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SLE loop measures

Abstract

We use Minkowski content (i.e., natural parametrization) of SLE to construct several types of \(\hbox {SLE}_\kappa \) loop measures for \(\kappa \in (0,8)\). First, we construct rooted \(\hbox {SLE}_\kappa \) loop measures in the Riemann sphere \(\widehat{\mathbb {C}}\), which satisfy Möbius covariance, conformal Markov property, reversibility, and space-time homogeneity, when the loop is parametrized by its \((1+\frac{\kappa }{8})\)-dimensional Minkowski content. Second, by integrating rooted \(\hbox {SLE}_\kappa \) loop measures, we construct the unrooted \(\hbox {SLE}_\kappa \) loop measure in \(\widehat{\mathbb {C}}\), which satisfies Möbius invariance and reversibility. Third, we use Brownian loop measures to extend the rooted and unrooted \(\hbox {SLE}_\kappa \) loop measures from \(\widehat{\mathbb {C}}\) to subdomains of \(\widehat{\mathbb {C}}\), which respectively satisfy conformal covariance and conformal invariance, and then further use the conformal invariance to extend unrooted \(\hbox {SLE}_\kappa \) loop measures to some Riemann surfaces. Finally, using a similar approach, we construct \(\hbox {SLE}_\kappa \) bubble measures in simply/multiply connected domains rooted at a boundary point. The space-time homogeneity of rooted \(\hbox {SLE}_\kappa \) loop measures in \(\widehat{\mathbb {C}}\) confirms a conjecture by Greg Lawler on the existence of such measures.

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Notes

  1. Werner told the author privately that they were able to prove that the rooted loop measure is well defined and satisfies the conformal Markov property (CMP) as described in the current paper [Theorem 4.1 (ii)]. Given this fact, using the uniqueness statement [Theorem 4.1 (vii)], we see that the loop measures constructed in the current paper for \(\kappa \in (8/3,4]\) agree with Kemppainen–Werner’s measures.

References

  1. Ahlfors, L.V.: Conformal Invariants: Topics in Geometric Function Theory. McGraw-Hill Book Co., New York (1973)

    MATH  Google Scholar 

  2. Alberts, T., Sheffield, S.: The covariant measure of SLE on the boundary. Probab. Theory Relat. 149, 331–371 (2011)

    Article  MathSciNet  Google Scholar 

  3. Bauer, R.O., Friedrich, R.M.: Stochastic Loewner evolution in multiply connected domains. C. R. Acad. Sci. Paris Ser. I 339(8), 579–584 (2004)

    Article  MathSciNet  Google Scholar 

  4. Beffara, V.: The dimension of SLE curves. Ann. Probab. 36, 1421–1452 (2008)

    Article  MathSciNet  Google Scholar 

  5. Benoist, S., Dubédat, J.: An \(\text{ SLE}_2\) loop measure. Ann. I. H. Poincare-Pr. 52(3), 1406–1436 (2016)

    Article  Google Scholar 

  6. Benoist, S.: Classifying conformally invariant loop measures (2016). arXiv:1608.03950

  7. Benoist, S., Dubédat, J.: Building \(\text{ SLE}_\kappa \) loop measures for \(\kappa <4\) (in preparation)

  8. Dubédat, J.: SLE and the free field: partition functions and couplings. J. Am. Math. Soc. 22, 995–1054 (2009)

    Article  MathSciNet  Google Scholar 

  9. Field, L.S.: Two-sided radial SLE and length-biased chordal SLE (2016). arXiv:1601.03374

  10. Field, L.S., Lawler, G.F.: Reversed radial SLE and the Brownian loop measure. J. Stat. Phys. 150(6), 1030–1062 (2013)

    Article  MathSciNet  Google Scholar 

  11. Field, L.S., Lawler, G.F.: SLE loops rooted at an interior point (in preparation)

  12. Holden, N., Sun, X.: SLE as a mating of trees in Euclidean geometry. Commun. Math. Phys. 364(1), 171–201 (2018)

    Article  MathSciNet  Google Scholar 

  13. Kallenberg, O.: Foundations of Modern Probability. Springer, Berlin (2011)

    MATH  Google Scholar 

  14. Kassel, A., Kenyon, R.: Random curves on surfaces induced from the Laplacian determinant. Ann. Probab. 45(2), 932–964 (2017)

    Article  MathSciNet  Google Scholar 

  15. Kemppainen, A., Werner, W.: The nested simple conformal loop ensembles in the Riemann sphere. Probab. Theory Relat. Fields 165(3), 835–866 (2016)

    Article  MathSciNet  Google Scholar 

  16. Kontsevich, M., Suhov, Y.: On Malliavin measures, SLE, and CFT. Proc. Steklov Inst. Math. 258, 100–146 (2007)

    Article  MathSciNet  Google Scholar 

  17. Lawler, G.F.: Minkowski content of the intersection of a Schramm–Loewner evolution (SLE) curve with the real line. J. Math. Soc. Jpn. 67(4), 1631–1669 (2015)

    Article  MathSciNet  Google Scholar 

  18. Lawler, G.F.: Defining SLE in multiply connected domains with the Brownian loop measure (2011) (in preprint). arXiv:1108.4364

  19. Lawler, G.F.: Conformally Invariant Processes in the Plane. American Mathematical Society, Providence (2005)

    MATH  Google Scholar 

  20. Lawler, G.F., Rezaei, M.A.: Minkowski content and natural parametrization for the Schramm–Loewner evolution. Ann. Probab. 43(3), 1082–1120 (2015)

    Article  MathSciNet  Google Scholar 

  21. Lawler, G.F., Schramm, O., Werner, W.: Values of Brownian intersection exponents I: half-plane exponents. Acta Math. 187(2), 237–273 (2001)

    Article  MathSciNet  Google Scholar 

  22. Lawler, G.F., Schramm, O., Werner, W.: Conformal restriction: the chordal case. J. Am. Math. Soc. 16(4), 917–955 (2003)

    Article  MathSciNet  Google Scholar 

  23. Lawler, G.F., Sheffield, S.: A natural parametrization for the Schramm–Loewner evolution. Ann. Probab. 39, 1896–1937 (2011)

    Article  MathSciNet  Google Scholar 

  24. Lawler, G.F., Viklund, F.: Convergence of loop-erased random walk in the natural parametrization (2016). arXiv:1603.05203

  25. Lawler, G.F., Werner, W.: The Brownian loop soup. Probab. Theory Relat. Fields 128(4), 565–588 (2004)

    Article  MathSciNet  Google Scholar 

  26. Lawler, G.F., Zhou, W.: SLE curves and natural parametrization. Ann. Probab. 41(3A), 1556–1584 (2013)

    Article  MathSciNet  Google Scholar 

  27. Malliavin, P.: The canonic diffusion above the diffeomorphism group of the circle. C. R. Acad. Sci. Paris Ser. I 329(4), 325–329 (1999)

    Article  MathSciNet  Google Scholar 

  28. Miller, J., Sheffield, S.: Imaginary Geometry IV: interior rays, whole-plane reversibility, and space-filling trees. Probab. Theory Relat. Fields 169(3–4), 729–869 (2017)

    Article  MathSciNet  Google Scholar 

  29. Miller, J., Sheffield, S.: Imaginary Geometry III: reversibility of \(\text{ SLE}_\kappa \) for \(\kappa \in (4, 8)\). Ann. Math. 184(2), 455–486 (2016)

