Reformulation of Laplacian-$b$ motion in terms of stochastic Komatu-Loewner evolution in the chordal case

TL;DR

This paper establishes a connection between Laplacian-$b$ motion and stochastic Komatu-Loewner evolution (SKLE) in multiply connected domains, showing they are equivalent under certain conditions and analyzing their finite-time explosion behavior.

Contribution

It reformulates Laplacian-$b$ motion in terms of SKLE, providing new insights into their relationship and demonstrating finite-time blow-up for the case corresponding to SLE_6.

Findings

SKLE with a specific stochastic differential equation matches Laplacian-$b$ motion after a time change

Finite time explosion occurs for SKLE related to Laplacian-$0$ motion (SLE_6)

Solution to the Komatu-Loewner equation for SLE_6 blows up in finite time

Abstract

We investigate the relation between the Laplacian-$b$ motion and stochastic Komatu-Loewner evolution (SKLE) on multiply connected subdomains of the upper half-plane, both of which are analogues to SLE. In particular, we show that, if the driving function of an SKLE is given by a certain stochastic differential equation, then this SKLE is the same as a time-changed Laplacian-$b$ motion. As an application, we prove the finite time explosion of SKLE corresponding to Laplacian-$0$ motion, or $\mathrm{SLE_6}$, in the sense that the solution to the Komatu-Loewner equation for the slits blows up.

Equations

$$\mathbf{s}$$

$$\in (0,\infty {)}^N\times {\mathbb{R}}^{2N}$$

$$b_l({\xi }_0,\mathbf{s})=\{\begin{array}{ll}-2\pi \Im {\Psi }_{D(\mathbf{s})}(z_l,{\xi }_0)& (1\le l\le N)\\ -2\pi \Re {\Psi }_{D(\mathbf{s})}(z_{l-N},{\xi }_0)& (N+1\le l\le 2N)\\ -2\pi \Re {\Psi }_{D(\mathbf{s})}(z_{l-2N}^r,{\xi }_0)& (2N+1\le l\le 3N),\end{array}$$

$$\frac{d}{dt}{\mathbf{s}}_l(t)$$

$$\frac{d}{dt}{g}_t(z)$$

$$\bigcap _{\delta >0}\overline{g_t(F_{t+\delta }\setminus F_t)}=\{\xi (t)\}.$$

$${\Psi }_{\mathbb{H}}(z,{\xi }_0)=-\frac{1}{\pi }\frac{1}{z-{\xi }_0},$$

$$d\xi (t)=\alpha (\mathbf{s}(t)-\hat{\xi }(t))d{B}_t+\beta (\mathbf{s}(t)-\hat{\xi }(t))dt.$$

$$\hat{P}(t,z,dw):=\frac{\Im w}{\Im z}{P}^{\mathbb{H}}(t,z,dw),t\ge 0,z\in \mathbb{H}.$$

$$Q(z,D^{\prime }):={\hat{\mathbb{P}}}_z\left(\hat{Z}(0,\infty )\subset D^{\prime }\right),z\in D^{\prime }\cup \partial \mathbb{H}.$$

$$Q(z,D^{\prime })=1-{\hat{\mathbb{P}}}_z\left({\hat{\tau }}_{D^{\prime }}<\infty \right)=1-\frac{{\mathbb{E}}_z^{\mathbb{H}}\left[\Im Z_{{\tau }_{D^{\prime }}}^{\mathbb{H}};{\tau }_{D^{\prime }}<\infty \right]}{\Im z},z\in D^{\prime },$$

$$X_t:=\frac{6-\kappa }{2\sqrt{\kappa }}\frac{\partial \log Q}{\partial {\xi }_0}(U(t),D_t^{\prime })1_{\{t<T_{D^{\prime }}\}}\text{and}{X}_t^{(n)}:=X_t{1}_{\{t<T_n\}},$$

$$M_t^{(n)}:=\exp \left({\int }_0^t{X}_s^{(n)}d{B}_s-\frac{1}{2}{\int }_0^t|X_s^{(n)}{|}^{2}ds\right)$$

$$M_{t\wedge S_{n,m}}^{(n)}=\exp \left({\int }_0^{t\wedge S_{n,m}}{X}_{s}d{B}_s-\frac{1}{2}{\int }_0^{t\wedge S_{n,m}}|X_s{|}^{2}ds\right)$$

$${\mathbf{Q}}^{(n,m)}(A):={\mathbf{E}}^{\mathbf{P}}\left[1_A\underset{t\to \infty }{\lim }{M}_{t\wedge S_{n,m}}^{(n)}\right],A\in {\mathcal{F}}_{\infty },$$

$$B_t^{(n,m)}:=B_t-{\int }_0^{t\wedge S_{n,m}}{X}_{s}ds,t\in [0,\infty ),$$

$$dU(t)=\frac{6-\kappa }{2}\frac{\partial \log Q}{\partial {\xi }_0}(U(t),D_t^{\prime })dt+\sqrt{\kappa }d{B}_t^{(n,m)},0\le t\le S_{n,m}.$$

