Controlled Loewner-Kufarev Equation Embedded into the Universal Grassmannian

Takafumi Amaba\flat\flat$\flat$ This work was supported by JSPS KAKENHI Grant
Number 15K17562. 1 and Roland Friedrich\sharp\sharp$\sharp$Supported by the
ERC advanced grant “Noncommutative distributions in free probability”.2

1Fukuoka University, 8-19-1 Nanakuma, Jônan-ku, Fukuoka, 814-0180, Japan.

2Saarland University, Faculty of Mathematics, D-66123 Saarbruecken, Germany

> Abstract. We introduce the notion of controlled Loewner-Kufarev equations
> and discuss aspects of the algebraic nature of the equation embedded into
> the (Sato)-Segal-Wilson Grassmannian. Further, we relate it to conformal
> field theory (CFT) and free probability.

000 2010 Mathematics Subject Classification. Primary 35Q99; Secondary 30F10,
35C10, 58J65 000 Key words and phrases. Loewner-Kufarev equation,
Grassmannian, Conformal field theory, Witt algebra, Free probability theory,
Faber polynomial, Grunsky coefficient, Signature

##  1\. Introduction

C. Loewner [24] and P. P. Kufarev [25] initiated a theory which was then
further extended by C. Pommerenke [31], which shows that given any
continuously increasing family of simply connected domains containing the
origin in the complex plane, the inverses of the Riemann mappings associated
to the domains are described by a partial differential equation, the so-called
Loewner-(Kufarev) equation

$\frac{\partial}{\partial t}f_{t}(z)=zf_{t}^{\prime}(z)p(t,z)$

(see Section 2.2 for details). More recently, I. Markina and A. Vasil’ev [27,
29] considered the so-called alternate Loewner-Kufarev equation, which
describes not necessarily increasing chains of domains.

In this paper, we further generalise it and introduce a class of controlled
Loewner-Kufarev equations,

$\mathrm{d}f_{t}(z)=zf_{t}^{\prime}(z)\\{\mathrm{d}x_{0}(t)+\mathrm{d}\xi(\mathbf{x},z)_{t}\\},\quad
f_{0}(z)\equiv z\in\mathbb{D}_{0},$

where $\mathbb{D}_{0}$ is the unit disc in the complex plane centred at zero,
$x_{0},x_{1},x_{2},\cdots$ are given functions which will be called the
driving functions, $\mathbf{x}=(x_{1},x_{2},\cdots)$ and
$\xi(\mathbf{x},z)_{t}=\sum_{n=1}^{\infty}x_{n}(t)z^{n}$.

In writing down the controlled Loewner-Kufarev equation, the right hand side
becomes $-\sum_{n=0}^{\infty}(L_{n}f)(z)\mathrm{d}x_{n}$, where
$L_{n}=-z^{n+1}\partial/(\partial z)$ for $n\in\mathbb{Z}$, the $L_{n}$ are
exactly the generators of the non-negative part of the Witt algebra, or put
differently, central charge zero Virasoro algebra. Therefore, we are going to
consider an extension of [9], where the second author established and studied
the role of Lie vector fields, boundary variations and the Witt algebra in
connection with the Loewner-Kufarev equation.

So, let us start first with the classical work of A. A. Kirillov and D. V.
Yuriev [17] / G.B. Segal and G. Wilson [34] / N. Kawamoto, Y. Namikawa, A.
Tsuchiya and Y. Yamada [16] which will be also fundamental and basic in the
present context, in particular in understanding the appearance of the Virasoro
algebra with nontrivial central charge.

A. A. Kirillov and D. V. Yuriev [17], constructed a highest weight
representation of the Virasoro algebra, where the representation space is
given by the space of all holomorphic sections of an analytic line bundle over
the orientation-preserving diffeomorphism group $\mathrm{Diff}_{+}S^{1}$ of
the unit circle $S^{1}$ (modulo rotations). They also gave an embedding of
$(\mathrm{Diff}_{+}S^{1})/S^{1}$ into the infinite dimensional Grassmannian.
In fact, this embedding is an example of a construction of solutions to the
KdV hierarchy found by I.Krichever [21], which we address in Section 3.2.

If we embed a univalent function on the unit disc $\mathbb{D}_{0}$ into the
infinite dimensional Grassmannian, by the methods of Kirillov-Yuriev [17],
Krichever [21], or Segal-Wilson[34], then one needs to track the Faber
polynomials and Grunsky coefficients associated to the univalent function. In
general, it is not easy to calculate them from the definition.

One of our main results is, however, the following.

######  Theorem 1.1 (see Proposition 2.10 and Proposition 2.12).

The Faber polynomials and Grunsky coefficients associated to solutions of the
controlled Loewner-Kufarev equation satisfy linear differential equations, and
the Grunsky coefficients can be explicitly calculated.

R. Friedrich [9] proposed to lift the embedded Loewner-Kufarev equation to the
determinant line bundle over the Sato-Segal-Wilson Grassmannian
$\mathrm{Gr}(H)$, as a natural extension of the “Virasoro Uniformisation”
approach by M. Kontsevich [18] / R. Friedrich and J. Kalkkinen [8] to
construct generalised stochastic / Schramm-Loewner evolutions [33] on
arbitrary Riemann surfaces, which would also yield a connection with conformal
field theory in the spirit of [34, 16]. Let us also mention the work of B.
Doyon [3], who uses conformal loop ensembles (CLE), which can be nicely
related with the content of the present article.

In [29], I. Markina and A. Vasil’ev established basic parts of this program,
by considering embedded solutions to the Loewner-Kufarev equation into the
Segal-Wilson Grassmannian and related the dynamics therein with the
representation of the Virasoro algebra, as discussed by Kirillov-Yuriev [17].

Further, they considered the tau-function associated to the embedded solution
as a lift to the determinant line bundle.

As observed and briefly discussed in [18, 8], the generator of the stochastic
Loewner equation is hypo-elliptic.

This observation was more recently worked out and detailed by J. Dubédat [4,
5]. Also, I. Markina, I. Prokhorov and A. Vasil’ev [26] observed and discussed
the sub-Riemannian nature of the coefficients of univalent functions. This
connects nicely with the general theory of hypo-elliptic flows, as explained
e.g. in the book by F. Baudoin [1], and which led to propose a connection of
the global geometry of (stochastic) Loewner-Kufarev equations with rough
paths.

Now, in the theory of rough paths (see e.g., the introduction in [23]), one of
the central objects of consideration is the following controlled differential
equation:

(1.1) $\mathrm{d}Y_{t}=\varphi(Y_{t})\mathrm{d}X_{t},$

where $X_{t}$ is a continuous path in a normed space $V$, called the input of
(1.1). On the other hand, the path $Y_{t}$ is called the output of (1.1). When
we deal with this equation, an important object is the signature of the input
$X_{t}$, with values in the (extended) tensor algebra associated with $V$ and
which is written in the following form:

$S(X)_{s,t}:=(1,X_{s,t}^{1},X_{s,t}^{2},\cdots,X_{s,t}^{n},\cdots),\quad
s\leqslant t.$

If $X_{t}$ has finite variation with respect to $t$, then each $X_{s,t}^{n}$
is the $n$th iterated integral of $X_{t}$ over the interval $[s,t]$. With this
object, a combination of the Magnus expansion and the Chen-Strichartz
expansion theorem (see e.g., [1, Section 1.3]) tells us that the output
$Y_{t}$ is given as the result of the action of $S(X)_{0,t}$ applied to
$Y_{0}$.

Heuristically, we may say that a ‘group element’ $S(X)$ in some big ‘group’
acts on some element in the (extended) tensor algebra $T((V))$ which gives the
output $Y_{t}$, or it might be better to say that the vector field $\varphi$
defines how the ‘group element’ acts on the algebra. In this spirit, we would
like to describe such a picture in the context of controlled Loewner-Kufarev
equations.

For this, we extract the algebraic structure of the controlled Loewner-Kufarev
equation. If we regard the driving functions $x_{0},x_{1},x_{2},\cdots$ just
as letters in an alphabet then it turns out that explicit expressions for the
associated Grunsky coefficients are given by the algebra of formal power
series, where the space of coefficients is given by words over this alphabet.
It is worth mentioning that the action of the words over this alphabet will be
actually given by the negative part of the Witt generators. Thus the action of
the signature encodes many actions of such negative generators. This can be
used to derive a formula for $f_{t}(z)$ as the signature ‘applied’ to the
initial data $f_{0}(z)\equiv z$ (see Theorem 3.8).

However, the story so far lets us ask how the signature associated with the
driving functions describes the corresponding tau-function rather than the
$f_{t}$ itself.

######  Theorem 1.2 (see Theorem 3.9).

Along the solution of the controlled Loewner-Kufarev equation, the associated
tau-function can be written as the determinant of a quadratic form of the
signature.

Let us now summarise the structure of the paper. In Section 2, we formulate
solutions $f_{t}(z)$ to controlled Loewner-Kufarev equations. We add also a
brief review of the classical Loewner-Kufarev equation, and then explain how
the classical one is recovered by the controlled Loewner-Kufarev equation. We
track the variation of the Taylor-coefficients of $f_{t}$ and also the Faber
polynomials and Grunsky coefficients. In Section 3, we first recall briefly
basics of the Segal-Wilson Grassmannian and Krichever’s construction. After
that, we describe how a univalent function on $\mathbb{D}_{0}$ is embedded
into the Grassmannian. We extract the algebraic structure of the controlled
Loewner-Kufarev equations in order to obtain Theorem 3.9. Finally, in Section
4, we establish and briefly discuss connections of the material considered so
far with CFT and free probability. All fundamental things about Faber
polynomials and Grunsky coefficients are wrapped up in Appendix A. Several
proofs of propositions are also relegated to Appendix A, because they tend to
be rather long (in particular, the proof of Proposition 2.9 will be very long)
as we intended not to interrupt the flow of the principal story.

##  2\. Controlled Loewner-Kufarev equation

###  2.1. Definition of solutions to controlled Loewner-Kufarev equations

Given functions $x_{1},x_{2},\cdots:[0,T]\to\mathbb{C}$, we will write
$\mathbf{x}:=(x_{1},x_{2},\cdots)$ and

$\xi(\mathbf{x},z)_{t}:=\sum_{n=1}^{\infty}x_{n}(t)z^{n}\quad\text{for
$z\in\mathbb{C}$}$

if it converges. If $A:[0,T]\to\mathbb{C}$ is of bounded variation, we write
$\mathrm{d}A$ or $A(\mathrm{d}t)$ (when emphasising the coordinate $t$ on
$[0,T]$) the associated complex-valued Lebesgue-Stieltjes measure on $[0,T]$,
and the total variation measure will be denoted by $|\mathrm{d}A|$.

######  Definition 2.1.

Let $T>0$. Suppose that $x_{0}:[0,T]\to\mathbb{R}$,
$x_{1},x_{2},\cdots:[0,T]\to\mathbb{C}$ are continuous and of bounded
variations, and $x_{0}(0)=0$. Let $f_{t}:\mathbb{D}_{0}\to\mathbb{C}$ be
conformal mappings for $0\leqslant t\leqslant T$. We say
$\\{f_{t}\\}_{0\leqslant t\leqslant T}$ is a solution to

(2.1)
$\begin{split}\left\\{\begin{array}[]{l}\displaystyle\mathrm{d}f_{t}(z)=zf_{t}^{\prime}(z)\\{\mathrm{d}x_{0}(t)+\mathrm{d}\xi(\mathbf{x},z)_{t}\\},\\\
f_{0}(z)\equiv z\in\mathbb{D}_{0}\end{array}\right.\end{split}$

if

  * (1)

$f_{0}(z)\equiv z$ for $z\in\mathbb{D}_{0}$;

  * (2)

$\sum_{n=1}^{\infty}n\int_{[0,T]}|\mathrm{d}x_{n}|(t)z^{n}$ converges
uniformly in $z$ on each compact set in $\mathbb{D}_{0}$;

  * (3)

for each compact set $K\subset\mathbb{D}_{0}$, the mapping $[0,T]\ni t\mapsto
f_{t}^{\prime}|_{K}\in C(K)$ is continuous with respect to the uniform norm on
$K$;

  * (4)

it holds that

$\begin{split}f_{t}(z)-z=\int_{0}^{t}zf_{s}^{\prime}(z)\big{\\{}\mathrm{d}x_{0}(s)+\mathrm{d}\xi(\mathbf{x},z)_{s}\big{\\}},\quad(t,z)\in[0,T]\times\mathbb{D}_{0}.\end{split}$

In the sequel, we refer to equation (2.1) as a controlled Loewner-Kufarev
equation (with driving paths $x_{0}$ and $\mathbf{x}:=(x_{1},x_{2},\cdots)$).

######  Remark 2.1.

  * (a)

We do not put the univalency condition into the notion of solution. However,
the condition (1) and (3) assure that $f_{t}$, $0\leqslant t\leqslant T$ are
univalent on some open neighbourhood of $0$ for some $T>0$.

  * (b)

Under the conditions of Definition 2.1–(2), the function
$t\mapsto\xi(\mathbf{x},z)_{t}$ is continuous and of finite variation for each
$z\in\mathbb{D}_{0}$.

  * (c)

Let $\phi:\mathbb{N}\times[0,T]\to\mathbb{R}$ be a bounded measurable function
and $r\in[0,1)$. Under the conditions of Definition 2.1–(2),
$\mu([s,t]):=\sum_{k=1}^{\infty}\int_{[s,t]}\phi(k,u)r^{k}\mathrm{d}x_{k}(u)$
defines a finite signed measure $\mu$. We will denote it by
$\mu(\mathrm{d}t)=:\sum_{k=1}^{\infty}\phi(k,t)r^{k}\mathrm{d}x_{k}(t)$. One
can check that the signed measure
$\sum_{k=1}^{N}\phi(k,t)r^{k}\mathrm{d}x_{k}(t)$ converges to
$\sum_{k=1}^{\infty}\phi(k,t)r^{k}\mathrm{d}x_{k}(t)$ weakly as $N\to\infty$,
and hence we may write for each $z\in\mathbb{D}_{0}$,

$\mathrm{d}\xi(\mathbf{x},z)_{t}=\sum_{k=1}^{\infty}z^{k}\mathrm{d}x_{k}(t).$

######  Remark 2.2.

Let $\\{f_{t}\\}_{0\leqslant t\leqslant T}$ be a solution to the controlled
Loewner-Kufarev equation (2.1). Then the following holds:

  * (a)

For each $w\in\mathbb{C}$ and a compact set $K\subset\mathbb{D}_{0}$, the
mapping $t\mapsto\sup_{z\in K}|f_{t}^{\prime}(z)-w|$ is continuous since we
have

$\begin{split}&\Big{|}\sup_{z\in K}|f_{t}^{\prime}(z)-w|-\sup_{z\in
K}|f_{s}^{\prime}(z)-w|\Big{|}\leqslant\sup_{z\in
K}|f_{t}^{\prime}(z)-f_{s}^{\prime}(z)|\end{split}$

and the condition in Definition 2.1–(3).

  * (b)

The mapping $t\mapsto f_{t}$ is continuous with respect to the uniform norm on
each compact set in $\mathbb{D}_{0}$. In fact, by using Definition 2.1–(4), we
have for each compact set $K\subset\mathbb{D}_{0}$,

$\begin{split}&\sup_{z\in K}|f_{t}(z)-f_{s}(z)|\leqslant\sup_{0\leqslant
u\leqslant
T}\|f_{u}^{\prime}|_{K}\|_{\infty}\int_{s}^{t}\big{\\{}|\mathrm{d}x_{0}(u)|+\sum_{n=1}^{\infty}|z_{0}|^{n}|\mathrm{d}x_{n}(u)|\big{\\}},\end{split}$

where $z_{0}\in K$ is such that $|z_{0}|=\sup_{z\in K}|z|$. By Definition
2.1–(3) (or the above remark (a)), we find that $\sup_{0\leqslant u\leqslant
T}\|f_{u}^{\prime}|_{K}\|_{\infty}<+\infty$. Therefore, by using the dominated
convergence theorem with Definition 2.1–(2), the right-hand side goes to zero
as $|t-s|\to 0$.

######  Definition 2.2.

We say $\\{f_{t}\\}_{0\leqslant t\leqslant T}$ is a univalent solution to the
controlled Loewner-Kufarev equation if it is a solution to (2.1) and $f_{t}$
is a univalent function on $\mathbb{D}_{0}$ for each $0\leqslant t\leqslant
T$.

###  2.2. Loewner-Kufarev equation as a controlled Loewner-Kufarev equation

######  Definition 2.3.

Suppose that $\Omega(t)\subset\mathbb{C}$ is given for each $0\leqslant
t\leqslant T$. $\\{\Omega(t)\\}_{0\leqslant t\leqslant T}$ is called a Loewner
subordination chain if

  * (1)

$0\in\Omega(s)\subsetneq\Omega(t)$ for each $0\leqslant s<t\leqslant T$;

  * (2)

$\Omega(t)$ is a simply connected domain (i.e., open, connected and simply
connected) for each $t\in[0,T]$;

  * (3)

(Continuity in the sense of Carathéodory, under the conditions (1) and (2)):
For each $t\in[0,T]$ and any sequence $0\leqslant t_{n}\uparrow t$,
$\cup_{n=1}^{\infty}\Omega(t_{n})=\Omega(t)$.

For the following Definition 2.4, cf. specifically [31, Chapter 6, Section
6.1, pp. 156–157; Chapter 2, Section 2.1, p. 35 and Lemma 2.1].

######  Definition 2.4 ([31]).

Let $f_{t}:\mathbb{D}_{0}\to\mathbb{C}$ be given for $0\leqslant t\leqslant
T$. Then $\\{f_{t}\\}_{0\leqslant t\leqslant T}$ is called a Loewner chain if

  * (1)

$f_{t}$ is analytic and univalent on $\mathbb{D}_{0}$ for each $0\leqslant
t\leqslant T$;

  * (2)

$f_{t}(z)=\mathrm{e}^{t}z+a_{2}(t)z^{2}+\cdots$ for $z\in\mathbb{D}_{0}$;

  * (3)

$f_{s}(\mathbb{D}_{0})\subset f_{t}(\mathbb{D}_{0})$ for each $0\leqslant
s<t\leqslant T$.

The above chains $\\{\Omega(t)\\}$ and $\\{f_{t}\\}$ are known to be in one-
to-one correspondence via the relation $\Omega(\tau)=f_{t}(\mathbb{D}_{0}),$
where $t=\log f_{\tau}^{\prime}(0)$ is a time-reparametrisation to satisfy
Definition 2.4–(2) (see [31, Chapter 6, Section 6.1]).

######  Theorem 2.5 ([31, Theorem 6.2]).

Let $f_{t}:\mathbb{D}_{0}\to\mathbb{C}$ be given for $0\leqslant t\leqslant
T$. Then $\\{f_{t}\\}_{0\leqslant t\leqslant T}$ is a Loewner chain if and
only if there exist constants $r_{0},K_{0}>0$ and a function $p(t,z)$,
analytic in $z\in\mathbb{D}_{0}$, and measurable in $t\in[0,T]$ such that

  * (i)

for each $0\leqslant t\leqslant T$, the function
$f_{t}(z)=\mathrm{e}^{t}z+\cdots$ is analytic in $|z|<r_{0}$, the mapping
$[0,T]\ni t\mapsto f_{t}(z)$ is absolutely continuous for each $|z|<r_{0}$,
and

$|f_{t}(z)|\leqslant K_{0}\mathrm{e}^{t}\quad\text{for all $|z|<r_{0}$ and
$t\in[0,T]$.}$

  * (ii)

$\mathrm{Re}\\{p(t,z)\\}>0$ for all $(t,z)\in[0,T]\times\mathbb{D}_{0}$ and

(2.2) $\begin{split}\frac{\partial}{\partial
t}f_{t}(z)=zf_{t}^{\prime}(z)p(t,z)\end{split}$

for all $|z|<r_{0}$ and for almost all $t\in[0,T]$.

According to the terminology in [2] we call the equation (2.2) the Loewner-
Kufarev equation (if we regard $p(t,z)$ as given and $f_{t}(z)$ as unknown).

Because of the equation (2.2), it holds that $p(t,0)=\lim_{z\to
0}(\frac{\partial}{\partial t}f_{t}(z))/(zf_{t}^{\prime}(z))=1$, and hence the
‘Herglotz Representation Theorem’ applies, which permits to conclude that, for
every $t\in[0,T]$, there exists a probability measure $\nu_{t}$ on
$S^{1}=\partial\mathbb{D}_{0}$ (which is naturally identified with $[0,2\pi]$
as measurable spaces, and then the induced probability measure is still
denoted by $\nu_{t}$) such that

$\begin{split}p(t,z)=\int_{0}^{2\pi}\frac{\mathrm{e}^{i\theta}+z}{\mathrm{e}^{i\theta}-z}\nu_{t}(\mathrm{d}\theta)\quad\text{for
$z\in\mathbb{D}_{0}$.}\end{split}$

Substituting this into (2.2), the Loewner-Kufarev equation becomes

(2.3) $\begin{split}\frac{\partial f_{t}}{\partial
t}(z)=zf_{t}^{\prime}(z)\int_{0}^{2\pi}\frac{\mathrm{e}^{i\theta}+z}{\mathrm{e}^{i\theta}-z}\nu_{t}(\mathrm{d}\theta).\end{split}$

Assuming that $\nu_{t}(\mathrm{d}\theta)=:\nu_{t}(\theta)\mathrm{d}\theta$, is
sufficiently regular, we write the Fourier series of $\nu_{t}(\theta)$ as

$\nu_{t}(\theta)=\frac{1}{2\pi}\Big{\\{}a_{0}(t)+\sum_{k=1}^{\infty}\big{(}a_{k}(t)\cos(k\theta)+b_{k}(t)\sin(k\theta)\big{)}\Big{\\}}.$

We temporally introduce the notations
$x_{0}(t):=\int_{0}^{t}a_{0}(s)\mathrm{d}s$ and

$\begin{split}u_{k}(t):=\int_{0}^{t}a_{k}(s)\mathrm{d}s,\quad
v_{k}(t):=-\int_{0}^{t}b_{k}(s)\mathrm{d}s\end{split}$

for $k=1,2,\cdots$. By using the relations

$\frac{1}{2\pi}\int_{0}^{2\pi}\frac{\mathrm{e}^{i\theta}+z}{\mathrm{e}^{i\theta}-z}\cos(k\theta)\mathrm{d}\theta=z^{k},\quad\frac{1}{2\pi}\int_{0}^{2\pi}\frac{\mathrm{e}^{i\theta}+z}{\mathrm{e}^{i\theta}-z}\sin(k\theta)\mathrm{d}\theta=-iz^{k},$

for $k=1,2,\cdots$ and $|z|<1$, equation (2.3) assumes the following form:

$\begin{split}\frac{\partial f_{t}}{\partial
t}(z)=zf_{t}^{\prime}(z)\Big{\\{}\dot{x}_{0}(t)+\sum_{k=1}^{\infty}\big{(}\dot{u}_{k}(t)+i\dot{v}_{k}(t)\big{)}z^{k}\Big{\\}}.\end{split}$

This can be rewritten as the following controlled differential equation

(2.4)
$\begin{split}\mathrm{d}f_{t}(z)=zf_{t}^{\prime}(z)\\{\mathrm{d}x_{0}(t)+\mathrm{d}\xi(\mathbf{x},z)_{t}\\}\end{split}$

where $x_{k}(t)=u_{k}(t)+iv_{k}(t)$ for $k\geq 1$, and
$\xi(\mathbf{x},z)_{t}=\sum_{k=1}^{\infty}x_{k}(t)z^{k}$.

