Limits of radial multiple SLE and a Burgers-Loewner differential equation

TL;DR

This paper studies the behavior of multiple radial SLE curves as their number grows large, revealing a connection to Burgers-type Loewner equations and exploring their implications in probability and complex analysis.

Contribution

It establishes conditions for the tightness of processes in the infinite-curve limit and links the limiting behavior to Burgers differential equations within the Loewner framework.

Findings

Infinite-slit limit described by a Burgers-type Loewner equation

Conditions for tightness of multiple radial SLE processes

Connection between Burgers equation and free probability measures

Abstract

We consider multiple radial SLE as the number of curves tends to infinity. We give conditions that imply the tightness of the associated processes given by the Loewner equation. In the case of equal weights, the infinite-slit limit is described by a Loewner equation whose Herglotz vector field is given by a Burgers differential equation. Furthermore, we investigate a more general form of the Burgers equation. On the one hand, it appears in connection with semigroups of probability measures on the unit circle with respect to free convolution. On the other hand, the Burgers equation itself is also a Loewner differential equation for certain subordination chains.

Equations

$$\frac{\partial }{\partial t}{f}_t(z)=-zh(z,t)\cdot \frac{\partial }{\partial z}{f}_t(z)\text{for almost every t≥0, f0=F0.}$$

$$\frac{\partial }{\partial t}{g}_t(z)=g_t(z)h(g_t(z),t)\text{for almost every t∈[0,T] and z∈fT(D).}$$

$$\frac{\partial }{\partial t}{h}_t(z)=-h_t(z)h(h_t(z),T-t)\text{for almost every t∈[0,T], z∈D, and h0(z)≡z.}$$

$$\frac{\partial }{\partial t}{g}_t(z)=g_t(z)\frac{e^{i\sqrt{\kappa }{B}_t}+g_t(z)}{e^{i\sqrt{\kappa }{B}_t}-g_t(z)},g_0(z)=z\in \mathbb{D}.$$

$$\frac{\partial }{\partial t}{g}_t(z)=g_t(z)\frac{V(t)+g_t(z)}{V(t)-g_t(z)},g_0(z)=z\in \mathbb{D},$$

$$dV(t)=-\frac{\kappa }{2}V(t)dt+i\sqrt{\kappa }V(t)d{B}_t,V(0)=p\in \mathbb{T}.$$

$$dV(t)=-\frac{\kappa }{2}V(t)dt+\sum _{j=1}^m\frac{{\varrho }_j}{2}V(t)\frac{X_j(t)+V(t)}{X_j(t)-V(t)}dt+i\sqrt{\kappa }V(t)d{B}_t,V(0)=p,$$

$$d{X}_j(t)=X_j(t)\frac{V(t)+X_j(t)}{V(t)-X_j(t)}dt,X_j(0)=x_j.$$

$$\frac{d}{dt}{g}_t(z)=\frac{2}{g_t(z)-U(t)},g_0(z)=z\in \mathbb{H},$$

$$dU(t)=\sqrt{\kappa }d{B}_t+\sum _{j=1}^m\frac{{\varrho }_j}{U(t)-Y_j(t)}dt+(\kappa -6-\sum _{j=1}^m{\varrho }_j)\mathsf{Re}\left(\frac{1}{U(t)-S(t)}\right)dt,U(0)=C(p)\in \mathbb{R},$$

$$d{Y}_j(t)=\frac{2}{Y_j(t)-U(t)}dt,Y_j(0)=C(x_j)\in \mathbb{R},$$

$$dS(t)=\frac{2}{S(t)-U(t)}dt,S(0)=i=C(0).$$

$$\text{Vj(t)=gt(“force point”) or Vj(t)=gt(“tip of the growing curve”)}.$$

$$d{V}_k(t)=V_k(t)\frac{V_j(t)+V_k(t)}{V_j(t)-V_k(t)}dt$$

$$d{V}_k(t)=-\frac{\kappa }{2}{V}_k(t)dt+\sum _{j\ne k}{V}_k(t)\frac{V_j(t)+V_k(t)}{V_j(t)-V_k(t)}dt+i\sqrt{\kappa }{V}_k(t)d{B}_t,$$

$$\frac{d}{dt}{g}_t(z)=g_t(z)\sum _{k=1}^N{\lambda }_k\frac{X_k(t)+g_t(z)}{X_k(t)-g_t(z)},g_0(z)=z\in \mathbb{D}.$$

$$d{X}_k(t)=-\frac{\kappa {\lambda }_k}{2}{X}_k(t)dt+\sum _{j\ne k}({\lambda }_k+{\lambda }_j)X_k(t)\frac{X_j(t)+X_k(t)}{X_j(t)-X_k(t)}dt+i\sqrt{\kappa {\lambda }_k}{X}_k(t)d{W}_k(t),X_k(0)=p_k,$$

$$d{X}_k(t)$$

$$+i\sqrt{\kappa }{X}_k(t)d{B}_k({\lambda }_{k}t)+\sum _{j\ne k}{X}_k(t)\frac{X_j(t)+X_k(t)}{X_j(t)-X_k(t)}d({\lambda }_{j}t)$$

$$=-\frac{\kappa {\lambda }_k}{2}{X}_k(t)dt+\sum _{j\ne k}({\lambda }_k+{\lambda }_j)X_k(t)\frac{X_j(t)+X_k(t)}{X_j(t)-X_k(t)}dt+i\sqrt{\kappa {\lambda }_k}{X}_k(t)d{W}_k(t),$$

$$d{V}_{N,k}(t)=V_{N,k}(t)\left(\sum _{j\ne k}({\lambda }_{N,k}+{\lambda }_{N,j})\frac{V_{N,j}(t)+V_{N,k}(t)}{V_{N,j}(t)-V_{N,k}(t)}-\frac{\kappa {\lambda }_{N,k}}{2}\right)dt+i{V}_{N,k}\sqrt{\kappa {\lambda }_{N,k}}d{B}_{N,k}(t),$$

$$\frac{d}{dt}{g}_{N,t}(z)=g_{N,t}(z)\sum _{k=1}^N{\lambda }_{N,k}\frac{V_{N,k}(t)+g_{N,t}(z)}{V_{N,k}(t)-g_{N,t}(z)},g_{N,0}(z)=z\in \mathbb{D},$$

$$d{V}_{N,k}(t)=V_{N,k}(t)\sum _{j\ne k}\frac{2}{N}\frac{V_{N,j}(t)+V_{N,k}(t)}{V_{N,j}(t)-V_{N,k}(t)}dt-\frac{\kappa }{2N}{V}_{N,k}dt+i{V}_{N,k}\sqrt{\kappa /N}d{B}_{N,k}(t),V_{N,k}(0)=x_{N,k}.$$

$$\frac{d}{dt}{g}_{N,t}=g_{N,t}{\int }_{\partial \mathbb{D}}\frac{x+g_{N,t}}{x-g_{N,t}}{\alpha }_{N,t}(dx).$$

$$\underset{k\in \{1,...,N\}}{\max }{\lambda }_{N,k}\le \frac{C}{N}.$$

$$L_N(x)\to L(x)\text{uniformly on [0,1] as N→∞.}$$

$${\mu }_{N,0}⟶\mu \text{as N→∞}$$

$$F_{N,t}(x)=L_N(G_{N,t}(x)).$$

$$F_t(x)=L\circ G_t(x),$$

$$\frac{d}{dt}\left({\int }_{\mathbb{T}}f(x){\mu }_t(dx)\right)=-\int {\int }_{{\mathbb{T}}^2}\frac{x{f}^{\prime }(x)-y{f}^{\prime }(y)}{x-y}(x+y){\mu }_t(dx){\alpha }_t(dy),{\mu }_0=\mu ,$$

$${\left\{{\int }_{\mathbb{T}}f(x){\mu }_{N,t}(dx)\right\}}_{N\in \mathbb{N}}$$

$$d\left({\int }_{\mathbb{T}}f(x){\mu }_{N,t}(dx)\right)=d\left(\sum _{k=1}^N\frac{1}{N}f(V_{N,k}(t))\right)$$

$$=\sum _{k=1}^N\frac{V_{N,k}(t)}{N}\left(f^{\prime }(V_{N,k}(t))\left(\sum _{j\ne k}({\lambda }_{N,k}+{\lambda }_{N,j})\frac{V_{N,j}(t)+V_{N,k}(t)}{V_{N,j}(t)-V_{N,k}(t)}-\frac{\kappa {\lambda }_{N,k}}{2}\right)-\frac{f^{\prime \prime }(V_{N,k}(t))\kappa {\lambda }_{N,k}}{2}\right)dt$$

Full text

Furthermore, we investigate a more general form of the Burgers equation. On the one hand, it appears in connection with semigroups of probability measures on $\mathbb{T}$ with respect to free convolution. On the other hand, the Burgers equation itself is also a Loewner differential equation for certain subordination chains.

Keywords: Schramm-Loewner evolution, multiple SLE, complex Burgers equation, free probability, infinitely divisible distributions.