    Article  MathSciNet  Google Scholar 

  30. Miller, J., Sheffield, S.: Imaginary geometry I: intersecting SLEs. Probab. Theory Relat. Fields 164(3), 553–705 (2016)

    Article  Google Scholar 

  31. Revuz, D., Yor, M.: Continuous Martingales and Brownian Motion. Springer, Berlin (1991)

    Book  Google Scholar 

  32. Rezaei, M.A., Zhan, D.: Green’s function for chordal SLE curves. Probab. Theory Relat. Fields 171(3), 1093–1155 (2018)

    Article  MathSciNet  Google Scholar 

  33. Rohde, S., Schramm, O.: Basic properties of SLE. Ann. Math. 161, 879–920 (2005)

    Article  MathSciNet  Google Scholar 

  34. Schramm, O.: Scaling limits of loop-erased random walks and uniform spanning trees. Isr. J. Math. 118, 221–288 (2000)

    Article  MathSciNet  Google Scholar 

  35. Schramm, O., Wilson, D.B.: SLE coordinate changes. N. Y. J. Math. 11, 659–669 (2005)

    MathSciNet  MATH  Google Scholar 

  36. Sheffield, S., Werner, W.: Conformal loop ensembles: the Markovian characterization and the loop-soup construction. Ann. Math. 176(3), 1827–1917 (2012)

    Article  MathSciNet  Google Scholar 

  37. Werner, W.: The conformally invariant measure on self-avoiding loops. J. Am. Math. Soc. 21, 137–169 (2008)

    Article  MathSciNet  Google Scholar 

  38. Zhan, D.: Optimal Hölder continuity and dimension properties for SLE with Minkowski content parametrization. Probab. Theory Relat. Fields 175, 447–466 (2019)

    Article  Google Scholar 

  39. Zhan, D.: Decomposition of Schramm–Loewner evolution along its curve. Stoch. Proc. Appl. 129(1), 129–152 (2019)

    Article  MathSciNet  Google Scholar 

  40. Zhan, D.: Ergodicity of the tip of an SLE curve. Probab. Theory Relat. Fields 164(1), 333–360 (2016)

    Article  MathSciNet  Google Scholar 

  41. Zhan, D.: Reversibility of whole-plane SLE. Probab. Theory Relat. 161(3), 561–618 (2015)

    Article  MathSciNet  Google Scholar 

  42. Zhan, D.: Reversibility of chordal SLE. Ann. Probab. 36(4), 1472–1494 (2008)

    Article  MathSciNet  Google Scholar 

  43. Zhan, D.: Random Loewner chains in Riemann surfaces. Ph.D Dissertation, Caltech (2004)

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Acknowledgements

The author would like to thank Greg Lawler and Wendelin Werner for inspiring discussions, thank Laurie Field and an anonymous referee for pointing out mistakes in earlier drafts and providing important suggestions, thank Stéphane Benoist for explaining MKS loop measures, and thank Wei Wu and Yiling Wang for some useful comments. The author acknowledges the support from the National Science Foundation (DMS-1056840) and from the Simons Foundation (#396973). The author also thanks the Institut Mittag-Leffler and Columbia University, where part of this work was carried out during workshops held there.

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Correspondence to Dapeng Zhan.

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Research partially supported by NSF Grant DMS-1056840 and Simons Foundation Grant #396973.

Appendices

Chordal SLE in multiply connected domains

In the appendix, we review the definition of chordal SLE in multiply connected domains for \(\kappa \in (0,8)\). First, we review hulls, Loewner chains and chordal Loewner equations, which define chordal SLE in simply connected domains. The reader is referred to [19] for details.

A subset K of \({\mathbb {H}}\) is called an \({\mathbb {H}}\)-hull if it is bounded, relatively closed in \({\mathbb {H}}\), and \({\mathbb {H}}{\setminus }K\) is simply connected. For every \({\mathbb {H}}\)-hull K, there is are a unique \(c\ge 0\) and a unique \(g_K:{\mathbb {H}}{\setminus }K{\mathop {\twoheadrightarrow }\limits ^{{\mathrm{Conf}}}}{\mathbb {H}}\) such that \(g_K(z)=z+\frac{c}{z}+O(\frac{1}{z^2})\) as \(z\rightarrow \infty \). The number c is called the \({\mathbb {H}}\)-capacity of K, and is denoted by \({{\,\mathrm{hcap}\,}}(K)\).

If \(K_1\subset K_2\) are two \({\mathbb {H}}\)-hulls, we define \(K_2/K_1=g_{K_1}(K_2{\setminus }K_1)\). Then \(K_2/ K_1\) is also an \({\mathbb {H}}\)-hull, and we have \({{\,\mathrm{hcap}\,}}(K_2/K_1)={{\,\mathrm{hcap}\,}}(K_2)-{{\,\mathrm{hcap}\,}}(K_1)\).

The following proposition is essentially Lemma 2.8 in [21].

Proposition A.1

Let W be a conformal map defined on a neighborhood of \(x_0\in {\mathbb {R}}\) such that an open real interval containing \(x_0\) is mapped into \({\mathbb {R}}\). Then

$$\begin{aligned} \lim _{H\rightarrow z_0} \frac{{{\,\mathrm{hcap}\,}}(W(H))}{{{\,\mathrm{hcap}\,}}(H)}=|W'(z_0)|^2, \end{aligned}$$

where \(H\rightarrow z_0\) means that \({{\,\mathrm{diam}\,}}(H\cup \{z_0\})\rightarrow 0\) with H being a nonempty \({\mathbb {H}}\)-hull.

Let \(T\in (0,\infty ]\) and \(\lambda \in C([0,T),{\mathbb {R}})\). The chordal Loewner equation driven by \(\lambda \) is

$$\begin{aligned} \partial _t g_t(z)=\frac{2}{g_t(z)-\lambda _t},\quad 0\le t<T;\quad g_0(z)=z. \end{aligned}$$
(A.1)

For each \(z\in \mathbb {C}\), let \(\tau _z\) be such that the maximal interval for \(t\mapsto g_t(z)\) is \([0,\tau _z)\). Let \(K_t=\{z\in {\mathbb {H}}:\tau _z\le t\}\), i.e., the set of \(z\in {\mathbb {H}}\) such that \(g_t(z)\) is not defined. Then \(g_t\) and \(K_t\), \(0\le t<T\), are called the chordal Loewner maps and hulls driven by \(\lambda \). It is known that \((K_t)\) is an increasing family of \({\mathbb {H}}\)-hulls with \({{\,\mathrm{hcap}\,}}(K_t)=2t\) and \(g_t=g_{K_t}\) for \(0\le t<T\). At \(t=0\), \(K_0=\emptyset \) and \(g_0={{\,\mathrm{id}\,}}\).

