩ A ck )] = 0 forevery n ∈ N , and so P [ E ] = 0. This contradiction completes the proof. In this subsection, we are going to use orthogonal polynomials to derive the transition densityof R ( t ) = ( R + ( t ) , R − ( t )), t ≥
0, against the Lebesgue measure restricted to [0 , . A similarapproach was first used in [24, Appendix B] to calculate the transition density of radial Besselprocesses, where one-variable orthogonal polynomials were used. Two-variable orthogonal poly-nomials were used in [21, Section 5] to calculate the transition density of a two-dimensionaldiffusion process. Here we will use another family of two-variable orthogonal polynomials tocalculate the transition density of the ( R ) here. In addition, we are going to derive the invariantdensity of ( R ), and estimate the convergence of the transition density to the invariant density.Let X = R + − R − and Y = 1 − R + R − . Since R + and R − satisfy (4.15) throughout R + , X and Y then satisfy (4.16,4.17,4.18) throughout R + . Moreover, we have ( X, Y ) ∈ ∆ \ { (0 , } ,where ∆ = { ( x, y ) ∈ R : 0 < | x | < y < } . We will first find the transition density of( X ( t ) , Y ( t )). Assume that the transition density p ( t, ( x, y ) , ( x ∗ , y ∗ )) exists, and is smooth in( x, y ), then it should satisfies the Kolmogorov’s backward equation: − ∂ t p + L p = 0 , (4.19)where L is the second order differential operator defined by L = κ y − x ) ∂ x + κx (1 − y ) ∂ x ∂ y + κ y (1 − y ) ∂ y [( ρ + + ρ − + ρ + 6) x + ( ρ + − ρ − )] ∂ x − [( ρ + + ρ − + ρ + 6) y − ( ρ + + ρ − + 4)] ∂ y . We perform a change of coordinate ( x, y ) ( r, h ) by x = rh and y = h at y = 0. Directcalculation shows that ∂ r = h∂ x , ∂ h = r∂ x + ∂ y , ∂ r = h ∂ x , ∂ h = r ∂ x + 2 r∂ x ∂ y + ∂ y , ∂ r ∂ h = rh∂ x + h∂ x ∂ y . Define α = 2 κ ( ρ + 2) − , α ± = 2 κ ( ρ ± + 2) − , β = α + + α − + 1; L ( r ) = (1 − r ) ∂ r − [( α + + α − + 2) r + ( α + − α − )] ∂ r ; L ( h ) = h (1 − h ) ∂ h − [( α + β + 2) h − ( β + 1)] ∂ h . Then in the ( r, h )-coordinate, L = κ [ L ( h ) + h L ( r ) ]. Let λ n = − n ( n + α + β + 1) , λ ( r ) n = − n ( n + β ) , n ≥ . Direct calculation shows that[ L ( h ) + 1 h λ ( r ) n ] h n = h n [ L ( h ) − n ( h − ∂ h + λ n ] , (4.20)where each h n in the formula is understood as a multiplication by the n -th power of h . From(2.5) we know that Jacobi polynomials P ( α + ,α − ) n ( r ), n ≥
0, satisfy that L ( r ) P ( α + ,α − ) n ( r ) = λ ( r ) n P ( α + ,α − ) n ( r ) , n ≥
0; (4.21)and the functions P ( α ,β +2 n ) m (2 h − m ≥
0, satisfy that( L ( h ) − n ( h − ∂ h + λ n ) P ( α ,β +2 n ) m (2 h −
1) = λ m + n P ( α ,β +2 n ) m (2 h − , m ≥ . (4.22)For n ≥
0, define a homogeneous two-variable polynomial Q ( α + ,α − ) n ( x, y ) of degree n suchthat Q ( α + ,α − ) n ( x, y ) = y n P ( α + ,α − ) n ( x/y ) if y = 0. It has nonzero coefficient for x n . For everypair of integers n, m ≥
0, define a two-variable polynomial v n,m ( x, y ) of degree n + m by v n,m ( x, y ) = P ( α ,β +2 n ) m (2 y − Q ( α + ,α − ) n ( x, y ) . Then v n,m is also a polynomial in r, h with the expression: v n,m ( r, h ) = h n P ( α ,β +2 n ) m (2 h − P ( α + ,α − ) n ( r ) . (4.23)By (4.20,4.21,4.22), on R \ { y = 0 } ,2 κ L v n,m = 2 κ [ L ( h ) + 1 h L ( r ) ] v n,m = [ L ( h ) + 1 h λ ( r ) n ]( h n P ( α ,β +2 n ) m (2 h − P ( α + ,α − ) n ( r ))= h n [ L ( h ) − n ( h − ∂ h + λ n ]( P ( α ,β +2 n ) m (2 h − P ( α + ,α − ) n ( r )) = λ n + m v n,m . v n,m is a polynomial in x, y , by continuity the above equation holds throughout R . Thus,for every n, m ≥ v n,m ( x, y ) e κ λ n + m t solves (4.19), and the same is true for any linear combina-tion of such functions. From (4.23) we get an upper bound of k v n,m k ∞ := sup ( x,y ) ∈ ∆ | v n,m ( x, y ) | : k v n,m k ∞ ≤ k P ( α ,β +2 n ) m k ∞ k P ( α + ,α − ) n k ∞ , (4.24)where the supernorm of the Jacobi polynomials are taken on [ − ,
1] as in (2.6,2.7).Since P ( α + ,α − ) n ( r ), n ≥
0, are mutually orthogonal w.r.t. the weight function Ψ ( α + ,α − ) ( r ),and for any fixed n ≥ P ( α ,β +2 n ) m (2 h − m ≥
0, are mutually orthogonal w.r.t. the weightfunction Ψ ( α ,β +2 n ) (2 h −
1) = 2 α + β +2 n (0 , ( h )(1 − h ) α h β +2 n , using a change of coordinateswe conclude that v n,m ( x, y ), n, m ∈ N ∪ { } , are mutually orthogonal w.r.t. the weight functionΨ( x, y ) := ∆ ( x, y )( y − x ) α + ( y + x ) α − (1 − y ) α . Moreover, we have k v n,m k = 2 − ( α + β +2 n +1) k P ( α ,β +2 n ) m k ( α ,β +2 n ) · k P ( α + ,α − ) n k ( α + ,α − ) . (4.25)Let f ( x, y ) be a polynomial in two variables. Then f can be expressed by a linear combina-tion f ( x, y ) = P ∞ n =0 P ∞ m =0 a n,m v n,m ( x, y ), where a n,m := h f, v ( n,m ) i Ψ / k v n,m k are zero for allbut finitely many ( n, m ). In fact, every polynomial in x, y of degree ≤ k can be expressed asa linear combination of v n,m with n + m ≤ k . In fact, the number of such v n,m is ( k +1)( k +2)2 .Define f ( t, ( x, y )) = ∞ X n =0 ∞ X m =0 a n,m v n,m ( x, y ) e κ λ n + m t = ∞ X n =0 ∞ X m =0 h f, v n,m i Ψ k v n,m k · v n,m ( x, y ) e κ λ n + m t . Then f ( t, ( x, y )) solves (4.19). Let ( X ( t ) , Y ( t )) be a diffusion process in ∆, which solves(4.16,4.17,4.18) with initial value ( x, y ). Fix t > M t = f ( t − t, ( X ( t ) , Y ( t ))),0 ≤ t ≤ t . By Itˆo’s formula, M is a bounded martingale, which implies that E [ f ( X ( t ) , Y ( t ))] = E [ M t ] = M = f ( t , ( x, y ))= ∞ X n =0 ∞ X m =0 Z Z ∆ f ( x ∗ , y ∗ )Ψ( x ∗ , y ∗ ) v n,m ( x ∗ , y ∗ ) v n,m ( x, y ) k v n,m k · e κ λ n + m t dx ∗ dy ∗ . (4.26)For t >
0, ( x, y ) ∈ ∆, and ( x ∗ , y ∗ ) ∈ ∆, define p t (( x, y ) , ( x ∗ , y ∗ )) = Ψ( x ∗ , y ∗ ) ∞ X n =0 ∞ X m =0 v n,m ( x, y ) v n,m ( x ∗ , y ∗ ) k v n,m k · e κ λ n + m t . (4.27)Let p ∞ ( x ∗ , y ∗ ) = ∆ ( x ∗ , y ∗ )Ψ( x ∗ , y ∗ ) / k k . Note that λ = 0 and v , ≡ P α ,β = P α + ,α − ≡
1. So p ∞ ( x ∗ , y ∗ ) corresponds to the first term in the series.45 emma 4.14. For any t > , the series in (4.27) (without the factor Ψ( x ∗ , y ∗ ) ) convergesuniformly on [ t , ∞ ) × ∆ × ∆ , and there is C t ∈ (0 , ∞ ) depending only on κ , ρ , and t suchthat for any ( x, y ) ∈ ∆ and ( x ∗ , y ∗ ) ∈ ∆ , | p t (( x, y ) , ( x ∗ , y ∗ )) − p ∞ ( x ∗ , y ∗ ) | ≤ C t e − ( ρ + + ρ − + ρ +6) t Ψ( x ∗ , y ∗ ) , t ≥ t . (4.28) Moreover, for any t > and ( x ∗ , y ∗ ) ∈ ∆ , p ∞ ( x ∗ , y ∗ ) = Z Z ∆ p ∞ ( x, y ) p t (( x, y ) , ( x ∗ , y ∗ )) dxdy. (4.29) Proof.
