Chordal Komatu–Loewner equation
for a family of continuously growing hulls
In this paper, we discuss the chordal Komatu–Loewner equation on standard slit domains in a manner applicable not just to a simple curve but also a family of continuously growing hulls. As applications, we first give a characterization of the explosion time for the Komatu–Loewner evolution that is originated from the observation by Bauer and Friedrich (2008). We then give a new treatment of the Laplacian- motion, which was introduced by Lawler (2006), by regarding it as a special case of the stochastic Komatu–Loewner evolution.
keywords:Komatu–Loewner equation, continuously growing hulls, kernel convergence, explosion time, Laplacian- motion, stochastic Komatu–Loewner evolution
Msc: Primary 60J67, Secondary 30C20, 60J70, 60H10
The Komatu–Loewner equation is an extension of the celebrated Loewner equation to multiply connected domains. In this paper, we shall give a systematic treatment of this equation for a family of growing hulls which are not necessarily induced by a simple curve and discuss some applications. In order to describe mathematical details, we begin to recall the Loewner theory briefly. The reader can consult La05 () for further detail.
We denote by the upper half-plane . Let be a simple curve with and . For each , there exists a unique conformal map from onto with the hydrodynamic normalization
for some constant . This is a version of Riemann’s mapping theorem. If we reparametrize so that (as mentioned later in Section 3.1), then we obtain the chordal Loewner equation
where . We call the driving function of .
Since (1.1) is an ODE satisfying the local Lipschitz condition, the solution to (1.1) uniquely exists up to its explosion time . If we set , then must be the complement of the domain of definition of , that is, . Thus the information on the curve is fully encoded into the driving function via the Loewner equation. More generally, we can consider (1.1) driven by any continuous function . Even in this case, the solution defines a unique conformal map with the hydrodynamic normalization, though the resulting family is not necessarily a simple curve but a family of bounded sets called growing hulls. , or the couple is called the Loewner evolution driven by . In the theory of conformal maps, is usually called the Loewner chain.
Schramm Sc00 () used the Loewner equation (1.1) to define the stochastic Loewner evolution (SLE). For , is the random Loewner evolution driven by , where is the one-dimensional standard Brownian motion (BM). Schramm’s original aim was to describe the scaling limit of two-dimensional lattice models in statistical physics. was actually proven to be the scaling limit of some models according to the value of . For individual models, we refer the reader to (Ka15, , Section 2.5) and the references therein. In addition, recent studies such as Fr10 () reveal the relation between the Loewner equation and integrable systems. We therefore have much interest in the Loewner theory from various points of view.
As seen, for example, from the usage of Riemann’s mapping theorem above, the simple connectivity of is crucial to the Loewner theory. Thus it is not straightforward to extend the Loewner equation to multiply connected domains (or to Riemann surfaces). This problem was originally proposed by Komatu Ko50 (). He claimed that a corresponding equation exists on a multiply connected domain but did not give the explicit form of that equation. After more than fifty years, Bauer and Friedrich BF08 () established its concrete expression in a standard way of complex analysis, using the Green function and harmonic measures. Lawler La06 () then gave a probabilistic comprehension of the equation in view of the excursion reflected Brownian motion (ERBM). The idea provided in La06 () was implemented by Drenning Dr11 () in detail. Motivated by La06 () and Dr11 (), Chen, Fukushima and Rohde CFR16 () introduced the Brownian motion with darning (BMD) to fill missing arguments in the existing proofs.
We now describe the framework where our domain has multiple connectivity. Fix a positive integer . Let , , be mutually disjoint horizontal slits, that is, segments parallel to the real axis. We call a standard slit domain. We note that any -connected domain is conformally equivalent to some standard slit domain. (See (BF08, , Section 2) and references therein for this fact.)
Let be a simple curve with and . For each , there exists a unique conformal map from onto another standard slit domain with the hydrodynamic normalization. After the same reparametrization of , satisfies the chordal Komatu–Loewner equation ((BF08, , Theorem 3.1), (CFR16, , Theorem 9.9))
where . , , is the conformal map on defined in Section 2.1.
Here (1.2) differs from (1.1) in that the image differs from and varies as time passes. Let be the -th slit of so that . The left and right endpoints of are denoted by and , respectively. These endpoints then satisfy the Komatu–Loewner equation for slits ((BF08, , Theorem 4.1), (CF18, , Theorem 2.3))
Hence the motion of is described by (1.3) in terms of those of the slits .