2010 Mathematics Subject Classification: 37L05, 46L54, 60J67.

1. 1 Introduction

2. 2 Subordination chains and the Loewner equation

3. 3 Infinite-slit limits of radial multiple SLE

4. 3.1 Radial single SLE

5. 3.2 Radial SLE($\kappa,\varrho$)

6. 3.3 Radial multiple SLE

7. 3.4 Limits of radial multiple SLE

8. 3.5 Tightness

9. 3.6 The simultaneous case

10. 4 A general Burgers-Loewner equation

11. 4.1 Free semigroups

12. 4.2 Properties of the Burgers-Loewner equation

13. A Appendix

14. A.1 Free additive semigroups

15. A.2 Radial multiple SLE in the upper half-plane

1 Introduction

The Schramm-Loewner evolution SLE($\kappa$) has been proved to describe the scaling limit of various curves from stochastic geometry and statistical physics, see [Law05]. There are several approaches to generalize this framework to “multiple SLE”, which concerns the simultaneous distribution of $N\in\mathbb{N}$ SLE curves within a given domain, see [Kar19] and the references therein for the historical development and the recent progress. Also multiple SLE can be motivated by scaling limits of curves from well known random models. One might think of the critical Ising model which produces several interface curves between clusters of spin -1 and +1, see [Koz09].

In the present article, we consider a multiple radial SLE which consists of $N$ non-intersecting simple SLE($\kappa$) curves ($\kappa\in[0,4]$) connecting $N$ points $x_{1},...,x_{N}$ on the unit circle $\mathbb{T}$ to [math] within the unit disc $\mathbb{D}$. These curves can be generated by a version of Loewner’s differential equation, where we also need to specify weights $\lambda_{1},...,\lambda_{N}\in(0,1)$ with $\lambda_{1}+...+\lambda_{N}=1$. The weight $\lambda_{k}$ can be thought of as the speed of the growth of the $k$-th curve.

We address the following question: Which conditions on the points $x_{1},...,x_{N}$ and the weights $\lambda_{1},...,\lambda_{N}$ imply the existence of a limit growth process as $N$ goes to infinity?

The following pictures show numerical simulations of such curves.

We find conditions on the points and the weights that guarantee tightness of the associated stochastic processes (Section 3.5). If all the weights are equal ($\lambda_{k}=1/N$), we obtain a simple limit equation involving the a certain complex Burgers equation (Section 3.6).

These results can be seen as the radial analogue of the works [dMS16, dMHS18, HK18], which considered this question for chordal multiple SLE. The assumptions in our main result (Theorem 3.6) are basically the same as in [dMHS18, Theorem 2.5]. One difference between the radial and the chordal case is the compactness of $\mathbb{T}$, and thus the compactness of the space of all probability measures on $\mathbb{T}$ endowed with the topology induced by weak convergence, versus the non-compactness of $\mathbb{R}$. The chordal case requires an additional tightness assumption for the starting points on $\mathbb{R}$ (condition (c) in [dMHS18, Theorem 2.5]).

In addition to these works, we point out that the Burgers equation for the unit disc is an instance of a more general evolution equation for free semigroups of probability measures on the unit circle (Section 4). From this perspective, the Burgers equation describes the distributions of a free unitary Brownian motion.

We can use the probabilistic interpretation to obtain a geometric property of the hulls described by the Burgers equation, see Theorem 3.15.

As the literature more often focuses on the chordal case, we also explain the chordal analogue of the free semigroup equation in the appendix.

Our paper is organized as follows: In Section 2 we briefly review the notion of subordination chains and Loewner’s differential equation. In Section 3, we investigate the limit behaviour of radial multiple SLE as the number of curves converges to infinity. Finally, in Section 4, we identify the Burgers equation as an example of a more general differential equation arising naturally in free probability theory.

2 Subordination chains and the Loewner equation

We let $\mathbb{D}=\{z\in\mathbb{C}\,|\,|z|<1\}$ be the unit disc, $\mathbb{T}=\partial\mathbb{D}$ the unit circle, and $RH=\{z\in\mathbb{C}\,|\,\mathsf{Re}(z)>0\}$ the right half plane.

Definition 2.1**.**

Let $\{f_{t}\}_{t\geq 0}$ be a family of holomorphic functions on $\mathbb{D}$ with $f_{t}(0)=0$. Then,

• •

$\{f_{t}\}_{t\geq 0}$ is called a (decreasing and normalized) Loewner chain if every $f_{t}$ is univalent, $f_{t}(\mathbb{D})\subset f_{s}(\mathbb{D})$ whenever $0\leq s\leq t$, $f_{0}(z)\equiv z$, and $f^{\prime}_{t}(0)=e^{-ta}$ for some $a\in RH$.

• •

The family $\{f_{t}\}_{t\geq 0}$ is called a (decreasing and normalized) subordination chain if $f_{t}=f_{0}\circ g_{t}$ for all $t\geq 0$, where $\{g_{t}\}_{t\geq 0}$ is a (decreasing and normalized) Loewner chain.

Usually, the literature focuses on increasing Loewner chains where $f_{s}(\mathbb{D})\subset f_{t}(\mathbb{D})$ whenever $0\leq s\leq t$. Clearly, if $\{f_{t}\}_{t\in[0,T]}$ is an increasing Loewner chain, then $\{f_{T-t}\}_{t\in[0,T]}$ is (a part of) a decreasing Loewner chain.

Loewner chains have been introduced by C. Loewner in [Loe23] in order to attack the Bieberbach conjecture, a coefficient problem for univalent functions. We refer the reader to [Pom75, Chapter 6] and [Dur83, Chapter 3] for the classical treatment of Loewner chains, and to [AB10] for a summary of more recent developments.

Increasing subordination chains (with $f_{t}^{\prime}(0)\not=0$) have been considered by Pommerenke in [Pom65, Satz 4]. The case of decreasing Loewner chains and subordination chains is handled in [CDMG14]. Subordination chains of the above form are related to Loewner’s partial differential equation. In our situation we have the following relationship.

Theorem 2.2**.**

Let $\{F_{t}\}_{t\geq 0}$ be a subordination chain. Then there exists a Herglotz vector field $h:\mathbb{D}\times[0,\infty)\to\mathbb{C}$, i.e. $z\mapsto h(z,t)$ is holomorphic with $h(0,t)=a\in RH$, $\mathsf{Re}(h(z,t))>0$ for all $z\in\mathbb{D}$ and all $t\geq 0$ and $t\mapsto h(z,t)$ is measurable for all $z\in\mathbb{D}$, such that $\{F_{t}\}_{t\geq 0}$ is a solution of the initial value problem

[TABLE]

Conversely, if $h$ is a Herglotz vector field, then (2.1) has a unique solution $\{f_{t}\}_{t\geq 0}$, which is a subordination chain.

Proof.

Let ${F_{t}}$ be a subordination chain with $F_{t}=F_{0}\circ g_{t}$ for all $t\geq 0$, where $\{g_{t}\}$ is a Loewner chain. Note that if $F_{0}$ is constant (i.e., $F_{0}(z)=0$ for all $z$), then $F_{t}=F_{0}$ for all $t\geq 0$ and $F_{t}$ satisfies (2.1) for any Herglotz vector field.

Now it is known that $g_{t}$ satisfies (2.1) with initial value $f_{0}(z)\equiv z$, see [CDMG14, Theorem 3.2]. The form of the right side and the normalization $h(0,t)=a$ follows from the normalization $g_{t}(0)=0$, $g^{\prime}_{t}(0)=e^{-ta}$. A simple computation shows that $\{F_{t}\}_{t\geq 0}$ satisfies (2.1).

The converse statement follows from [CDMG14, Theorem 3.2], [CDMG14, Theorem 1.11]. ∎

Remark 2.3**.**

*Assume that $F_{0}(z)\equiv z$. Then the solution to (2.1) is a Loewner chain. Note that two different Herglotz vector fields $h_{1}$ and $h_{2}$ can yield the same Loewner chain, but if they do, they must coincide for almost every $t\geq 0$; see the comment on “essential uniqueness” in [CDMG14, Theorem 3.2].

Thus, Loewner chains are in a one-to-one correspondence with Herglotz vector fields modulo changes on null subsets of the time domain $[0,\infty)$.

Also, note that the constant subordination chain $f_{t}(z)=0$ for all $z\in\mathbb{D}$, $t\geq 0$, trivially satisfies (2.1) for every Herglotz vector field.*

Now consider a decreasing Loewner chain $\{f_{t}\}_{t\in[0,T]}$. Then $t\mapsto g_{t}(z):=f_{t}^{-1}(z)$ satisfies the following ordinary differential equation (see again [CDMG14, Theorem 3.2])

[TABLE]

In general, we can solve this equation also for $z\in f_{0}(\mathbb{D})$, but the solution might not be defined for all $t\in[0,T]$ then.

In some cases, it might also be useful to regard the time-reverse version of (2.2). Let $T>0$ and consider

[TABLE]

This equation has a unique solution for every $z\in\mathbb{D}$ and each $h_{t}:\mathbb{D}\to\mathbb{D}$ is a univalent function. We have $h_{T}=g_{T}^{-1}$ (but in general, $h_{t}\not=g_{t}^{-1}$). In the literature, $(h_{t})$ is also called the associated inverse, reverse, or backward Loewner flow, see, e.g., [Law05, p. 95].