The estimate (4.28) and the uniform convergence of the series in (4.27) both follows fromStirling’s formula, (4.24,4.25,2.4,2.7), and the facts that 0 > λ = − κ ( ρ + + ρ − + ρ + 6) > λ n for any n > λ n ≍ − n for big n . Formula (4.29) follows from the orthogonality of v n,m w.r.t. h· , ·i Ψ and the uniform convergence of the series in (4.27). Lemma 4.15.
The process ( X ( t ) , Y ( t )) has a transition density, which is p t (( x, y ) , ( x ∗ , y ∗ )) ,and an invariant density, which is p ∞ ( x ∗ , y ∗ ) .Proof. Fix ( x, y ) ∈ ∆ \ { (0 , } . Let ( X ( t ) , Y ( t )) be the process that satisfies (4.16,4.17,4.18)with initial value ( x, y ). It suffices to show that, for any continuous function f on ∆, we have E [ f ( X ( t ) , Y ( t ))] = Z Z ∆ p t (( x, y ) , ( x ∗ , y ∗ )) f ( x ∗ , y ∗ ) dx ∗ dy ∗ . (4.30)By Stone-Weierstrass theorem, f can be approximated by a polynomial in two variables uni-formly on ∆. Thus, it suffices to show that (4.30) holds whenever f is a polynomial in x, y ,which follows immediately from (4.26). The statement on p ∞ ( x ∗ , y ∗ ) follows from (4.29).Since X = R + − R − , Y = 1 − R + R − , and the Jacobian of the transformation is − ( R + + R − ),we arrive at the following statement. Corollary 4.16.
The process ( R ( t )) has a transition density: p Rt ( r, r ∗ ) := p t (( r + − r − , − r + r − ) , ( r ∗ + − r ∗− , − r ∗ + r ∗− )) · ( r ∗ + + r ∗− ) , and an invariant density: p R ∞ ( r ∗ ) := p ∞ ( r ∗ + − r ∗− , − r ∗ + r ∗− ) · ( r ∗ + + r ∗− ) ; and for any t > , thereis C t ∈ (0 , ∞ ) depending only on κ , ρ , and t such that for any r ∈ [0 , and r ∗ ∈ (0 , , | p Rt ( r, r ∗ ) − p R ∞ ( r ∗ ) | ≤ C t e − ( ρ + + ρ − + ρ +6) t p R ∞ ( r ∗ ) , t ≥ t . Commuting Pair of hSLE Curves
In this section, we study three commuting pairs of hSLE κ curves. It turns out that each ofthem is “locally” absolutely continuous w.r.t. a commuting pair of chordal SLE κ (2 , ρ ) curvesfor some suitable force values. So the results in the previous section can be applied here. Fix κ ∈ (0 ,
8) and v − < w − < w + < v + ∈ R . We write w = ( w + , w − ) and v = ( v + , v − ). For ρ = ( ρ + , ρ − ) that satisfies the conditions in Section 4.1, let P ρw ; v denote the law of the drivingfunctions of a commuting pair of choral SLE κ (2 , ρ ) curves started from ( w ; v ). -SLE κ Suppose that ( b η + , b η − ) is a 2-SLE κ in H with link pattern ( w + → v + ; w − → v − ). By [19,Proposition 6.10], for σ ∈ { + , −} , b η σ is an hSLE κ curve in H from w σ to v σ with force points w − σ and v − σ . Stopping b η σ at the first time that it disconnects ∞ from any of its force points,and parametrizing the stopped curve by H -capacity, we get a chordal Loewner curve η σ ( t ),0 ≤ t < T σ , which is an hSLE κ curve in the chordal coordinate. Let b w σ and K σ ( · ) be respectivelythe chordal Loewner driving function and hulls for η σ ; and let F σ be the filtration generatedby η σ . Let F be the separable R -indexed filtration generated by F + and F − .For σ ∈ { + , −} , if τ − σ is an F − σ -stopping time, then conditionally on F − στ − σ and the event { τ − σ < T − σ } , the whole η σ and the part of b η − σ after η ( τ − σ ) together form a 2-SLE κ in H \ K − σ ( τ − σ ) with link pattern ( w σ → v σ ; η − σ ( τ − σ ) → v − σ ). This in particular implies that theconditional law of b η σ is that of an hSLE κ curve from w σ to v σ in H \ K − σ ( τ − σ ) with forcepoints η − σ ( τ − σ ) and v − σ . Since f K − σ ( τ − σ ) maps H conformally onto H \ K − σ ( τ − σ ), and sends b w − σ ( τ − σ ), g K − σ ( τ − σ ) ( w σ ) and g K − σ ( τ − σ ) ( v ν ), ν ∈ { + , −} , respectively to η − σ ( τ − σ ), w σ and v ν , ν ∈ { + , −} , we see that there a.s. exists a chordal Loewner curve η τ − σ σ with some speed suchthat η σ = f K − σ ( τ − σ ) ◦ η σ,τ − σ , and the conditional law of the normalization of η σ,τ − σ given F − στ − σ is that of an hSLE κ curve in H from g K − σ ( τ − σ ) ( w σ ) to g K − σ ( τ − σ ) ( v σ ) with force points b w − σ ( τ − σ )and g K − σ ( τ − σ ) ( v − σ ), in the chordal coordinate.Thus, ( η + , η − ) a.s. satisfies the conditions in Definition 3.2 with I σ = [0 , T σ ), I ∗ σ = I σ ∩ Q , σ ∈ { + , −} , and D := I + × I − . By discarding a null event, we assume that ( η + , η − ; D ) isalways a commuting pair of chordal Loewner curves, and call ( η + , η − ; D ) a commuting pairof hSLE κ curves in the chordal coordinate started from ( w ; v ). We adopt the functions fromSection 3. Define a function M on D by M = G ( W + , W − ; V + , V − ), where G is given by(1.2). Since F is continuous and positive on [0 , | W σ − V ν | ≤ | V + − V − | for σ, ν ∈ { + , −} , and κ − , κ >
0, there is a constant
C > κ such that M ≤ C | V + − V − | κ − min σ ∈{ + , −} {| W σ − V σ |} κ − ≤ C | V + − V − | κ − . (5.1)Note that M > D because | W σ − V σ | > σ ∈ { + , −} , on D . We will prove that M extends to an F -martingale on R , which acts as Radon-Nikodym derivatives between twomeasures. We first need some deterministic properties of M . Lemma 5.1. M a.s. extends continuously to R with M ≡ on R \ D . roof. It suffices to show that for σ ∈ { + , −} , as t σ ↑ T σ , M → t − σ ∈ [0 , T − σ ).By symmetry, we may assume that σ = +. Since the union of (the whole) η + and η − isbounded, by (3.15) | V + − V − | is bounded (by random numbers) on D . For a fixed t − ∈ [0 , T − ),as t + ↑ T + , η + ( t + ) tends to either some point on [ v + , ∞ ) or some point on ( −∞ , v − ). By (5.1),it suffices to show that when η + terminates at [ v + , ∞ ) (resp. at ( −∞ , v − )), W + − V + → W − − V − →