Once we get (1.2) and (1.3), the initial value problem for them, as done for (1.1), is a natural question. Namely, for a given continuous function , we look for the solution to (1.2) and (1.3) and then obtain a family of growing hulls. We shall explain the actual procedure in Section 2.2. As a result, (1.2) generates a family of conformal maps and of growing hulls. They are called the Komatu–Loewner evolution driven by . Let us call the Komatu–Loewner chain as well in this paper. In addition, Chen and Fukushima CF18 () defined the stochastic Komatu–Loewner evolution (SKLE) with the random driving function given by the SDE (2.7), based on the discussion in (BF08, , Section 5). Its relation to SLE was also examined by Chen, Fukushima and Suzuki CFS17 ().
In such a research on SKLE, the trouble often arises concerning the “transformation of the Komatu–Loewner chains.” Here by the term “transformation” we mean the following situation: Let be the Komatu–Loewner evolution in a standard slit domain and be another slit domain. There is then a unique conformal map from onto a slit domain with the hydrodynamic normalization. We expect to be the Komatu–Loewner evolution in , that is, generated by the equation (modulo time-change). This fact needs proof since we have deduced the equation only for a simple curve, not for a family of growing hulls. From this standpoint, we can say that CFS17 () established exactly the transformation of chains with by the hitting time analysis for BM. This method is successful but not applicable to general and , and thus some problems mentioned in (CFS17, , Section 5) remain open.
One of the major purpose of this paper is to settle down these circumstances. To be more precise, we shall deduce the Komatu–Loewner equation for a family of “continuously” growing hulls in Section 3. The continuity of growing hulls is introduced in Definition 3.9 via the kernel convergence of domains, which is a key concept in this paper. The continuity of hulls and the existence condition (2.6) of driving function prove to be a complete characteristic of the Komatu–Loewner evolution in Theorem 3.12. Our definition of the continuity is moreover independent of the domain and conformally invariant, and thus the chains can be transformed for any domains (Proposition 3.13 and Theorem 3.14). This systematic treatment of the Komatu–Loewner equation is our main result. In Corollary 3.15, we further show that our result implies the locality of chordal in a full generality, which is an answer to an open problem in (CFS17, , Section 5).
As an application of Theorem 3.14, we discuss the characterization of the explosion time for the Komatu–Loewner evolution in Section 4. It was commented in (BF08, , Theorem 4.1) that the explosion time for (1.3) is characterized by the property
for some . This observation seems quite natural. Indeed, as described in Section 2.2, for an interior point if and only if reaches by the time . If moves in the same way as does, then (1.4) should be true. In theory, however, the explosion may happen in different manners illustrated after Example 4.17. Its verification is thus worthwhile, because it enables us to tell whether the explosion occurs or not by examining only the quantity . To justify (1.4), we shall prove
in Theorem 4.18. The property (1.5) implies (1.4) and is consistent to our intuition. The proof of (1.5) is based on the transformation of the Komatu–Loewner chain into the Loewner one, together with a general theory of complex analysis.
Another application of Theorems 3.14 and 4.19 concerns the extension of SLE to multiply connected domains. We have already introduced SKLE as an extension of SLE, but there are other approaches trying to extend SLE, one of which was done by Lawler La06 (). He introduced the Laplacian- motion as a candidate for the scaling limit of the Laplacian- walk in a multiply connected domain. This is also a possible extension of SLE. Important is that his definition was based only on the Loewner equation in , not on the Komatu–Loewner one. He employed the Girsanov transform instead to define the Laplacian- motion. In Section 5, we shall offer another way to examine the Laplacian- motion using the Komatu–Loewner equation, namely, regarding it as SKLE by the appropriate choice of the parameters. We show this to be possible in Section 5.1 by transforming the Loewner chain into the Komatu–Loewner one. As a result, Theorem 4.19, the stochastic version of (1.5), can be applied to the Laplacian- motion. In (La06, , Section 4.6), the question arose whether the Laplacian- motion hits the holes of the domain in finite time. We relate Theorem 4.19 to this question in terms of Remark 4.20 and especially give an affirmative answer when the exponent is zero in Section 5.2.
Throughout this paper, we use basic estimates on univalent functions. They are summarized in Appendix A for the reader’s convenience.
This section is devoted to the preliminaries to the subsequent sections. The relation between BMD and conformal maps is summarized in Section 2.1. The initial value problem for the Komatu–Loewner equation is then explained in Section 2.2. In Section 2.3, we discuss the convergence of a sequence of univalent functions based on the notion of kernel.