We say that \(\lambda \) generates a chordal Loewner curve \(\gamma \) if

$$\begin{aligned} \gamma (t)\,{:=}\,\lim _{{\mathbb {H}}\ni z\rightarrow \lambda (t)} g_t^{-1}(z)\in \overline{{\mathbb {H}}} \end{aligned}$$

exists for \(0\le t<T\), and \(\gamma \) is a continuous curve. We call such \(\gamma \) the chordal Loewner curve driven by \(\lambda \). If the such \(\gamma \) exists, then for each t, \({\mathbb {H}}{\setminus }K_t\) is the unbounded component of \({\mathbb {H}}{\setminus }\gamma ([0,t])\), and \(g_t^{-1}\) extends continuously from \({\mathbb {H}}\) to \({\mathbb {H}}\cup {\mathbb {R}}\). Since \({{\,\mathrm{hcap}\,}}(K_t)=2t\) for all t, we say that \(\gamma \) is parametrized by half-plane capacity.

Another way to characterize the chordal Loewner hulls \((K_t)\) is using the notation of \({\mathbb {H}}\)-Loewner chain. A family of \({\mathbb {H}}\)-hulls: \(K_t\), \(0\le t<T\), is called an \({\mathbb {H}}\)-Loewner chain if

  1. 1.

    \(K_0=\emptyset \) and \(K_{t_1}\subsetneqq K_{t_2}\) whenever \(0\le t_1<t_2<T\);

  2. 2.

    for any fixed \(a\in [0,T)\), the diameter of \(K_{t+\varepsilon }/ K_t\) tends to 0 as \(\varepsilon \rightarrow 0^+\), uniformly in \(t\in [0,a]\).

An \({\mathbb {H}}\)-Loewner chain \((K_t)\) is said to be normalized if \({{\,\mathrm{hcap}\,}}(K_t)=2t\) for each t. The following proposition is a result in [21].

Proposition A.2

Let \(T\in (0,\infty ]\). The following are equivalent.

  1. (i)

    \(K_t\), \(0\le t<T\), are chordal Loewner hulls driven by some \(\lambda \in C([0,T))\).

  2. (ii)

    \(K_t\), \(0\le t<T\), is a normalized \({\mathbb {H}}\)-Loewner chain.

If either of the above holds, we have

$$\begin{aligned} \{ \lambda (t)\}=\bigcap _{\varepsilon >0}\overline{K_{t+\varepsilon }/K_t}, \quad 0\le t<T. \end{aligned}$$
(A.2)

If \(K_t\), \(0\le t<T\), is any \({\mathbb {H}}\)-Loewner chain, then the function \(u(t){:}{=}{{\,\mathrm{hcap}\,}}(K_t)/2\), \(0\le t<T\), is continuous and strictly increasing with \(u(0)=0\), which implies that \(K_{u^{-1}(t)}\), \(0\le t<u(T)\), is a normalized \({\mathbb {H}}\)-Loewner chain.

For \(\kappa >0\), chordal \(\hbox {SLE}_\kappa \) is defined by solving the chordal Loewner equation with \(\lambda (t)=\sqrt{\kappa }B(t)\), where B(t) is a Brownian motion. The chordal Loewner curve \(\gamma \) driven by this driving function a.s. exists, and satisfies \(\lim _{t\rightarrow \infty }\gamma (t)=\infty \). So it is called a chordal \(\hbox {SLE}_\kappa \) curve in \({\mathbb {H}}\) from 0 to \(\infty \). It satisfies that, if \(\kappa \in (0,4]\), \(\gamma \) is simple, and \(K_t=\gamma ((0,t])\); if \(\kappa \ge 8\), \(\gamma \) is space-filling, i.e., visits every point in \(\overline{{\mathbb {H}}}\); if \(\kappa \in (4,8)\), \(\gamma \) is neither simple nor space-filling, and every bounded subset of \(\overline{{\mathbb {H}}}\) is contained in \(K_t\) for some finite \(t> 0\).

Via conformal maps, we may define an \(\hbox {SLE}_\kappa \) curve in any simply connected domain D from one prime end a to another prime end b. Recall that we use \(\mu ^\#_{D;a\rightarrow b}\) to denote the law of such a curve (modulo a time change).

Now we review the definition of chordal SLE in multiply connected domains in [18]. The laws of such SLE are no longer probability measures, but finite or \(\sigma \)-finite measures. We will use the following notation. Suppose D is a simply connected domain with two distinct prime ends a and b. Let \(U\subset D\) be an open neighborhood of both a and b in D. We define

$$\begin{aligned} \mu ^D_{U;a\rightarrow b}=\mathbf{1}_{\{\cdot \cap (D {\setminus }U)=\emptyset \}}e^{\frac{{{\,\mathrm{c}\,}}}{2} \mu ^{lp}({{\mathcal {L}}}_{D}(\cdot , D{\setminus }U))}\cdot \mu ^\#_{D;a\rightarrow b}. \end{aligned}$$
(A.3)

Proposition A.3

Let U and V be open neighborhoods of \({\mathbb {R}}\cup \{\infty \}\) in \({\mathbb {H}}\). Suppose \( W: U{\mathop {\twoheadrightarrow }\limits ^{{\mathrm{Conf}}}}V\) extends conformally across \({\mathbb {R}}\cup \{\infty \}\) such that \(W({\mathbb {R}})={\mathbb {R}}\) and \(W(\infty )=\infty \). Then for any \(x\in {\mathbb {R}}\),

$$\begin{aligned} \mu ^{{\mathbb {H}}}_{ V; W(x)\rightarrow \infty }=| W'(x)\cdot W'(\infty ) |^{-\frac{6-\kappa }{2\kappa }} W(\mu ^{{\mathbb {H}}}_{ U;x\rightarrow \infty }), \end{aligned}$$

where \( W'(\infty ){:}{=}(J\circ W\circ J)'(0)\) with \(J(z){:}{=}-1/z\).

Proof

This proposition was proved in [18, Section 4.1] for \(\kappa \in (0,4]\) by considering simply connected subdomains of U. In this proof, we assume that \(\kappa \in (4,8)\). The proof is similar to those of Theorem 5.1 and Lemma 3.4, and uses a standard argument that originated in [22]. WLOG, we may assume that \(x=0\) and \(W(0)=0\). Let \(P_a\) denote the multiplication map \(z\mapsto az\). By conformal invariance of chordal SLE and Brownian loop measure, we know that \(\mu ^{{\mathbb {H}}}_{ P_a(V); 0\rightarrow \infty }={P}_a(\mu ^{{\mathbb {H}}}_{ V; 0\rightarrow \infty })\) for any \(a>0\). Since \((aW)'(0)\cdot (aW)'(\infty )=W'(0)\cdot W'(\infty )\), we may assume that \(W'(\infty )=1\) by replacing W with aW for some \(a>0\).

Let \(\gamma \) be a chordal \(\hbox {SLE}_\kappa \) curve in \({\mathbb {H}}\) from 0 to \(\infty \) with driving function \(\lambda _t=\sqrt{\kappa }B_t\). Let \(g_t\) and \(K_t\), \(0\le t<\infty \), be the chordal Loewner maps and hulls, respectively, driven by \(\lambda \).

