1 Introduction

The Schramm–Loewner evolution \({{\mathrm{SLE}}}_{\kappa }\), introduced by Oded Schramm, generates random curves in plane domains which are the scaling limits of a number of critical two dimensional lattice models. Many work have been done to prove the convergence of various discrete models to SLE with different parameters \(\kappa \). It is also interesting to study the geometric properties of the SLE curves.

The current paper focuses on studying the tips of two versions of SLE: chordal SLE and radial SLE at some fixed capacity time. There were previous work on the tips of SLE, e.g., [3], in which the multifractal spectrum of the SLE tip is studied. This paper studies the ergodic property of the SLE near its tip. Now we explain it.

Consider a chordal or radial \(\hbox {SLE}_\kappa (\kappa \in (0,4))\) curve \(\beta \), which is parameterized by the half-plane or disc capacity. Let \(h_{t}\) denote the harmonic measure of the left side of \(\beta [t,1]\) in \(\widehat{\mathbb {C}}{\setminus }\beta [t,1]\) as seen form \(\infty \) (ignoring the real line and the rest of the curve). Let \(v(t)\) be the (logarithm) capacity of \(\beta ([t,1])\). Then as \(\tau \rightarrow -\infty , h_{v^{-1}(\tau )}\rightarrow h\) in distribution, where the law of \(h\) is given explicitly. Moreover, for nicely-behaved functions \(f\) on \([0,1]\), the averages of \(f(h_{v^{-1}(\tau )})\) over \(\tau \) converge to \(\mathbb {E}[f(h)]\).

We will use results about backward SLE derived in [13]. The traditional chordal or radial \(\hbox {SLE}_\kappa \) is defined by solving a chordal or radial Loewner equation driven by \(\sqrt{\kappa }B(t)\). Adding a minus sign to the (forward) Loewner equations, we get the backward Loewner equations. The backward chordal or radial \(\hbox {SLE}_\kappa \) is then defined by solving a backward chordal or radial Loewner equation driven by \(\sqrt{\kappa }B(t)\).

The backward radial \(\hbox {SLE}(\kappa ;\rho )\) processes resemble the forward radial \(\hbox {SLE}(\kappa ;\rho )\) processes, and play an important role in this paper. If \(\kappa \in (0,4]\) and \(\rho \le -\frac{\kappa }{2}-2\), a backward radial \(\hbox {SLE}(\kappa ;\rho )\) process induces a random welding \(\phi \) which is an involution (an auto homeomorphism whose inverse is itself) of the unit disc with exactly two fixed points such that for \(w\ne z, w=\phi (z)\) iff \(f_{t}(z)=f_{t}(w)\) when \(t\) is big enough, where \((f_{t})\) are the solutions of the backward Loewner equation. It is proven in [13] that, for \(\kappa \in (0,4]\), there is a coupling of two different backward radial \(\hbox {SLE}(\kappa ;-\kappa -6)\) processes which induce the same welding.

In Sect. 4 of this paper, we use a limit procedure to define a normalized backward radial \(\hbox {SLE}(\kappa ;\rho )\) trace, and prove that, up to a reflection about the unit circle, it agrees with the forward whole-plane \(\hbox {SLE}(\kappa ;-4-\rho )\) curve (Theorem 4.6). Using the symmetry of backward radial \(\hbox {SLE}(\kappa ;-\kappa -6)\) welding together with the conformal removability of \(\hbox {SLE}_\kappa \) curves, we prove in Sect. 5 that, for \(\kappa \in (0,4)\), a whole-plane \(\hbox {SLE}(\kappa ;\kappa +2)\) curve stopped at the time \(0\) satisfies reversibility (Theorem 5.1). One should keep in mind that a whole-plane \(\hbox {SLE}(\kappa ;\rho )\) trace grows from \(0\) with time interval \([-\infty ,\infty )\), and the time \(0\) is when the curve reaches the capacity of the closed unit disc.

This reversibility is different from the reversibility of whole-plane \(\hbox {SLE}_\kappa (\kappa \le 4)\) derived in [18], or more generally, the reversibility of whole-plane \(\hbox {SLE}_\kappa (\rho ) (\kappa \le 8, \rho >-2\) and \(\rho \ge \frac{\kappa }{2}-4)\) derived in [8], where the trace does not stop in the middle, but goes all the way to \(\infty \). The methods in [8, 18] used couplings of two SLE processes and couplings of an SLE process with a Gaussian free field, respectively, which can not be used to derive the reversibility here. In fact, the reversibility here does not hold if \(\kappa +2\) is replaced by any other number.

This reversibility of the stopped whole-plane \(\hbox {SLE}(\kappa ;\kappa +2)\) is then used to prove that, for \(\kappa \in (0,4)\), a forward chordal \(\hbox {SLE}_\kappa \) curve stopped at a fixed capacity time can be mapped conformally to an initial segment of a whole-plane \(\hbox {SLE}(\kappa ;\kappa +2)\) curve, and the same is true up to a change of the probability measure for a forward radial \(\hbox {SLE}_\kappa \) (Theorems 5.3 and 5.4). In Sect. 6, we use the above conformal relations to derive ergodic properties of a chordal or radial \(\hbox {SLE}_\kappa \) curves at a fixed capacity time (Theorem 6.6).

Throughout this paper, we use the following symbols and notation. Let \(\widehat{\mathbb {C}}={\mathbb {C}}\cup \{\infty \}, \mathbb {D}=\{z\in {{\mathbb {C}}}:|z|<1\}, \mathbb {D}^*=\widehat{{\mathbb {C}}}{\setminus }\overline{\mathbb {D}}, \mathbb {T}=\{z\in {{\mathbb {C}}}:|z|=1\}\), and \(\mathbb {H}=\{z\in {\mathbb {C}}:\hbox {Im}\,z>0\}\). Let \(\cot _{2}(z)=\cot (z/2)\) and \(\sin _{2}(z)=\sin (z/2)\). Let \(I_\mathbb {T}(z)=1/\overline{z}\) be the reflections about \(\mathbb {T}\). By an interval on \(\mathbb {T}\), we mean a connected subset of \(\mathbb {T}\). We use \(B(t)\) to denote a standard real Brownian motion. We use \(C(J)\) to denote the space of real valued continuous functions on \(J\). By \(f:D\mathop {\twoheadrightarrow }\limits ^\mathrm{Conf}E\) we mean that \(f\) maps \(D\) conformally onto \(E\). By \(f_n\mathop {\longrightarrow }\limits ^\mathrm{l.u.}f\) in \(U\) we mean that \(f_n\) converges to \(f\) locally uniformly in \(U\).

2 Loewner equations

2.1 Forward equations

We review the definitions and basic facts about (forward) Loewner equations. The reader is referred to [4] for details.

A set \(K\) is called an \(\mathbb {H}\)-hull if it is a bounded relatively closed subset of \(\mathbb {H}\), and \(\mathbb {H}{\setminus }K\) is simply connected. For every \(\mathbb {H}\)-hull \(K\), there is a unique \(g_K:\mathbb {H}{\setminus }K\mathop {\twoheadrightarrow }\limits ^\mathrm{Conf}\mathbb {H}\) such that \(g_K(z)-z\rightarrow 0\) as \(z\rightarrow \infty \). The number \({{\mathrm{hcap}}}(K):=\lim _{z\rightarrow \infty } z(g_K(z)-z)\) is always nonnegative, and is called the half plane capacity of \(K\). A set \(K\) is called a \(\mathbb {D}\)-hull if it is a relatively closed subset of \(\mathbb {D}\), does not contain \(0\), and \(\mathbb {D}{\setminus }K\) is simply connected. For every \(\mathbb {D}\)-hull \(K\), there is a unique \(g_K:\mathbb {D}{\setminus }K\mathop {\twoheadrightarrow }\limits ^\mathrm{Conf}\mathbb {D}\) such that \(g_K(0)=0\) and \(g_K'(0)>0\). The number \({{\mathrm{dcap}}}(K):=\log (g_K'(0))\) is always nonnegative, and is called the disc capacity of \(K\). A set \(K\) is called a \({\mathbb {C}}\)-hull if it is a connected compact subset of \({\mathbb {C}}\) such that \({\mathbb {C}}{\setminus }K\) is connected. For every \({\mathbb {C}}\)-hull with more than one point, \(\widehat{\mathbb {C}}{\setminus }K\) is simply connected, and there is a unique \(g_K:\widehat{\mathbb {C}}{\setminus }K\mathop {\twoheadrightarrow }\limits ^\mathrm{Conf}\mathbb {D}^*\) such that \(g_K(\infty )=\infty \) and \(g_K'(\infty ):=\lim _{z\rightarrow \infty } z/g_K(z)>0\). The real number \({{\mathrm{cap}}}(K):=\log (g_K'(\infty ))\) is called the whole-plane capacity of \(K\). In either of the three cases, let \(f_K=g_K^{-1}\).

Let \(\lambda \in C([0,T))\), where \(T\in (0,\infty ]\). The chordal Loewner equation driven by \(\lambda \) is

$$\begin{aligned} \partial _{t} g_{t}(z)=\frac{2}{g_{t}(z)-\lambda (t)},\quad 0\le t<T;\quad \quad g_{0}(z)=z. \end{aligned}$$

The radial Loewner equation driven by \(\lambda \) is

$$\begin{aligned} \partial _{t} g_{t}(z)=g_{t}(z)\frac{e^{i\lambda (t)}+g_{t}(z)}{e^{i\lambda (t)}-g_{t}(z)},\quad 0\le t<T;\quad g_{0}(z)=z. \end{aligned}$$

Let \(g_{t}, 0\le t<T\), be the solutions of the chordal (resp. radial) Loewner equation. For each \(t\in [0,T)\), let \(K_{t}\) be the set of \(z\in \mathbb {H}\,(\hbox {resp.}\ \in \mathbb {D})\) at which \(g_{t}\) is not defined. Then for each \(t, K_{t}\) is an \(\mathbb {H}\,(\hbox {resp.}\ \mathbb {D})\)-hull with \({{\mathrm{hcap}}}(K_{t})=2t \,(\hbox {resp.}\ {{\mathrm{dcap}}}(K_{t})=t)\) and \(g_{K_{t}}=g_{t}\). We call \(g_{t}\) and \(K_{t}, 0\le t<T\), the chordal (resp. radial) Loewner maps and hulls driven by \(\lambda \). We say that the process generates a chordal (resp. radial) trace \(\beta \) if each \(g_{t}^{-1}\) extends continuously to \(\overline{\mathbb {H}}\,(\hbox {resp.}\ \overline{\mathbb {D}})\), and \(\beta (t):= g_{t}^{-1}(\lambda (t)) \,(\hbox {resp.}\ :=g_{t}^{-1}(e^{i\lambda (t)}))\), \(0\le t<T\), is a continuous curve in \(\overline{\mathbb {H}}\,(\hbox {resp.}\ \overline{\mathbb {D}})\). If the chordal (resp. radial) trace \(\beta \) exists, then for each \(t, K_{t}\) is the \(\mathbb {H}\)-hull generated by \(\beta ([0,t])\), i.e., \(\mathbb {H}{\setminus }K_{t} \,(\hbox {resp.}\ \mathbb {D}{\setminus }K_{t})\) is the component of \(\mathbb {H}{\setminus }\beta ([0,t]) \,(\hbox {resp.}\ \mathbb {D}{\setminus }\beta ([0,t]))\) which is unbounded (resp. contains \(0\)). Note that \(\beta (0)=\lambda (0)\in \mathbb {R}\,(\hbox {resp.}\ =e^{i\lambda (0)}\in \mathbb {T})\). The trace \(\beta \) is called \(\mathbb {H}\)-simple (resp. \(\mathbb {D}\)-simple) if it has no self-intersections and intersects \(\mathbb {R}\,(\hbox {resp.}\ \mathbb {T})\) only at its one end point, in which case we have \(K_{t}=\beta ((0,t])\) for \(0\le t<T\). Since \({{\mathrm{hcap}}}(K_{t})=2t \,(\hbox {resp.}\ {{\mathrm{dcap}}}(K_{t})=t)\) for all \(t\), we say that the chordal (resp. radial) trace is parameterized by the half-plane (resp. disc) capacity.

A simple property of the chordal (resp. radial) Loewner process is the translation (resp. rotation) symmetry. Let \(C\in \mathbb {R}\) and \(\lambda ^*=\lambda +C\). Let \(g^*_{t}\) and \(K^*_{t}\) be the chordal (resp. radial) Loewner maps and hulls driven by \(\lambda ^*\). Then \(K^*_{t}=C+K_{t}\) and \(g^*_{t}(z)=C+g_{t}(z-C) \,(\hbox {resp.}\ K^*_{t}=e^{iC} K_{t}\) and \(g^*_{t}(z)=e^{iC}g_{t}(z/e^{iC}))\). If \(\lambda \) generates a chordal (resp. radial) trace \(\beta \), then \(\lambda ^*\) also generates a chordal (resp. radial) trace \(\beta ^*\) such that \(\beta ^*=C+\beta \,(\hbox {resp.}\ =e^{iC}\beta )\).

Let \(\kappa >0\). The chordal (resp. radial) \(\hbox {SLE}_\kappa \) is defined by solving the chordal (resp. radial) Loewner equation with \(\lambda (t)=\sqrt{\kappa }B(t)\), and the process a.s. generates a chordal (resp. radial) trace, which is \(\mathbb {H}\)(resp. \(\mathbb {D}\))-simple if \(\kappa \in (0,4]\).