2.1 Brownian motion with darning and conformal maps on multiply connected domains
Fix a positive integer and a simply connected domain . Let , , be mutually disjoint compact continua such that each is connected. Here, a continuum means a connected closed set consisting of more than one point. The domain is then -connected. We “darn” each hole as follows: Regarding each as one point , we define the quotient topological space by . BMD is defined on by (CFR16, , Definition 2.1). The harmonicity on BMD is then defined by (CFR16, , (3.2)). The next proposition shows that the BMD-harmonicity is a stronger condition than the usual harmonicity on the absorbing BM (ABM):
The following are equivalent for a continuous function :
is BMD-harmonic on .
There is a holomorphic function on whose real or imaginary part is ;
In particular, a function on satisfying Condition (ii) extends to a BMD-harmonic function on if it takes a constant value on each , .
By Proposition 2.1, we can apply the Poisson integral representation (2.2) or other properties of BMD-harmonic functions to holomorphic functions which take constant values on the boundaries. Such a relation between BMD and holomorphic functions was also used to prove (CFR16, , Theorem 7.2), which we state below as Proposition 2.2. Here, a set is called a compact -hull if is bounded, , and is simply connected. We say simply a hull for such a set because we deal with only this type of hulls.
Let and be as above. Suppose that is a hull contained in . Then, there are a unique standard slit domain and a unique conformal map with the hydrodynamic normalization
where is a positive constant.
We refer to as the canonical map from onto and as the half-plane capacity of relative to . If itself is a standard slit domain , then by (CF18, , (A.20)), the half-plane capacity has an expression
for any . Here , and denotes the expectation with respect to starting at .
In addition to these propositions, we need the (complex) Poisson kernel of BMD to discuss the Komatu–Loewner equation. Let be a hull with piecewise smooth boundary, and be as above. We denote by the 0-order resolvent kernel of . Taking the normal derivative, we get the Poisson kernel of
where is the outward unit normal at . The kernels and can be expressed by the classical Green function and harmonic measures. See Sections 4 and 5 in CFR16 () for their concrete expressions.
We can write Poisson’s integral formula using . Suppose that a BMD-harmonic function on vanishes at infinity, extends continuously to and has a compact support on . Then by (CFR16, , (5.5)) and the proof of (CFR16, , Theorem 6.4), satisfies
where is the lifetime of . Note that the former equality holds even if is not smooth.
When is a horizontal slit for each , we can further define the complex Poisson kernel of by (CFR16, , Lemma 6.1). Namely, there is a unique holomorphic function , , such that and . This is equal to in (BF08, , Section 2.2) by their construction. Consequently, is a unique conformal map from onto another standard slit domain with and
This fact also follows directly from (CFR16, , Theorem 11.2) and (CF18, , Lemma 5.6). Let be the mirror reflection with respect to the real axis. By Schwarz’s reflection and (2.3), extends to a conformal map from onto .
2.2 Initial value problem for the Komatu–Loewner equation
Fix and let , be mutually disjoint horizontal slits. We denote the left and right endpoints of the -th slit by and , respectively. Then, the -tuple of the slits are identified with an element in . We define the open subset of consisting of all such elements by
We denote by (resp. ) the -th slit (resp. the standard slit domain) corresponding to . is the BMD complex Poisson kernel of .
For and , we put
where and , , are the left and right endpoints of the -th slit , respectively. The function , , has an invariance under horizontal translations, that is,
where denotes the vector in whose first entries are zero and last entries are . (CF18 () called this property the homogeneity in -direction.) We can easily check this invariance since .
The Komatu–Loewner equation for slits (1.3) is now written as
where is the -th entry of . Since is locally Lipschitz on for each by (CF18, , Lemma 4.1), (2.4) is solved up to its explosion time . Here we note that Condition (L) on a function appearing in (CF18, , Lemma 4.1) is equivalent to each of the following conditions:
the local Lipschitz continuity of in ,
the local Lipschitz continuity of in .
Therefore we simply say that is locally Lipschitz if one of these conditions holds.
In this context, we introduce a few more notations. For a function , we denote by and regard it as a function on with the invariance under horizontal translations. Conversely, for a function with the invariance , we denote by and regard it as a function on .