3 Infinite-slit limits of radial multiple SLE

In the literature the generalization of SLE to multiple SLE curves is discussed in two approaches, a global and a local approach. While the local approach describes these curves in a framework of partition functions as solutions to a system of partial differential equations, the global approach directly constructs the probability space of the curves by means of the Brownian loop measure.

There are several works on the relation between the two approaches and we refer to [Kar19] and the references therein for the recent progress.

In the simple situation we are looking at, both approaches can be used and we will refer to the work [Dub07]. Here, we need to restrict ourselves to the case $\kappa\in[0,4]$, which guarantees that the SLE curves are indeed globally defined, see [Dub07, Section 8].

3.1 Radial single SLE

Fix some $\kappa\in[0,4]$. The radial Schramm-Loewner evolution SLE($\kappa$) is defined as the evolution of a random simple curve $\gamma:[0,\infty)\to\overline{\mathbb{D}}$ with $\gamma(0)=1$ and $\gamma(0,\infty)\subset\mathbb{D}\setminus\{0\}$. This curve is defined as follows:

Let $B_{t}$ be a standard one-dimensional Brownian motion and consider the radial Loewner equation (of the form (2.2)) with driving function $e^{i\sqrt{\kappa}B_{t}}$, i.e.

[TABLE]

Then $g_{t}$ is a conformal mapping of the form $g_{t}:\mathbb{D}\setminus\gamma[0,t]\to\mathbb{D}$, with $g_{t}(0)=0$ and $g_{t}^{\prime}(0)=e^{t}$, for a random simple curve $\gamma$. As $t\to\infty$ we have $\gamma(t)\to 0$, i.e. $\gamma$ connects the boundary point $p=1$ to the interior point $q=0\in\mathbb{D}$. By conformal invariance, this construction can easily be transferred to any Jordan domain $D$ and any $p\in\partial D$, $q\in D$.

Remark 3.1**.**

If we replace the point $1$ by some $p\in\mathbb{T}$, we simply rotate the driving function $e^{i\sqrt{\kappa}B_{t}}$ to $V(t)=e^{i\sqrt{\kappa}B_{t}}p$. By Itō’s formula, the resulting differential equation can also be written as

[TABLE]

with

[TABLE]

The driving function $V$ is simply the image of the tip of the generated curve $\gamma$ under the mapping $g_{t}$, i.e. $V(t)=g_{t}(\gamma(t))$.

Remark 3.2**.**

For $\kappa>4$, equation (3.1) still yields random conformal mappings of the form $g_{t}:\mathbb{D}\setminus K_{t}\to\mathbb{D}$, but $K_{t}$ is not a simple curve anymore. For the Loewner equation and SLE, we refer the reader to [Law05].

3.2 Radial SLE($\kappa,\varrho$)

SLE can be generalized by adding force points, which leads to SLE($\kappa,\varrho$). Again we consider $\kappa\in[0,4]$. Let $m\in\mathbb{N}$ and $\varrho_{1},...,\varrho_{m}\in\mathbb{R}$. Furthermore, let $x_{1},...,x_{m}\in\mathbb{T}$, $p\in\mathbb{T}\setminus\{x_{1},...,x_{m}\}$, and let $B_{t}$ be a Brownian motion. Then radial SLE($\kappa,\varrho_{1},...,\varrho_{m}$) with force points $x_{1},...,x_{m}$ and starting point $p$ is defined via (3.2) and the driving function $V$ is given by the following generalization of (3.3):

[TABLE]

Note that $X_{j}(t)$ simply solves the Loewner equation, i.e. $X_{j}(t)=g_{t}(x_{j})$. The solution $g_{t}$ describes the evolution of a random simple curve (because SLE($\kappa,\varrho$) can be represented as a weighted SLE($\kappa$) $\gamma$ and we chose $\kappa\in[0,4]$, see [SW05, Section 5]), but $g_{t}$ may not be defined for all $t\geq 0$. In the first place, it is only defined as long as $V(t)-X_{j}(t)\not=0$ for all $j$.

Remark 3.3**.**

*Radial SLE($\kappa,\varrho$) can also be considered for force points within $\mathbb{D}$. In particular, this more general framework unifies radial and chordal SLE, see [SW05, Theorem 3]. This explains the factor $\frac{1}{2}$ for the weights $\frac{\varrho_{j}}{2}$ in the above SDE.

Let $C:\mathbb{D}\to\mathbb{H}=\{z\in\mathbb{C}\,|\,\mathsf{Im}(z)>0\},$ $C(z)=i\frac{1+z}{1-z}$, be the Cayley transform. The chordal Loewner equation*

[TABLE]

*where $U$ is real-valued, yields conformal mappings of the form $g_{t}:\mathbb{H}\setminus K_{t}\to\mathbb{H}$.

Assume that $p,x_{1},...,x_{m}\in\mathbb{T}\setminus\{1\}$. By [SW05, Theorem 3], the sets $K_{t}$ correspond (up to a time change) to $C(\gamma[0,t])$ if $U$ satisfies*

[TABLE]

Note that $Y_{j}(t)=g_{t}(C(x_{j}))$ and $S(t)=g_{t}(i)$.

3.3 Radial multiple SLE

Our setting of radial multiple SLE follows the works [Car03] and [Dub07].

Let $p_{1},...,p_{N}$ be $N$ points on $\mathbb{T}$ and fix again $\kappa\in[0,4]$. We now generate $N$ pairwise disjoint simple curves starting at $p_{1},...,p_{N}$ and growing to [math] via the Loewner equation.

First, fix a time interval $[0,T]$ and some coefficients $\lambda_{1},...,\lambda_{N}\in(0,1)$ with $\lambda_{1}+...+\lambda_{N}=1.$

For $t\in[0,\lambda_{1}T]$, we use the Loewner equation (3.2) to generate a curve $\gamma_{1}$ which is an SLE$(\kappa,2,...,2)$ with starting point $p_{1}$ and force points $p_{2},...,p_{N}$. Denote the driving function $V$ by $V_{1}$ and let $V_{j}(t)=g_{t}(p_{j})$ for $j=2,...,N$.

For $t\in[\lambda_{1}T,(\lambda_{1}+\lambda_{2})T]$, we generate an SLE$(\kappa,2,...,2)$ curve with starting point $V_{2}(\lambda_{1}T)=g_{\lambda_{1}T}(p_{2})$ and force points $V_{1}(\lambda_{1}T)=g_{\lambda_{1}T}(\gamma_{1}(\lambda_{1}T)),V_{3}(\lambda_{1}T)=g_{\lambda_{1}T}(p_{3}),...,V_{N}(\lambda_{1}T)$ via (3.2) (and we use a Brownian motion which is independent from the one used for the interval $[0,T]$). Then $g_{(\lambda_{1}+\lambda_{2})T}$ maps $\mathbb{D}\setminus(\Gamma_{1}\cup\Gamma_{2})$ conformally onto $\mathbb{D}$, where $\Gamma_{1}$ and $\Gamma_{2}$ are two disjoint simple curves starting at $p_{1}$ and $p_{2}$ respectively. We extend $V_{2}$ to $[\lambda_{1}T,(\lambda_{1}+\lambda_{2})T]$ by letting $V_{2}$ be the corresponding driving function, and we extend $V_{1},V_{3},...,V_{N}$ via $V_{j}(t)=g_{t}(p_{j})$. In other words, every $V_{j}$ has the form

[TABLE]

We extend this construction for the time interval $[0,T]$ and we obtain $N$ disjoint simple curves $\Gamma_{1},...,\Gamma_{N}$ starting from $p_{1},...,p_{N}$. Now we can repeat the procedure for the interval $[T,2T]$, $[2T,3T]$, etc., and thus the growth of $N$ random curves $\gamma_{1},...,\gamma_{N}$, connecting $p_{1},...,p_{N}$ to [math], is defined for all $t\in[0,\infty)$.

It has been shown in [Dub07] that this procedure does in fact not depend on $\lambda_{1},...,\lambda_{N}$, $T$, and the order of $(p_{1},...,p_{N})$, i.e. the distribution of $(\gamma_{1},...,\gamma_{N})$ is the same for all possible choices of these parameters (see the case (ii) in [Dub07, Section 8.2] and [Dub07, Theorem 7]). This defines radial multiple SLE($\kappa$) (from $p_{1},...,p_{N}\in\mathbb{T}$ to [math]).

Note that $V_{k}$ satisfies

[TABLE]

if $j\not=k$ and the $j$-th curve is growing, and

[TABLE]

when the $k$-th curve is growing.

In contrast to the above construction, we can also generate the curves simultaneously. Now we use the coefficients $\lambda_{1},...,\lambda_{N}$ as “speeds” in the multi-slit Loewner equation

[TABLE]

One can now derive a commutation relation for the infinitesimal generators for $X_{1},...,X_{n}$ (and use the fact that $s\mapsto\gamma_{1}[0,t]\cup...\cup\gamma_{j}[0,t+s]\cup...\cup\gamma_{N}[0,t]$ describes an SLE($\kappa,2,...,2$) process in $\mathbb{D}\setminus(\gamma_{1}[0,t]\cup...\cup\gamma_{N}[0,t])$), which leads to the following differential equations (see [Dub07, Section 8.2]):

[TABLE]

where $W_{1},...,W_{N}$ are independent Brownian motions. (See also [Gra07, Section 5] for the corresponding calculations in the chordal case.)