Let \(\tau _U\) be the first time that \(\gamma \) exits U. Then \(\beta (t){:}{=}W(\gamma (t))\) is well defined for \(0\le t<\tau _U\). For each \(0\le t<\tau _U\), let \(L_t\) be the \({\mathbb {H}}\)-hull such that \({\mathbb {H}}{\setminus }L_t\) is the unbounded connected component of \({\mathbb {H}}{\setminus }\beta ([0,t])\). If \(K_t\subset U\), then \(L_t=W(K_t)\). Since \(\kappa \in (4,8)\), \(K_t\) may swallow some relatively clopen subset of \({\mathbb {H}}{\setminus }U\) before the time \(\tau _U\), and \(W(K_t)\) is not defined at that time. Using the conformal invariance of extremal length, we can see that \((L_t)\) is an \({\mathbb {H}}\)-Loewner chain (even after \(K_t\) intersects \({\mathbb {H}}{\setminus }U\)). From Proposition A.2, we may reparametrize the family \((L_t)\) using the function \(u(t)={{\,\mathrm{hcap}\,}}(L_t)/2\) to get a family of chordal Loewner hulls. Let \(\sigma _s\), \(0\le s<S{:}{=}u(\tau _U)\), be the driving function for the normalized \((L_s)\). Let \(h_s\), \(0\le s<S\), be the corresponding chordal Loewner maps. We also reparametrize \(\beta \) using u. Then \(\beta \) is the chordal Loewner curve driven by \(\sigma \), and \(\beta _{u(t)}=W(\gamma (t))\), \(0\le t<\tau _U\).

For \(0\le t<\tau _U\), define \( U_t=g_t( U{\setminus }K_t)\), \( V_t=h_{u(t)}( V{\setminus }L_{u(t)})\), and \(W_t=h_{u(t)}\circ W\circ g_t^{-1}\). Then \( U_t\) and \( V_t\) are open neighborhoods of \({\mathbb {R}}\cup \{\infty \}\) in \({\mathbb {H}}\), \( W_t: U_t{\mathop {\twoheadrightarrow }\limits ^{{\mathrm{Conf}}}}V_t\), and satisfies that, if \(z\in U_t\) tends to \({\mathbb {R}}\) or \(\infty \), then \( W_t\) tends to \({\mathbb {R}}\) or \(\infty \), respectively. By Schwarz reflection principle, \( W_t\) extends conformally across \({\mathbb {R}}\), and maps \({\mathbb {R}}\) onto \({\mathbb {R}}\). Since \(W,g_t,h_{u(t)}\) all fix \(\infty \), and have derivative 1 at \(\infty \), \(W_t\) also satisfies this property.

By the continuity of \( g_t\) and \( h_{u(t)}\) in t and the maximal principle, we know that the extended \( W_t\) is continuous in t (and z). Fix \(0\le t<\tau _U\). Let \(\varepsilon \in (0,\tau _U-t)\). Now \(K_{t+\varepsilon }/K_t\) is an \({\mathbb {H}}\)-hull with \({\mathbb {H}}\)-capacity being \(2\varepsilon \); and \(L_{u(t+\varepsilon )}/ L_{u(t)}\) is an \({\mathbb {H}}\)-hull with \({\mathbb {H}}\)-capacity being \(2u(t+\varepsilon )-2u(t)\). Since \( W_t(K_{t+\varepsilon }/K_t)=L_{u(t+\varepsilon )}/ L_{u(t)}\), using Propositions A.1 and A.2, we get

$$\begin{aligned} \sigma _{u(t)}= W_t(\lambda _t), \end{aligned}$$
(A.4)

and \(u_+'(t)= W_t'({\lambda _t})^2\). Using the continuity of \(W_t\) in t, we get

$$\begin{aligned} u'(t)= W_t'(\lambda _t)^2. \end{aligned}$$
(A.5)

Thus, \( h_{u(t)}\) satisfies the equation

$$\begin{aligned} \partial _t h_{u(t)}(z)= \frac{2 W_t'(\lambda _t)^2}{ h_{u(t)}(z)-\sigma _{u(t)}}. \end{aligned}$$
(A.6)

From the definition of \( W_t\), we get the equality

$$\begin{aligned} W_t\circ g_t(z)=h_{u(t)}\circ W(z), \quad z \in U{\setminus }K_t. \end{aligned}$$
(A.7)

Differentiating this equality w.r.t. t and using (A.1, A.6), we get

$$\begin{aligned} \partial _t W_t( g_t(z))+\frac{2 W_t'( g_t(z))}{ g_t(z)-\lambda _t}=\frac{2 W_t'(\lambda _t)^2}{ h_{u(t)}\circ W(z)-\sigma _{u(t)}},\quad z\in U{\setminus }K_t. \end{aligned}$$

Combining this formula with (A.4, A.7) and replacing \(g_t(z)\) with w, we get

$$\begin{aligned} \partial _t W_t(w)=\frac{2 W_t'(\lambda _t)^2}{ W_t(w)- W_t(\lambda _t)} -\frac{2 W_t'(w)}{w-\lambda _t},\quad w\in U_t. \end{aligned}$$
(A.8)

Letting \( U_t\ni w\rightarrow \lambda _t\) in (A.8), we get

$$\begin{aligned} \partial _t W_t(\lambda _t)=-3 W_t''(\lambda _t). \end{aligned}$$
(A.9)

Differentiating (A.8) w.r.t. w and letting \( U_t\ni w\rightarrow \lambda _t\), we get

$$\begin{aligned} \frac{\partial _t W_t'(\lambda _t)}{ W_t'(\lambda _t)}=\frac{1}{2}\left( \frac{ W_t''(\lambda _t)}{ W_t'(\lambda _t)}\right) ^2-\frac{4}{3}\frac{ W_t'''(\lambda _t)}{ W_t'(\lambda _t)}. \end{aligned}$$
(A.10)

Combining (A.4, A.9), and using Itô’s formula and that \(\lambda _t=\sqrt{\kappa }B_t\), we see that \(\sigma _{u(t)}\) satisfies the SDE

$$\begin{aligned} d\sigma _{u(t)}= W_t'(\lambda _t)\sqrt{\kappa }dB_t +\left( \frac{\kappa }{2}-3\right) W_t''(\lambda _t)dt. \end{aligned}$$
(A.11)

Combining (A.10) with \(\lambda _t=\sqrt{\kappa }B_t\) and using Itô’s formula, we get

$$\begin{aligned} \frac{d W_t'(\lambda _t)}{ W_t'(\lambda _t)}=\frac{ W_t''(\lambda _t)}{ W_t'(\lambda _t)}\sqrt{\kappa }dB_{t} +\frac{1}{2}\left( \frac{ W_t''(\lambda _t)}{ W_t'(\lambda _t)}\right) ^2dt+\left( \frac{\kappa }{2}-\frac{4}{3}\right) \frac{ W_t'''(\lambda _t)}{ W_t'(\lambda _t)}\,dt . \end{aligned}$$
(A.12)

Let \((Sf)(z)=\frac{f'''(z)}{f'(z)}-\frac{3}{2}(\frac{f''(z)}{f'(z)})^2\) be the Schwarzian derivative of f. Using (A.12) and Itô’s formula, we see that

$$\begin{aligned} \frac{d W_t'(\lambda _t)^{\frac{6-\kappa }{2\kappa }}}{ W_t'(\lambda _t)^{\frac{6-\kappa }{2\kappa }}}=\frac{6-\kappa }{2 } \frac{ W_t''(\lambda _t)}{ W_t'(\lambda _t)} \frac{dB_t}{\sqrt{\kappa }} +\frac{{{\,\mathrm{c}\,}}}{6} S( W_t)(\lambda _t)dt.\ \end{aligned}$$
(A.13)