(2.5) has a unique solution up to by Theorem 5.5 (i) of CF18 (). By Theorems 5.5, 5.8 and 5.12 of CF18 (), is the canonical map from onto where , , and is a family of growing (i.e. strictly increasing) hulls satisfying
for all . The family , or here is called the Komatu–Loewner evolution driven by . In the present paper, we also refer to as the Komatu–Loewner chain.
Since , , the equations (2.4) and (2.5) generate downward flows. The points whose image eventually reach are contained in the hull . Thus, the flow of and the continuity of strongly affect the shape of . There are further visual and detailed descriptions in the case of the Loewner equation in , for example, in (Ka15, , Chapter 2). Such a visual comprehension might help the reader to understand the examples in Section 4.1.
In the same manner, we introduce the stochastic Komatu–Loewner evolution (SKLE) as we defined SLE in Section 1. We say that a function is homogeneous with degree if, for any ,
Take two functions and homogeneous with degree and , respectively, and suppose that both of them satisfy the local Lipschitz condition. We consider the following SDEs:
where is the one-dimensional standard BM. The second equation (2.8) is the same as (2.4), though we regard it as a part of the system of SDEs instead of a single ODE. By the local Lipschitz condition, this system has a unique strong solution up to its explosion time ((CF18, , Theorem 4.2)). The above-mentioned properties also holds for this solution . We designate the resulting random evolution as .
2.3 Convergence of a sequence of univalent functions
The term univalent means meromorphic and injective. If a univalent function is surjective, then it is called a conformal map. To discuss the convergence of a sequence of univalent functions, the notion of kernel has been employed classically in univalent function theory. In this subsection, we summarize the basic theory and apply it to multiply connected domains in .
We shall use the following notations throughout this paper: We denote by the extended complex plane, or Riemann sphere. is the open disk with center and radius . stands for the exterior of . We put and . The linear fractional transformation on is defined by . We note that and that for . The classes and of univalent functions are defined in Appendix A. denotes the reflection with respect to the real axis .
We start with the definition of the kernel, following Section 5 of Chapter V in Go69 (). Let be a sequence of domains in containing the point at infinity . The kernel of (with respect to ) is defined as the largest domain containing such that each compact subset is included by all except for finitely many . If such does not exist, we say that does not have the kernel. By definition, has the kernel if and only if every but for finitely many contains a common neighborhood of . Next suppose that has the kernel , and let each be a function on . We say as usual that converges to a function uniformly on compacta if, for every compact subset of , converges to uniformly on . This definition makes sense because is included by eventually. We note that the convergence around can be interpreted as that around the origin by composing the inversion map appropriately. We designate the above convergence as u.c. on . We further say that converges to a domain in the sense of kernel convergence if is the kernel of every subsequence of . We denote it simply by .
With these definitions, we state a version of Carathéodory’s kernel theorem. Although the convergence in the latter part of Proposition 2.5 is not mentioned explicitly in (Go69, , Theorem V.5.1), the proof is not difficult and similar statements can be found in (Go69, , Theorem II.5.1) and (Co95, , Theorem 15.4.7).
Proposition 2.5 ((Go69, , Theorem V.5.1)).
Assume that is a sequence of domains containing with and that each is univalent with and . We put . Then the following conditions are equivalent:
There exists a univalent function on such that u.c. on ;
There exists a domain such that .
If one of these conditions happens, then , and u.c. on .
We now proceed to modify the notion of kernel and kernel convergence for multiply connected domains of a specific form in . Note that the usage of terms introduced below are conventions only in the present paper, not in common.
Let where is a hull for each . Let , , be mutually disjoint compact continua in which are disjoint from as well. We put . Suppose that for and are all contained in a common bounded region. We define the kernel of (with respect to ) under this setting as the largest unbounded domain such that each compact subset is included by all except for finitely many . This definition is the same as the above one, except that is a boundary point of in this case. By the assumption that are all contained in a bounded region, the kernel always exists. We also define the uniform convergence of functions on compacta and the kernel convergence of in the same way.
Now consider the sets , and having the same properties as , and , and let be a conformal map with the hydrodynamic normalization
extends to a conformal map from onto . It is easily seen that the kernel of is nothing but the intersection of and the kernel of . We can thus obtain the following version of kernel theorem:
Suppose . Then the following conditions are equivalent:
There exists a univalent function on such that u.c. on ;
There exists a domain such that .
If one of these conditions happens, then , and u.c. on .