Remark 3.4**.**

Informally, we can consider the limit $T\to 0$ for $V_{k}$ in (3.5), (3.6), which leads to a simultaneous growth of the curves, and $dX_{k}$ is obtained as a convex combination of the $N$ terms with weights $\lambda_{1},...,\lambda_{N}$, i.e. we obtain

[TABLE]

where $B_{1},...,B_{N}$ are independent Brownian motions and we used that $B_{k}(ct)=\sqrt{c}W_{k}(t)$ for another standard Brownian motion $W_{k}$.

3.4 Limits of radial multiple SLE

Let $N\in\mathbb{N}$ and $x_{N,1},...,x_{N,N}$ be $N$ points on $\mathbb{T}$, ordered in counter-clockwise direction such that $1$ lies on the circular arc $(x_{N,N},x_{N,1}]$. Furthermore, choose $\lambda_{N,1},...,\lambda_{N,N}\in(0,1)$ such that $\sum_{k=1}^{N}\lambda_{N,k}=1.$

We now grow $N$ radial multiple SLE curves simultaneously for these parameters, i.e. we define $N$ random processes $V_{N,1},...,V_{N,N}$ on $\mathbb{T}$ as the solution of the SDE system

[TABLE]

with $V_{N,k}(0)=x_{N,k}$ and $B_{N,1},...,B_{N,N}$ are $N$ independent standard Brownian motions.

The corresponding $N$-slit Loewner equation

[TABLE]

describes the growth of $N$ multiple SLE curves growing from $x_{N,1},...,x_{N,N}$ to [math].

The function $z\mapsto g_{N,t}(z)$ is a conformal mapping from $\mathbb{D}\setminus(\gamma_{N,1}[0,t]\cup...\cup\gamma_{N,N}[0,t])$ onto $\mathbb{D}$, where the curves $\gamma_{N,k}:[0,\infty)\to\overline{\mathbb{D}}\setminus\{0\}$ are non-intersecting simple curves with $\gamma_{N,k}(0)=x_{N,k}$ and $g_{N,t}$ has the normalization $g_{N,t}(0)=0,$ $g_{N,t}^{\prime}(0)=e^{t}$.

The special case $\lambda_{N,k}=\frac{1}{N}$ for all $k=1,...,N$ (simultaneous growth) leads to

[TABLE]

Problem: We are interested in the limit $N\to\infty$ of the growing curves, i.e. the convergence of $\gamma_{N,1}[0,t]\cup...\cup\gamma_{N,N}[0,t]$ to a set $K_{t}$, more precisely:

Fix some $t>0.$ Under which conditions does the sequence $\mathbb{D}\setminus(\gamma_{N,1}[0,t]\cup...\cup\gamma_{N,N}[0,t])$ of domains converge to a (simply connected) domain $\mathbb{D}\setminus K_{t}$ with respect to kernel convergence? According to Carathéodory’s kernel theorem, this is equivalent to asking for locally uniform convergence of the mappings $g_{N,t}$ to a conformal mapping $g_{t}:\mathbb{D}\setminus K_{t}\to\mathbb{D}.$ Also, we would like to be able to describe $g_{t}$ again by a Loewner equation.

In Section 3.5, we consider the general case (3.7) and obtain a tightness result under certain assumptions on the coefficients $\lambda_{N,k}$.

In Section 3.6, we consider the case (3.9) and we prove convergence of $g_{N,t}$. The processes (3.9) are related to Dyson’s model for unitary random matrices, whose limit behaviour is investigated in [CL01].

3.5 Tightness

Definition 3.5**.**

Fix $T>0$ and let ${\rm\bf P}(\mathbb{T})$ be the space of probability measures on $\mathbb{T}$ endowed with the topology of weak convergence. Note that ${\rm\bf P}(\mathbb{T})$ is a metric space due to the well-known Lévy-Prokhorov metric. We denote by $\mathcal{M}(T)=C([0,T],{\rm\bf P}(\mathbb{T}))$ the space of all continuous measure-valued processes on $[0,T]$ endowed with the topology of uniform convergence.

Let $\delta_{x}$ be the Dirac mass at $x$ and let $\alpha_{N,t}=\sum_{k=1}^{N}\lambda_{N,k}\delta_{V_{N,k}(t)}$. Then equation (3.8) can be written as

[TABLE]

For every $N\in\mathbb{N},$ $\alpha_{N,t}$ can be regarded as a random element from $\mathcal{M}(T)$. We now make three assumptions that allow us to prove tightness of the sequence $\{\alpha_{N,t}\}_{N}$.

The first assumption: There exists $C>0$ such that for every $N\in\mathbb{N}:$

[TABLE]

Next, let $L_{N}:[0,1]\to[0,1]$ be defined by $L_{N}(k/N)=\sum_{j=1}^{k}\lambda_{N,j}$ for $k=0,...,N$ and define $L_{N}(x)$ by linear interpolation for all $x\in[0,1].$ The family $\{L_{N}\}_{N\in\mathbb{N}}$ is uniformly bounded by $1$ and equicontinuous by (a). The Arzelà–Ascoli theorem implies that it is precompact. We will assume that the limit exists:

[TABLE]

While the conditions (a) and (b) define a “convergence of the weights”, we also need a precise meaning of “the convergence of the initial points”. For this purpose, we introduce the “empirical distribution” $\mu_{N,t}=\sum_{k=1}^{N}\frac{1}{N}\delta_{V_{N,k}(t)}.$ Our third assumption is

[TABLE]

for some probability measure $\mu$ on $\mathbb{T}$.

For $x\in\mathbb{T}$, let $[1,x]$ be the circular arc on $\mathbb{T}$ from $1$ to $x$ in counter-clockwise direction. Let $F_{N,t}(x)=\alpha_{N,t}([1,x])$ and $G_{N,t}(x)=\mu_{N,t}([1,x])$ be the cumulative distribution functions. Then, $G_{N,t}$ and $F_{N,t}$ are related as follows:

[TABLE]

Due to (b), assumption (c) also implies the convergence of $\alpha_{N,0}$.

We let $C^{2}(\mathbb{T},\mathbb{C})$ be the space of all twice continuously differentiable functions $f:\mathbb{T}\to\mathbb{C}$.

Theorem 3.6**.**

*Assume that (a), (b), and (c) hold. Let $T>0$. Then the sequences $\{\mu_{N,t}\}_{N}$ and $\{\alpha_{N,t}\}_{N}$ are tight with respect to $\mathcal{M}(T)$.

Furthermore, let $\{\mu_{N_{k},t}\}_{k}$ be a converging subsequence with limit $\mu_{t}.$ Then $\{\alpha_{N_{k},t}\}_{k}$ converges to $\alpha_{t}$ which is defined by*

[TABLE]

where $F_{t}$ and $G_{t}$ are the cumulative distribution functions of the measures $\alpha_{t}$ and $\mu_{t}$ respectively. Finally, $\mu_{t}$ satisfies the (distributional) differential equation

[TABLE]

for all $f\in C^{2}(\mathbb{T},\mathbb{C}).$

The proof is similar to the one of [CL01, Theorem 4.1] (diffusion processes on $\mathbb{T}$) and [dMHS18, Theorem 2.5] (chordal multiple SLE).

Proof.

Proving tightness of the measure-valued processes $\{\mu_{N,t}\}_{N}$ can be reduced to proving tightness of stochastic complex-valued processes, see [RS93, Section 3] and also [Gär88, Section 1.3]:

The sequence $\{\mu_{N,t}\}_{N}$ is tight if

[TABLE]

are tight sequences (with respect to the space space $C([0,T],\mathbb{C})$ with uniform convergence) for all $f\in C^{2}(\mathbb{T},\mathbb{C})$.

First, let $f\in C^{2}(\mathbb{T},\mathbb{C})$. Itō’s formula gives

[TABLE]

Now we write

[TABLE]

Hence we arrive at

[TABLE]

As $f^{\prime}$ and $f^{\prime\prime}$ are bounded and $\lambda_{N,k}\leq C/N$, we see that the sums $B_{N}(t),C_{N}(t)$, and $D_{N}(t)$ converge absolutely (and uniformly on $[0,T]$) to [math] with probability $1$ as $N\to\infty$.

Furthermore, as $xf^{\prime}(x)$ is continuously differentiable, the term $A_{N}(t)$ is uniformly bounded with respect to $N$ and $t$. By the stochastic Arzelà-Ascoli theorem ([Bil99, Thm. 7.3]), we conclude that $\left\{\int_{\mathbb{T}}f(x)\mu_{N,t}(dx)\right\}_{N\in\mathbb{N}}$ is tight. Hence, $\{\mu_{N,t}\}_{N}$ is tight and each limit process satisfies equation (3.13).