If we omit the condition $\mathrm{Re}\\{p(t,z)\\}>0$, that is, we allow the
real part of $p(t,z)$ to have arbitrary signs, then the equation (2.2) is
called alternate Loewner-Kufarev equation, as considered by I. Markina and A.
Vasil’ev [27]. Intuitively, this describes evolutions of conformal mappings
whose images of $\mathbb{D}_{0}$ are not necessary increasing, i.e. not strict
subordinations. It appears that the general theory with respect to the
existence and uniqueness of solutions is not yet fully developed. However, our
controlled Loewner-Kufarev equations (2.1) deal with this alternate case
because we have not assumed that
$p(t,z):=\frac{\mathrm{d}}{\mathrm{d}t}(x_{0}(t)+\xi(\mathbf{x},z)_{t})$ has a
positive real part.

######  Remark 2.3.

Readers focusing on radial Loewner equations might feel puzzled by the
heuristic assumption that the Radon-Nikodym density
$\frac{\nu_{t}(\mathrm{d}\theta)}{\mathrm{d}\theta}=\nu_{t}(\theta)$ exists,
because the radial Loewner equation describes the case
$\nu_{t}(\mathrm{d}\theta)=\delta_{\mathrm{e}^{iw(t)}}(\mathrm{d}\theta)$
where $w(t)$ is a continuous path in $\mathbb{R}$, so that there does not
exist a Radon-Nikodym density. However, several explicit examples of Loewner-
Kufarev equations within this setting, are presented with simulations in Sola
[35].

###  2.3. Taylor coefficients along the controlled Loewner-Kufarev equation

Suppose that $x_{0}:[0,+\infty)\to\mathbb{R}$,
$x_{1},x_{2},\cdots:[0,+\infty)\to\mathbb{C}$ are continuous and of bounded
variation.

Let $\\{f_{t}\\}_{0\leqslant t\leqslant T}$ be a solution to the controlled
Loewner-Kufarev equation (2.1). We parametrise $f_{t}$ as

(2.5)
$\begin{split}f_{t}(z)=C(t)\big{(}z+c_{1}(t)z^{2}+c_{2}(t)z^{3}+c_{3}(t)z^{4}+\cdots\big{)},\end{split}$

with the convention $c_{0}(t)\equiv 1$.

The dynamics of the coefficients $(c_{1},c_{2},\cdots)$ has been previously
studied by Vasil’ev and his co-authors [13, 26, 27, 28]. The
(stochastic/Schramm)-Loewner (equation/evolution) (SLE) case is discussed by
Friedrich [9]. In our framework, we obtain similarly the following.

######  Proposition 2.6.

Let $\\{f_{t}\\}_{0\leq t\leq T}$ be a solution to the controlled Loewner-
Kufarev equation (2.1) with the parametrisation (2.5). Then we have

(2.6) $\begin{split}\mathrm{d}C(t)=C(t)\mathrm{d}x_{0}(t)\end{split}$

and

(2.7)
$\begin{split}\left\\{\begin{array}[]{l}\mathrm{d}c_{1}(t)=\mathrm{d}x_{1}(t)+c_{1}(t)\mathrm{d}x_{0}(t),\\\
\mathrm{d}c_{2}(t)=\mathrm{d}x_{2}(t)+2c_{1}(t)\mathrm{d}x_{1}(t)+2c_{2}(t)\mathrm{d}x_{0}(t),\\\
\hskip 85.35826pt\vdots\\\
\displaystyle\mathrm{d}c_{n}(t)=\mathrm{d}x_{n}(t)+\sum_{k=1}^{n-1}(k+1)c_{k}(t)\mathrm{d}x_{n-k}(t)+nc_{n}(t)\mathrm{d}x_{0}(t),\\\
\hskip 199.16928pt\text{for $n\geq 2$,}\end{array}\right.\end{split}$

with the initial conditions $C(0)=1$ and $c_{1}(0)=c_{2}(0)=\cdots=0$. In
particular, $C=\\{C(t)\\}_{0\leq t\leq T}$ takes its values in $\mathbb{R}$.

Since then $f_{t}^{\prime}(0)=C(t)=\mathrm{e}^{x_{0}(t)-x_{0}(0)}\neq 0$, we
obtain

######  Corollary 2.7.

Let $\\{f_{t}\\}_{0\leq t\leq T}$ be a solution to the controlled Loewner-
Kufarev equation (2.1). Then $f_{t}$ is univalent in a neighbourhood of $0$
for each $0\leqslant t\leqslant T$.

By applying variation of constants to (2.7), we obtain the following recurence
relation

$\begin{split}c_{n}(t)=\mathrm{e}^{nx_{0}(t)}\int_{0}^{t}\mathrm{e}^{-nx_{0}(s)}\mathrm{d}x_{n}(s)+\sum_{k=1}^{n-1}(k+1)\mathrm{e}^{nx_{0}(t)}\int_{0}^{t}\mathrm{e}^{-nx_{0}(s)}c_{k}(s)\mathrm{d}x_{n-k}(s)\end{split}$

for $n\geqslant 2$, and we get

######  Theorem 2.8.

Let $\\{f_{t}\\}_{0\leq t\leq T}$ be a solution to the controlled Loewner-
Kufarev equation (2.1). Then for each $n\in\mathbb{N}$, the coefficient
$c_{n}$ in (2.5) is given by

$\begin{split}c_{n}(t)&=\sum_{p=1}^{n}\sum_{\begin{subarray}{c}i_{1},\cdots,i_{p}\in\mathbb{N}:\\\
i_{1}+\cdots+i_{p}=n\end{subarray}}\widetilde{w}(n)_{i_{1},\cdots,i_{p}}\mathrm{e}^{nx_{0}(t)}\\\
&\hskip 28.45274pt\times\int_{0\leqslant s_{1}<s_{2}<\cdots<s_{p}\leqslant
t}\mathrm{e}^{-i_{1}x_{0}(s_{1})}\mathrm{d}x_{i_{1}}(s_{1})\mathrm{e}^{-i_{2}x_{0}(s_{2})}\mathrm{d}x_{i_{2}}(s_{2})\cdots\mathrm{e}^{-i_{p}x_{0}(s_{p})}\mathrm{d}x_{i_{p}}(s_{p})\end{split}$

where

$\begin{split}\widetilde{w}(n)_{i_{1},\cdots,i_{p}}:=\big{\\{}(n-i_{1})+1\big{\\}}\big{\\{}(n-(i_{1}+i_{2}))+1\big{\\}}\cdots\big{\\{}(n-(i_{1}+i_{2}+\cdots+i_{p-1}))+1\big{\\}},\end{split}$

and where $n=i_{1}+\cdots+i_{p}$.

The proof is similar to the one of Proposition 2.12 below and hence omitted.

###  2.4. Variation of Grunsky coefficients induced by a Loewner-Kufarev
equation

Here we recall that for each holomorphic function $f$, defined on an open
neighbourhood $U$ of the origin, the Faber polynomials and hence Grunsky
coefficients are defined once $f$ is univalent on an open neighbourhood
$V\subset U$ of the origin (see Appendix A–Definition A.1).

######  Proposition 2.9.

Let $\\{f_{t}\\}_{0\leqslant t\leqslant T}$ be a solution to the controlled
Loewner-Kufarev equation (2.1). Then there exists an open neighbourhood $U$ of
the origin such that

  * (i)

$\overline{U}\subset\mathbb{D}_{0}$;

  * (ii)

$f_{t}|_{U}$ is univalent for each $t\in[0,T]$;

  * (iii)

$V:=\cap_{0\leqslant t\leqslant T}f_{t}(U)$ is an open neighbourhood of the
origin;

  * (iv)

For each $\zeta\in V$, $[0,T]\ni t\mapsto f_{t}^{-1}(\zeta)$ is continuous and
of bounded variation;

  * (v)

For each $\zeta\in V$, with $f^{-1}(t,\zeta):=f_{t}^{-1}(\zeta)$ and
$\mathrm{d}f_{t}^{-1}(\zeta):=f^{-1}(\mathrm{d}t,\zeta)$, we have

$\mathrm{d}f_{t}^{-1}(\zeta)=-f_{t}^{-1}(\zeta)\Big{\\{}\mathrm{d}x_{0}(t)+\sum_{k=1}^{\infty}(f_{t}^{-1}(\zeta))^{k}\mathrm{d}x_{k}(t)\Big{\\}}$

as Lebesgue-Stieltjes measures on $[0,T]$.

The proof of Proposition 2.9 is based on classical techniques to prove inverse
function theorems however a little bit involved. Therefore we shall postpone
it to Appendix A.2.

Let $\\{f_{t}\\}_{0\leq t\leq T}$ be a solution to the controlled Loewner-
Kufarev equation (2.1).

Because of Corollary 2.7, associated to $f_{t}(z)$ are Faber polynomials and
Grunsky coefficients (see Appendix A–Definition A.1) which will be denoted by
$Q_{n}(t,w)$ and $b_{-n,-m}(t)$ respectively.

######  Proposition 2.10.

  * (i)

(Variation of Faber polynomials): We have for each $n\in\mathbb{N}$,

$\begin{split}\mathrm{d}Q_{n}(t,w)=n\mathrm{d}x_{n}(t)+n\sum_{k=1}^{n}Q_{k}(t,w)\mathrm{d}x_{n-k}(t).\end{split}$

  * (ii)

(Variation of Grunsky coefficients): For each $n,m\in\mathbb{N}$,

(2.8)
$\begin{split}\mathrm{d}b_{-n,-m}(t)&=-\mathrm{d}x_{n+m}(t)+\sum_{\begin{subarray}{c}k,l\in\mathbb{Z}_{\geqslant
0};\\\ k+l=m-1\end{subarray}}(k+1)b_{-n,-(k+1)}(t)\mathrm{d}x_{l}(t)\\\
&\hskip 156.49014pt+\sum_{\begin{subarray}{c}k,l\in\mathbb{Z}_{\geqslant
0};\\\ k+l=n-1\end{subarray}}(k+1)b_{-m,-(k+1)}\mathrm{d}x_{l}(t),\end{split}$

with the initial conditions $b_{-n,-m}(0)=0$ for all $n,m\in\mathbb{N}$.

###### Proof.

(i) Let $n\in\mathbb{N}$. Let $U$ and $V$ be as in Proposition 2.9. Then
$f_{t}^{-1}(\zeta)$, $\zeta\in V$ satisfies the equation

$\mathrm{d}f_{t}^{-1}(\zeta)=-f_{t}^{-1}(\zeta)\Big{\\{}\mathrm{d}x_{0}(t)+\sum_{k=1}^{\infty}(f_{t}^{-1}(\zeta))^{k}\mathrm{d}x_{k}(t)\Big{\\}}.$

Let $X_{0}\subset V$ be an open disc centred at $0$. By using Cauchy’s
integral formula, we have for $w\in X_{0}$,

$\begin{split}&\mathrm{d}Q_{n}(t,w)=\frac{1}{2\pi i}\int_{\partial
X_{0}}\frac{\mathrm{d}\zeta}{\zeta}\frac{\mathrm{d}(f_{t}^{-1}(\zeta))^{-n}}{1-\zeta
w^{-1}}\\\ &=\frac{1}{2\pi i}\int_{\partial
X_{0}}(-n)\frac{(f_{t}^{-1}(\zeta))^{-n-1}}{1-\zeta
w^{-1}}(-f_{t}^{-1}(\zeta))\sum_{k=0}^{\infty}(f_{t}^{-1}(\zeta))^{k}\mathrm{d}x_{k}(t)\frac{\mathrm{d}\zeta}{\zeta}\\\
&=\sum_{k=0}^{n}\frac{n}{2\pi i}\Big{(}\int_{\partial
X_{0}}\frac{(f_{t}^{-1}(\zeta))^{-n+k}}{1-\zeta
w^{-1}}\frac{\mathrm{d}\zeta}{\zeta}\Big{)}\mathrm{d}x_{k}(t)\\\
&=\frac{n\mathrm{d}x_{n}(t)}{2\pi i}\int_{\partial X_{0}}\frac{1}{1-\zeta
w^{-1}}\frac{\mathrm{d}\zeta}{\zeta}+n\sum_{k=0}^{n-1}Q_{n-k}(t,w)\mathrm{d}x_{k}(t).\end{split}$

Noting that the orientation of $\partial X_{0}$ is anti-clockwise, we get

$\frac{1}{2\pi i}\int_{\partial X_{0}}\frac{1}{1-\zeta
w^{-1}}\frac{\mathrm{d}\zeta}{\zeta}=1,$

and hence the result.

(ii) Since $f_{t}(z)$ satisfies the controlled Loewner-Kufarev equation, by
putting
$p(\mathrm{d}t,z):=\mathrm{d}x_{0}(t)+\mathrm{d}\xi(\mathbf{x},z)_{t}$, we
have

$\begin{split}\mathrm{d}Q_{n}(t,f_{t}(z))&=Q_{n}(\mathrm{d}t,f_{t}(z))+Q_{n}^{\prime}(t,f_{t}(z))\mathrm{d}f_{t}(z)\\\
&=Q_{n}(\mathrm{d}t,f_{t}(z))+Q_{n}^{\prime}(t,f_{t}(z))\big{\\{}zf_{t}^{\prime}(z)p(\mathrm{d}t,z)\big{\\}}\\\
&=Q_{n}(\mathrm{d}t,f_{t}(z))+z\big{[}\partial_{z}Q_{n}(t,f_{t}(z))\big{]}p(\mathrm{d}t,z),\end{split}$

so that

(2.9)
$\begin{split}\mathrm{d}Q_{n}(t,f_{t}(z))=Q_{n}(\mathrm{d}t,f_{t}(z))+z\big{[}\partial_{z}Q_{n}(t,f_{t}(z))\big{]}p(\mathrm{d}t,z).\end{split}$

Recalling that
$Q_{n}(t,f_{t}(z))=z^{-n}+n\sum_{m=1}^{\infty}b_{-n,-m}(t)z^{m}$,

(2.10) $\begin{split}(\text{LHS of (\ref{t-der})})_{\geqslant 1}=(\text{LHS of
(\ref{t-der})})=n\sum_{m=1}^{\infty}z^{m}\mathrm{d}b_{-n,-m}(t).\end{split}$

On the other hand, by Proposition 2.10–(i), we have

$\begin{split}&\mathrm{d}Q_{n}(t,f_{t}(z))=n\mathrm{d}x_{n}(t)+n\sum_{k=1}^{n}Q_{k}(t,f_{t}(z))\mathrm{d}x_{n-k}(t)\\\
&=n\mathrm{d}x_{n}(t)+n\sum_{k=1}^{n}\Big{(}z^{-k}+k\sum_{m=1}^{\infty}b_{-k,-m}(t)z^{m}\Big{)}\mathrm{d}x_{n-k}(t)\\\
&=n\mathrm{d}x_{n}(t)+n\sum_{k=1}^{n}z^{-k}\mathrm{d}x_{n-k}(t)+n\sum_{m=1}^{\infty}\Big{(}\sum_{k=1}^{n}kb_{-k,-m}(t)\mathrm{d}x_{n-k}(t)\Big{)}z^{m}\end{split}$

so that

(2.11) $\begin{split}\Big{(}\mathrm{d}Q_{n}(t,f_{t}(z))\Big{)}_{\geqslant
1}=n\sum_{m=1}^{\infty}\Big{(}\sum_{k=1}^{n}kb_{-k,-m}\mathrm{d}x_{n-k}(t)\Big{)}z^{m}.\end{split}$

We further have

$\begin{split}&z\big{[}\partial_{z}Q_{n}(t,f_{t}(z))\big{]}p(\mathrm{d}t,z)\\\
&=z\Big{(}-nz^{-n-1}+n\sum_{k=1}^{\infty}kb_{-n,-k}z^{k-1}\Big{)}\Big{(}\mathrm{d}x_{0}(t)+\sum_{l=1}^{\infty}\mathrm{d}x_{l}(t)z^{l}\Big{)}\\\
&=n\Big{(}-\mathrm{d}x_{0}(t)z^{-n}-\sum_{m=1-n}^{\infty}\mathrm{d}x_{m+n}(t)z^{m}\\\
&\hskip
28.45274pt+\sum_{m=1}^{\infty}mb_{-n,-m}(t)\mathrm{d}x_{0}(t)z^{m}+\sum_{m=2}^{\infty}\sum_{\begin{subarray}{c}k,l\geqslant
1;\\\
k+l=m\end{subarray}}kb_{-n,-k}(t)\mathrm{d}x_{l}(t)z^{m}\Big{)},\end{split}$

from which we conclude

(2.12)
$\begin{split}&\Big{(}z\big{[}\partial_{z}Q_{n}(t,f_{t}(z))\big{]}p(\mathrm{d}t,z)\Big{)}_{\geqslant
1}=n\sum_{m=1}^{\infty}\Big{(}-\mathrm{d}x_{n+m}(t)+\sum_{\begin{subarray}{c}k\geqslant
1,\ l\geqslant 0;\\\
k+l=m\end{subarray}}kb_{-n,-k}(t)\mathrm{d}x_{l}(t)\Big{)}z^{m}.\end{split}$

Combining (2.11) and (2.12), we obtain

$\begin{split}&(\text{RHS of (\ref{t-der})})_{\geqslant 1}\\\
&=n\sum_{m=1}^{\infty}\Big{(}-\mathrm{d}x_{n+m}(t)+\sum_{\begin{subarray}{c}k,l\in\mathbb{Z}_{\geqslant
0};\\\ k+l=m-1\end{subarray}}(k+1)b_{-n,-(k+1)}(t)\mathrm{d}x_{l}(t)\\\
&\hskip 156.49014pt+\sum_{\begin{subarray}{c}k,l\in\mathbb{Z}_{\geqslant
0};\\\
k+l=n-1\end{subarray}}(k+1)b_{-m,-(k+1)}\mathrm{d}x_{l}(t)\Big{)}z^{m},\end{split}$

and then by comparing with (2.10), we get the result.

The initial condition is obviously derived from $f_{0}(z)\equiv z$. ∎

To derive an explicit formula for general $b_{-n,-m}(t)$, we shall introduce
some notation.

######  Definition 2.11.

Let $p,q\in\mathbb{N}$.

  * (1)

A bijection $\sigma:\\{1,2,\cdots,p+q\\}\to\\{1,2,\cdots,p+q\\}$ is called a
$(p,q)$-shuffle if it holds that $\sigma(1)<\sigma(2)<\cdots<\sigma(p)$ and
$\sigma(p+1)<\sigma(p+2)<\cdots<\sigma(p+q)$.

  * (2)

Suppose that $x_{1},x_{2},\cdots,x_{p+q}:[0,T]\to\mathbb{C}$ are continuous
and of bounded variations. Then for each $0\leqslant t\leqslant T$, we set

$\begin{split}&\big{(}(x_{1}\cdots x_{p})\shuffle(x_{p+1}\cdots
x_{p+q})\big{)}(t)\\\ &:=:\int_{0\leqslant s_{q}\leqslant\cdots\leqslant
s_{1}\leqslant t_{p}\leqslant\cdots\leqslant t_{1}\leqslant
t}\big{(}\mathrm{d}x_{1}(t_{1})\cdots\mathrm{d}x_{p}(t_{p})\big{)}\shuffle\big{(}\mathrm{d}x_{p+1}(s_{1})\cdots\mathrm{d}x_{p+q}(s_{q})\big{)}\\\
&:=\sum_{\text{$\sigma$:
$(p,q)$-shuffle}}\int_{0}^{t}\mathrm{d}x_{\sigma(1)}(t_{1})\int_{0}^{t_{1}}\mathrm{d}x_{\sigma(2)}(t_{2})\cdots\int_{0}^{t_{p-1}}\mathrm{d}x_{\sigma(p)}(t_{p})\\\
&\hskip
28.45274pt\times\int_{0}^{t_{p}}\mathrm{d}x_{\sigma(p+1)}(s_{1})\int_{0}^{s_{1}}\mathrm{d}x_{\sigma(p+2)}(s_{2})\cdots\int_{0}^{s_{q-1}}\mathrm{d}x_{\sigma(p+q)}(s_{q}).\end{split}$

The general formula for the Grunsky-coefficients along the controlled Loewner-
Kufarev equation (2.1) is given by the following. The proof is given in
Section A.3.