We only have to prove that (i) implies the u.c. convergence of on . It holds that u.c. on by Schwarz’s reflection. Since for sufficiently large by the expansion
maps bounded regions to bounded ones by Koebe’s one-quarter theorem A.33. Vitali’s theorem (cf. (Go69, , Theorem I.1.2)) then implies that converges uniformly on every bounded set intersecting . We next take a sufficiently large so that is a neighborhood of in . By the uniform convergence of on , is uniformly bounded on by the maximal value principle. Vitali’s theorem again tells us that converges u.c. on and so does . ∎
3 Komatu–Loewner equation for a family of growing hulls
3.1 Deduction of the Komatu–Loewner equation
Based on the preliminaries, we discuss the Komatu–Loewner equation for a family of growing hulls in this subsection. First of all, we state a key estimate which was originally given in (Dr11, , Proposition 6.12) in terms of ERBM.
Let be a standard slit domain and suppose and satisfy . Take a hull contained in such that . Then, there exists a locally bounded function depending only on , and such that, for all with ,
Here denotes the canonical map on .
Since we use BMD instead of ERBM, we prove Proposition 3.7 in Appendix B for the sake of completeness. Proposition 3.7 is a generalization of (LSW01, , Lemma 2.7), and Drenning Dr11 () used it to obtain the Komatu–Loewner equation for a simple curve in the right derivative sense. He then discussed the left differentiability by some probabilistic methods based on the fact that the hull at issue was a simple curve. We also establish the right differentiability in Proposition 3.8 by Proposition 3.7 as he did, but the subsequent argument is completely different. We employ the kernel convergence condition instead of his methods to examine the left differentiability for a family of “continuously” growing hulls.
We now mention our basic setting. Let be a family of growing hulls in a fixed standard slit domain . For each , let be the canonical map, and correspond to the slits of . is sometimes denoted by as well. We further define, for ,
Clearly is a hull, and is the canonical map on .
In what follows, several conditions are imposed on . If there exists a function such that (2.6) holds for any , then we call the driving function of . The condition (2.6) is sometimes called the right continuity of and employed in the existing literature, for example, (La05, , Section 4.1), (La06, , Section 4) and (CF18, , Section 6). One reason is that, for a family of growing hulls having this property, we can obtain the Komatu–Loewner equation in the right derivative sense as in Proposition 3.8. However, it should be noted that we mean a weaker condition than (2.6) by the “right continuity” in Definition 3.9.
Let be a family of growing hulls in with driving function .
The half-plane capacity is strictly increasing and right continuous in .
u.c. on as for any .
For each , is right differentiable in , and
Here denotes the right derivative of with respect to .
(i) Let . We can easily observe that
The left continuity of and left differentiability of do not follow from (2.6). To proceed further, we define the continuity of as the continuity of in the sense of kernel convergence.
is said to be (left/right) continuous in at if as approaches (from left/right).
Such a continuity condition did not appear in the recent studies BF08 (); La06 (); Dr11 (); CFR16 (); CF18 (), but it is not new in complex analysis. Indeed, a similar condition was imposed when Pommerenke established a version of the radial Loewner equation in (Po75, , Section 6.1). Below we show that Definition 3.9 works well even when the domain has multiple connectivity.
If satisfies (2.6) for some at , then it is right continuous in at .
We first show that is locally bounded over . Actually for a fixed , we can take so that . Then for by the same computation as (2.9). By Koebe’s one-quarter theorem A.33, we have and thus . This means that . Hence is bounded in .
Suppose that is left continuous in at . Then as , that is, is left continuous at . Moreover, u.c. on as , and is left continuous at .
Suppose that is right continuous in at . Then as , that is, is right continuous at . Moreover, u.c. on as , and is right continuous at .
By the local boundedness of in the proof of 3.10, we can take a sequence with so that exists in . Though is not necessary in , it is obvious from definition that converges to a slit domain . Some of the slits of may degenerate. Since by the left continuity of , we can apply the kernel theorem 2.6 to to obtain the limiting conformal map . Then, all the slits of must not degenerate, and must be the canonical map on , which yields and by the uniqueness in Proposition 2.2. In particular, this limit is independent of the choice of . We therefore conclude that as .
The equivalence between the left continuity of and that of can be checked easily from definition, and so we omit it.
Since and as , the kernel theorem 2.6 implies u.c., which in turn yields u.c. as . To show the left continuity of , we regard as an element of by Schwarz’s reflection for large enough. Writing the Laurent series expansion around infinity as
we get, from the Cauchy–Schwarz inequality and the area theorem A.30,
Since for any