# The skew Brownian permuton: a new universality class for random constrained
permutations

Jacopo Borga<EMAIL_ADDRESS>Department of Mathematics, Stanford
University

###### Abstract

We construct a new family of random permutons, called _skew Brownian permuton_
, which describes the limits of several models of random constrained
permutations. This family is parametrized by two real parameters.

For a specific choice of the parameters, the skew Brownian permuton coincides
with the Baxter permuton, i.e. the permuton limit of Baxter permutations. We
prove that for another specific choice of the parameters, the skew Brownian
permuton coincides with the biased Brownian separable permuton, a one-
parameter family of permutons previously studied in the literature as the
limit of uniform permutations in substitution-closed classes. This brings two
different limiting objects under the same roof, identifying a new larger
universality class.

The skew Brownian permuton is constructed in terms of flows of solutions of
certain stochastic differential equations (SDEs) driven by two-dimensional
correlated Brownian excursions in the non-negative quadrant. We call these
SDEs _skew perturbed Tanaka equations_ because they are a mixture of the
perturbed Tanaka equations and the equations encoding skew Brownian motions.
We prove existence and uniqueness of (strong) solutions for these new SDEs.

In addition, we show that some natural permutons arising from Liouville
quantum gravity spheres decorated with two Schramm-Loewner evolution curves
are skew Brownian permutons and such permutons cover almost the whole range of
possible parameters. Some connections between constrained permutations and
decorated planar maps have been investigated in the literature at the discrete
level; this paper establishes this connection directly at the continuum level.
Proving the latter result, we also give an SDE interpretation of some
quantities related to SLE-decorated Liouville quantum gravity spheres.

###### Contents

  1. 1 Introduction
    1. 1.1 Limits of random permutations
    2. 1.2 The biased Brownian separable permuton
    3. 1.3 The Baxter permuton
    4. 1.4 A new family of universal permutons: the skew Brownian permuton
      1. 1.4.1 Definitions and construction
      2. 1.4.2 The skew perturbed Tanaka equations and correctness of the definition
      3. 1.4.3 Relations with the biased Brownian separable permuton and the Baxter permuton
      4. 1.4.4 A model of random permutations converging to the skew Brownian permuton with non-degenerate parameters
    5. 1.5 The skew Brownian permuton in Liouville quantum gravity theory
    6. 1.6 Open problems
  2. 2 The skew perturbed Tanaka equations
    1. 2.1 The skew pure perturbed Tanaka equation
      1. 2.1.1 The Brownian motion case
      2. 2.1.2 The Brownian excursion case
    2. 2.2 The skew Tanaka equation
  3. 3 The skew Brownian permuton
    1. 3.1 The skew Brownian permuton is well-defined
    2. 3.2 The biased Brownian separable permuton is a particular case of the skew Brownian permuton
  4. 4 The skew Brownian permuton and SLE-decorated Liouville quantum gravity spheres
  5. A Absolute continuity between correlated Brownian excursions in cones and correlated Brownian motions
  6. B Simulations of the skew Brownian permuton

##  1 Introduction

###  1.1 Limits of random permutations

The study of limits of random permutations is a classical topic in probability
theory. The typical question is to determine the behavior of a large random
permutation when its size tends to infinity. For many years, the main approach
to answering this question has been to study the convergence of relevant
statistics, such as the number of inversions, the length of the longest
increasing subsequence, the number of cycles, and many others. In the last
decade, a more geometric approach has been investigated, mainly to study
limits of _non-uniform_ models of random permutations. In this case, the goal
is to directly determine the limit of the permutation itself from a global
perspective. This theory goes under the name of _permuton limits_. For a
complete introduction to the theory of permuton limits, we refer the reader to
[Bor21, Section 2.1]. Here we only recall some basic definitions.

A Borel probability measure $\mu$ on the unit square $[0,1]^{2}$ is called a
_permuton_ if its marginals are uniform, i.e.
$\mu([0,1]\times[x,y])=\mu([x,y]\times[0,1])=x-y$ for all $0\leq x\leq y\leq
1$.

Presutti and Stromquist [PS10] first used permutons, calling them _normalized
measures_ , to investigate the packing density of some specific patterns. A
more theoretical approach to the study of deterministic permutons (again
without using this terminology) was developed by Hoppen, Kohayakawa, Moreira,
Rath, and Sampaio [HKM+13]. The word _permuton_ appeared for the first time in
a work of Glebov, Grzesik, Klimošová, and Král [GGKK15]. They adopt this
terminology by analogy with _graphon_ in the theory of random graphs [Lov12].
Various results appearing in [HKM+13] were generalized to the case of random
permutons by Bassino, Bouvel, Féray, Gerin, Maazoun, and Pierrot [BBF+20].

To a permutation $\sigma$ of size $n\geq 1$, it is possible to associate a
natural permuton $\mu_{\sigma}$, given by the sum of Lebesgue area measures

$\mu_{\sigma}(A)=n\sum_{i=1}^{n}\operatorname{Leb}\big{(}[(i-1)/n,i/n]\times[(\sigma(i)-1)/n,\sigma(i)/n]\cap
A\big{)},$ (1)

where $A$ is a Borel measurable set of $[0,1]^{2}$. Note that $\mu_{\sigma}$
corresponds to the normalized diagram of $\sigma$, where each dot has been
replaced with a square of dimension $1/n\times 1/n$ and total mass $1/n$. See
the example given in Fig. 1.

Figure 1: Left: The diagram of the permutation $\sigma=3142$. Right: The
corresponding permuton. Every small square of the diagram containing a dot is
endowed with Lebesgue measure of total mass $1/4$. With this procedure we
obtain a probability measure on the unit square with uniform marginals, that
is, the permuton $\mu_{\sigma}$.

Let $\mathcal{M}$ be the set of permutons endowed with the weak topology. A
sequence of permutons $(\mu_{n})_{n\in\mathbb{Z}_{>0}}$ converges weakly to
$\mu$, and we write $\mu_{n}\to\mu$, if

$\int_{[0,1]^{2}}fd\mu_{n}\rightarrow\int_{[0,1]^{2}}fd\mu,$ (2)

for every continuous function $f:[0,1]^{2}\to\mathbb{R}$. When a sequence of
permutons $(\mu_{n})_{n\in\mathbb{Z}_{>0}}$ converges weakly to $\mu$ we will
often say that $(\mu_{n})_{n\in\mathbb{Z}_{>0}}$ converges to $\mu$ in the
permuton sense.

In the past years, permuton limits of various models of non-uniform random
permutations have been investigated in the literature. These models can be
divided into two main different classes:

  * •

The models that exhibit a _random fractal limiting permuton_ : here the
classical examples are pattern-avoiding permutations (see below for a more
detailed discussion).

  * •

The models that exhibit a _deterministic limiting permuton_ : here some
examples are Erdös–Szekeres permutations [Rom06], Mallows permutations
[Sta09], random sorting networks [Dau21], square111To be precise, we remark
that the permuton limits of square and almost square permutations with a
finite number of internal points are random, but their randomness can be
simply expressed in terms of a beta distributed random variable. On the
contrary, almost square permutations with an infinite number of internal
points exhibit the same deterministic permuton limit. For these reasons we
prefer to classify square and almost square permutations as permutations that
exhibit a deterministic permuton limit. and almost square permutations [BS20,
BDS21], and permutations sorted with the _runsort_ algorithm [ADK22].

The present paper focuses on models that exhibit a limiting random fractal
permuton. Our main goal is to introduce a new family of limiting permutons
that unifies the various instances of random fractal limiting permutons that
appear in the literature and some new ones.

We now quickly review the literature on models of _pattern-avoiding
permutations_ that exhibit a random fractal permuton in the limit. If the
reader is not familiar with the terminology related to pattern-avoiding
permutations, he/she can find a quick introduction in [Bor21, Section 1.6.1].

Figure 2: Left: An instance of the Brownian separable permuton. Middle: An
instance of the biased Brownian separable permuton. Right: An instance of the
Baxter permuton.More explanations on these limiting permutons will be given in
the next sections of the introduction.

  * •

In [BBF+18], Bassino, Bouvel, Féray, Gerin and Pierrot prove that a sequence
of uniform random separable permutations222Separable permutations are the
permutations avoiding the patterns $2413$ and $3142$. They are one of the most
studied families of pattern-avoiding permutations, see for instance [BBL98,
AN81, SS91, AHP15]. converges to the _Brownian separable permuton_ (see the
picture on the left-hand side of Fig. 2). This is the first work where a
random fractal limiting permuton appears in the literature.

  * •

In a second work [BBF+20] (see also [BBFS20]) the authors prove that the
Brownian separable permuton is _universal_ : they consider uniform random
permutations in proper substitution-closed classes333For an introduction to
proper substitution-closed classes we refer the reader to [Bor21, Section
3.2]. and show that their limits in the permuton sense are a one-parameter
deformation of the Brownian separable permuton, called _biased Brownian
separable permuton_ (see the picture in the middle of Fig. 2).

  * •

These new universal permutons are later investigated by Maazoun [Maa20].

  * •

In [BBF+22] the authors investigate permuton limits for permutations in
classes having a finite combinatorial specification for the substitution
decomposition. The limit depends on the structure of the specification
restricted to families with the largest growth rate. When the specification is
strongly connected, two cases occur. If the associated system of equations is
linear, the limiting permuton is a deterministic $X$-shape. Otherwise, the
limiting permuton is the biased Brownian separable permuton.

  * •

Finally, in [BM22], Maazoun and the author of the present paper show that the
permuton limit of Baxter permutations444Baxter permutations were introduced by
Glen Baxter in 1964 [Bax64] to study fixed points of commuting functions.
Baxter permutations are permutations avoiding the vincular patterns
$2\underbracket{41}3$ and $3\underbracket{14}2$, i.e. permutations $\sigma$
such that there are no indices $i<j<k$ such that
$\sigma(j+1)<\sigma(i)<\sigma(k)<\sigma(j)$ or
$\sigma(j)<\sigma(k)<\sigma(i)<\sigma(j+1)$. These permutations are deeply
studied in combinatorics, see for instance [Boy67, CGHK78, Mal79, FFNO11,
BGRR18] and references therein. is a new random fractal limiting permuton,
called _the Baxter permuton_ (see the picture in the right-hand side of Fig.
2), that is not included in the biased Brownian separable permuton
universality class.

The next two sections review the constructions of the biased Brownian
separable permuton and the Baxter permuton. These constructions are
fundamental to understand later in Section 1.4 our definition of the new
family of limiting permutons mentioned before.

###  1.2 The biased Brownian separable permuton

We introduce the biased Brownian separable permuton following [Maa20, Sections
1.3-4]. Consider a one-dimensional Brownian excursion555Here and throughout
the paper we denote random quantities using bold characters.
$(\bm{e}(t))_{t\in[0,1]}$ on $[0,1]$ and a parameter $p\in[0,1]$. Conditional
on $\bm{e}$, consider an i.i.d. sequence
$(\bm{s}(\ell))_{\ell}\in\\{+1,-1\\}^{\mathbb{Z}_{>0}}$ indexed by the local
minima666For the technicalities involved in indexing an i.i.d. sequence by
this random countable set, see [Maa20, Section 2.2]. of $\bm{e}$ and with
distribution $\mathbb{P}(\bm{s}(\ell)=+1)=p=1-\mathbb{P}(\bm{s}(\ell)=-1)$. We
denote by $(\widetilde{\bm{e}},p)$ the pair $(\bm{e},(\bm{s}(\ell))_{\ell})$.
We define the following random relation
$\vartriangleleft_{\widetilde{\bm{e}},p}$: conditional on $\bm{e}$, if
$x,y\in[0,1]$ and $x<y$ and $\min_{[x,y]}\bm{e}$ is reached at a unique point
which is a strict local minimum $\ell\in[x,y]$ then

$\begin{cases}x\vartriangleleft_{\widetilde{\bm{e}},p}y,&\quad\text{if}\quad\bm{s}(\ell)=+1,\\\
y\vartriangleleft_{\widetilde{\bm{e}},p}x,&\quad\text{if}\quad\bm{s}(\ell)=-1.\\\
\end{cases}$ (3)

Maazoun showed that there exists a random set $\bm{A}\subset[0,1]^{2}$ of a.s.
zero Lebesgue measure, i.e. $\mathbb{P}(\operatorname{Leb}(\bm{A})=0)=1$, such
that for every $x,y\in[0,1]^{2}\setminus\bm{A}$ with $x<y$ then
$\min_{[x,y]}\bm{e}$ is reached at a unique point which is a strict local
minimum. In particular, the restriction of
$\vartriangleleft_{\widetilde{\bm{e}},p}$ to $[0,1]^{2}\setminus\bm{A}$ is a
total order. Setting

$\psi_{\widetilde{\bm{e}},p}(t)\coloneqq\operatorname{Leb}\left(\big{\\{}x\in[0,1]|x\vartriangleleft_{\widetilde{\bm{e}},p}t\big{\\}}\right),\quad
t\in[0,1],$ (4)

then the biased Brownian separable permuton is defined (see [Maa20, Theorem
1.3]) as the push-forward of the Lebesgue measure on $[0,1]$ via the mapping
$(\operatorname{Id},\psi_{\widetilde{\bm{e}},p})$, that is

$\bm{\mu}^{S}_{p}(\cdot)\coloneqq(\operatorname{Id},\psi_{\widetilde{\bm{e}},p})_{*}\operatorname{Leb}(\cdot)=\operatorname{Leb}\left(\\{t\in[0,1]|(t,\psi_{\widetilde{\bm{e}},p}(t))\in\cdot\,\\}\right).$

Heuristically, $\psi_{\widetilde{\bm{e}},p}$ is the “ _continuum permutation_
” of the elements in the interval $[0,1]$ induced by the order
$\vartriangleleft_{\widetilde{\bm{e}},p}$ and $\bm{\mu}^{S}_{p}$ is the
diagram of $\psi_{\widetilde{\bm{e}},p}$.

###  1.3 The Baxter permuton

We now introduce the Baxter permuton following [BM22]. To do that, we first
define the _continuous coalescent-walk process_ driven by a two-dimensional
Brownian excursion.

We start by recalling that a _two-dimensional Brownian motion_ _of
correlation_ $\rho\in[-1,1]$, denoted
$({\bm{\mathcal{W}}}_{\rho}(t))_{t\in\mathbb{R}_{\geq
0}}=({\bm{\mathcal{X}}}_{\rho}(t),{\bm{\mathcal{Y}}}_{\rho}(t)))_{t\in\mathbb{R}_{\geq
0}}$, is a continuous two-dimensional Gaussian process such that the
components ${\bm{\mathcal{X}}}_{\rho}$ and ${\bm{\mathcal{Y}}}_{\rho}$ are
standard one-dimensional Brownian motions, and
$\mathrm{Cov}({\bm{\mathcal{X}}}_{\rho}(t),{\bm{\mathcal{Y}}}_{\rho}(s))=\rho\cdot\min\\{t,s\\}$.
We also recall that a _two-dimensional Brownian excursion_
$({\bm{\mathcal{E}}}_{\rho}(t))_{t\in[0,1]}$ _of correlation_ 777We highlight
that we excluded the case $\rho=-1$ in the definition of the two-dimensional
Brownian excursion. Indeed, if $\rho=-1$ it is not meaningful to condition a
two-dimensional Brownian motion of correlation $\rho=-1$ to stay in the non-
negative quadrant. $\rho\in(-1,1]$ _in the non-negative quadrant_ (here simply
called a _two-dimensional Brownian excursion of correlation_ $\rho$) is a two-
dimensional Brownian motion of correlation $\rho$ conditioned to stay in the
non-negative quadrant $\mathbb{R}_{\geq 0}^{2}$ and to end at the origin, i.e.
${\bm{\mathcal{E}}}_{\rho}(1)=(0,0)$. The latter process was formally
constructed in various works (see for instance [MS19, Section 3] and [DW20]).

Let $({\bm{\mathcal{E}}}_{-1/2}(t))_{t\in[0,1]}$be a two-dimensional Brownian
excursion of correlation $-1/2$. Consider the (strong) solutions – which exist
and are unique thanks to Theorem 4.6 in [BM22] – of the following family of
stochastic differential equations (SDEs) indexed by $u\in[0,1]$ and driven by
${\bm{\mathcal{E}}}_{-1/2}(t)=({\bm{\mathcal{X}}}_{-1/2}(t),{\bm{\mathcal{Y}}}_{-1/2}(t))$:

$\begin{cases}d{\bm{\mathcal{Z}}}^{(u)}(t)=\operatorname{\mathds{1}}_{\\{{\bm{\mathcal{Z}}}^{(u)}(t)>0\\}}d{\bm{\mathcal{Y}}}_{-1/2}(t)-\operatorname{\mathds{1}}_{\\{{\bm{\mathcal{Z}}}^{(u)}(t)\leq
0\\}}d{\bm{\mathcal{X}}}_{-1/2}(t),&t\in(u,1),\\\
{\bm{\mathcal{Z}}}^{(u)}(t)=0,&t\in[0,u].\end{cases}$ (5)

######  Definition 1.1.

The _continuous coalescent-walk process driven by_ ${\bm{\mathcal{E}}}_{-1/2}$
is the collection of stochastic processes
$\left\\{{\bm{\mathcal{Z}}}^{(u)}\right\\}_{u\in[0,1]}$ defined by the SDEs in
Eq. 5.