$$d\xi (t)$$

$$\displaystyle\phantom{=}{}\biggl{.}-h_{t}^{\prime}(U(t))^{2}b_{\mathrm{BMD}}(\xi(t),\mathbf{s}(t))\biggr{\}}\,dt+\sqrt{\kappa}h_{t}^{\prime}(U(t))\,dB^{(n,m)}_{t}$$

$$\frac{\partial \log Q}{\partial {\xi }_0}(U(t),D_t^{\prime })=h_t^{\prime }(U(t))\frac{\partial \log Q}{\partial {\xi }_0}(\xi (t),D_t)+\frac{h_t^{\prime \prime }(U(t))}{h_t^{\prime }(U(t))}.$$

$$d\xi (t)$$

$$+\sqrt{\kappa }{h}_t^{\prime }(U(t))d{B}_t^{(n,m)},t\le S_{n,m},$$

$$d\check{\xi }(t)$$

$$+\sqrt{\kappa }d{\check{B}}_t^{(n,m)},t<{\check{S}}_{n,m},$$

$$\alpha (\mathbf{s}):=\sqrt{\kappa }\text{and}\beta (\mathbf{s}):=\frac{6-\kappa }{2}\frac{\partial \log Q}{\partial {\xi }_0}(0,\mathbf{s})-b_{\mathrm{BMD}}(0,\mathbf{s}),$$

$$Q({\xi }_0,\mathbf{s})$$

$$=1+\frac{\partial }{\partial {\mathbf{n}}_{{\xi }_0}}\sum _{j=1}^N{\int }_{{\partial }_p{C}_j}\Im w{K}_D({\xi }_0,w)|dw|$$

$$=1+\sum _{j=1}^N{y}_j\frac{\partial }{\partial {\mathbf{n}}_{{\xi }_0}}{\phi }_{\mathbf{s}}^{(j)}({\xi }_0).$$

$$\frac{\partial \log Q}{\partial {\xi }_0}({\xi }_0,\mathbf{s})={\left(1+\sum _{j=1}^N{y}_j\frac{\partial }{\partial {\mathbf{n}}_{{\xi }_0}}{\phi }_{\mathbf{s}}^{(j)}({\xi }_0)\right)}^{-1}\sum _{j=1}^N{y}_j\frac{\partial }{\partial {\xi }_0}\frac{\partial }{\partial {\mathbf{n}}_{{\xi }_0}}{\phi }_{\mathbf{s}}^{(j)}({\xi }_0).$$

$$R({\xi }_0,\mathbf{s}):=\underset{1\le j\le N}{\min }\mathrm{dist}(C_j(\mathbf{s}),{\xi }_0)$$

$$Q(z,D)={\hat{\mathbb{E}}}_z\left[Q({\hat{Z}}_{{\hat{\sigma }}_{{\mathbb{D}}_{+}}},D)\right]={\int }_0^{\pi }Q(e^{i\theta },D)K_{{\mathbb{D}}_{+}}(z,e^{i\theta })\frac{\sin \theta }{\Im z}d\theta$$

$$Q({\xi }_0,\mathbf{s})=-{\int }_0^{\pi }Q(e^{i\theta },D)\frac{\partial }{\partial {\mathbf{n}}_{{\xi }_0}}{K}_{{\mathbb{D}}_{+}}({\xi }_0,e^{i\theta })\sin \theta d\theta .$$

$$\frac{\partial \log Q}{\partial {\xi }_0}({\xi }_0,\mathbf{s})=\frac{\frac{\partial }{\partial {\xi }_0}Q({\xi }_0,\mathbf{s})}{Q({\xi }_0,\mathbf{s})}=\frac{{\int }_0^{\pi }Q(e^{i\theta },D)\frac{\partial }{\partial {\xi }_0}\frac{\partial }{\partial {\mathbf{n}}_{{\xi }_0}}{K}_{{\mathbb{D}}_{+}}({\xi }_0,e^{i\theta })\sin \theta d\theta }{{\int }_0^{\pi }Q(e^{i\theta },D)\frac{\partial }{\partial {\xi }_0}{K}_{{\mathbb{D}}_{+}}({\xi }_0,e^{i\theta })\sin \theta d\theta }.$$

$$\frac{\partial \log Q}{\partial {\xi }_0}(0,\mathbf{s})=\frac{2{\int }_0^{\pi }Q(e^{i\theta },D){\sin }^2\theta \cos \theta d\theta }{{\int }_0^{\pi }Q(e^{i\theta },D){\sin }^2\theta d\theta }.$$