######  Proposition 2.12.

For $n,m\in\mathbb{N}$ and $t\geqslant 0$,

(2.13) $\begin{split}&b_{-m,-n}(t)\\\
&=-\mathrm{e}^{(n+m)x_{0}(t)}\int_{0}^{t}\mathrm{e}^{-(n+m)x_{0}(s)}\mathrm{d}x_{m+n}(s)\\\
&-\sum_{k=2}^{n+m-2}\sum_{\begin{subarray}{c}1\leqslant i<m;\\\ 1\leqslant
j<n:\\\
i+j=k\end{subarray}}\sum_{p=1}^{m-i}\sum_{q=1}^{n-j}\sum_{\begin{subarray}{c}i_{1},\cdots,i_{p}\in\mathbb{N}:\\\
i_{1}+\cdots+i_{p}=m-i\end{subarray}}\sum_{\begin{subarray}{c}j_{1},\cdots,j_{q}\in\mathbb{N}:\\\
j_{1}+\cdots+j_{q}=n-j\end{subarray}}w(i,j)_{i_{1},\cdots,i_{p};j_{1},\cdots,j_{q}}\\\
&\hskip 28.45274pt\times\mathrm{e}^{(m+n)x_{0}(t)}\int_{0\leqslant
u_{q}\leqslant\cdots\leqslant u_{1}\leqslant s_{q}\leqslant\cdots\leqslant
s_{1}\leqslant
t}\big{(}\mathrm{e}^{-i_{1}x_{0}(s_{1})}\mathrm{d}x_{i_{1}}(s_{1})\cdots\mathrm{e}^{-i_{p}x_{0}(s_{p})}\mathrm{d}x_{i_{p}}(s_{p})\big{)}\\\
&\hskip
142.26378pt\shuffle\big{(}\mathrm{e}^{-j_{1}x_{0}(u_{1})}\mathrm{d}x_{j_{1}}(u_{1})\cdots\mathrm{e}^{-j_{q}x_{0}(u_{q})}\mathrm{d}x_{j_{q}}(u_{q})\big{)}\int_{0}^{u_{q}}\mathrm{e}^{-kx_{0}(s)}\mathrm{d}x_{k}(s)\\\
&-\sum_{k=m+1}^{n+m-1}\sum_{q=1}^{n+m-k}\sum_{\begin{subarray}{c}j_{1},\cdots,j_{q}\in\mathbb{N}:\\\
j_{1}+\cdots+j_{q}=n+m-k\end{subarray}}w(k-m)_{\varnothing;j_{1},\cdots,j_{q}}\\\
&\hskip 28.45274pt\times\mathrm{e}^{(m+n)x_{0}(t)}\int_{0\leqslant
s_{q}\leqslant\cdots\leqslant s_{1}\leqslant
t}\big{(}\mathrm{e}^{-j_{1}x_{0}(s_{1})}\mathrm{d}x_{j_{1}}(s_{1})\cdots\mathrm{e}^{-j_{q}x_{0}(s_{q})}\mathrm{d}x_{j_{q}}(s_{q})\big{)}\int_{0}^{s_{q}}\mathrm{e}^{-kx_{0}(s)}\mathrm{d}x_{k}(s)\\\
&-\sum_{k=n+1}^{n+m-1}\sum_{p=1}^{m+n-k}\sum_{\begin{subarray}{c}i_{1},\cdots,i_{p}\in\mathbb{N}:\\\
i_{1}+\cdots+i_{p}=m+n-k\end{subarray}}w(k-n)_{i_{1},\cdots,i_{p};\varnothing}\\\
&\hskip 28.45274pt\times\mathrm{e}^{(m+n)x_{0}(t)}\int_{0\leqslant
u_{p}\leqslant\cdots\leqslant u_{1}\leqslant
t}\big{(}\mathrm{e}^{-i_{1}x_{0}(u_{1})}\mathrm{d}x_{i_{1}}(u_{1})\cdots\mathrm{e}^{-i_{p}x_{0}(u_{p})}\mathrm{d}x_{i_{p}}(u_{p})\big{)}\int_{0}^{u_{p}}\mathrm{e}^{-kx_{0}(u)}\mathrm{d}x_{k}(u),\end{split}$

where, for $m=i_{1}+\cdots+i_{p}+r$ and $n=j_{1}+\cdots+j_{q}+s$, we have put

$\begin{split}w(r)_{i_{1},\cdots,i_{p};\varnothing}&=(m-i_{1})(m-(i_{1}+i_{2}))\cdots(m-(i_{1}+i_{2}+\cdots+i_{p})),\\\
w(s)_{\varnothing;j_{1},\cdots,j_{q}}&=(n-j_{1})(n-(j_{1}+j_{2}))\cdots(n-(j_{1}+j_{2}+\cdots+j_{q})),\end{split}$

and
$w(r,s)_{i_{1},\cdots,i_{p};j_{1},\cdots,j_{q}}:=w(r)_{i_{1},\cdots,i_{p};\varnothing}w(s)_{\varnothing;j_{1},\cdots,j_{q}}$.

##  3\. Controlled Loewner-Kufarev equation embedded into the Segal-Wilson
Grassmannian

###  3.1. Segal-Wilson Grassmannian

Let $H:=L^{2}(S^{1},\mathbb{C})$ be the Hilbert space which consists of all
square-integrable complex functions on the unit circle $S^{1}$. It decomposes
orthogonally into $H=H_{+}\oplus H_{-}$, where $H_{+}$ and $H_{-}$ are the
closure of $\mathrm{span}\\{z^{k}:k\geqslant 0\\}$ and
$\mathrm{span}\\{z^{k}:k<0\\}$, respectively.

######  Definition 3.1 (G. Segal and G. Wilson [34, Section 2]).

The Segal-Wilson Grassmannian $\mathrm{Gr}:=\mathrm{Gr}(H)$ is the set of all
closed subspace $W$ of $H$ satisfying the following:

  * (1)

The orthogonal projection $\mathrm{pr}_{+}:W\to H_{+}$ is Fredholm;

  * (2)

The orthogonal projection $\mathrm{pr}_{-}:W\to H_{-}$ is compact.

The Fredholm index of the orthogonal projection $\mathrm{pr}_{+}:W\to H_{+}$
is called the virtual dimension of $W$. For $d\in\mathbb{Z}$, we set

$\mathrm{Gr}({\textstyle\frac{\infty}{2}+d},\infty):=\\{W\in\mathrm{Gr}:\text{the
virtual dimension of $W$ is $d$}\\}$

and
$\mathrm{Gr}({\textstyle\frac{\infty}{2}},\infty):=\mathrm{Gr}({\textstyle\frac{\infty}{2}+0},\infty)$.

If we take $W=H_{+}$, then the corresponding projections are given by
$\mathrm{pr}_{+}=\mathrm{id}_{H_{+}}$ and $\mathrm{pr}_{-}=0$ which are
Fredholm and compact operators, respectively. Therefore we have
$H_{+}\in\mathrm{Gr}(\frac{\infty}{2},\infty)$.

######  Definition 3.2 ([34, Section 5]).

We denote by $\Gamma_{+}$ the set of all continuous functions
$g:S^{1}\to\mathbb{C}^{*}$ such that
$g(z)=\mathrm{e}^{\text{{\tiny$\sum_{k=1}^{\infty}t_{k}z^{k}$}}}$, $z\in
S^{1}$ for some $\mathbf{t}=(t_{1},t_{2},t_{3},\cdots)$.

The set $\Gamma_{+}$ acts on $H$ by pointwise multiplication. In particular,
$\Gamma_{+}$ forms a group. This action induces the action of $\Gamma_{+}$ on
$\mathrm{Gr}$: $\Gamma_{+}\times\mathrm{Gr}\ni(g,W)\mapsto gW\in\mathrm{Gr}$
(see [34, Lemma 2.2 and Proposition 2.3]), where $gW=\\{gf:f\in W\\}$. For any
$g=\mathrm{e}^{\text{{\tiny$\sum_{k=1}^{\infty}t_{k}z^{k}$}}}\in\Gamma_{+}$,
the action of $g$ on $H$ is of the form

$\begin{split}g=\left(\begin{array}[]{cc}a&b\\\
0&d\end{array}\right)\quad\text{along $H=H_{+}\oplus H_{-}$,}\end{split}$

where $a:H_{+}\to H_{+}$ is invertible and $b:H_{-}\to H_{+}$ is of trace
class (see [34, Proposition 2.3]).

Let $\mathcal{U}$ be the set of all
$W\in\mathrm{Gr}({\textstyle\frac{\infty}{2}},\infty)$ such that the
orthogonal projection $W\to H_{+}$ is an isomorphism. Then, associated to each
$W\in\mathcal{U}$ is the tau-function $\tau_{W}(\mathbf{t})$ of $W$, where
$\mathbf{t}=(t_{1},t_{2},\cdots)$, satisfying the following:

######  Proposition 3.3 ([34, Proposition 3.3]).

Let $W\in\mathcal{U}$. For
$g=\mathrm{e}^{\text{{\tiny$\sum_{n=1}^{\infty}t_{n}z^{n}$}}}\in\Gamma_{+}$,
we have

$\tau_{W}(\mathbf{t})=\mathrm{det}(1+a^{-1}bA),$

where $\mathbf{t}=(t_{1},t_{2},t_{3},\cdots)$,

$g^{-1}=\left(\begin{array}[]{cc}a&b\\\ 0&d\end{array}\right)\quad\text{along
$H=H_{+}\oplus H_{-}$}$

and $A:H_{+}\to H_{-}$ is the linear operator such that $\mathrm{graph}(A)=W$.

###  3.2. Krichever’s construction

In connection with algebraic geometry and infinite-dimensional integrable
systems, a fundamental observation / construction of Krichever [19, 20, 21]
states that, associated to each non-singular algebraic curve with some
additional data (which are called algebro-geometric data) is a solution of the
KdV equation. This construction has been developed further, and after a remark
by Mumford [30], it was formalised by Segal-Wilson [34] as follows.

In this context, an algebro-geometric datum
$(X,\mathcal{L},x_{\infty},z,\varphi)$ consists of a complete irreducible
complex algebraic curve $X$, a rank-one torsion-free coherent sheaf
$\mathcal{L}$ over $X$, a non-singular point $x_{\infty}\in X$, a closed
neighbourhood $X_{\infty}$ of $x_{\infty}$, a local parameter
$1/z:X_{\infty}\to\overline{\mathbb{D}_{0}}\subset\widehat{\mathbb{C}}$ which
sends $x_{\infty}$ to $0$, and
$\varphi:\mathcal{L}|_{X_{\infty}}\to\overline{\mathbb{D}_{0}}\times\mathbb{C}$
is a trivialisation of $\mathcal{L}|_{X_{\infty}}$. Each section of
$\mathcal{L}|_{X_{\infty}}$ is identified with a complex function on
$\overline{\mathbb{D}_{0}}$ under $\varphi$. Let $X_{0}:=X\setminus(\text{the
interior of $X_{\infty}$})$ and then the closed sets $X_{0}$ and $X_{\infty}$
cover $X$, and $X_{0}\cap X_{\infty}$ is identified with $S^{1}$ under $z$.

Given this algebro-geometric datum, one can associate a closed subspace
$W\subset H$ consisting of all analytic functions $S^{1}\to\mathbb{C}$ which,
under the above identification, extend to a holomorphic section of
$\mathcal{L}$ on an open neighbourhood of $X_{0}$. More explicitly, one can
write

$W=\overline{\Big{\\{}\begin{array}[]{c}\text{the second component}\\\
\text{of $\varphi\circ
s\circ(1/z)^{-1}|_{S^{1}}$}\end{array}:\begin{array}[]{l}\text{$s$ is a
holomorphic section}\\\ \text{on a neighbourhood of
$X_{0}$}\end{array}\Big{\\}}}^{H},$

where $(1/z)^{-1}:\overline{\mathbb{D}_{0}}\to X_{\infty}$ is the inverse
function of $1/z$.

It is known that $W\in\mathrm{Gr}$ (see [34, Proposition 6.1]) and further, in
the class where $X$ is a compact Riemann surface (then $\mathcal{L}$ is
automatically a complex line bundle, hence a maximal torsion-free sheaf), this
correspondence $(X,\mathcal{L},x_{\infty},z,\varphi)\mapsto W\in\mathrm{Gr}$
is one-to-one (see [34, Proposition 6.2]).

###  3.3. The appearance of Faber polynomials and Grunsky coefficients

Let $f:\mathbb{D}_{0}\to\mathbb{C}$ be a univalent function such that $f(0)=0$
and $f(\mathbb{D}_{0})$ is bounded by a Jordan curve. We set
$\beta:\widehat{\mathbb{C}}\to\widehat{\mathbb{C}}$ by $\beta(w):=1/w$. For a
subset $A\subset\widehat{\mathbb{C}}$, we shall write $A^{-1}:=\beta(A)$. Let
$\widehat{\mathbb{D}}_{\infty}:=\widehat{\mathbb{C}}\setminus\overline{\mathbb{D}_{0}}$.
We obtain an algebro-geometric datum $(X,\mathcal{L},x_{\infty},z,\varphi)$ by
setting $X=\widehat{\mathbb{C}}$,
$\mathcal{L}=\widehat{\mathbb{C}}\times\mathbb{C}$, $x_{\infty}:=\infty$,

$\begin{split}&X_{\infty}:=f(\overline{\mathbb{D}_{0}})^{-1},\\\
&z:=\beta\circ
f^{-1}\circ\beta^{-1}:X_{\infty}\to\widehat{\mathbb{D}}_{\infty},\end{split}$

and $\varphi=(1/z)\times\mathrm{id}_{\mathbb{C}}$. Accordingly we have
$X_{0}=\widehat{\mathbb{C}}\setminus(f(\mathbb{D}_{0})^{-1})$. Here we note
that $z$ extends continuously to $X_{\infty}$ by virtue of the Caratheodory’s
extension theorem for $z$.

Therefore we can embed $f$ into the Grassmannian by assigning a Hilbert space
$W=W_{f}$ as above. In this case, we have
$\widehat{\mathbb{C}}\setminus(f(\mathbb{D}_{0})^{-1})$, and hence

$W_{f}=\overline{\Big{\\{}F\circ(1/z)^{-1}|_{S^{1}}:\begin{array}[]{l}\text{$F$
is a holomorphic function}\\\ \text{on a neighbourhood of
$\widehat{\mathbb{C}}\setminus(f(\mathbb{D}_{0})^{-1})$}\end{array}\Big{\\}}}^{H}.$

In order to start this paper’s main calculation, let us specify this more
explicitly.

For a closed subset $V$ in $\widehat{\mathbb{C}}$, we denote by
$\mathcal{O}(V)$ the space of all holomorphic functions defined on an open
neighbourhood of $V$.

For a univalent function
$g:\widehat{\mathbb{D}}_{\infty}\to\widehat{\mathbb{C}}$ with
$g(\infty)=\infty$ and for each $h\in\mathcal{O}(\overline{\mathbb{D}_{0}})$,
we call

$(\mathcal{F}h)(z):=\frac{1}{2\pi i}\int_{\partial(\mathbb{C}\setminus
g(\mathbb{D}_{\infty}))}\frac{h(g^{-1}(\xi))}{\xi-z}\mathrm{d}\xi,\quad
z\in\mathbb{C}\setminus\overline{g(\mathbb{D}_{\infty})}$

the Faber transform of $\phi$ (with respect to $g$). When the boundary
$\partial(\mathbb{C}\setminus g(\mathbb{D}_{\infty}))$ is analytic, it is
known that $h\in\mathcal{O}(\overline{\mathbb{D}_{0}})$ iff
$\mathcal{F}h\in\mathcal{O}(\mathbb{C}\setminus g(\mathbb{D}_{\infty}))$ (see
[14, Theorem 1]) and
$\mathcal{F}:\mathcal{O}(\overline{\mathbb{D}_{0}})\to\mathcal{O}(\mathbb{C}\setminus
g(\mathbb{D}_{\infty}))$ is bijective.

In our case, we put

$g:=(1/z)^{-1}=\beta\circ f\circ\beta^{-1}:\widehat{\mathbb{D}}_{\infty}\to
f(\mathbb{D}_{0})^{-1},$

and then we can describe $\mathcal{O}(X_{0})$ by
$\mathcal{O}(\widehat{\mathbb{D}}_{\infty})$ through the transformation

$\mathcal{F}\circ(\beta^{-1})^{*}=(\beta^{-1})^{*}\circ\mathrm{Ad}_{\beta^{*}}(\mathcal{F}):\mathcal{O}(\widehat{\mathbb{D}}_{\infty})\to\mathcal{O}(X_{0})$

where
$\mathrm{Ad}_{\beta^{*}}(\mathcal{F}):=\beta^{*}\circ\mathcal{F}\circ(\beta^{-1})^{*}:\mathcal{O}(\widehat{\mathbb{D}}_{\infty})\to\mathcal{O}(\widehat{\mathbb{C}}\setminus
f(\mathbb{D}_{0}))$.

For each
$h(\eta)=\sum_{k=0}^{\infty}a_{k}\eta^{-k}\in\mathcal{O}(\widehat{\mathbb{D}}_{\infty})$,
a direct calculation shows that

(3.1) $(\mathrm{Ad}_{\beta^{*}}(\mathcal{F})h)(w)=\frac{1}{2\pi
i}\int_{\partial f(\mathbb{D}_{0})}\frac{h(f^{-1}(\zeta))}{1-\zeta
w^{-1}}\frac{\mathrm{d}\zeta}{\zeta},\quad w\in\widehat{\mathbb{C}}\setminus
f(\mathbb{D}_{0}).$

As a result, $(\mathrm{Ad}_{\beta^{*}}(\mathcal{F})\phi)(w)$ is a power series
in $1/w$. Actually, in view of the Cauchy integral formula

$\frac{1}{2\pi
i}\int_{S^{1}}\frac{\zeta^{n}}{1-\zeta\eta^{-1}}\frac{\mathrm{d}\zeta}{\zeta}=\left\\{\begin{array}[]{ll}\eta^{n}&\text{if
$n\leqslant 0$},\\\ 0&\text{if $n\geqslant
1$},\end{array}\right.\quad\eta\in\mathbb{D}_{\infty}$

where $S^{1}=\partial\mathbb{D}_{0}$, we have

$\begin{split}(\mathrm{Ad}_{\beta^{*}}(\mathcal{F})\phi)(w)=\sum_{k=0}^{n}\frac{a_{k}}{2\pi
i}\int_{\partial X_{0}}\frac{(f^{-1}(\zeta))^{-k}}{1-\zeta
w^{-1}}\frac{\mathrm{d}\zeta}{\zeta}=\sum_{k=0}^{n}a_{k}[(f^{-1}(w))^{-k}]_{\leqslant
0}\end{split}$

where $[(f^{-1}(w))^{-k}]_{\leqslant 0}$ denotes the constant-part plus the
principal-part of the Laurent series for $(f^{-1}(w))^{-k}=(1/f^{-1}(w))^{k}$.
In particular, every element in $\mathcal{O}(\widehat{\mathbb{C}}\setminus
f(\mathbb{D}_{0}))$ can be written as a series of $1$ and
$Q_{k}(w):=[(f^{-1}(w))^{-k}]_{\leqslant 0}$ for $k\geqslant 1$. Each $Q_{k}$
is called the $k$-th Faber polynomial associated to the domain
$\mathbb{C}\setminus\overline{f(\mathbb{D}_{0})}$ (or just simply $f$). By
definition, $Q_{k}(w)$ is a polynomial of degree $n$ in $1/w$.

Then we conclude that
$[(\beta^{-1})^{*}\circ\mathrm{Ad}_{\beta^{*}}(h)]\circ(1/z)^{-1}=[\mathrm{Ad}_{\beta^{*}}(h)]\circ
f\circ\beta^{-1}$ and hence

$W_{f}=\overline{\mathrm{span}\big{(}\\{1\\}\cup\\{Q_{n}\circ
f\circ(1/z)|_{S^{1}}\\}_{n\geqslant 1}\big{)}}^{H},$

where $z$ is the identity map on $\widehat{\mathbb{D}}_{\infty}$. We note that
$W_{f}=H_{+}$ in the case of $f(z)\equiv z$.

######  Remark 3.1.

The Faber polynomials appeared first (with a different formalism, but
equivalent to our presentation) in the context of approximations of functions
in one complex variable by analytic functions (see [7] and [6]). Since then,
they also play an important role in the theory of univalent functions (see
[32]). We introduced the Faber polynomials in a slightly non-standard way in
order to have them in a form which is suitable for embedding univalent
functions into the Grassmannian by using Faber polynomials.

###  3.4. Action of words

Let $X=\\{x_{1},x_{2},x_{3},\cdots\\}$ be a countable set of letters. The free
monoid $X^{*}$ on $X$ is the set of all words in the letters $X$, including
the empty word $\emptyset$. We denote by

$\mathbb{C}\langle X\rangle:=\bigoplus_{w\in
X^{*}}\mathbb{C}w=\mathbb{C}\oplus\bigoplus_{n=1}^{\infty}\mathbb{C}\langle
X\rangle_{n}$

the free associative and unital $\mathbb{C}$-algebra on $X$. The unit of this
algebra is the empty word which we will denote by $1:=\emptyset$. The set
$\mathbb{C}\langle X\rangle_{n}$ stands for $\oplus_{w}\mathbb{C}w$ where the
summation is taken over all words $w$ of length $n$.