We can now formally introduce the Baxter permuton (a heuristic explanation is
given after 1.2). Consider the following stochastic process:

$\varphi_{{\bm{\mathcal{Z}}}}(t)\coloneqq\operatorname{Leb}\left(\big{\\{}x\in[0,t)|{\bm{\mathcal{Z}}}^{(x)}(t)<0\big{\\}}\cup\big{\\{}x\in[t,1]|{\bm{\mathcal{Z}}}^{(t)}(x)\geq
0\big{\\}}\right),\quad t\in[0,1],$ (6)

where $\operatorname{Leb}(\cdot)$ denotes the one-dimensional Lebesgue
measure.

######  Definition 1.2.

The _Baxter permuton_ $\bm{\mu}^{B}$ is the push-forward of the Lebesgue
measure on $[0,1]$ via the mapping
$(\operatorname{Id},\varphi_{{\bm{\mathcal{Z}}}})$, that is,

$\bm{\mu}^{B}(\cdot)\coloneqq(\operatorname{Id},\varphi_{{\bm{\mathcal{Z}}}})_{*}\operatorname{Leb}(\cdot)=\operatorname{Leb}\left(\\{t\in[0,1]|(t,\varphi_{{\bm{\mathcal{Z}}}}(t))\in\cdot\,\\}\right).$

The Baxter permuton $\bm{\mu}^{B}$ is a random measure on the unit square
$[0,1]^{2}$ and it has uniform marginals (Lemma 5.5 in [BM22]), hence it is a
permuton. Informally, as in the case of the biased Brownian separable
permuton, $\varphi_{{\bm{\mathcal{Z}}}}$ is the “ _continuum permutation_ ” of
the elements of $[0,1]$ induced by the order

$t\preccurlyeq_{{\bm{\mathcal{Z}}}}s\qquad\text{if and only
if}\qquad{\bm{\mathcal{Z}}}^{(t)}(s)<0,$ (7)

where $0\leq t<s\leq 1$, and $\bm{\mu}^{B}$ is the diagram of
$\varphi_{{\bm{\mathcal{Z}}}}$. More details are given in Section 3.1.

###  1.4 A new family of universal permutons: the skew Brownian permuton

_Note added in revision:_ In this paper, we called the family of permutons
introduced in 1.4 the _skew Brownian permuton_ , but we now believe that it
would be more appropriate to refer to them as the _skew Brownian permutons_ ,
as done for instance in [BGS22].

In the previous two sections, we introduced the biased Brownian separable
permuton and the Baxter permuton. A natural question is to explain what is the
connection between the two. The main goal of the present paper is to answer
this question by constructing a new family of permutons, called _skew Brownian
permuton_ , that includes both the biased Brownian separable permuton and the
Baxter permuton. This brings two different limiting objects under the same
roof, identifying a new larger universality class.

####  1.4.1 Definitions and construction

Let $({\bm{\mathcal{E}}}_{\rho}(t))_{t\in[0,1]}$ be a two-dimensional Brownian
excursion of correlation $\rho\in(-1,1]$ and let $q\in[0,1]$ be a further
parameter. Consider the solutions (see Section 1.4.2 for a discussion on
existence and uniqueness) to the following family of SDEs indexed by
$u\in[0,1]$ and driven by
${\bm{\mathcal{E}}}_{\rho}=({\bm{\mathcal{X}}}_{\rho},{\bm{\mathcal{Y}}}_{\rho})$:

$\begin{cases}d{\bm{\mathcal{Z}}}_{\rho,q}^{(u)}(t)=\operatorname{\mathds{1}}_{\\{{\bm{\mathcal{Z}}}_{\rho,q}^{(u)}(t)>0\\}}d{\bm{\mathcal{Y}}}_{\rho}(t)-\operatorname{\mathds{1}}_{\\{{\bm{\mathcal{Z}}}_{\rho,q}^{(u)}(t)\leq
0\\}}d{\bm{\mathcal{X}}}_{\rho}(t)+(2q-1)\cdot
d{\bm{\mathcal{L}}}^{{\bm{\mathcal{Z}}}_{\rho,q}^{(u)}}(t),&t\in(u,1)\\\
{\bm{\mathcal{Z}}}_{\rho,q}^{(u)}(t)=0,&t\in[0,u],\end{cases}$ (8)

where ${\bm{\mathcal{L}}}^{{\bm{\mathcal{Z}}}_{\rho,q}^{(u)}}(t)$ denotes the
symmetric local-time process at zero of the process
${\bm{\mathcal{Z}}}_{\rho,q}^{(u)}$, i.e.

${\bm{\mathcal{L}}}^{{\bm{\mathcal{Z}}}^{(u)}_{\rho,q}}(t)=\lim_{\varepsilon\to
0}\frac{1}{2\varepsilon}\int_{0}^{t}\operatorname{\mathds{1}}_{\\{{\bm{\mathcal{Z}}}^{(u)}_{\rho,q}(s)\in[-\varepsilon,\varepsilon]\\}}ds.$

######  Definition 1.3.

For all $\rho\in(-1,1]$ and $q\in[0,1]$, we call _continuous coalescent-walk
process driven by_ $({\bm{\mathcal{E}}}_{\rho},q)$ the collection of
stochastic processes
${\bm{\mathcal{Z}}}_{\rho,q}=\left\\{{\bm{\mathcal{Z}}}^{(u)}_{\rho,q}\right\\}_{u\in[0,1]}$.

We then consider the following stochastic process:

$\varphi_{{\bm{\mathcal{Z}}}_{\rho,q}}(t)\coloneqq\operatorname{Leb}\left(\big{\\{}x\in[0,t)|{\bm{\mathcal{Z}}}_{\rho,q}^{(x)}(t)<0\big{\\}}\cup\big{\\{}x\in[t,1]|{\bm{\mathcal{Z}}}_{\rho,q}^{(t)}(x)\geq
0\big{\\}}\right),\quad t\in[0,1].$ (9)

######  Definition 1.4.

Fix888For a discussion on the possible construction of the skew Brownian
permuton when $\rho=-1$ we refer the reader to the open problems in Section
1.6. $\rho\in(-1,1]$ and $q\in[0,1]$. The _skew Brownian permuton_ of
parameters $\rho,q$, denoted $\bm{\mu}_{\rho,q}$, is the push-forward of the
Lebesgue measure on $[0,1]$ via the mapping
$(\operatorname{Id},\varphi_{{\bm{\mathcal{Z}}}_{\rho,q}})$, that is

$\bm{\mu}_{\rho,q}(\cdot)\coloneqq(\operatorname{Id},\varphi_{{\bm{\mathcal{Z}}}_{\rho,q}})_{*}\operatorname{Leb}(\cdot)=\operatorname{Leb}\left(\\{t\in[0,1]|(t,\varphi_{{\bm{\mathcal{Z}}}_{\rho,q}}(t))\in\cdot\,\\}\right).$
(10)

######  Remark 1.5.

Note that when $\rho=-1/2$ and $q=1/2$ the SDEs in Eq. 8 are exactly the SDEs
considered in Eq. 5. Therefore the Baxter permuton $\bm{\mu}^{B}(\cdot)$
coincides with the skew Brownian permuton $\bm{\mu}_{-1/2,1/2}$.

######  Remark 1.6.

We highlight that for different values of $u\in[0,1]$, the SDEs in Eq. 8 are
defined using _the same_ two-dimensional Brownian excursion
$({\bm{\mathcal{E}}}_{\rho}(t))_{t\in[0,1]}$. Therefore the coupling of
${\bm{\mathcal{Z}}}_{\rho,q}^{(u)}$ for different values of $u\in[0,1]$ is
highly nontrivial.

To guarantee that the skew Brownian permuton is well-defined for all
$\rho\in(-1,1]$ and $q\in[0,1]$, we need first to discuss existence and
uniqueness of solutions to the SDEs in Eq. 8 and then to check that
$\bm{\mu}_{\rho,q}$ is indeed a permuton, that is, it has uniform marginals.

####  1.4.2 The skew perturbed Tanaka equations and correctness of the
definition

We now focus on the SDEs in Eq. 8, which we call _skew perturbed Tanaka
equations_. The terminology is inspired by the names of some SDEs already
studied in the literature. Indeed, when $q=1/2$, i.e. when the local time term
cancels in Eq. 8, we obtain the _perturbed Tanaka equations_ studied in999To
be precise the articles [Pro13, cHK18] study the perturbed Tanaka equations
driven by a correlated Brownian motion instead of a correlated Brownian
excursion as in Eq. 8. In [Pro13, cHK18] it is proved that there exists a
unique (strong) solution to the perturbed Tanaka equation driven by a
correlated Brownian motion. Then in [BM22, Theorem 4.6] pathwise uniqueness
and existence of a strong solution to the SDE in Eq. 8 was deduced from the
results in [Pro13, cHK18] using absolute continuity arguments. [Pro13, cHK18].
Since here we are adding a local time term, we adopt the terminology _skew
perturbed Tanaka equations_ by analogy with the case of skew Brownian motions:
A _skew Brownian motion_ of parameter $q\in[0,1]$ is a standard one-
dimensional Brownian motion where each excursion is flipped independently to
the positive side with probability $q$ (see for instance [Lej06, Theorem 6]).
It is shown in [HS81] that the unique strong solution to the SDE

${\bm{\mathcal{Z}}}(t)={\bm{\mathcal{B}}}(t)+(2q-1)d{\bm{\mathcal{L}}}^{{\bm{\mathcal{Z}}}}(t),\qquad
t\in\mathbb{R}_{\geq 0},$ (11)

where ${\bm{\mathcal{B}}}(t)$ is a standard one-dimensional Brownian motion,
is a skew Brownian motion of parameter $q$.

It is also convenient to distinguish the following two cases:

  * •

when $\rho\in(-1,1)$ (and $q\in[0,1]$), we will refer to the SDEs in Eq. 8 as
_skew pure perturbed Tanaka equations_ ;

  * •

when $\rho=1$ (and $q\in[0,1]$), we will refer to the SDEs in Eq. 8 simply as
_skew Tanaka equations_.

The skew pure perturbed Tanaka equations have not been investigated so far in
the literature. Our first result guarantees existence and uniqueness of a
strong solution.

######  Theorem 1.7.

Fix $\rho\in(-1,1)$ and $q\in[0,1]$. Pathwise uniqueness and existence of a
strong solution to the SDEs in Eq. 8 hold for all $u\in(0,1)$.

More precisely, denoting by $\mathbb{P}_{{\bm{\mathcal{E}}}_{\rho}}$ the law
of ${\bm{\mathcal{E}}}_{\rho}$, we consider the sigma-algebra
$\mathcal{F}^{(u)}_{t}$ generated by
${\bm{\mathcal{E}}}_{\rho}(s)-{\bm{\mathcal{E}}}_{\rho}(u)$ for $s\in[u,1]$
and completed by negligible events of
$\mathbb{P}_{{\bm{\mathcal{E}}}_{\rho}}$. For every $u\in(0,1)$, there exists
a continuous $\mathcal{F}^{(u)}_{t}$-adapted stochastic process
${\bm{\mathcal{Z}}}_{\rho,q}^{(u)}$ on $[u,1)$, such that

  1. 1.

the function $(\omega,u)\mapsto{\bm{\mathcal{Z}}}_{\rho,q}^{(u)}$ is jointly
measurable.

  2. 2.

For every $u,r\in(0,1)$, $u<r$, ${\bm{\mathcal{Z}}}_{\rho,q}^{(u)}$ satisfies
Eq. 8 a.s. on the interval $[u,r]$.

  3. 3.

If $u,r\in(0,1)$, $u<r$, and $\widetilde{{\bm{\mathcal{Z}}}}$ is a continuous
$\mathcal{F}^{(u)}_{t}$-adapted stochastic process that satisfies Eq. 8 a.s.
on the interval $[u,r]$, then
$\widetilde{{\bm{\mathcal{Z}}}}={\bm{\mathcal{Z}}}_{\rho,q}^{(u)}$ a.s. on
$[u,r]$.

1.7 guarantees that the continuous coalescent-walk process
${\bm{\mathcal{Z}}}_{\rho,q}=\left\\{{\bm{\mathcal{Z}}}^{(u)}_{\rho,q}\right\\}_{u\in[0,1]}$
introduced in 1.3 is well-defined for all $(\rho,q)\in(-1,1)\times[0,1]$ in
the following sense: By Tonelli’s theorem, we have that for almost every
$\omega,$ ${\bm{\mathcal{Z}}}^{(u)}_{\rho,q}$ is a solution for almost every
$u$. Since the definition in Eq. 9 does not depend on a negligible subset of
$[0,1]$, the latter issue causes no problems in defining the skew Brownian
permuton (1.4).

######  Remark 1.8.

We remark that 1.7 is also a fundamental tool to prove permuton convergence
towards the skew Brownian permuton for various models of random constrained
permutations, as shown in [Bor22]. For more details see for instance the proof
of [Bor22, Theorem 4.4].

We highlight that the case $\rho=1$ is excluded in 1.7. Indeed, when $\rho=1$,
existence of a strong solution to the SDEs in Eq. 8 fails (see Section 2.2
below for a precise discussion). Note that when $\rho=1$, the two-dimensional
Brownian excursion $({\bm{\mathcal{E}}}_{1}(t))_{t\in[0,1]}$ rewrites as
${\bm{\mathcal{E}}}_{1}(t)=(\bm{e}(t),\bm{e}(t))$, where
$(\bm{e}(t))_{t\in[0,1]}$ is a one-dimensional Brownian excursion on $[0,1]$.
Therefore the SDEs in Eq. 8 take the simplified form

$\begin{cases}d{\bm{\mathcal{Z}}}^{(u)}(t)=\operatorname{sgn}({\bm{\mathcal{Z}}}^{(u)}(t))d\bm{e}(t)+(2q-1)\cdot
d{\bm{\mathcal{L}}}^{{\bm{\mathcal{Z}}}^{(u)}}(t),&t\in[u,1),\\\
{\bm{\mathcal{Z}}}^{(u)}(t)=0,&t\in[0,u],\end{cases}$ (12)

where
$\operatorname{sgn}(x)\coloneqq\operatorname{\mathds{1}}_{\\{x>0\\}}-\operatorname{\mathds{1}}_{\\{x\leq
0\\}}$.

Solutions to the SDEs in Eq. 12 are not measurable functions of the driving
process $(\bm{e}(t))_{t\in[0,1]}$ but can be constructed using some external
randomness as follows. Conditional on $\bm{e}$, consider an i.i.d. sequence of
random variables $(\bm{s}(\ell))_{\ell}$ indexed by the local minima101010For
the technicalities involved in indexing an i.i.d. sequence by this random
countable set, see again [Maa20, Section 2.2]. of $\bm{e}$ and with
distribution $\mathbb{P}(\bm{s}(\ell)=+1)=q=1-\mathbb{P}(\bm{s}(\ell)=-1)$.
For $0\leq u\leq t\leq 1$, set $\bm{m}^{(u)}(t)\coloneqq\inf_{[u,t]}\bm{e}$
and
$\bm{\varepsilon}_{q}^{(u)}(t)\coloneqq\bm{s}\left(\sup\\{r\in[u,t]:\bm{e}(r)=\bm{m}^{(u)}(t)\\}\right)$.
Then, define for $0\leq u\leq t\leq 1$,

${\bm{\mathcal{Z}}}_{q}^{(u)}(t)\coloneqq(\bm{e}(t)-\bm{m}^{(u)}(t))\cdot\bm{\varepsilon}_{q}^{(u)}(t).$
(13)

######  Proposition 1.9.

The family $\left\\{{\bm{\mathcal{Z}}}_{q}^{(u)}(t)\right\\}_{u\in[0,1]}$
defined in Eq. 13 is a family of solutions to the SDEs in Eq. 12.

######  Remark 1.10.

The construction in Eq. 13 is inspired by the works of Hajri [Haj11] and Le
Jan and Raimond [LJR04, LJR20] on stochastic flow of maps. We will discuss
this connection in Section 2.2. For a discussion on the uniqueness of the
solution in 1.9, we refer the reader to 2.8.

1.7 and 1.9 therefore imply that the continuous coalescent-walk process
${\bm{\mathcal{Z}}}_{\rho,q}=\left\\{{\bm{\mathcal{Z}}}^{(u)}_{\rho,q}\right\\}_{u\in[0,1]}$
introduced in 1.3 is well-defined for all $(\rho,q)\in(-1,1]\times[0,1]$.
Checking that the marginals of skew Brownian permuton $\bm{\mu}_{\rho,q}$ are
uniform (as done in Section 3.1), we obtain the following result.

######  Theorem 1.11.

The skew Brownian permuton $\bm{\mu}_{\rho,q}$ is well-defined for all
$(\rho,q)\in(-1,1]\times[0,1]$. That is, $\bm{\mu}_{\rho,q}$ is a permuton for
all $(\rho,q)\in(-1,1]\times[0,1]$.

Simulations of the skew Brownian permuton $\bm{\mu}_{\rho,q}$ for various
values of $(\rho,q)\in(-1,1]\times[0,1]$ can be found in Fig. 3. How these
simulations were obtained is explained in Appendix B.