Let \(T\in \mathbb {R}\) and \(\lambda \in C((-\infty ,T])\). The whole-plane Loewner equation driven by \(\lambda \) is

$$\begin{aligned} \left\{ \begin{array}{ll} \partial _{t} g_{t}(z)=g_{t}(z)\frac{e^{i\lambda (t)}+g_{t}(z)}{e^{i\lambda (t)}-g_{t}(z)},&{}\quad t\le T;\\ \lim _{t\rightarrow -\infty } e^t g_{t}(z)=z,&{}\quad z\ne 0. \end{array}\right. \end{aligned}$$

It turns out that the family \((g_{t})\) always exists, and is uniquely determined by \((e^{i\lambda (t)})\). Moreover, there is an increasing family of \({\mathbb {C}}\)-hulls \((K_{t})_{-\infty <t\le T}\) in \({\mathbb {C}}\) with \(\bigcap _{t} K_{t}=\{0\}\) such that \({{\mathrm{cap}}}(K_{t})=t\) and \(g_{K_{t}}=g_{t}\). We call \(g_{t}\) and \(K_{t}, -\infty <t<T\), the whole-plane Loewner maps and hulls driven by \(\lambda \). We say that the process generates a whole-plane trace \(\beta \) if each \(g_{t}^{-1}\) extends continuously to \(\overline{\mathbb {D}^*}\), and \(\beta (t):= g_{t}^{-1}(e^{i\lambda (t)}), -\infty < t<T\), is a continuous curve in \({\mathbb {C}}\). If the whole-plane trace \(\beta \) exists, then it extends continuously to \([-\infty ,T]\) with \(\beta (-\infty )=0\), and for every \(t, {\mathbb {C}}{\setminus }K_{t}\) is the unbounded component of \({\mathbb {C}}{\setminus }\beta ([-\infty ,t])\). If \(\beta \) is a simple curve, then \(K_{t}=\beta ([-\infty ,t])\) for every \(t\). So we say that the whole-plane trace is parameterized by the whole-plane capacity.

2.2 Backward equations

Now we review the definitions and basic facts about backward Loewner equations. The reader is referred to [13] for details.

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

$$\begin{aligned} \partial _{t} f_{t}(z)=-\frac{2}{f_{t}(z)-\lambda (t)},\quad 0\le t<T; \quad f_{0}(z)=z. \end{aligned}$$

The backward radial Loewner process driven by \(\lambda \) is

$$\begin{aligned} \partial _{t} f_{t}(z)=-f_{t}(z)\frac{e^{i\lambda (t)}+f_{t}(z)}{e^{i\lambda (t)}-f_{t}(z)},\quad 0\le t<T;\quad f_{0}(z)=z. \end{aligned}$$

Let \(f_{t}, 0\le t<T\), be the solutions of the backward chordal (resp. radial) Loewner equation. Let \(L_{t}=\mathbb {H}{\setminus }f_{t}(\mathbb {H}) \,(\hbox {resp.}\ \mathbb {D}{\setminus }f_{t}(\mathbb {D})), 0\le t<T\). Then every \(L_{t}\) is an \(\mathbb {H}\,(\hbox {resp.}\ \mathbb {D})\)-hull with \({{\mathrm{hcap}}}(L_{t})=2t \,(\hbox {resp.}\ {{\mathrm{dcap}}}(L_{t})=t)\) and \(f_{L_{t}}=f_{t}\). We call \(f_{t}\) and \(L_{t}, 0\le t<T\), the backward chordal (resp. radial) Loewner maps and hulls driven by \(\lambda \).

Define a family of maps \(f_{t_{2},t_{1}}, t_{1},t_{2}\in [0,T)\), such that, for any fixed \(t_{1}\in [0,T)\) and \(z\in \widehat{\mathbb {C}}{\setminus }\{\lambda (t_{1})\}\), the function \(t_{2}\mapsto f_{t_{2},t_{1}}(z)\) is the solution of the first (resp. second) equation below (with the maximal definition interval):

$$\begin{aligned} \partial _{t_{2}} f_{t_{2},t_{1}}(z)=-\frac{2}{f_{t_{2},t_{1}}(z)-\lambda (t_{2})},\quad f_{t_{1},t_{1}}(z)=z; \end{aligned}$$
$$\begin{aligned} \partial _{t_{2}} f_{t_{2},t_{1}}(z)=-f_{t_{2},t_{1}}(z)\frac{e^{i\lambda (t_{2})}+f_{t_{2},t_{1}}(z)}{e^{i\lambda (t_{2})}-f_{t_{2},t_{1}}(z)},\quad f_{t_{1},t_{1}}(z)=z. \end{aligned}$$
(2.1)

We call \((f_{t_{2},t_{1}})\) the backward chordal (resp. radial) Loewner flow driven by \(\lambda \). Note that we allow that \(t_{2}\) to be smaller than \(t_{1}\) if \(t_{1}>0\). If \(t_{2}\ge t_{1}, f_{t_{2},t_{1}}\) is defined on the whole \(\mathbb {H}\,(\hbox {resp.}\ \mathbb {D})\); and this is not the case if \(t_{2}<t_{1}\). The following lemma is obvious.

Lemma 2.1

  1. (i)

    For any \(t_{1},t_{2},t_3\in [0,T), f_{t_3,t_{2}}\circ f_{t_{2},t_{1}}\) is a restriction of \(f_{t_3,t_{1}}\). In particular, this implies that \(f_{t_{1},t_{2}}=f_{t_{2},t_{1}}^{-1}\).

  2. (ii)

    For any fixed \(t_{0}\in [0,T), f_{t_{0}+t,t_{0}}, 0\le t<T-t_{0}\), are the backward chordal (resp. radial) Loewner maps driven by \(\lambda (t_{0}+t), 0\le t<T-t_{0}\). Especially, \(f_{t,0}=f_{t}, 0\le t<T\).

  3. (iii)

    For any fixed \(t_{0}\in [0,T), f_{t_{0}-t,t_{0}}, 0\le t\le t_{0}\), are the forward chordal (resp. radial) Loewner maps driven by \(\lambda (t_{0}-t), 0\le t\le t_{0}\).

We say that a backward chordal (resp. radial) Loewner process driven by \(\lambda \in C([0,T))\) generates a family of backward chordal (resp. radial) traces \(\beta _{t}, 0\le t\le T\), if for each fixed \(t_{0}\in (0,T)\), the forward chordal (resp. radial) Loewner process driven by \(\lambda (t_{0}-t), 0\le t\le t_{0}\), generates a chordal (resp. radial) trace, which is \(\beta _{t_{0}}(t_{0}-t), 0\le t\le t_{0}\). Equivalently, this means that, for each \(t_{0}, \beta _{t_{0}}:[0,t_{0}]\rightarrow \overline{\mathbb {H}} \,(\hbox {resp.}\ \overline{\mathbb {D}})\) is continuous, and or any \(t_{2}\ge t_{1}\ge 0, f_{t_{2},t_{1}}\) extends continuously to \(\overline{\mathbb {H}}\,(\hbox {resp.}\ \overline{\mathbb {D}})\) such that \(\beta _{t_{2}}(t_{1})=f_{t_{2},t_{1}}({\lambda (t_{1})}) \,(\hbox {resp.}\ f_{t_{2},t_{1}}(e^{i\lambda (t_{1})}))\). Taking \(t_{2}=t_{1}=t\), we get \(\beta _{t}(t)=\lambda (t)\in \mathbb {R}\) (resp. \(=e^{i\lambda (t)}\in \mathbb {T})\). Moreover, the equality \(f_{t_{2},t_{1}}\circ f_{t_{1},t_{0}}=f_{t_{2},t_{0}}, t_{2}\ge t_{1}\ge t_{0}\ge 0\), holds after the continuation, and so we have

$$\begin{aligned} f_{t_{2},t_{1}}(\beta _{t_{1}}(t))=\beta _{t_{2}}(t),\quad t_{2}\ge t_{1}\ge t\ge 0. \end{aligned}$$
(2.2)

The backward chordal (resp. radial) \(\hbox {SLE}_\kappa \) is defined to be the backward chordal (resp. radial) Loewner process driven by \(\sqrt{\kappa }B(t), 0\le t<\infty \). The existence of the forward chordal (resp. radial \(\hbox {SLE}_\kappa )\) trace together with Lemma 2.1 and the translation (resp. rotation) symmetry implies that the backward chordal (resp. radial) \(\hbox {SLE}_\kappa \) process generates a family of backward chordal (resp. radial) traces, which are \(\mathbb {H}\,(\hbox {resp.}\ \mathbb {D})\)-simple, if \(\kappa \le 4\).

Remark

One should keep in mind that each \(\beta _{t}\) is a continuous function defined on \([0,t], \beta _{t}(0)\) is the tip of \(\beta _{t}\), and \(\beta _{t}(t)\) is the root of \(\beta _{t}\), which lies on \(\mathbb {R}\). The parametrization is different from a forward chordal Loewner trace.

For every \(\mathbb {H}\,(\hbox {resp.}\ \mathbb {D})\)-hull \(L, g_L\) extends analytically to \(\mathbb {R}{\setminus }\overline{L} \,(\hbox {resp.}\ \mathbb {T}\setminus \overline{L})\), and maps \(\mathbb {R}\setminus \overline{L} \,(\hbox {resp.}\ \mathbb {T}\setminus \overline{L})\) to an open subset of \(\mathbb {R}\,(\hbox {resp.}\ \mathbb {T})\). The set \(S_L:=\mathbb {R}\setminus g_L(\mathbb {R}\setminus \overline{L}) \,(\hbox {resp.}\ :=\mathbb {T}\setminus g_L(\mathbb {T}\setminus \overline{L}))\) is a compact subset of \(\mathbb {R}\,(\hbox {resp.}\ \mathbb {T})\), and is called the support of \(L\). The map \(f_L\) then extends analytically to \(\mathbb {R}\setminus S_L \,(\hbox {resp.}\ \mathbb {T}\setminus S_L)\). If \((L_{t})_{0\le t<T}\) are \(\mathbb {H}\,(\hbox {resp.}\ \mathbb {D})\)-hulls generated by a backward chordal (resp. radial) Loewner process, then each \(S_{L_{t}}\) is an interval on \(\mathbb {R}\,(\hbox {resp.}\ \mathbb {T})\), and \(S_{L_{t_{1}}}\subset S_{L_{t_{2}}}\) if \(t_{1}<t_{2}\) (c.f. Lemmas 2.7 and 3.3 in [13]). The following is Lemma 3.5 in [13].

Lemma 2.2

Let \(L_{t}, 0\le t<\infty \), be \(\mathbb {D}\)-hulls generated by a backward radial Loewner process. Then \(\bigcup _{t} S_{L_{t}}\) is equal to either \(\mathbb {T}\) or \(\mathbb {T}\) without a single point.

Now we review the welding induced by a backward Loewner process. See Section 3.5 of [13] for details.

Suppose \(L=\beta \) is an \(\mathbb {H}\,(\hbox {resp.}\ \mathbb {D})\)-simple curve. Then \(S_\beta \) is the union of two intervals on \(\mathbb {R}\,(\hbox {resp.}\ \mathbb {T})\), which intersects at one point, and \(f_\beta \) extends continuously to \(S_\beta \), and maps the two intervals onto the two sides of \(\beta \). Every point on \(\beta \) except the tip point has two preimages. The welding \(\phi _\beta \) induced by \(\beta \) is the involution of \(S_\beta \) with exactly one fixed point which is the \(f_\beta \)-pre-image of the tip of \(\beta \), such that for \(x\ne y\in S_\beta , y=\phi _\beta (x)\) if and only if \(f_{\beta }(x)=f_{\beta }(y)\).

Suppose a backward chordal (resp. radial) Loewner process generates a family of \(\mathbb {H}\,(\hbox {resp.}\ \mathbb {D})\)-simple traces \((\beta _{t})_{0\le t<T}\). Then for any \(t_{1}<t_{2}, S_{\beta _{t_{1}}}\) is contained in the interior of \(S_{\beta _{t_{2}}}\), and \(\phi _{\beta _{t_{1}}}\) is a restriction of \(\phi _{\beta _{t_{2}}}\). The latter can be seen from \(f_{t_{2},t_{1}}\circ f_{t_{1}}=f_{t_{2}}\). So the process naturally induces a welding \(\phi \) which is an involution of the open interval \(\bigcup _{0\le t<T} S_{\beta _{t}}\) on \(\mathbb {R}\,(\hbox {resp.}\ \mathbb {T})\) such that \(\phi |_{S_{\beta _{t}}}=\phi _{\beta _{t}}\) for each \(t\). The welding has only one fixed point: \(\lambda (0)\in \mathbb {R}\,(\hbox {resp.}\ e^{i\lambda (0)}\in \mathbb {T})\). Consider the radial case and suppose \(T=\infty \). Lemma 2.2 and the properties of \(S_{\beta _{t}}\)’s imply that \(\mathbb {T}\setminus \bigcup _{0\le t<\infty } S_{\beta _{t}}\) contains exactly one point, say \(w_{0}\). We call \(w_{0}\) the joint point of the process, which is the only point such that \(f_{t}(w_{0})\in \mathbb {T}\) for all \(t\ge 0\). In this case we extend \(\phi \) to an involution of \(\mathbb {T}\) with exactly two fixed points: \(e^{i\lambda (0)}\) and \(w_{0}\).

3 \(\hbox {SLE}(\kappa ;\rho )\) processes

In this section, we review the definitions of the forward and backward radial \(\hbox {SLE}(\kappa ;\rho )\) processes, respectively, as well as the whole-plane \(\hbox {SLE}(\kappa ;\rho )\) process.

Let \(\kappa >0\) and and \(\rho \in \mathbb {R}\). Let \(\sigma \in \{1,-1\}\). The case \(\sigma =1 \,(\hbox {resp.}\ =-1)\) corresponds to the forward (resp. backward) process. Let \(z\ne w\in \mathbb {T}\). Choose \(x,y\in \mathbb {R}\) such that \(e^{ix}=z, e^{iy}=w\), and \(0<x-y<2\pi \). Let \(\lambda (t)\) and \(q(t), 0\le t<T\), be the solution of the system of SDE:

$$\begin{aligned} \left\{ \begin{array}{ll} d\lambda (t)=\sqrt{\kappa }dB(t)+\sigma \frac{\rho }{2}\cot _{2}(\lambda (t)-q(t))dt,&{}\quad \lambda (0)=x;\\ dq(t)=\sigma \cot _{2}(q(t)-\lambda (t))dt,&{}\quad q(0)=y. \end{array}\right. \end{aligned}$$
(3.1)

If \(\sigma =1 \,(\hbox {resp.}\ =-1)\), the forward (resp. backward) radial Loewner process driven by \(\lambda \) is called a forward (resp. backward) \(\hbox {SLE}(\kappa ;\rho )\) process started from \((z;w)\). Recall that \(\cot _{2}(z)=\cot (z/2)\). The appearance of \(\cot _{2}\) comes from the covering forward and backward radial Loewner equations. Since \(\cot _{2}\) has period \(2\pi \), it is easy to see that the definition does not depend on the choice of \(x,y\).