0) as t + ↑ T + , uniformly in [0 , T − ).For any t = ( t + , t − ) ∈ D , neither η + [0 , t + ] nor η − [0 , t − ] hit ( −∞ , v − ] ∪ [ v + , ∞ ), whichimplies that v + , v − K ( t ) and V ± ( t ) = g K ( t ) ( v ± ). Suppose that η + terminates at x ∈ [ v + , ∞ ). Since SLE κ is not boundary-filling for κ ∈ (0 , x , η − ) >
0. Let r = min {| w + − v + | , dist( x , η − ) } >
0. Fix ε ∈ (0 , r ). Since x = lim t ↑ T + η + ( t ), there is δ > | η + ( t ) − x | < ε for t ∈ ( T + − δ, T + ). Fix t + ∈ ( T + − δ, T + ) and t − ∈ [0 , T − ). Let J be the connected component of {| z − x | = ε } ∩ ( H \ K ( t )) whose closure contains x + ε . Then J disconnects v + and η + ( t + , T + ) ∩ ( H \ K ( t )) from ∞ in H \ K ( t ). Thus, g K ( t ) ( J ) disconnects V + ( t ) and W + ( t ) from ∞ . Since η + ∪ η − is bounded, there is a (random) R ∈ (0 , ∞ ) suchthat η + ∪ η − ⊂ {| z − x | < R } . Let ξ = {| z − x | = 2 R } ∩ H . By comparison principle, theextremal length ([1]) of the family of curves in H \ K ( t ) that separate J from ξ is ≤ π log( R/ε ) . Byconformal invariance, the extremal length of the family of curves in H that separate g K ( t ) ( J )from g K ( t ) ( ξ ) is also ≤ π log( R/ε ) . Now g K ( t ) ( ξ ) and g K ( t ) ( J ) are crosscuts of H such that theformer encloses the latter. Let D denote the subdomain of H bounded by g K ( t ) ( ξ ). FromProposition 2.3 we know that D ⊂ {| z − x | ≤ R } . So the Euclidean area of D is less than13 πR . By the definition of extremal length, there is a curve γ in D that separates g K ( t ) ( J )from g K ( t ) ( ξ ) with Euclidean distance less than 2 q πR ∗ π log( R/ε ) < πR ∗ log( R/ε ) − / .Since g K ( t ) ( J ) disconnects V + ( t ) and W + ( t ) from ∞ , γ also separates V + ( t ) and W + ( t ) from ∞ . Thus, | W + ( t ) − V + ( t ) | < πR ∗ log( R/ε ) − / if t + ∈ ( T + − δ, T + ) and t − ∈ [0 , T − ). Thisproves the uniform convergence of lim t + ↑ T + | W + − V + | = 0 in t − ∈ [0 , T − ) in the case thatlim t + ↑ T + η + ( t + ) ∈ [ v + , ∞ ). The proof of the uniform convergence of lim t + ↑ T + | W − − V − | = 0 in t − ∈ [0 , T − ) in the case that lim t + ↑ T + η + ( t + ) ∈ ( −∞ , v − ) is similar.From now on, we view M as a continuous stochastic process defined on R with constantzero on R \ D . For σ ∈ { + , −} and R > | v + − v − | /
2, let τ σR be the first time that | η σ ( t ) − ( v + + v − ) / | = R if such time exists; otherwise τ σR = T σ . Let τ R = ( τ + R , τ − R ). Note that τ + R , τ − R ≤ m( τ R ) ≤ R / K ( τ R ) ⊂ { z ∈ H : | z − ( v + + v − ) / | ≤ R } . Lemma 5.2.
For every
R > , the M ( · ∧ τ R ) is an R -indexed martingale w.r.t. the filtration ( F + t + ∧ τ + R ∨ F − t − ∧ τ − R ) ( t + ,t − ) ∈ R closed by M ( τ R ) . Moreover, if the underlying probability measureis weighted by M ( τ R ) /M (0) , then the new law of the driving functions ( b w + , b w − ) agrees with P (2 , w ; v on the σ -algebra F + τ + R ∨ F − τ − R .Proof. Let
R > σ ∈ { + , −} , t − σ ≥
0, and τ − σ = t − σ ∧ τ − σR . Since W σ | − στ − σ , W − σ | − στ − σ and V ν | − στ − σ , ν ∈ { + , −} , are all ( F σt ∨ F − στ − σ ) t ≥ -adapted, and are driving function and force pointfunctions for an hSLE κ curves with some speeds in the chordal coordinate conditional on F − στ − σ ,48y Proposition 2.16 (with a time-change), M | − στ − σ ( t ), 0 ≤ t < T σ , is an ( F σt ∨ F − στ − σ ) t ≥ -localmartingale. Since M is uniformly bounded on [0 , τ R ] and τ ± R ≤ R / M | − στ − σ ( · ∧ τ σR ) is an( F + t + ∧ τ + R ∨ F − t − ∧ τ − R ) t σ ≥ -martingale closed by M | − στ − σ ( τ σR ). Applying this result twice respectivelyfor σ = + and − , we obtain the martingale property of M ( · ∧ τ R )Let P denote the underlying probability measure. By weighting P by M ( τ R ) /M (0), weget another probability measure, denoted by P . To describe the restriction of P to F τ R , westudy the new marginal law of η − up to τ − R and the new conditional law of η + up to τ + R giventhat part of η − . We may first weight P by N := M (0 , τ − R ) /M (0 ,
0) to get a new probabilitymeasure P . , and then weight P . by N := M ( τ + R , τ − R ) /M (0 , τ − R ) to get P .By Proposition 2.16, the η − up to τ − R under P . is a chordal SLE κ (2 , ,
2) curve in H startedfrom w − with force points v − , w + , v + , respectively, up to τ − R . Since N depends only on η − ,the conditional law of η + given any part of η − under P . agrees with that under P . Since M (0 , τ − R ) = 0 implies that N = 0, and P . -a.s. N >
0, we see that N is P . -a.s. welldefined. Since E [ N |F − τ − R ] = 1, the law of η − up to τ − R under P agrees with that under P . .To describe the conditional law of η + up to τ + R = τ + R ( η + ) given η − up to τ − R , it suffices toconsider the conditional law of η τ − R + up to τ + R ( η + ) since we may recover η + from η τ − R + using η + = f K − ( τ − R ) ◦ η τ − R + . By Proposition 2.16 again, the conditional law of the normalization of η τ − R + up to τ + R ( η + ) under P is that of a chordal SLE κ (2 , ,
2) curve in H started from W + (0 , τ − R ) withforce points at V + (0 , τ − R ), W − (0 , τ − R ) and V − (0 , τ − R ), respectively. Thus, under P the joint lawof η + up to τ + R and η − up to τ − R agrees with that of a commuting pair of SLE κ (2 , ,
2) curvesstarted from ( w ; v ) respectively up to τ + R and τ − R . So the proof is completeWe let P denote the joint law of the driving functions b w + and b w − here, and let P = P (2 , w ; v .From the lemma, we find that, for any t = ( t + , t − ) ∈ R and R > d P | ( F + t + ∧ τ + R ∨ F − t − ∧ τ − R ) d P | ( F + t + ∧ τ + R ∨ F − t − ∧ τ − R ) = M ( t ∧ τ R ) M (0) . (5.2) Lemma 5.3.