Because of relation (3.11) and our assumption (b) it follows that the subsequence $\{\alpha_{N_{k},t}\}_{k}$ converges provided that $\{\mu_{N_{k},t}\}_{k}$ converges, and that relation (3.12) holds for the limit processes. ∎

Remark 3.7**.**

*The assumptions in Theorem 3.6 are basically the same as in the chordal case [dMHS18, Theorem 2.5]. One difference between the radial and the chordal case is the compactness of $\mathbb{T}$ (and thus the compactness of ${\rm\bf P}(\mathbb{T})$) versus the non-compactness of $\mathbb{R}$. In [dMHS18, Example 2.16], the authors describe an example of chordal multiple SLE processes where the corresponding family $\{\alpha_{N,t}\}_{N}$ of measure-valued processes is not tight. Even though $\mu_{N,0}$ and $\alpha_{N,0}$ converge as $N\to\infty$, the rightmost driving function $V_{N,N}(t)$ satisfies $V_{N,N}(t)\to\infty$ as $N\to\infty$ for every $t>0$.

Question: Is there an example of radial multiple SLE data $\{x_{N,k}\}$ and $\{\lambda_{N,k}\}$ such that the process $\{\alpha_{N,t}\}_{N}$ is not tight?*

Next we will show that if $\{\mu_{N_{k},t}\}_{k}$ is a converging subsequence, then $g_{N_{k},t}$ converges as $k\to\infty$.

Let $\lambda$ be the Lebesgue measure on $[0,T]$ and let $\mathcal{N}(T)$ be the space of all finite measures on $\mathbb{T}\times[0,T]$ endowed with the topology of weak convergence. We will need the following control-theoretic result. A proof can be found in [JVST12, Proposition 1] or [MS13, Theorem 1.1].

Theorem 3.8**.**

Let $\{\beta_{N,t}\}_{N\in\mathbb{N}}$ be a sequence of processes from $\mathcal{M}(T)$ and assume that $\beta_{N,t}\lambda(dt)$ converges to $\beta_{t}\lambda(dt)$ within the space $\mathcal{N}(T)$. Denote by $h_{N,t}$ the solution to the Loewner equation

[TABLE]

Then $h_{N,t}$ converges locally uniformly to $h_{t}$ for every $t\in[0,T]$ as $N\to\infty,$ where $(h_{t})_{t\in[0,T]}$ is the unique solution to

[TABLE]

Let $\mathcal{C}$ be the set of all $M(z)=\int_{\mathbb{T}}\frac{x+z}{x-z}\beta(dx)$, where $\beta$ is a probability measure. The measure $\beta$ can be recovered from $M$ by a version of the Stieltjes-Perron inversion formula. Denote the distribution function of $\beta$ by $F(x)$. Then $L\circ F(x)$ is also a distribution function, which corresponds to a measure $\hat{\beta}.$ In this way, we obtain a map $\mathcal{L}:\mathcal{C}\to\mathcal{C},$

[TABLE]

The limit of the Loewner equation can now be described as follows.

Corollary 3.9**.**

Assume that (a), (b), and (c) hold. Let $\{\mu_{N_{k},t}\}_{k}$ be a converging subsequence with limit $\mu_{t}$. Then, as $k\to\infty,$ $g_{N_{k},t}$ converges in distribution w.r.t. locally uniform convergence to $g_{t},$ the solution of the Loewner equation

[TABLE]

where $M_{t}=\int_{\mathbb{T}}\frac{x+z}{x-z}\mu_{t}(dx)$ solves the (abstract) differential equation

[TABLE]

Proof.

For $z\in\mathbb{D}$ we consider $f(x)=\frac{x+z}{x-z}\in C^{2}(\mathbb{T},\mathbb{C})$. Then $\frac{\partial f}{\partial x}=\frac{-2z}{(x-z)^{2}}$, $\frac{\partial f}{\partial z}=\frac{2x}{(x-z)^{2}}$, and we obtain

[TABLE]

Let $M_{t}(z)=\int_{\mathbb{T}}\frac{x+z}{x-z}\,\mu_{t}(dx)$. Then

[TABLE]

Furthermore, let $g_{t}$ be the solution to

[TABLE]

Fix some $t\geq 0$. The canonical mapping $\mathcal{M}(t)\ni\alpha_{s}\mapsto\alpha_{s}\lambda(ds)\in\mathcal{N}(t)$ is continuous. Hence, the Continuous Mapping Theorem (see [Bil99], p. 20) implies that $\alpha_{N_{k},s}\lambda(ds)$ converges in distribution with respect to weak convergence to $\alpha_{s}\lambda(ds)$.

Hence, Theorem 3.8 and again the Continuous Mapping Theorem imply that $g_{N_{k},t}$, which is the solution to (3.10), converges in distribution to $g_{t}$ with respect to locally uniform convergence. ∎

Remark 3.10**.**

The convergence of $\alpha_{N,t}$ and $g_{N,t}$ would follow immediately if we knew that equation (3.15), or equivalently (3.13), had a unique solution. If $\lambda_{N,k}=\frac{1}{N}$, then (3.15) is a standard PDE and uniqueness can be shown easily, see Section 3.6.

3.6 The simultaneous case

Theorem 3.11**.**

Let $\lambda_{N,k}=\frac{1}{N}$ for all $k$ and $N$. Assume $\mu_{N,0}$ converges weakly to a probability measure $\mu.$ Then $\mu_{N,t}$ converges to the deterministic process $\mu_{t}$ defined as the unique solution to the equation

[TABLE]

*for all $f\in C^{2}(\mathbb{T},\mathbb{C}).$

The conformal mappings $g_{N,t}$ converge locally uniformly to $g_{t},$ the solution of the system*

[TABLE]

[TABLE]

Proof.

Clearly, (3.16) and (3.18) are the special cases of (3.13) and (3.15). It only remains to show that equation (3.18) (and hence also equation (3.16)) has a unique solution. This is proven in Theorem 4.5 (for an even more general situation).

∎

Equation (3.18) is in fact the (inviscid) Burgers’ equation, which becomes clear after a change of variables. For $\mathsf{Im}(z)>0$ define $V_{t}(z)=-iM_{t}(e^{iz})$. Then $V_{t}$ maps the upper half-plane into the lower half-plane and we obtain

[TABLE]

Remark 3.12**.**

Let $h_{t}:\mathbb{H}\to\mathbb{H}$ be a lift of $g_{t}(e^{iz}):\mathbb{H}\to\mathbb{D}\setminus\{0\}$ with respect to $e^{iz}:\mathbb{H}\to\mathbb{D}\setminus\{0\}$, i.e. $g_{t}(e^{iz})=e^{ih_{t}(z)}$. Suppose that $t\mapsto h_{t}$ is continuous and $h_{0}(z)\equiv z$. Locally we can write $h_{t}(z)=-i\log(g_{t}(e^{iz}))$ and we obtain the differential equation

[TABLE]

The system (3.19), (3.20) is now the same as in [dMS16, Theorem 1.1], the only difference being that the initial value for $V_{0}$ is not a Cauchy transform in our setting, but of the form

[TABLE]

for a probability measure $\mu^{\prime}$ on $[0,2\pi]$.

Example 3.13**.**

Let $\mu$ be the uniform distribution on $\mathbb{T}$. Then $M_{t}(z)=1$ solves (3.18) and the conformal mapping $g_{t}$ is given by $g_{t}(z)=e^{t}z$, which maps $e^{-t}\mathbb{D}$ conformally onto $\mathbb{D}$.

Example 3.14**.**

We can now determine the limit hulls of the curves in Figure 1, where $\lambda_{N,k}=1/N$ and $\mu_{N,0}\to\delta_{1}$. (The corresponding chordal case is considered in [HK18].) A numerical solution of the system (3.17), (3.18) with $M_{0}(z)=\frac{1+z}{1-z}$ yields:

Recall that every $M_{t}$ has the form

[TABLE]

Theorem 3.15**.**

There exists $T>0$ such that $M_{t}$ and $g_{t}^{-1}$ both extend analytically to $\overline{\mathbb{D}}$, $\mathsf{Re}(M_{t}(x))>0$ for all $x\in\mathbb{T}$, and $\operatorname{supp}\mu_{t}=\mathbb{T}$ for all $t\geq T$.

Proof.

Due to Theorem 4.7, there exists $T>0$ such that, for all $t\geq T$, $M_{t}$ extends analytically to $\overline{\mathbb{D}}$ with $\mathsf{Re}(M_{t}(x))>0$ for all $x\in\mathbb{T}$ and $\operatorname{supp}\mu_{t}=\mathbb{T}$.

Fix any $T_{2}>T$. Instead of (3.17), we consider the time-reverse equation (cf. equation (2.3))

[TABLE]

Recall that we have $f_{T_{2}}=g_{T_{2}}^{-1}$.

For $t\in[0,T_{2}-T]$, the above equation can also be considered on $\mathbb{T}$ and we see that $f_{t}$ can be extended analytically to $\overline{\mathbb{D}}$ with $f_{t}(\mathbb{T})\subset\mathbb{D}$ for all $t\in(0,T_{2}-T]$. Fix $t_{0}\in(0,T_{2}-T]$. As $f_{T_{2}}=\varphi\circ f_{t_{0}}$ for some holomorphic $\varphi:\mathbb{D}\to\mathbb{D}$, we see that $f_{T_{2}}$ can also be extended analytically to $\overline{\mathbb{D}}$ with $f_{T_{2}}(\mathbb{T})\subset\mathbb{D}$.