Let \(Z_{t}=\lambda (t)-q(t)\). Then \((\frac{1}{2}Z_{\frac{4}{\kappa }t})\) is a radial Bessel process of dimension \(\delta :=\frac{4}{\kappa }\sigma (\frac{\rho }{2}+1)+1\) (see “Appendix B”). Thus, \(T=\infty \) if \(\delta \ge 2; T<\infty \) if \(\delta <2\).

Lemma 3.1

Let \(\kappa >0\) and \(\rho \le -\frac{\kappa }{2}-2\). Let \(L_{t}, 0\le t<\infty \), be \(\mathbb {D}\)-hulls generated by a backward radial \(\hbox {SLE}(\kappa ;\rho )\) process started from \((z;w)\). Then \(\bigcup _{t\ge 0} S_{L_{t}}=\mathbb {T}{\setminus }\{w\}\).

Proof

Since \(\sigma =-1\) for the backward equation, \(\rho \le -\frac{\kappa }{2}-2\) implies that \(\delta \ge 2\), and so \(T=\infty \). Let \(f_{t}, 0\le t<\infty \), be the conformal maps generated by the backward radial \(\hbox {SLE}(\kappa ;\rho )\) process. Formula (3.1) in the case \(\sigma =-1\) implies that \(e^{iq(t)}=f_{t}(w), 0\le t<\infty \). This means that \(w\not \in S_{L_{t}}, 0\le t<\infty \). The conclusion then follows from Lemma 2.2. \(\square \)

Assume that \(\delta \ge 2\), which means that \(\rho \ge \frac{\kappa }{2}-2\) if \(\sigma =1\) and \(\rho \le -\frac{\kappa }{2}-2\) if \(\sigma =-1\). From Corollary 8.2, \((Z_{t})\) has a unique stationary distribution \(\mu _\delta \) which has a density proportional to \(\sin _{2}(x)^{\delta -1}\), and the stationary process is reversible. Let \((\bar{Z}_{t})_{t\in \mathbb {R}}\) denote the stationary process. Let \(\bar{y}\) be a random variable with uniform distribution \(U_{[0,2\pi )}\) on \([0,2\pi )\) such that \(\bar{y}\) is independent of \((\bar{Z}_{t})\). Let \(\bar{q}(t)={\bar{y}}-\sigma \int _{0}^t \cot _{2}(\bar{Z}_s)ds\) and \(\bar{\lambda }(t)=\bar{q}(t)+\bar{Z}_{t}, t\in \mathbb {R}\). If \(\sigma =1 \,(\hbox {resp.}\ =-1)\), the forward (resp. backward) radial Loewner process driven by \(\bar{\lambda }(t), 0\le t<\infty \), is called a stationary forward (resp. backward) radial \(\hbox {SLE}(\kappa ;\rho )\) process. Equivalently, a stationary forward (resp. backward) radial \(\hbox {SLE}(\kappa ;\rho )\) process is a forward (resp. backward) radial \(\hbox {SLE}(\kappa ;\rho )\) process started from a random pair \((e^{i\bar{x}},e^{i\bar{y}})\) with \((\bar{x},\bar{x}-\bar{y})\sim U_{[0,2\pi )}\times \mu _\delta \). If \(\sigma =1\), the whole-plane Loewner process driven by \(\bar{\lambda }(t), t\in \mathbb {R}\), is called a whole-plane \(\hbox {SLE}(\kappa ;\rho )\) process.

It is easy to verify the following Markov-type relation between a whole-plane \(\hbox {SLE}(\kappa ;\rho )\) process and a forward radial \(\hbox {SLE}(\kappa ;\rho )\) process. Recall that \(I_\mathbb {T}(z)=1/\bar{z}\) is the reflection about \(\mathbb {T}\). Let \(g_{t}\) and \(K_{t}, t\in \mathbb {R}\), be maps and hulls generated by a whole-plane \(\hbox {SLE}(\kappa ;\rho )\) process. Let \(t_{0}\in \mathbb {R}\). Then \(I_\mathbb {T}\circ g_{t_{0}+t}\circ g_{t_{0}}^{-1}\circ I_\mathbb {T}\) and \(I_\mathbb {T}\circ g_{t_{0}}(K_{t_{0}+t}{\setminus } K_{t_{0}}), t\ge 0\), are maps and hulls generated by a stationary forward radial \(\hbox {SLE}(\kappa ;\rho )\) process.

Using the reversibility of the stationary radial Bessel processes of dimension \(\delta \ge 2\), we obtain the following lemma.

Lemma 3.2

Let \(\kappa >0\) and \(\rho \le -\frac{\kappa }{2}-2\). Let \(\lambda (t), t\ge 0\), be a driving function of a stationary backward radial \(\hbox {SLE}(\kappa ;\rho )\) process. Then for any \(t_{0}>0, \lambda (t_{0}-t), 0\le t\le t_{0}\), is a driving function up to time \(t_{0}\) of a stationary forward radial \(\hbox {SLE}(\kappa ;-4-\rho )\) process; and \(\lambda (-t), -\infty < t\le 0\), is a driving function up to time \(0\) of a whole-plane \(\hbox {SLE}(\kappa ;-4-\rho )\) process.

Girsanov’s theorem implies that many properties of forward or backward radial \(\hbox {SLE}_\kappa \) process carry over to radial \(\hbox {SLE}(\kappa ;\rho )\) processes. For example, a forward (resp. backward) radial \(\hbox {SLE}(\kappa ;\rho )\) process generates a forward radial trace (resp. a family of backward radial traces). If \(\kappa \le 4\) and \(\rho \le -\frac{\kappa }{2}-2\), then a backward radial \(\hbox {SLE}(\kappa ;\rho )\) process induces a welding, say \(\phi \), of \(\mathbb {T}\) with two fixed points. Suppose the process is started from \((z;w)\). From \(e^{i\lambda (0)}=e^{i(q(0)+Z_{0})}=e^{ix}=z\) we see that \(z\) is one fixed point of \(\phi \). Lemma 3.1 implies that \(w\) is the joint point of the process, and so is the other fixed point of \(\phi \).

Corollary 3.3

Let \(\kappa >0\) and \(\rho \le -\frac{\kappa }{2}-2\). Let \((\beta _{t})\) be a family of backward radial traces generated by a stationary backward radial \(\hbox {SLE}(\kappa ;\rho )\) process. Let \(\beta \) be a stationary forward radial \(\hbox {SLE}(\kappa ;-4-\rho )\) trace. Then for every fixed \(t_{0}\in (0,\infty ), \beta _{t_{0}}(t), 0\le t\le t_{0}\), has the same distribution as \(\beta (t_{0}-t), 0\le t\le t_{0}\).

Remark

One special value of \(\rho \) is \(-4\). Theorem 6.8 in [13] implies that, if \(\kappa \in (0,4]\), a stationary backward radial \(\hbox {SLE}(\kappa ;-4)\) process is a stationary backward radial \(\hbox {SLE}_\kappa \) process, i.e., the process driven by \(\lambda (t)=\bar{x}+\sqrt{\kappa } B(t)\), where \(\bar{x}\) is a random variable uniformly distributed on \([0,2\pi )\) and independent of \(B(t)\). So the above corollary provides a connection between a family of stationary backward radial \(\hbox {SLE}_\kappa \) traces and a stationary forward radial \(\hbox {SLE}_\kappa \) trace.

We are especially interested in the backward radial \(\hbox {SLE}(\kappa ;-\kappa -6)\) processes. The proposition below is Corollary 4.8 in [13].

Proposition 3.4

Let \(\kappa >0\) and \(z_{0}\ne z_\infty \in \mathbb {T}\). Let \(f_{t}\) and \(L_{t}, 0\le t<\infty \), be the backward radial \(\hbox {SLE}(\kappa ;-\kappa -6)\) maps and hulls started from \((z_{0},z_\infty )\). Let \(W\) be a Möbius transformation with \(W(\mathbb {D})=\mathbb {H}, W(z_{0})=0\), and \(W(z_\infty )=\infty \). Then there is a strictly increasing function \(v\) with \(v([0,\infty ))=[0,\infty )\) such that \(W^\mathcal{H}(L_{v(t)}), 0\le t<\infty \), are the \(\mathbb {H}\)-hulls driven by a backward chordal \(\hbox {SLE}_\kappa \) process.

That the range of \(v\) is \([0,\infty )\) is a part of the statement of Corollary 4.8 in [13]: up to a time change, \(W^\mathcal{H}(L_{t})\) is a (complete) backward chordal \(\hbox {SLE}_\kappa \) process. See the end of the proof of a similar proposition: Theorem 4.6 in [13].

The symbol \(W^\mathcal{H}(L)\) is defined in Section 2.3 of [13]. Theorem 2.20 in [13] ensures that for a \(\mathbb {D}\)-hull \(L\) and a Möbius transformation \(W\) from \(\mathbb {D}\) onto \(\mathbb {H}\) with \(W^{-1}(\infty )\not \in S_L\), there is a unique Möbius transformation \(W^L\) from \(\mathbb {D}\) onto \(\mathbb {H}\) such that \(W^L(L)\) is an \(\mathbb {H}\)-hull, and \(W^L\circ f^\mathbb {D}_L=f^{\mathbb {H}}_{W^L(L)}\circ W\) holds in \(\mathbb {D}\). The \(W^\mathcal{H}(L)\) is then defined to be the \(\mathbb {H}\)-hull \(W^L(L)\). Since \(z_\infty \) is the joint point of the process, \(W^{-1}(\infty )=z_\infty \not \in S_{L_{t}}\) for each \(t\), and so \(W^{L_{t}}\) and \(W^\mathcal{H}(L_{t})\) are well defined.

Write \(W_{t}=W^{L_{t}}, 0\le t< \infty \). Let \(\lambda \) be the driving function for the backward radial Loewner process \((L_{t})\). Let \(\widehat{\lambda }\) be the driving function for the backward chordal process \((W^\mathcal{H}(L_{v(t)})=W_{v(t)}(L_{v(t)}))\). Then (4.10) in [13] implies that \(W_{t}(e^{i\lambda (t)})=\widehat{\lambda }(v(t))\). In fact, in (4.10) of [13], the \(\widetilde{W}\) satisfies that \(e^{i\widetilde{W}(z)}=W(e^{iz})\), and the \(\lambda ^*(t)\) corresponds to the \(\widehat{\lambda }(v(t))\) here. Let \(f_{t} \,(\hbox {resp.}\ \widehat{f}_{t})\), \(f_{t_{2},t_{1}} \,(\hbox {resp.}\ \widehat{f}_{t_{2},t_{1}})\), and \((\beta _{t}) \,(\hbox {resp.}\ \widehat{\beta }_{t}), 0\le t<\infty \), be the backward radial (resp. chordal) Loewner maps, flows, and traces driven by \(\lambda \,(\hbox {resp.}\ \widehat{\lambda })\). Then we have \(W_{t}\circ f_{t}=\widehat{f}_{v^{-1}(t)}\circ W\) in \(\mathbb {D}\) for any \(t\ge 0\). Applying this equality to \(t=t_{2}\) and \(t=t_{1}\), where \(t_{2}\ge t_{1}\ge 0\), and using Lemma 2.1, we get \(W_{t_{2}}\circ f_{t_{2},t_{1}}\circ f_{t_{1}}=\widehat{f}_{v^{-1}(t_{2}),v^{-1}(t_{1})}\circ W_{t_{1}}\circ f_{t_{1}}\) in \(\mathbb {D}\), which implies that \(W_{t_{2}}\circ f_{t_{2},t_{1}}=\widehat{f}_{v^{-1}(t_{2}),v^{-1}(t_{1})}\circ W_{t_{1}}\) in \(\mathbb {D}\), and so

$$\begin{aligned} \widehat{\beta }_{t_{2}}(t_{1})&= \widehat{f}_{t_{2},t_{1}}(\widehat{\lambda }(t_{1}))=\widehat{f}_{t_{2},t_{1}}\circ W_{v(t_{1})}(e^{i\lambda (v(t_{1}))})\\&= W_{v(t_{2})}\circ f_{v(t_{2}),v(t_{1})}(e^{i\lambda (v(t_{1}))})=W_{v(t_{2})}(\beta _{v(t_{2})}(v(t_{1}))). \end{aligned}$$

Thus, the proposition above implies the following corollary.

Corollary 3.5

Let \(\kappa >0\) and \(z_{0}\ne z_\infty \in \mathbb {T}\). Let \(\beta _{t}, 0\le t<\infty \), be the backward radial \(\hbox {SLE}(\kappa ;-\kappa -6)\) traces started from \((z_{0},z_\infty )\). Then there exist a strictly increasing function \(v\) with \(v([0,\infty ))=[0,\infty )\), and a family of Möbius transformations \((W_{t})_{t\ge 0}\) with \(W_{t}(\mathbb {D})=\mathbb {H}\), such that \(\widehat{\beta }_{t}:=W_{v(t)}\circ \beta _{v(t)}\circ v, 0\le t<\infty \), are backward chordal traces generated by a backward chordal \(\hbox {SLE}_\kappa \) process.

The following proposition is Theorem 6.1 in [13].

Proposition 3.6

Let \(\kappa \in (0,4]\). Let \(z_{1}\ne z_{2}\in \mathbb {T}\). There is a coupling of two backward radial \(\hbox {SLE}(\kappa ;-\kappa -6)\) processes, one started from \((z_{1};z_{2})\), the other started from \((z_{2};z_{1})\), such that the two processes induce the same welding.