So we get the following positive continuous local martingale

$$\begin{aligned} M_t{:}{=}W_t'(\lambda _t)^{\frac{6-\kappa }{2\kappa }}\exp \left( -\int _0^t \frac{{{\,\mathrm{c}\,}}}{6} S( W_s)(\lambda _s)ds\right) , \end{aligned}$$
(A.14)

which satisfies the SDE

$$\begin{aligned} \frac{d M_t}{ M_t}= \frac{6-\kappa }{2 } \frac{ W_t''(\lambda _t)}{ W_t'(\lambda _t)} \frac{dB_t}{\sqrt{\kappa }},\quad 0\le t<\tau _U. \end{aligned}$$
(A.15)

We claim that the following equality holds: for any \(0\le T<\tau _U\),

$$\begin{aligned} \int _{0}^T \frac{1}{6} S(W_t)(\lambda _t)dt=\frac{1}{2}\mu ^{{{\,\mathrm{lp}\,}}}({{\mathcal {L}}}_{{\mathbb {H}}}(\beta ([0,u(T)],{\mathbb {H}}{\setminus }V))-\frac{1}{2}\mu ^{{{\,\mathrm{lp}\,}}}({{\mathcal {L}}}_{{\mathbb {H}}}(\gamma ([0,T]), {\mathbb {H}}{\setminus }U)).\nonumber \\ \end{aligned}$$
(A.16)

Note that this is similar to Lemma 3.4. To prove (A.16), we use the Brownian bubble analysis of Brownian loop measure. Let \(\mu ^{{{\,\mathrm{bb}\,}}}_{ {x_0}}\) denote the Brownian bubble measure in \({\mathbb {H}}\) rooted at \(x_0\in {\mathbb {R}}\) as defined in [25], from which we know, for any \(0\le T<\tau _U\),

$$\begin{aligned} \frac{1}{2} \mu ^{{{\,\mathrm{lp}\,}}}({{\mathcal {L}}}_{{\mathbb {H}}}(\gamma ([0,T]), {\mathbb {H}}{\setminus }U))&=\int _{0}^T \mu ^{{{\,\mathrm{bb}\,}}}_{\lambda _t}({{\mathcal {L}}}({\mathbb {H}}{\setminus }U_t))dt; \end{aligned}$$
(A.17)
$$\begin{aligned} \frac{1}{2} \mu ^{{{\,\mathrm{lp}\,}}}({{\mathcal {L}}}_{{\mathbb {H}}}(\beta ([0,u(T)],{\mathbb {H}}{\setminus }V))&=\int _{0}^{u(T)} \mu ^{{{\,\mathrm{bb}\,}}}_{\sigma _s}({{\mathcal {L}}}({\mathbb {H}}{\setminus }V_s))ds \nonumber \\&=\int _{0}^T W_t'(\lambda _t)^2 \mu ^{{{\,\mathrm{bb}\,}}}_{\sigma _{u(t)}}({{\mathcal {L}}}({\mathbb {H}}{\setminus }V_{u(t)}))dt. \end{aligned}$$
(A.18)

If \(U^*\) is a subdomain of \({\mathbb {H}}\) that contains a neighborhood of \({\mathbb {R}}\cup \{\infty \}\) in \({\mathbb {H}}\), we let \(P^{U^*}_{x_0}\) denote the Poisson kernel in \(U^*\) with the pole at \(x_0\in {\mathbb {R}}\). Especially, \(P^{{\mathbb {H}}}_{x_0}(z)={{\,\mathrm{Im }\,}}\frac{-1/\pi }{z-x_0}\). From [25] we know

$$\begin{aligned} \mu ^{{{\,\mathrm{bb}\,}}}_{{\lambda _t}}({{\mathcal {L}}}({{\mathbb {H}}{\setminus }U_t})) =\lim _{U_t\ni z\rightarrow {\lambda _t}}\frac{1}{|z-{\lambda _t}|^2}\Big (1-\frac{P^{U_t}_{\lambda _t} (z)}{P^{{\mathbb {H}}}_{\lambda _t}(z)}\Big ) \end{aligned}$$

Similarly, using (A.4) and that \(W_t:U_t{\mathop {\twoheadrightarrow }\limits ^{{\mathrm{Conf}}}}V_{u(t)}\), we get

$$\begin{aligned} \mu ^{{{\,\mathrm{bb}\,}}}_{\sigma _{u(t)}}({{\mathcal {L}}}({{\mathbb {H}}{\setminus }V_{u(t)}}))&= \lim _{V_{u(t)}\ni w\rightarrow {\sigma _{u(t)}}}\frac{1}{|w-{\sigma _{u(t)}}|^2}\Big (1-\frac{P^{V_{u(t)}}_{\sigma _{u(t)}}(w)}{ P^{{\mathbb {H}}}_{\sigma _{u(t)}}(w)}\Big )\\&= \lim _{U_{t}\ni z\rightarrow {\lambda _t}} \frac{1}{| W_t(z) - W_t(\lambda _t)|^2}\Big (1-\frac{ P^{V_{u(t)}}_{\sigma _{u(t)}}\circ W_t(z)}{P^{{\mathbb {H}}}_{\sigma _{u(t)}}\circ W_t(z)}\Big )\\&= \lim _{ U_{t}\ni z\rightarrow {\lambda _t}} \frac{ W_t'(\lambda _t)^{-2}}{|z-\lambda _t|^2}\Big (1-\frac{ W_t'(\lambda _t)^{-1} P^{U_t}_{\lambda _t}(z) }{P^{{\mathbb {H}}}_{\sigma _{u(t)}}\circ W_t(z)}\Big ). \end{aligned}$$

Combining the above two formulas and using some tedious but straightforward computation involving power series expansions, we get

$$\begin{aligned} W_t'(\lambda _t)^2 \mu ^{{{\,\mathrm{bb}\,}}}_{{\sigma _{u(t)}}}({{\mathcal {L}}}({{\mathbb {H}}{\setminus }V_{u(t)}}))-\mu ^{{{\,\mathrm{bb}\,}}}_{{\lambda _t}}({{\mathcal {L}}}({{\mathbb {H}}{\setminus }U_t}))=\frac{1}{6} S(W_t)(\lambda _t). \end{aligned}$$

This together with (A.17, A.18) completes the proof of (A.16).

Since \(\gamma \) is continuous and tends to \(\infty \), from (2.1, A.16), we see that, on the event that \(\gamma \cap ({\mathbb {H}}{\setminus }U)=\emptyset \), the improper integral \(\int _0^\infty \frac{1}{6} S(W_s)(\lambda _s)ds\) converges to \(\frac{1}{2}\mu ^{{{\,\mathrm{lp}\,}}}({{\mathcal {L}}}_{{\mathbb {H}}}(\beta ,{\mathbb {H}}{\setminus }V))-\frac{1}{2}\mu ^{{{\,\mathrm{lp}\,}}}({{\mathcal {L}}}_{{\mathbb {H}}}(\gamma , {\mathbb {H}}{\setminus }U))\).