######  Definition 3.4.

We define

$\xi(\mathbf{x},z):=\sum_{n=1}^{\infty}x_{k}z^{k}\in\mathbb{C}\langle
X\rangle[\\![z]\\!]$

and a distinguished element $S(\xi(\mathbf{x},z))\in\mathbb{C}\langle
X\rangle[\\![z]\\!]$ by

$S(\xi(\mathbf{x},z)):=1+\sum_{n=1}^{\infty}z^{n}\sum_{p=1}^{n}\sum_{\begin{subarray}{c}i_{1},\cdots,i_{p}\in\mathbb{N}:\\\
i_{1}+\cdots+i_{p}=n\end{subarray}}x_{i_{1}}\cdots x_{i_{p}}.$

######  Definition 3.5.

Let $x_{0}:[0,+\infty)\to\mathbb{R}$ and
$x_{1},x_{2},\cdots:[0,+\infty)\to\mathbb{C}$ be continuous and of bounded
variation. For $0\leqslant s\leqslant t$, we define $[\int 1]_{s,t}:=1$ and

$\begin{split}&\big{[}\int(x_{i_{p}}\cdots
x_{i_{2}}x_{i_{1}})\big{]}_{s,t}:=\int_{s\leqslant
u_{1}<u_{2}<\cdots<u_{p}\leqslant
t}\mathrm{e}^{-i_{1}x_{0}(u_{1})}\mathrm{d}x_{i_{1}}(u_{1})\mathrm{e}^{-i_{2}x_{0}(u_{2})}\mathrm{d}x_{i_{2}}(u_{2})\cdots\mathrm{e}^{-i_{p}x_{0}(u_{p})}\mathrm{d}x_{i_{p}}(u_{p}).\end{split}$

The action of $\int$ naturally extends to $\mathbb{C}\langle
X\rangle[\\![z]\\!]$ and then we call

$\begin{split}S(\xi(\mathbf{x},z))_{s,t}:=\big{[}\int
S(\xi(\mathbf{x},z))\big{]}_{s,t}\end{split}$

the signature of $\xi(\mathbf{x},z)$.

######  Definition 3.6.

We define a $\mathbb{C}$-bilinear map

$T:\mathbb{C}\langle X\rangle(\\!(z^{-1})\\!)\times\mathbb{C}\langle
X\rangle\to\mathbb{C}\langle X\rangle(\\!(z^{-1})\\!)$

by extending $\mathbb{C}$-bilinearly the pairing

$\begin{split}&T(f,1):=f,\\\ &T(f,x_{i_{p}}\cdots
x_{i_{1}}):=(L_{-i_{1}}\cdots L_{-i_{p}}f)x_{i_{p}}\cdots
x_{i_{1}}\end{split}$

for $f\in\mathbb{C}\langle X\rangle(\\!(z^{-1})\\!)$, $p\geqslant 1$ and
$i_{1},\cdots,i_{p}\in\mathbb{N}$, where $L_{k}:=-z^{k+1}\partial/(\partial
z)$ for $k\leqslant-1$. Here, $\partial/(\partial z)$ is a formal derivation
on $\mathbb{C}\langle X\rangle(\\!(z^{-1})\\!)$.

For $f\in\mathbb{C}\langle X\rangle(\\!(z^{-1})\\!)$ and
$x\in\mathbb{C}\langle X\rangle$, $T(f,x)$ will be denoted by $f._{z}x$ in the
sequel. The following is clear by definition:

######  Proposition 3.7.

$T$ defines an action of the $\mathbb{C}$-algebra $\mathbb{C}\langle X\rangle$
on $\mathbb{C}\langle X\rangle(\\!(z^{-1})\\!)$ from the right.

The right action $T$ can be extended to the right action

(3.2) $\mathbb{C}\langle X\rangle(\\!(w^{-1})\\!)\times\mathbb{C}\langle
X\rangle[\\![z]\\!]\to\mathbb{C}\langle X\rangle(\\!(w^{-1})\\!)[\\![z]\\!],$

under which, the image of $(f,z^{n}x_{i_{p}}\cdots x_{i_{1}})$ is mapped to
$z^{n}(f._{w}x_{i_{p}}\cdots x_{i_{1}})=:f._{w}(z^{n}x_{i_{p}}\cdots
x_{i_{1}})$. Note that now the notation $f._{w}S(\mathbf{x})$ makes sense.

######  Theorem 3.8.

Let $\\{f_{t}\\}_{t\geqslant 0}$ be a solution to the Loewner-Kufarev
equation. Then

$\begin{split}f_{t}(z)=\big{[}\int\underset{w=0}{\operatorname{Res\
}}\Big{(}\frac{\mathrm{e}^{x_{0}(t)}z}{1-zw}\big{(}w^{-1}._{w}S(\xi(\mathbf{x},\mathrm{e}^{x_{0}(t)}))\big{)}\Big{)}\big{]}_{0,t}.\end{split}$

###### Proof.

By setting

$\begin{split}\widetilde{w}(n)_{i_{1},\cdots,i_{p}}&:=\big{\\{}(n-i_{1})+1\big{\\}}\big{\\{}(n-(i_{1}+i_{2}))+1\big{\\}}\cdots\big{\\{}\big{(}n-(i_{1}+i_{2}+\cdots+i_{p-1})\big{)}+1\big{\\}},\end{split}$

where $n=i_{1}+\cdots+i_{p}$, we have

$\begin{split}w^{-1}._{w}1&=w^{-1},\\\ w^{-1}._{w}x_{i_{p}}\cdots
x_{i_{1}}&=\widetilde{w}(n)_{i_{1},\cdots,i_{p}}x_{i_{p}}\cdots
x_{i_{1}}w^{-(i_{1}+\cdots+i_{p}+1)}.\end{split}$

Therefore $\underset{w=0}{\operatorname{Res\
}}\big{(}\sum_{m=0}^{\infty}z^{m}w^{m}(w^{-1}._{w}1)\big{)}=1=(\text{the empty
word})$, and

$\begin{split}\underset{w=0}{\operatorname{Res\
}}\Big{(}\sum_{m=0}^{\infty}z^{m}w^{m}(w^{-1}._{w}x_{i_{p}}\cdots
x_{i_{1}})\Big{)}=z^{(i_{1}+\cdots+i_{p})}\widetilde{w}(n)_{i_{1},\cdots,i_{p}}x_{i_{p}}\cdots
x_{i_{1}}.\end{split}$

Hence we get

$\begin{split}&\underset{w=0}{\operatorname{Res\
}}\Big{(}\frac{\mathrm{e}^{x_{0}(t)}z}{1-zw}\big{(}w^{-1}._{w}S(\xi(\mathbf{x},\mathrm{e}^{x_{0}(t)}))\big{)}\Big{)}\\\
&=\mathrm{e}^{x_{0}(t)}z+\sum_{n=1}^{\infty}\mathrm{e}^{(n+1)x_{0}(t)}z^{n+1}\sum_{p=1}^{n}\sum_{\begin{subarray}{c}i_{1},\cdots,i_{p}\in\mathbb{N}:\\\
i_{1}+\cdots+i_{p}=n\end{subarray}}\widetilde{w}(n)_{i_{1},\cdots,i_{p}}x_{i_{p}}\cdots
x_{i_{1}}.\end{split}$

Now, in view of Theorem 2.8, we obtain the result. ∎

By tensoring the right action (3.2) it gives rise to

$\big{(}\mathbb{C}\langle X\rangle(\\!(w^{-1})\\!)\otimes\mathbb{C}\langle
X\rangle(\\!(u^{-1})\\!)\big{)}\times\big{(}\mathbb{C}\langle
X\rangle[\\![z]\\!]\otimes\mathbb{C}\langle
X\rangle[\\![z]\\!]\big{)}\to\mathbb{C}\langle
X\rangle(\\!(w^{-1})\\!)\otimes\mathbb{C}\langle X\rangle(\\!(u^{-1})\\!),$

under which, the image of $(f\otimes g,x\otimes y)$ will be denoted by
$(f._{w}x)\otimes(g._{u}y)$ in the sequel.

We recall that the tau-function corresponding to $W\in\mathrm{Gr}$, is given
by

$\begin{split}\tau(\mathbf{t})=\det(w_{+})=\det(1+a^{-1}bA)\end{split}$

(see [34, Proposition 3.3 and p. 50–51]) up to a multiplicative constant,
where $w_{+}:\mathrm{e}^{\xi(\mathbf{t},z)}W\to H_{+}$ is the orthogonal
projection, $\mathrm{e}^{\xi(\mathbf{t},z)}:H\to H$ is the multiplication
operator by $\mathrm{e}^{\xi(\mathbf{t},z)}$ with matrix representation

$\mathrm{e}^{-\xi(\mathbf{t},z)}=\left(\begin{array}[]{cc}a&b\\\
0&d\end{array}\right)\quad\text{along $H=H_{+}\oplus H_{-}$},$

and $A:H_{+}\to H_{-}$ is such that $\mathrm{graph}(A)=W$.

Given a bounded univalent function $f:\mathbb{D}_{0}\to\mathbb{C}$ with
$f(0)=0$, we denote by $A_{f}:H_{+}\to H_{-}$ the linear map such that
$\mathrm{graph}(A_{f})=W_{f}$.

######  Theorem 3.9.

Let $\\{f_{t}\\}_{0\leqslant t\leqslant T}$ be a univalent solution to the
Loewner-Kufarev equation such that $f_{t}(\mathbb{D}_{0})$ is bounded for
every $t\in[0,T]$. Then for each $h\in H_{+}$ and $|z|>1$, we have

$\begin{split}(A_{f_{t}}h)(z)=\big{[}\int\underset{\begin{subarray}{c}w=0,\\\
u=0\end{subarray}}{\operatorname{Res\
}}\Big{(}\frac{h^{\prime}(u)}{w-z}\sum_{r,s=1}^{\infty}\mathrm{e}^{(r+s)x_{0}(t)}x_{r+s}(w^{-r}._{w}S(\xi(\mathbf{x},\mathrm{e}^{x_{0}(t)})))\shuffle(u^{-s}._{u}S(\xi(\mathbf{x},\mathrm{e}^{x_{0}(t)})))\Big{)}\big{]}_{0,t}.\end{split}$

The proof can be found in Section A.4

##  4\. Connections with Conformal Field Theory and Free Probability

In this last section, we would like to briefly discuss aspects of the relation
the material developed so far has with free probability and conform field
theory. Let us first recall D. V. Voiculescu’s [38] basic definitions and
notions.

A pair $(\mathcal{A},\varphi)$ consisting of an unital algebra $\mathcal{A}$
and a linear functional $\varphi:\mathcal{A}\to\mathbb{C}$ with
$\varphi(1_{\mathcal{A}})=1$, is called a non-commutative probability space,
and then any element in $\mathcal{A}$ is called a random variable.

For any random variable $a\in\mathcal{A}$ with $\mu_{a}(1_{\mathcal{A}}):=1$,
the $\mathbb{C}$-linear function $\mu_{a}:\mathbb{C}[\\![X]\\!]\to\mathbb{C}$
defined by

$\mu_{a}(X^{n}):=\varphi(a^{n}),\quad n\in\mathbb{N}$

is called the distribution of $a$. More generally, any element in the set
$\Sigma:=\\{\mu:\mathbb{C}[X]\to\mathbb{C}|\text{ $\mu$ is linear and
$\mu(1)=1$}\\}$ is called a distribution. We denote by
$\Sigma_{1}^{\times}:=\\{\mu\in\Sigma:\mu(X)=1\\}$ the family of distributions
of mean one. Then, D. V. Voiculescu [38] introduced two operations $\boxplus$
and $\boxtimes$ as follows.

For each $\mu\in\Sigma$,

$\mathcal{R}_{\mu}(z):=G^{-1}(z)-\frac{1}{z}\in\mathbb{C}[\\![z]\\!]$

is called the $\mathcal{R}$-transform of $\mu$, where $G^{-1}(z)$ is the
compositional inverse of the Cauchy transform
$G(z):=\sum_{n=0}^{\infty}\mu(X^{n})z^{-(n+1)}$ of $\mu$. Then
$R_{\mu}(z):=z\mathcal{R}_{\mu}(z)\in z\mathbb{C}[\\![z]\\!]$ is called the
(modified) $R$-transform of $\mu$. For two distributions $\mu,\nu\in\Sigma$, a
new distribution $\mu\boxplus\nu\in\Sigma$ can be defined by the following
formula

$\mathcal{R}_{\mu\boxplus\nu}(z)=\mathcal{R}_{\nu}(z)+\mathcal{R}_{\nu}(z)$

(then we also have $R_{\mu\boxplus\nu}(z)=R_{\nu}(z)+R_{\nu}(z)$) and it is
called the additive free convolution of $\mu$ and $\nu$ ([38, Definition 3.1.1
and Remark 3.1.2]).

On the other hand, for any distribution $\mu\in\Sigma^{\times}$,

$S_{\mu}(z):=\frac{1+z}{z}M_{\mu}^{-1}(z)\in\mathbb{C}[\\![z]\\!]^{\times}$

is called the $S$-transform of $\mu$, where $M_{\mu}^{-1}(z)$ is the
compositional inverse of the associated moment series of $\mu$:
$M_{\mu}(z):=\sum_{n=1}^{\infty}\mu(X^{n})z^{n}$. Then for any two
distribution $\mu,\nu\in\Sigma^{\times}$, a new distribution
$\mu\boxtimes\nu\in\Sigma^{\times}$ can be determined by the relation

$S_{\mu\boxtimes\nu}(z)=S_{\nu}(z)\cdot S_{\nu}(z)$

which is called the multiplicative free convolution of $\mu$ and $\nu$ ([38,
Definition 3.6.1 and Remark 3.6.2]).

These two sets $(\Sigma,\boxplus)$ and $(\Sigma^{\times},\boxtimes)$ with
operations form commutative groups ([38, Theorem 3.2.3 and Theorem 3.6.3]).
Moreover, it is a result by R. Friedrich and J. McKay [11] that there exists a
group isomorphism $\mathrm{EXP}:\Sigma\to\Sigma^{\times}$ such that the
following diagram commutes.

$\textstyle{\Sigma_{1}^{\times}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$S$-transform$\mathrm{LOG}$$\textstyle{\Lambda(\mathbb{C})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$z\frac{\mathrm{d}}{\mathrm{d}z}\log$$\textstyle{\Sigma\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$R$-transform$\mathrm{EXP}$$\textstyle{z\mathbb{C}[\\![z]\\!]}$

Here, we use that $\mathrm{LOG}=\mathrm{EXP}^{-1}$.

Given a Hilbert space $H$, the pair of $(B(\mathcal{F}(H)),\tau_{\Omega})$ of
the unital algebra of all bounded linear operators on the full Fock space

$\mathcal{F}(H):=\bigoplus_{n=0}^{\infty}H^{\otimes
n}=(\mathbb{C}\Omega)\oplus\bigoplus_{n=1}^{\infty}H^{\otimes n}$

(where $\Omega:=1$ and $\otimes$ is understood in the sense of
$\otimes_{\mathbb{C}}$) of $H$, and the vacuum expectation

$\tau_{\Omega}:B(\mathcal{F}(H))\ni a\mapsto\langle a\rangle:=\langle
a(\Omega),\Omega\rangle_{\mathcal{A}}\in\mathbb{C}$

is a non-commutative probability space.

For each unit vector $e\in H$, we define $l(e),l^{*}(e)\in B(\mathcal{F}(H))$
by $l(e)(x):=e\otimes x$, $x\in\mathcal{F}(H)$ and

$\begin{split}l^{*}(e)(\Omega)&:=0\\\ l^{*}(e)(x_{1}\otimes\cdots\otimes
x_{n})&:=\left\\{\begin{array}[]{ll}\langle e,x_{1}\rangle&\text{if $n=1$,}\\\
\langle e,x_{1}\rangle_{H}x_{2}\otimes\cdots\otimes x_{n}&\text{if $n\geqslant
2$.}\end{array}\right.\end{split}$

If one fixes a unit vector $e$ and writes $l:=l(e)$ and $l^{*}:=l^{*}(e)$,
then the pair $(\widetilde{\mathcal{E}}_{1},\widetilde{\tau}_{1})$ of the
unital $\mathbb{C}$-algebra
$\widetilde{\mathcal{E}}_{1}:=\mathbb{C}[\\![l,l^{*}]\\!]$ and
$\widetilde{\tau}_{1}:\widetilde{\mathcal{E}}_{1}\ni\alpha(l,l^{*})\mapsto\alpha(0,0)\in\mathbb{C}$
is also a non-commutative probability space.

Now, U. Haagerup [12] showed the following: For any distribution
$\nu\in\Sigma^{\times}$, there exists a unique
$f\in\mathbb{C}[\\![z]\\!]^{\times}$ such that the distribution (with respect
to $\widetilde{\tau}_{1}$) of the random variable
$a:=(1+l)f(l^{*})\in\widetilde{\mathcal{E}}_{1}$ coincides with $\nu$.
Moreover, we have then $S_{\mu_{a}}(z)=1/f(z)$.

If one restricts this result to $\nu\in\Sigma_{1}^{\times}$, then it follows
from results by R. Friedrich and J. McKay [11], that the following diagram
commutes:

$\textstyle{\\{a\in\widetilde{\mathcal{E}_{1}}:\widetilde{\tau}_{1}(a)=1\\}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\begin{array}[]{c}\text{{\tiny
taking}}\vspace{-2mm}\\\
\text{{\tiny$\widetilde{\tau}_{1}$-distribution}}\end{array}}$$\textstyle{\Lambda(\mathbb{C})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\begin{array}[]{l}\text{{\tiny$(1+l)$}}\vspace{-2mm}\\\
\text{{\tiny$\times\text{(evaluation}$}}\vspace{-2mm}\\\ \text{{\tiny\hskip
19.91692pt$\text{at $z=l^{*}$)}$}}\vspace{-2mm}\\\
\text{{\tiny(Haagerup)}}\end{array}}$$\textstyle{\Sigma_{1}^{\times}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{id}}$$\scriptstyle{\begin{array}[]{l}\text{{\tiny$S$-transform}}\vspace{-2mm}\\\
\text{{\tiny(Voiculescu)}}\end{array}}$$\textstyle{\Lambda(\mathbb{C})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$inversion$z\frac{\mathrm{d}}{\mathrm{d}z}\log$$\textstyle{\Sigma\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\begin{array}[]{l}\text{{\tiny$R$-transform}}\vspace{-2mm}\\\
\text{{\tiny(Voiculescu)}}\end{array}}$$\scriptstyle{\begin{array}[]{c}\text{{\tiny$\mathrm{EXP}$}}\vspace{-2mm}\\\
\text{{\tiny(Friedrich-
McKay)}}\end{array}}$$\textstyle{z\mathbb{C}[\\![z]\\!]}$

From this point of view, the solution $f_{t}(z)$ to the Loewner-Kufarev
equation lives in $z\mathbb{C}[\\![z]\\!]$ in the above diagram. This, via the
diagram, is realised as a random variable of the free probability space
$(\widetilde{\mathcal{E}}_{1},\widetilde{\tau}_{1})$. This would imply that we
would have a Bosonic expression of the Loewner-Kufarev equation. Thus, the
Loewner-Kufarev equation acts on the Bosonic Fock space. Here we shall recall
that we have the positive part of the Witt generators in the expression of the
Loewner-Kufarev equation. One reasonable questions might be, what is the
meaning of these Witt generators in this Bosonic expression.

In 2017, R. Friedrich111Talk delivered at the conference “Stochastic Analysis
and Related Topics 2017”, Ritsumeikan University (BKC), 18. October 2017,
pointed out that any distribution of mean one corresponds to a tau-function of
the KP hierarchy, extending results obtained with J. McKay [10]. This can be
seen as follows: A sequence
$K=\\{K_{n}(x_{1},\cdots,x_{n})\\}_{n\in\mathbb{N}}$ of polynomials
$K_{n}(x_{1},\cdots,x_{n})\in\mathbb{C}[x_{1},\cdots,x_{n}]$ determines a
mapping

$K:\Lambda(\mathbb{C})\ni
1+\sum_{k=1}^{\infty}p_{k}z^{k}\mapsto\sum_{n=0}^{\infty}K_{k}(p_{1},\cdots,p_{k})z^{k}\in\mathbb{C}[\\![z]\\!],$

where $\Lambda(A):=1+z\mathbb{C}[\\![z]\\!]$ and $K_{0}:=1$. The sequence
$K=\\{K_{n}(x_{1},\cdots,x_{n})\\}_{n\in\mathbb{N}}$ is called a (Hirzebruch)
multiplicative sequence (or m-sequence over $\mathbb{C}$) if the above mapping
is a homomorphism between the monoid $\Lambda(A)$ (where the product is
defined by pointwise multiplication) and the ring of formal power series
$\mathbb{C}[\\![z]\\!]$ (but considered as a monoid). A result by T. Katsura,
Y. Shimizu and K. Ueno [15] says that, if one defines $b_{n}\in\mathbb{C}$ as
the coefficient of $x_{n}$ in the polynomial $K_{n}(x_{1},\cdots,x_{n})$, then

$\tau_{T}(\mathbf{t}):=\exp\Big{(}\sum_{k=1}^{\infty}(-1)^{k+1}b_{k}t_{k}\Big{)},$

where $\mathbf{t}=(t_{1},t_{2},\cdots)$, is a tau-function of the KP hierarchy
([15, Theorem 5.3, 1)]). Furthermore, the corresponding wave operator belongs
to the class
$1+\partial_{t_{1}}^{-1}\mathbb{C}[\\![\partial_{t_{1}}^{-1}]\\!]$ (i.e., all
the coefficients are constants, and hence the corresponding Lax operator is
given by $L=\partial_{t_{1}}$). In particular, in [10, Theorem 4.3] this was
related to the infinite Lie group $(\Sigma_{1},\boxtimes)$.