Under P , M is an F -martingale; and for any F -stopping time τ , d P |F τ ∩ { τ ∈ R } d P |F τ ∩ { τ ∈ R } = M ( τ ) M (0) . (5.3) Proof.
For t ∈ R and R >
0, since F + t + ∧ τ + R ∨ F − t − ∧ τ − R agrees with F + t + ∨ F − t − = F t on { t ≤ τ R } ,by (5.2), d P | ( F t ∩ { t ≤ τ R } ) d P | ( F t ∩ { t ≤ τ R } ) = M ( t ) M (0) . Sending R → ∞ , we get d P |F t /d P |F t = M ( t ) /M (0) for all t ∈ R . So M is an F -martingaleunder P . Let τ be an F -stopping time. Fix A ∈ F τ ∩ { τ ∈ R } . Let t ∈ R . Define the49 -stopping time τ t as in Proposition 2.22. Then A ∩ { τ < t } = A ∩ { τ < τ t } ∈ F τ t ⊂ F t . Sowe get P [ A ∩ { τ < t } ] = E h A ∩{ τ For any F -stopping time τ , d P |F τ ∩ { τ ∈ D } d P |F τ ∩ { τ ∈ D } = M (0) M ( τ ) . Proof. This follows from Lemma 5.3 and the fact that M > D .Assume now that v := ( v + + v − ) / ∈ [ w − , w + ]. We understand v as w − σσ if ( v + + v − ) / w σ , σ ∈ { + , −} . Let V be the force point function started from v . By Section 3.4, we maydefine the time curve u : [0 , T u ) → D such that V σ ( u ( t )) − V ( u ( t )) = e t ( v σ − v ), 0 ≤ t < T u , σ ∈ { + , −} , and u can not be extended beyond T u with such property. We follow the notationthere: for every X defined on D , we use X u to denote the function X ◦ u defined on [0 , T u ).We also define the processes R σ = W uσ − V u V uσ − V u ∈ [0 , σ ∈ { + , −} , and R = ( R + , R − ). Since T σ isan F σ -stopping time for σ ∈ { + , −} , D = [0 , T + ) × [0 , T − ) is an F -stopping region. As beforewe extend u to R + such that if s ≥ T u then u ( s ) = lim t ↑ T u u ( t ). By Proposition 3.22, for any t ≥ u ( t ) is an F -stopping time.Let I = v + − v = v − v − and define G ∗ on [0 , by G ∗ ( r + , r − ) = G ( r + , − r − ; 1 , − M u ( t ) = ( e t I ) α G ∗ ( R ( t )) for t ∈ [0 , T u ), where α = 2( κ − 1) is as in Theorem 1.1We now derive the transition density of the process ( R ( t )) ≤ t Lemma 5.5. Let p t ( r, r ∗ ) be the transition density p Rt ( r, r ∗ ) given in Corollary 4.16 with ρ = 0 and ρ + = ρ − = 2 . Then under P , the transition density of ( R ) is e p t ( r, r ∗ ) := e − α t p t ( r, r ∗ ) G ∗ ( r ) /G ∗ ( r ∗ ) . .2 Opposite pair of hSLE κ curves, the generic case Second, we consider another pair of random curves. Let w = ( w + , w − ) and v = ( v + , v − ) be asbefore. Let ( η w , η v ) be a 2-SLE κ in H with link pattern ( w + ↔ w − ; v + ↔ v − ). For σ ∈ { + , −} ,let b η σ be the curve η w oriented from w σ to w − σ and parametrized by the capacity viewed from w − σ , which is an hSLE κ curve in H from w σ to w − σ with force points v σ and v − σ . Then b η + and b η − are time-reversal of each other.For σ ∈ { + , −} , parametrizing the part of b η σ up to the time that it disconnects w − σ from ∞ by H -capacity, we get a chordal Loewner curve: η σ ( t ), 0 ≤ t < T σ , which is an hSLE κ curvein the chordal coordinate. Let b w σ and K σ ( · ) denote the chordal Loewner driving function andhulls for η σ . Let K ( t + , t − ) = Hull( K + ( t + ) ∪ K − ( t − )), ( t + , t − ) ∈ [0 , T + ) × [0 , T − ), and define anHC region: D = { t ∈ [0 , T + ) × [0 , T − ) : K ( t ) $ Hull( η w ) } . (5.4)For σ ∈ { + , −} , let F σ be the filtration generated by η σ . Let τ − be an F − -stopping time.Conditionally on F − τ − and the event { τ − < T − } , the part of b η w between η − ( τ − ) and w + andthe whole η v form a 2-SLE κ in H \ K − ( τ − ) with link pattern ( w + ↔ η − ( τ − ); v + ↔ v − ). So theconditional law of the part of b η + up to hitting η − ( τ − ) is that of an hSLE κ curve in H \ K − ( τ − )from w + to η − ( τ − ) with force points v + , v − , up to a time-change. This implies that there is arandom curve b η τ − + such that the f K − ( τ − ) -image of b η τ − + is the above part of b η + , and the conditionallaw of a time-change of b η τ − + is that of an hSLE κ curve in H from g K − ( τ − ) ( w + ) to b w − ( τ − ) withforce points g K − ( τ − ) ( v + ) , g K − ( τ − ) ( v − ). By the definition of D , the part of η + up to T D + ( τ − ) isa time-change of the part of b η + up to the first time that it hits η − ( τ − ) or separates η − ( τ − ) from ∞ , which is then the f K − ( τ − ) -image of the part of b η τ − + up to the first time that it hits b w − ( τ − )or separates b w − ( τ − ) from ∞ . So there is a random curve η τ − + such that the f K − ( τ − ) -image of η τ − + is the part of η + up to T D + ( τ − ), and the conditional law of a time-change of η τ − + is that ofan hSLE κ curve in H from g K − ( τ − ) ( w + ) to b w − ( τ − ) with force points g K − ( τ − ) ( v + ) , g K − ( τ − ) ( v − ),in the chordal coordinate. A similar statement holds with “+” and “ − ” swapped.Taking the stopping times in the previous paragraph to be deterministic numbers, we findthat ( η + , η − ; D ) a.s. satisfies the conditions in Definition 3.2 with I ± = [0 , T ± ) and I ∗± = I ± ∩ Q .By removing a null event, we may assume that ( η + , η − ; D ) is always a commuting pair ofchordal Loewner curves. We call ( η + , η − ; D ) a commuting pair of hSLE κ curves in the chordalcoordinate started from ( w + ↔ w − ; v + , v − ).Let F be the separable R -indexed filtration generated by F + and F − , and let F be theright-continuous augmentation of F . Then D is an F -stopping region because by Lemma 3.4, { t ∈ D } = lim Q ∋ s ↓ t ( { s < ( T + , T − ) } ∩ { K ( t ) $ K ( s ) } ) ∈ F t , ∀ t ∈ R . Define M : D → R + by M = G ( W + , W − ; V + , V − ), where G is given by (1.3). Since V + ≥ W + ≥ W − ≥ V − , and F is uniformly positive on [0 , C > κ such that M ≤ C | W + − W − | κ − | V + − V − | κ − ≤ C | V + − V − | κ (12 − κ ) . (5.5)51 emma 5.6. M a.s. extends continuously to R with M ≡ on R \ D .Proof. Since for σ ∈ { + , −} , η σ a.s. extends continuously to [0 , T σ ], by Remark 3.8, W + and