We claim that \(\lim _{t\rightarrow \infty } W_t'(\lambda _t)=1\) on the event that \(\gamma \cap {\mathbb {H}}{\setminus }U=\emptyset \). Since \(\kappa \in (4,8)\), there is \(t_0\in (0,\infty )\) such that \({\mathbb {H}}{\setminus }U\subset K_{t_0}\). Then for \(t\ge t_0\), \(U{\setminus }K_t={\mathbb {H}}{\setminus }K_t\), and so \(U_t={\mathbb {H}}\). Similarly, \(V_t={\mathbb {H}}\) for \(t\ge t_0\). Thus, for \(t\ge t_0\), \(W_t:({\mathbb {H}};\infty ){\mathop {\twoheadrightarrow }\limits ^{{\mathrm{Conf}}}}({\mathbb {H}};\infty )\) and \(W_t'(\infty )=1\), which implies that \(W_t'(\lambda _t)=1\). So the claim is proved.

From the above we see that \(M_\infty {:}{=}\lim _{t\rightarrow \infty } M_t=e^{\frac{{{\,\mathrm{c}\,}}}{2} \mu ^{{{\,\mathrm{lp}\,}}}({{\mathcal {L}}}_{{\mathbb {H}}}(\gamma , {\mathbb {H}}{\setminus }U))}/e^{\frac{{{\,\mathrm{c}\,}}}{2} \mu ^{{{\,\mathrm{lp}\,}}}({{\mathcal {L}}}_{{\mathbb {H}}}(W(\gamma ) ,{\mathbb {H}}{\setminus }V))}\) on the event that \(\gamma \cap ({\mathbb {H}}{\setminus }U)=\emptyset \). Thus, \(M_t\), \(0\le t<\infty \), is bounded on this event.

For \(n\in {\mathbb {N}}\), let \(T_n\) be the first time that \(\gamma \) hits \({\mathbb {H}}{\setminus }U\) or \(M_t\ge n\), whichever happens first. Then \(T_n\) is a stopping time, and \(M_t\) up to \(T_n\) is bounded by n. Thus, \({\mathbb {E}}[M_{T_n}]=M_0=W'(0)^{\frac{6-\kappa }{2\kappa }}\). Weighting the underlying probability measure by \(M_{T_n}/M_0\), we get a new probability measure. By Girsanov Theorem and (A.15), we find that

$$\begin{aligned} \widehat{B} _t \,{:=}\, B_t-\frac{1}{\sqrt{\kappa }} \int _0^t \frac{6-\kappa }{2} \frac{ W_s''(\lambda _s)}{ W_s'(\lambda _s)}ds,\quad 0\le t< T_n, \end{aligned}$$

is a Brownian motion under the new probability measure. From (A.11), we get

$$\begin{aligned} d\sigma _{u(t)}= W_t'(\lambda _t)\sqrt{\kappa }d\widehat{B}_t,\quad 0\le t< T_n. \end{aligned}$$

From (A.5) we see that, under the new probability measure, \(\sigma _s/\sqrt{\kappa }\), \(0\le s< u(T_n)\), is a Brownian motion, and so \(\beta _s\), \(0\le s\le u(T_n)\), is a chordal \(\hbox {SLE}_\kappa \) curve in \({\mathbb {H}}\) from 0 to \(\infty \), stopped at \(u(T_n)\). let \(E_n\) denote the event that \(\gamma \cap ({\mathbb {H}}{\setminus }U)=\emptyset \) and \(M_t\le n\) for \(0\le t<\infty \); and let \(F_n\) denote the event that \(W^{-1}(\beta )\in E_n\). Then on the event \(E_n\), \(T_n=u(T_n)=\infty \), and \(M_{T_n}/M_0=M_\infty /W'(0)^{\frac{6-\kappa }{2\kappa }}\). From the above argument, we get

$$\begin{aligned} \mathbf{1}_{F_n}\cdot \mu ^\#_{{\mathbb {H}};0\rightarrow \infty }=W(W'(0)^{-\frac{6-\kappa }{2\kappa }}e^{\frac{{{\,\mathrm{c}\,}}}{2} \mu ^{{{\,\mathrm{lp}\,}}}({{\mathcal {L}}}_{{\mathbb {H}}}(\cdot , {\mathbb {H}}{\setminus }U ))}/e^{\frac{{{\,\mathrm{c}\,}}}{2} \mu ^{{{\,\mathrm{lp}\,}}}({{\mathcal {L}}}_{{\mathbb {H}}}(W(\cdot ),{\mathbb {H}}{\setminus }V))} \mathbf{1}_{E_n}\cdot \mu ^\#_{{\mathbb {H}};0\rightarrow \infty } ). \end{aligned}$$

Since \(\mu ^\#_{{\mathbb {H}};0\rightarrow \infty }\)-a.s. \(\bigcup E_n=\{\cdot \cap {\mathbb {H}}{\setminus }U=\emptyset \}\) and \(\bigcup F_n=\{\cdot \cap {\mathbb {H}}{\setminus }V=\emptyset \}\), the above formula holds with \(E_n\) and \(F_n\) replaced by \(\{\cdot \cap {\mathbb {H}}{\setminus }U=\emptyset \}\) and \(\{\cdot \cap {\mathbb {H}}{\setminus }V=\emptyset \}\), respectively. The proposition now follows from this formula since we assumed that \(W'(\infty )=1\). \(\square \)

Remark A.4

The above proof also works for \(\kappa \in (0,4]\) except that the proof of the limit \(\lim _{t\rightarrow \infty } W_t'(\lambda _t)=1\) on the event \(\gamma \cap ({\mathbb {H}}{\setminus }U)=\emptyset \) requires a little bit more work.

Lemma A.5

Let K and L be two non-degenerate interior hulls. Let \(U,V\subset \widehat{\mathbb {C}}\) be open neighborhoods of K and L, respectively. Suppose \(W:(U;K){\mathop {\twoheadrightarrow }\limits ^{{\mathrm{Conf}}}}(V;L)\). Let a and b be distinct prime ends of \(\widehat{\mathbb {C}}{\setminus }K\). Then W(a) and W(b) are distinct prime ends of \(\widehat{\mathbb {C}}{\setminus }L\). Let \(g_K:\widehat{\mathbb {C}}{\setminus }K{\mathop {\twoheadrightarrow }\limits ^{{\mathrm{Conf}}}}{\mathbb {D}}^*\) and \(g_L:\widehat{\mathbb {C}}{\setminus }L{\mathop {\twoheadrightarrow }\limits ^{{\mathrm{Conf}}}}{\mathbb {D}}^*\). Suppose \(g_K(a)=e^{i\lambda }\), \(g_K(b)=e^{iq}\), \(g_L(W(a))=e^{i\sigma }\), and \(g_L(W(b))=e^{ip}\) for some \(\lambda ,q,\sigma ,p\in {\mathbb {R}}\). Let \(W_K=g_L\circ W\circ g_K^{-1}\). Extend \(W_K\) conformally across \({\mathbb {T}}\). Then we have

$$\begin{aligned} \mu ^{\widehat{C}{\setminus }L}_{V{\setminus }L;W(a)\rightarrow W(b)}= \Big |\frac{\sin _2(\sigma -p)}{\sin _2(\lambda -q)}\Big |^{\frac{6}{\kappa }-1}\cdot |W_K'(e^{i\sigma })W_K'(e^{iq}) |^{-\frac{6-\kappa }{2\kappa }} \cdot W(\mu ^{\widehat{\mathbb {C}}{\setminus }K}_{U{\setminus }K;a\rightarrow b}). \end{aligned}$$