$\rho=$

$-0.995$

$q\approx 0$

$q<0.5$

$q=0.5$

$q>0.5$

$q\approx 1$

${\bm{\mathcal{E}}}_{\rho}=({\bm{\mathcal{X}}}_{\rho},{\bm{\mathcal{Y}}}_{\rho})$

$\rho=$

$-0.5$

$\rho=0$

$\rho=$

$0.5$

$\rho=$

$0.995$

$\rho=1$

Figure 3: We collect here some simulations of the skew Brownian permuton
$\bm{\mu}_{\rho,q}$ for various values of $(\rho,q)\in(-1,1]\times[0,1]$. In
every row, there are five simulations of $\bm{\mu}_{\rho,q}$ and at the end of
each row, there is the corresponding two-dimensional Brownian excursion of
correlation $\rho$ in the non-negative quadrant (the specific value of $\rho$
is indicated at the beginning of every row). Each column corresponds to
different values of the parameter $q$ (the specific value of $q$ is indicated
at the top of every column). We highlight that permutons in the same row are
driven by _the same_ Brownian excursion plotted at the end of the row and so
they are coupled. Note that when $\rho=1$ (this is the case of the last row)
the corresponding two-dimensional Brownian excursion is simply a one-
dimensional Brownian excursion and it is plotted using the standard diagram
for real-valued functions. We also remark that the plots of the two-
dimensional Brownian excursions
$({\bm{\mathcal{X}}}_{\rho},{\bm{\mathcal{Y}}}_{\rho})$ on the right-hand side
have different scales for different values of $\rho$ because
$\operatorname{Var}({\bm{\mathcal{X}}}_{\rho}+{\bm{\mathcal{Y}}}_{\rho})$ is
of order $1+\rho$.

####  1.4.3 Relations with the biased Brownian separable permuton and the
Baxter permuton

As already highlighted in 1.5, the Baxter permuton coincides with the skew
Brownian permuton of parameters $\rho=-1/2$ and $q=1/2$. The relation between
the skew Brownian permuton and the biased Brownian separable permuton is less
trivial and will be investigated in Section 3.2, where we will prove the
following result.

######  Theorem 1.12.

For all $p\in[0,1]$, the biased Brownian separable permuton $\bm{\mu}^{S}_{p}$
has the same distribution as the skew Brownian permuton $\bm{\mu}_{1,1-p}$.

As mentioned in the abstract, our new construction of the skew Brownian
permuton brings two different limiting objects under the same roof,
identifying a new larger universality class and explaining the connection
between the Baxter permuton and the biased Brownian separable permuton.

The unsatisfactory feature of the instances of the skew Brownian permuton
mentioned above is that either $\rho=1$ (biased Brownian separable permuton)
or $q=1/2$ (Baxter permuton), and in both cases the SDEs in Eq. 8 take a
simplified form: either the driving process is a one-dimensional Brownian
excursion, as in Eq. 12, or the local time term cancels, as in Eq. 5 (and so
we simply have a perturbed Tanaka equation instead of a skew perturbed Tanaka
equation). In a companion paper [Bor22], we show that this is not the case for
the family of _strong-Baxter permutations_. We recall this result in the next
section.

####  1.4.4 A model of random permutations converging to the skew Brownian
permuton with non-degenerate parameters

We start by defining _strong-Baxter permutations_.

######  Definition 1.13.

_Strong-Baxter permutations_ are permutations avoiding the three vincular
patterns $2\underbracket{41}3$, $3\underbracket{14}2$ and
$3\underbracket{41}2$, i.e. permutations $\sigma$ such that there are no
indices $1\leq i<j<k-1<n$ such that
$\sigma(j+1)<\sigma(i)<\sigma(k)<\sigma(j)$ or
$\sigma(j)<\sigma(k)<\sigma(i)<\sigma(j+1)$ or
$\sigma(j+1)<\sigma(k)<\sigma(i)<\sigma(j)$.

In [Bor22] we proved the following permuton convergence result.

######  Theorem 1.14 ([Bor22, Theorem 1.6]).

Let $\bm{\sigma}_{n}$ be a uniform strong-Baxter permutation of size $n$. The
following convergence in distribution in the permuton sense holds:

$\bm{\mu}_{\bm{\sigma}_{n}}\xrightarrow{d}\bm{\mu}_{\rho,q},$ (14)

where $\rho\approx-0.2151$ is the unique real root of the polynomial

$1+6\rho+8\rho^{2}+8\rho^{3},$ (15)

and $q\approx 0.3008$ is the unique real root of the polynomial

$-1+6q-11q^{2}+7q^{3}.$ (16)

In the same paper, we also investigated the permuton limit of _semi-Baxter
permutations_. We first recall their definition.

######  Definition 1.15.

_Semi-Baxter permutations_ are permutations avoiding the vincular pattern
$2\underbracket{41}3$, i.e. permutations $\sigma$ such that there are no
indices $1\leq i<j<k-1<n$ such that
$\sigma(j+1)<\sigma(i)<\sigma(k)<\sigma(j)$.

######  Theorem 1.16 ([Bor22, Theorem 1.8]).

Let $\bm{\sigma}_{n}$ be a uniform semi-Baxter permutation of size $n$. The
following convergence in distribution in the permuton sense holds:

$\bm{\mu}_{\bm{\sigma}_{n}}\xrightarrow{d}\bm{\mu}_{\rho,q},$ (17)

where

$\rho=-\frac{1+\sqrt{5}}{4}\approx-0.8090\quad\text{and}\quad q=\frac{1}{2}.$
(18)

A summarizing table of the various models of random constrained permutations
that are currently known to converge to the skew Brownian permuton is given in
Fig. 4. We hope that this table will be enlarged in future research projects
(see possible research directions in Section 1.6).

Figure 4: A table with the various models of random constrained permutations
that are known to converge to the skew Brownian permuton. The last column
shows the values of the parameters $\gamma$ and $\kappa^{\prime}$ for the
corresponding SLE-decorated LQG spheres. These values are obtained using the
relations in Eq. 19 and 1.17 below. For the case $\rho=1$ we conjecture that
the corresponding $\gamma$-parameter is $\gamma=2$; for further explanations
see 1.18.

###  1.5 The skew Brownian permuton in Liouville quantum gravity theory

In the previous section, we saw that the skew Brownian permuton is the
permuton limit of various families of constrained permutations. In this
section, we explain how the skew Brownian permuton also arises from a
Liouville quantum gravity (LQG) sphere decorated with two coupled Schramm-
Loewner evolution (SLE) curves. This result will explain _directly at the
continuum level_ various connections between decorated planar maps and
constrained permutations investigated in the literature _at the discrete
level_. More precisely, in the combinatorial literature, several bijections
between families of constrained permutations and decorated planar maps have
been investigated, see for instance [DGW96, Bón97, BBMF11, KSSU13, FNS21].

We highlight that the connection between these two (apparently unrelated)
objects has proven to be a relevant tool to establish some results on the skew
Brownian permuton using SLE/LQG techniques (see [BHSY23, BGS22]). In addition,
we believe that building on this new connection, some convergence results
available for discrete models of random permutations might be transferred to
convergence results for the corresponding discrete models of planar maps (we
refer to Section 1.6 for more details on open problems).

Finally, we mention that our results give a new description of the
interactions between two SLE-curves (coupled in the _imaginary geometry sense_
; see below for more details) decorating an LQG cone or sphere in terms of the
skew perturbed Tanaka equations (see Lemmas 4.2 and 4.3 below). To the best of
our knowledge, this connection with SDEs was not been explicitly exploited so
far in the Liouville quantum gravity literature.

We introduce various classical objects related to Liouville quantum gravity.
We do not give precise definitions of all these objects, but we provide
precise references for each of them.111111Most of the references in this
section point to the self-contained survey paper [GHS19], where the authors
chose to focus (in the finite-volume case) on _quantum disks with boundary
length_ $\ell$. In the present paper, we need to consider _quantum spheres_
instead of _quantum disks_. We remark that a quantum sphere can be viewed as a
quantum disk with zero boundary length. If the reader is still not satisfied,
we point out that all the definitions used in this section can be also found
in [MS19].

We fix $\gamma\in(0,2)$ and we introduce the following parameters defined in
terms of $\gamma$,

$\kappa=\gamma^{2},\qquad\kappa^{\prime}=16/\gamma^{2},\qquad\rho=-\cos(\pi\gamma^{2}/4),\qquad\chi=2/\gamma-\gamma/2.$
(19)

We start by recalling the construction of space-filling
SLE${}_{\kappa^{\prime}}$ curves on $\mathbb{C}\cup\\{\infty\\}$. One way of
constructing them (as proved in [MS17]) is from the flow lines of the vector
field $e^{i(\hat{\bm{h}}/\chi+\theta)}$, where $\hat{\bm{h}}$ is a whole-plane
Gaussian free field (modulo a global additive multiple of $2\pi\chi$) and
$\theta\in[0,\pi]$. Flow lines of the vector field
$e^{i(\hat{\bm{h}}/\chi+\theta)}$ are constructed in the imaginary geometry
sense and are shown to be whole-plane SLE${}_{\kappa}(2-\kappa)$ processes
from the starting point to $\infty$. As shown in [MS17, Theorem 1.9], flow
lines of $e^{i(\hat{\bm{h}}/\chi+\theta)}$ started at different points of
$\mathbb{Q}^{2}$ merge into each other upon intersecting and form a tree. The
_space-filling_ SLE${}_{\kappa^{\prime}}$ _counterflow line from $\infty$ to
$\infty$ generated by_ $\hat{\bm{h}}$ _and of angle_ $\theta$, denoted
$\bm{\eta}_{\theta}$, is the Peano curve of this tree, i.e. it is the curve
which visits the points of $\mathbb{C}$ in chronological order, where a point
$x\in\mathbb{C}$ is visited before a point $y\in\mathbb{C}$ if the flow line
of angle $\theta$ from $x$ merges into the flow line of angle $\theta$ from
$y$ on the left side of the latter flow line. For $\theta\in[0,\pi]$, we
consider the pair $(\bm{\eta}_{0},\bm{\eta}_{\theta})$ of space-filling
SLE${}_{\kappa^{\prime}}$ counterflow lines from $\infty$ to $\infty$
generated by the same Gaussian free field $\hat{\bm{h}}$ of angle $0$ and
$\theta$, respectively.

Let now $({\mathbb{C}}\cup\\{\infty\\},\bm{h},\infty)$ be a $\gamma$-LQG
sphere, independent of $\hat{\bm{h}}$, with quantum area one and one marked
point at $\infty$ (see [GHS19, Definition 3.20]). We parametrize the pair
$(\bm{\eta}_{0},\bm{\eta}_{\theta})$ by the $\mu_{\bm{h}}$-LQG area measure
(see [GHS19, Section 3.3]) so that
$\bm{\eta}_{0}(0)=\bm{\eta}_{0}(1)=\bm{\eta}_{\theta}(0)=\bm{\eta}_{\theta}(1)=\infty$
and
$\mu_{\bm{h}}(\bm{\eta}_{0}([0,t]))=\mu_{\bm{h}}(\bm{\eta}_{\theta}([0,t]))=t$
for each $t\in[0,1]$.

######  Theorem 1.17.

Fix $\gamma\in(0,2)$ and $\theta\in[0,\pi]$. Let
$({\mathbb{C}}\cup\\{\infty\\},\bm{h},\infty)$ and
$(\bm{\eta}_{0},\bm{\eta}_{\theta})$ be the $\gamma$-LQG sphere and the pair
of space-filling SLE${}_{\kappa^{\prime}}$ introduced above. For $t\in[0,1]$,
let $\bm{\psi}_{\gamma,\theta}(t)\in[0,1]$ denote the first time when
$\bm{\eta}_{\theta}$ hits the point $\bm{\eta}_{0}(t)$.121212We recall that
space-filling SLE curves have multiple points. Nevertheless, for each
$z\in\mathbb{C}$, a.s. $z$ is not a multiple point of $\bm{\eta}_{\theta}$,
i.e., $\bm{\eta}_{\theta}$ hits $z$ exactly once. Since $\bm{h}$ is
independent from $(\bm{\eta}_{0},\bm{\eta}_{\theta})$ and $\bm{\eta}_{0}$ and
$\bm{\eta}_{\theta}$ are parametrized by $\mu_{\bm{h}}$-mass, a.s. the set of
times $t\in[0,1]$ such that $\bm{\eta}_{0}$ is a multiple point of
$\bm{\eta}_{\theta}$ has zero Lebesgue measure. Then the random measure

$(\operatorname{Id},\bm{\psi}_{\gamma,\theta})_{*}\operatorname{Leb}$

is a skew Brownian permuton of parameter $\rho=-\cos(\pi\gamma^{2}/4)$ and
$\overline{q}=\overline{q}_{\gamma}(\theta)\in[0,1]$.

######  Remark 1.18.

The explicit expression of the function $\overline{q}_{\gamma}(\theta)$ is
unknown. It holds that $\overline{q}_{\gamma}(\pi/2)=1/2$. Moreover, for every
fixed $\gamma\in(0,2)$, the function

$\overline{q}_{\gamma}(\theta):[0,\pi]\to[0,1]$

is a homeomorphism and therefore has an inverse function
$\theta_{\gamma}(\overline{q})$. Finally, for all $\theta\in[0,\pi/2)$, it
holds that
$\overline{q}_{\gamma}(\theta)+\overline{q}_{\gamma}(\pi-\theta)=1$. All these
properties are consequences131313The fact that the function
$\overline{q}_{\gamma}(\theta)$ in 1.17 is the same function considered in
[LSW17, Lemma 4.3] follows from 4.2 below. of [LSW17, Lemma 4.3]. We highlight
that the fact that $\rho=-\cos(\pi\gamma^{2}/4)$ and
$\overline{q}_{\gamma}(\theta)$ is a homeomorphism guarantees that the range
of parameters for the skew Brownian permuton appearing in 1.17 is
$\rho\in(-1,1)$ and $q\in[0,1]$. That is, all the possible skew Brownian
permutons appear in 1.17, except the biased Brownian separable permuton
$\bm{\mu}_{1,q}$ for $q\in[0,1]$. We conjecture that the latter case can be
obtained from the critical $2$-LQG since there are several analogies between
the construction of $\bm{\mu}_{1,q}$ and the results in [AHPS21].

The proof of 1.17 is given in Section 4 building on results of [DMS21] and
[GHS16]. We also invite the curious reader to look at [BGS22, Lemma 2.8] where
a simple consequence of 1.17 is derived (giving another description of the
skew Brownian permuton in terms of SLEs and LQG) and where it is also shown
that our specific choice of $\bm{\psi}_{\gamma,\theta}(t)\in[0,1]$ in the
statement of 1.17 can be replaced by any measurable function
$\widetilde{\bm{\psi}}_{\gamma,\theta}(t)\in[0,1]$ such that
$\bm{\eta}_{\theta}(\widetilde{\bm{\psi}}_{\gamma,\theta}(t))=\bm{\eta}_{0}(t)$
for all $t\in[0,1]$.

###  1.6 Open problems

In this final section of the introduction, we collect a list of open questions
and problems that we plan to address in future research projects.

  1. 1.

Intensity measure and Hausdorff dimension of the skew Brownian permuton. The
skew Brownian permuton $\bm{\mu}_{\rho,q}$ is a new fractal random measure of
the unit square and we plan to investigate some of its properties in the
future. For instance, two natural questions are:

    1. (a)

What is the density of the intensity measure $\mathbb{E}[\bm{\mu}_{\rho,q}]$?
How does it depend on the two parameters $\rho$ and $q$?

    2. (b)

What is the Hausdorff dimension of the support of $\bm{\mu}_{\rho,q}$? And
again, how does it depend on the two parameters $\rho$ and $q$?

We highlight that the questions above were answered in [Maa20] for the biased
Brownian separable permuton $\bm{\mu}_{1,q}$. For instance, it was shown that
almost surely, the support of $\bm{\mu}_{1,q}$ is totally disconnected, and
its Hausdorff dimension is 1 (with one-dimensional Hausdorff measure bounded
above by $\sqrt{2}$). For an expression for the intensity measure see [Maa20,
Theorem 1.7].

We believe that a key tool to answer the two questions above would be the new
connection with SLE-decorated LQG spheres described in 1.17.

  2. 2.