Remark

If \(\delta =\frac{4}{\kappa }\sigma (\frac{\rho }{2}+1)+1\in (1,2)\), we may define a forward (resp. backward) radial \(\hbox {SLE}(\kappa ;\rho )\) process in the case \(\sigma =1 \,(\hbox {resp.}\ \sigma =-1)\) such that the time interval of the process is \([0,\infty )\). First, the second remark in “Appendix B” says that a radial Bessel process \((X_{t})\) of dimension \(\delta >0\) started from \((x-y)/2\) can be defined for all \(t\ge 0\). Second, the transition density of \((X_{t})\) given by Proposition (8.1) (which is also true in the case \(\delta \in (0,2))\) shows that, if \(\delta >1\), then \(\cot (X_{t}), 0\le t<\infty \), is locally integrable. Thus, if \(\delta >1\), we may let \(q(t)=y-\sigma \int _{0}^t \cot _{2}(Z_s)ds\) and \(\lambda (t)=q(t)+Z_{t}, 0\le t<\infty \), where \(Z_{t}=2X_{\frac{\kappa }{4} t}\), and use \(\lambda \) as the driving function to define a forward (resp. backward) radial \(\hbox {SLE}(\kappa ;\rho )\) process. The corresponding stationary processes are similarly defined. Lemma 3.2 still holds thanks to the reversibility of the stationary radial Bessel process in the case \(\delta \in (1,2)\). But Girsanov’s theorem does not apply beyond the time that \(\lambda (t)-q(t)\) hits \(\{0,2\pi \}\).

4 Normalized backward radial Loewner trace

In general, a backward chordal (resp. radial) Loewner process does not naturally generate a single curve even if the backward chordal (resp. radial) traces \((\beta _{t})\) exist, because they may not satisfy \(\beta _{t_{1}}\subset \beta _{t_{2}}\) when \(t_{1}\le t_{2}\). A normalization method was introduced in [13] to define a normalized backward chordal Loewner trace (under certain conditions). In this section we will define a normalized backward radial Loewner trace.

Lemma 4.1

Let \(\lambda \in C([0,\infty ))\), and \((f_{t_{2},t_{1}})\) be the backward radial Loewner flow driven by \(\lambda \). Define \(F_{t_{2},t_{1}}=e^{t_{2}} f_{t_{2},t_{1}}, t_{2}\ge t_{1}\ge 0\). Then for every fixed \(t_{0}\in [0,\infty ), F_{t,t_{0}}\) converges locally uniformly in \(\mathbb {D}\) as \(t\rightarrow \infty \) to a conformal map, denoted by \(F_{\infty ,t_{0}}\), which satisfies that \(F_{\infty ,t_{0}}(0)=0, F_{\infty ,t_{0}}'(0)=e^{t_{0}}\), and

$$\begin{aligned} F_{\infty ,t_{2}}\circ f_{t_{2},t_{1}}=F_{\infty ,t_{1}},\quad t_{2}\ge t_{1}\ge 0. \end{aligned}$$
(4.1)

Moreover, let \(G_s=I_{\mathbb {T}}\circ F_{\infty ,-s}^{-1}\circ I_{\mathbb {T}}\) and \(K_s={\mathbb {C}}{\setminus } I_\mathbb {T}\circ F_{\infty ,-s}(\mathbb {D}), -\infty <s\le 0\). Then \(G_s\) and \(K_s\) are whole-plane Loewner maps and hulls driven by \(\lambda (-s), -\infty <s\le 0\).

Proof

Lemma 2.1(ii) implies that, if \(t_{2}\ge t_{1}\ge 0\), then \(f_{t_{2},t_{1}}\) is a conformal map on \(\mathbb {D}\) with \(f_{t_{2},t_{1}}(0)=0\) and \(f_{t_{2},t_{1}}'(0)=e^{-(t_{2}-t_{1})}\). Thus, every \(F_{t_{2},t_{1}}\) is a conformal map on \(\mathbb {D}\) that satisfies \(F_{t_{2},t_{1}}(0)=0\) and \(F_{t_{2},t_{1}}'(0)=e^{t_{1}}\). Koebe’s distortion theorem (c.f. [1]) implies that, for every fixed \(t_{1}, (F_{t_{2},t_{1}})_{t_{2}\ge t_{1}}\) is a normal family. Let \(S\) be a countable unbounded subset of \([0,\infty )\), and write \(S_{\ge t}=\{x\in S:x\ge t\}\) for every \(t\ge 0\). Using a diagonal argument, we can find a positive sequence \(t_n\rightarrow \infty \) such that for any \(x\in S, (F_{t_n,x})\) converges locally uniformly in \(\mathbb {D}\). Let \(F_{\infty ,x}\) denote the limit. Lemma implies that \(F_{\infty ,x}\) is a conformal map on \(\mathbb {D}\), and satisfies \(F_{\infty ,x}(0)=0\) and \(F_{\infty ,x}'(0)=e^{x}\).

Let \(x_{2}\ge x_{1}\in S\). From \(f_{t_n,x_{2}}\circ f_{x_{2},x_{1}}=f_{t_n,x_{1}}\) we conclude that \(F_{\infty ,x_{2}}\circ f_{x_{2},x_{1}}=F_{\infty ,x_{1}}\). For \(t\in [0,\infty )\), choose \(x\in S_{\ge t}\) and define the conformal map \(F_{\infty ,t}=F_{\infty ,x}\circ f_{x,t}\) on \(\mathbb {D}\). Lemma 2.1(i) and \(F_{\infty ,x_{2}}\circ f_{x_{2},x_{1}}=F_{\infty ,x_{1}}\) for \(x_{2}\ge x_{1}\in S\) imply that the definition of \(F_{\infty ,t}\) does not depend on the choice of \(x\in S_{\ge t}\), and (4.1) holds.

From (2.1) we see that \(f_{t_{2},t_{1}}\) commutes with the reflection \(I_\mathbb {T}(z)=1/\bar{z}\). Since \(f_{t_{2},t_{1}}^{-1}=f_{t_{1},t_{2}}\), using (4.1) we get \(G_{s_{1}}=f_{-s_{1},-s_{2}}\circ G_{s_{2}}\) if \(s_{1}\le s_{2}\le 0\). From (2.1) we see that \(G_{s}\) satisfies the equation

$$\begin{aligned} \partial _s G_{s}(z)=G_{s}(z)\frac{e^{i\lambda (-s)}+G_{s}(z)}{e^{i\lambda (-s)}-G_{s}(z)},\quad -\infty <s\le 0. \end{aligned}$$
(4.2)

Let \(\widehat{F}_{\infty ,t}(z)=F_{\infty ,t}(e^{-t} z), t\ge 0\). Then each \(\widehat{F}_{\infty ,t}\) is a conformal map defined on \(e^t\mathbb {D}\), and satisfies \(\widehat{F}_{\infty ,t}(0)=0\) and \(\widehat{F}_{\infty ,t}'(0)=1\). As \(t\rightarrow \infty , e^t\mathbb {D}\mathop {\longrightarrow }\limits ^\mathrm{Cara}{\mathbb {C}}\) (c.f. Definition 7.1). Koebe’s distortion theorem implies that \(|\widehat{F}_{\infty ,t}(z)|\le \frac{|z|}{(1-e^{-t}|z|)^2}\) for \(z\in e^t\mathbb {D}\). Thus, for every \(r>0\), there exists \(t_{0}\in \mathbb {R}\) such that, if \(t\ge t_{0}\), then \(|\widehat{F}_{\infty ,t}|\le 2r\) on \(\{|z|\le r\}\). Therefore, every sequence \((t_n)\), which tends to \(\infty \), contains a subsequence \((t_{n_k})\) such that \(\widehat{F}_{\infty ,t_{n_k}}\) converges locally uniformly in \({\mathbb {C}}\). Applying Lemma , we see that the limit function is a conformal map on \({\mathbb {C}}\), which fixes \(0\) and has derivative \(1\) at \(0\). Such conformal map must be the identity. Hence \(\widehat{F}_{\infty ,t}\mathop {\longrightarrow }\limits ^\mathrm{l.u.}{{\mathrm{id}}}\) in \({\mathbb {C}}\) as \(t\rightarrow \infty \). Applying Lemma again, we see that \(e^t F_{\infty ,t}^{-1}(z)\mathop {\longrightarrow }\limits ^\mathrm{l.u.}{{\mathrm{id}}}\) in \({\mathbb {C}}\) as \(t\rightarrow \infty \). Thus, \(\lim _{s\rightarrow -\infty } e^{s} G_{s}(z)=z\) for any \(z\in {\mathbb {C}}{\setminus }\{0\}\), which together with (4.2) implies that \(G_s, -\infty <s\le 0\), are whole-plane Loewner maps driven by \(\lambda (-s)\). The \(K_s\) are the corresponding hulls because \(K_s={\mathbb {C}}{\setminus } G_s^{-1}(\mathbb {D}^*)\).

It remains to show that, for any \(t\in [0,\infty ), F_{x,t}\mathop {\longrightarrow }\limits ^\mathrm{l.u.}F_{\infty ,t}\) in \(\mathbb {D}\) as \(x\rightarrow \infty \). Assume that this is not true for some \(t_{0}\in [0,\infty )\). Since \((F_{x,t_{0}})_{x\ge t_{0}}\) is a normal family, there exists \(x_n\rightarrow \infty \) such that \(F_{x_n,t_{0}}\) converges locally uniformly in \(\mathbb {D}\) to a function other than \(F_{\infty ,t_{0}}\). Let \(\widetilde{F}_{\infty ,t_{0}}\) denote the limit. Let \(S=\mathbb {N}\cup \{t_{0}\}\). By passing to a subsequence, we may assume that, for every \(t\in S, F_{x_n,t}\mathop {\longrightarrow }\limits ^\mathrm{l.u.}\widetilde{F}_{\infty ,t}\) in \(\mathbb {D}\). Now we may repeat the above construction to define \(\widetilde{F}_{\infty ,t}\) for every \(t\in [0,\infty )\). The previous argument shows that \(I_\mathbb {T}\circ \widetilde{F}_{\infty ,-t}^{-1}\circ I_\mathbb {T}, -\infty <t\le 0\), are the whole-plane Loewner maps driven by \(\lambda (-t), -\infty <t\le 0\). Since the same is true for \(I_{\mathbb {T}}\circ F_{\infty ,-t}^{-1}\circ I_{\mathbb {T}}\), we get \(\widetilde{F}_{\infty ,t}=F_{\infty ,t}\) for every \(t\), which contradicts that \(\widetilde{F}_{\infty ,t_{0}}\ne F_{\infty ,t_{0}}\). Thus, \(F_{x,t}\mathop {\longrightarrow }\limits ^\mathrm{l.u.}F_{\infty ,t}\) in \(\mathbb {D}\) as \(x\rightarrow \infty \). \(\square \)

Lemma 4.2

Let \(\lambda \in C([0,\infty ))\). Let \((F_{\infty ,t})_{t\ge 0}\) be given by the above lemma. Suppose the backward radial Loewner process driven by \(\lambda \) generates a family of backward radial Loewner traces \(\beta _{t}, 0\le t<\infty \), and

$$\begin{aligned} \forall t_{0}\in [0,\infty ),\quad \exists t_{1}\in (t_{0},\infty ),\quad \beta _{t_{1}}([0,t_{0}])\subset \mathbb {D}. \end{aligned}$$
(4.3)

Then every \(F_{\infty ,t}\) extends to a continuous function \(\overline{\mathbb {D}}\rightarrow \widehat{\mathbb {C}}\), and there is a continuous curve \(\beta (t), 0\le t<\infty \), with \(\lim _{t\rightarrow \infty }\beta (t)=\infty \) such that

$$\begin{aligned} \beta (t)=F_{\infty ,t_{0}}(\beta _{t_{0}}(t)),\quad t_{0}\ge t\ge 0; \end{aligned}$$
(4.4)

and for any \(t\ge 0, F_{\infty ,t}(\mathbb {D})\) is the component of \({\mathbb {C}}{\setminus } \beta ([t,\infty ))\) that contains \(0\). Furthermore, \(\gamma (s):=I_\mathbb {T}(\beta (-s)), -\infty <s\le 0\), is the whole-plane Loewner trace driven by \(\lambda (-s)\).

Proof

For every \(t_{0}\in [0,\infty )\), using (4.3) we may pick \(t_{1}\in (t_{0},\infty )\) such that \(\beta _{t_{1}}([0,t_{0}])\subset \mathbb {D}\), and define \( \beta (t)=F_{\infty ,t_{1}}\circ \beta _{t_{1}}(t), t\in [0,t_{0}]\). From (2.2) and (4.1) we see that the definition of \(\beta \) does not depend on \(t_{0}\) and \(t_{1}\), and \(\beta \) is continuous on \([0,\infty )\).

Let \(L_{t_{2},t_{1}}=\mathbb {D}{\setminus } f_{t_{2},t_{1}}(\mathbb {D}), t_{2}\ge t_{1}\ge 0\). Then \(L_{t_{2},t_{1}}\) is the \(\mathbb {D}\)-hull generated by \( \beta _{t_{2}}([t_{1},t_{2}])\), i.e., \(\mathbb {D}{\setminus } L_{t_{2},t_{1}}\) is the component of \(\mathbb {D}{\setminus } \beta _{t_{2}}([t_{1},t_{2}])\) that contains \(0\). Hence \(\partial L_{t_{2},t_{1}}\cap \mathbb {D}\subset \beta _{t_{2}}([t_{1},t_{2}])\).

Let \(G_s\) and \(K_s, -\infty <s\le 0\), be given by the previous lemma. Then \((K_s)\) is an increasing family with \(\bigcap _{s\le 0} K_s=\{0\}\). If \(s_{2}\le s_{1}\le 0\), from \(F_{\infty ,-s_{1}}=F_{\infty ,-s_{2}}\circ f_{-s_{2},-s_{1}}\) and \(f_{-s_{2},-s_{1}}(\mathbb {D})=\mathbb {D}{\setminus } L_{-s_{2},-s_{1}}\), we see that \(K_{s_{1}}{\setminus } K_{s_{2}}=I_{\mathbb {T}}\circ F_{\infty , -s_{2}}(L_{-s_{2},-s_{1}})\).