Full text

1. Introduction

Ever since Schramm [14] introduced the stochastic Loewner evolution with parameter $\kappa>0$ (abbreviated as $\mathrm{SLE}_{\kappa}$), numerous studies have been conducted for identifying the scaling limits of several two-dimensional lattice models in statistical physics, such as loop-erased random walk, percolation exploration process and critical Ising interface. Most of these results are established on simply connected planar domains such as the upper half-plane $\mathbb{H}=\{z\in\mathbb{C};\Im z>0\}$, since the definition of $\mathrm{SLE}_{\kappa}$ depends on the theory of conformal maps on simply connected domains, especially on the chordal Loewner equation. It is thus a non-trivial problem to extend $\mathrm{SLE}_{\kappa}$ to multiply connected domains, and the way of extension is not unique. For example, Lawler [10] introduced the Laplacian-$b$ motion $\mathrm{LM}_{b}$ with $b=(6-\kappa)/(2\kappa)$ 111Zhan [15] considered a similar object independently, where he called it the harmonic random Loewner chain and stood in a viewpoint different from that of Lawler. as a candidate of the scaling limit of Laplacian-$b$ random walk. Its definition is based on the chordal Loewner equation and the Maruyama–Girsanov transform of the driving process associated with $\mathrm{SLE}_{\kappa}$. On the other hand, Chen and Fukushima [4] introduced the stochastic Komatu–Loewner evolution $\mathrm{SKLE}_{\alpha,\beta}$ as a process satisfying the domain Markov property and invariance under linear conformal maps, both of which are typical properties of $\mathrm{SLE}_{\kappa}$. (Here, we use $\beta$ instead of $b$ in [4, Eq. (3.32)] to avoid a conflict with the exponent $b$ above.) Its definition is motivated by Bauer and Friedrich [1, 2, 3] and based on the chordal Komatu–Loewner equations.

In this paper, we investigate the relationship between the two processes above by means of Chen, Fukushima and Suzuki [6] and the author [11]. After the definitions of $\mathrm{SKLE}_{\alpha,\beta}$ and $\mathrm{LM}_{b}$ are reviewed in Sections 2.1 and 2.2, respectively, we show in Section 3.1 that $\mathrm{LM}_{b}$ can be regarded as $\mathrm{SKLE}_{\alpha,\beta}$ with appropriate $\alpha$ and $\beta$ modulo time-change. In particular, the relation between $\mathrm{SKLE}_{\alpha,\beta}$ and $\mathrm{LM}_{b}$ becomes rather simple in the case $b=0$ ($\kappa=6$). By using this identification, we prove in Section 3.2 that the finite time explosion of the solution to the Komatu–Loewner equation for the slits [4, Eq. (3.32) and (3.33)] corresponding to $\mathrm{LM}_{0}$. This result provides a non-trivial example that satisfies the assumption of [12, Theorem 3.2]. We note that, although each of the results in Section 3 is simple, they together illustrate a possible way to apply SKLE to the theory of SLE on multiply connected domains.

2. Preliminaries

2.1. Stochastic Komatu–Loewner evolution

Let us recall the notation in [4, 11]. Let $N$ be a positive integer.

• •

$\mathsf{Slit}$ is the set of all elements

[TABLE]

with $x_{j}<x^{r}_{j}$ for each $j=1,\ldots,N$ such that either $x^{r}_{j}<x_{k}$ or $x^{r}_{k}<x_{j}$ holds if $y_{j}=y_{k}$ for two distinct numbers $j,k\in\{1,\ldots,N\}$.

• •

$C_{j}(\mathbf{s})$ is the segment whose endpoints are $z_{j}=x_{j}+iy_{j}$ and $z^{r}_{j}=x^{r}_{j}+iy_{j}$ for $\mathbf{s}\in\mathsf{Slit}$ and $1\leq j\leq N$.

• •

$D(\mathbf{s})$ is the standard slit domain $\mathbb{H}\setminus\bigcup_{j=1}^{N}C_{j}(\mathbf{s})$ for $\mathbf{s}\in\mathsf{Slit}$.

• •

$b_{l}\colon\mathbb{R}\times\mathsf{Slit}\to\mathbb{R}$, $1\leq l\leq 3N$, are the functions defined by

[TABLE]

where $\Psi_{D}(\mathbf{s})$ is the complex Poisson kernel of Brownian motion with darning (BMD) for $D(\mathbf{s})$ [5, Lemma 4.1].

We also recall that a set $F\subset\mathbb{H}$ is called a (compact $\mathbb{H}$-)hull if $F$ is bounded and relatively closed in $\mathbb{H}$ and if $\mathbb{H}\setminus F$ is still simply connected. For any hull $F$ in a standard slit domain $D$, there exists a unique conformal map $f_{F}$ from $D\setminus F$ onto another standard slit domain $\tilde{D}$, which is called the canonical map, such that the hydrodynamic normalization at infinity $f_{F}(z)=z+a/z+o(z^{-1})$ as $z\to\infty$ holds. The positive constant $\operatorname{hcap}^{D}(F):=a$ is called the (BMD) half-plane capacity relative to $D$. See [11, Proposition 2.3].