Summarising the above, we have the following isomorphisms of commutative
groups and their embedding into the universal Grassmannian $\mathrm{UGM}$:

$\textstyle{\Sigma_{1}^{\times}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\begin{array}[]{c}\text{{\tiny$S$-transform}}\vspace{-2mm}\\\
\text{{\tiny(Voiculescu)}}\\\
\cong\end{array}}$$\textstyle{\Lambda(\mathbb{C})}$$\textstyle{\big{\\{}{\begin{array}[]{c}\text{{\small
m-sequence}}\vspace{-1.5mm}\\\ \text{{\small over
$\mathbb{C}$}}\end{array}}\big{\\}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\begin{array}[]{c}\text{{\tiny
assigning the}}\vspace{-2mm}\\\ \text{{\tiny characteristic}}\vspace{-2mm}\\\
\text{{\tiny power series}}\vspace{-2mm}\\\ \text{{\tiny(Hirzebruch)}}\\\
\cong\end{array}}$$\scriptstyle{\begin{array}[]{c}\text{{\tiny Katsura-
Shimizu-Ueno}}\vspace{-2mm}\\\ \text{{\tiny\cite[cite]{[\@@bibref{}{KSU}{}{},
Theorem~{}5.3--3)]}}}\vspace{-2mm}\\\
\text{\phantom{a}}\end{array}}$$\textstyle{\mathrm{UGM}}$

Hence as pointed out by the second author, any distribution of mean one is
embeddable into the universal Grassmannian.

Now, it would be interesting to interpret the structure of the convolution
semigroup (or flow) in $(\Sigma_{1}^{\times},\boxtimes)$ as a submanifold of
UGM. A further question is, how the central limit theorem in
$\Sigma_{1}^{\times}$ is described in the Grassmannian in relation with the
tau-function of the KdV hierarchy associated to
$\triangle=L^{2}=\partial_{t_{1}}^{2}$?

In this picture, the solution to the Loewner-Kufarev equation lives in
$\Sigma_{1}^{\times}$ and thus can be again embedded into the Grassmannian.
So, how does the positive part of the Witt generators in the Loewner-Kufarev
equation act in this context?

##  Appendix A

###  A.1. Elements of the theory of Faber polynomials and Grunsky coefficients

######  Definition A.1.

Let $f:\mathbb{D}_{0}\to\mathbb{C}$ be a univalent function with $f(0)=0$.

  * (i)

For each $n\in\mathbb{N}$, $Q_{n}(w):=[(f^{-1}(w))^{-n}]_{\leqslant 0}$ (the
principal plus constant part of the Laurent series for $(1/f^{-1}(w))^{n}$) is
called the $n$-th Faber polynomial associated with the domain
$\mathbb{C}\setminus\overline{f(\mathbb{D}_{0})}$ (or just simply $f$).

  * (ii)

The coefficients $(b_{-m,-n})_{m,n=1}^{\infty}$ in the series expansion

(A.1)
$\begin{split}\log\frac{f(z)-f(\zeta)}{z-\zeta}=-\sum_{m=0}^{\infty}\sum_{n=0}^{\infty}b_{-m,-n}z^{m}\zeta^{n}\end{split}$

at $(z,\zeta)=(0,0)$, are called the Grunsky coefficients of $f$.

This definition still works for any analytic functions $f$ defined on an open
neighbourhood of the origin (not only $\mathbb{D}_{0}$) of the form
$f(z)=a_{1}z+a_{2}z^{2}+\cdots$ with $a_{1}\neq 0$, and then the
$\\{Q_{n}\\}_{n\geqslant 1}$ and $(b_{-m,-n})_{m,n\geqslant 1}$ are also
called the Faber polynomials and Grunsky coefficients of $f$ respectively.
Also, the following Proposition A.2 extends obviously to this situation if we
replace $\mathbb{D}_{0}$ by ‘some open neighbourhood of the origin’.

The Grunsky coefficients are, by definition, symmetric: $b_{-m,-n}=b_{-n,-m}$
for $n,m\in\mathbb{Z}_{\geqslant 0}$. Note also that

$\log\frac{f(z)}{z}=-\sum_{m=0}^{\infty}b_{-m,0}z^{m}\quad\text{for
$z\in\mathbb{D}_{0}$.}$

######  Proposition A.2.

Let $f:\mathbb{D}_{0}\to\mathbb{C}$ be a univalent function with $f(0)=0$ and
$n\in\mathbb{N}$. Then the following are equivalent:

  * (i)

$Q_{n}(w)$ is the $n$-th Faber polynomial associated to the domain
$\mathbb{C}\setminus\overline{f(\mathbb{D}_{0})}$, i.e.,
$Q_{n}(w)=[(1/f^{-1}(w))^{n}]_{\leqslant 0}$, the principal plus constant part
of the Laurent series for $(1/f^{-1}(w))^{n}$.

  * (ii)

We have

(A.2)
$\begin{split}\log\frac{w-f(z)}{w}=\log\frac{f(z)}{f^{\prime}(0)z}-\sum_{n=1}^{\infty}\frac{Q_{n}(w)}{n}z^{n}\end{split}$

for $z\in\mathbb{D}_{0}$ and
$w\in\mathbb{C}\setminus\overline{f(\mathbb{D}_{0})}$.

  * (iii)

$Q_{n}(w)$ is a polynomial of degree $n$ in $1/w$ such that

$Q_{n}(f(z))=z^{-n}+n\sum_{m=1}^{\infty}b_{-n,-m}z^{m}.$

  * (iv)

$Q_{n}(w)$ is a polynomial of degree $n$ in $1/w$ such that

$Q_{n}(f(z))-z^{-n}\to 0\quad\text{as $z\to 0$.}$

See [36, Chapter 2, Section 2] or [37, Section 2.2] for details.

###  A.2. Proof of Proposition 2.9

######  Lemma A.3.

Let $\mu$ be a finite measure on $[0,T]$ without atoms. Then the nondecreasing
function $h:[0,T]\to[0,+\infty)$ defined by

$\begin{split}h(t)&:=\sup_{0\leqslant a\leqslant
T}\int_{a}^{\min\\{a+t,T\\}}\mathrm{d}\mu,\quad t\in[0,T]\end{split}$

is uniformly continuous.

###### Proof.

Let $\varepsilon>0$ be arbitrary. Since $\mathbb{R}\ni a\mapsto\min\\{a,T\\}$
and $\phi:[0,T]\ni a\mapsto\int_{0}^{a}\mathrm{d}\mu$ are uniformly
continuous, there exists $\delta>0$ such that, for each $a,b\in[0,T]$ with
$|a-b|<\delta$, it holds that
$|\int_{0}^{\min\\{a,T\\}}\mathrm{d}\mu-\int_{0}^{\min\\{b,T\\}}\mathrm{d}\mu|<\varepsilon/2$.
Now, let $s,t\in[0,T]$ be arbitrary with $|t-s|<\delta$. By the continuity of
$\phi$, we can take $a,b\in[0,T]$ such that
$h(t)=\int_{a}^{\min\\{a+t,T\\}}\mathrm{d}\mu$ and
$h(s)=\int_{b}^{\min\\{b+t,T\\}}\mathrm{d}\mu$. If $s\leqslant t$, then
$|h(t)-h(s)|=h(t)-h(s)\leqslant\int_{a}^{\min\\{a+t,T\\}}\mathrm{d}\mu-\int_{a}^{\min\\{a+s,T\\}}\mathrm{d}\mu=\int_{0}^{\min\\{a+t,T\\}}\mathrm{d}\mu-\int_{0}^{\min\\{a+s,T\\}}\mathrm{d}\mu<\varepsilon$.
Similarly, if $s\geqslant t$ then
$|h(t)-h(s)|=h(s)-h(t)\leqslant\int_{b}^{\min\\{b+s,T\\}}\mathrm{d}\mu-\int_{b}^{\min\\{b+t,T\\}}\mathrm{d}\mu<\varepsilon$.
∎

######  Proposition A.4.

Let $\\{f_{t}\\}_{0\leqslant t\leqslant T}$ be a solution to the controlled
Loewner-Kufarev equation (2.1). Then there exists an open neighbourhood $D$ of
$0\in\mathbb{C}$ such that

  * (i)

$f_{t}|_{D}$ is univalent for each $t\in[0,T]$;

  * (ii)

For each $\zeta\in\cap_{0\leqslant t\leqslant T}f_{t}(D)$, the mapping
$[0,T]\ni t\mapsto f_{t}^{-1}(\zeta)\in D$ is continuous and of bounded
variation.

###### Proof.

(i) Consider the mapping

$\begin{split}\varphi:[0,T]\times\mathbb{D}_{0}\ni(t,z)\mapsto(t,f_{t}(z))\in[0,T]\times\mathbb{C}.\end{split}$

We will show that, there exists an open neighbourhood $D$ of $0\in\mathbb{C}$
such that $\varphi|_{[0,T]\times D}$ is injective. We divide the proof into
two parts.

(i–a) We first prepare the necessary estimates. Define
$g:[0,T]\times\mathbb{D}_{0}\to[0,T]\times\mathbb{C}$ by

$g(t,z):=\varphi(t,z)-(t,f_{t}^{\prime}(0)z)=(0,f_{t}(z)-f_{t}^{\prime}(0)z).$

Then, for each $(t,z),(s,w)\in[0,T]\times\mathbb{D}_{0}$, by writing
$z-w=|z-w|\mathrm{e}^{i\theta}$ and
$\mu(\mathrm{d}u,w):=\mathrm{d}x_{0}(u)+\mathrm{d}\xi(\mathbf{x},w)_{u}$, we
have

$\begin{split}&g(t,z)-g(s,w)\\\
&=\\{g(t,z)-g(t,w)\\}+\\{g(t,w)-g(s,w)\\}=(0,I+I\\!I+I\\!I\\!I),\end{split}$

where, by writing $z=w+|z-w|\mathrm{e}^{i\theta}$ and
$\eta=w+a\mathrm{e}^{i\theta}$ for $0\leqslant a\leqslant|z-w|$,

$\begin{split}I&:=f_{t}(z)-f_{s}(z),\quad\quad
I\\!I:=(f_{s}^{\prime}(0)-f_{t}^{\prime}(0))z,\\\
I\\!I\\!I&:=\big{\\{}f_{s}(z)-f_{s}^{\prime}(0)z\big{\\}}-\big{\\{}f_{s}(w)-f_{s}^{\prime}(0)w\big{\\}}=\int_{0}^{|z-w|}(f_{s}^{\prime}(\eta)-f_{s}^{\prime}(0))\mathrm{d}a.\end{split}$

By Proposition 2.6, we have $f_{t}^{\prime}(0)=\mathrm{e}^{x_{0}(t)}$.

Let $r\in(0,\frac{1}{2}]$ be arbitrary. It is straightforward to get the
following estimates: If $0\leqslant s\leqslant t\leqslant T$ and $z,w\in
r\mathbb{D}_{0}$, then

$\begin{split}|I|&\leqslant
c_{1}(r)\int_{s}^{t}\big{\\{}|\mathrm{d}x_{0}(u)|+\sum_{n=1}^{\infty}2^{-n}|\mathrm{d}x_{n}(u)|\big{\\}},\\\
|I\\!I|&\leqslant
c_{2}(r)\int_{s}^{t}|\mathrm{d}x_{0}(u)|\quad\text{and}\quad|I\\!I\\!I|\leqslant
c_{3}(r)|z-w|,\end{split}$

where
$c_{1}(r):=r\sup_{u\in[0,T]\times\overline{(\frac{1}{2}\mathbb{D}_{0})}}|f_{u}^{\prime}(z)|$,
$c_{2}(r):=r\exp(\|x_{0}\|_{\infty})$ and

$c_{3}(r):=\sup_{(u,\eta)\in[0,T]\times\overline{(r\mathbb{D}_{0})}}|f_{u}^{\prime}(\eta)-f_{u}^{\prime}(0)|.$

We also note the following inequality: For any $z,w\in r\mathbb{D}_{0}$ and
$t,s\in[0,T]$,

$\begin{split}|z-w|\leqslant\mathrm{e}^{\|x_{0}\|_{\infty}}\Big{\\{}|f_{t}^{\prime}(0)z-f_{s}^{\prime}(0)w|+r|f_{t}^{\prime}(0)-f_{s}^{\prime}(0)|\
\Big{\\}},\end{split}$

which follows from $f_{t}^{\prime}(0)=\mathrm{e}^{x_{0}(t)}$ and the trivial
identity
$z-w=\frac{1}{f_{t}^{\prime}(0)}(f_{t}^{\prime}(0)z-f_{s}^{\prime}(0)w)+\frac{1}{f_{t}^{\prime}(0)}w(f_{s}^{\prime}(0)-f_{t}^{\prime}(0))$.

(i–b) Define $h:\mathbb{R}\times\mathbb{C}\to[0,+\infty)$ by

$\begin{split}h(t,z):=|z|+|t|+\sup_{0\leqslant u\leqslant
T}\int_{u}^{\min\\{u+|t|,T\\}}\big{\\{}|\mathrm{d}x_{0}(u)|+\sum_{k=1}^{\infty}2^{-k}|\mathrm{d}x_{k}(u)|\big{\\}}.\end{split}$

Since the measure
$\mu(\mathrm{d}s,a):=|\mathrm{d}x_{0}(s)|+\sum_{n=1}^{\infty}a^{n}|\mathrm{d}x_{n}(s)|$
on $[0,T]$ doesn’t have atoms for each $a\in[0,1)$, $h$ is continuous by
virtue of Lemma A.3. It is easy to see that for each
$(t,z),(s,w)\in\mathbb{R}\times\mathbb{C}$,

  * (1)

$h(t,z)=0\Longleftrightarrow(t,z)=(0,0)$;

  * (2)

$h(t+s,z+w)\leqslant h(t,z)+h(s,w)$.

Also note that for each $(t,z),(s,w)\in[0,T]\times\mathbb{D}_{0}$, we have

(A.3) $h((t,z)-(s,w))\leqslant|z-w|+h(\varphi(t,z)-\varphi(s,w)).$

Let $(t,z),(s,w)\in[0,T]\times r\mathbb{D}_{0}$ be arbitrary with $s\leqslant
t$. By using the relation $(t,f_{t}^{\prime}(0)z)=\varphi(t,z)-g(t,z)$ and
estimates in (a), we obtain

$\begin{split}&|z-w|\\\
&\leqslant\mathrm{e}^{\|x_{0}\|_{\infty}}\left[h\Big{(}\big{\\{}\varphi(t,z)-g(t,z)\big{\\}}-\big{\\{}\varphi(s,w)-g(s,w)\big{\\}}\Big{)}+c_{2}(r)\int_{s}^{t}|\mathrm{d}x_{0}(u)|\right]\\\
&\leqslant\mathrm{e}^{\|x_{0}\|_{\infty}}\left[h\Big{(}\varphi(t,z)-\varphi(s,w)\Big{)}+h\Big{(}g(t,z)-g(s,w)\Big{)}+c_{2}(r)\int_{s}^{t}|\mathrm{d}x_{0}(u)|\right]\\\
&\leqslant\mathrm{e}^{\|x_{0}\|_{\infty}}h\Big{(}\varphi(t,z)-\varphi(s,w)\Big{)}+2\mathrm{e}^{\|x_{0}\|_{\infty}}c_{4}(r)h\big{(}(t,z)-(s,w)\big{)},\end{split}$

where $c_{4}(r):=c_{1}(r)+c_{2}(r)+c_{3}(r)$. Then, by using (A.3), we get

$\begin{split}|z-w|\leqslant\mathrm{e}^{\|x_{0}\|_{\infty}}(1+2c_{4}(r))h\big{(}\varphi(t,z)-\varphi(s,w)\big{)}+2\mathrm{e}^{\|x_{0}\|_{\infty}}c_{4}(r)|z-w|.\end{split}$

Then, by taking $r\in(0,\frac{1}{2}]$ small so that
$2\mathrm{e}^{\|x_{0}\|_{\infty}}c_{4}(r)<\frac{1}{2}$ and putting
$D:=r\mathbb{D}_{0}$, we reached: For every $(t,z),(s,w)\in[0,T]\times D$, it
holds that

(A.4)
$\begin{split}\frac{1}{2}|z-w|\leqslant\mathrm{e}^{\|x_{0}\|_{\infty}}(1+2c_{4}(r))h\big{(}\varphi(t,z)-\varphi(s,w)\big{)},\end{split}$

which implies that $\varphi|_{[0,T]\times D}$ is injective.

(ii) Let $D$ be as in the proof of (i). If $\zeta\in\cap_{0\leqslant
t\leqslant T}f_{t}(D)$, then for each $t\in[0,T]$, we have
$(t,\zeta)\in\varphi([0,T]\times D)$, and then
$\varphi^{-1}(t,\zeta)=(t,f_{t}^{-1}(\zeta))$. Now the assertion follows by
(A.4). ∎

######  Definition A.5.

Let $X$ and $Y$ be nonempty subsets of $\mathbb{C}$.

  * (1)

For each $x\in\mathbb{C}$, we set $d(x,Y):=\inf\\{|x-y|:y\in Y\\}$.

  * (2)

The Hausdorff distance between $X$ and $Y$ is defined by

$d_{H}(X,Y):=\inf\\{\varepsilon>0:\text{$X\subset Y_{\varepsilon}$ and
$Y\subset X_{\varepsilon}$}\\},$

where $X_{\varepsilon}:=\cup_{x\in X}\\{z\in\mathbb{C}:|z-x|<\varepsilon\\}$.

######  Definition A.6.

Let $\gamma\subset\mathbb{C}$ be a Jordan curve and $\delta>0$. We call

$T_{\delta}(\gamma):=\\{z\in\mathbb{C}:d(z,\gamma)<\delta\\}$

a tubular $\delta$-neighbourhood of $\gamma$ if for every
$\varepsilon\in(0,\delta]$,

  * (1)

there is a homeomorphism between $T_{\varepsilon}(\gamma)$ and $(-1,1)\times
S^{1}$, under which $\gamma\cong\\{0\\}\times S^{1}$;

  * (2)

the boundary $\partial T_{\varepsilon}(\gamma)$ is homeomorphic to a disjoint
union of two circles.

######  Remark A.1.

  * (a)

If the Jordan curve $\gamma$ is $C^{\infty}$, there exists a tubular
$\delta$-neighbourhood of $\gamma$ for sufficiently small $\delta>0$ (Tubular
neighbourhood theorem).

  * (b)

For two Jordan curves $\gamma_{1},\gamma_{2}\subset\mathbb{C}$, it is easy to
see that

(A.5)
$\begin{split}d_{H}(\gamma_{1},\gamma_{2})\leqslant\inf_{\begin{subarray}{c}\text{{\rm
homeomorphisms}}\\\ \phi_{i}:S^{1}\to\gamma_{i};\\\
i=1,2\end{subarray}}\|\phi_{1}-\phi_{2}\|_{\infty}.\end{split}$

######  Lemma A.7.

Let $\\{f_{t}\\}_{0\leqslant t\leqslant T}$ be a solution to the controlled
Loewner-Kufarev equation (2.1). Then for each $t_{0}\in[0,T]$, we can take
$r^{\prime}\in(0,1)$ satisfying the following property: For each
$r\in(0,r^{\prime})$, there exist $\delta>0$ and an interval $I\ni t_{0}$
which is open in $[0,T]$ such that, by putting $D:=r\mathbb{D}_{0}$, we have
for any $s,t\in I$,

  * (i)

the tubular $\delta$-neighbourhood $T_{\delta}(f_{t}(\partial D))$ exists;

  * (ii)

$f_{s}(\partial D)\subset T_{\delta}(f_{t}(\partial D))$.

###### Proof.

Let $t_{0}\in[0,T]$ be arbitrary. By Proposition A.4, its proof and the fact
that $f_{t_{0}}^{\prime}(0)>0$, we can take $r\in(0,\frac{1}{4})$ and an
interval $J^{\prime}$ containing $t_{0}$, which is open in $[0,T]$, so that
$f_{t}|_{(2r\overline{\mathbb{D}_{0}})}$ is injective for every $t\in
J^{\prime}$, (A.4) holds on $J^{\prime}\times(2r\mathbb{D}_{0})$ and

$c_{0}:=\inf_{(t,z)\in
J^{\prime}\times(2r\overline{\mathbb{D}_{0}})}\mathrm{Re}(f_{t}^{\prime}(z))>0.$

We put

$\mathbf{n}(t,z):=\frac{zf_{t}^{\prime}(z)}{|zf_{t}^{\prime}(z)|},\quad(t,z)\in[0,T]\times\partial(r\mathbb{D}_{0}),$

which is the outer normal unit vector of $f_{t}(\partial(r\mathbb{D}_{0}))$ at
$f_{t}(z)$. We then consider the mapping
$\varphi:[0,T]\times(-r,r)\times\partial(r\mathbb{D}_{0})\to[0,T]\times\mathbb{C}$
defined by

$\varphi(t,\varepsilon,z):=(t,f_{t}(z)+\varepsilon\mathbf{n}(t,z))$

for $(t,\varepsilon,z)\in[0,T]\times(-r,r)\times\partial(r\mathbb{D}_{0})$.

We define $h:\mathbb{R}\times\mathbb{C}\to[0,+\infty)$ by

$\begin{split}h(t,z)&:=|z|+|t|+\sup_{0\leqslant a\leqslant
T}\int_{a}^{\min\\{a+t,T\\}}\big{\\{}|\mathrm{d}x_{0}(s)|+\sum_{k=1}^{\infty}k2^{-k}|\mathrm{d}x_{k}(s)|\big{\\}}.\end{split}$

Since the measure
$\mu(\mathrm{d}s,a):=|\mathrm{d}x_{0}(s)|+\sum_{n=1}^{\infty}na^{n}|\mathrm{d}x_{n}(s)|$
on $[0,T]$ doesn’t have atoms, $h$ is continuous (see Lemma A.3) and it is
easy to see that for each $(t,z),(s,w)\in[0,T]\times\mathbb{C}$,

  * (1)

$h(t,z)=0\Longleftrightarrow(t,z)=(0,0)$;

  * (2)

$h(t+s,z+w)\leqslant h(t,z)+h(s,w)$.

We shall divide the proof into three parts.

(i–a) Firstly, we show that

> for each $z_{0}\in\partial(r\mathbb{D}_{0})$, there exists an open
> neighbourhood $U\subset[0,T]\times(-r,r)\times\partial(r\mathbb{D}_{0})$ of
> $(t_{0},0,z_{0})$ such that, for any
> $(t,\varepsilon,z),(s,\varepsilon^{\prime},w)\in U$, it holds that
>
> (A.6)
> $h(\varphi(t,\varepsilon,z)-\varphi(s,\varepsilon^{\prime},w))\geqslant\frac{1}{2}h\Big{(}(t,f_{t}(r_{0}(\varepsilon)z))-(s,f_{s}(r_{0}(\varepsilon^{\prime})w))\Big{)},$
>
> where
> $r_{0}(\varepsilon):=1+\frac{\varepsilon}{r|f_{t_{0}}^{\prime}(z_{0})|}$,
> and hence $\varphi|_{U}$ is injective.