Fix \(t_{2}\ge t_{1}\ge 0\). Choose \(T>t_{2}\) such that \(\beta _T([0,t_{2}])\subset \mathbb {D}\). Then \(\beta (t_{2})=F_{\infty ,T}\circ \beta _T(t_{2})\). Since \(f_{T,t_{1}}:\mathbb {D}\mathop {\twoheadrightarrow }\limits ^\mathrm{Conf}\mathbb {D}{\setminus } L_{T,t_{1}}, L_{T,t_{1}}\) is the \(\mathbb {D}\)-hull generated by \(\beta _T([t_{1},T])\), and \(t_{2}\in [t_{1},T]\), we see that \(\beta _T(t_{2})\not \in f_{T,t_{1}}(\mathbb {D})\). So \(\beta (t_{2})\not \in F_{\infty ,T}\circ f_{T,t_{1}}(\mathbb {D})=F_{\infty ,t_{1}}(\mathbb {D})\). This implies that, if \(s_{2}\le s_{1}\le 0\), then \(\gamma (s_{2})=I_\mathbb {T}(\beta (-s_{2}))\in {\mathbb {C}}{\setminus } I_\mathbb {T}\circ F_{\infty ,-s_{1}}(\mathbb {D})=K_{s_{1}}\). Thus, \(\gamma ((-\infty ,s])\subset K_s\) for every \(s\le 0\). Since \(\bigcap _{s\le 0} K_s=\{0\}\), we get \(\lim _{s\rightarrow -\infty }\gamma (s)=0\).

Define \(\gamma (-\infty )=0\). Let \(s\le 0\). Let \(z_{0}\in \partial K_s\). If \(z_{0}=0\), then \(z_{0}=\gamma (-\infty )\in \gamma ([-\infty ,s])\). Now suppose \(z_{0}\ne 0\). Since \((K_s)\) is increasing and \(\bigcap _{s\le 0} K_s=\{0\}\), there is \(s_{0}<s\) such that \(z_{0}\not \in K_{s_{0}}\). Thus, \(z_{0}\in K_s{\setminus } K_{s_{0}}=I_{\mathbb {T}}\circ F_{\infty ,-s_{0}}(L_{-s_{0},-s})\). From \(z_{0}\in \partial K_s\) we see that \(w_{0}:=F_{\infty ,-s_{0}}^{-1}\circ I_\mathbb {T}(z_{0})\in \partial L_{-s_{0},-s}\cap \mathbb {D}\). Since \(L_{-s_{0},-s}\) is the \(\mathbb {D}\)-hull generated by \(\beta _{-s_{0}}([-s,-s_{0}])\), there is \(t_{1}\in [-s,-s_{0}]\) such that \(w_{0}=\beta _{-s_{0}}(t_{1})\). Thus, \(z_{0}=I_\mathbb {T}\circ F_{\infty ,-s_{0}}(\beta _{-s_{0}}(t_{1}))=\gamma (-t_{1})\in \gamma ([-\infty ,s])\). Thus, \(\partial K_s\subset \gamma ([-\infty ,s])\), which implies that \(\partial K_s\) is locally connected. Since \(I_\mathbb {T}\circ F_{\infty ,-s}\circ I_\mathbb {T}:\mathbb {D}^*\mathop {\twoheadrightarrow }\limits ^\mathrm{Conf}\widehat{\mathbb {C}}{\setminus } K_s\), we see that \(F_{\infty ,t}\) extends continuously to \(\overline{\mathbb {D}}\) for each \(t\ge 0\) (c.f. [10]). The equality (4.1) holds after continuation, which together with (2.2) and the definition of \(\beta \) implies (4.4). Setting \(t_{1}=t=-s\), we see that \(\gamma (s)=I_\mathbb {T}\circ F_{\infty , t}(e^{i\lambda (t)})=G_s^{-1}(e^{i\lambda (-s)})\). Thus, \(\gamma (s), -\infty \le s\le 0\), is the whole-plane Loewner trace driven by \(\lambda (-s), -\infty <s\le 0\). This implies that \(\lim _{t\rightarrow \infty }\beta (t)=I_\mathbb {T}(\lim _{s\rightarrow -\infty }\gamma (s))=\infty \).

Finally, from the properties of the whole-plane Loewner trace, we see that for any \(s\ge 0, G_s^{-1}(\mathbb {D}^*)\) is the component of \(\widehat{\mathbb {C}}{\setminus } \gamma ([-\infty ,s])\) that contains \(0\). Since \(G_{-t}=I_\mathbb {T}\circ F_{\infty ,t}^{-1}\circ I_\mathbb {T}\) and \(\gamma (-t)=I_\mathbb {T}(\beta (t))\), we see that, for any \(t\ge 0, F_{\infty , t}(\mathbb {D})\) is the component of \({\mathbb {C}}{\setminus } \beta ([t,\infty ))\) that contains \(I_\mathbb {T}(\infty )=0\). \(\square \)

Definition 4.3

The \(\beta (t), 0\le t<\infty \), given by the lemma is called the normalized backward radial Loewner trace driven by \(\lambda \).

If the backward radial Loewner traces \(\beta _{t}\) are all \(\mathbb {D}\)-simple traces, then (4.3) clearly holds because we may always choose \(t_{1}=t_{0}+1\). Moreover, (4.4) implies that for any \(t_{0}>0, \beta \) restricted to \([0,t_{0})\) is simple. Thus, the whole curve \(\beta \) is simple. This implies further that \(F_{\infty ,t}(\mathbb {D})={\mathbb {C}}{\setminus }\beta ([t,\infty ))\) for any \(t\ge 0\). In particular, \(F_{\infty ,0}\) maps two arcs on \(\mathbb {T}\) with two common end points onto the two sides of \(\beta \). Let \(\phi \) be the welding induced by the process. The equality \(F_{\infty ,0}=F_{\infty ,t}\circ f_{t}\) implies that, if \(y=\phi (x)\) then \(F_{\infty ,0}(x)=F_{\infty ,0}(y)\in \beta \). The two fixed points of \(\phi \) are mapped to the two ends of \(\beta \) such that \(e^{i\lambda (0)}\) is mapped to \(\beta (0)\in {\mathbb {C}}\), and the joint point is mapped to \(\infty \).

We will prove that (4.3) holds in some other cases. We say that an \(\mathbb {H}\,(\hbox {resp.}\ \mathbb {D})\)-hull \(K\) is nice if \(S_K\) is an interval on \(\mathbb {R}\,(\hbox {resp.}\ \mathbb {D})\), and \(f_K\) extends continuously to \(S_K\) and maps the interior of \(S_K\) into \(\mathbb {H}\,(\hbox {resp.}\ \mathbb {D})\). This means that \(\partial K\cap \mathbb {H}\,(\hbox {resp.}\ \partial K\cap \mathbb {D})\) is the image of an open curve in \(\mathbb {H}\,(\hbox {resp.}\ \mathbb {D})\), whose two ends approach \(\mathbb {R}\,(\hbox {resp.}\ \mathbb {T})\). It is easy to see that, if \(K\) is a nice \(\mathbb {H}\)-hull, and \(W\) is a Möbius transformation such that \(W(\mathbb {H})=\mathbb {D}\) and \(0\not \in W(K)\), then \(W(K)\) is a nice \(\mathbb {D}\)-hull.

Lemma 4.4

Let \(\kappa >4\) and \(\rho \le -\frac{\kappa }{2}-2\). Let \((L_{t})\) be \(\mathbb {D}\)-hulls generated by a backward radial \(\hbox {SLE}(\kappa ;\rho )\) process. Then for every fixed \(t_{0}\in (0,\infty )\), a.s. \(L_{t_{0}}\) is nice.

Proof

Theorem 6.1 in [17] shows that, if \((H_{t})\) are \(\mathbb {H}\)-hulls generated by a (forward) chordal \(\hbox {SLE}_\kappa \) process, then for any stopping time \(T\in (0,\infty )\), a.s. \(H_T\) is a nice \(\mathbb {H}\)-hull. From the equivalence between chordal \(\hbox {SLE}_\kappa \) and radial \(\hbox {SLE}_\kappa \) (Proposition 4.2 in [6]), we conclude that, if \((K_{t})\) are \(\mathbb {D}\)-hulls generated by a forward radial \(\hbox {SLE}_\kappa \) process, then for any deterministic point \(z_{0}\in \mathbb {T}\) and any stopping time \(T\in (0,\infty )\) such that \(z_{0}\not \in \overline{K_T}\), a.s. \(K_T\) is a nice \(\mathbb {D}\)-hull. This further implies that, for any stopping time \(T\in (0,\infty )\), on the event that \(\mathbb {T}\not \subset \overline{K_T}\), a.s. \(K_T\) is a nice \(\mathbb {D}\)-hull. Let \((L^0_{t})\) be \(\mathbb {H}\)-hulls generated by a backward radial \(\hbox {SLE}_\kappa \) process. The above result in the case that \(T\) is a deterministic time together with Lemma 2.1 and the rotation symmetry of radial Loewner processes implies that, for any fixed \(t_{0}\in (0,\infty )\), on the event that \(S_{L^0_{t_{0}}}\ne \mathbb {T}\), a.s. \(L^0_{t_{0}}\) is a nice \(\mathbb {D}\)-hull.

By rotation symmetry, we may assume that the backward radial \(\hbox {SLE}(\kappa ;\rho )\) process which generates \((L_{t})\) is started from \((1;w_{0})\). Fix \(t_{0}\in (0,\infty )\). Girsanov’s theorem implies that the distribution of \((L_{t})_{0\le t\le t_{0}}\) is absolutely continuous w.r.t. that of \((L^0_{t})_{0\le t\le t_{0}}\) given by the last paragraph conditioned on the event that \(f^0_{t}(w_{0})\in \mathbb {T}\) for \(0\le t\le t_{0}\). Since \(f^0_{t_{0}}(w_{0})\in \mathbb {T}\) is equivalent to \(w_{0}\in \mathbb {T}{\setminus } S_{L_{t_{0}}}\), which implies that \(S_{L_{t_{0}}}\ne \mathbb {T}\), the proof is completed. \(\square \)

Proposition 4.5

Let \(\kappa >0\) and \(\rho \le -\frac{\kappa }{2}-2\). Then condition (4.3) almost surely holds for a backward radial \(\hbox {SLE}(\kappa ;\rho )\) process.

Proof

The result is clear if \(\kappa \le 4\) since the traces are \(\mathbb {D}\)-simple. Now assume that \(\kappa >4\). Suppose the process is started from \((z_{0};w_{0})\). Lemma 3.1 implies that \(S_{L_{t_{0}}}\subset \mathbb {T}{\setminus }\{w_{0}\}\). So \(f_{t_{0}}(w_{0})\not \in \overline{L_{t_{0}}}\). Since \(L_{t_{0}}\) is the \(\mathbb {D}\)-hull generated by \(\beta _{t_{0}}\), we have \(f_{t_{0}}(w_{0})\not \in \beta _{t_{0}}([0,t_{0}])\). The Markov property of Brownian motion and the fact that \(e^{i q(t)}=f_{t}(w_{0})\) for all \(t\) imply that, conditioned on \(\lambda (t), 0\le t\le t_{0}\), the maps \(f_{t_{0}+t,t_{0}}, t\ge 0\), are generated by a backward radial \(\hbox {SLE}(\kappa ;\rho )\) process started from \((e^{i\lambda (t_{0})};f_{t_{0}}(w_{0}))\). Let \(L_{t_{0}+t,t_{0}}=\mathbb {D}{\setminus } f_{L_{t_{0}+t,t_{0}}}(\mathbb {D})\). Lemma 4.4 implies that, for every \(t_{1}>t_{0}\), a.s. \(L_{t_{1},t_{0}}\) is nice. Lemma 3.1 implies that the probability that \(\beta _{t_{0}}([0,t_{0}])\cap \mathbb {T}\) is contained in the interior of \(S_{L_{t_{1},t_{0}}}\) tends to \(1\) as \(t_{1}\rightarrow \infty \).

If \(L_{t_{1},t_{0}}\) is nice and \(\beta _{t_{0}}([0,t_{0}])\cap \mathbb {T}\) is contained in the interior of \(S_{L_{t_{1},t_{0}}}\), then

$$\begin{aligned} \beta _{t_{1}}([0,t_{0}])=f_{t_{1},t_{0}}(\beta _{t_{0}}([0,t_{0}]))=f_{L_{t_{1},t_{0}}}(\beta _{t_{0}}([0,t_{0}]))\subset \mathbb {D}. \end{aligned}$$

In fact, if \(z\in \beta _{t_{1}}([0,t_{0}])\cap \mathbb {D}\), then obviously \(f_{L_{t_{1},t_{0}}}(z)\in \mathbb {D}\); if \(z\in \beta _{t_{0}}([0,t_{0}])\cap \mathbb {T}\), then \(f_{L_{t_{1},t_{0}}}(z)\in \mathbb {D}\) follows from that \(L_{t_{1},t_{0}}\) is nice and \(z\) lies in the interior of \(S_{L_{t_{1},t_{0}}}\). Thus, as \(t_{1}\rightarrow \infty \), the probability that \(\beta _{t_{1}}([0,t_{0}])\subset \mathbb {D}\) tends to \(1\). This means that, for every fixed \(t_{0}>0\), a.s. there exists a (random) \(t_{1}>t_{0}\) such that \(\beta _{t_{1}}([0,t_{0}])\subset \mathbb {D}\). Thus, on an event with probability \(1\), (4.3) holds for every \(t_{0}\in \mathbb {N}\). Since \(\beta _{t_{1}}([0,t_{0}])\subset \beta _{t_{1}}([0,n])\subset \mathbb {D}\) if \(t_{0}<n\in \mathbb {N}\), we see that (4.3) holds on that event. This completes the proof. \(\square \)

Thus, a normalized backward radial \(\hbox {SLE}(\kappa ;\rho )\) trace can be well defined for any \(\kappa >0\) and \(\rho \le -\frac{\kappa }{2}-2\). Combining Lemmas 3.2 and 4.2, we obtain the following theorem.