Now, fix $t_{0}\in(0,\infty]$. Suppose that $a_{t}$ is a strictly increasing differentiable function of $t\in[0,t_{0})$ with $a_{0}=0$ and that $\xi(t)$ be an $\mathbb{R}$-valued continuous function on the same interval. We consider the following ordinary differential equations:

[TABLE]

Here, the dot on $a_{t}$ stands for the $t$-derivative. We call (2.2) the chordal Komatu–Loewner equation [3, 5] and (2.1) the Komatu–Loewner equation for the slits [3, 4]. We use the symbol $\zeta$ to denote the explosion time of the solution $\mathbf{s}(t)$ to (2.1). Moreover, we define $F_{t}:=\{z\in D;t_{z}\leq t\}$ for $t\in[0,\zeta)$, where $t_{z}:=\zeta\wedge\sup\{t>0;\lvert g_{t}(z)-\xi(t)\rvert>0\}$ for $z\in D=D(\mathbf{s}(0))$. Then $\{F_{t}\}_{t\in[0,\zeta)}$ is a family of continuously growing hulls [11, Definition 4.2] with $\operatorname{hcap}^{D}(F_{t})=a_{t}$, $g_{t}$ is the canonical map from $D\setminus F_{t}$ onto $D(\mathbf{s}(t))$ for each $t\in[0,\zeta)$, and it follows that

[TABLE]

We call a function $\xi(t)$ satisfying (2.3) the driving function of $\{F_{t}\}$. See [4, Section 5] for the proof of these facts. We remark that they are valid also in the case $N=0$, except that (2.1) does not appear. Hence we may put $\zeta=\infty$. In this case, the complex Poisson kernel is

[TABLE]

and (2.2) is called the chordal Loewner equation.

Let $\alpha$ be a non-negative function on $\mathsf{Slit}$ homogeneous with degree 0. Here, a function $f(\mathbf{s})$ is said to be homogeneous with degree $\delta\in\mathbb{R}$ if $f(c\mathbf{s})=c^{\delta}f(\mathbf{s})$ holds for all $c>0$ and $\mathbf{s}\in\mathsf{Slit}$. Let $\beta$ be a function on $\mathsf{Slit}$ homogeneous with degree $-1$. We further suppose that both $\alpha$ and $\beta$ are locally Lipschitz continuous. $\mathrm{SKLE}_{\alpha,\beta}$ on a standard slit domain $D$ [4, Section 5] is defined as the random continuously growing hulls $F_{t}$ in $D$ that are obtained via the procedure above with $a_{t}=2t$ and $\xi(t)$ given by the following stochastic differential equation (SDE)

[TABLE]

Here, $\widehat{\xi}(t)$ stands for the $3N$-dimensional vector whose first $N$ entries are zero and last $2N$ entries are $\xi(t)$. In this case, we should regard (2.1) and (2.4) together as a system of SDEs, and $\zeta$ above is replaced by the explosion time of the solution $(\xi(t),\mathbf{s}(t))$ to this system. $\mathrm{SLE}_{\kappa}$ is a special case of this definition where $N=0$, $\alpha=\sqrt{\kappa}$ and $\beta=0$.

Since it is sometimes convenient to regard $\alpha$ and $\beta$ in (2.4) as functions on $\mathbb{R}\times\mathsf{Slit}$, we introduce the notation $f(\xi_{0},\mathbf{s}):=f(\mathbf{s}-\widehat{\xi_{0}})$ for a function $f$ on $\mathsf{Slit}$. The function $f(\xi_{0},\mathbf{s})$ so defined has the invariance under horizontal translation $f(\xi_{0},\mathbf{s})=f(0,\mathbf{s}-\widehat{\xi_{0}})$. Conversely, we define $\tilde{f}(\mathbf{s}):=\tilde{f}(0,\mathbf{s})$ if a function $\tilde{f}$ on $\mathbb{R}\times\mathsf{Slit}$ has this invariance.

2.2. Laplacian-$b$ motion

Let $Z^{\mathbb{H}}=(Z^{\mathbb{H}}_{t},\mathbb{P}^{\mathbb{H}}_{z_{0}})$ be an absorbing Brownian motion in $\mathbb{H}$ starting at $z_{0}\in\mathbb{H}$ and $P^{\mathbb{H}}(t,z,dw)$ be the transition probability of $Z^{\mathbb{H}}$. Doob’s $h$-transform $\hat{Z}=(\hat{Z}_{t},\hat{\mathbb{P}}_{z_{0}})$ with harmonic function $h(z)=\Im z$ is called an $\mathbb{H}$-excursion ([9, Section 5.3], [10, Section 3.5]). In other words, $\hat{Z}$ is a strong Markov process with transition probability

[TABLE]

(See [7] or [8] for a general definition of $h$-transform.) We can observe from (2.5) that $\Re\hat{Z}$ and $\Im\hat{Z}$ are independent, that $\Re\hat{Z}$ is just a standard Brownian motion and that $\Im\hat{Z}$ is the $h$-transform of a standard Brownian motion with $h(x)=x$. By Chapter VI, Section 3 of [13], it is a three-dimensional Bessel process. Therefore, we can define an $\mathbb{H}$-excursion $\hat{Z}$ starting at a boundary point $z_{0}\in\partial\mathbb{H}$ as well, and the lifetime of $\hat{Z}$ is infinite $\hat{\mathbb{P}}_{z_{0}}$-almost surely.