W − a.s. extend continuously to D . From (3.15) we know that a.s. | V + − V − | is bounded on D .Thus, by (5.5) it suffices to show that the continuations of W + and W − agree on ∂ D ∩ R .Define A σ = { t σ e σ + T D − σ ( t σ ) e − σ : t σ ∈ (0 , T σ ) } , σ ∈ { + , −} . Then A + ∪ A − is dense in ∂ D ∩ (0 , ∞ ) . By symmetry, it suffices to show that W + and W − agree on A + . If this is nottrue, then there exists ( s + , s − ) ∈ D such that W + ( s + , · ) > W − ( s + , · ) on [ s − , T D − ( s + )].Let K s − ( t ) = K ( s + , s − + t ) /K ( s ) = K s + − ( s − + t ) /K s + − ( s − ), 0 ≤ t < T ′ := T D − ( s + ) − s − . Since K s + − ( t − ), 0 ≤ t − < T D − ( s + ), are chordal Loewner hulls driven by W − ( s + , · ) with speed m( s + , · ), K s − ( t ), 0 ≤ t < T ′ , are chordal Loewner hulls driven by W − ( s + , s − + · ) with speed m( s + , s − + · ).By Lemma 3.12 and Proposition 2.11, W + ( s + , s − + t ) = g W − ( s ) K s − ( t ) ( W + ( s )), 0 ≤ t < T ′ . Since W + ( s + , s − + · ) > W − ( s + , s − + · ) on [0 , T ′ ), we have dist( W + ( s ) , K s − ( t )) > ≤ t < T ′ .Since W + ( s + , s − + · ), W − ( s + , s − + · ) and m( s + , s − + · ) all extend continuously to [0 , T ′ ], and W + ( s + , s − + T ′ ) > W − ( w + , s − + T ′ ), the chordal Loewner process driven by W − ( s + , s − + t ),0 ≤ t ≤ T ′ , with speed m( s + , s − + · ) does not swallow W + ( s ) at the time T ′ , which impliesthat dist( W + ( s ) , Hull( S ≤ t 0, be as defined before Lemma 5.2. Lemma 5.7. For any R > , M ( · ∧ τ R ) is an ( F + t + ∧ τ + R ∨ F − t − ∧ τ − R ) ( t + ,t − ) ∈ R -martingale closedby M ( τ R ) , and if the underlying probability measure is weighted by M ( τ R ) /M (0) , then thenew law of ( b w + , b w − ) agrees with the probability measure P (2 , w ; v on F + τ + R ∨ F − τ − R .Proof. We follow the argument in the proof of Lemma 5.2, where Proposition 2.16 is the keyingredient, except that here we use (5.5) instead of (5.1).Let P denote the joint law of the driving functions b w + and b w − here, and let P = P (2 , w ; v .Following the proof of Lemma 5.3 and using Lemma 5.7, we get the following lemma. Lemma 5.8. A revision of Lemma 5.3 holds with all subscripts “ ” replaced by “ ” and thefiltration F replaced by F . Lemma 5.9. For any F -stopping time τ , M ( τ ) is P -a.s. positive on { τ ∈ D } . roof. Let τ be an F -stopping time. Let A = { τ ∈ D } ∩ { M ( τ ) = 0 } . We are going to showthat P [ A ] = 0. Since D is an F -stopping region, we have { τ ∈ D } ∈ F τ , and A ∈ F τ . Since M ( τ ) = 0 on A , by Lemma 5.8, P [ A ] = 0. For any t ∈ Q , since A ∈ F τ + t , by Lemma 5.8, P -a.s M ( τ + t ) = 0 on A . Thus, on the event A , P -a.s. M ( τ + t ) = 0 for any t ∈ Q , whichimplies by the continuity that M ≡ τ + R , which further implies that W + ≡ W − on( τ + R ) ∩ D , which in turn implies by Lemma 3.6 that η + ( τ + + t + ) = η − ( τ − + t − ) for any t = ( t + , t − ) ∈ R such that τ + t ∈ D . This is impossible since it implies (by setting t − = 0)that η + stays constant on [ τ + , T D + ( τ − )). So we have P [ A ] = 0. Remark 5.10. We do not have M > D if there is ( t + , t − ) ∈ D such that η + ( t + ) = η − ( t − ), which almost surely happens when κ ∈ (4 , Lemma 5.11. A revision of Lemma 5.4 holds with all subscripts “ ” replaced by “ ” and thefiltration F replaced by F .Proof. This follows from Lemmas 5.8 and 5.9.Assume that v := ( v + + v − ) / ∈ [ w − , w + ], and let V be the force point function startedfrom v . Here if v = w σ for some σ ∈ { + , −} , we treat it as w − σσ . We may define the timecurve u : [0 , T u ) → D and the processes R σ ( t ), σ ∈ { + , −} , and R ( t ) as in Section 3.4, andextend u to R + such that u ( s ) = lim t ↑ T u u ( t ) for s ≥ T u . Since D is an F -stopping region, byProposition 3.22, for any t ≥ u ( t ) is an F -stopping time.Define G ∗ on [0 , by G ∗ ( r + , r − ) = G ( r + , − r − ; 1 , − M u ( t ) = ( e t I ) α G ∗ ( R ( t ))for t ∈ [0 , T u ), where α = 2( κ − 1) is as in Theorem 1.1. Applying Lemma 5.11 to u ( t ), weget the following lemma, which is similar to Lemma 5.5. Lemma 5.12. Let p t ( r, r ∗ ) be the transition density p Rt ( r, r ∗ ) given in Corollary 4.16 with ρ = 0 and ρ + = ρ − = 2 . Then under P , the transition density of ( R ) is e p t ( r, r ∗ ) := e − α t p t ( r, r ∗ ) G ∗ ( r ) /G ∗ ( r ∗ ) . κ curves, a limit case Let w − < w + < v + ∈ R . Let ( η w , η v ) be a 2-SLE κ in H with link pattern ( w + ↔ w − ; v + ↔ ∞ ).For σ ∈ { + , −} , let b η σ be the curve η w oriented from w σ to w − σ and parametrized by thecapacity viewed from w − σ , which is an hSLE κ curve in H from w σ to w − σ . Then b η + and b η − are time-reversal of each other.For σ ∈ { + , −} , parametrizing the part of b η σ up to the time that it disconnects w − σ from ∞ by H -capacity, we get a chordal Loewner curve: η σ ( t ), 0 ≤ t < T σ , which is an hSLE κ curvefrom w σ to w − σ in the chordal coordinate. Define D using (5.4) for the ( η + , η − ) here. Then( η + , η − ; D ) is a.s. a commuting pair of chordal Loewner curves. Define W ± , V + and F for the( η + , η − ) here in the same way as in the previous subsection. Then D is an F -stopping region.53e call ( η + , η − ; D ) a commuting pair of hSLE κ curves in the chordal coordinate started from( w + ↔ w − ; v + ).Define M : D → R + by M = G ( W + , W − ; V + ), where G is given by (1.4). Since V + ≥ W + ≥ W − , we have M ≤ C | W + − W − | κ − | V + − V − | κ ≤ C | V + − V − | κ − for someconstant C > κ . Then the exactly same proof of Lemma 5.6 can be used hereto prove the following lemma. Lemma 5.13. M a.s. extends continuously to R with M ≡ on R \ D . Let P denote the joint law of the driving functions b w + and b w − here, and let P be thejoint law of the driving functions for a commuting