Theorem 4.6

Let \(\kappa >0\) and \(\rho \le -\frac{\kappa }{2}-2\). Let \(\beta (t), 0\le t<\infty \), be a normalized stationary backward radial \(\hbox {SLE}(\kappa ;\rho )\) trace. Then \(\gamma (s):=I_\mathbb {T}(\beta (-s)), -\infty <s\le 0\), is a whole-plane \(\hbox {SLE}(\kappa ;-4-\rho )\) trace stopped at time \(0\).

5 Conformal images of the tips

Theorem 5.1

Let \(\kappa \in (0,4)\). Let \(\gamma (s), -\infty \le s\le 0\), be a whole-plane \(\hbox {SLE}(\kappa ;\kappa +2)\) trace stopped at time \(0\). Then after an orientation reversing time change, the curve \(\gamma (s)-\gamma (0), -\infty \le s\le 0\), has the same distribution as \(\gamma (s), -\infty \le s\le 0\).

Proof

Theorem 4.6 shows that \(\beta (t):=I_\mathbb {T}(\gamma (-t)), 0\le t\le \infty \), is a normalized stationary backward radial \(\hbox {SLE}(\kappa ;-\kappa -6)\) trace, which is a simple curve with \( \beta (\infty )=\infty \), and there is \(F_{\infty ,0}:\mathbb {D}\mathop {\twoheadrightarrow }\limits ^\mathrm{Conf}{\mathbb {C}}{\setminus }\beta \) such that \(F_{\infty ,0}(0)=0, F_{\infty ,0}'(0)=1\), and \(F_{\infty ,0}(x)=F_{\infty ,0}(y)\) implies that \(y=x\) or \(y=\phi (x)\), where \(\phi \) is the welding induced by the stationary backward radial \(\hbox {SLE}(\kappa ;-\kappa -6)\) process. Proposition 3.6 implies that this process can be coupled with another stationary backward radial \(\hbox {SLE}(\kappa ;-\kappa -6)\) process, which induces the same welding, but has a different joint point. Let \(\widetilde{\beta }\) and \(\widetilde{F}_{\infty ,0}\) be the normalized trace and map for the second process. Let \(\widetilde{\gamma }(s)=I_\mathbb {T}(\widetilde{\beta }(-s)), -\infty \le s\le 0\). Theorem 4.6 implies that \(\widetilde{\gamma }\) is also a whole-plane \(\hbox {SLE}(\kappa ;\kappa +2)\) trace stopped at time \(0\).

Define \(W=I_\mathbb {T}\circ \widetilde{F}_{\infty ,0}\circ F_{\infty ,0}^{-1}\circ I_\mathbb {T}\). Then \(W:\widehat{\mathbb {C}}{\setminus } \gamma \mathop {\twoheadrightarrow }\limits ^\mathrm{Conf}\widehat{\mathbb {C}}{\setminus } \widetilde{\gamma }\) and satisfies that \(W(\infty )=\infty \) and \(W'(\infty )=1\). Since the two backward radial \(\hbox {SLE}(\kappa ;-\kappa -6)\) processes induce the same welding, we see that \(F_{\infty ,0}(x)=F_{\infty ,0}(y)\) iff \(\widetilde{F}_{\infty ,0}(x)=\widetilde{F}_{\infty ,0}(y)\). Thus, \(W\) extends continuously to \(\gamma \). The work in [2] shows that the boundary of a Hölder domain is conformally removable; while the work in [12] shows that, for \(\kappa \in (0,4)\), a chordal \(\hbox {SLE}_\kappa \) trace is the boundary of a Hölder domain, which together with the Girsanov’s theorem and the equivalence between chordal \(\hbox {SLE}_\kappa \) and radial \(\hbox {SLE}_\kappa \) implies that a radial \(\hbox {SLE}(\kappa ;\rho )\) trace is conformally removable for \(\kappa \in (0,4)\) and \(\rho \ge \frac{\kappa }{2}-2\) (which is true if \(\rho =\kappa +2)\). The Markov-type relation between whole-plane \(\hbox {SLE}(\kappa ;\kappa +2)\) and radial \(\hbox {SLE}(\kappa ;\kappa +2)\) processes implies that \(\gamma ([t_{0},0])\) is conformally removable for any \(t_{0}\in (-\infty ,0)\), and so is the whole curve \(\gamma =\gamma ([-\infty ,0])\). Thus, \(W\) extends to a conformal map defined on \(\widehat{\mathbb {C}}\) such that \(W(\gamma )=\widetilde{\gamma }\). Since \(W(\infty )=\infty \) and \(W'(\infty )=1\), we have \(W(z)=z+C\) for some constant \(C\in {\mathbb {C}}\). This means that \(\widetilde{\gamma }=\gamma +C\), where both curves are viewed as sets. Since both curves are simple, \(W\) maps end points of \(\gamma \) to end points of \(\widetilde{\gamma }\). Now \(0\) is an end point of both curves. Since \(F_{\infty ,0}\) and \(\widetilde{F}_{\infty ,0}\) map the joint points of the two processes, respectively, to \(\infty \), and the two joints points are different, \(W\) does not fixed \(0\). So \(W\) maps the other end point of \(\gamma \): \(\gamma (0)\) to \(0\), which implies that \(C=-\gamma (0)\) and the orientations of \(\widetilde{\gamma }\) and \(W(\gamma )=\gamma -\gamma (0)\) are opposite to each other. Thus, the whole-plane \(\hbox {SLE}(\kappa ;\kappa +2)\) trace \(\widetilde{\gamma }\) up to time \(0\) is an orientation reversing time-change of \(\gamma -\gamma (0)\) up to time \(0\), which completes the proof. \(\square \)

Remark

This theorem says that a whole-plane \(\hbox {SLE}(\kappa ;\kappa +2) (\kappa \in (0,4))\) trace stopped at whole-plane capacity time \(0\) satisfies reversibility. So a tip segment of the trace at time \(0\) has the same shape as an initial segment of the trace.

Lemma 5.2

Let \(\kappa >0\). Let \(\beta \) be a forward chordal \(\hbox {SLE}_\kappa \) trace. Let \(t_{0}\in (0,\infty )\) be fixed. Then there is a whole-plane \(\hbox {SLE}(\kappa ;\kappa +2)\) process, which generates hulls \((K_s)\) and a trace \(\gamma \), and a random conformal map \(W\) defined on \(\mathbb {H}\) such that \(W(\mathbb {H})=\widehat{\mathbb {C}}{\setminus } K_{s_{0}}\) for some random \(s_{0}<0\) and \(W(\beta (t))= \gamma (v(t)), 0\le t\le t_{0}\), where \(v\) is a random strictly increasing function with \(v([0,t_{0}])=[s_{0},0]\).

Proof

Let \(\lambda \) be the driving function for \(\beta \). Lemma 2.1 and the translation symmetry implies that there is a backward chordal \(\hbox {SLE}_\kappa \) process, which generates backward chordal traces \((\widetilde{\beta }_{t})\) such that \(\widetilde{\beta }_{t_{0}}(t_{0}-t)=\beta (t)-\lambda (t_{0}), 0\le t\le t_{0}\). Corollary 3.5 implies that there exist a stationary backward radial \(\hbox {SLE}(\kappa ;-\kappa -6)\) process generating backward radial traces \((\widehat{\beta }_{t})\), a family of Möbius transformations \((V_{t})\) with \(V_{t}(\mathbb {H})=\mathbb {D}\) for each \(t\), and a strictly increasing function \(u\) with \(u([0,\infty ))=[0,\infty )\), such that \(V_{t_{1}}(\widetilde{\beta }_{t_{1}}(t))=\widehat{\beta }_{u(t_{1})}(u(t))\) for any \(t_{1}\ge t\ge 0\). In particular, it follows that \(V_{t_{0}}(\beta (t)-\lambda (t_{0}))=\widehat{\beta }_{u(t_{0})}(u(t_{0}-t)), 0\le t\le t_{0}\).

Let \(\widehat{\beta }\) be the normalized backward radial trace generated by that stationary backward radial \(\hbox {SLE}(\kappa ;-\kappa -6)\) process, which exists thanks to Proposition 4.5. Lemmas 4.1 and 4.2 state that there exists a family of conformal maps \(F_{\infty ,t}, t\ge 0\), defined on \(\mathbb {D}\), with continuation to \(\overline{\mathbb {D}}\), such that \(\widehat{\beta }(t)=F_{\infty ,t_{1}}(\beta _{t_{1}}(t))\) for any \(t_{1}\ge t\ge 0\). In particular, we have

$$\begin{aligned}&F_{\infty ,u(t_{0})}(V_{t_{0}}(\beta (t)-\lambda (t_{0})))=F_{\infty ,u(t_{0})}(\widehat{\beta }_{u(t_{0})}(u(t_{0}-t)))=\widehat{\beta }(u(t_{0}-t)),\quad \\&0\le t\le t_{0}. \end{aligned}$$

Theorem 4.6 states that \(\gamma (s):=I_\mathbb {T}(\widehat{\beta }(-s)), -\infty <s\le 0\), is a whole-plane \(\hbox {SLE}(\kappa ;\kappa +2)\) trace stopped at time \(0\). Lemma 4.1 states that \(K_s:={\mathbb {C}}{\setminus } I_\mathbb {T}\circ F_{\infty ,-s}(\mathbb {D})\) are the corresponding hulls. Then we have \(I_\mathbb {T}(F_{\infty ,u(t_{0})}(V_{t_{0}}(\beta (t)-\lambda (t_{0}))))=\gamma (-u(t_{0}-t)) , 0\le t\le t_{0}\). Now it is easy to check that \(W(z):=I_\mathbb {T}(F_{\infty ,u(t_{0})}(V_{t_{0}}(z-\lambda (t_{0})))), v(t):=-u(t_{0}-t)\), and \(s_{0}:=-u(t_{0})\) satisfy the desired properties. \(\square \)

Theorem 5.3

Let \(\kappa \in (0,4)\) and \(t_{0}\in (0,\infty )\). Let \(\beta (t), t\ge 0\), be a forward chordal \(\hbox {SLE}_\kappa \) trace (parameterized by the half-plane capacity). Then there is a random conformal map \(V\) defined on \(\mathbb {H}\) such that \(V(\beta (t_{0}))=0\), and \(V(\beta (t_{0}-t)), 0\le t\le t_{0}\), is an initial segment of a whole-plane \(\hbox {SLE}(\kappa ;\kappa +2)\) trace, up to a time change.

Proof

Lemma 5.2 states that we can map \(\beta (t_{0}-t), 0\le t\le t_{0}\), conformally to a tip segment of a whole-plane \(\hbox {SLE}(\kappa ;\kappa +2)\) trace at time \(0\). Then we may apply Theorem 5.1. \(\square \)

We may derive a similar but weaker result for radial SLE.

Theorem 5.4

Let \(\kappa \in (0,4)\) and \(t_{0}\in (0,\infty )\). Let \(\beta (t), t\ge 0\), be a forward radial \(\hbox {SLE}_\kappa \) trace (parameterized by the disc capacity). Then there is a random conformal map \(V\) defined on \(\mathbb {D}\) such that \(V(\beta (1))=0\), and up to a time change, \(V(\beta (t_{0}-t)), 0\le t\le t_{0}\), has a distribution, which is absolutely continuous w.r.t. an initial segment of a whole-plane \(\hbox {SLE}(\kappa ;\kappa +2)\) trace.

Proof

From Theorem 5.1, it suffices to prove the theorem with “an initial segment” replaced by “a tip segment at time \(0\)”. By rotation symmetry, we may assume that \(\beta \) is a forward stationary radial \(\hbox {SLE}(\kappa ;0)\) trace. By Corollary 3.3, \(\beta (t_{0}-t), 0\le t\le t_{0}\), has the distribution of a backward stationary radial \(\hbox {SLE}(\kappa ;-4)\) trace at time \(t_{0}\), say \(\widetilde{\beta }_{t_{0}}\). Girsanov’s theorem implies that the distribution of \(\widetilde{\beta }_{t_{0}}\) is absolutely continuous w.r.t. a backward stationary radial \(\hbox {SLE}(\kappa ;-\kappa -6)\) trace at time \(t_{0}\). This backward stationary radial \(\hbox {SLE}(\kappa ;-\kappa -6)\) trace at \(t_{0}\) can then be mapped conformally to a tip segment of the normalized trace generated by the process. Finally, the reflection \(I_\mathbb {T}\) maps that tip segment to a tip segment of a whole-plane \(\hbox {SLE}(\kappa ;\kappa +2)\) trace at time \(0\) thanks to Theorem 4.6. \(\square \)

6 Ergodicity

We will apply Theorems 5.3 and 5.4 to study some ergodic behavior of the tip of a chordal or radial \(\hbox {SLE}_\kappa (\kappa \in (0,4))\) trace at a deterministic half plane or disc capacity time.