Let $D^{\prime}:=\mathbb{H}\setminus\bigcup_{j=1}^{N}A_{j}$, where $A_{j}\subset\mathbb{H}$, $1\leq j\leq N$, are mutually disjoint, compact continua with smooth boundaries. Put

[TABLE]

It then follows from (2.5) that

[TABLE]

where $\hat{\tau}_{D^{\prime}}$ and $\tau_{D^{\prime}}$ are the exiting times from $D^{\prime}$ of $\hat{Z}$ and $Z^{\mathbb{H}}$, respectively. $Q(\xi_{0},D^{\prime})$ for $\xi_{0}\in\partial\mathbb{H}$ is the limit of (2.6) as $z\to\xi_{0}$. We see later that $Q(\xi_{0},D^{\prime})$ is a differentiable function of $\xi_{0}\in\mathbb{R}$. In particular, we can consider $(\partial\log Q/\partial\xi_{0})(\xi_{0},D^{\prime})$.

Based on the function $Q$, the construction of $\mathrm{LM}_{b}$ for $b>-1/2$ is done as follows: Let $\kappa:=6/(2b-1)$ and $(g^{0}_{t},F^{0}_{t})_{t\geq 0}$ be an $\mathrm{SLE}_{\kappa}$ driven by $U(t)=\sqrt{\kappa}B_{t}$, where $B=(B_{t})_{t\geq 0}$ is a standard Brownian motion on a filtered probability space $(\Omega,\mathcal{F}_{\infty},(\mathcal{F}_{t})_{t\geq 0},\mathbf{P})$ with the usual conditions. We put $T_{D^{\prime}}:=\inf\{t>0;F^{0}_{t}\nsubseteq D^{\prime}\}$ and $D^{\prime}_{t}:=g^{0}_{t}(D^{\prime}\setminus F^{0}_{t})$, and define the processes $X$ and $X^{(n)}$ for $n\in\mathbb{N}$ by

[TABLE]

where $T_{n}:=T_{D^{\prime}}\wedge\inf\{t>0;\lvert X_{t}\rvert\geq n\}$. We further define the exponential local martingale

[TABLE]

and the stopping times $S_{n,m}:=T_{n}\wedge\inf\{t>0;\lvert M^{(n)}_{t}\rvert\geq m\}$, $m\in\mathbb{N}$. The stopped local martingale

[TABLE]

is then bounded and thus a uniformly integrable martingale. If we define the measure $\mathbf{Q}^{(n,m)}$ on $\mathcal{F}_{\infty}$ by

[TABLE]

then by the Maruyama–Girsanov theorem

[TABLE]

is a standard Brownian motion under the measure $\mathbf{Q}^{(n,m)}$. In other words, the driving function $U(t)$ of $\{F^{0}_{t}\}$ satisfies the following SDE under $\mathbf{Q}^{(n,m)}$:

[TABLE]

(The law of) $\{F^{0}_{t}\}_{0\leq t\leq S_{n,m}}$ under $\mathbf{Q}^{(n,m)}$ is regarded as $\mathrm{LM}_{b}$ stopped by $S_{n,m}$. Motivated by (2.7), Lawler [10] defined $\mathrm{LM}_{b}$ as the random Loewner evolution driven by a solution to (2.7) with $B^{(n,m)}$ replaced by a Brownian motion independent of $n$ and $m$.

Remark 2.1**.**

The exponent $b$ implicitly appears in (2.7) in the sense that $(6-\kappa)/2=\kappa b$. Since the chordal Loewner equation that we consider in this paper is the linear time-change of [10, Eq. (4.16)], the SDE (2.7) is the time-change of the original one given in [10, Section 4.4]. The exponential martingale $M^{(n)}_{t}$ is originally obtained in a different fashion as well.

In this article, we only refer to Lawler [10, Section 4] and do not mention the reason why the random evolution defined as above is a candidate of the scaling limit of Laplacian-$b$ random walk. However, let us comment that his argument is remarkable in that we can extend $\mathrm{SLE}_{\kappa}$ to multiply connected domains by the Maruyama–Girsanov transformation.