For this, we further set
$g:[0,T]\times(-r,r)\times\partial(r\mathbb{D}_{0})\to[0,T]\times\mathbb{C}$
by

$\begin{split}g(t,\varepsilon,z)&:=\varphi(t,\varepsilon,z)-\left(t,f_{t}(r_{0}(\varepsilon)z)\right)=\big{(}0,\varepsilon
G(t,\varepsilon,z)\big{)},\end{split}$

where

$\begin{split}G(t,\varepsilon,z)&:=\varepsilon^{-1}\big{\\{}f_{t}(z)-f_{t}(r_{0}(\varepsilon)z)\big{\\}}+\mathbf{n}(t,z)=\int_{0}^{1}\Big{\\{}\mathbf{n}(t,z)-\frac{zf_{t}^{\prime}(r_{0}(\varepsilon
a)z)}{|z_{0}f_{t_{0}}^{\prime}(z_{0})|}\Big{\\}}\mathrm{d}a.\end{split}$

For
$(t,\varepsilon,z),(s,\varepsilon^{\prime},z^{\prime})\in[0,T]\times(-r,r)\times\partial(r\mathbb{D}_{0})$,
we have

(A.7) $\begin{split}&\varepsilon
G(t,\varepsilon,z)-\varepsilon^{\prime}G(s,\varepsilon^{\prime},w)=(\varepsilon-\varepsilon^{\prime})G(s,\varepsilon^{\prime},w)+\varepsilon\big{(}G(t,\varepsilon,z)-G(s,\varepsilon^{\prime},w)\big{)},\end{split}$

where, for the last term, we have

(A.8)
$\begin{split}G(t,\varepsilon,z)-G(s,\varepsilon^{\prime},w)=I+I\\!I+I\\!I\\!I+I\\!V,\end{split}$

where

$\begin{split}I&:=\mathbf{n}(t,z)-\mathbf{n}(t,w),\quad
I\\!I:=\mathbf{n}(t,w)-\mathbf{n}(s,w),\\\
I\\!I\\!I&:=\frac{1}{|z_{0}f_{t_{0}}^{\prime}(z_{0})|}\int_{0}^{1}\Big{\\{}wf_{s}^{\prime}\Big{(}w+\frac{\varepsilon^{\prime}wa}{r|f_{t_{0}}^{\prime}(z_{0})|}\Big{)}-zf_{s}^{\prime}\Big{(}z+\frac{\varepsilon
za}{r|f_{t_{0}}^{\prime}(z_{0})|}\Big{)}\Big{\\}}\mathrm{d}a,\\\
I\\!V&:=\frac{1}{|z_{0}f_{t_{0}}^{\prime}(z_{0})|}\int_{0}^{1}\Big{\\{}zf_{s}^{\prime}\Big{(}z+\frac{\varepsilon
za}{r|f_{t_{0}}^{\prime}(z_{0})|}\Big{)}-zf_{t}^{\prime}\Big{(}z+\frac{\varepsilon
za}{r|f_{t_{0}}^{\prime}(z_{0})|}\Big{)}\Big{\\}}\mathrm{d}a.\end{split}$

Let $B_{r/2}(z_{0}):=\\{z\in\mathbb{C}:|z-z_{0}|<\frac{r}{2}\\}$ and
$r^{\prime}:=\min\\{r,|f_{t_{0}}^{\prime}(z_{0})|(\frac{1}{4}-r)r\\}$. Suppose
that $s,t\in J^{\prime}$ with $s\leqslant t$,
$\varepsilon,\varepsilon^{\prime}\in(-r^{\prime},r^{\prime})$ and
$z,w\in\partial(r\mathbb{D}_{0})\cap B_{r/2}(z_{0})$. Then, by writing
$z=w+|z-w|\mathrm{e}^{i\theta}$ and $\zeta=w+a\mathrm{e}^{i\theta}$ for
$a\in[0,1]$, we have

$\begin{split}I=\int_{0}^{|z-w|}\Big{\\{}\mathrm{e}^{i\theta}\frac{f_{t}^{\prime}(\zeta)+\zeta
f_{t}^{\prime\prime}(\zeta)}{|\zeta f_{t}^{\prime}(\zeta)|}-\frac{\zeta
f_{t}^{\prime}(\zeta)}{|\zeta
f_{t}^{\prime}(\zeta)|^{3}}\mathrm{Re}\Big{(}\mathrm{e}^{i\theta}(f_{t}^{\prime}(\zeta)+\zeta
f_{t}^{\prime\prime}(\zeta))\overline{\zeta
f_{t}^{\prime}(\zeta)}\Big{)}\Big{\\}}\mathrm{d}a,\end{split}$

so that $|I|\leqslant c_{1}|z-w|$, where

$\begin{split}c_{1}:=\sup_{(t,\zeta)\in
J^{\prime}\times\overline{B_{r/2}(z_{0})}}\Big{\\{}\frac{1}{|\zeta|}+\frac{|f_{t}^{\prime\prime}(\zeta)|}{|f_{t}^{\prime}(\zeta)|}+\frac{|f_{t}^{\prime}(\zeta)|+|\zeta
f_{t}^{\prime\prime}(\zeta)|}{|\zeta
f_{t}^{\prime}(\zeta)|}\Big{\\}}<+\infty.\end{split}$

For the second term, we have

$\begin{split}I\\!I=\int_{s}^{t}\frac{w}{|wf_{u}^{\prime}(w)|}f^{\prime}(\mathrm{d}u,w)-\int_{s}^{t}wf_{u}^{\prime}(w)\mathrm{Re}\Big{(}\frac{f^{\prime}(\mathrm{d}u,w)}{|f_{u}^{\prime}(w)|f_{u}^{\prime}(w)}\Big{)}.\end{split}$

Note that $|w|\leqslant\frac{3}{4}r<\frac{3}{16}<2^{-1}$. We shall note also
that, as measures on $[0,T]$, it holds that

$\begin{split}f^{\prime}(\mathrm{d}u,w)&=(f_{u}^{\prime}(w)+wf_{u}^{\prime\prime}(w))\\{\mathrm{d}x_{0}(u)+\mathrm{d}\xi(\mathbf{x},w)_{u}\\}+f_{u}^{\prime}(w)\sum_{n=1}^{\infty}nw^{n}\mathrm{d}x_{n}(u).\end{split}$

By using this, we have $|I\\!I|\leqslant
c_{2}\int_{s}^{t}\mu(\mathrm{d}u,2^{-1}),$ where

$c_{2}:=\sup_{(u,w)\in
J^{\prime}\times\overline{B_{r/2}(z_{0})}}4\Big{(}1+\frac{|f_{u}^{\prime\prime}(w)|}{|f_{u}^{\prime}(w)|}\Big{)}<+\infty.$

For the third term, we have

$\begin{split}&|z_{0}f_{t_{0}}^{\prime}(z_{0})|I\\!I\\!I\\\
&=\int_{0}^{1}\Big{\\{}wf_{s}^{\prime}\Big{(}w+\frac{\varepsilon^{\prime}wa}{r|f_{t_{0}}^{\prime}(z_{0})|}\Big{)}-zf_{s}^{\prime}\Big{(}z+\frac{\varepsilon^{\prime}za}{r|f_{t_{0}}^{\prime}(z_{0})|}\Big{)}\Big{\\}}\mathrm{d}a\\\
&\hskip
28.45274pt+\int_{0}^{1}\Big{\\{}zf_{s}^{\prime}\Big{(}z+\frac{\varepsilon^{\prime}za}{r|f_{t_{0}}^{\prime}(z_{0})|}\Big{)}-zf_{s}^{\prime}\Big{(}z+\frac{\varepsilon
za}{r|f_{t_{0}}^{\prime}(z_{0})|}\Big{)}\Big{\\}}\mathrm{d}a\\\
&=\int_{0}^{1}\int_{0}^{|z-w|}-\mathrm{e}^{i\theta}\Big{\\{}f_{s}^{\prime}\Big{(}\eta+\frac{\varepsilon^{\prime}a\eta}{r|f_{t_{0}}^{\prime}(z_{0})|}\Big{)}+\Big{(}\eta+\frac{\varepsilon^{\prime}a\eta}{r|f_{t_{0}}^{\prime}(z_{0})|}\Big{)}f_{s}^{\prime\prime}\Big{(}\eta+\frac{\varepsilon^{\prime}a\eta}{r|f_{t_{0}}^{\prime}(z_{0})|}\Big{)}\Big{\\}}\mathrm{d}b\mathrm{d}a\\\
&\hskip
42.67912pt+\int_{0}^{1}\frac{z^{2}a}{r|f_{t_{0}}^{\prime}(z_{0})|}\int_{0}^{\varepsilon^{\prime}-\varepsilon}f_{s}^{\prime\prime}\Big{(}z+\frac{cza}{r|f_{t_{0}}^{\prime}(z_{0})|}\Big{)}\mathrm{d}c\mathrm{d}a\end{split}$

where $z=w+|z-w|\mathrm{e}^{i\theta}$, $\eta=z+b\mathrm{e}^{i(\theta+\pi)}$
and $0\leqslant b\leqslant|z-w|$. Therefore, by noting that
$|r_{0}(\varepsilon^{\prime})\eta|\leqslant(1+\frac{r^{\prime}}{r|f_{t_{0}}^{\prime}(z_{0})|})(|z|+|z-w|)\leqslant(1+\frac{r^{\prime}}{r|f_{t_{0}}^{\prime}(z_{0})|})(r+r)\leqslant\frac{1}{2}$,
we find that $|I\\!I\\!I|\leqslant
c_{3}\big{(}|\varepsilon-\varepsilon^{\prime}|+|z-w|\big{)}$, where

$\begin{split}&c_{3}:=\frac{1}{|z_{0}f_{t_{0}}^{\prime}(z_{0})|}\sup_{(u,\eta)\in
J^{\prime}\times\frac{1}{2}\overline{\mathbb{D}_{0}}}\Big{(}|f_{u}^{\prime}(\eta)|+|\eta
f_{u}^{\prime\prime}(\eta)|\Big{)}.\end{split}$

For the fourth term, we have

$\begin{split}&|z_{0}f_{t_{0}}^{\prime}(z_{0})|I\\!V=-z\int_{0}^{1}\int_{s}^{t}r_{0}(\varepsilon
a)zf_{u}^{\prime}\big{(}r_{0}(\varepsilon
a)z\big{)}\big{\\{}\mathrm{d}x_{0}(u)+\mathrm{d}\xi\big{(}\mathbf{x},r_{0}(\varepsilon
a)z\big{)}_{u}\big{\\}}\end{split}$

Therefore, $|I\\!V|\leqslant
c_{4}\int_{s}^{t}\mu\big{(}\mathrm{d}u,2^{-1}\big{)}$, where

$\begin{split}c_{4}:=\frac{1}{|z_{0}f_{t_{0}}^{\prime}(z_{0})|}\sup_{(u,\eta)\in
J^{\prime}\times\frac{1}{2}\overline{\mathbb{D}_{0}}}\frac{1}{2}|f_{u}^{\prime}(\eta)|.\end{split}$

Combining all the above, we obtained

$\begin{split}&|\varepsilon
G(t,\varepsilon,z)-\varepsilon^{\prime}G(s,\varepsilon^{\prime},w)|\\\
&\leqslant\Big{(}|G(t,\varepsilon,z)|+|\varepsilon|\sum_{i=1}^{4}c_{i}\Big{)}\Big{(}|\varepsilon-\varepsilon^{\prime}|+|z-w|+\int_{s}^{t}\mu(\mathrm{d}u,2^{-1})\Big{)}\end{split}$

We also note that for
$(\varepsilon,z),(\varepsilon^{\prime},w)\in(-\frac{r|f_{t_{0}}^{\prime}(z_{0})|}{2},\frac{r|f_{t_{0}}^{\prime}(z_{0})|}{2})\times\partial(r\mathbb{D}_{0})$,

$\begin{split}|\varepsilon-\varepsilon^{\prime}|+|z-w|\leqslant\Big{(}|f_{t_{0}}^{\prime}(z_{0})|+4\Big{)}|r_{0}(\varepsilon)z-r_{0}(\varepsilon^{\prime})w|,\end{split}$

where
$r_{0}(\varepsilon):=1+\frac{\varepsilon}{r|f_{t_{0}}^{\prime}(z_{0})|}>\frac{1}{2}$.
In fact, since $|z|=|w|=r$, we find that
$|\varepsilon-\varepsilon^{\prime}|=r|f_{t_{0}}^{\prime}(z_{0})||r_{0}(\varepsilon)-r_{0}(\varepsilon^{\prime})|=r|f_{t_{0}}^{\prime}(z_{0})|\times\big{|}|(r_{0}(\varepsilon)/r)z|-|(r_{0}(\varepsilon^{\prime})/r)w|\big{|}\leqslant|f_{t_{0}}^{\prime}(z_{0})|\times|r_{0}(\varepsilon)z-r_{0}(\varepsilon^{\prime})w|$,
and then
$|z-w|=|r_{0}(\varepsilon)^{-1}(r_{0}(\varepsilon)z-r_{0}(\varepsilon^{\prime})w)+(\frac{r_{0}(\varepsilon^{\prime})}{r_{0}(\varepsilon)}-1)w|\leqslant
r_{0}(\varepsilon)^{-1}|r_{0}(\varepsilon)z-r_{0}(\varepsilon^{\prime})w|+\frac{r}{r_{0}(\varepsilon)}|r_{0}(\varepsilon)-r_{0}(\varepsilon^{\prime})|=r_{0}(\varepsilon)^{-1}|r_{0}(\varepsilon)z-r_{0}(\varepsilon^{\prime})w|+(r_{0}(\varepsilon)|f_{t_{0}}^{\prime}(z_{0})|)^{-1}|\varepsilon-\varepsilon^{\prime}|\leqslant
2(r_{0}(\varepsilon))^{-1}|r_{0}(\varepsilon)z-r_{0}(\varepsilon^{\prime})w|$.

Moreover, for any
$(\varepsilon,z),(\varepsilon^{\prime},w)\in(-r^{\prime},r^{\prime})\times\partial(r\mathbb{D}_{0})$,
by noting that $|r_{0}(\varepsilon)z|,|r_{0}(\varepsilon^{\prime})w|<2r$, we
obtain

$\begin{split}|f_{t}(r_{0}(\varepsilon)z)-f_{t}(r_{0}(\varepsilon^{\prime})w)|&=\Big{|}\int_{0}^{|r_{0}(\varepsilon)z-r_{0}(\varepsilon^{\prime})w|}f_{t}^{\prime}(\zeta)\mathrm{d}r\Big{|}\\\
&\geqslant\Big{|}\int_{0}^{|r_{0}(\varepsilon)z-r_{0}(\varepsilon^{\prime})w|}\mathrm{Re}(f_{t}^{\prime}(\zeta))\mathrm{d}r\Big{|}\geqslant
c_{0}|r_{0}(\varepsilon)z-r_{0}(\varepsilon^{\prime})w|.\end{split}$

Furthermore,

$\begin{split}&|f_{t}(r_{0}(\varepsilon)z)-f_{t}(r_{0}(\varepsilon^{\prime})w)|\\\
&\leqslant|f_{t}(r_{0}(\varepsilon)z)-f_{s}(r_{0}(\varepsilon^{\prime})w)|+|f_{s}(r_{0}(\varepsilon^{\prime})w)-f_{t}(r_{0}(\varepsilon^{\prime})w)|\\\
&\leqslant|f_{t}(r_{0}(\varepsilon)z)-f_{s}(r_{0}(\varepsilon^{\prime})w)|+\Big{(}\sup_{(u,\zeta)\in
J^{\prime}\times(2r)\overline{\mathbb{D}_{0}}}|f_{u}^{\prime}(\zeta)|\Big{)}\int_{s}^{t}\mu(\mathrm{d}u,2^{-1}).\end{split}$

Thus we obtain

$\begin{split}&|\varepsilon
G(t,\varepsilon,z)-\varepsilon^{\prime}G(s,\varepsilon^{\prime},w)|\\\
&\leqslant\Big{(}|G(t,\varepsilon,z)|+|\varepsilon|\sum_{i=1}^{4}c_{i}\Big{)}(2c_{0}^{-1})\Big{(}4+\sup_{(u,\zeta)\in
J^{\prime}\times(2r)\overline{\mathbb{D}_{0}}}|f_{u}^{\prime}(\zeta)|\Big{)}^{2}\\\
&\hskip
56.9055pt\times\Big{(}|f_{t}(r_{0}(\varepsilon))-f_{s}(r_{0}(\varepsilon^{\prime}))|+\int_{s}^{t}\mu(\mathrm{d}u,2^{-1})\Big{)},\end{split}$

which implies that

$\begin{split}&h(g(t,\varepsilon,z)-g(s,\varepsilon^{\prime},w))\\\ &\leqslant
c_{5}\Big{(}|G(t,\varepsilon,z)|+|\varepsilon|\sum_{i=1}^{4}c_{i}\Big{)}h\big{(}(t,f_{t}(r_{0}(\varepsilon)z))-(s,f_{s}(r_{0}(\varepsilon^{\prime})w))\big{)},\end{split}$

where $c_{5}:=(2c_{0}^{-1})(4+\sup_{(u,\zeta)\in
J^{\prime}\times(2r)\overline{\mathbb{D}_{0}}}|f_{u}^{\prime}(\zeta)|)^{2}$.

Now, take an interval $J\subset J^{\prime}$ containing $t_{0}$, which is open
in $[0,T]$, $r^{\prime\prime}>0$ so that

$\sup_{(u,\varepsilon,z)\in
J\times(-r^{\prime\prime},r^{\prime\prime})\times\overline{B_{r^{\prime\prime}}(z_{0})}}|G(t,\varepsilon,z)|\leqslant(4c_{5})^{-1}$

and then, we put
$r^{\prime\prime\prime}:=\min\\{r^{\prime\prime},(4c_{5}\sum_{i=1}^{4}c_{i})^{-1}\\}$
and

$U:=J\times(-r^{\prime\prime\prime},r^{\prime\prime\prime})\times(\partial(r\mathbb{D}_{0})\cap
B_{r^{\prime\prime\prime}}(z_{0})).$

Then for each $(t,\varepsilon,z),(s,\varepsilon^{\prime},w)\in U$, it holds
that

$\begin{split}&h(g(t,\varepsilon,z)-g(s,\varepsilon^{\prime},w))\leqslant\frac{1}{2}h\big{(}(t,f_{t}(r_{0}(\varepsilon)z))-(s,f_{s}(r_{0}(\varepsilon^{\prime})w))\big{)}.\end{split}$

On the other hand, since it holds that
$(t,f_{t}(r_{0}(\varepsilon)z))-(s,f_{s}(r_{0}(\varepsilon^{\prime})w))=\\{\varphi(t,\varepsilon,z)-g(t,\varepsilon,z)\\}-\\{\varphi(s,\varepsilon^{\prime},w)-g(s,\varepsilon^{\prime},w)\\}$,
we have

$\begin{split}&h\big{(}(t,f_{t}(r_{0}(\varepsilon)z))-(s,f_{s}(r_{0}(\varepsilon^{\prime})w))\big{)}\\\
&\leqslant
h\big{(}\varphi(t,\varepsilon,z)-\varphi(s,\varepsilon^{\prime},w)\big{)}+h\big{(}g(t,\varepsilon,z)-g(s,\varepsilon^{\prime},w)\big{)}\\\
&\leqslant
h\big{(}\varphi(t,\varepsilon,z)-\varphi(s,\varepsilon^{\prime},w)\big{)}+\frac{1}{2}h\big{(}(t,f_{t}(r_{0}(\varepsilon)z))-(s,f_{s}(r_{0}(\varepsilon^{\prime})w))\big{)},\end{split}$

which proves (A.6).

(i–b) By compactness of $\partial(r\mathbb{D}_{0})$ and (a), there exist an
interval $I$, containing $t_{0}$ which is open in $[0,T]$, $m\in\mathbb{N}$,
$z_{1},z_{2},\cdots,z_{m}\in\partial(r\mathbb{D}_{0})$ and
$r_{1},r_{2},\cdots,r_{m}\in(0,r)$ such that
$\cup_{i=1}^{m}B_{r_{i}}(z_{i})\supset\partial(r\mathbb{D}_{0})$ and, with by
putting anew $r^{\prime}:=\min\\{r_{i}:i=1,2,\cdots,m\\}$ and
$V_{i}^{\prime}:=I\times(-r^{\prime},r^{\prime})\times[B_{r_{i}}(z_{i})\cap\partial(r\mathbb{D}_{0})]$,
$\varphi|_{V_{i}^{\prime}}$ is injective for each $i=1,2,\cdots,m$. Take
$\delta^{\prime}>0$ so that
$T_{\delta^{\prime}}(\partial(r\mathbb{D}_{0}))\subset\cup_{i=1}^{m}B_{r_{i}}(z_{i})$.
Take further $\delta^{\prime\prime}\in(0,\delta^{\prime})$ such that for any
$z,w\in\partial(r\mathbb{D}_{0})$, $|z-w|<\delta^{\prime\prime}$ implies
$z,w\in B_{r_{i}}(z_{i})$ for some $i$. By the way, note that we have (A.4),
which implies that the mapping from a compact set, $\\{(t,\zeta):\text{$t\in
I$, $\zeta\in
f_{t}(\partial(r\mathbb{D}_{0}))$}\\}\ni(t,\zeta)\mapsto(t,f_{t}^{-1}(\zeta))\in
I\times\partial(r\mathbb{D}_{0})$, is continuous, in particular, uniformly
continuous. Therefore, there exists
$\delta\in(0,\min\\{\delta^{\prime\prime}/2,r^{\prime}\\})$ such that for any
$t\in I$ and $z,w\in\partial(r\mathbb{D}_{0})$, $|f_{t}(z)-f_{t}(w)|<2\delta$
implies $|z-w|<\delta^{\prime\prime}$, so that $z,w\in B_{r_{i}}(z_{i})$ for
some $i$. Then $\varphi$ is injective on
$V:=I\times(-\delta,\delta)\times\partial(r\mathbb{D}_{0})$. In fact, if
$(t,\varepsilon,z),(s,\varepsilon^{\prime},w)\in V$ and
$\varphi(t,\varepsilon,z)=\varphi(s,\varepsilon^{\prime},w)$, then $t=s$ and
$f_{t}(z)+\varepsilon\mathbf{n}(t,z)=f_{s}(w)+\varepsilon\mathbf{n}(s,w)=:\zeta$.
Then we have $|f_{t}(z)-f_{s}(w)|\leqslant|f_{t}(z)-\zeta|+|\zeta-
f_{s}(w)|=2|\varepsilon|<2\delta$. Therefore
$(t,\varepsilon,z),(s,\varepsilon^{\prime},w)\in V_{i}$ for some $i$, on which
$\varphi$ is injective. Hence $(t,\varepsilon,z)=(s,\varepsilon^{\prime},w)$.