pair of chordal SLE κ (2 , 2) started from( w + , w − ; v + ). Then similar arguments as in the previous subsection give the following lemma. Lemma 5.14. Revision of Lemmas 5.9 and 5.11 hold with all subscripts “ ” replaced by “ ”. Introduce two new points: v = ( w + + w − ) / v − = 2 v − v + . Let V and V − berespectively the force point functions started from v and v − . Since v = ( v + + v − ) / 2, we maydefine the time curve u : [0 , T u ) → D and the processes R σ ( t ), σ ∈ { + , −} , and R ( t ) as inSection 3.4. Let G ∗ ( r + , r − ) = G ( r + , r − ; 1). Then M u ( t ) = ( e t I ) α G ∗ ( R ( t )) for t ∈ [0 , T u ),where α = κ − u ( t ), we get the followinglemma, which is similar to Lemma 5.5. Lemma 5.15. Let p t ( r, r ∗ ) be the transition density p Rt ( r, r ∗ ) given in Corollary 4.16 with ρ = ρ − = 0 and ρ + = 2 . Then under P , the transition density of ( R ) is e p t ( r, r ∗ ) := e − α t p t ( r, r ∗ ) G ( r ) /G ( r ∗ ) . For j = 1 , , 3, using Lemmas 5.5, 5.12, and 5.15, we can obtain a quasi-invariant densityof R under P j as follows. Let G ∗ j ( r + , r − ) = G j ( r + , − r − ; 1 , − j = 1 , 2, and G ∗ ( r + , r − ) = G ( r + , − r − ; 1). Let p j ∞ be the invariant density p R ∞ of R under P j given by Corollary 4.16,where P = P = P (2 , r + , − r − ;1 , − and P = P (2) r + , − r − ;1 . Define Z j = Z (0 , p j ∞ ( r ∗ ) G ∗ j ( r ∗ ) dr ∗ , e p j ∞ = 1 Z j p j ∞ G ∗ j , j = 1 , , . (5.6)It is straightforward to check that Z j ∈ (0 , ∞ ), j = 1 , , 3. To see this, we compute p j ∞ ( r + , r − ) ≍ (1 − r + ) κ − (1 − r − ) κ − ( r + r − ) κ − for j = 1 , 2, and ≍ (1 − r + ) κ − (1 − r − ) κ − ( r + r − ) κ − for j = 3; G ∗ j ( r + , r − ) ≍ (1 − r + ) κ − (1 − r − ) κ − for j = 1, and ≍ ( r + + r − ) κ − for j = 2 , Lemma 5.16. The following statements hold. i) For any j ∈ { , , } , t > and r ∗ ∈ (0 , , R [0 , e p j ∞ ( r ) e p jt ( r, r ∗ ) dr = e − α j t e p j ∞ ( r ∗ ) .This means, under the law P j , if the process ( R ) starts from a random point in (0 , with density e p j ∞ , then for any deterministic t ≥ , the density of (the survived) R ( t ) is e − α j t e p j ∞ . So we call e p Rj a quasi-invariant density for ( R ) under P j .(ii) Let β = β = 10 and β = 8 . For j ∈ { , , } and r ∈ (0 , , if R starts from r , then P j [ T u > t ] = Z j G ∗ j ( r ) e − α j t (1 + O ( e − β j t )); (5.7) e p Rj ( t, r, r ∗ ) = P j [ T u > t ] e p j ∞ ( r ∗ )(1 + O ( e − β j t )) . (5.8) Here we emphasize that the implicit constants in the O symbols do not depend on r .Proof. Part (i) follows easily from (4.29). For part (ii), suppose R starts from r . Using Corollary4.16, Lemmas 5.5, 5.12, and 5.15, and formulas (5.6), we get P j [ T u > t ] = Z (0 , e p jt ( r, r ∗ ) dr ∗ = Z (0 , e − α j t p jt ( r, r ∗ ) G ∗ j ( r ) G ∗ j ( r ∗ ) dr ∗ = Z (0 , e − α j t p j ∞ ( r ∗ )(1 + O ( e − β j t )) G ∗ j ( r ) G ∗ j ( r ∗ ) dr ∗ = Z j G ∗ j ( r ) e − α j t (1 + O ( e − β j t )) , which is (5.7); and e p jt ( r, r ∗ ) = e − α j t p j ∞ ( r ∗ )(1 + O ( e − β j t )) G ∗ j ( r ) G ∗ j ( r ∗ ) = e − α j t Z j e p j ∞ ( r ∗ )(1 + O ( e − β j t )) G ∗ j ( r ) , which together with (5.7) implies (5.8).We will need the following lemma, which follows from the argument in [20, Appendix A]. Lemma 5.17. For j = 1 , , , the ( η + , η − ; D j ) in the three subsections satisfies the two-curve DMP as described in Lemma 4.1 except that the conditional law of the normalizationof ( e η + , e η − ; e D j ) has the law of a commuting pair of hSLE κ curves in the chordal coordinaterespectively started from ( W + , W − ; V + , V − ) | τ , ( W + ↔ W − ; V + , V − ) | τ , and ( W + ↔ W − ; V + ) | τ . We are going to prove the main theorem in this section. Lemma 6.1. For j = 1 , , let U j be a simply connected subdomain of the Riemann sphere b C ,which contains ∞ but no , and let f j be a conformal map from D ∗ := b C \ {| z | ≤ } onto U j ,which fixes ∞ . Let a j = lim z →∞ | f j ( z ) | / | z | > , j = 1 , , and a = a /a . If R > a , then {| z | > R } ⊂ U , and {| z | > aR + 4 a } ⊂ f ◦ f − ( {| z | > R } ) ⊂ {| z | ≥ aR − a } . roof. By scaling we may assume that a = a = 1. Let f = f ◦ f − . That {| z | > } ⊂ U follows from Koebe’s 1 / J ◦ f ◦ J , where J ( z ) := 1 /z . Fix z ∈ U . Let z = f − ( z ) ∈ D ∗ and z = f ( z ) ∈ U . Let r j = | z j | , j = 0 , , 2. Applying Koebe’s distortiontheorem to J ◦ f j ◦ J , we find that r + r − ≤ r j ≤ r + r + 2, j = 1 , 2, which implies that | r − r | ≤ 4. Thus, for R > f ( {| z | > R } ) ⊂ {| z | > R − } , and f ( {| z | = R } ) ⊂ {| z | ≤ R + 4 } .The latter inclusion implies that f ( {| z | > R } ) ⊃ {| z | > R + 4 } . Theorem 6.2. Let v − < w − < w + < v + ∈ R be such that ∈ [ v − , v + ] . Let ( b η + , b η − ) be a -SLE κ in H with link pattern ( w + ↔ v + ; w − ↔ v − ) . Let α = 2( κ − , β ′ = , and G ( w ; v ) be as in (1.2). Then there is a constant C > depending only on κ such that, P [ b η σ ∩ {| z | > L } 6 = ∅ , σ ∈ { + , −} ] = CL − α G ( w ; v )(1 + O ( | v + − v − | /L ) β ′ ) , (6.1) as L → ∞ , where the implicit constants in the O ( · ) symbol depend only on κ .Proof. Let p ( w ; v ; L ) denote the LHS of (6.1). Construct the random commuting pair of chordalLoewner curves ( η + , η − ; D ) from b η + and b η − as in Section 5.1, where D = [0 , T + ) × [0 , T − ),and T σ is the lifetime of η σ , σ ∈ { + , −} . We adopt the symbols from Sections 3.1. Note that,when L > | v + | ∨ | v − | , b η + and b η − both intersect {| z | > L } if and only if η + and η − both intersect {| z | > L } . In fact, for any σ ∈ { + , −} , η σ either disconnects v j from ∞ , or disconnects v − j from ∞ . If η σ does not intersect {| z | > L } , then in the former case, b η σ grows in a bounded connectedcomponent of H \ η σ after the end of η σ , and so can not hit {| z | > L } ; and in the latter