Properties of the stochastic process $\varphi_{{\bm{\mathcal{Z}}}_{\rho,q}}$.
Recall that the stochastic process $\varphi_{{\bm{\mathcal{Z}}}_{\rho,q}}$ was
defined in Eq. 9 as follows

$\varphi_{{\bm{\mathcal{Z}}}_{\rho,q}}(t)\coloneqq\operatorname{Leb}\left(\big{\\{}x\in[0,t)|{\bm{\mathcal{Z}}}_{\rho,q}^{(x)}(t)<0\big{\\}}\cup\big{\\{}x\in[t,1]|{\bm{\mathcal{Z}}}_{\rho,q}^{(t)}(x)\geq
0\big{\\}}\right),\quad t\in[0,1].$ (20)

Heuristically speaking, the skew Brownian permuton $\bm{\mu}_{\rho,q}$ is the
graph of the function $\varphi_{{\bm{\mathcal{Z}}}_{\rho,q}}$. Therefore,
determining some properties of the process
$\varphi_{{\bm{\mathcal{Z}}}_{\rho,q}}$ might be a useful step in the
investigation of the permuton $\bm{\mu}_{\rho,q}$. Our intuition suggests that
$\varphi_{{\bm{\mathcal{Z}}}_{\rho,q}}$ might be a particular (conditioned)
_fragmentation process_ in the sense of [Ber17]. We also believe that a key
step to understanding the process $\varphi_{{\bm{\mathcal{Z}}}_{\rho,q}}$ will
be to investigate the joint law of two processes
${\bm{\mathcal{Z}}}_{\rho,q}^{(x)}$ and ${\bm{\mathcal{Z}}}_{\rho,q}^{(y)}$.
We remark that similar questions have been investigated in [BK04] for pair of
solutions to the SDEs

$\begin{cases}d{\bm{\mathcal{Z}}}^{(u)}(t)=d{\bm{\mathcal{B}}}(t)+(2q-1)\cdot
d{\bm{\mathcal{L}}}^{{\bm{\mathcal{Z}}}^{(u)}}(t),&t\in\mathbb{R}_{>u},\\\
{\bm{\mathcal{Z}}}^{(u)}(t)=0,&t\in[0,u],\end{cases}$ (21)

where ${\bm{\mathcal{B}}}(t)$ is a standard one-dimensional Brownian motion.

  3. 3.

Proportions of patterns in the skew Brownian permuton. We first define the
permutation induced by $k$ points in the square $[0,1]^{2}$. Take a sequence
of $k$ points $(X,Y)=((x_{1},y_{1}),\dots,(x_{k},y_{k}))$ in $[0,1]^{2}$ with
distinct $x$ and $y$ coordinates. The _$x$ -reordering_ of $(X,Y)$ is the
unique reordering of the sequence $(X,Y)$ such that $x_{(1)}<\cdots<x_{(k)}$,
and is denoted by $\left((x_{(1)},y_{(1)}),\dots,(x_{(k)},y_{(k)})\right)$.
The values $(y_{(1)},\ldots,y_{(k)})$ are then in the same relative order as
the values of a unique permutation of size $k$, that we call the _permutation
induced by_ $(X,Y)$.

Let $\mu$ be a permuton and $((\bm{X}_{i},\bm{Y}_{i}))_{i\in\mathbb{Z}_{>0}}$
be an i.i.d. sequence with distribution $\mu$. We denote by
$\operatorname{Perm}_{k}(\mu)$ the random permutation induced by
$((\bm{X}_{i},\bm{Y}_{i}))_{i\in\llbracket 1,k\rrbracket}$.

Finally, if $\sigma$ is a permutation of size $n$ and $\pi$ a pattern141414We
refer the reader to [Bor21, Section 1.6.1] for an introduction to permutation
patterns. of size $k\leq n$, then we denote by
$\operatorname{occ}(\pi,\sigma)$ the number of occurrences of $\pi$ in
$\sigma$. Moreover, we denote by $\operatorname{\widetilde{occ}}(\pi,\sigma)$
the proportion of occurrences of $\pi$ in $\sigma,$ that is,
$\operatorname{\widetilde{occ}}(\pi,\sigma)\coloneqq\frac{\operatorname{occ}(\pi,\sigma)}{\binom{n}{k}}$.

######  Question 1.19.

Let $\bm{\mu}_{\rho,q}$ be a skew Brownian permuton with parameters
$(\rho,q)\in(-1,1]\times[0,1]$. For all $k\in\mathbb{Z}_{>0}$ and patterns
$\pi$ of size $k$ is it possible to determine the limiting distribution of the
random variables

$\lim_{n\to\infty}\operatorname{\widetilde{occ}}(\pi,\operatorname{Perm}_{n}(\bm{\mu}_{\rho,q}))=\mathbb{P}(\operatorname{Perm}_{k}(\bm{\mu}_{\rho,q})=\pi|\bm{\mu}_{\rho,q})?$

We point out that the fact that the latter limit exists and satisfies the
expression above is a consequence of the fact that
$\operatorname{Perm}_{n}(\bm{\mu}_{\rho,q})$ converges in distribution in the
permuton sense to $\bm{\mu}_{\rho,q}$ (see [BBF+20, Lemma 2.3 and Theorem
2.5]).

A simpler question is to compute, for all $k\in\mathbb{Z}_{>0}$ and patterns
$\pi$ of size $k$, the probabilities

$\mathbb{P}(\operatorname{Perm}_{k}(\bm{\mu}_{\rho,q})=\pi)=\lim_{n\to\infty}\mathbb{E}\left[\operatorname{\widetilde{occ}}(\pi,\operatorname{Perm}_{n}(\bm{\mu}_{\rho,q}))\right].$

Our intuition suggests the following statement.

######  Conjecture 1.20.

For all $(\rho,q)\in(-1,1)\times(0,1)$ and for all $k\in\mathbb{Z}_{>0}$ and
patterns $\pi$ of size $k$, it holds that

$\mathbb{P}(\operatorname{Perm}_{k}(\bm{\mu}_{\rho,q})=\pi)>0.$

We remark that since $\bm{\mu}_{1,q}$ can be obtained as the limit of pattern-
avoiding permutations then we know that
$\mathbb{P}(\operatorname{Perm}_{k}(\bm{\mu}_{\rho,q})=\pi)$ is zero for some
$q\in(0,1)$ and some pattern $\pi$.

  4. 4.

The length of the longest increasing subsequence in the skew Brownian
permuton.

This question is motivated by the recent work [BBD+21]. The authors show that
the length of the longest increasing subsequence in a sequence of permutations
converging to the biased Brownian separable permuton has sublinear size. Their
proof builds on some self-similarity properties of the biased Brownian
separable permuton.

######  Question 1.21.

Let $(\bm{\sigma}_{n})_{n\in\mathbb{Z}_{\geq 0}}$ be a sequence of random
permutations converging in distribution in the permuton sense to the skew
Brownian permuton $\bm{\mu}_{\rho,q}$. What is the length of the longest
increasing subsequence in $\bm{\sigma}_{n}$?

We expect, under some regularity conditions on the sequence $\bm{\sigma}_{n}$,
a formula only depending on $\rho,q$.

  5. 5.

Properties of the parameters $\rho$ and $q$ defining the skew Brownian
permuton. The skew Brownian permuton $\bm{\mu}_{\rho,q}$ is defined in terms
of the two parameters $\rho\in(-1,1]$ and $q\in[0,1]$. It would be interesting
to find some natural statistics on permutons (i.e. a map from the space of
permutons $\mathcal{M}$ to $\mathbb{R}$ describing some natural quantity)
determined by these two parameters. For instance, we expect that $q$ controls
the proportion of inversions in $\bm{\mu}_{\rho,q}$. More precisely, we
conjecture the following.

######  Conjecture 1.22.

Set $\mathbb{P}(\operatorname{Perm}_{2}(\bm{\mu}_{\rho,q})=21)=f(\rho,q)$.
Then for every fixed $\rho\in(-1,1]$ the function $f(\rho,q)$ is increasing in
$q$.

It would be even more interesting to derive an explicit expression for
$f(\rho,q)$. It would be also remarkable to find some natural statistics
$\operatorname{stc}(\cdot)$ on permutons such that
$\operatorname{stc}(\bm{\mu}_{\rho,q})=g(\rho,q)$ a.s., for some function $g$.
This would imply that the laws of $\bm{\mu}_{\rho,q}$ are singular for
different values of the parameters $\rho\in(-1,1]$ and $q\in[0,1]$.

  6. 6.

The skew Brownian permuton for $\rho=-1$. The skew Brownian permuton
$\bm{\mu}_{-1,q}$ has not been defined yet, because when $\rho=-1$, it is
meaningless to condition a two-dimensional Brownian motion of correlation
$\rho=-1$ to stay in the non-negative quadrant. It would be interesting to
investigate if there is a natural way of constructing $\bm{\mu}_{-1,q}$. A
possible way would be to consider the limit of the permutons
$\bm{\mu}_{\rho_{n},q}$ for an appropriate sequence $\rho_{n}\to-1$. Looking
at the simulations in Fig. 3, we also believe that a natural candidate model
for defining $\bm{\mu}_{-1,q}$ is the Mallows permuton [Sta09, SW18].

  7. 7.

Critical 2-LQG and biased Brownian separable permuton. 1.17 connects the skew
Brownian permuton $\bm{\mu}_{\rho,q}$ to SLE-decorated LQG spheres in the
range of parameters $(\rho,q)\in(-1,1)\times[0,1]$. It would be interesting to
show that the biased Brownian separable permuton $\bm{\mu}_{1,q}$ for
$q\in[0,1]$ arises from the critical $2$-LQG sphere (in the same spirit of
1.17). Here some new ideas are needed, as it is not clear how to define two
SLE curves decorating a $2$-LQG sphere (see [AHPS21]).

  8. 8.

Models of planar maps associated with semi-Baxter permutations and separable
permutations. We believe that there are two natural models of decorated planar
maps associated with semi-Baxter permutations and separable permutations that
might converge to SLE-decorated LQG spheres. This belief is justified by 1.17
and the results in [BM22]. More precisely,

     * •

We conjecture that _bipolar posets_ (introduced in [FNS21]) are in bijection
with semi-Baxter permutations and converge to a $\gamma$-LQG sphere decorated
with two SLE curves of angle $\theta(q)$, where $\gamma$ and $\theta(q)$ are
related to the parameters $\rho,q$ appearing in 1.16 through the relations
given in 1.17.

     * •

We further conjecture that _rooted series-parallel maps_ (studied in [BBMF11,
Proposition 6], where it is also shown that they are in bijection with
separable permutations) converge to the critical $2$-LQG sphere. This would be
the first discrete model of planar maps shown to converge to the critical
$2$-LQG sphere.

We point out that the results in this paper give three different possible
approaches to answer the questions above: the _permutation approach_ , the
_SDE approach_ , and the _SLE-decorated LQG approach_.151515 _Note added in
revision:_ The SLE-decorated LQG approach has been first used in [BHSY23] to
compute the intensity measure of the Baxter permuton (giving a partial answer
to the Open Problem 1a) and to prove Conjectures 1.20 and 1.22; and recently
in [BGS22] to solve the Open Problem 1b and to give a partial answer to 1.21.

## Acknowledgements

During the realization of this long-term project, I had the opportunity to
exchange various ideas with several people. Here is an alphabetically ordered
list of the ones who participated in some discussions and to whom I would like
to express my gratitude: thanks to Mathilde Bouvel, Valentin Féray, Ewain
Gwynne, Hatem Hajri, Nina Holden, Antoine Lejay, Mickaël Maazoun, Grégory
Miermont, Vilmos Prokaj, Kilian Raschel, and Xin Sun. We thank the anonymous
referees for all their precious and useful comments.

##  2 The skew perturbed Tanaka equations

In this section we focus on the skew perturbed Tanaka equations introduced in
Eq. 8, p. 8, and Eq. 12, p. 12. In particular, in the following two
subsections, we prove 1.7 and 1.9. In both cases, we will first investigate
the SDEs in Eqs. 8 and 12 when they are driven by a correlated two-dimensional
Brownian motion instead of a correlated two-dimensional Brownian excursion,
and then we will transfer the results to our specific cases using absolute
continuity arguments.

These results are fundamental to then show in Section 3.1 that the skew
Brownian permuton is well-defined, proving 1.11.

###  2.1 The skew pure perturbed Tanaka equation

####  2.1.1 The Brownian motion case

Let $({\bm{\mathcal{W}}}_{\rho}(t))_{t\in\mathbb{R}_{\geq
0}}=({\bm{\mathcal{X}}}_{\rho}(t),{\bm{\mathcal{Y}}}_{\rho}(t))_{t\in\mathbb{R}_{\geq
0}}$ be a two-dimensional Brownian motion of correlation $\rho\in(-1,1)$ and
let $q\in[0,1]$ be a parameter. We consider the following SDE

$d{\bm{\mathcal{Z}}}_{\rho,q}(t)=\operatorname{\mathds{1}}_{\\{{\bm{\mathcal{Z}}}_{\rho,q}(t)>0\\}}d{\bm{\mathcal{Y}}}_{\rho}(t)-\operatorname{\mathds{1}}_{\\{{\bm{\mathcal{Z}}}_{\rho,q}(t)\leq
0\\}}d{\bm{\mathcal{X}}}_{\rho}(t)+(2q-1)\cdot
d{\bm{\mathcal{L}}}^{{\bm{\mathcal{Z}}}_{\rho,q}}(t),\quad
t\in\mathbb{R}_{\geq 0},$ (22)

where we recall that ${\bm{\mathcal{L}}}^{{\bm{\mathcal{Z}}}_{\rho,q}}(t)$ is
the symmetric local-time process at zero of ${\bm{\mathcal{Z}}}_{\rho,q}$.
From now on, to simplify notation, we write all the involved processes
forgetting the indexes $\rho$ and $q$. We also denote by $\mathcal{C}(I)$ the
set of continuous functions from an interval $I$ of $\mathbb{R}_{\geq 0}$ to
$\mathbb{R}$. Recall also the definition of skew Brownian motion from the
beginning of Section 1.4.2. We prove the following result.

######  Theorem 2.1.

Fix $\rho\in(-1,1)$ and $q\in[0,1]$. Pathwise uniqueness and existence of a
strong solution to the SDE in Eq. 22 hold. In addition, the solution is a skew
Brownian motion of parameter $q$.

More precisely, if $(\Omega,\mathcal{F},(\mathcal{F}_{t})_{t},\mathbb{P})$ is
a filtered probability space satisfying the usual conditions, and assuming
that ${\bm{\mathcal{W}}}_{\rho}$ is an $\mathcal{F}_{t}$-Brownian motion of
correlation $\rho$,

  1. 1.

if ${\bm{\mathcal{Z}}},\widetilde{{\bm{\mathcal{Z}}}}$ are two
$\mathcal{F}_{t}$-adapted continuous processes that solve Eq. 22 a.s., then
${\bm{\mathcal{Z}}}=\widetilde{{\bm{\mathcal{Z}}}}$ a.s.

  2. 2.

There exists an $\mathcal{F}_{t}$-adapted continuous process
${\bm{\mathcal{Z}}}$ which solves Eq. 22 a.s.

In particular, for every $t\in\mathbb{R}_{\geq 0}$ there exists a measurable
_solution map_ $F_{t}:\mathcal{C}([0,t))\to\mathcal{C}([0,t))$ such that

  1. 3.

$F_{t}({\bm{\mathcal{W}}}_{\rho}|_{[0,t)})$ satisfies Eq. 22 a.s. on the
interval $[0,t)$.

  2. 4.

For every $s,t\in\mathbb{R}_{\geq 0}$ with $s\leq t$, then
$F_{t}({\bm{\mathcal{W}}}_{\rho}|_{[0,t)})|_{[0,s)}=F_{s}({\bm{\mathcal{W}}}_{\rho}|_{[0,s)})$
a.s.

From now on we fix $\rho\in(-1,1)$ and $q\in(0,1)$. The case $q\in\\{0,1\\}$
will be considered at the end of this section. We introduce the function
$r(x)=x/(1-q)\cdot\mathds{1}_{x>0}+x/q\cdot\mathds{1}_{x\leq 0}$ and the SDE

$d{\bm{\mathcal{R}}}(t)=(1-q)\operatorname{\mathds{1}}_{\\{{\bm{\mathcal{R}}}(t)>0\\}}d{\bm{\mathcal{Y}}}(t)-q\operatorname{\mathds{1}}_{\\{{\bm{\mathcal{R}}}(t)\leq
0\\}}d{\bm{\mathcal{X}}}(t),\quad t\in\mathbb{R}_{\geq 0}.$ (23)

We have the following result.

######  Proposition 2.2.

Let ${\bm{\mathcal{R}}}=({\bm{\mathcal{R}}}(t))_{t\in\mathbb{R}_{\geq 0}}$ and
${\bm{\mathcal{Z}}}=({\bm{\mathcal{Z}}}(t))_{t\in\mathbb{R}_{\geq 0}}$ be two
stochastic processes such that
${\bm{\mathcal{Z}}}(t)=r({\bm{\mathcal{R}}}(t))$ for all $t\in\mathbb{R}_{\geq
0}$. The process ${\bm{\mathcal{R}}}$ is a strong solution to Eq. 23 if and
only if the process ${\bm{\mathcal{Z}}}$ is a strong solution to Eq. 22.

Thanks to 2.2, pathwise uniqueness and existence of a strong solution to the
SDEs in Eqs. 22 and 23 are equivalent. Thanks to the Yamada–Watanabe
theorem161616If further explanations are needed, the reader can look at the
discussion at the beginning of [cHK18, Section 2.1]. (see [KS91, Proposition
5.3.20 and Corollary 5.3.23]), to show pathwise uniqueness and existence of a
strong solution to the SDE in Eq. 23, it is enough to show pathwise uniqueness
and existence of a weak solution to the SDE in Eq. 23.

######  Proposition 2.3.

Pathwise uniqueness holds for Eq. 23.