Now we show that

> $\varphi$ is a homeomorphism between $V$ and the open set
>
> $\mathcal{V}:=\big{\\{}(t,\zeta)\in
> I\times\mathbb{C}:d(\zeta,f_{t}(\partial(r\mathbb{D}_{0})))<\delta\big{\\}}.$
>
> (the openness of $\mathcal{V}$ follows from the continuity $t\mapsto
> f_{t}(\partial(r\mathbb{D}_{0}))$ which is because of Definition 2.1–(3) and
> (A.5))

Suppose that $(t,\zeta)\in\mathcal{V}$. It is clear that $t\in I$. By
compactness of $f_{t}(\partial(r\mathbb{D}))$, we can take
$w\in\partial(r\mathbb{D})$ such that
$\delta>\min_{z\in\partial(r\mathbb{D}_{0})}|f_{t}(z)-\zeta|=|f_{t}(w)-\zeta|=:\varepsilon$.
Note that $w\in\partial(r\mathbb{D})$ is uniquely determined, otherwise it
would contradict the the injectivity of $\varphi|_{V}$. Hence, by writing
$w=r\mathrm{e}^{i\theta_{0}}$, we have
$\left.\frac{\mathrm{d}}{\mathrm{d}\theta}\right|_{\theta=\theta_{0}}f_{t}(r\mathrm{e}^{i\theta})=iwf_{t}(w)$
and
$0=\left.\frac{\mathrm{d}}{\mathrm{d}\theta}\right|_{\theta=\theta_{0}}|f_{t}(r\mathrm{e}^{i\theta})-\zeta|^{2}=-2\mathrm{Re}(iwf_{t}(w)\overline{f_{t}(w)-\zeta})=\langle
iwf_{t}(w),f_{t}(w)-\zeta\rangle_{\mathbb{R}^{2}}$, the segment joining
$\zeta$ and $f_{t}(w)$ is perpendicular to $f_{t}(\partial(r\mathbb{D}_{0}))$
at $f_{t}(w)$. Therefore, it holds that
$\zeta=f_{t}(w)+(\kappa\varepsilon)\mathbf{n}(t,w)$ where $\kappa=-1$ or $+1$.
Then we have

$\begin{split}\delta&>d(\zeta,f_{t}(\partial(r\mathbb{D}_{0})))=|f_{t}(w)-\zeta|=\varepsilon.\end{split}$

Hence $(t,\kappa\varepsilon,w)\in V$ and we established a mapping
$\psi:\mathcal{V}\ni(t,\zeta)\mapsto(t,\kappa\varepsilon,w)\in V$. By the
construction, we have $\varphi|_{V}\circ\psi=\mathrm{id}_{\mathcal{V}}$, which
shows $\varphi|_{V}$ is surjective, in particular, it follows that
$\varphi(V)\subset\mathcal{V}$.

We know so far that $\varphi|_{V}:V\to\mathcal{V}$ is injective and
surjective. It remains to prove the continuity of $(\varphi|_{V})^{-1}$, but
this is clear from (A.6) and the fact that $h$ is continuous.

(i–c) According to the notation in (b), we find that
$(-\delta,\delta)\times(\partial(r\mathbb{D}_{0}))\cong
T_{\delta}(f_{t}(\partial(r\mathbb{D}_{0})))$ for each $t\in I$, and hence the
assertion (1) is proved.

(2) Let $I$, $\delta$ and $r$ be as in (c). By the condition in Definition
2.1–(3), we can take an interval $I^{\prime}\subset I$ containing $t_{0}$,
which is open in $[0,T]$ so that for any $s,t\in I^{\prime}$, $\sup_{s,t\in
I^{\prime}}\|f_{t}|_{\partial(r\mathbb{D}_{0})}-f_{s}|_{\partial(r\mathbb{D}_{0})}\|_{\infty}<\delta$.
Now we fix $t\in I$, arbitrarily. Then for each $s\in I$ and
$z\in\partial(r\mathbb{D}_{0})$, we have

$\begin{split}d(f_{s}(z),f_{t}(\partial(r\mathbb{D}_{0})))\leqslant\|f_{t}|_{\partial(r\mathbb{D}_{0})}-f_{s}|_{\partial(r\mathbb{D}_{0})}\|_{\infty}<\delta,\end{split}$

which implies $f_{s}(\partial(r\mathbb{D}_{0}))\subset
T_{\delta}(f_{t}(\partial(r\mathbb{D}_{0})))$. ∎

Let $\gamma_{1},\gamma_{2}\subset\mathbb{C}$ be two Jordan curves. According
to the Jordan curve theorem, we can decompose $\mathbb{C}\setminus\gamma_{i}$
into a disjoint union

$\mathbb{C}\setminus\gamma_{i}=C_{i}\cup C_{i}^{\prime},\quad\text{for
$i=1,2$,}$

where $C_{i}$ is the bounded connected component of
$\mathbb{C}\setminus\gamma_{i}$ and $C_{i}^{\prime}$ is the unbounded
connected component of $\mathbb{C}\setminus\gamma_{i}$. By this notation, we
have:

######  Lemma A.8.

Suppose that both of $\gamma_{1}$ and $\gamma_{2}$ are surrounding the origin,
and there exists $\delta>0$ such that

  * (i)

$T_{\delta}(\gamma_{i})$ is the tubular $\delta$-neighbourhood of $\gamma_{i}$
for $i=1,2$;

  * (ii)

$\gamma_{1}\subset T_{\delta}(\gamma_{2})$ and $\gamma_{2}\subset
T_{\delta}(\gamma_{1})$.

Then it holds that $d_{H}(C_{1}^{\prime},C_{2}^{\prime})\leqslant
d_{H}(\gamma_{1},\gamma_{2})$.

###### Proof.

Let $d_{H}(\gamma_{1},\gamma_{2})=:\varepsilon_{0}<\delta$ and
$\varepsilon\in(\varepsilon_{0},\delta)$ be arbitrary. For each subset
$A\subset\mathbb{C}$, we write
$A_{\varepsilon}:=\\{z\in\mathbb{C}:d(z,A)<\varepsilon\\}$.

Since $T_{\varepsilon}(\gamma_{i})$ is a tubular neighbourhood of $\gamma_{i}$
for each $i=1,2$, we can decompose

$\partial T_{\varepsilon}(\gamma_{i})=\eta_{i}\cup\eta_{i}^{\prime},\quad
i=1,2$

where $\eta_{i}\subset C_{i}$ and $\eta_{i}^{\prime}\subset C_{i}^{\prime}$
are homeomorphic to $S^{1}$. Since $\gamma_{2}\subset
T_{\varepsilon}(\gamma_{1})$, we have $\eta_{1}\cap\gamma_{2}=\emptyset$, so
that $\eta_{1}\subset C_{2}\cup C_{2}^{\prime}$. Since $C_{2}$ and
$C_{2}^{\prime}$ are connected and $\eta_{1}$ is homeomorphic to $S^{1}$, it
holds that $\eta_{1}\subset C_{2}$ or $\eta_{1}\subset C_{2}^{\prime}$.
Similarly, we have $\eta_{2}\subset C_{1}$ or $\eta_{2}\subset
C_{1}^{\prime}$. Totally, we may have the following cases:

  * (1)

$\eta_{1}\subset C_{2}$ and $\eta_{2}\subset C_{1}$.

  * (2)

$\eta_{1}\subset C_{2}$ and $\eta_{2}\subset C_{1}^{\prime}$.

  * (3)

$\eta_{1}\subset C_{2}^{\prime}$ and $\eta_{2}\subset C_{1}$.

  * (4)

$\eta_{1}\subset C_{2}^{\prime}$ and $\eta_{2}\subset C_{1}^{\prime}$.

Here we shall observe that $\eta_{2}\subset C_{1}^{\prime}$ does not occur. In
fact, suppose that this is the case. Since $\eta_{2}\cap\gamma_{1}=\emptyset$,
we have only two cases: $C_{1}\subset(\text{inner domain of $\eta_{2}$})$ or
$C_{1}\cap(\text{inner domain of $\eta_{2}$})=\emptyset$. The first case can
not occur since then contradicts to $\gamma_{1}\subset
T_{\varepsilon}(\gamma_{2})$. The second case are also impossible because
$\gamma_{1}$ and $\gamma_{2}$ surround the origin. Similarly, $\eta_{1}\subset
C_{2}^{\prime}$ does not occur.

Therefore, (1) is the only case. It is clear that
$C_{1}^{\prime}\cap(C_{2}^{\prime})_{\varepsilon}$ is nonempty and open in
$C_{1}^{\prime}$. We shall show that it is also closed in $C_{1}^{\prime}$.
Suppose that $z_{n}\in C_{1}^{\prime}\cap(C_{2}^{\prime})_{\varepsilon}$,
$n\in\mathbb{N}$ converges to a point $z\in C_{1}^{\prime}$. By (1), we have
$\partial(C_{2}^{\prime})_{\varepsilon}=\eta_{2}\subset C_{1}$, so that it
must be $z\in(C_{2}^{\prime})_{\varepsilon}$. Therefore
$C_{1}^{\prime}\cap(C_{2}^{\prime})_{\varepsilon}$ is closed in
$C_{1}^{\prime}$. Since $C_{1}^{\prime}$ is connected, we have
$C_{1}^{\prime}\cap(C_{2}^{\prime})_{\varepsilon}=C_{1}^{\prime}$, namely,
$C_{1}^{\prime}\subset(C_{2}^{\prime})_{\varepsilon}$.

Similarly, we get $C_{2}^{\prime}\subset(C_{1}^{\prime})_{\varepsilon}$, and
hence the result. ∎

######  Lemma A.9.

Let $\\{f_{t}\\}_{0\leqslant t\leqslant T}$ be a solution to the controlled
Loewner-Kufarev equation (2.1). Then there exists $r^{\prime}\in(0,1)$ such
that for any $r\in(0,r^{\prime})$, the open set $U:=r\mathbb{D}_{0}$ satisfies
the following.

  * (i)

$f_{t}|_{U}$ is univalent for each $t\in[0,T]$;

  * (ii)

$\cap_{0\leqslant t\leqslant T}f_{t}(U)$ is an open neighbourhood of the
origin.

###### Proof.

Let $r^{\prime}\in(0,1)$ be as in Lemma A.7. Take
$r^{\prime\prime}\in(0,r^{\prime})$ small so that
$D:=r^{\prime\prime}\mathbb{D}_{0}$ satisfies Lemma A.4-(i, ii). Now, for each
$r\in(0,r^{\prime\prime})$, put $U:=r\mathbb{D}_{0}$. Then (i) is clearly
satisfied.

(ii) Let $x\in\cap_{t\in[0,T]}f_{t}(U)$ and $t_{0}\in[0,T]$ be arbitrary.
Clearly we have $x\in f_{t_{0}}(U)$. By the Open Mapping Theorem,
$f_{t_{0}}(D)$ is an open set, which implies that $d(x,\mathbb{C}\setminus
f_{t_{0}}(U))>0$.

By Lemma A.7 and Lemma A.8, we find that $t\mapsto\mathbb{C}\setminus
f_{t}(U)$ is continuous with respect to the Hausdorff distance. Since then

$|d(x,\mathbb{C}\setminus f_{t}(U))-d(x,\mathbb{C}\setminus
f_{t_{0}}(U))|\leqslant d_{H}(\mathbb{C}\setminus
f_{t_{0}}(U),\mathbb{C}\setminus f_{t}(U))\to 0$

as $t\to t_{0}$, there exists an interval $I_{0}\ni t_{0}$ which is open in
$[0,T]$, such that

$r_{0}:=\inf_{t\in I_{0}}d(x,\mathbb{C}\setminus f_{t}(U))>0.$

Thus we have $B_{r_{0}}(x)=\\{z\in\mathbb{C}:|z-x|<r_{0}\\}\subset\cap_{t\in
I_{0}}f_{t}(U)$.

Now, we can cover $[0,T]$ by a family of open sets $I$ in $[0,T]$ for which
$\cap_{t\in I}f_{t}(U)$ contains an open ball centred at $x$. Hence by
compactness of $[0,T]$, there exists $r>0$ such that
$B_{r}(x)\subset\cap_{t\in[0,T]}f_{t}(U)$. ∎

Before entering the proof of Proposition 2.9, it would be better to recall
Remark 2.1–(c).

###### Proof of Proposition 2.9.

Take $D$ in Lemma A.4 and $U$ in Lemma A.9 so that $U\subset D$. Then (i),
(ii), (iii) and (iv) are now clear. We shall turn to (v). We have

$\begin{split}0&=f_{t}(f_{t}^{-1}(\zeta))-\zeta\\\
&=\int_{0}^{t}f_{s}^{\prime}(f_{s}^{-1}(\zeta))f^{-1}(\mathrm{d}s,\zeta)+\int_{0}^{t}f(\mathrm{d}s,f_{s}^{-1}(\zeta))\end{split}$

(see e.g., [25, Theorem 1.1]). For the last term, we have by Definition
2.1–(4) that for each $t\in[0,T]$ and $\zeta\in V$,

$\begin{split}\int_{0}^{t}f(\mathrm{d}s,f_{s}^{-1}(\zeta))=f_{s}^{-1}(\zeta)\int_{0}^{t}f_{s}^{\prime}(f_{s}^{-1}(\zeta))\big{\\{}\mathrm{d}x_{0}(s)+\sum_{k=1}^{\infty}(f_{s}^{-1}(\zeta))^{k}\mathrm{d}x_{k}(s)\big{\\}}.\end{split}$

Since $f_{s}^{-1}(\zeta)\in D$ and $f_{s}$ is univalent on $D$, it must be
$f_{s}^{\prime}(f_{s}^{-1}(\zeta))\neq 0$. Now combining the above, we get the
result. ∎

###  A.3. Proof of Proposition 2.12

By applying variation of constants to (2.8), we have

$\begin{split}&b_{-m,-n}(t)\\\
&=-\mathrm{e}^{(n+m)x_{0}(t)}\int_{0}^{t}\mathrm{e}^{-(n+m)x_{0}(t)}\mathrm{d}x_{n+m}(s)\\\
&\hskip
14.22636pt+\mathrm{e}^{(n+m)x_{0}(t)}\int_{0}^{t}\big{\\{}(n-1)b_{-m,-(n-1)}(s)\mathrm{d}x_{1}(s)+\cdots+b_{-m,-1}(s)\mathrm{d}x_{n-1}(s)\big{\\}}\\\
&\hskip
14.22636pt+\mathrm{e}^{(n+m)x_{0}(t)}\int_{0}^{t}\big{\\{}(m-1)b_{-(m-1),-n}(s)\mathrm{d}x_{1}(s)+\cdots+b_{-1,-n}(s)\mathrm{d}x_{m-1}(s)\big{\\}}.\end{split}$

Henceforth, by taking anew $x_{1},x_{2},\cdots$ as non-commutative
indeterminates, and $b_{-m,-n}$’s as polynomials in $x_{i}$’s, we shall
consider the following equation:

(A.9)
$\begin{split}b_{-m,-n}&=-x_{n+m}+\Big{\\{}(n-1)b_{-m,-(n-1)}x_{1}+\cdots+2b_{-m,-2}x_{n-2}+b_{-m,-1}x_{n-1}\Big{\\}}\\\
&\hskip
14.22636pt+\Big{\\{}(m-1)b_{-(m-1),-n}x_{1}+\cdots+2b_{-2,-n}x_{m-2}+b_{-1,-n}x_{m-1}\Big{\\}},\end{split}$

(roughly speaking, the polynomial $b_{-m,-n}$ means
$\mathrm{e}^{-(m+n)x_{0}(t)}b_{-m,-n}(t)$ and ‘applying the indeterminate
$x_{k}$ from the right’ means ‘applying
$\int_{0}^{t}\mathrm{e}^{-kx_{0}(s)}\mathrm{d}x_{k}(s)\times$ to functions of
$s$’) and then we shall make some observations about the equation (A.9) and
introduce some notations: If we apply (A.9) to $b_{-m,-n}$, we get

  * (a)

The terms $(n-k)b_{-m,-(n-k)}x_{k}$ and $(m-k)b_{-(m-k),-n}x_{k}$ for each
$k$. We shall denote these phenomena by

$b_{-m,-n}\stackrel{{\scriptstyle\text{{\tiny$(n-k)x_{k}\times$}}}}{{\to}}b_{-m,-(n-k)}\quad\text{and}\quad\begin{array}[]{cc}b_{-m,-n}\\\
\text{{\tiny$(m-k)x_{k}\times$}}\downarrow\phantom{\text{{\tiny$(m-1)x_{k}\times$}}}\\\
b_{-(m-k),-n}\end{array}$

respectively (Note that the multiplication by the $x_{*}$’s must sit just
right to the next $b_{*,*}$’s).

  * (b)

The term $-x_{n+m}$, to which we can not apply (A.9) anymore. This means,
consider the situation that we apply (A.9) iteratively to $b_{*,*}$’s appeared
in the previous stage. Suppose we have the term $b_{-m,-n}$ at some stage.
Then chasing the term multiplied by $-x_{*}$ which arose from the first term
on the right-hand side in (A.9), makes us get out of the loop of iterations.
We shall denote this phenomenon by

$b_{-m,-n}\stackrel{{\scriptstyle\text{{\tiny$-x_{n+m}\times$}}}}{{\rightrightarrows}}\text{end}\quad\text{or}\quad\begin{array}[]{cc}b_{-m,-n}\\\
\text{{\tiny$-x_{n+m}\times$}}\downdownarrows\phantom{\text{{\tiny$-x_{n+m}\times$}}}\\\
\text{end}\end{array}.$

Note that the multiplication by $x_{*}$’s must be from the left. Hence in
particular, to get the term of the form $x_{k}(...)$ in the polynomial
expression of $b_{-m,-n}$ in the $x_{i}$’s, we have to escape the loop by
passing to the phenomena

$b_{-i,-j}\stackrel{{\scriptstyle\text{{\tiny$-x_{k}\times$}}}}{{\rightrightarrows}}\text{end}\quad\text{or}\quad\begin{array}[]{cc}b_{-i,-j}\\\
\text{{\tiny$-x_{k}\times$}}\downdownarrows\phantom{\text{{\tiny$-x_{k}\times$}}}\\\
\text{end}\end{array},$

where $i,j\in\mathbb{N}$ with $i+j=k$.

  * (c)

If we have $b_{-1,-1}$, applying (A.9) does not produce $b_{*,*}$’s. Namely we
must have

$b_{-1,-1}\stackrel{{\scriptstyle\text{{\tiny$-x_{2}\times$}}}}{{\rightrightarrows}}\text{end}\quad\text{or}\quad\begin{array}[]{cc}b_{-1,-1}\\\
\text{{\tiny$-x_{2}\times$}}\downdownarrows\phantom{\text{{\tiny$-x_{2}\times$}}}\\\
\text{end}\end{array}.$

Again, the multiplication by $x_{2}$ must be from the left. In particular,
$b_{-m,-n}$ does not contain the term $x_{1}(...)$ and hence $b_{-m,-n}$ is a
linear combination of $x_{k}(...)$’s for $k\geqslant 2$, though the factor
$(...)$ may involve $x_{1}$.