∎

Remark 3.16**.**

*Assume that $\operatorname{supp}\mu_{0}\not=\mathbb{T}$. Then we can compute the smallest time $t$ with $\operatorname{supp}\mu_{t}=\mathbb{T}$ explicitly.

Recall formula (3.21). Let $x\in\mathbb{R}$ with $x+2k\pi\not\in\operatorname{supp}\mu^{\prime}$ for all $k\in\mathbb{Z}$. Then $V_{0}$ can be extended analytically to a neighbourhood of $x$ and*

[TABLE]

We obtain

[TABLE]

The theory of the real (inviscid) Burgers equation, see [Mil06, p. 77, 78], implies that $\exp(ix)$ belongs to $\operatorname{supp}\mu_{t}$ the first time at $t=T(x)=\frac{-1}{2V_{0}^{\prime}(x)}$. Now put $m=\min_{x\in\mathbb{R}}|V^{\prime}_{0}(x)|>0$. Then $T(x)\leq\frac{1}{2m}$ and we see that $\operatorname{supp}\mu_{t}=\mathbb{T}$ the first time for $t=\frac{1}{2m}$.

Example 3.17**.**

Let $\mu_{0}=\delta_{1}$, i.e. $M_{0}(z)=\frac{1+z}{1-z}$. In this case we can choose $\mu^{\prime}=\delta_{0}$ and we have

[TABLE]

Thus $m=\frac{1}{2}$ and $T=\frac{1}{2m}=1$ is the smallest time with $\operatorname{supp}\mu_{T}=\mathbb{T}$.

Finally, we note that the simultaneous case can also be handled by switching the setting to the upper half-plane where the Loewner equation becomes quite similar to the one describing the evolution of trajectories of certain quadratic differentials. Then one can use a result from [dMHS18] to obtain the limit equation in the upper half-plane, see Appendix A.2.

4 A general Burgers-Loewner equation

Let $\alpha$ be a probability measure on $\mathbb{T}$ and let $S$ be a holomorphic mapping from the right half-plane $RH$ into itself.

The differential equation (3.18) is a special case of the following partial differential equation

[TABLE]

This equation can be interpreted in at least three different ways:

• (A)

Each $M_{t}$ can be written as $M_{t}(z)=\int_{\mathbb{T}}\frac{x+z}{x-z}\mu_{t}(dx)$ for some probability measure $\mu_{t}$ on $\mathbb{T}$. These measures are related to a semigroup with respect to free convolution.

In Section 4.2, we explain this view in more detail. In particular, we see that (4.1) always has a unique solution.

• (B)

As in Section 3.6, the solution $\{M_{t}\}$ can be seen as a Herglotz vector field for the Loewner equation

[TABLE]

The case $S_{\mu}(z)=2z$ describes the infinite-slit limit of radial multiple SLE with equal weights. The measures from (A) in this special case describe the distributions of a (time changed) free unitary Brownian motion, see [AWZ14]. (In particular, see [AWZ14, p. 3490], which gives the $\Sigma$-transforms of these measures. The $\Sigma$-transform is defined below.)

• (C)

Equation (4.1) can also be regarded as a special case of Loewner’s partial differential equation. In case $M_{0}$ is univalent, $\{M_{t}\}_{t\geq 0}$ is a decreasing Loewner chain. In general, the solution is a decreasing subordination chain.

4.1 Free semigroups

Let $\mu$ be a probability measure on $\mathbb{T}$. The moment generating function is a holomorphic function on $\mathbb{D}$ defined by

[TABLE]

The classical independence of random variables leads to the classical convolution, or Hadamard convolution, $\mu\star\nu$, with $\psi_{\mu\star\nu}=\sum_{n=1}^{\infty}\left(\int_{\mathbb{T}}x^{n}\,\mu({\rm d}x)\right)\left(\int_{\mathbb{T}}x^{n}\,\nu({\rm d}x)\right)z^{n}$. Other notions of independence from non-commutative probability theory lead to further convolutions. First, we need to define the $\eta$-transform of $\mu$. Let

[TABLE]

Lemma 4.1** (See Proposition 3.2 in [BB05]).**

Let $\psi\colon\mathbb{D}\to\mathbb{C}$ be holomorphic. The following conditions are equivalent.

1. (1)

There exists a probability measure $\mu$ on $\mathbb{T}$ such that $\psi=\psi_{\mu}$. 2. (2)

$\psi(0)=0$* and $\mathsf{Re}(\psi(z))\geq-\frac{1}{2}$ for all $z\in\mathbb{D}$.*

Let $\eta\colon\mathbb{D}\to\mathbb{C}$ be holomorphic. The following conditions are equivalent.

1. (3)

There exists a probability measure $\mu$ on $\mathbb{T}$ such that $\eta=\eta_{\mu}$. 2. (4)

$\eta(0)=0$* and $\eta$ maps $\mathbb{D}$ into $\mathbb{D}$.*

Recall that we denote by ${\rm\bf P}(\mathbb{T})$ the set of all probability measures $\mu$ on $\mathbb{T}$ (Definition 3.5). We let ${\rm\bf P}_{\times}(\mathbb{T})$ be the set of all $\mu\in{\rm\bf P}(\mathbb{T})$ with $\eta_{\mu}^{\prime}(0)\not=0$, i.e. the first moment of $\mu$ is $\not=0$. Then we can invert $\eta_{\mu}$ in a neighbourhood of [math]. Denote this locally defined function by $\eta_{\mu}^{-1}$. (For the inversion with respect to multiplication, we will always write $\frac{1}{\eta_{\mu}}$.) The $\Sigma$-transform of $\mu$ is defined by

[TABLE]

For two probability measures $\mu,\nu\in{\rm\bf P}_{\times}(\mathbb{T})$, Voiculescu [Voi87] characterized multiplicative free convolution $\boxtimes$ by

[TABLE]

in a neighborhood of [math]. The multiplicative free convolution arises from the notion of free independence of unitary operators. An introduction to free probability theory can be found in [NS06].

Remark 4.2**.**

Another notion of independence of unitary operators, the so called monotone independence, leads to the multiplicative monotone convolution $\mu\rhd\nu$ defined by the composition of the $\eta$-transforms, i.e. $\eta_{\mu\rhd\nu}=\eta_{\mu}\circ\eta_{\nu}$. Monotone independence is closely related to univalent functions. The following two statements are equivalent:

1. (1)

$\eta_{\mu}$* is univalent on $\mathbb{D}$.* 2. (2)

There exists a quantum process $\{X_{t}\}_{t\geq 0}$ of unitary operators with monotonically independent increments such that $\mu$ is the distribution of $X_{1}$.

The precise definitions of the “distribution of an operator”, “quantum processes” and “monotone independence” can be found in [FHS]. In particular, the proof of the above statement is given in [FHS, Sections 5.1-5.4].

$\mu\in{\rm\bf P}_{\times}(\mathbb{T})$ is called (freely) infinitely divisible if for every $n\in\mathbb{N}$ there exists $\mu_{n}\in{\rm\bf P}_{\times}(\mathbb{T})$ such that $\mu=\mu_{n}\boxtimes\cdots\boxtimes\mu_{n}$ ($n$-fold convolution). Freely infinitely divisible distributions can be characterized in the following way.

Theorem 4.3** (See Lemma 6.6 in [BV92]).**

Let $\mu\in{\rm\bf P}_{\times}(\mathbb{T})$. The following three statements are equivalent.

1. (1)

$\mu$* is freely infinitely divisible.* 2. (2)

There exists a weakly continuous $\boxtimes$-convolution semigroup $\{\mu_{t}\}_{t\geq 0}$ (i.e. $\mu_{t+s}=\mu_{t}\boxtimes\mu_{s}$ for all $s,t\geq 0$ and $t\mapsto\mu_{t}$ is continuous with respect to weak convergence) such that $\mu_{0}=\delta_{1}$ and $\mu_{1}=\mu$. 3. (3)

There exists an analytic map $u_{\mu}\colon\mathbb{D}\to\mathbb{C}$ with $\mathsf{Re}(u_{\mu})\geq 0$ such that $\Sigma_{\mu}(z)=\exp(u_{\mu}(z))$.

Moreover, the analytic map $u_{\mu}$ in (3) can be characterized by the Herglotz representation

[TABLE]

where $\alpha\in\mathbb{R}$ and $\rho$ is a finite, non-negative measure on $\mathbb{T}$.

Conversely, for any analytic map $u\colon\mathbb{D}\to\mathbb{C}$ with $\mathsf{Re}(u)\geq 0$, the function $\exp(u(z))$ is the $\Sigma$-transform of some freely infinitely divisible $\mu$.

Remark 4.4**.**

The multiplicative free convolution can also be considered for probability measures $\mu$ with zero mean. Such a measure is infinitely divisible if and only if $\mu$ is the uniform distribution on $\mathbb{T}$, i.e. $\psi_{\mu}(z)\equiv 0$, see [BV92, Lemma 6.1].