Let \(\gamma (t), a\le t\le b\), be a simple curve in \({\mathbb {C}}\) such that \(\gamma (a)=0\). We may reparameterize \(\gamma \) using the whole-plane capacity. Let \(T={{\mathrm{cap}}}(\gamma )\). Define \(v\) on \([a,b]\) such that \(v(a)=-\infty \) and \(v(t)={{\mathrm{cap}}}(\gamma ([a,t])), a<t\le b\). Then \(v\) is a strictly increasing function with \(v([a,b])=[-\infty ,T]\). It turns out that (c.f. [4]) \(\gamma ^v(t):=\gamma (v^{-1}(t)), -\infty \le t\le T\), is a whole-plane Loewner trace driven by some \(\lambda \in C((-\infty ,T])\). Let \(g_{t}, -\infty <t\le T\), be the corresponding maps. Then each \(g_{t}^{-1}\) extends continuously to \(\overline{\mathbb {D}^*}\) and maps \(\mathbb {T}\) onto \(\gamma ^v([-\infty ,t])\). At time \(t\), there are two special points on \(\mathbb {T}\), which are mapped by \(g_{t}^{-1}\) to the two ends of \(\gamma ^v([-\infty ,t])\). One is \(e^{i\lambda (t)}\), which is mapped to \(\gamma ^v(t)\). Let \(z(t)\) denote the point on \(\mathbb {T}\) which is mapped to \(\gamma ^v(-\infty )=0\). Then \(z(t)\) satisfies the equation \(z'(t)=z(t)\frac{e^{i\lambda (t)}+z(t)}{e^{i\lambda (t)}-z(t)}, -\infty <t\le T\). There exists a unique \(q\in C((-\infty ,T])\) such that \(z(t)=e^{iq(t)}\) and \(0<\lambda (t)-q(t)<2\pi , -\infty <t\le T\). Then \(q(t)\) satisfies the equation \(q'(t)=\cot _{2}(q(t)-\lambda (t)), -\infty <t\le T\). The number \(\lambda (t)-q(t)\in (0,2\pi )\) has a geometric meaning. It is equal to \(2\pi \) times the harmonic measure viewed from \(\infty \) of the right side of \(\gamma ^v([-\infty ,t])\) in \(\widehat{\mathbb {C}}{\setminus } \gamma ^v([-\infty ,t])\).

Let \(\kappa \le 4\) and \(\rho \ge \frac{\kappa }{2}-2\). A whole-plane \(\hbox {SLE}(\kappa ;\rho )\) process generates a simple trace, say \(\gamma (t), -\infty \le t<\infty \), which is parameterized by whole-plane capacity. Recall the definition in Sect. 3. There are \(\lambda ,q\in C(\mathbb {R})\) such that \(\lambda \) is the driving function, \(q(t)\) satisfies the equation \(q'(t)=\cot _{2}(q(t)-\lambda (t))\), and \(Z(t):=\lambda (t)-q(t)\in (0,2\pi ), -\infty <t<\infty \), is a reversible stationary diffusion process with SDE: \(dZ(t)=\sqrt{\kappa }dB(t)+(\frac{\rho }{2}+1)\cot _{2}(Z(t))dt\). Let \(\mu _{\kappa ;\rho }\) denote the invariant distribution for \((Z(t))\). Corollary 8.2 shows that \(\mu _{\kappa ;\rho }\) has a density, which is proportional to \(\sin _{2}(x)^{\frac{4}{\kappa }(\frac{\rho }{2}+1)}\). Corollary 8.3 shows that \((Z(t))\) is ergodic. Thus, for any \(t_{0}\in \mathbb {R}\) and \(f\in L^1(\mu _{\kappa ;\rho })\), almost surely

$$\begin{aligned} \lim _{t\rightarrow -\infty } \frac{1}{t_{0}-t} \int _{t}^{t_{0}} f(Z(s))ds=\int f(x) d\mu _{\kappa ;\rho }(x). \end{aligned}$$
(6.1)

We will prove that this property is preserved under conformal maps fixing \(0\), as long as \(f\) is uniformly continuous. The following lemma is obvious.

Lemma 6.1

Let \(T_{1},T_{2}\in \mathbb {R}\). Let \(Z_j\in C((-\infty ,T_j)), j=1,2\). Suppose that there is an increasing differentiable function \(v\) defined on \((-\infty ,T_{1})\) such that \(v((-\infty ,T_{1}])=(-\infty ,T_{2}], v'(t)\rightarrow 1\) and \(Z_{2}(v(t))-Z_{1}(t)\rightarrow 0\) as \(t\rightarrow -\infty \). Let \(f\in C(\mathbb {R})\) be uniformly continuous. Then

$$\begin{aligned} \lim _{t\rightarrow -\infty } \frac{1}{t_{0}-t} \int _{t}^{t_{0}} f(Z_{1}(s))ds=\lim _{t\rightarrow -\infty } \frac{1}{t_{0}-t} \int _{t}^{t_{0}} f(Z_{2}(s))ds \end{aligned}$$

as long as either limit exists and lies in \(\mathbb {R}\) for some/every \(t_{0}\in (-\infty , T_{1}\wedge T_{2})\).

We will need some properties of \({\mathbb {C}}\)-hulls. Let \(K\) be a \({\mathbb {C}}\)-hull such that \(\{0\}\subsetneqq K\). The following well-known fact follows from Schwarz lemma and Koebe’s \(1/4\) theorem (c.f. [1]):

$$\begin{aligned} e^{{{\mathrm{cap}}}(K)}\le \max _{z\in K} |z|\le 4e^{{{\mathrm{cap}}}(K)}. \end{aligned}$$
(6.2)

Lemma 6.2

For the above \(K, |e^{{{\mathrm{cap}}}(K)} g_K(z)-z|\le 5e^{{{\mathrm{cap}}}(K)}\) for any \(z\in {\mathbb {C}}{\setminus } K\).

Proof

Since the derivative of \(e^{{{\mathrm{cap}}}(K)} g_K(z)\) at \(\infty \) is \(1, e^{{{\mathrm{cap}}}(K)} g_K(z)-z\) extends analytically to \(\widehat{\mathbb {C}}{\setminus } K\). Applying the maximum modulus principle, we see that \(\sup _{z\in {\mathbb {C}}{\setminus } K}|e^{{{\mathrm{cap}}}(K)} g_K(z)-z|\) is approached by a sequence \((z_n)\) in \({\mathbb {C}}{\setminus } K\) that tends to \(K\). We have \(|e^{{{\mathrm{cap}}}(K)} g_K(z_n)|\rightarrow e^{{{\mathrm{cap}}}(K)}\) and \(\limsup |z_n|\le \max _{z\in K}|z|\). The proof is completed by (6.2) \(\square \)

Let \(W\) be a conformal map, whose domain \(\Omega \) contains \(0\). Let \(K\) be a \({\mathbb {C}}\)-hull such that \(\{0\}\subsetneqq K\subset \Omega \). Let \(\Omega _K=g_K(\Omega {\setminus } K)\), and define \(W_K(z)=g_{W(K)}\circ W\circ g_K^{-1}(z)\) for \(z\in \Omega _K\). Now \(\Omega _K\) contains a neighborhood of \(\mathbb {T}\) in \(\mathbb {D}^*\), and as \(z\rightarrow \mathbb {T}\) in \(\Omega _K, W_K(z)\rightarrow \mathbb {T}\) as well. Let \(\Omega _K^\dagger =\Omega _K\cup \mathbb {T}\cup I_\mathbb {T}(\Omega _K)\). Schwarz reflection principle implies that \(W_K\) extends to a conformal map on \(\Omega _K^\dagger \) such that \(W_K(\mathbb {T})=\mathbb {T}\).

Lemma 6.3

There are real constants \(C_{0}<0\) and \(C_{1},C_{2}>0\) depending only on \(\Omega \) and \(W\) such that if \(K\) is a \({\mathbb {C}}\)-hull with \(\{0\}\subsetneqq K\) and satisfies \({{\mathrm{cap}}}(K)\le C_{1}\), then

$$\begin{aligned} |{{\mathrm{cap}}}(W(K))-{{\mathrm{cap}}}(K)-\log |W'(0)||\le C_{1} e^{\frac{1}{2}{{\mathrm{cap}}}(K)}; \end{aligned}$$
(6.3)
$$\begin{aligned} \log |W_K'(z)|\le C_{2} e^{\frac{1}{2}{{\mathrm{cap}}}(K)}/|{{\mathrm{cap}}}(K)|,\quad z\in \mathbb {T}. \end{aligned}$$
(6.4)

Proof

Since \(W(0)=0\) and \(W'(0)\ne 0\), there is \(V\) analytic in a neighborhood \(\Omega '\subset \Omega \) of \(0\) such that \(V(0)=0\) and \(W(z)=W'(0)ze^{V(z)}\) in \(\Omega '\). There exist positive constants \(C\ge 1\) and \(\delta \le \frac{1}{10}\) such that \(|z|\le \delta \) implies that \(z\in \Omega '\) and \(|V(z)|\le C|z|\). Thus,

$$\begin{aligned} |W(z)|\ge |W'(0)||z|e^{-C|z|},\quad |W(z)-W'(0)z|\le |W'(0)||z|(e^{C|z|}-1),\quad |z|\le \delta . \end{aligned}$$
(6.5)

Suppose \(K\) is a \({\mathbb {C}}\)-hull with \(\{0\}\subsetneqq K\), and satisfies \(e^{{{\mathrm{cap}}}(K)}\le \delta ^2\wedge \frac{1}{(320C)^2}\). From (6.2) we see that \(K\subset \{|z|\le 4\delta ^2\}\subset \{|z|\le \delta \}\subset \Omega \). So \(W(K)\) and \(W_K\) are well defined. Using (6.2) and the connectedness of \(K\), we may choose \(z_{0}\in K\) such that \(|z_{0}|=e^{{{\mathrm{cap}}}(K)}\). Using (6.5) we get

$$\begin{aligned} |W(z_{0})|\ge |W'(0)||z_{0}|e^{-C|z_{0}|}\ge |W'(0)|e^{{{\mathrm{cap}}}(K)}e^{-1/5}\ge \frac{4}{5} |W'(0)|e^{{{\mathrm{cap}}}(K)}. \end{aligned}$$

Since \(W(z_{0})\in W(K)\), using (6.2) again, we get \({{\mathrm{cap}}}(W(K))\ge \frac{1}{4}|W(z_{0})|\ge \frac{1}{5} |W'(0)|e^{{{\mathrm{cap}}}(K)}\). Let \(\alpha =\alpha _{W,K}=W'(0)e^{{{\mathrm{cap}}}(K)-{{\mathrm{cap}}}(W(K))}\). Then we have \(|\alpha |\le 5\).

Let \(R=\frac{1}{2}e^{-\frac{1}{2}{{\mathrm{cap}}}(K)}, z_{1}\in \{|z|=R\}\), and \(z_{2}=g_K^{-1}(z_{1})\). From Lemma 6.2, we get

$$\begin{aligned} |z_{2}-e^{{{\mathrm{cap}}}(K)} z_{1}|\le 5 e^{{{\mathrm{cap}}}(K)}. \end{aligned}$$

Since \(R\ge \frac{1}{2}(\delta ^2)^{-1/2}\ge 5\), we have

$$\begin{aligned} |z_{2}|\le (R+5)e^{{{\mathrm{cap}}}(K)}\le 2Re^{{{\mathrm{cap}}}(K)} =e^{\frac{1}{2}{{\mathrm{cap}}}(K)}\le \delta \wedge \frac{1}{360C}. \end{aligned}$$

Let \(J\) denote the Jordan curve \(g_K^{-1}(\{|z|=R\})\), and \(U_J\) denote its interior. Then \(J\subset \{|z|\le \delta \}\), which implies that \(U_J\subset \{|z|\le \delta \}\subset \Omega \). Since \(g_K^{-1}\) maps the annulus \(\{1<|z|\le R\}\) conformally onto \((J\cup U_J){\setminus } K\subset \Omega {\setminus } K\), we see that \(\{1<|z|\le R\}\subset \Omega _K\), and so \(\{1/R\le |z|\le R\}\subset \Omega _K^\dagger \). Let \(z_3=W(z_{2})\). Using (6.5) and \(0\le C|z_{2}|\le 1\), we get

$$\begin{aligned} |z_3-W'(0)z_{2}|\le |W'(0)||z_{2}|(e^{C|z_{2}|}-1)\le 2C |W'(0)||z_{2}|^2\le 2C|W'(0)| e^{{{\mathrm{cap}}}(K)}. \end{aligned}$$

Let \(z_4=g_{W(K)}(z_3)\). From Lemma 6.2 we get

$$\begin{aligned} |z_4-e^{-{{\mathrm{cap}}}(W(K))}z_3|\le 5. \end{aligned}$$

Combining the above four displayed formulas and that \(|\alpha |\le 5\), we get

$$\begin{aligned} |z_4-\alpha z_{1}|\le 5+2 C |\alpha |+5|\alpha |\le 30+10C\le 40C. \end{aligned}$$

Note that \(z_4=W_K(z_{1})\). So we get

$$\begin{aligned} |W_K(z)-\alpha z|\le 40C,\quad |z|&= R.\end{aligned}$$
(6.6)
$$\begin{aligned} |\alpha |R-40C\le |W_K(z)|\le |\alpha |R+40C,\quad |z|&= R. \end{aligned}$$
(6.7)

We may find \(R'>R\) such that \(A:=\{1/R'<|z|<R'\}\subset \Omega _K^\dagger \). Then \(W_K\) is analytic in \(A\). Since \(W_K\) is an orientation preserving auto homeomorphism of \(\mathbb {T}\), there is an analytic function \(V_K\) such that \(W_K(z)=e^{V_K(z)}z\) in \(A\). We have \(\hbox {Re}\,V_K(z)=\log |W_K(z)|-\log |z|\). Thus, \(\hbox {Re}\,V_K\equiv 0\) on \(\mathbb {T}\). Cauchy’s theorem implies that \(\oint _{|z|=1} \frac{V_K(z)}{z}dz=\oint _{|z|=R} \frac{V_K(z)}{z}dz\), which means that \(\int _{0}^{2\pi } V_K(e^{i\theta })d\theta =\int _{0}^{2\pi } V_K(Re^{i\theta })d\theta \). So we get

$$\begin{aligned} 0&= \int _{0}^{2\pi } \hbox {Re}\,V_K(e^{i\theta })d\theta =\int _{0}^{2\pi } \hbox {Re}\,V_K(Re^{i\theta })d\theta \\ {}&= \int _{0}^{2\pi } (\log |W_K(Re^{i\theta })|-\log R)d\theta . \end{aligned}$$

Using (6.7), we get \( |\alpha |R-40C\le R\le |\alpha |R+40C\), which implies that \(|1-|\alpha ||\le \frac{40C}{R}\). This implies (6.3) since \(\log |\alpha |=\log |W'(0)|+{{\mathrm{cap}}}(K)-{{\mathrm{cap}}}(W(K))\) and \(1/R=O(e^{\frac{1}{2}{{\mathrm{cap}}}(K)})\).