######  Proposition 2.4.

There exists a weak solution
${\bm{\mathcal{R}}}=({\bm{\mathcal{R}}}(t))_{t\in\mathbb{R}_{\geq 0}}$ to Eq.
23 such that $r({\bm{\mathcal{R}}})$ is a skew Brownian motion of parameter
$q$.

Note that the last two propositions together with 2.2 prove 2.1 for
$\rho\in(-1,1)$ and $q\in(0,1)$. We now proceed with the proof of these three
propositions.

We start by recalling the _symmetric Itô–Tanaka formula_ for convex functions
(see for instance [Lej06, Section 5.1]) since we will use it repeatedly in
this section. Let $f$ be a function from $\mathbb{R}$ to $\mathbb{R}$ which is
the difference of two convex functions. Then

$f^{\prime}_{r}(x)\coloneqq\lim_{\varepsilon\to
0,\varepsilon>0}\varepsilon^{-1}(f(x+\varepsilon)-f(x))\quad\text{and}\quad
f^{\prime}_{\ell}(x)\coloneqq\lim_{\varepsilon\to
0,\varepsilon<0}\varepsilon^{-1}(f(x+\varepsilon)-f(x))$

exist for almost every $x\in\mathbb{R}$. In addition, there exists a signed
measure $\nu$ on $\mathbb{R}$, called the second derivative measure, such that

$\int_{\mathbb{R}}g(x)\nu(dx)=-\int_{\mathbb{R}}g^{\prime}(x)f^{\prime}_{\ell}(x)dx$

for every piecewise $\mathcal{C}^{1}$ function $g$ with compact support on
$\mathbb{R}$. If $f$ has a second derivative, then $f^{\prime\prime}$ is the
density of $\nu$ with respect to the Lebesgue measure.

Let ${\bm{\mathcal{M}}}=({\bm{\mathcal{M}}}(t))_{t\in\mathbb{R}_{\geq 0}}$ be
a real-valued semi-martingale. The _symmetric Itô–Tanaka formula_ for
$f({\bm{\mathcal{M}}})$ states that

$f({\bm{\mathcal{M}}}(t))=f({\bm{\mathcal{M}}}(0))+\int_{0}^{t}\tfrac{1}{2}\Big{(}f^{\prime}_{r}({\bm{\mathcal{M}}}(s))+f^{\prime}_{\ell}({\bm{\mathcal{M}}}(s))\Big{)}d{\bm{\mathcal{M}}}(s)+\frac{1}{2}\int_{\mathbb{R}}{\bm{\mathcal{L}}}_{x}^{{\bm{\mathcal{M}}}}(t)\nu(dx),\quad
t\in\mathbb{R}_{\geq 0},$ (24)

where ${\bm{\mathcal{L}}}_{x}^{{\bm{\mathcal{M}}}}(t)$ denotes the symmetric
local time process at $x$ of ${\bm{\mathcal{M}}}$, that is

${\bm{\mathcal{L}}}^{{\bm{\mathcal{M}}}}_{x}(t)=\lim_{\varepsilon\to
0}\frac{1}{2\varepsilon}\int_{0}^{t}\operatorname{\mathds{1}}_{\\{{\bm{\mathcal{M}}}(s)\in[x-\varepsilon,x+\varepsilon]\\}}ds.$

We can now proceed with the proof of 2.2.

###### Proof of 2.2.

We start by recalling that

$r(x)=x/(1-q)\cdot\mathds{1}_{x>0}+x/q\cdot\mathds{1}_{x\leq 0}.$

We also introduce some additional functions that are used in the proof. We set

$\displaystyle
r^{\prime}(x)=1/(1-q)\cdot\mathds{1}_{x>0}+1/(2q(1-q))\cdot\mathds{1}_{x=0}+1/q\cdot\mathds{1}_{x<0},$
(25) $\displaystyle s(x)=(1-q)x\cdot\mathds{1}_{x>0}+qx\cdot\mathds{1}_{x\leq
0},$ (26) $\displaystyle
s^{\prime}(x)=(1-q)\cdot\mathds{1}_{x>0}+\frac{1}{2}\cdot\operatorname{\mathds{1}}_{x=0}+q\cdot\mathds{1}_{x<0}.$
(27)

Note that $r(s(x))=s(r(x))=x$. We first assume that the process
${\bm{\mathcal{R}}}$ is a strong solution to Eq. 23 and we show that the
process ${\bm{\mathcal{Z}}}$, defined by
${\bm{\mathcal{Z}}}(t)=r({\bm{\mathcal{R}}}(t))$, is a strong solution to Eq.
22. Applying the Itô–Tanaka formula (Eq. 24) we have that

$d{\bm{\mathcal{Z}}}(t)=r^{\prime}({\bm{\mathcal{R}}}(t))d{\bm{\mathcal{R}}}(t)+\frac{\gamma}{2}d{\bm{\mathcal{L}}}^{{\bm{\mathcal{R}}}}(t),$
(28)

where $\gamma\coloneqq 1/(1-q)-1/q$. Using Eq. 23 we obtain that

$d{\bm{\mathcal{Z}}}(t)=\operatorname{\mathds{1}}_{\\{{\bm{\mathcal{R}}}(t)>0\\}}d{\bm{\mathcal{Y}}}(t)-\operatorname{\mathds{1}}_{\\{{\bm{\mathcal{R}}}(t)<0\\}}d{\bm{\mathcal{X}}}(t)-\frac{1}{2(1-q)}\operatorname{\mathds{1}}_{\\{{\bm{\mathcal{R}}}(t)=0\\}}d{\bm{\mathcal{X}}}(t)+\frac{\gamma}{2}d{\bm{\mathcal{L}}}^{{\bm{\mathcal{R}}}}(t)\\\
=\operatorname{\mathds{1}}_{\\{{\bm{\mathcal{Z}}}(t)>0\\}}d{\bm{\mathcal{Y}}}(t)-\operatorname{\mathds{1}}_{\\{{\bm{\mathcal{Z}}}(t)\leq
0\\}}d{\bm{\mathcal{X}}}(t)+\frac{\gamma}{2}d{\bm{\mathcal{L}}}^{{\bm{\mathcal{R}}}}(t),$
(29)

where in the last equality we used that
${\bm{\mathcal{Z}}}(t)=r({\bm{\mathcal{R}}}(t))$ and that
$\int_{0}^{t}\operatorname{\mathds{1}}_{\\{{\bm{\mathcal{R}}}(s)=0\\}}d{\bm{\mathcal{X}}}(s)$
is identically zero. Indeed, this stochastic integral has zero mean and

$\operatorname{Var}\left[\int_{0}^{t}\operatorname{\mathds{1}}_{\\{{\bm{\mathcal{R}}}(s)=0\\}}d{\bm{\mathcal{X}}}(s)\right]=\mathbb{E}\left[\int_{0}^{t}\operatorname{\mathds{1}}_{\\{{\bm{\mathcal{R}}}(s)=0\\}}ds\right]=0,$
(30)

since
$\int_{0}^{t}\operatorname{\mathds{1}}_{\\{{\bm{\mathcal{R}}}(s)=0\\}}ds=0$ by
[KS91, Exercise 3.7.10].

It remains to find a relation between
${\bm{\mathcal{L}}}^{{\bm{\mathcal{R}}}}(t)$ and
${\bm{\mathcal{L}}}^{{\bm{\mathcal{Z}}}}(t)$. Define
$\widetilde{\operatorname{sgn}}(x)\coloneqq\operatorname{\mathds{1}}_{x>0}-\operatorname{\mathds{1}}_{x<0}$
(note that $\widetilde{\operatorname{sgn}}(0)=0$ by definition). By Itô–Tanaka
formula (Eq. 24) and Eq. 29,

$|{\bm{\mathcal{Z}}}(t)|=\int_{0}^{t}\widetilde{\operatorname{sgn}}({\bm{\mathcal{Z}}}(s))d{\bm{\mathcal{Z}}}(s)+{\bm{\mathcal{L}}}^{{\bm{\mathcal{Z}}}}(t)\\\
=\int_{0}^{t}\widetilde{\operatorname{sgn}}({\bm{\mathcal{Z}}}(s))(\operatorname{\mathds{1}}_{\\{{\bm{\mathcal{Z}}}(s)>0\\}}d{\bm{\mathcal{Y}}}(s)-\operatorname{\mathds{1}}_{\\{{\bm{\mathcal{Z}}}(s)\leq
0\\}}d{\bm{\mathcal{X}}}(s))+\frac{\gamma}{2}\int_{0}^{t}\widetilde{\operatorname{sgn}}({\bm{\mathcal{Z}}}(s))d{\bm{\mathcal{L}}}^{{\bm{\mathcal{R}}}}(s)+{\bm{\mathcal{L}}}^{{\bm{\mathcal{Z}}}}(t)\\\
=\int_{0}^{t}\widetilde{\operatorname{sgn}}({\bm{\mathcal{Z}}}(s))(\operatorname{\mathds{1}}_{\\{{\bm{\mathcal{Z}}}(s)>0\\}}d{\bm{\mathcal{Y}}}(s)-\operatorname{\mathds{1}}_{\\{{\bm{\mathcal{Z}}}(s)\leq
0\\}}d{\bm{\mathcal{X}}}(s))+{\bm{\mathcal{L}}}^{{\bm{\mathcal{Z}}}}(t),$ (31)

where in the last equality we used that
$\int_{0}^{t}\widetilde{\operatorname{sgn}}({\bm{\mathcal{Z}}}(s))d{\bm{\mathcal{L}}}^{{\bm{\mathcal{R}}}}(s)=\int_{0}^{t}\widetilde{\operatorname{sgn}}({\bm{\mathcal{R}}}(s))d{\bm{\mathcal{L}}}^{{\bm{\mathcal{R}}}}(s)=0$
because $\widetilde{\operatorname{sgn}}(0)=0$ and
${\bm{\mathcal{L}}}^{{\bm{\mathcal{R}}}}(s)$ increases only when
${\bm{\mathcal{R}}}(s)=0$. Again by Itô–Tanaka formula (Eq. 24), setting
$\gamma^{\prime}\coloneqq 1/(1-q)+1/q$, we have that

$|r({\bm{\mathcal{R}}}(t))|=\int_{0}^{t}\widetilde{\operatorname{sgn}}(r({\bm{\mathcal{R}}}(s)))r^{\prime}({\bm{\mathcal{R}}}(s))d{\bm{\mathcal{R}}}(s)+\frac{\gamma^{\prime}}{2}{\bm{\mathcal{L}}}^{{\bm{\mathcal{R}}}}(t)\\\
=\int_{0}^{t}\widetilde{\operatorname{sgn}}({\bm{\mathcal{Z}}}(s))(\operatorname{\mathds{1}}_{\\{{\bm{\mathcal{Z}}}(s)>0\\}}d{\bm{\mathcal{Y}}}(s)-\operatorname{\mathds{1}}_{\\{{\bm{\mathcal{Z}}}(s)\leq
0\\}}d{\bm{\mathcal{X}}}(s))+\frac{\gamma^{\prime}}{2}{\bm{\mathcal{L}}}^{{\bm{\mathcal{R}}}}(t),$
(32)

where in the last equality we used that
$\widetilde{\operatorname{sgn}}(r({\bm{\mathcal{R}}}(s)))=\widetilde{\operatorname{sgn}}({\bm{\mathcal{Z}}}(s))$
and Eq. 23. Comparing Eqs. 31 and 32, we obtain that
${\bm{\mathcal{L}}}^{{\bm{\mathcal{Z}}}}(t)=\frac{\gamma^{\prime}}{2}{\bm{\mathcal{L}}}^{{\bm{\mathcal{R}}}}(t)$.
Substituting the latter expression in Eq. 29 we can conclude that
${\bm{\mathcal{Z}}}$ is a strong solution to Eq. 22.

Assume now that the process ${\bm{\mathcal{Z}}}$ is a strong solution to Eq.
22. Recalling that $r(s(x))=s(r(x))=x$, we show that the process
${\bm{\mathcal{R}}}$, defined by
${\bm{\mathcal{R}}}(t)=s({\bm{\mathcal{Z}}}(t))$ is a strong solution to Eq.
23. By Itô–Tanaka formula (Eq. 24),

$d{\bm{\mathcal{R}}}(t)=s^{\prime}({\bm{\mathcal{Z}}}(t))d{\bm{\mathcal{Z}}}(t)+\frac{1-2q}{2}d{\bm{\mathcal{L}}}^{{\bm{\mathcal{Z}}}}(t).$
(33)

From Eq. 22, using similar arguments as before, we obtain that

$\displaystyle d{\bm{\mathcal{R}}}(t)$
$\displaystyle=s^{\prime}({\bm{\mathcal{Z}}}(t))(\operatorname{\mathds{1}}_{\\{{\bm{\mathcal{Z}}}(t)>0\\}}d{\bm{\mathcal{Y}}}(t)-\operatorname{\mathds{1}}_{\\{{\bm{\mathcal{Z}}}(t)\leq
0\\}}d{\bm{\mathcal{X}}}(t))+\left((2q-1)s^{\prime}(0)+\frac{1-2q}{2}\right)d{\bm{\mathcal{L}}}^{{\bm{\mathcal{Z}}}}(t)$
(34)
$\displaystyle=(1-q)\operatorname{\mathds{1}}_{\\{{\bm{\mathcal{R}}}(t)>0\\}}d{\bm{\mathcal{Y}}}(t)-\frac{1}{2}\operatorname{\mathds{1}}_{\\{{\bm{\mathcal{R}}}(t)=0\\}}d{\bm{\mathcal{X}}}(t)-q\operatorname{\mathds{1}}_{\\{{\bm{\mathcal{Z}}}(t)<0\\}}d{\bm{\mathcal{X}}}(t).$
(35)

The latter SDE is equivalent to the SDE in Eq. 23 because as before
$\int_{0}^{t}\operatorname{\mathds{1}}_{\\{{\bm{\mathcal{R}}}(s)=0\\}}d{\bm{\mathcal{X}}}(s)$
is identically zero. ∎

We now move to the proof of 2.3.

###### Proof of 2.3.

Our strategy to show that pathwise uniqueness holds for the SDE

$d{\bm{\mathcal{R}}}(t)=(1-q)\operatorname{\mathds{1}}_{\\{{\bm{\mathcal{R}}}(t)>0\\}}d{\bm{\mathcal{Y}}}(t)-q\operatorname{\mathds{1}}_{\\{{\bm{\mathcal{R}}}(t)\leq
0\\}}d{\bm{\mathcal{X}}}(t)$ (36)

is to apply [FIKP13, Theorem 8.1] that guarantees pathwise uniqueness in the
following setting: Let $f:\mathbb{R}\to\mathbb{R}$ be a function of finite
variation. Consider two continuous local martingales
$({\bm{\mathcal{M}}}(t))_{t\in\mathbb{R}_{\geq 0}}$ and
$({\bm{\mathcal{N}}}(t))_{t\in\mathbb{R}_{\geq 0}}$ started at zero, which
almost surely satisfy for some constant $c>0$,

$\langle{\bm{\mathcal{M}}},{\bm{\mathcal{N}}}\rangle(t)=0,\quad
d\langle{\bm{\mathcal{M}}}\rangle(t)\leq c\cdot
d\langle{\bm{\mathcal{N}}}\rangle(t),\qquad t\in\mathbb{R}_{\geq 0}.$ (37)

Then pathwise uniqueness holds for the SDE

$d{\bm{\mathcal{Z}}}(t)=f({\bm{\mathcal{Z}}}(t))d{\bm{\mathcal{M}}}(t)+d{\bm{\mathcal{N}}}(t).$

The rest of the proof is devoted to rewriting the SDE in Eq. 36 in the form
described above. We first construct the continuous local martingales
$({\bm{\mathcal{M}}}(t))_{t\in\mathbb{R}_{\geq 0}}$ and
$({\bm{\mathcal{N}}}(t))_{t\in\mathbb{R}_{\geq 0}}$. It is straightforward to
check that defining $x$ such that $\sin(2x)=\rho$, then the process

$\begin{pmatrix}\cos(x)&-\sin(x)\\\
-\sin(x)&\cos(x)\end{pmatrix}\begin{pmatrix}{\bm{\mathcal{X}}}(t)\\\
{\bm{\mathcal{Y}}}(t)\end{pmatrix}$

is a standard two-dimensional Brownian motion, in particular, it has
uncorrelated coordinates.