Let $k\in\mathbb{N}$ be such that $2\leqslant k\leqslant n+m$. We shall find
the term of the form $x_{k}(...)$ in the polynomial expression of $b_{-m,-n}$
in the $x_{i}$’s. For this, we shall fix $i\in\\{1,\cdots,m\\}$ and
$j\in\\{1,\cdots,n\\}$ such that $i+j=k$. Suppose that $p,q\in\mathbb{N}$ and
$i_{1},\cdots,i_{p},j_{1},\cdots,j_{q}\in\mathbb{N}$ satisfy
$i_{1}+\cdots+i_{p}=m-i$ and $j_{1}+\cdots+j_{q}=n-j$. We then put
$a_{r}:=m-(i_{1}+\cdots+i_{r})$ for $r=1,\cdots,p$ and
$c_{s}:=n-(j_{1}+\cdots+j_{s})$ for $s=1,\cdots,q$. Note that $a_{p}=i$ and
$c_{q}=j$. According to this notation, we can divide the situation into the
following three cases:

(1) If there exist such $p$, $q$, $(i_{1},\cdots,i_{p})$ and
$(j_{1},\cdots,j_{q})$, then we can consider the following diagram:

$\textstyle{b_{-m,-n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a_{1}x_{i_{1}}\times}$$\scriptstyle{c_{1}x_{j_{1}}\times}$$\textstyle{b_{-m,-c_{1}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a_{1}x_{i_{1}}\times}$$\scriptstyle{c_{2}x_{j_{2}}\times}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{c_{q}x_{j_{q}}\times}$$\textstyle{b_{-m,-c_{q}}=b_{-m,-j}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a_{1}x_{i_{1}}\times}$$\textstyle{b_{-a_{1},-n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a_{2}x_{i_{2}}\times}$$\scriptstyle{c_{1}x_{j_{1}}\times}$$\textstyle{b_{-a_{1},-c_{1}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a_{2}x_{i_{2}}\times}$$\scriptstyle{c_{2}x_{j_{2}}\times}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{c_{q}x_{j_{q}}\times}$$\textstyle{b_{-a_{1},-c_{q}}=b_{-a_{1},-j}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a_{2}x_{i_{2}}\times}$$\textstyle{\vdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a_{p}x_{i_{p}}\times}$$\textstyle{\vdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a_{p}x_{i_{p}}\times}$$\textstyle{\vdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a_{p}x_{i_{p}}\times}$$\textstyle{b_{-i,-n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{c_{1}x_{j_{1}}\times}$$\textstyle{b_{-i,-c_{1}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{c_{2}x_{j_{2}}\times}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{c_{q}x_{j_{q}}\times}$$\textstyle{b_{-i,-c_{q}}=b_{-i,-j}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(-1)x_{k}\times}$end

During the loop of iterations of (A.9), we have
$\binom{p+q}{p}=\binom{p+q}{q}$-paths from $b_{-m,-n}$ to the ‘end’ in the
above diagram, each of which produces terms

$-w_{i_{1},\cdots,i_{p};j_{1},\cdots,j_{q}}x_{k}(...)$’s

where

$\begin{split}&w_{i_{1},\cdots,i_{p};j_{1},\cdots,j_{q}}=a_{1}a_{2}\cdots
a_{p}b_{1}b_{2}\cdots b_{q}\\\
&=(m-i_{1})(m-(i_{1}+i_{2}))\cdots(m-(i_{1}+i_{2}+\cdots+i_{p}))\\\ &\hskip
14.22636pt\times(n-j_{1})(n-(j_{1}+j_{2}))\cdots(n-(j_{1}+j_{2}+\cdots+j_{q})),\end{split}$

(note that $w_{i_{1},\cdots,i_{p};j_{1},\cdots,j_{q}}$ depends only on
$i_{1},\cdots,i_{p}$ and $j_{1},\cdots,j_{q}$ but not on the choice of paths
in the diagram) and $(...)$ is a monomial consisting of
$x_{i_{p}},x_{i_{p-1}},\cdots,x_{i_{1}}$ and
$x_{j_{q}},x_{j_{q-1}}\cdots,x_{j_{1}}$, which is interlacing according to a
riffle shuffle permutation (Note that we should distinguish, for example
$x_{i_{1}}x_{j_{1}}$ and $x_{j_{1}}x_{i_{1}}$ even if $i_{1}=j_{1}$ ). Hence
all paths produce in total

$-w_{i_{1},\cdots,i_{p};j_{1},\cdots,j_{q}}x_{k}\big{(}(x_{i_{p}}x_{i_{p-1}}\cdots
x_{i_{1}})\shuffle(x_{j_{q}}x_{j_{q-1}}\cdots x_{j_{1}})\big{)}.$

(2) If there exist such $p$ and $(i_{1},\cdots,i_{p})$ but not for $q$ and
$(j_{1},\cdots,j_{q})$ (then we have $j=n$), then the diagram which we can
have is the following:

$\textstyle{b_{-m,-n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a_{1}x_{i_{1}}\times}$$\textstyle{b_{-a_{1},-n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a_{2}x_{i_{2}}\times}$$\textstyle{\vdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a_{p}x_{i_{p}}\times}$$\textstyle{b_{-i,-n}=b_{-i,-j}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(-1)x_{k}\times}$end

Hence we have a single path from $b_{-m,-n}$ to the ‘end’ in the above
diagram. This path produces the term

$-w_{i_{1},\cdots,i_{p}}x_{k}(x_{i_{p}}\cdots x_{i_{2}}x_{i_{1}})$

where $w_{i_{1},\cdots,i_{p};j_{1},\cdots,j_{q}}=a_{1}a_{2}\cdots
a_{p}=(m-i_{1})(m-(i_{1}+i_{2}))\cdots(m-(i_{1}+i_{2}+\cdots+i_{p}))$.

(3) If there exist such $q$ and $(j_{1},\cdots,j_{q})$ but not for $p$ and
$(i_{1},\cdots,i_{p})$ (then we have $i=m$), then the diagram which we can
have is the following:

$\textstyle{b_{-m,-n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{c_{1}x_{j_{1}}\times}$$\textstyle{b_{-m,-c_{1}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{c_{2}x_{j_{2}}\times}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{c_{q}x_{j_{q}}\times}$$\textstyle{b_{-m,-c_{q}}=b_{-i,-j}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(-1)x_{k}\times}$end

Hence we have a single path from $b_{-m,-n}$ to the ‘end’ in the above
diagram. This path produces the term

$-w_{j_{1},\cdots,j_{q}}x_{k}(x_{j_{q}}\cdots x_{j_{2}}x_{j_{1}})$

where

$\begin{split}&w_{j_{1},\cdots,j_{q}}=c_{1}c_{2}\cdots c_{q}\\\
&=(n-j_{1})(n-(j_{1}+j_{2}))\cdots(n-(j_{1}+j_{2}+\cdots+j_{q})).\end{split}$

Now by reinterpreting it in the language of paths $x_{k}(t)$’s, we conclude
the result.

###  A.4. Proof of Theorem 3.9

Since $\\{u^{n}\\}_{n\geqslant 1}$ forms a basis of $H_{+}$, it is enough to
show

(A.10) $\begin{split}&n\big{[}\int\underset{\begin{subarray}{c}w=0,\\\
u=0\end{subarray}}{\operatorname{Res\
}}\Big{(}\frac{u^{n-1}}{w-z}\sum_{r,s=1}^{\infty}\mathrm{e}^{(r+s)x_{0}(t)}x_{r+s}(w^{-r}._{w}S(\xi(\mathbf{x},\mathrm{e}^{x_{0}(t)})))\shuffle(u^{-s}._{u}S(\xi(\mathbf{x},\mathrm{e}^{x_{0}(t)})))\Big{)}\big{]}_{t}\\\
&=n\sum_{m=1}^{\infty}b_{-n,-m}(t)z^{-m},\end{split}$

where $b_{-n,-m}(t)$ are the Grunsky coefficients associated with $f_{t}$.

According to the decomposition

$\begin{split}S(\xi(\mathbf{x},\mathrm{e}^{x_{0}(t)}))=1+\sum_{m^{\prime}=1}^{\infty}\mathrm{e}^{m^{\prime}x_{0}(t)}\sum_{p=1}^{m^{\prime}}\sum_{\begin{subarray}{c}i_{1},\cdots,i_{p}\in\mathbb{N}:\\\
i_{1}+\cdots+i_{p}=m^{\prime}\end{subarray}}x_{i_{1}}x_{i_{2}}\cdots
x_{i_{p}},\end{split}$

we have

$\begin{split}&\big{(}w^{-r}._{w}S(\xi(\mathbf{x},\mathrm{e}^{x_{0}(t)}))\big{)}\shuffle\big{(}u^{-s}._{u}S(\xi(\mathbf{x},\mathrm{e}^{x_{0}(t)}))\big{)}\\\
&=(w^{-r}._{w}1)\shuffle(u^{-s}._{u}1)+F_{r,s}(w,u)+G_{r,s}(w,u)+H_{r,s}(w,u),\end{split}$

where

$\begin{split}F_{r,s}(w,u)&:=\big{[}w^{-r}._{w}\big{(}S(\xi(\mathbf{x},\mathrm{e}^{x_{0}(t)}))-1\big{)}\big{]}\shuffle\big{[}u^{-s}._{u}\big{(}S(\xi(\mathbf{x},\mathrm{e}^{x_{0}(t)}))-1\big{)}\big{]}\\\
&=\sum_{m^{\prime}=1}^{\infty}\sum_{n^{\prime}=1}^{\infty}\sum_{p=1}^{m^{\prime}}\sum_{q=1}^{n^{\prime}}\sum_{\begin{subarray}{c}i_{1},\cdots,i_{p}\in\mathbb{N}:\\\
i_{1}+\cdots+i_{p}=m^{\prime}\end{subarray}}\sum_{\begin{subarray}{c}j_{1},\cdots,j_{q}\in\mathbb{N}:\\\
j_{1}+\cdots+j_{q}=n^{\prime}\end{subarray}}\mathrm{e}^{(m^{\prime}+n^{\prime})x_{0}(t)}\\\
&\hskip 85.35826pt\times(w^{-r}._{w}x_{i_{1}}x_{i_{2}}\cdots
x_{i_{p}})\shuffle(u^{-s}._{u}x_{j_{1}}x_{j_{2}}\cdots x_{j_{q}}),\\\
G_{r,s}(w,u)&:=(w^{-r}._{w}1)\shuffle\big{[}u^{-s}._{u}\big{(}S(\xi(\mathbf{x},\mathrm{e}^{x_{0}(t)}))-1\big{)}\big{]}\\\
&=\sum_{n^{\prime}=1}^{\infty}\mathrm{e}^{n^{\prime}x_{0}(t)}\sum_{q=1}^{n^{\prime}}\sum_{\begin{subarray}{c}j_{1},\cdots,j_{q}\in\mathbb{N}:\\\
j_{1}+\cdots+j_{q}=n^{\prime}\end{subarray}}(w^{-r}._{w}1)\shuffle(u^{-s}._{u}x_{j_{1}}x_{j_{2}}\cdots
x_{j_{q}}),\\\
H_{r,s}(w,u)&:=\big{[}w^{-r}._{w}\big{(}S(\xi(\mathbf{x},\mathrm{e}^{x_{0}(t)}))-1\big{)}\big{]}\shuffle(u^{-s}._{u}1)\\\
&=\sum_{m^{\prime}=1}^{\infty}\mathrm{e}^{m^{\prime}x_{0}(t)}\sum_{p=1}^{m^{\prime}}\sum_{\begin{subarray}{c}i_{1},\cdots,i_{p}\in\mathbb{N}:\\\
i_{1}+\cdots+i_{p}=m^{\prime}\end{subarray}}(w^{-r}._{w}x_{i_{1}}x_{i_{2}}\cdots
x_{i_{p}})\shuffle(u^{-s}._{u}1).\end{split}$

Since $w^{-r}._{w}1=w^{-r}$, we get
$(w^{-r}._{w}1)\shuffle(u^{-s}._{u}1)=w^{-r}u^{-s}$. Then, by using

$\frac{1}{w-z}=-\sum_{m=1}^{\infty}z^{-m}w^{m-1}\quad\text{for $|z|>|w|$,}$

we have

$\begin{split}&\underset{\begin{subarray}{c}w=0;\\\
u=0\end{subarray}}{\operatorname{Res\
}}\Big{(}\frac{u^{n-1}}{w-z}\sum_{r,s=1}^{\infty}\mathrm{e}^{(r+s)x_{0}(t)}x_{r+s}\big{(}(w^{-r}._{w}1)\shuffle(u^{-s}._{u}1)\big{)}\Big{)}\\\
&=-\underset{\begin{subarray}{c}w=0;\\\ u=0\end{subarray}}{\operatorname{Res\
}}\Big{(}u^{n-1}\sum_{m=1}^{\infty}z^{-m}w^{m-1}\sum_{r,s=1}^{\infty}\mathrm{e}^{(r+s)x_{0}(t)}x_{r+s}w^{-r}u^{-s}\Big{)}=-\sum_{m=1}^{\infty}\mathrm{e}^{(m+n)x_{0}(t)}x_{m+n}.\end{split}$

Let us set

$\begin{split}w(r)_{i_{1},\cdots,i_{p};\varnothing}&:=r(i_{p}+r)(i_{p}+i_{p-1}+r)\cdots(i_{p}+i_{p-1}+\cdots+i_{2}+r)\\\
&=\big{(}m-(i_{1}+\cdots+i_{p})\big{)}\cdots\big{(}m-(i_{1}+i_{2})\big{)}(m-i_{1})\\\
&\hskip 142.26378pt\text{where $m=i_{1}+\cdots+i_{p}+r$},\\\
w(s)_{\varnothing;j_{1},\cdots,j_{q}}&:=s(j_{q}+s)(j_{q}+j_{q-1}+s)\cdots(j_{q}+j_{q-1}+\cdots+j_{2}+s)\\\
&=\big{(}n-(j_{1}+\cdots+j_{q})\big{)}\cdots\big{(}n-(j_{1}+j_{2})\big{)}(n-j_{1})\\\
&\hskip 142.26378pt\text{where $n=j_{1}+\cdots+j_{q}+s$,}\end{split}$

and
$w(r,s)_{i_{1},\cdots,i_{p};j_{1},\cdots,j_{q}}:=w(r)_{i_{1},\cdots,i_{p};\varnothing}w(s)_{\varnothing;j_{1},\cdots,j_{q}}$.

For $F_{r+s}(w,u)$, we first observe that

$\begin{split}&w^{-r}._{w}x_{i_{p}}\cdots x_{i_{2}}x_{i_{1}}\\\
&=x_{i_{p}}\cdots x_{i_{2}}x_{i_{1}}L_{-i_{1}}L_{-i_{2}}\cdots
L_{-i_{p}}w^{-r}\\\ &=w(r)_{i_{1},\cdots,i_{p};\varnothing}x_{i_{p}}\cdots
x_{i_{2}}x_{i_{1}}w^{-(i_{1}+i_{2}+\cdots+i_{p}+r)},\end{split}$

and similarly

$u^{-s}._{u}x_{j_{q}}\cdots
x_{j_{2}}x_{j_{1}}=w(s)_{\varnothing;j_{1},\cdots,j_{q}}x_{j_{q}}\cdots
x_{j_{2}}x_{j_{1}}u^{-(j_{1}+j_{2}+\cdots+j_{p}+s)}.$

Therefore we have

$\begin{split}&\underset{\begin{subarray}{c}w=0;\\\
u=0\end{subarray}}{\operatorname{Res\
}}\Big{(}\frac{u^{n-1}}{w-z}x_{r+s}\big{(}(w^{-r}._{w}x_{i_{1}}x_{i_{2}}\cdots
x_{i_{p}})\shuffle(u^{-s}._{u}x_{i_{1}}x_{i_{2}}\cdots
x_{i_{p}})\big{)}\Big{)}\\\ &=-1_{\\{1\leqslant
n-s=j_{1}+\cdots+j_{q}\\}}\sum_{m=1}^{\infty}z^{-m}1_{\\{1\leqslant
m-r=i_{1}+\cdots+i_{p}\\}}\\\ &\hskip 56.9055pt\times
w(r,s)_{i_{1},\cdots,i_{p};j_{1},\cdots,j_{q}}x_{r+s}[(x_{i_{p}}\cdots
x_{i_{2}}x_{i_{1}})\shuffle(x_{j_{q}}\cdots x_{j_{2}}x_{j_{1}})],\end{split}$

so that

$\begin{split}&\underset{\begin{subarray}{c}w=0;\\\
u=0\end{subarray}}{\operatorname{Res\
}}\Big{(}\frac{u^{n-1}}{w-z}x_{r+s}F_{r,s}(w,u)\Big{)}\\\
&=-\sum_{m=1}^{\infty}z^{-m}\sum_{m^{\prime}=1}^{\infty}\sum_{n^{\prime}=1}^{\infty}\mathrm{e}^{(m^{\prime}+n^{\prime})x_{0}(t)}\sum_{p=1}^{m^{\prime}}\sum_{q=1}^{n^{\prime}}\sum_{\begin{subarray}{c}i_{1},\cdots,i_{p}\in\mathbb{N}:\\\
i_{1}+\cdots+i_{p}=m^{\prime}\end{subarray}}1_{\\{1\leqslant
m-r=i_{1}+\cdots+i_{p}\\}}\sum_{\begin{subarray}{c}j_{1},\cdots,j_{q}\in\mathbb{N}:\\\
j_{1}+\cdots+j_{q}=n^{\prime}\end{subarray}}1_{\\{1\leqslant
n-s=j_{1}+\cdots+j_{q}\\}}\\\ &\hskip 28.45274pt\times
w(r,s)_{i_{1},\cdots,i_{p};j_{1},\cdots,j_{q}}x_{r+s}[(x_{i_{p}}\cdots
x_{i_{2}}x_{i_{1}})\shuffle(x_{j_{q}}\cdots x_{j_{2}}x_{j_{1}})]\\\
&=-1_{\\{1\leqslant
n-s\\}}\sum_{m=1}^{\infty}z^{-m}\mathrm{e}^{((m-r)+(n-s))x_{0}(t)}1_{\\{1\leqslant
m-r\\}}\sum_{p=1}^{m-r}\sum_{q=1}^{n-s}\sum_{\begin{subarray}{c}i_{1},\cdots,i_{p}\in\mathbb{N}:\\\
i_{1}+\cdots+i_{p}=m-r\end{subarray}}\sum_{\begin{subarray}{c}j_{1},\cdots,j_{q}\in\mathbb{N}:\\\
j_{1}+\cdots+j_{q}=n-s\end{subarray}}w(r,s)_{i_{1},\cdots,i_{p};j_{1},\cdots,j_{q}}\\\
&\hskip 56.9055pt\times x_{r+s}[(x_{i_{p}}\cdots
x_{i_{2}}x_{i_{1}})\shuffle(x_{j_{q}}\cdots x_{j_{2}}x_{j_{1}})].\end{split}$

Hence we reached

$\begin{split}&\underset{\begin{subarray}{c}w=0;\\\
u=0\end{subarray}}{\operatorname{Res\
}}\Big{(}\frac{u^{n-1}}{w-z}\sum_{r,s=1}^{\infty}\mathrm{e}^{(r+s)x_{0}(t)}x_{r+s}F_{r,s}(w,u)\Big{)}\\\
&=-\sum_{m=1}^{\infty}z^{-m}\mathrm{e}^{(m+n)x_{0}(t)}\sum_{k=2}^{m+n-2}\sum_{\begin{subarray}{c}1\leqslant
r<m;\\\ 1\leqslant s<n:\\\
r+s=k\end{subarray}}\sum_{p=1}^{m-r}\sum_{q=1}^{n-s}\sum_{\begin{subarray}{c}i_{1},\cdots,i_{p}\in\mathbb{N}:\\\
i_{1}+\cdots+i_{p}=m-r\end{subarray}}\sum_{\begin{subarray}{c}j_{1},\cdots,j_{q}\in\mathbb{N}:\\\
j_{1}+\cdots+j_{q}=n-s\end{subarray}}w(r,s)_{i_{1},\cdots,i_{p};j_{1},\cdots,j_{q}}\\\
&\hskip 56.9055pt\times x_{k}[(x_{i_{p}}\cdots
x_{i_{2}}x_{i_{1}})\shuffle(x_{j_{q}}\cdots x_{j_{2}}x_{j_{1}})].\end{split}$

Similarly, we find that

$\begin{split}&\underset{\begin{subarray}{c}w=0;\\\
u=0\end{subarray}}{\operatorname{Res\
}}\Big{(}\frac{u^{n-1}}{w-z}\sum_{r,s=1}^{\infty}\mathrm{e}^{(r+s)x_{0}(t)}x_{r+s}G_{r,s}(w,u)\Big{)}\\\
&=-\sum_{m=1}^{\infty}z^{-m}\sum_{r,s=1}^{\infty}\mathrm{e}^{(r+s)x_{0}(t)}\sum_{n^{\prime}=1}^{\infty}\mathrm{e}^{n^{\prime}x_{0}(t)}\sum_{q=1}^{n^{\prime}}\sum_{\begin{subarray}{c}j_{1},\cdots,j_{q}\in\mathbb{N}:\\\
j_{1}+\cdots+j_{q}=n^{\prime}\end{subarray}}1_{\\{1\leqslant
n-s=j_{1}+\cdots+j_{q}\\}}1_{\\{m=r\\}}w(s)_{\varnothing;j_{1},\cdots,j_{q}}x_{r+s}(x_{j_{q}}\cdots
x_{j_{1}})\\\
&=-\sum_{m=1}^{\infty}z^{-m}\mathrm{e}^{(m+n)x_{0}(t)}\sum_{k=m+1}^{m+n-1}\sum_{q=1}^{n+m-k}\sum_{\begin{subarray}{c}j_{1},\cdots,j_{q}\in\mathbb{N}:\\\
j_{1}+\cdots+j_{q}=n+m-k\end{subarray}}w(k-m)_{\varnothing;j_{1},\cdots,j_{q}}x_{k}(x_{j_{q}}\cdots
x_{j_{1}}),\end{split}$

and

$\begin{split}&\underset{\begin{subarray}{c}w=0;\\\
u=0\end{subarray}}{\operatorname{Res\
}}\Big{(}\frac{u^{n-1}}{w-z}\sum_{r,s=1}^{\infty}\mathrm{e}^{(r+s)x_{0}(t)}x_{r+s}H_{r,s}(w,u)\Big{)}\\\
&=-\sum_{m=1}^{\infty}z^{-m}\mathrm{e}^{(m+n)x_{0}(t)}\sum_{k=n+1}^{m+n-1}\sum_{p=1}^{n+m-k}\sum_{\begin{subarray}{c}i_{1},\cdots,i_{p}\in\mathbb{N}:\\\
i_{1}+\cdots+i_{p}=m+n-k\end{subarray}}w(k-n)_{i_{1},\cdots,i_{p};\varnothing}x_{k}(x_{i_{p}}\cdots
x_{i_{1}}).\end{split}$

Now, in view of Theorem 2.13, we obtain (A.10), and hence the result.

### Acknowledgment

Both authors thank Theo Sturm for the hospitality he granted to T.A. at the
University of Bonn. T.A. thanks Roland Speicher for the hospitality offered
him in Saarbrücken. R.F. thanks Fukuoka University and the MPI in Bonn for its
hospitality, Roland Speicher for his support from which this paper
substantially profited.

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