Proof

Let \(\phi (z)=i\frac{z+e^{iq}}{z-e^{iq}}\) and \(\psi (z)=i\frac{z+e^{ip}}{z-e^{ip}}\). Then \(\phi :({\mathbb {D}}^*;e^{i\lambda },e^{iq}){\mathop {\twoheadrightarrow }\limits ^{{\mathrm{Conf}}}}({\mathbb {H}};\cot _2(\lambda -q),\infty )\) and \(\psi :({\mathbb {D}}^*;e^{i\sigma },e^{ip}){\mathop {\twoheadrightarrow }\limits ^{{\mathrm{Conf}}}}({\mathbb {H}};\cot _2(\sigma -p),\infty )\). Let \(U_K=g_K(U{\setminus }K)\) and \(V_L=g_L(V{\setminus }L)\). Then \(U_K\) and \(V_L\) are open neighborhoods of \({\mathbb {T}}\) in \({\mathbb {D}}^*\), \(W_K:U_K{\mathop {\twoheadrightarrow }\limits ^{{\mathrm{Conf}}}}V_L\), and can be extended conformally across \({\mathbb {T}}\). The extended \(W_K\) maps \({\mathbb {T}}\) onto \({\mathbb {T}}\), and maps \(e^{i\lambda }\) and \(e^{iq}\) to \(e^{i\sigma }\) and \(e^{ip}\), respectively. Let \(\widehat{U}_K= \phi (U_K)\), \(\widehat{V}_L=\psi (V_L)\), and \(\widehat{W}_K=\psi \circ W_K\circ \phi ^{-1}\). Then \(\widehat{U}_K\) and \(\widehat{V}_L\) are open neighborhoods of \({\mathbb {R}}\cup \{\infty \}\) in \({\mathbb {H}}\), and \(\widehat{W}_K:(\widehat{U}_K;{\mathbb {R}},\cot _2(\lambda -q),\infty ){\mathop {\twoheadrightarrow }\limits ^{{\mathrm{Conf}}}}(\widehat{V}_K;{\mathbb {R}},\cot _2(\sigma -p),\infty )\). From Proposition A.3, we have

$$\begin{aligned} \mu ^{{\mathbb {H}}}_{\widehat{V}_L;\cot _2(\sigma -p)\rightarrow \infty } = |\widehat{W}_K'(\cot _2(\lambda -q))\widehat{W}'(\infty )|^{-\frac{6-\kappa }{2\kappa }} \widehat{W}_K(\mu ^{{\mathbb {H}}}_{\widehat{U}_K;\cot _2(\lambda -q)\rightarrow \infty }). \end{aligned}$$

We have \(\phi \circ g_K:(\widehat{\mathbb {C}}{\setminus }K,U{\setminus }K;a,b){\mathop {\twoheadrightarrow }\limits ^{{\mathrm{Conf}}}}({\mathbb {H}},\widehat{U}_K;\cot _2(\lambda -q),\infty )\) and \(\psi \circ g_L:(\widehat{\mathbb {C}}{\setminus }L,V{\setminus }L;W(a),W(b)){\mathop {\twoheadrightarrow }\limits ^{{\mathrm{Conf}}}}({\mathbb {H}},\widehat{V}_L;\cot _2(\sigma -p),\infty )\). From the conformal invariance of chordal SLE and Brownian loop measure, we have

$$\begin{aligned}&\phi \circ g_K(\mu ^{\widehat{\mathbb {C}}{\setminus }K}_{U{\setminus }K;a\rightarrow b})= \mu ^{{\mathbb {H}}}_{\widehat{U}_K;\cot _2(\lambda -q)\rightarrow \infty },\quad \\&\psi \circ g_L(\mu ^{\widehat{\mathbb {C}}{\setminus }L}_{V{\setminus }L;W(a)\rightarrow W(b)}) = \mu ^{{\mathbb {H}}}_{\widehat{V}_L;\cot _2(\sigma -p)\rightarrow \infty }. \end{aligned}$$

Combining the above displayed formulas and the fact that \(\widehat{W}_K=\psi \circ g_L\circ W\circ g_K^{-1}\circ \phi ^{-1}\), we see that it suffices to prove that

$$\begin{aligned} \Big |\frac{\sin _2(\sigma -p)}{\sin _2(\lambda -q)}\Big |^{-2}\cdot |W_K'(e^{i\sigma })W_K'(e^{iq}) | = |\widehat{W}_K'(\cot _2(\lambda -q))\widehat{W}_K'(\infty )| . \end{aligned}$$

To see this, one may check that \(|\phi '(e^{i\lambda })|=|\sin _2(\lambda -q)|^{-2}/2\), \(|\psi '(e^{i\sigma })|=|\sin _2(\sigma -p)|^{-2}/2\); and with \(J(z){:}{=}-1/z\), \(|(J\circ \phi )'(e^{iq})|=|(J\circ \psi )'(e^{ip})|=1/2\). \(\square \)

Lemma A.6

Let K and L be two \({\mathbb {H}}\)-hulls. Let U and V be open neighborhoods of \( {\mathbb {R}}\cup \{\infty \}\) in \({\mathbb {H}}\) such that \(K\subset U\) and \(L\subset V\). Suppose \(W:(U;{\mathbb {R}},\infty ,K){\mathop {\twoheadrightarrow }\limits ^{{\mathrm{Conf}}}}(V;{\mathbb {R}},\infty ,L)\). Let a and b be distinct prime ends of \({\mathbb {H}}{\setminus }K\) that lie on \(\partial K\). Then W(a) and W(b) are distinct prime ends of \({\mathbb {H}}{\setminus }L\) that lie on \(\partial L\). Let \(g_K:{\mathbb {H}}{\setminus }K{\mathop {\twoheadrightarrow }\limits ^{{\mathrm{Conf}}}}{\mathbb {H}}\) and \(g_L:{\mathbb {H}}{\setminus }L{\mathop {\twoheadrightarrow }\limits ^{{\mathrm{Conf}}}}{\mathbb {H}}\). Suppose \(g_K(a)={\lambda }\), \(g_K(b)={q}\), \(g_L(W(a))={\sigma }\), and \(g_L(W(b))={p}\) for some \(\lambda ,q,\sigma ,p\in {\mathbb {R}}\). Let \(W_K=g_L\circ W\circ g_K^{-1}\). Extend \(W_K\) conformally across \({\mathbb {R}}\). Then we have

$$\begin{aligned} \mu ^{{\mathbb {H}}{\setminus }L}_{V{\setminus }L;W(a)\rightarrow W(b)}= \Big |\frac{ \sigma -p}{ \lambda -q }\Big |^{\frac{6}{\kappa }-1}\cdot |W_K'({\sigma })W_K'({q}) |^{-\frac{6-\kappa }{2\kappa }} \cdot W(\mu ^{{\mathbb {H}}{\setminus }K}_{U{\setminus }K;a\rightarrow b}). \end{aligned}$$

Proof

The proof is similar to that of Lemma A.5 except that here we use the functions \(\phi (z)=-\frac{z+q}{z-q}\) and \(\psi (z)=-\frac{z+p}{z-p}\), which map \({\mathbb {H}}\) conformally onto \({\mathbb {H}}\). \(\square \)

Image of an interior hull under a conformal map

Proposition B.1

Let \({{\mathcal {H}}}_0\) denote the set of all interior hulls that contain 0. For \(K\in {{\mathcal {H}}}_0\), let \(r_K\) denote \(e^{{{\,\mathrm{cap}\,}}(K)}\). Suppose W is a conformal map defined on a neighborhood U of 0, and \(W(0)=0\). If \(K\in {{\mathcal {H}}}_0\) is such that \(r_K\) is small enough, then \(W(K)\in {{\mathcal {H}}}_0\). Moreover, we have

$$\begin{aligned} r_{W(K)}=|W'(0)|r_K(1+O(r_K^{1/2})),\quad \text{ as } r_K\rightarrow 0, \end{aligned}$$
(B.1)

where the implicit constants depend only on W and U.