4.2 Properties of the Burgers-Loewner equation

Let $\eta_{t}=\eta_{\mu_{t}}$, where $\{\mu_{t}\}$ is a semigroup as in (2) of Theorem 4.3. Furthermore, assume that $u_{\mu}$ does not have the form $u_{\mu}(z)\equiv xi$ for some $x\in\mathbb{R}$. We exclude this simple case which only leads to simple rotations of point measures. Then by (3) of Theorem 4.3, we have

[TABLE]

This yields the differential equation

[TABLE]

We put $M_{t}(z)=\frac{1+\eta_{t}(z)}{1-\eta_{t}(z)}=1+2\psi_{\mu_{t}}$. Then we obtain the partial differential equation

[TABLE]

with $S_{\mu}(z)=u_{\mu}(\frac{1-z}{1+z})$. Note that $S_{\mu}$ maps the right half-plane $RH$ holomorphically into itself. We now consider this equation, the Burgers-Loewner equation, with an arbitrary initial value.

Theorem 4.5**.**

Let $S:RH\to RH$ be holomorphic and $\alpha\in{\rm\bf P}(\mathbb{T})$. Then there exists exactly one solution $\{M_{t}\}_{t\geq 0}$ of holomorphic mappings $M_{t}:\mathbb{D}\to RH$ of

[TABLE]

There exists a $\boxtimes$-semigroup $(\nu_{t})_{t\geq 0}\subset{\rm\bf P}(\mathbb{T})$ such that $M_{t}=M_{0}\circ\eta_{\nu_{t}}$ for all $t\geq 0$.

In Appendix A.1, we briefly describe the “chordal” or “additive” case, which corresponds to the free convolution of probability measures on $\mathbb{R}$.

Proof.

For every $t\geq 0$, the function $\exp(tS(M_{0}(z))$ is the $\Sigma$-transform of a freely infinitely divisible measure $\nu_{t}$ by Theorem 4.3. Thus $\eta_{\nu_{t}}^{-1}(z)=z\exp(tS(M_{0}(z))$ and

[TABLE]

This yields

[TABLE]

Let $N_{t}=M_{0}\circ\eta_{\nu_{t}}$. Then

[TABLE]

We see that equation (4.6) has at least one solution.

Let $M_{t}$ be an arbitrary solution and put $\eta_{t}=\frac{1-M_{t}}{1+M_{t}}$. Then $\eta_{t}$ maps $\mathbb{D}$ holomorphically into $\mathbb{D}$ and satisfies

[TABLE]

where $u=S(\frac{1-z}{1+z})$ maps $\mathbb{D}$ into $RH$. Putting $z=0$ yields that $\eta_{t}(0)=0$ for all $t\geq 0$. We now consider the power series expansions $\eta_{t}(z)=\sum_{n=1}^{\infty}a_{n}(t)z^{n}$, $\eta_{\alpha}(z)=\sum_{n=1}^{\infty}c_{n}z^{n}$, and $u(z)=\sum_{n=0}^{\infty}b_{n}z^{n}$.

The above differential equation yields

[TABLE]

and we obtain a recursive system of differential equations for the coefficients $a_{n}(t)$, namely

[TABLE]

We conclude that each function $t\mapsto a_{n}(t)$ is uniquely determined and thus also $(\eta_{t})_{t\geq 0}$ and $(M_{t})_{t\geq 0}$ is uniquely determined. We conclude $M_{t}=N_{t}$ for all $t\geq 0$. ∎

By the Herglotz representation formula, every $M_{t}$ can be written as

[TABLE]

for a probability measure $\mu_{t}$.

Remark 4.6**.**

*We have $M_{t}=M_{0}\circ\eta_{\nu_{t}}$, i.e. $\psi_{\mu_{t}}=\psi_{\mu_{0}}\circ\eta_{\nu_{t}}$. In other words: $\mu_{t}=\mu_{0}\rhd\nu_{t}$.

Now let us consider instead $\alpha_{t}:=\mu_{0}\boxtimes\nu_{t}$. We have $\eta_{\alpha_{t}}^{-1}(z)=\frac{1}{z}\eta_{\mu_{0}}^{-1}(z)\eta_{\nu_{t}}^{-1}(z)$ and locally*

[TABLE]

This yields

[TABLE]

Let $\hat{M}_{t}=1+2\psi_{\alpha_{t}}(z)=F(\eta_{\alpha_{t}})$ with $F(z)=\frac{1-z}{1+z}$. (Note that $F^{-1}(z)=F(z)$.) Put $\hat{S}=S\circ M_{0}\circ F$, which maps $RH$ into $RH$. Then $\hat{M}_{t}$ satisfies an equation of the same type as (4.6):

[TABLE]

For $\mathsf{Im}(z)>0$ define $V_{t}(z)=-iM_{t}(e^{iz})$. Then $V_{t}$ maps the upper half-plane into the lower half-plane and we obtain

[TABLE]

with $U(z)=iS(iz)$, which maps $-\mathbb{H}$ into $\mathbb{H}$.

Theorem 4.7**.**

Let $\{M_{t}\}_{t\geq 0}$ be the solution to (4.6).

• (a)

$\{M_{t}\}_{t\geq 0}$* is a (decreasing and normalized) subordination family.*

• (b)

There exists $T>0$ such that, for all $t\geq T$, $\operatorname{supp}\mu_{t}=\mathbb{T}$ and $M_{t}$ can be extended analytically to $\overline{\mathbb{D}}$ with $\mathsf{Re}(M_{t}(x))>0$ for all $x\in\mathbb{T}$. Furthermore, $\mu_{t}$ is absolutely continuous for all $t\geq T$.

• (c)

We have $M_{t}(z)\to 1$ locally uniformly as $t\to\infty$. Equivalently, $\mu_{t}$ converges weakly to the uniform distribution on $\mathbb{T}$ as $t\to\infty$.

Remark 4.8**.**

The fact that free convolution also leads to subordination was observed by Voiculescu in [Voi93, Proposition 4.4] for the case of compactly supported measures. The general case was considered in [Bia98, Theorem 3.1].

Proof.

We start with (a). Equation (4.6) is Loewner’s partial differential equation (2.1) with Herglotz vector field $h(z,t)=S(M_{t}(z))$. We have $h(0,t)=S(M_{t}(0))=S(1)=:a\in RH$. Theorem 2.2 implies that $\{M_{t}\}_{t\geq 0}$ is a normalized subordination family. This can also be seen directly from the proof of Theorem 4.5.

Next we prove (b) and we use the function $V_{t}$ satisfying (4.8).

Let $z_{0}\in\mathbb{H}$ and denote by $z(t)=z(t;z_{0})$ the solution to

[TABLE]

A simple calculation shows that $\frac{d}{dt}V_{t}(z(t))=0$. Hence $V_{t}(z(t))=V_{0}(z_{0})$ for all $t\geq 0$ and $\ddot{z}(t)=0$, so

[TABLE]

Note that $\mathsf{Im}(-U(V_{0}(z_{0})))<0$. So $z(t)$ will hit the real axis at some $x(z_{0})$ at a certain time $t=t_{0}(z_{0})$. Clearly, $t_{0}(z_{0})>0$ as $z_{0}\in\mathbb{H}$. Furthermore, as $V_{t}(z)=V_{t}(z+2\pi)$ for all $z\in\mathbb{H}$ and $t\geq 0$, we have $z(t;z_{0}+2\pi)=z(t;z_{0})+2\pi$ and thus $t_{0}(z_{0}+2\pi)=t_{0}(z_{0})$ for all $z_{0}\in\mathbb{H}$.

Next we note that $z(t)$ is defined for all $t\geq 0$. This shows that $V_{t_{0}}(z_{0})$ can be extended analytically to a neighbourhood of $x(z_{0})$. Now consider $T=\sup_{x\in\mathbb{R}}t_{0}(i+x)$. Clearly, $T>0$ and we have $T<\infty$ since $x\mapsto t_{0}(i+x)$ is $2\pi$-periodic.

Denote the holomorphic function $z_{0}\mapsto z(t;z_{0})$ by $f_{t}(z_{0})$. We have $V_{t}\circ f_{t}=V_{0}$.

For $t\geq T$, $f_{t}$ maps a subdomain of $\mathsf{Im}(z)>1$ onto the upper half-plane. Hence, $V_{t}$ can be extended analytically to $\mathbb{R}$ and $\mathsf{Im}(V_{t}(x))<0$ for all $x\in\mathbb{R}$. This shows that $M_{t}$ can be extended analytically to $\overline{\mathbb{D}}$ and $\mathsf{Re}(M_{t}(x))>0$ for all $x\in\mathbb{T}$ and $t\geq T$.

Let $t\geq T$. The analytic extension of $\mathsf{Re}(M_{t})$ to the boundary $\mathbb{T}$ and the fact that $\mathsf{Re}(M_{t}(x))>0$ on $\mathbb{T}$ imply that $\operatorname{supp}\mu_{t}=\mathbb{T}$. This follows from the inversion formula (see [Akh65, p.91])

[TABLE]

for all $0\leq\alpha\leq\beta\leq 2\pi$. Hence $\operatorname{supp}\mu_{t}=\mathbb{T}$ for all $t\geq T$. Furthermore, the function $[0,2\pi]\ni x\mapsto\mu_{t}([1,e^{ix}])$ is continuously differentiable and thus absolutely continuous. Hence $\mu_{t}$ is absolutely continuous (with respect to the uniform distribution on $\mathbb{T}$) for all $t\geq T$.