case η − σ grows in a bounded connected component of H \ η σ , and can not hit {| z | > L } . We firstconsider a very special case: v ± = ± w ± = ± r ± , where r ± ∈ [0 , v = 0. Let V ν bethe force point function started from v ν , ν ∈ { , + , −} , as before. Since | v + − v | = | v − v − | ,we may define a time curve u : [0 , T u ) → D as in Section 3.4 and adopt the symbols from there.Define p ( r ; L ) = p ( r + , − r − ; 1 , − L ).Suppose L > e , and so log( L/ > 3. Let t ∈ [3 , log( L/ η + and η − intersect {| z | > L } , then there is some t ′ ∈ [0 , T u ) such that either η + ◦ u + [0 , t ′ ] or η − ◦ u − [0 , t ′ ] intersects {| z | > L } , which by (3.34) implies that L ≤ e t ′ , and so T u > t ′ ≥ log( L/ / > t . Thus, { η σ ∩ {| z | > L } 6 = ∅ , σ ∈ { + , −}} ⊂ { T u > t } . By (3.34) again, rad ( η σ [0 , u σ ( t )]) ≤ e t < L .So η σ ◦ u σ [0 , t ], σ ∈ { + , −} , do not intersect {| z | > L } .Let b g ut ( z ) = ( g K ( u ( t )) ( z ) − V u ( t )) /e t . Then b g ut maps C \ ( K ( u ( t )) doub ∪ [ v − , v + ]) con-formally onto C \ [ − , ∞ with b g ut ( z ) /z → e − t as z → ∞ . From V u − ≤ v − < V u + ≥ v + > 0, and V u = ( V u + + V u − ) / 2, we get | V u ( t ) | ≤ | V u + ( t ) − V u − ( t ) | / e t . ApplyingLemma 6.1 to f ( z ) = ( z + 1 /z ) / a = 1 / f = ( b g ut ) − ◦ f and a = e t / 2, and using that L > e t , we get {| z | > L } ⊂ C \ ( K ( u ( t )) doub ∪ [ v − , v + ]) and {| z | > L/e t − } ⊃ b g ut ( {| z | > L } ) ⊃ {| z | > L/e t + 2 } . (6.2)Note that both η + and η − intersect {| z | > L } if and only if T u > t and the b g ut -image ofthe parts of η σ after u σ ( t ), σ ∈ { + , −} , both intersect the b g ut -image of {| z | > L } . By Lemma5.17 for j = 1, conditionally on F u ( t ) and the event { T u > t } , the b g ut -image of the parts of η σ after u σ ( t ), σ ∈ { + , −} , after normalization, form a commuting pair of hSLE κ curves in56he chordal coordinate started from ( R + ( t ) , − R − ( t ); 1 , − η σ ( u σ ( t )) η − σ [0 , u − σ ( t )], σ ∈ { + , −} , is a.s. satisfied on { T u > t } ,which follows from Lemma 3.17 andthe fact that a.s. R σ ( t ) = ( W uσ ( t ) − V u ( t )) / ( V uσ ( t ) − V u ( t )) > σ ∈ { + , −} , on { T u > t } (because of the transition density of ( R ) vanishes outside (0 , ). From (6.2) we get P [ η σ ∩ {| z | > L } 6 = ∅ , σ ∈ { + , −}|F u ( t ) , T u > t ] R p ( R ( t ); L/e t ± . (6.3)Here when we choose + (resp. − ) in ± , the inequality holds with ≥ (resp. ≤ ).We use the approach of [6] to prove the convergence of lim L →∞ L α p ( r, L ). We first estimate p ( L ) := R (0 , p ( r ; L ) e p ∞ ( r ) dr , where e p ∞ is the quasi-invariant density for the process ( R ) under P given in Lemma 5.16. This is the probability that the two curves in a 2-SLE κ in H with linkpattern ( r + ↔ − r − ↔ − 1) both hit {| z | > L } , where ( r + , r − ) is a random point in (0 , thatfollows the density e p ∞ . From Lemma 5.16 we know that, for the deterministic time t , P [ T u >t ] = e − α t , and the law of ( R ( t )) conditionally on { T u > t } still has density e p ∞ . Thus, theconditional joint law of the b g ut -images of the parts of b η σ after η σ ( u σ ( t )), σ ∈ { + , −} given F ut and { T u > t } agrees with that of ( b η + , b η − ). From (6.3) we get P ( L ) R e − α t p ( L/e t ± q ( L ) = L α p ( L ). Then (if t ≥ L > e t ) q ( L ) R (1 ± e t /L ) − α q ( L/e t ± . (6.4)Suppose L > L ≥ e ( L + 2). Let t ± = log( L/ ( L ∓ / 2. Then L/e t ± ± L , t + > t − ≥ L = ( L − e t + > e t + > e t − . From (6.4) (applied here with t ± in placeof t ) we get q ( L ) R (1 ∓ /L ) α q ( L ) , if L ≥ e ( L + 2) and L > . (6.5)From (3.34) we know that T u > t implies that both η + and η − intersect {| z | > e t / } . Since P [ T u > t ] = e − α t > t ≥ 0, we see that p is positive on [0 , ∞ ), and so is q . From(6.5) we see that lim L →∞ q ( L ) converges to a point in (0 , ∞ ). Denote it by q ( ∞ ). By fixing L ≥ L → ∞ in (6.5), we get p ( L ) R q ( ∞ ) L − α (1 ∓ /L ) − α , if L ≥ . (6.6)Now we estimate p ( r ; L ) for a fixed deterministic r ∈ [0 , \ { (0 , } . The process ( R )starts from r and has transition density e p t given by Lemma 5.5. Fix L > e and choose t ∈ [3 , log( L/ / η + and η − intersect {| z | > L } implies that T u > t . Let β = 10. From Lemma 5.16 we know that P [ T u > t ] = Z G ∗ ( r ) e − α t (1 + O ( e − β t )) and thelaw of R ( t ) conditionally on { T u > t } has a density on (0 , , which equals e p ∞ · (1+ O ( e − β t )),where β = 10. Using Lemma 5.17 and (6.3,6.6) we get p ( r ; L ) = Z q ( ∞ ) G ∗ ( r ) e − α t ( L/e t ) − α (1 + O ( e − β t ))(1 + O ( e t /L )) . For L > e , by choosing t > e t = L / (2+ β ) and letting C = Z q ( ∞ ), we get p ( r ; L ) = C G ∗ ( r ) L − α (1 + O ( L − β ′ )). Here we note that β ′ = β / ( β + 2).Since G ∗ ( r + , r − ) = G ( r + , − r − ; 1 , − v ± = ± w + ∈ [0 , w − ∈ ( − , G ( aw + + b, aw − + b ; av + + b, av − + b ) = a − α G ( w + , w − ; v + , v − ) for any a > b ∈ R , by a translation and a dilation, we get (6.1) in the case that ( v + + v − ) / ∈ [ w − , w + ].Here we use the assumption that 0 ∈ [ v − , v + ] to control the amount of translation.Finally, we consider all other cases, i.e., ( v + + v − ) / [ w − , w + ]. By symmetry, we mayassume that ( v + + v − ) / < w − . Let v = ( w + + w − ) / 2. Then v + > w + > v > w − > v − , but v + − v < v − v − . We still let V ν be the force point functions started from v ν , ν ∈ { , + , −} .By (3.20), V ν satisfies the PDE ∂ + V ν ae = W , V ν − W + on D disj1 as defined in Section 3.3. Thus, on D disj1 , for any ν = ν ∈ { + , − , } , ∂ + log | V ν − V ν | ae = − W , ( V ν − W + )( V ν − W + ) , which implies that ∂ + ( V + − V V − V − ) ∂ + log( V + − V − ) = V + − V W + − V · V + − V − V − V − > . (6.7)The displayed formula means that