Proof

Since \(r_{aK}=|a| r_K\), we may assume that \(W'(0)=1\). Then there exist \(r,C\in (0,\infty )\) such that \(\{|z|\le r\}\subset U\), and for any \(|z|\le r\), \(|W(z)-z|\le C|z|^2\). We may assume that \(r\le 1\) and \(rC\le \frac{1}{2}\). Then \(|z|/2\le |W(z)|\le 2|z|\) when \(|z|\le r\). Recall that by Koebe 1/4 theorem, for \(K\in {{\mathcal {H}}}_0\), \(r_K\le \max \{|z|:z\in K\}\le 4 r_K\). Thus, if \(K\in {{\mathcal {H}}}_0\) and \(r_K\le r/4\), then \(K\subset \{|z|\le r\}\), and so \(W(K)\in {{\mathcal {H}}}_0\), and \(r_K/8\le r_{W(K)}\le 8r_K\).

Let \(K\in {{\mathcal {H}}}_0\). Applying Koebe distortion theorem to \(r_K*J\circ g_K^{-1}\circ J\), where \(J(z){:}{=}1/z\), we find that for any \(z\in \mathbb {C}\) with \(|z|>1\),

$$\begin{aligned} | {r_K}^{-1/2}{|g_K^{-1}(z)|}^{1/2}- {|z|}^{1/2}|\le |z|^{-1/2}. \end{aligned}$$
(B.2)

Suppose that \(r_K\le r^2/16\) and \(1<|z|\le r_K^{-1/2}\). We get by (B.2)

$$\begin{aligned} {|g_K^{-1}(z)|} \le r_K (|z|^{-1/2}+|z|^{1/2})^2\le 4 r_K^{1/2}\le r. \end{aligned}$$
(B.3)

So \(g_K^{-1}(\{1<|z|\le r_K^{-1/2}\})\subset \{|z|\le r\}\subset U\). Let \(W_K= g_{W(K)}\circ W\circ g_K^{-1}\). Then \(W_K\) is well defined on \(\{1<|z|\le r_K^{-1/2}\}\), and maps the annulus \(A{:}{=}\{1<|z|< r_K^{-1/2}\}\) conformally onto the doubly connected domain D bounded by \(\{|z|=1\}\) and \(J{:}{=}W_K(\{|z|=r_K^{-1/2}\})\), where J is a Jordan curve surrounding \(\{|z|=1\}\). Since D has the same modulus as A, J must intersect the circle \(\{|z|=r_K^{-1/2}\})\). So there is \(z_0\) with \(|z_0|=r_K^{-1/2}\) such that \(|W_K(z_0)|=r_K^{-1/2}\). Let \(z_0'=W_K(z_0)\), \(z_K=g_K^{-1}(z_0)\), and \(z_W'=W(z_K)=g_{W(K)}^{-1}(z_0')\). Applying (B.2) to \((K,z_0)\) and \((W(K),z_0')\), respectively, we get

$$\begin{aligned} |r_K^{-1/2} |z_K|^{1/2}-r_K^{-1/4}|\le r_K^{1/4},\quad |r_{W(K)}^{-1/2} |z_W'|^{1/2}-r_K^{-1/4}|\le r_K^{1/4}. \end{aligned}$$
(B.4)

The latter inequality implies that

$$\begin{aligned} r_{W(K)}^{-1/2} |z_W'|^{1/2}\ge r_K^{-1/4}-r_K^{1/4}\ge r_K^{-1/4}/2. \end{aligned}$$
(B.5)

Since \(z_W'=W(z_K)\) and by (B.3), \(|z_K|\le 4 r_K^{1/2}\le r\), we get \(|z_W'-z_K|\le C|z_K|^2\), which implies that \(||z_W'|^{1/2}-|z_K|^{1/2}|\le C|z_K|^{3/2}\le 8C r_K^{3/4}\). Thus, \(|r_K^{-1/2} |z_K|^{1/2}-r_{K}^{-1/2} |z_W'|^{1/2}|\le 8C r_K^{1/4}\). Combining this with (B.4), we get \(|r_{W(K)}^{-1/2} |z_W'|^{1/2}-r_K^{-1/2} |z_W'|^{1/2}|\le (2+8C) r_K^{1/4}\), which combined with (B.5) implies that

$$\begin{aligned} |(r_{W(K)} /r_K)^{1/2}-1|\le r_{W(K)} ^{1/2}|z_W'|^{-1/2} (2+8C) r_K^{1/4}\le 2(2+8C) r_K^{1/2}, \end{aligned}$$

when \(r_K\le r^2/16\). So we get the estimate (B.1). \(\square \)

Remark B.2

This proposition resembles [21, Lemma 2.8], which concerns the growth rates of expanding \({\mathbb {H}}\)-hulls. The estimate (B.1) is not sharp, but is sufficient for the application in (3.7). We at least have the following estimate: \(r_{W(K)}=|W'(0)|r_K(1+O(r_K |\log (r_K)|))\). Here is a sketchy proof. By approximation we may assume that there is a family of whole-plane Loewner hulls \(K_t\), \(-\infty <t\le a=\log (r_K)\), driven by \(\lambda \), such that \(K=K_a\). Recall that \({{\,\mathrm{cap}\,}}(K_t)=t\). Then \(W(K_t)\), \(-\infty <t\le a\), is a time-change of a family of whole-plane Loewner hulls. Using a radial counterpart of [21, Lemma 2.8], we know that \(\frac{d}{dt} {{\,\mathrm{cap}\,}}(W(K_t))=|W_t'(e^{i\lambda _t})|^2\), where \(W_t{:}{=}g_{W(K_t)}\circ W\circ g_{K_t}^{-1}\). By [41, Lemma 4.4], \(W_t'(e^{i\lambda _t})=1+O(|t|e^t)\). Thus,

$$\begin{aligned} ({{\,\mathrm{cap}\,}}(W(K))-{{\,\mathrm{cap}\,}}(W(K_s)))-(a-s)=\int _s^a O(|t|e^t)dt=O(|a| e^a). \end{aligned}$$

By Proposition B.1, \({{\,\mathrm{cap}\,}}(W(K_s))=\log |W'(0)|+s+O(e^{s/2})\). Sending \(s\rightarrow -\infty \) in the displayed formula, we get \({{\,\mathrm{cap}\,}}(W(K))=a+O(|a|e^a)\), which implies the improved estimate.

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Zhan, D. SLE loop measures. Probab. Theory Relat. Fields 179, 345–406 (2021). https://doi.org/10.1007/s00440-020-01011-7

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  • 30C