3. Relationship between LM and SKLE

3.1. Reformulation of Laplacian-$b$ motion as SKLE

In this section, we observe the relationship between LM and SKLE. We take $\mathrm{LM}_{b}$ as our starting point. Let $D^{\prime}$ and $(g^{0}_{t},F^{0}_{t})$ be as in Section 2.2. To move our domain from $D^{\prime}$ to a standard slit domain as in Section 2.1, we perform a preparatory transformation on $(g^{0}_{t},F^{0}_{t})$ as follows: Let $h$ be a conformal map hydrodynamically normalized from $D^{\prime}$ to a standard slit domain $D$, whose existence and uniqueness are ensured by [11, Proposition 2.3]. For each $t$, we denote the canonical map of the hull $F_{t}:=h(F^{0}_{t})$ by $g_{t}\colon D\setminus F_{t}\to D_{t}$ and put $h_{t}:=g_{t}\circ h\circ(g^{0}_{t})^{-1}$. Then by [11, Theorem 4.8], $\{F_{t}\}_{t\leq S_{n,m}}$ is a family of continuously growing hulls in $D$, the map $g_{t}$ satisfies (2.2) with $\dot{a}_{t}=2h_{t}^{\prime}(U(t))^{2}$ and $\xi(t)=h_{t}(U(t))$, and the solution $\mathbf{s}(t)$ to (2.1) with these $a_{t}$ and $\xi(t)$ enjoys $D(\mathbf{s}(t))=D_{t}$. Moreover, it follows from [11, Eq. (4.7)] that

[TABLE]

for $t\leq S_{n,m}$ under the measure $\mathbf{Q}^{(n,m)}$. Here, $b_{\mathrm{BMD}}$ is the BMD domain constant, a locally Lipschitz function on $\mathbb{R}\times\mathsf{Slit}$ which is invariant under horizontal translation and homogeneous with degree $-1$. See Eq. (6.1) and Lemma 6.1 of [4]. The aim of this subsection is to rewrite (3.1) in the form of (2.4) and to check that its coefficients satisfy the conditions that are required in Section 2.1 to define $\mathrm{SKLE}_{\alpha,\beta}$.

To deform the expression (3.1) into a form independent of $D^{\prime}_{t}$, we utilize the conformal transformation rule [10, Eq. (3.12)] for $\partial\log Q/\partial\xi_{0}$:

[TABLE]

Substituting (3.2) into (3.1) yields

[TABLE]

where $Q(\xi_{0},\mathbf{s}):=Q(\xi_{0},D(\mathbf{s}))$ for $\xi_{0}\in\mathbb{R}$ and $\mathbf{s}\in\mathsf{Slit}$.

We now reparametrize $\{F_{t}\}$ by the half-plane capacity relative to $D$, that is, $\check{F}_{t}:=F_{a^{-1}(2(t\wedge\check{S}_{n,m}))}$ with $\check{S}_{n,m}:=a(S_{n,m})/2$. By this time-change, we have $\operatorname{hcap}^{D}(\check{F}_{t})=2t$ for all $t\in[0,\check{S}_{n,m})$. All the other quantities that are reparametrized in the same manner are indicated by adding check mark symbol as well. Brownian motion $(\check{B}^{(n,m)}_{t})_{t\geq 0}$ then exists on some enlargement of the filtered probability space $(\Omega,(\check{\mathcal{F}}_{t})_{t\geq 0},\mathbf{Q}^{(n,m)})$, satisfying

[TABLE]

by [13, Theorem V.1.7].

The expression (3.3) suggests the relation between $\mathrm{LM}_{b}$ and $\mathrm{SKLE}_{\alpha,\beta}$. Namely, if we can define $\mathrm{SKLE}_{\alpha,\beta}$ on $D$ with

[TABLE]

and pull it back to $D^{\prime}$ by the map $h\colon D^{\prime}\to D$, then the resulting random evolution is a time-changed $\mathrm{LM}_{b}$ on $D^{\prime}$. Indeed, we can do the reverse procedure, that is, start at (3.3) to get (2.7) by a similar computation and enlargement of the underlying probability space using the inverse map $h^{-1}$. The remaining thing is thus to check $\alpha$ and $\beta$ of (3.4) satisfies the conditions in Section 2.1:

Proposition 3.1**.**

$(\partial\log Q/\partial\xi_{0})(\xi_{0},\mathbf{s})$* is invariant under horizontal translations, homogeneous with degree $-1$ and locally Lipschitz continuous.*

Proof.

The translation invariance and homogeneity with degree $-1$ are obvious from the conformal transformation rule (3.2). We therefore prove only the local Lipschitz continuity.

We put $D:=D(\mathbf{s})$ and $C_{j}:=C_{j}(\mathbf{s})$ for $\mathbf{s}\in\mathsf{Slit}$, and denote by $K_{D}$ the Poisson kernel of absorbing Brownian motion in $D$. We also denote by $\varphi_{\mathbf{s}}^{(j)}(z)$ the harmonic measure of $C_{j}$. In other words, $\varphi_{\mathbf{s}}^{(j)}$ is a unique bounded harmonic function on $D$ with boundary values $1$ on $C_{j}$ and [math] on $\partial\mathbb{H}\cup\bigcup_{k\neq j}C_{k}$. The quantity $Q(\xi_{0},\mathbf{s})$ for $\xi_{0}\in\partial\mathbb{H}$ is computed by using (2.6) and $K_{D}$ as follows:

[TABLE]