V + − V V − V − | − is increasing faster than log( V + − V − ) | − . From theassumption, V + (0) − V (0) V (0) − V − (0) = v + − v v − v − ∈ (0 , τ + be the first t such that V + ( t, − V ( t, V ( t, − V − ( t, = 1; ifsuch time does not exist, then set τ + = T + . Then τ + is an F + -stopping time, and from (6.7) weknow that, for any 0 ≤ t < τ + , | V + ( t, − V − ( t, | < e | v + − v − | , which implies by (3.15) thatdiam([ v − , v + ] ∪ η + [0 , t ]) < e | v + − v − | . Let L = e | v + − v − | . From 0 ∈ [ v − , v + ] we get τ + ≤ τ + L .Here and below, we write W and V for ( W + , W − ) and ( V + , V − ), respectively. From Lemma5.2 we know that M ( · ∧ τ + L , 0) is a martingale closed by M ( τ + L , M = G ( W ; V ) and M ( t, 0) = 0 for t ≥ T + , we get E [ { τ + 0) = G ( w ; v ) . (6.8)Using the same argument as in the proof of (6.3) with ( τ + , 0) in place of u ( t ) and g K ( τ + , inplace of b g ut , we get P [ η σ ∩ {| z | = L } 6 = ∅ , σ ∈ { + , −}|F + τ + , τ + < T + ] R p (( W ; V ) | ( τ + , ; L ± ( V + − V − ) | ( τ + , ) . (6.9)Suppose τ + < T + . Then the middle point of [ V − ( τ + , , V + ( τ + , V ( τ + , W − ( τ + , , W + ( τ + , ∈ [ V − ( τ + , , V + ( τ + , V ± ( τ + , R v ± R 0. Let L ± = L ± ( V + − V − ) | ( τ + , . We may apply the result in the particular case to get p (( W ; V ) | ( τ + , ; L ± ) = C G ( W ; V ) | ( τ + , · L − α ± (1 + O (( V + − V − ) | ( τ + , /L ± ) β ′ )= C G ( W ; V ) | ( τ, · L − α (1 + O ( | v + − v − | /L ) β ′ ) . (6.10)Here in the last step we used ( V + − V − ) | ( τ + , ≤ e | v + − v − | and L ± /L = 1 + O ( | v + − v − | /L ).Plugging (6.10) into (6.9), taking expectation on both sides of (6.9), and using the fact that { η + ∩ {| z | = L } 6 = ∅} ⊂ { τ + < T + } , we get p ( w ; v ; L ) = C E [ { τ 8) is used to guarantee that the probability of all event are positive for any L > Theorem 6.4. Let v − < w − < w + < v + ∈ R be such that ∈ [ v − , v + ] . Let η w be an hSLE κ curve in H connecting w + and w − with force points v + and v − . Let α = κ (12 − κ ) , β ′ = , and G be as in (1.3. Then there is a constant C > depending only on κ such that, as L → ∞ , P [ b η w ∩ {| z | > L } 6 = ∅ ] = CL − α G ( w ; v )(1 + O ( | v + − v − | /L ) β ′ ) , where the implicit constants in the O ( · ) symbol depend only on κ .Proof. Let ( η + , η − ; D ) be the random commuting pair of chordal Loewner curves as defined inSection 5.2. Then for L > max {| v + | , | v − |} , b η w ∩ {| z | > L } 6 = ∅ if and only if η σ ∩ {| z | > L } 6 = ∅ for σ ∈ { + , −} . The rest of the proof follows that of Theorem 6.2 except that we now applyLemmas 5.16 and 5.17 with j = 2 and use Lemma 5.7 in place of Lemma 5.2. Theorem 6.5. Let w − < w + < v + ∈ R be such that ∈ [ w − , v + ] . Let η w be an hSLE κ in H connecting w + and w − with force points v + and ∞ . Let α = κ − , β ′ = , and G ( w ; v + ) beas in (1.4). Then there is a constant C > depending only on κ such that, as L → ∞ , P [ b η w ∩ {| z | > L } 6 = ∅ ] = CL − α G ( w ; v + )(1 + O ( | w + − v − | /L ) β ′ ) , where the implicit constants in the O ( · ) symbol depend only on κ .Proof. The proof follows those of Theorems 6.4 and 6.2 except that we now introduce v :=( w + + w − ) / v − := 2 v − v + as in Section 5.3. Then we can define the time curve u as inSection 3.4 and apply Lemmas 5.16 and 5.17 with j = 3. Proof of Theorem 1.1. By conformal invariance of 2-SLE κ , we may assume that D = H and z = ∞ . Case (A1) follows immediately from Theorem 6.2. Cases (A2) and (B) respectivelyfollow from Theorems 6.4 and 6.5 since we only need to consider the Green’s function for thecurve connecting w + and w − , which is an hSLE κ curve. Remark 6.6. The hSLE κ curve is a special case of the intermediate SLE κ ( ρ ) (iSLE κ ( ρ ) forshort) curves in [25] with ρ = 2. An iSLE κ ( ρ ) curve is defined using Definition 2.13 with F := F (1 − κ , ρκ ; ρ +4 κ ; · ) and e G := κ F ′ F + ρ . The curve is well defined for κ ∈ (0 , 8) and ρ > min {− , κ − } , and satisfies reversibility when κ ∈ (0 , 4] and ρ > − κ ∈ (4 , 8) and ρ ≥ κ − κ ( ρ ) satisfies reversibility, we can obtain a commuting pair of59SLE κ ( ρ ) curves in the chordal coordinate started from ( w + ↔ w − ; v + , v − ) or ( w + ↔ w − ; v + ) forgiven points v − < w − < w + < v + , which satisfy two-curve DMP. Following similar arguments,we find that Theorems 6.4 and 6.5 respectively hold for iSLE κ ( ρ ) curves with α = ρ +2 κ ( ρ +4 − κ ), α = κ ( ρ + 4 − κ ), β ′ = ρ +62 ρ +8 , β ′ = ρ +6 ρ +8 , and (with F = F (1 − κ , ρκ ; ρ +4 κ ; · )) G ( w ; v ) = | w + − w − | κ − | v + − v − | ρ (2 ρ +4 − κ )2 κ Y σ ∈{ + , −} | w σ − v − σ | ρκ F (cid:16) ( v + − w + )( w − − v − )( w + − v − )( v + − w − ) (cid:17) − ,G ( w ; v + ) = | w + − w − | κ − | v + − w − | ρκ F (cid:16) v + − w + w + − w − (cid:17) − . The proofs use the estimate on the transition density of R under P ( ρ,ρ ) w ; v and P ( ρ ) w ; v (Corollary 4.16)and revisions of Lemmas 5.11 and 5.14) with P and P now respectively representing P ( ρ,ρ ) w ; v and P ( ρ ) w ; v , P and P now respectively representing the joint law of the driving functions for a com-muting pair of iSLE κ ( ρ ) curves in the chordal coordinate started from ( w + ↔ w − ; v + , v − ) andfrom ( w + ↔ w − ; v + ), and M and M replaced by G ( W + , W − ; V + , V − ) and G ( W + , W − ; V + )for the current G and G .The revision of Theorem 6.4 (resp. 6.5) also holds in the degenerate case: v + = w ++ , inwhich the η w oriented from w = to w + is a chordal SLE κ ( ρ ) curve in H from w − to w + withthe force point at v − (resp. ∞ ). After a conformal map, we then obtain the boundary Green’sfunction for a chordal SLE κ ( ρ ) curve in H from 0 to ∞ with the force point v > z ∈ ( v, ∞ ) or at z = v . 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