Finally we show (c). The coefficient function $a_{m}(t)$ in the proof of Theorem 4.5 satisfies a differential equation of the form $\dot{a}_{m}(t)=-mb_{0}a_{m}(t)+f_{m}(t)$ and $a_{1}(t)=c_{1}e^{-b_{0}t}$. We have $b_{0}=S(1)\in RH$. By induction we see that $f_{m}(t)$ is a linear combination of decreasing exponential functions and thus also $a_{m}(t)$ is a linear combination of decreasing exponential functions and we conclude that $\lim_{t\to\infty}a_{m}(t)=0$ for all $m\geq 1$.

This implies that $\frac{1-M_{t}(z)}{1+M_{t}(z)}\to 0$ and thus $M_{t}(z)\to 1$ locally uniformly. Note that the constant function $1$ is the Herglotz function of the uniform distribution, i.e. $1\equiv\int_{0}^{2\pi}\frac{e^{ix}+z}{e^{ix}-z}\frac{dx}{2\pi}$ on $\mathbb{D}$. Finally, locally uniform convergence of Herglotz functions is equivalent to weak convergence of the probability measures, see [FHS, Lemma 2.11]. ∎

Recall that $M_{t}$ has the form $M_{t}=M_{0}\circ\eta_{\nu_{t}}$, where $(\eta_{\nu_{t}})_{t\geq 0}$ is a decreasing Loewner chain.

Due to Theorem 4.3, each $f_{t}:=\eta_{\nu_{t}}^{-1}$ extends analytically to $\mathbb{D}$. So one might wonder whether $(f_{t})_{t\geq 0}$ is an increasing Loewner chain on $\mathbb{D}$.

This is the case in the simple example $\eta_{\nu_{t}}(z)=e^{-t}z$, but the following example shows that it is not true in general.

Example 4.9**.**

Assume that $\eta_{\mu}$ extends to a conformal mapping from $\mathbb{C}\setminus(-\infty,-1]$ onto $\mathbb{D}$ and that $\eta_{\mu}^{\prime}(0)>0$. Then $f:=\eta_{\mu}^{-1}$ is $4$ times the Koebe function, $f(z)=\frac{4z}{(1-z)^{2}}$.

Due to Theorem 4.3, $\mu$ is freely infinitely divisible and $u_{\mu}(z)=\log(f(z)/z)=\log 4-2\log(1-z)$ maps $\mathbb{D}$ into $RH$.

Now let $(\eta_{\nu_{t}})_{t\geq 0}$ be the decreasing Loewner chain from Theorem 4.5 with $S(z)=z$ and $M_{0}(z)=u_{\mu}(z)$. Due to the proof of Theorem 4.5 we have

[TABLE]

and $f_{1}=f$.

Now assume that $(f_{t})_{t\geq 0}$ is an increasing Loewner chain. Then, for $t>1$, the function $f_{t}$ maps $\mathbb{D}$ conformally onto a simply connected domain that contains $\mathbb{C}\setminus(-\infty,-1]$ with $f_{t}(0)=0$ and $f^{\prime}_{t}(0)=4^{t}$. This implies that $f_{t}(\mathbb{D})=\mathbb{C}\setminus(-\infty,-4^{t-1}]$ and $f_{t}=4^{t-1}f_{1}$. But (4.9) shows that $t\mapsto f_{t}/4^{t-1}$ is not constant on $[1,\infty)$. Hence, $(f_{t})_{t\geq 0}$ is not an increasing Loewner chain.

The figure below illustrates $f_{t}$ and it seems that $(f_{t})_{0\leq t\leq 1}$ is indeed (a part of) an increasing Loewner chain.

Appendix A Appendix

A.1 Free additive semigroups

We briefly describe the “chordal” (or “additive”) analogue of equation (4.6).

Let $\mu$ be a probability measure on $\mathbb{R}$. The Cauchy transform (or Stieltjes transform) of a probability measure $\mu$ on $\mathbb{R}$ is given by

[TABLE]

The $F$-transform of $\mu$ is simply defined by $F_{\mu}(z)=1/G_{\mu}(z)$ for $z\in\mathbb{H}=\{z\in\mathbb{C}\,|\,\mathsf{Im}(z)>0\}$.

In some truncated cone in $\mathbb{H}$, we can invert $G_{\mu}$ (with respect to composition). Denote the inverse by $G_{\mu}^{-1}(z)$. The $R$-transform $R_{\mu}$ is defined by $R_{\mu}(z)=G_{\mu}^{-1}(z)-\frac{1}{z}$. Instead of $R_{\mu}$, one might also regard the Voiculescu transform $\varphi_{\mu}(z)$ defined by $\varphi_{\mu}(z)=F_{\mu}^{-1}(z)-z=G_{\mu}^{-1}(1/z)-z=R_{\mu}(1/z)$.

For two probability measures $\mu$ and $\nu$ on $\mathbb{R}$, the additive free convolution $\boxplus$ is defined by

[TABLE]

A probability measure $\mu$ on $\mathbb{R}$ is called (freely) infinitely divisible if for every $n\in\mathbb{N}$ there exists $\mu_{n}$ such that $\mu=\mu_{n}\boxplus\cdots\boxplus\mu_{n}$ ($n$-fold convolution).

Theorem A.1** (Theorem 5.10 in [BV93]).**

For a probability measure $\mu$ on $\mathbb{R}$, the following statements are equivalent.

1. (1)

$\mu$* is infinitely divisible with respect to free convolution $\boxplus$.* 2. (2)

$\mu=\mu_{1}$* for a weakly continuous $\boxplus$-semigroup $\{\mu_{t}\}_{t\geq 0}$.* 3. (3)

For any $t>0$, there exists a probability measure $\mu^{\boxplus t}$ with the property $\varphi_{\mu^{\boxplus t}}(z)=t\varphi_{\mu}(z).$ 4. (4)

$R_{\mu}$* extends to a Pick function, i.e. an analytic map of $\mathbb{H}$ into $\mathbb{H}\cup\mathbb{R}$.* 5. (5)

There exist $\gamma\in\mathbb{R}$ and a finite, non-negative measure $\rho$ on $\mathbb{R}$ such that

[TABLE]

The pair $(\gamma,\rho)$ is unique.

Conversely, given $\gamma\in\mathbb{R}$ and a finite, non-negative measure $\rho$ on $\mathbb{R}$, there exists a unique $\boxplus$-infinitely divisible distribution $\mu$ which has the Voiculescu transform of the form (5).

Let $\{\mu_{t}\}_{0\leq t}$ be a $\boxplus$-semigroup and let $G_{t}=G_{\mu_{t}}$. Then

[TABLE]

This is a special case of Loewner’s partial differential equation on the upper half-plane $\mathbb{H}$ due to property (4) of the above theorem.

Put $F_{t}=F_{\mu_{t}}=1/G_{t}$. Then

[TABLE]

This equation corresponds to (4.5) and we obtain the analogue of (4.6) by regarding an arbitrary initial value $F_{0}=F_{\alpha}$ for some probability measure $\alpha$.

A.2 Radial multiple SLE in the upper half-plane

By Remark 3.3, we can regard radial multiple SLE also in $\mathbb{H}$ by using the Cayley transform $C$ (provided that $x_{N,k}$ is not $1$) and switching to the chordal multi-slit Loewner equation

[TABLE]

with

[TABLE]

If we drop the second $dt$-term for $U_{N,k}$, we obtain chordal multiple SLE. Questions concerning the limit $N\to\infty$ for this chordal case are discussed in [dMHS18].

However, we can also use [dMHS18] to obtain a statement about the limit equation for the radial multiple SLE mappings if $\lambda_{N,k}=1/N$ for all $k$ and $N$. In this case, the evolution of the curves is quite similar to the evolution of trajectories of a certain quadratic differential and the proof of [dMHS18, Theorem 3.3] can be easily adjusted to get the following result.

In contrast to Theorem 3.11, however, we only obtain a convergent subsequence and we need an additional condition on the starting points of the SLE curves.

Theorem A.2**.**

Let $\lambda_{N,k}=1/N$ and $y_{N,k}=C(x_{N,k})$. Assume that there exists $M>0$ such that $y_{N,k}\in[-M,M]$ for all $N$ and $k$. Finally, assume that

[TABLE]

with respect to weak convergence, where $\mu$ is a probability measure on $\mathbb{R}$. Then there exists a $T>0$ and a subsequence $\{g_{N_{k},t}\}_{k}$ which converges for every $t\in[0,T]$ in distribution with respect to locally uniform convergence and the limit process $g_{t}$ satisfies

[TABLE]

Proof.

By comparing the SDE for $U_{N,k}$ with the one for $V_{N,k}$ in [dMHS18, Section 3], we see that our setting corresponds to $M_{N}=1$, $\alpha_{N,1}=(\kappa-4-2N)/2$ and $s_{N,1}=i$, i.e. $\sigma_{N,0}=\frac{\kappa-4-2N}{2N}\delta_{i}\to-\delta_{i}=\sigma$. Furthermore, $2(\lambda_{N,k}+\lambda_{N,j})=4/N$ leads to an additional factor $2$ in front of $\frac{\partial}{\partial z}M_{t}(z)\cdot M_{t}(z)$. The proof is now analogous to the proof of [dMHS18, Theorem 3.3]. ∎