Now, setting171717The parameters $p,q,t,u$ are chosen so that the process
$({{\bm{\mathcal{M}}}}(t),\widetilde{{\bm{\mathcal{N}}}}(t))$ defined in Eq.
40 is a standard two-dimensional Brownian motion and Eq. 41 holds.
$\Theta=(1+2(q-1)q-2(-1+q)q\sin(2x))^{-1/2}$ and

$\displaystyle p$ $\displaystyle=(-q\cos(x)+(q-1)\sin(x))\Theta,\qquad
t=((q-1)\cos(x)-q\sin(x))\Theta,$ (38) $\displaystyle q$
$\displaystyle=(\cos(x)-q\cos(x)+q\sin(x))\Theta,\qquad
u=(-q\cos(x)+(q-1)\sin(x))\Theta,$ (39)

it is simple to check that the matrix $\begin{pmatrix}p&t\\\ q&u\end{pmatrix}$
is orthogonal. We finally define $A,B,C,D$ such that $\begin{pmatrix}A&B\\\
C&D\end{pmatrix}=\begin{pmatrix}p&t\\\
q&u\end{pmatrix}\begin{pmatrix}\cos(x)&-\sin(x)\\\
-\sin(x)&\cos(x)\end{pmatrix}$. With these definitions, the process

$({{\bm{\mathcal{M}}}}(t),\widetilde{{\bm{\mathcal{N}}}}(t))\coloneqq\begin{pmatrix}A&B\\\
C&D\end{pmatrix}\begin{pmatrix}{\bm{\mathcal{X}}}(t)\\\
{\bm{\mathcal{Y}}}(t)\end{pmatrix}$ (40)

is a standard two-dimensional Brownian motion. Indeed, it is a linear
isometric transformation of another standard two-dimensional Brownian motion.
Additionally, it holds that

$A(1-q)=Bq.$ (41)

Note that setting $S=(AD-BC)^{-1}$ we have that

${\bm{\mathcal{X}}}(t)=S(D{{\bm{\mathcal{M}}}}(t)-B\widetilde{{\bm{\mathcal{N}}}}(t))\quad\text{and}\quad{\bm{\mathcal{Y}}}(t)=S(A\widetilde{{\bm{\mathcal{N}}}}(t)-C{{\bm{\mathcal{M}}}}(t)).$
(42)

Therefore the SDE
$d{\bm{\mathcal{R}}}(t)=(1-q)\operatorname{\mathds{1}}_{\\{{\bm{\mathcal{R}}}(t)>0\\}}d{\bm{\mathcal{Y}}}(t)-q\operatorname{\mathds{1}}_{\\{{\bm{\mathcal{R}}}(t)\leq
0\\}}d{\bm{\mathcal{X}}}(t)$ can be written in the following equivalent form

$d{\bm{\mathcal{R}}}(t)=S(1-q)\operatorname{\mathds{1}}_{\\{{\bm{\mathcal{R}}}(t)>0\\}}(A\cdot
d\widetilde{{\bm{\mathcal{N}}}}(t)-C\cdot
d{{\bm{\mathcal{M}}}}(t))-Sq\operatorname{\mathds{1}}_{\\{{\bm{\mathcal{R}}}(t)\leq
0\\}}(D\cdot d{{\bm{\mathcal{M}}}}(t)-B\cdot
d\widetilde{{\bm{\mathcal{N}}}}(t))\\\
=S(-C(1-q)\operatorname{\mathds{1}}_{\\{{\bm{\mathcal{R}}}(t)>0\\}}-qD\operatorname{\mathds{1}}_{\\{{\bm{\mathcal{R}}}(t)\leq
0\\}})d{{\bm{\mathcal{M}}}}(t)+qSBd\widetilde{{\bm{\mathcal{N}}}}(t),$ (43)

where in the last equality we used Eq. 41. Note that we can write the latter
SDE as follows

$d{\bm{\mathcal{R}}}(t)=f({\bm{\mathcal{R}}}(t))d{{\bm{\mathcal{M}}}}(t)+d{{\bm{\mathcal{N}}}}(t),$
(44)

where
$f(x)=S(-C(1-q)\operatorname{\mathds{1}}_{\\{x>0\\}}-qD\operatorname{\mathds{1}}_{\\{x\leq
0\\}})$ is a bounded variation function and
${{\bm{\mathcal{N}}}}(t)=qSB\widetilde{{\bm{\mathcal{N}}}}(t)$. In addition,
the continuous martingales ${{\bm{\mathcal{M}}}}(t)$ and
${{\bm{\mathcal{N}}}}(t)$ satisfy all properties required to apply [FIKP13,
Theorem 8.1]. Therefore we can conclude that pathwise uniqueness holds for the
SDE in Eq. 36. ∎

We finally prove 2.4.

###### Proof of 2.4.

Let $({\bm{\mathcal{B}}}(t))_{t\in\mathbb{R}_{\geq 0}}$ be a standard one-
dimensional Brownian motion. Consider the SDE

$d{\bm{\mathcal{R}}}(t)=((1-q)\operatorname{\mathds{1}}_{\\{{\bm{\mathcal{R}}}(t)>0\\}}-q\operatorname{\mathds{1}}_{\\{{\bm{\mathcal{R}}}(t)\leq
0\\}})d{\bm{\mathcal{B}}}(t).$ (45)

Since we assumed $q\in(0,1)$, according to a result due to S. Nakao [Nak72]
(see also [LG83, LG84]), the latter SDE has a unique strong solution. Note
that

$d\langle{\bm{\mathcal{R}}}\rangle(t)=((1-q)^{2}\operatorname{\mathds{1}}_{\\{{\bm{\mathcal{R}}}(t)>0\\}}+q^{2}\operatorname{\mathds{1}}_{\\{{\bm{\mathcal{R}}}(t)\leq
0\\}})dt.$ (46)

We now consider two standard one-dimensional Brownian motions
$(\widetilde{{\bm{\mathcal{X}}}}(t))_{t\in\mathbb{R}_{\geq 0}},$
$(\widetilde{{\bm{\mathcal{Y}}}}(t))_{t\in\mathbb{R}_{\geq 0}}$ such that

$\langle{\bm{\mathcal{B}}},\widetilde{{\bm{\mathcal{X}}}}\rangle(t)=\langle{\bm{\mathcal{B}}},\widetilde{{\bm{\mathcal{Y}}}}\rangle(t)=\rho
t.$ (47)

This is possible (for instance) by considering two additional independent
Brownian motions ${\bm{\mathcal{B}}}_{1}$ and ${\bm{\mathcal{B}}}_{2}$, both
also independent of ${\bm{\mathcal{B}}}$, and setting
$\widetilde{{\bm{\mathcal{X}}}}=\sqrt{\rho}{\bm{\mathcal{B}}}+\sqrt{1-\rho}{\bm{\mathcal{B}}}_{1}$
and
$\widetilde{{\bm{\mathcal{Y}}}}=\sqrt{\rho}{\bm{\mathcal{B}}}+\sqrt{1-\rho}{\bm{\mathcal{B}}}_{2}$.
We define the following two processes

$\displaystyle\overline{{\bm{\mathcal{X}}}}(t)$
$\displaystyle\coloneqq\int_{0}^{t}\operatorname{\mathds{1}}_{\\{{\bm{\mathcal{R}}}(s)>0\\}}d\widetilde{{\bm{\mathcal{Y}}}}(s)-\frac{1}{q}\int_{0}^{t}\operatorname{\mathds{1}}_{\\{{\bm{\mathcal{R}}}(s)\leq
0\\}}d{{\bm{\mathcal{R}}}}(s),$ (48)
$\displaystyle\overline{{\bm{\mathcal{Y}}}}(t)$
$\displaystyle\coloneqq\frac{1}{1-q}\int_{0}^{t}\operatorname{\mathds{1}}_{\\{{\bm{\mathcal{R}}}(s)>0\\}}d{{\bm{\mathcal{R}}}}(s)-\int_{0}^{t}\operatorname{\mathds{1}}_{\\{{\bm{\mathcal{R}}}(s)\leq
0\\}}d\widetilde{{\bm{\mathcal{X}}}}(s).$ (49)

Note that

$(1-q)\operatorname{\mathds{1}}_{\\{{\bm{\mathcal{R}}}(t)>0\\}}d\overline{{\bm{\mathcal{Y}}}}(t)-q\operatorname{\mathds{1}}_{\\{{\bm{\mathcal{R}}}(t)\leq
0\\}}d\overline{{\bm{\mathcal{X}}}}(t)=d{\bm{\mathcal{R}}}(t),$ (50)

and so the triplet
$({\bm{\mathcal{R}}},\overline{{\bm{\mathcal{X}}}},\overline{{\bm{\mathcal{Y}}}})$
solves Eq. 23. To show that
$({\bm{\mathcal{R}}},\overline{{\bm{\mathcal{X}}}},\overline{{\bm{\mathcal{Y}}}})$
is a weak solution to Eq. 23, it remains to show that
$(\overline{{\bm{\mathcal{X}}}},\overline{{\bm{\mathcal{Y}}}})$ is a two-
dimensional Brownian motion with correlation $\rho$. By definition,
$\overline{{\bm{\mathcal{X}}}}$ and $\overline{{\bm{\mathcal{Y}}}}$ are
continuous local martingales. In addition, we have that

$\langle\overline{{\bm{\mathcal{X}}}}\rangle(t)=\int_{0}^{t}\operatorname{\mathds{1}}_{\\{{\bm{\mathcal{R}}}(s)>0\\}}ds+\frac{1}{q^{2}}\int_{0}^{t}\operatorname{\mathds{1}}_{\\{{\bm{\mathcal{R}}}(s)\leq
0\\}}d\langle{{\bm{\mathcal{R}}}}\rangle(s)=\int_{0}^{t}\operatorname{\mathds{1}}_{\\{{\bm{\mathcal{R}}}(s)>0\\}}ds+\int_{0}^{t}\operatorname{\mathds{1}}_{\\{{\bm{\mathcal{R}}}(s)\leq
0\\}}ds=t,$ (51)

where in the second inequality we used Eq. 46. Similarly,

$\langle\overline{{\bm{\mathcal{Y}}}}\rangle(t)=\tfrac{1}{(1-q)^{2}}\int_{0}^{t}\operatorname{\mathds{1}}_{\\{{\bm{\mathcal{R}}}(s)>0\\}}d\langle{{\bm{\mathcal{R}}}}\rangle(s)+\int_{0}^{t}\operatorname{\mathds{1}}_{\\{{\bm{\mathcal{R}}}(s)\leq
0\\}}ds=\int_{0}^{t}\operatorname{\mathds{1}}_{\\{{\bm{\mathcal{R}}}(s)>0\\}}ds+\int_{0}^{t}\operatorname{\mathds{1}}_{\\{{\bm{\mathcal{R}}}(s)\leq
0\\}}ds=t.$ (52)

Therefore, by Lévy’s characterization theorem ([KS91, Theorem 3.3.16]),
$\overline{{\bm{\mathcal{X}}}}$ and $\overline{{\bm{\mathcal{Y}}}}$ are
standard one-dimensional Brownian motions. It only remains to check that
$\overline{{\bm{\mathcal{X}}}}$ and $\overline{{\bm{\mathcal{Y}}}}$ have the
desired correlation. Note that

$\langle\overline{{\bm{\mathcal{X}}}},\overline{{\bm{\mathcal{Y}}}}\rangle(t)=\frac{1}{1-q}\int_{0}^{t}\operatorname{\mathds{1}}_{\\{{\bm{\mathcal{R}}}(s)>0\\}}d\langle\widetilde{{\bm{\mathcal{Y}}}},{{\bm{\mathcal{R}}}}\rangle(s)+\frac{1}{q}\int_{0}^{t}\operatorname{\mathds{1}}_{\\{{\bm{\mathcal{R}}}(s)\leq
0\\}}d\langle\widetilde{{\bm{\mathcal{X}}}},{{\bm{\mathcal{R}}}}\rangle(s).$
(53)

From Eq. 45 and Eq. 47, we have that

$d\langle\widetilde{{\bm{\mathcal{X}}}},{{\bm{\mathcal{R}}}}\rangle(s)=d\langle\widetilde{{\bm{\mathcal{Y}}}},{{\bm{\mathcal{R}}}}\rangle(s)=\rho\left((1-q)\operatorname{\mathds{1}}_{\\{{\bm{\mathcal{R}}}(s)>0\\}}ds+q\operatorname{\mathds{1}}_{\\{{\bm{\mathcal{R}}}(s)\leq
0\\}}ds\right),$ (54)

and substituting this expression in Eq. 53 we can conclude that
$\langle\overline{{\bm{\mathcal{X}}}},\overline{{\bm{\mathcal{Y}}}}\rangle(t)=\rho
t$, as desired.

The fact that $r({\bm{\mathcal{R}}}(t))_{t\in\mathbb{R}_{\geq 0}}$ is a skew
Brownian motion of parameter $q$ follows (for instance) using the same
arguments as in [Lej06, Section 5.2] and recalling that ${\bm{\mathcal{R}}}$
is a solution to Eq. 45. ∎

We complete the proof of 2.1, considering the case $\rho\in(-1,1)$ and $q=0$
(the case $q=1$ follows with similar arguments). We consider the functions
$g(x)=x\cdot\mathds{1}_{x\geq 0}$ and
$g^{\prime}(x)=\mathds{1}_{x>0}+\frac{1}{2}\cdot\operatorname{\mathds{1}}_{x=0}$.
We assume that ${\bm{\mathcal{Z}}}$ is a strong solution to Eq. 22 and we
define the process ${\bm{\mathcal{R}}}$ by
${\bm{\mathcal{R}}}(t)=g({\bm{\mathcal{Z}}}(t))$. By Itô–Tanaka formula (Eq.
24) and Eq. 22, we have that

$d{\bm{\mathcal{R}}}(t)=g^{\prime}({\bm{\mathcal{Z}}}(t))d{\bm{\mathcal{Z}}}(t)+\frac{1}{2}d{\bm{\mathcal{L}}}^{{\bm{\mathcal{Z}}}}(t)=\operatorname{\mathds{1}}_{\\{{\bm{\mathcal{R}}}(t)>0\\}}d{\bm{\mathcal{Y}}}(t)-\frac{1}{2}\operatorname{\mathds{1}}_{\\{{\bm{\mathcal{R}}}(t)=0\\}}d{\bm{\mathcal{X}}}(t).$
(55)

By definition the process ${\bm{\mathcal{R}}}$ is non-negative, continuous,
and started at zero. The last equation shows that ${\bm{\mathcal{R}}}$ is also
a martingale, and so ${\bm{\mathcal{R}}}$ is a.s. identically zero. This
implies that a.s. ${\bm{\mathcal{Z}}}(t)\leq 0$ for all $t\in\mathbb{R}_{\geq
0}$ and so ${\bm{\mathcal{Z}}}$ solves the SDE

$d{\bm{\mathcal{Z}}}(t)=-\operatorname{\mathds{1}}_{\\{{\bm{\mathcal{Z}}}(t)\leq
0\\}}d{\bm{\mathcal{X}}}(t)-d{\bm{\mathcal{L}}}^{{\bm{\mathcal{Z}}}}(t),\quad
t\in\mathbb{R}_{\geq 0}.$ (56)

From [HS81], we know that the latter SDE has a unique strong solution
${\bm{\mathcal{Z}}}(t)_{t\in\mathbb{R}_{\geq 0}}$ that is a skew Brownian
motion of parameter $0$. This completes the proof of 2.1.

####  2.1.2 The Brownian excursion case

Building on 2.1 it is straightforward to prove 1.7 using absolute continuity
arguments. We include the following (short) proof for making the article as
self-contained as we can, but we highlight that the arguments are similar to
the one used in [BM22, Theorem 4.6].

###### Proof of 1.7.

The strategy is to consider the solution mappings $F_{t}$ defined in 2.1 and
for all $r\in(u,1)$ to define the following measurable (with regards to
$\mathcal{F}_{r}^{(u)}$) process
${\bm{\mathcal{S}}}_{u,r}\in\mathcal{C}([u,r])$:

${\bm{\mathcal{S}}}_{u,r}(t)\coloneqq
F_{r-u}\Big{(}({\bm{\mathcal{E}}}_{\rho}({u+s})-{\bm{\mathcal{E}}}_{\rho}({u}))_{s\in[0,r-u]}\Big{)}(t-u),\quad
t\in[u,r].$ (57)

Since by A.1, the Brownian excursion
$(({\bm{\mathcal{E}}}_{\rho}({u+s})-{\bm{\mathcal{E}}}_{\rho}({u}))_{s\in[0,r-u]}$
is absolutely continuous with regards to a two-dimensional Brownian motion of
correlation $\rho$ on $[0,r-u]$, using the items 3-4 in 2.1 we have that

  1. 1.

${\bm{\mathcal{S}}}_{u,r}$ a.s. satisfies Eq. 8 on interval $[u,r]$;

  2. 2.

for $r,r^{\prime}\in(u,1)$, we have
${\bm{\mathcal{S}}}_{u,r}=({\bm{\mathcal{S}}}_{u,r^{\prime}})|_{[u,r]}$ almost
surely.

In addition, by construction, the map
$(u,r,\omega)\mapsto{\bm{\mathcal{S}}}_{u,r}$ is a measurable function. Now,
noting that the two statements above hold simultaneously for all
$r,r^{\prime}\in(u,1)\cap\mathbb{Q}$, we have that a.s. there exists a process
${\bm{\mathcal{Z}}}^{(u)}\in\mathcal{C}([u,1))$ such that
${\bm{\mathcal{Z}}}^{(u)}|_{[u,r]}={\bm{\mathcal{S}}}_{u,r}$ for every
$r\in\mathbb{(}u,1)\cap\mathbb{Q}$. Hence ${\bm{\mathcal{Z}}}^{(u)}$ is
$\mathcal{F}^{(u)}$-adapted and a.s. satisfies Eq. 8, proving the existence of
a strong solution.