Here, $\mathbf{n}_{\xi_{0}}$ stands for the outward unit normal vector at $\xi_{0}$, and $\partial_{p}C_{j}$ represents the boundary of $\mathbb{H}\setminus C_{j}$ in the path distance topology. Namely, it consists of the left and right endpoints $z_{j}$ and $z^{r}_{j}$, the upper side $C_{j}^{+}$ of the slit and the lower one $C_{j}^{-}$. It follows from (3.5) that

[TABLE]

The function $\partial_{\mathbf{n}_{\xi_{0}}}\varphi_{\mathbf{s}}^{(j)}(\xi_{0})$ is locally Lipschitz by [5, Eq. (9.24)]. We can mimic the argument of [5, Section 9], in which [5, Eq. (9.24)] was derived from [5, Eq. (9.12)], to show that $\partial_{\xi_{0}}\partial_{\mathbf{n}_{\xi_{0}}}\varphi_{\mathbf{s}}^{(j)}(\xi_{0})$ is also locally Lipschitz. We thus reach the desired conclusion. ∎

By Proposition 3.1, we can define $\mathrm{SKLE}_{\alpha,\beta}$ with $\alpha$ and $\beta$ given by (3.4) and thus obtain a time-changed $\mathrm{LM}_{b}$ by pulling it back to $D^{\prime}$. If $D^{\prime}$ itself is a standard slit domain, then $\mathrm{SKLE}_{\alpha,\beta}$ is exactly the same as the time-changed $\mathrm{LM}_{b}$ on $D^{\prime}$.

3.2. Explosion time of SKLE corresponding to $\mathrm{LM}_{b}$

In this subsection, we discuss the exit of $\mathrm{LM}_{b}$ from the domain $D^{\prime}$. Our problem is whether $\mathrm{LM}_{b}$ exit from $D^{\prime}$, i.e., the exit time $T_{D^{\prime}}$ in Section 2.2 is finite or not. We are interested in the behavior of $\mathrm{LM}_{b}$ around $T_{D^{\prime}}$ as well.

It is commented in [10, Section 4.6] that $\mathrm{LM}_{b}$ with $b=(6-\kappa)/(2\kappa)$ for $0<\kappa\leq 4$ is not likely to exit the domain $D^{\prime}$, that is, $T_{D^{\prime}}$ in Section 2.2 should be infinite. This observation is partly based on the fact that $\mathrm{SLE}_{\kappa}$ with $0<\kappa\leq 4$ is a simple curve with probability one [9, Proposition 6.9]. However, the proof of the property $T_{D^{\prime}}=\infty$ has not been known so far. One of the difficulties is that there may not exist a single probability measure $\mathbf{Q}$ on $(\Omega,\mathcal{F}_{\infty})$ under which (2.7) holds for all $t\in[0,T_{D^{\prime}})$ with $B^{(n,m)}$ replaced by a Brownian motion independent of $n$ and $m$. The problem itself may be “ill-posed” unless such a measure $\mathbf{Q}$ exists. We note that, since the SDE (2.7) is not closed with respect to the unknown variable $U(t)$ but contains another unknown variable $D^{\prime}_{t}$, the construction of $\mathbf{Q}$ via this SDE is not straightforward.

In what follows, the explosion problem of $\mathrm{SKLE}_{\alpha,\beta}$ with $\alpha$ and $\beta$ given by (3.4), which is always “well-posed”, is addressed instead of the original exit problem of $\mathrm{LM}_{b}$. In Section 2.1, $\mathrm{SKLE}_{\alpha,\beta}$ is defined only up to the explosion time $\zeta$ of the solution $W_{t}=(\xi(t),\mathbf{s}(t))$ to the system of SDEs (2.1) and (2.4). As described in [12, Section 1], $\zeta$ should correspond to the “exit time” of $\mathrm{SKLE}_{\alpha,\beta}$ from the standard slit domain $D$. On the basis of this observation, the author obtained the following asymptotic behavior of the solution to (2.1) around $\zeta$ in [12, Section 3]: Recall that

[TABLE]

is a function on $\mathbb{R}\times\mathsf{Slit}$ having the invariance under horizontal translation. We say that a function $f$ on $\mathsf{Slit}$ enjoys Condition (B’) if $f(\mathbf{s})$ is bounded on the set $\{\mathbf{s}\in\mathsf{Slit};R(\mathbf{s})=R(0,\mathbf{s})>r\}$ for every $r>0$. The previous result [12, Theorem 3.2] asserts that, under Condition (B’) on the coefficients in (2.4), $\lim_{t\to\zeta}R(W_{t})=0$ holds a.s. on the event $\{\zeta<\infty\}$. A intuitive picture of this result is that, if $\lim_{t\to\zeta}\operatorname{dist}(C_{j}(\mathbf{s}(t)),\xi(t))=0$, then the hull $F_{t}$ should “exit” $D$ from the corresponding slit $C_{j}$.

Now, we check that Condition (B’) holds in the case of $\mathrm{LM}_{b}$. Since $\alpha(\mathbf{s})=\sqrt{\kappa}$ clearly satisfies this condition, we only have to prove it for $\beta$ of (3.4):

Proposition 3.2**.**

$(\partial\log Q/\partial\xi_{0})(0,\mathbf{s})$* satisfies Condition (B’).*

Proof.