Let \(|z|=R\). From (6.6), we get \(|e^{V_K(z)}-\alpha |\le \frac{40C}{R}\). Since \(|\alpha |\ge 1-\frac{40C}{R}\), we have \(|e^{V_K(z)}|\ge 1-\frac{80C}{R}\ge \frac{1}{2}\) as \(R\ge 160C\). So there exists \(\widetilde{\alpha }\in {\mathbb {C}}\) with \(\alpha =e^{\widetilde{\alpha }}\) such that \(|V_K(z)-\widetilde{\alpha }|\le 2|e^{V_K(z)}-\alpha |\le \frac{80C}{R}\). From \(||\alpha |-1|\le \frac{40C}{R}\), we get \(|\hbox {Re}\,\widetilde{\alpha }|=|\log |\alpha ||\le \frac{80C}{R}\). Thus, \(|V_K(z)-i\hbox {Im}\,\widetilde{\alpha }|\le \frac{160C}{R}\) if \(|z|=R\). Let \(\widetilde{V}_K=V_K\circ \exp \). Then \(\widetilde{V}_K\) is analytic in the vertical strip \(\widetilde{A}:=\exp ^{-1}(A)=\{-\log R'<\hbox {Re}\,z<\log R'\}\), and is pure imaginary on \(i\mathbb {R}\). Thus, \(\widetilde{V}_K(-\overline{z})=-\overline{\widetilde{V}_K(z)}\). This implies that, on the two vertical lines \(\{\hbox {Re}\,z=\log R\}\) and \(\{\hbox {Re}\,z=-\log R\}, |\widetilde{V}_K(z)-i\hbox {Im}\,\widetilde{\alpha }|\le \frac{160C}{R}\). Since \(\widetilde{V}_K\) has period \(2\pi i\), the inequality holds in the strip \(\{-\log R\le \hbox {Re}\,z\le \log R\}\). We may apply Cauchy’s integral formula, and get \(|\widetilde{V}_K'(z)|\le \frac{160C}{R\log R}\) for \(z\in i\mathbb {R}\). Since \(\widetilde{V}_K(z)=V_K\circ \exp , e^{V_K(z)}=\frac{W_K(z)}{z}\) and \(W_K(\mathbb {T})=\mathbb {T}\), we get

$$\begin{aligned} \Big |W_K'(z)-\frac{W_K(z)}{z} \Big |=|\widetilde{V}_K'(\log z)|\le \frac{160C}{R\log R},\quad z\in \mathbb {T}. \end{aligned}$$

This implies (6.4) since \(\log R\ge |{{\mathrm{cap}}}(K)|/4\) and \(1/R=O(e^{\frac{1}{2}{{\mathrm{cap}}}(K)})\). \(\square \)

Now suppose \(\gamma (t), -\infty \le t<T\), is a simple whole-plane Loewner trace driven by \(\lambda \in C((-\infty ,T))\). Let \(\Omega \) be a domain that contains \(\gamma \). Let \(W\) be a conformal map defined on \(\Omega \) such that \(W(0)=0\). Let \(\beta (t)=W(\gamma (t)), -\infty \le t<T\). Define \(v\) on \([-\infty ,T)\) such that \(v(-\infty )=-\infty \) and \(v(t)={{\mathrm{cap}}}(\beta ([-\infty ,t]))\) for \(-\infty <t<T\). Let \(\widetilde{T}=v(T)\) and \(\widetilde{\gamma }(t)=\beta (v^{-1}(t)), -\infty \le t<\widetilde{T}\). Then \(\widetilde{\gamma }\) is a simple whole-plane Loewner trace, say driven by \(\widetilde{\lambda }\in C((-\infty ,\widetilde{T}))\). Let \((g_{t})\) and \((\widetilde{g}_{t})\) be the whole-plane Loewner maps driven by \(\lambda \) and \(\widetilde{\lambda }\), respectively. Then, \(g_{t}^{-1}(e^{i\lambda (t)})=\gamma (t)\) and \(\widetilde{g}_{t}^{-1}(e^{i\widetilde{\lambda }(t)})=\widetilde{\gamma }(t)\). Let \(z(t)\) and \(\widetilde{z}(t)\) be such that \(g_{t}^{-1}(z(t))=0\) and \(\widetilde{g}_{t}^{-1}(\widetilde{z}(t))=0\). Choose \(q\in C((-\infty ,T))\) and \(\widetilde{q}\in C((-\infty ,\widetilde{T}))\) such that \(z(t)=e^{iq(t)}, \widetilde{z}(t)=e^{i\widetilde{q}(t)}, \lambda (t)-q(t)\in (0,2\pi )\), and \(\widetilde{\lambda }(t)-\widetilde{q}(t)\in (0,2\pi )\). Let \(Z=\lambda -q\) and \(\widetilde{Z}=\widetilde{\lambda }-\widetilde{q}\).

Let \(K_{t}=\gamma ([-\infty ,t])\) and \(\widetilde{K}_{t}=\widetilde{\gamma }([-\infty ,t])\). Recall that \(g_{t}=g_{K_{t}}\) and \(\widetilde{g}_{t}=g_{\widetilde{K}_{t}}\). For \(-\infty < t<T\), let \(\Omega _{t}=\Omega _{K_{t}}, \Omega ^\dagger _{t}=\Omega ^\dagger _{K_{t}}\), and \(W_{t}=W_{K_{t}}\). Then \(W_{t}\) is a conformal map defined on \(\Omega ^\dagger _{t}\supset \mathbb {T}\) such that \(W_{t}(\mathbb {T})=\mathbb {T}\). Since \(W(K_{t})=\widetilde{K}_{v(t)}\), we have \(W_{t}=\widetilde{g}_{v(t)}\circ W\circ g_{t}^{-1}\) in \(\Omega _{t}\). Since \(g_{t}^{-1}(e^{i\lambda (t)})=\gamma (t)\) and \(\widetilde{g}_{v(t)}^{-1}(e^{i\widetilde{\lambda }(v(t))})=\widetilde{\gamma }(v(t))\) when both \(g_{t}^{-1}\) and \(\widetilde{g}_{v(t)}\) extends continuously to \(\mathbb {D}^*\cup \mathbb {T}\), and \(W(\gamma (t))=\widetilde{\gamma }(v(t))\), we get \(W_{t}(e^{i\lambda (t)})=e^{i\widetilde{\lambda }(v(t))}\). Similarly, since \(g_{t}^{-1}(e^{iq(t)})=0=\widetilde{g}_{v(t)}^{-1}(e^{i\widetilde{q}(v(t))})\) and \(W(0)=0\), we have \(W_{t}(e^{i q(t)})=e^{i\widetilde{q}(v(t))}\). Thus, we get

$$\begin{aligned} \widetilde{Z}(v(t))= \widetilde{\lambda }(v(t))-\widetilde{q}(v(t))=\int _{q(t)}^{\lambda (t)} |W_{t}'(e^{is})|ds. \end{aligned}$$
(6.8)

The following lemma is well known. For the proof, one may apply, e.g., Proposition 4.4(ii) in [13]. We now omit the details.

Lemma 6.4

For any \(t\in (-\infty ,T), v'(t)=|W_{t}'(e^{i\lambda (t)})|^2\).

Applying Lemma 6.3 to \(K=\gamma ([-\infty ,t])\) and using (6.8) and Lemma 6.4, we get

$$\begin{aligned} \lim _{t\rightarrow -\infty } |\widetilde{Z}(v(t))-Z(t)|=0,\quad \lim _{t\rightarrow -\infty } v'(t)=1,\quad \lim _{t\rightarrow -\infty } v(t)-t=\log |W'(0)|. \end{aligned}$$
(6.9)

Lemma 6.1 implies that, if \(f\) is continuous on \([0,2\pi ]\), then

$$\begin{aligned} \lim _{t\rightarrow -\infty } \frac{1}{t_{0}-t} \int _{t}^{t_{0}} f(Z(s))ds=\lim _{t\rightarrow -\infty } \frac{1}{t_{0}-t} \int _{t}^{t_{0}} f(\widetilde{Z}(s))ds,\quad t_{0}\in (-\infty , T\wedge \widetilde{T}), \end{aligned}$$

if either limit exists. Using (6.1) we obtain the following proposition.

Proposition 6.5

Let \(\kappa \le 4\) and \(\rho \ge \frac{\kappa }{2}-2\). Let \(\gamma (t), -\infty \le t<\infty \), be a whole-plane \(\hbox {SLE}(\kappa ;\rho )\) trace. Suppose that \(W\) is a random conformal map with (random) domain \(\Omega \ni 0\) such that \(W(0)=0\). Let \(T\) be such that \(\gamma ([-\infty ,T))\subset \Omega \). Let \(\widetilde{\gamma }\) be a reparametrization of \(W(\gamma (t)), -\infty \le t<T\), such that \(\widetilde{\gamma }(-\infty )=0\) and \({{\mathrm{cap}}}(\widetilde{\gamma }([-\infty ,t]))=t\) for \(-\infty <t<\widetilde{T}\). Let \(h(t)\in (0,1)\) denote the harmonic measure of the right side of \(\widetilde{\gamma }([-\infty ,t])\) in \(\widehat{\mathbb {C}}{\setminus } \widetilde{\gamma }([-\infty ,t])\) viewed from \(\infty \). Then for any \(f\in C([0,2\pi ])\) and \(t_{0}\in (-\infty ,\widetilde{T})\), almost surely

$$\begin{aligned} \lim _{t\rightarrow -\infty } \frac{1}{t_{0}\!-\!t} \int _{t}^{t_{0}} f(2\pi h(s))ds=\int _{0}^{2\pi } f(x)d\mu _{\kappa ;\rho }(x) \!=\!\frac{\int _{0}^{2\pi }f(x)\sin _{2}(x)^{\frac{4}{\kappa }(\frac{\rho }{2}+1)}dx }{\int _{0}^{2\pi }\sin _{2}(x)^{\frac{4}{\kappa }(\frac{\rho }{2}+1)}dx }. \end{aligned}$$

Combining the above proposition with Theorems 5.3 and 5.4, we obtain the following theorem.

Theorem 6.6

Let \(\kappa \in (0,4)\) and \(t_{0}\in (0,\infty )\). Let \(\beta \) be a chordal or radial \(\hbox {SLE}_\kappa \) trace. For \(0\le t< t_{0}\), let \(v(t)={{\mathrm{cap}}}(\beta ([t,t_{0}]))\) and \(h(t)\) be the harmonic measure of the left side of \(\beta ([t,t_{0}])\) in \(\widehat{\mathbb {C}}{\setminus } \beta ([t,t_{0}])\) viewed from \(\infty \). Then for any \(f\in C([0,1])\), almost surely

$$\begin{aligned} \lim _{t\rightarrow t_{0}^-} \frac{1}{v(t)-v(0)} \int _{0}^t f(h(s))dv(s) =\frac{\int _{0}^{2\pi }f(x)\sin _{2}(x)^{\frac{8}{\kappa }+2}dx }{\int _{0}^{2\pi }\sin _{2}(x)^{\frac{8}{\kappa }+2}dx }. \end{aligned}$$

Remark

  1. 1.

    We can now conclude that Theorem 5.1 does not hold with \(\kappa +2\) replaced by any other \(\rho \ge \frac{\kappa }{2}-2\). If this is not true, then Theorem 5.4 also holds with \(\kappa +2\) replaced by such \(\rho \). Then Theorem 6.6 holds in the radial case with the exponent \(\frac{8}{\kappa }+2\) replaced by \(\frac{4}{\kappa }(\frac{\rho }{2}+1)\), which is obviously impossible.

  2. 2.

    Fubini’s Theorem implies that Theorem 6.6 still holds if the deterministic number \(t_{0}\) is replaced by a positive random number \(\overline{t}_{0}\), whose distribution given \(\beta \) is absolutely continuous with respect to the Lebesgue measure. We do not expect that the theorem holds if the conditional distribution of \(\overline{t}_{0}\) does not have a density. In fact, if the conditional distribution of \(\overline{t}_{0}\) is absolutely continuous with respect to the natural parametrization introduced by Lawler and Sheffield [7], then we expect that \(\beta \) behaves like a two-sided radial \(\hbox {SLE}_\kappa \) process, which is a radial \(\hbox {SLE}(\kappa ;2)\) process, near \(\beta (\overline{t}_{0})\), and Theorem 6.6 is expected to hold with \(\frac{8}{\kappa }+2\) replaced by \(\frac{8}{\kappa }\).

Let \(\kappa \in (0,4]\). A whole-plane \(\hbox {SLE}(\kappa ;\rho )\) trace \(\gamma \) generates a simple curve. Combining the reversibility property derived in [18] with the Markov-type relation between whole-plane \(\hbox {SLE}_\kappa \) and radial \(\hbox {SLE}_\kappa \) processes, we see that, if \(\beta \) is a radial \(\hbox {SLE}_\kappa \), there is a conformal map \(V\) defined on \(\mathbb {D}\) with \(V(0)=0\), which maps \(\beta \) to an initial segment of a whole-plane \(\hbox {SLE}_\kappa \) trace. Applying Proposition 6.5, we obtain the following.

Theorem 6.7

Let \(\kappa \in (0,4]\). Let \(\beta \) be a radial \(\hbox {SLE}_\kappa \) trace. For \(0\le t<\infty \), let \(v(t)={{\mathrm{cap}}}(\beta ([t,\infty ]))\) and \(h(t)\) be the harmonic measure of the left side of \(\beta ([t,\infty ])\) in \(\widehat{\mathbb {C}}{\setminus } \beta ([t,\infty ])\) viewed from \(\infty \). Then for any \(f\in C([0,2\pi ])\), almost surely

$$\begin{aligned} \lim _{t\rightarrow \infty } \frac{1}{v(t)-v(0)} \int _{0}^t f(h(s))dv(s) =\frac{\int _{0}^{2\pi }f(x)\sin _{2}(x)^{\frac{4}{\kappa }}dx }{\int _{0}^{2\pi }\sin _{2}(x)^{\frac{4}{\kappa }}dx }. \end{aligned}$$