For pathwise uniqueness, let
${\bm{\mathcal{Z}}}^{(u)},\widetilde{{\bm{\mathcal{Z}}}}^{(u)}$ be two
$\mathcal{F}^{(u)}$-adapted solutions to the SDE in Eq. 8 and $r<1$. There
exist two functionals
$G,\widetilde{G}:\mathcal{C}([u,r])\to\mathcal{C}([u,r])$ such that a.s.,

${\bm{\mathcal{Z}}}^{(u)}=G\left(({\bm{\mathcal{E}}}_{\rho}({s})-{\bm{\mathcal{E}}}_{\rho}({u}))_{s\in[u,r]}\right)\quad\text{
and
}\quad\widetilde{{\bm{\mathcal{Z}}}}^{(u)}=\widetilde{G}\left(({\bm{\mathcal{E}}}_{\rho}({s})-{\bm{\mathcal{E}}}_{\rho}({u}))_{s\in[u,r]}\right).$
(58)

Using again the absolute continuity (in the other direction) in A.1, for a
two-dimensional Brownian motion ${\bm{\mathcal{W}}}_{\rho}$ of correlation
$\rho$, the stochastic processes $G({\bm{\mathcal{W}}}_{\rho})$ and
$\widetilde{G}({\bm{\mathcal{W}}}_{\rho})$ are two solutions to the SDE in Eq.
22. Since by 2.1 pathwise uniqueness holds for Eq. 22,
$G({\bm{\mathcal{W}}}_{\rho})=\widetilde{G}({\bm{\mathcal{W}}}_{\rho})$ a.s.
Therefore, by absolute continuity,
${\bm{\mathcal{Z}}}^{(u)}=\widetilde{{\bm{\mathcal{Z}}}}^{(u)}$ a.s. This ends
the proof. ∎

###  2.2 The skew Tanaka equation

The goal of this section is to prove 1.9. Fix $q\in[0,1]$. We recall that we
want to construct solutions to the following SDEs, defined for all $u\in[0,1]$
by

$\begin{cases}d{\bm{\mathcal{Z}}}^{(u)}(t)=\operatorname{sgn}({\bm{\mathcal{Z}}}^{(u)}(t))d\bm{e}(t)+(2q-1)\cdot
d{\bm{\mathcal{L}}}^{{\bm{\mathcal{Z}}}^{(u)}}(t),&t\in[u,1),\\\
{\bm{\mathcal{Z}}}^{(u)}(t)=0,&t\in[0,u].\end{cases}$ (59)

where
$\operatorname{sgn}(x)\coloneqq\operatorname{\mathds{1}}_{\\{x>0\\}}-\operatorname{\mathds{1}}_{\\{x\leq
0\\}}$, $(\bm{e}(t))_{t\in[0,1]}$ is a one-dimensional Brownian excursion on
$[0,1]$ and ${\bm{\mathcal{L}}}^{{\bm{\mathcal{Z}}}^{(u)}}(t)$ is the
symmetric local time at zero of ${\bm{\mathcal{Z}}}^{(u)}$.

Eq. 59 when $q=1/2$ and $(\bm{e}(t))_{t\in[0,1]}$ is replaced by a standard
one-dimensional Brownian motion $({\bm{\mathcal{B}}}(t))_{t\in\mathbb{R}_{\geq
0}}$ is the well-known Tanaka’s SDE:

$\begin{cases}d{\bm{\mathcal{Z}}}^{(u)}(t)=\operatorname{sgn}({\bm{\mathcal{Z}}}^{(u)}(t))d{\bm{\mathcal{B}}}(t),&t\in\mathbb{R}_{>u},\\\
{\bm{\mathcal{Z}}}^{(u)}(t)=0,&t\in[0,u].\end{cases}$ (60)

The striking feature of this equation is the absence of pathwise uniqueness:
solutions cannot be measurable functions of the driving process
${\bm{\mathcal{B}}}$ and must also incorporate additional randomness (see for
instance [KS91, Example 5.3.5]).

Similarly, there is absence of pathwise uniqueness also for the following SDEs
(for a proof see for instance the discussion between Eq. (4) and Eq. (6) in
[Haj11]):

$\begin{cases}d{\bm{\mathcal{Z}}}^{(u)}_{q}(t)=\operatorname{sgn}({\bm{\mathcal{Z}}}_{q}^{(u)}(t))d{\bm{\mathcal{B}}}(t)+(2q-1)\cdot
d{\bm{\mathcal{L}}}^{{\bm{\mathcal{Z}}}_{q}^{(u)}}(t),&t\in\mathbb{R}_{>u},\\\
{\bm{\mathcal{Z}}}_{q}^{(u)}(t)=0,&t\in[0,u].\end{cases}$ (61)

This absence of pathwise uniqueness (both for the SDEs in Eq. 60 and Eq. 61)
raises many questions when we want to couple solutions to Eq. 60 (or Eq. 61)
for several starting times $u\in\mathbb{R}$. An elegant solution was developed
by Le Jan and Raimond ([LJR04, LJR20]) using the notion of stochastic flow of
maps. The same authors proved in [LJR06, Theorem 2.1] that there exists a
stochastic flow of maps solving Eq. 60 and explicitly constructed this flow
(which was also studied in [Wat00]). Later, Hajri ([Haj11, Theorem 2])
extended the ideas of Le Jan and Raimond and explicitly constructed another
stochastic flow of maps solving181818We remark that the construction given in
[Haj11, Theorem 2] is presented in a much more general setting. Specifically,
Hajri considered general flow evolving on graphs and some generalized SDEs
(mixing Tanaka’s SDE and the skew Brownian motion SDE). Furthermore, he gave a
discrete approximation of this flow in [Haj12]. We do not need to consider
this general setting in the present paper. We just mention that our specific
case corresponds (following the notation in [Haj11, Theorem 2]) to $N=p=2$,
$\alpha_{1}=q$ and $\alpha_{2}=1-q$ (so that $\alpha^{+}=1$). Eq. 61.

Using our notation, the construction of Hajri gives the following solution
$\left\\{{\bm{\mathcal{Z}}}^{(u)}_{q}(t)\right\\}_{u\in\mathbb{R}_{\geq 0}}$
to the SDEs in Eq. 61. (We suggest comparing the following explanation with
Fig. 5.)

Conditional on ${\bm{\mathcal{B}}}$, consider an i.i.d. sequence
$(\bm{s}(\ell))_{\ell}$, indexed by the local minima of ${\bm{\mathcal{B}}}$,
and with distribution
$\mathbb{P}(\bm{s}(\ell)=+1)=q=1-\mathbb{P}(\bm{s}(\ell)=-1)$.

For $u,t\in[0,1]$ with $u\leq t$, set
${\bm{m}}^{(u)}(t)\coloneqq\inf_{[u,t]}{\bm{\mathcal{B}}}$ and
${\bm{\varepsilon}}^{(u)}_{q}(t)\coloneqq\bm{s}\left(\sup\\{r\in[u,t]:{\bm{\mathcal{B}}}(r)={\bm{m}}^{(u)}(t)\\}\right)$.
Then the solutions
$\left\\{{\bm{\mathcal{Z}}}^{(u)}_{q}(t)\right\\}_{u\in\mathbb{R}_{\geq 0}}$
are defined as follows

$\begin{cases}{\bm{\mathcal{Z}}}_{q}^{(u)}(t)\coloneqq({\bm{\mathcal{B}}}(t)-{\bm{m}}^{(u)}(t)){\bm{\varepsilon}}_{q}^{(u)}(t),&t\in\mathbb{R}_{>u},\\\
{\bm{\mathcal{Z}}}_{q}^{(u)}(t)=0,&t\in[0,u].\end{cases}$ (62)

Figure 5: A schema for the processes introduced in Eq. 62.

######  Remark 2.5.

Note that the solutions
$\left\\{{\bm{\mathcal{Z}}}^{(u)}_{q}(t)\right\\}_{u\in\mathbb{R}_{\geq 0}}$
defined in Eq. 62 are constructed using _the same_ Brownian motion
${\bm{\mathcal{B}}}$ and _the same_ sequence of i.i.d. signs
$(\bm{s}(\ell))_{\ell}$. In particular, there is a coupling between different
solutions.

######  Remark 2.6.

For every fixed $u\in\mathbb{R}_{\geq 0}$, the process
${\bm{\mathcal{Z}}}^{(u)}_{q}$ defined in Eq. 62 is a skew Brownian motion of
parameter $q$ (and in particular is measurable) as shown in191919We remark
again that it is enough to consider the case $N=2$, $\alpha_{1}=q$ and
$\alpha_{2}=1-q$ in [Haj11, Proposition 1] for our purposes. [Haj11,
Proposition 1].

The fact that the processes
$\left\\{{\bm{\mathcal{Z}}}_{q}^{(u)}(t)\right\\}_{u\in\mathbb{R}_{\geq 0}}$
defined in Eq. 62 form a family of solutions to the SDEs in Eq. 61 is a
consequence of the more general result stated in [Haj11, Theorem 2], as
already mentioned above. Since here we do not need such generality, we include
a simple self-contained proof of this result.

######  Proposition 2.7.

The family
$\left\\{{\bm{\mathcal{Z}}}_{q}^{(u)}(t)\right\\}_{u\in\mathbb{R}_{\geq 0}}$
defined in Eq. 62 is a family of solutions to the SDEs in Eq. 61. Moreover,
for every $u\in\mathbb{R}_{\geq 0}$,
$({\bm{\mathcal{Z}}}_{q}^{(u)}(t))_{t\in\mathbb{R}_{\geq u}}$ is a skew
Brownian motion of parameter $q$ started at zero at time $u$.

###### Proof.

Fix $u\in\mathbb{R}_{\geq 0}$. Recall from 2.6 that
${\bm{\mathcal{Z}}}^{(u)}_{q}(t)$ is a skew Brownian motion of parameter $q$.
From [HS81], there exists a one-dimensional Brownian motion
$({\bm{\mathcal{W}}}(t))_{t\in\mathbb{R}_{\geq u}}$ started at zero at time
$u$ such that

${\bm{\mathcal{Z}}}_{q}^{(u)}(t)={\bm{\mathcal{W}}}(t)+(2q-1){\bm{\mathcal{L}}}^{{\bm{\mathcal{Z}}}_{q}^{(u)}}(t),\qquad
t\in\mathbb{R}_{\geq u}.$ (63)

From Itô–Tanaka formula (Eq. 24), we have that

$\displaystyle\left|{\bm{\mathcal{Z}}}_{q}^{(u)}(t)\right|$
$\displaystyle=\int_{u}^{t}\left(\operatorname{\mathds{1}}_{\\{{\bm{\mathcal{Z}}}_{q}^{(u)}(s)>0\\}}-\operatorname{\mathds{1}}_{\\{{\bm{\mathcal{Z}}}_{q}^{(u)}(s)<0\\}}\right)d{\bm{\mathcal{Z}}}_{q}^{(u)}(s)+{\bm{\mathcal{L}}}^{{\bm{\mathcal{Z}}}_{q}^{(u)}}(t)$
(64)
$\displaystyle=\int_{u}^{t}\left(\operatorname{\mathds{1}}_{\\{{\bm{\mathcal{Z}}}_{q}^{(u)}(s)>0\\}}-\operatorname{\mathds{1}}_{\\{{\bm{\mathcal{Z}}}_{q}^{(u)}(s)<0\\}}\right)d{\bm{\mathcal{W}}}(s)+{\bm{\mathcal{L}}}^{{\bm{\mathcal{Z}}}_{q}^{(u)}}(t),$
(65)

where in the last equality we used Eq. 63 and the fact that

$\int_{u}^{t}\left(\operatorname{\mathds{1}}_{\\{{\bm{\mathcal{Z}}}_{q}^{(u)}(s)>0\\}}-\operatorname{\mathds{1}}_{\\{{\bm{\mathcal{Z}}}_{q}^{(u)}(s)<0\\}}\right)d{\bm{\mathcal{L}}}^{{\bm{\mathcal{Z}}}_{q}^{(u)}}(s)=0,$

because ${\bm{\mathcal{L}}}^{{\bm{\mathcal{Z}}}_{q}^{(u)}}(s)$ increases only
if ${\bm{\mathcal{Z}}}_{q}^{(u)}(s)=0$. Noting that
$\int_{u}^{t}\operatorname{\mathds{1}}_{\\{{\bm{\mathcal{Z}}}_{q}^{(u)}(s)=0\\}}d{\bm{\mathcal{W}}}(s)=0$
(this follows for instance using the same arguments as in Eq. 30), we obtain
that

$\left|{\bm{\mathcal{Z}}}_{q}^{(u)}(t)\right|=\int_{u}^{t}\operatorname{sgn}({\bm{\mathcal{Z}}}_{q}^{(u)}(s))d{\bm{\mathcal{W}}}(s)+{\bm{\mathcal{L}}}^{{\bm{\mathcal{Z}}}_{q}^{(u)}}(t).$
(66)

Now by construction (see Eq. 62) we also have that

$\left|{\bm{\mathcal{Z}}}_{q}^{(u)}(t)\right|={\bm{\mathcal{B}}}(t)-{\bm{m}}^{(u)}(t).$
(67)

Recall that every continuous semimartingale can be uniquely decomposed into a
continuous local martingale and a continuous finite variation process started
at zero. Therefore, comparing Eqs. 66 and 67, and noting that
${\bm{\mathcal{L}}}^{{\bm{\mathcal{Z}}}_{q}^{(u)}}(t)$ and
$-{\bm{m}}^{(u)}(t)$ are both increasing (and so with finite variation)
continuous processes started at zero, we obtain that

${\bm{\mathcal{L}}}^{{\bm{\mathcal{Z}}}_{q}^{(u)}}(t)=-{\bm{m}}^{(u)}(t)\qquad\text{and}\qquad{\bm{\mathcal{B}}}(t)=\int_{u}^{t}\operatorname{sgn}({\bm{\mathcal{Z}}}_{q}^{(u)}(s))d{\bm{\mathcal{W}}}(s).$
(68)

Using the equality on the right-hand side of the last equation, we get

$\int_{u}^{t}\operatorname{sgn}({\bm{\mathcal{Z}}}_{q}^{(u)}(s))d{\bm{\mathcal{B}}}(s)=\int_{u}^{t}\operatorname{sgn}({\bm{\mathcal{Z}}}_{q}^{(u)}(s))^{2}d{\bm{\mathcal{W}}}(s)=\int_{u}^{t}d{\bm{\mathcal{W}}}(s)={\bm{\mathcal{W}}}(t),$
(69)

where in the last equality we used that
$({\bm{\mathcal{W}}}(t))_{t\in\mathbb{R}_{\geq u}}$ is a Brownian motion
started at zero at time $u$. Substituting the last expression in Eq. 63, we
conclude that ${\bm{\mathcal{Z}}}_{q}^{(u)}(t)$ is a solution to the SDE in
Eq. 61. ∎

The result stated in 1.9 follows from 2.7 using standard absolute continuity
arguments between one-dimensional Brownian excursions and one-dimensional
Brownian motions. Since these arguments are very similar to the ones used for
the proof of 1.7 in Section 2.1.2, we skip the details here.

We conclude this section with a quick discussion on uniqueness of the
solutions to Eq. 61 given in 2.7.

######  Remark 2.8.

In the present paper, we decided to avoid using the formalism of stochastic
flow of maps and so we did not include any claim in 2.7 related to uniqueness
of the solutions to Eq. 61 defined in Eq. 62 (and we made the same choice for
1.9 in the introduction). Nevertheless, in [Haj11, Theorem 2] it is also
showed that the stochastic flow of maps corresponding to the processes
$\left\\{{\bm{\mathcal{Z}}}_{q}^{(u)}(t)\right\\}_{u\in\mathbb{R}_{\geq 0}}$
defined in Eq. 62 is _unique_ , that is, the unique flow adapted to the
filtration generated by the driving Brownian motion. We also remark that the
proof of this uniqueness result has been later simplified in [Haj15].

##  3 The skew Brownian permuton

In the previous section we investigated solutions to the skew perturbed Tanaka
equations driven by a two-dimensional correlated Brownian excursion. Building
on these results, we can now deduce that the skew Brownian permuton is well-
defined, proving 1.11. Later, we will also show that the biased Brownian
separable permuton is a particular case of the skew Brownian permuton, proving
1.12. These are the two goals of the next two sections.

###  3.1 The skew Brownian permuton is well-defined

We prove here 1.11, showing that the skew Brownian permuton
$\bm{\mu}_{\rho,q}$ is well-defined for all $(\rho,q)\in(-1,1]\times[0,1]$.
Before doing that we also give a slightly different (but equivalent)
definition of the skew Brownian permuton that will be also helpful later to
prove 1.12.