By the scaling property that is derived from (3.2), it suffices to prove that $(\partial\log Q/\partial\xi_{0})(0,\mathbf{s})$ is bounded when $R(\mathbf{s})>1$. By the strong Markov property of the $\mathbb{H}$-excursion $\hat{Z}_{t}$, we have

[TABLE]

for $\lvert z\rvert<1$. Here, $\hat{\sigma}_{\mathbb{D}_{+}}$ denotes the hitting time of $\hat{Z}$ to $\mathbb{D}_{+}=\mathbb{D}\cap\mathbb{H}$. Letting $z\to\xi_{0}\in\partial\mathbb{H}$ yields that

[TABLE]

Hence we have

[TABLE]

Computing the Poisson kernel $K_{\mathbb{D}_{+}}$ gives the following expression:

[TABLE]

The right-hand side is bounded by two. ∎

Since we have established Condition (B’), it is natural in view of the intuitive picture above that we interpret the exit problem of $\mathrm{LM}_{b}$ as follows: $\zeta=\infty$ holds a.s. if and only if $0<\kappa\leq 4$. However, this conjecture is still difficult to prove for general values of $\kappa$. In what follows, we look only at the special case $\kappa=6$ (i.e., $b=0$). The situation becomes much simpler in this case since the drift term of (2.7) vanishes. Hence $\mathrm{LM}_{0}$ is just $\mathrm{SLE}_{6}$, and especially no change of measures is necessary to obtain $\mathrm{LM}_{0}$. The reconstruction procedure in Section 3.1 thus provides another proof of [6, Theorem 4.2], which shows that the law of $\mathrm{SLE}_{6}$ coincides with $\mathrm{SKLE}_{\sqrt{6},-b_{\mathrm{BMD}}}$ modulo time-change until it exits $D$. The following result follows from the fact that the $\mathrm{SLE}_{6}$ hull is space-filling with probability one [9, Proposition 6.10]:

Theorem 3.3**.**

The explosion time $\zeta$ of the solution $W_{t}=(\xi(t),\mathbf{s}(t))$ to (2.1) and (2.4) whose coefficients are $\alpha(\mathbf{s})=\sqrt{6}$ and $\beta(\mathbf{s})=-b_{\mathrm{BMD}}(\mathbf{s})$ is finite with probability one.

Proof.

Let $\{F_{t}\}_{t\in[0,\infty)}$ be an $\mathrm{SLE}_{6}$ hull defined on a filtered probability space $(\Omega,(\mathcal{F}_{t})_{t\geq 0},\mathbf{Q})$ with the usual conditions. [9, Proposition 6.10] asserts that $\mathbf{Q}\left(\bigcup_{t\in[0,\infty)}F_{t}=\mathbb{H}\right)=1$, and hence $T_{D}:=\inf\{t>0;F_{t}\nsubseteq D\}<\infty$ $\mathbf{Q}$-a.s. We define

[TABLE]

$\{\check{F}_{t}\}_{t<\check{T}_{D}}$ is then the $\mathrm{SKLE}_{\sqrt{6},-b_{\mathrm{BMD}}}$ hull in $D$ defined on an enlargement of $(\Omega,(\mathcal{F}_{t})_{t\geq 0},\mathbf{Q})$. The pair of associated driving function $\check{\xi}(t)$ and slit vector $\check{\mathbf{s}}(t)$ is thus a weak solution of the SDEs (2.1) and (2.4) whose coefficients are the above $\alpha$ and $\beta$ up to $\check{T}_{D}$. The stopping time $\check{T}_{D}$ is its explosion time, because if the solution could be continued, then $F_{T_{D}}$ were included by $D$, which contradicts to the definition of $T_{D}$.

Because of the uniqueness in law of solutions to the SDEs (2.1) and (2.4), it suffices to show that $\check{T}_{D}$ is finite $\mathbf{Q}$-a.s. for completing the proof. We now take a sample $\{F_{t}\}$ such that $T_{D}<\infty$. Since $F_{T_{D}}$ is then bounded, there exists $R>0$ such that $F_{T_{D}}\subset B(0,R)\cap D$. By [4, Eq. (A.20)], we have

[TABLE]

As $T_{D}<\infty$ holds $\mathbf{Q}$-a.s., we reach the desired conclusion. ∎

We have two remarks on Theorem 3.3. First, this theorem is highly non-trivial from the form of SDE (2.4) while the proof is straightforward in terms of $\mathrm{SLE}_{6}$. Second, it is clear that the proof of Theorem 3.3 does not work for $\kappa\neq 6$. We have to deal with the quantity $\partial\log Q/\partial\xi_{0}$ and reveal how much repulsive force is exerted between the slit vector $\mathbf{s}(t)$ and driving function $\xi(t)$. This is yet to be investigated.

Acknowledgements

This work was supported by JSPS KAKENHI Grant Number JP19J13031.