We fix $(\rho,q)\in(-1,1]\times[0,1]$ and consider the continuous coalescent-
walk process202020Actually ${\bm{\mathcal{Z}}}_{\rho,q}^{(u)}$ was not defined
for $u\in\\{0,1\\}$ (see 1.1). As what happens on a negligible subset of
$[0,1]$ is irrelevant to the arguments to come, this causes no problems.
${\bm{\mathcal{Z}}}_{\rho,q}=\left\\{{\bm{\mathcal{Z}}}^{(u)}_{\rho,q}\right\\}_{u\in[0,1]}$.
We first define a random binary relation
$\preccurlyeq_{{\bm{\mathcal{Z}}}_{\rho,q}}$ on $[0,1]^{2}$ as follows:

$\begin{cases}t\preccurlyeq_{{\bm{\mathcal{Z}}}_{\rho,q}}t&\text{ for all
}t\in[0,1],\\\ t\preccurlyeq_{{\bm{\mathcal{Z}}}_{\rho,q}}s&\text{ for all
}0\leq t<s\leq 1\text{ s.t.\ }\ {\bm{\mathcal{Z}}}_{\rho,q}^{(t)}(s)<0,\\\
s\preccurlyeq_{{\bm{\mathcal{Z}}}_{\rho,q}}t,&\text{ for all }0\leq t<s\leq
1\text{ s.t.\ }\ {{\bm{\mathcal{Z}}}}_{\rho,q}^{(t)}(s)\geq 0.\end{cases}$
(70)

We note that the stochastic process $\varphi_{{\bm{\mathcal{Z}}}_{\rho,q}}(t)$
introduce in Eq. 9 satisfies

$\varphi_{{\bm{\mathcal{Z}}}_{\rho,q}}(t)=\operatorname{Leb}\left(\big{\\{}x\in[0,1]|x\preccurlyeq_{{\bm{\mathcal{Z}}}_{\rho,q}}t\big{\\}}\right),\qquad
t\in[0,1],$ (71)

and we recall that the skew Brownian permuton $\bm{\mu}_{\rho,q}$ is defined
by

$\bm{\mu}_{\rho,q}(\cdot)\coloneqq(\operatorname{Id},\varphi_{{\bm{\mathcal{Z}}}_{\rho,q}})_{*}\operatorname{Leb}(\cdot)=\operatorname{Leb}\left(\\{t\in[0,1]|(t,\varphi_{{\bm{\mathcal{Z}}}_{\rho,q}}(t))\in\cdot\,\\}\right).$
(72)

We restrict for the moment to the case $(\rho,q)\in(-1,1)\times[0,1]$ (the
case $\rho=1$ will be treated separately in Section 3.2). Using pathwise
uniqueness (Item 3 in 1.7) for the SDEs defining the continuous coalescent-
walk process ${\bm{\mathcal{Z}}}_{\rho,q}$, it is simple to obtain the
following result (for a proof see [BM22, Proposition 5.2]).

######  Proposition 3.1.

Fix $(\rho,q)\in(-1,1)\times[0,1]$. There exists a random set
$\bm{A}\subset[0,1]^{2}$ of a.s. zero Lebesgue measure, that is,
$\mathbb{P}(\operatorname{Leb}(\bm{A})=0)=1$, such that the restriction of the
relation $\preccurlyeq_{{\bm{\mathcal{Z}}}_{\rho,q}}$ to
$[0,1]^{2}\setminus\bm{A}$ is a.s. a total order.

We can now prove 1.11 (using similar ideas as in [BM22, Lemma 5.5]).

###### Proof of 1.11 for $\rho\neq 1$.

We start by proving that for $t,s\in(0,1)$ with $t<s$,
${\bm{\mathcal{Z}}}_{\rho,q}^{(t)}(s)\neq 0$ almost surely. Let
$\varepsilon>0$ be such that $s<1-\varepsilon<1$. Thanks to 2.1,
$({\bm{\mathcal{Z}}}_{\rho,q}^{(t)}(t+r))_{r\in[0,1-t-\varepsilon]}$ is
absolutely continuous with regards to a skew Brownian motion of parameter $q$
on $[0,1-t-\varepsilon]$. Since the time spent at zero by a skew Brownian
motion a.s. has zero Lebesgue measure, we can conclude that
${\bm{\mathcal{Z}}}_{\rho,q}^{(t)}(s)\neq 0$ a.s.

We can now prove that $\bm{\mu}_{\rho,q}$ is a permuton. By definition,
$\bm{\mu}_{\rho,q}$ is a random probability measure on the unit square, and
its first marginal is a.s. uniform (see Eq. 72). Therefore, it is enough to
verify that also its second marginal is a.s. uniform, i.e. that
$(\varphi_{{\bm{\mathcal{Z}}}_{\rho,q}})_{*}\operatorname{Leb}=\operatorname{Leb}$
a.s.

Let $(\bm{U}_{i})_{i\in\mathbb{Z}_{>0}}$ be a sequence of i.i.d. uniform
random variables on $[0,1]$. We define for $k\in\mathbb{Z}_{\geq 2}$,

$\bm{U}_{1,k}\coloneqq\tfrac{1}{k-1}{\\#\left\\{i\in\llbracket
2,k\rrbracket\middle|\bm{U}_{i}\preccurlyeq_{{\bm{\mathcal{Z}}}_{\rho,q}}\bm{U}_{1}\right\\}}=\tfrac{1}{k-1}\sum_{i\in\llbracket
2,k\rrbracket}\mathds{1}_{\left\\{\bm{U}_{i}\preccurlyeq_{{\bm{\mathcal{Z}}}_{\rho,q}}\bm{U}_{1}\right\\}}.$
(73)

The random variables
$\left(\mathds{1}_{\left\\{\bm{U}_{i}\preccurlyeq_{{\bm{\mathcal{Z}}}_{\rho,q}}\bm{U}_{1}\right\\}}\right)_{i\in\mathbb{Z}_{\geq
2}}$ are i.i.d. Bernoulli random variables of parameter
$\varphi_{{\bm{\mathcal{Z}}}_{\rho,q}}(\bm{U}_{1})$, conditionally on
$({\bm{\mathcal{E}}}_{\rho},\bm{U}_{1})$. Therefore, from the law of large
numbers, $\bm{U}_{1,k}$ a.s. converges to
$\varphi_{{\bm{\mathcal{Z}}}_{\rho,q}}(\bm{U}_{1})$ as $k$ tends to infinity.

On the other hand, by the exchangeability of the random variables
$(\bm{U}_{i})_{i\in\mathbb{Z}_{>0}}$, and using that
${\bm{\mathcal{Z}}}_{\rho,q}^{(t)}(s)\neq 0$ a.s., the random variable
$\bm{U}_{1,k}$ is uniform in
$\big{\\{}\frac{0}{k-1},\dots,\frac{k-1}{k-1}\big{\\}},$ conditionally on
${\bm{\mathcal{E}}}_{\rho}$. Therefore, $\bm{U}_{1,k}$ converges in
distribution to a uniform random variable on $[0,1]$ as $k$ tends to infinity.
Thus we can conclude that $\varphi_{{\bm{\mathcal{Z}}}_{\rho,q}}(\bm{U}_{1})$
is uniform on $[0,1]$ conditionally on ${\bm{\mathcal{E}}}_{\rho}$. This
concludes the proof. ∎

We state and prove a final lemma useful for later purposes.

######  Lemma 3.2.

Fix $(\rho,q)\in(-1,1)\times[0,1]$. Almost surely, for almost every $0\leq
t<s\leq 1$, either ${\bm{\mathcal{Z}}}_{\rho,q}^{(t)}(s)>0$ and
$\varphi_{{\bm{\mathcal{Z}}}_{\rho,q}}(s)<\varphi_{{\bm{\mathcal{Z}}}_{\rho,q}}(t)$,
or ${\bm{\mathcal{Z}}}_{\rho,q}^{(t)}(s)<0$ and
$\varphi_{{\bm{\mathcal{Z}}}_{\rho,q}}(s)>\varphi_{{\bm{\mathcal{Z}}}_{\rho,q}}(t)$.

###### Proof.

Consider two independent uniform random variables $\bm{U}$ and $\bm{V}$ on
$[0,1]$, also independent of ${\bm{\mathcal{E}}}_{\rho}$. It is immediate from
3.1 that if $\bm{U}<\bm{V}$ and
${\bm{\mathcal{Z}}}_{\rho,q}^{(\bm{U})}(\bm{V})>0$ then
$\varphi_{{\bm{\mathcal{Z}}}_{\rho,q}}(\bm{U})\geq\varphi_{{\bm{\mathcal{Z}}}_{\rho,q}}(\bm{V})$
a.s. Similarly, if $\bm{U}<\bm{V}$ and
${\bm{\mathcal{Z}}}_{\rho,q}^{(\bm{U})}(\bm{V})<0$ then
$\varphi_{{\bm{\mathcal{Z}}}_{\rho,q}}(\bm{U})\leq\varphi_{{\bm{\mathcal{Z}}}_{\rho,q}}(\bm{V})$
a.s. The case ${\bm{\mathcal{Z}}}_{\rho,q}^{(\bm{U})}(\bm{V})=0$ is a.s.
excluded by the first part of the previous proof, and the case
$\varphi_{{\bm{\mathcal{Z}}}_{\rho,q}}(\bm{U})=\varphi_{{\bm{\mathcal{Z}}}_{\rho,q}}(\bm{V})$
is a.s. excluded by the fact that
$\varphi_{{\bm{\mathcal{Z}}}_{\rho,q}}(\bm{U})$ and
$\varphi_{{\bm{\mathcal{Z}}}_{\rho,q}}(\bm{V})$ are two independent uniform
random variables, thanks to the second part of the previous proof. This is
enough to complete the proof of the lemma. ∎

###  3.2 The biased Brownian separable permuton is a particular case of the
skew Brownian permuton

Recalling the construction of the biased Brownian separable permuton from
Section 1.2 and the construction of the skew Brownian permuton presented in
Section 3.1, to prove 1.12 (and also to complete the proof of 1.11 when
$\rho=1$) it is enough to prove the following result.

######  Proposition 3.3.

Fix a one-dimensional Brownian excursion $(\bm{e}(t))_{t\in[0,1]}$ on $[0,1]$
and a parameter $p\in[0,1]$. Conditional on $\bm{e}$, consider an i.i.d.
sequence $(\bm{s}(\ell))_{\ell}\in\\{+1,-1\\}^{\mathbb{Z}_{>0}}$ indexed by
the local minima of $\bm{e}$ and with distribution
$\mathbb{P}(\bm{s}(\ell)=+1)=p=1-\mathbb{P}(\bm{s}(\ell)=-1)$. Let
$\vartriangleleft_{\widetilde{\bm{e}},p}$ be the relation defined in Eq. 3
constructed from the pair $(\bm{e}(t),(\bm{s}(\ell))_{\ell})$ and
$\preccurlyeq_{{\bm{\mathcal{Z}}}_{1,1-p}}$ be the relation defined in Eq. 70,
p. 70, constructed from the pair $(\bm{e}(t),(-\bm{s}(\ell))_{\ell})$ (note
that the two relations $\vartriangleleft_{\widetilde{\bm{e}},p}$ and
$\preccurlyeq_{{\bm{\mathcal{Z}}}_{1,1-p}}$ now have a specific coupling).

Then there exists a random set $\bm{A}\subset[0,1]^{2}$ of a.s. zero Lebesgue
measure, i.e. $\mathbb{P}(\operatorname{Leb}(\bm{A})=0)=1$, such that the
relations $\vartriangleleft_{\widetilde{\bm{e}},p}$ and
$\preccurlyeq_{{\bm{\mathcal{Z}}}_{1,1-p}}$ restricted to
$[0,1]^{2}\setminus\bm{A}$ are the same total order.

###### Proof.

Recall from Section 1.2 that there exists a random set
$\bm{A}\subset[0,1]^{2}$ of a.s. zero Lebesgue measure such that for every
$x,y\in[0,1]^{2}\setminus\bm{A}$ with $x<y$ then $\min_{[x,y]}\bm{e}$ is
reached at a unique point which is a strict local minimum. In addition,
$\vartriangleleft_{\widetilde{\bm{e}},p}$ is a random total order on
$[0,1]^{2}\setminus\bm{A}$.

Fix now $x,y\in[0,1]^{2}\setminus\bm{A}$ with $x<y$ and assume that
$\bm{s}(\ell)$ is the sign corresponding to the unique local minimum where
$\min_{[x,y]}\bm{e}$ is reached. By definition (see Eq. 3, p. 3),

$\begin{cases}x\vartriangleleft_{\widetilde{\bm{e}},p}y,&\quad\text{if}\quad\bm{s}(\ell)=+1,\\\
y\vartriangleleft_{\widetilde{\bm{e}},p}x,&\quad\text{if}\quad\bm{s}(\ell)=-1.\\\
\end{cases}$ (74)

Now note that since the excursion $\bm{e}$ has a unique local minimum on the
interval $[x,y]$, say at time $t$, then by construction (see Eq. 13) the
process ${\bm{\mathcal{Z}}}_{1,1-p}^{(x)}$ is positive at time $y$ if
$-s(\ell)=+1$ and negative if $-s(\ell)=-1$, therefore by definition (see Eq.
70, p. 70),

$\begin{cases}x\preccurlyeq_{{\bm{\mathcal{Z}}}_{1,1-p}}y,&\quad\text{if}\quad
s(\ell)=+1,\\\
y\preccurlyeq_{{\bm{\mathcal{Z}}}_{1,1-p}}x,&\quad\text{if}\quad
s(\ell)=-1.\\\ \end{cases}$ (75)

Comparing Eqs. 74 and 75 we can conclude that
$\vartriangleleft_{\widetilde{\bm{e}},p}$ and
$\preccurlyeq_{{\bm{\mathcal{Z}}}_{1,1-p}}$ restricted to
$[0,1]^{2}\setminus\bm{A}$ are the same total order. ∎

##  4 The skew Brownian permuton and SLE-decorated Liouville quantum gravity
spheres

This section is devoted to establishing the connection between the skew
Brownian permuton and SLE-decorated LQG spheres described in 1.17.

Recall that $({\mathbb{C}}\cup\\{\infty\\},\bm{h},\infty)$ and
$(\bm{\eta}_{0},\bm{\eta}_{\theta})$ denote the $\gamma$-LQG sphere and the
pair of space-filling SLE${}_{\kappa^{\prime}}$ introduced in Section 1.5.
1.17 immediately follows from the result in 4.1 below. To state this
proposition we need another construction.

For $t\in[0,1]$, let ${\bm{\mathcal{X}}}_{\rho}(t)$ be the $\nu_{\bm{h}}$-LQG
length measure (see [GHS19, Section 3.3]) of the left outer boundary of
$\bm{\eta}_{0}([0,t])$ and ${\bm{\mathcal{Y}}}_{\rho}(t)$ be the
$\nu_{\bm{h}}$-LQG length measure of the right outer boundary of
$\bm{\eta}_{0}([0,t])$. From [MS19, Theorem 1.1] (see also [GHS19, Theorem
4.10]) the process
${\bm{\mathcal{E}}}_{\rho}(t)=({\bm{\mathcal{X}}}_{\rho}(t),{\bm{\mathcal{Y}}}_{\rho}(t))$
defined above has (up to time reparametrization) the law of a two-dimensional
Brownian excursion of correlation $\rho$ in the non-negative quadrant. In
addition, the process $({\bm{\mathcal{E}}}_{\rho}(t))_{t\in[0,1]}$ a.s.
determines $(({\mathbb{C}}\cup\\{\infty\\},\bm{h},\infty),\bm{\eta}_{0})$ as a
curve-decorated quantum surface.

######  Proposition 4.1.

Fix $\gamma\in(0,2)$ and $\theta\in[0,\pi]$. Recall that
$\rho=-\cos(\pi\gamma^{2}/4)$. Let
${\bm{\mathcal{E}}}_{\rho}(t)=({\bm{\mathcal{X}}}(t),{\bm{\mathcal{Y}}}(t))$
be the two-dimensional Brownian excursion of correlation $\rho$ defined above
from the curve-decorated quantum surface
$(({\mathbb{C}}\cup\\{\infty\\},\bm{h},\infty),\bm{\eta}_{0})$. Recall that,
for $t\in[0,1]$, $\bm{\psi}_{\gamma,\theta}(t)\in[0,1]$ denotes the first time
when $\bm{\eta}_{\theta}$ hits the point $\bm{\eta}_{0}(t)$.

Let
${\bm{\mathcal{Z}}}_{\rho,q}=\left\\{{\bm{\mathcal{Z}}}_{\rho,q}^{(u)}\right\\}_{u\in\mathbb{R}}$
be the continuous coalescent-walk process driven by
$({\bm{\mathcal{E}}}_{\rho},q)$ and $\varphi_{{\bm{\mathcal{Z}}}_{\rho,q}}$ be
the associated stochastic process defined in Eq. 9. There exist a constant
$\overline{q}=\overline{q}_{\gamma}(\theta)\in[0,1]$ such that a.s. for almost
all $t\in(0,1)$

$\bm{\psi}_{\gamma,\theta}(t)=\varphi_{{\bm{\mathcal{Z}}}_{\rho,\overline{q}}}(t).$
(76)

The proof of 4.1 builds on the next lemma (see 4.2 below) whose proof is
postponed to the end of the section. We need some additional notation.

For $t\in[0,1]$, let $\hat{\bm{\phi}}_{\theta}^{(t)}$ denote the flow line of
the vector field $e^{i(\hat{\bm{h}}/\chi+\theta)}$ started at
$\bm{\eta}_{0}(t)$. We point out that there is a unique flow line of the
vector field $e^{i(\hat{\bm{h}}/\chi+\theta)}$ started from $\bm{\eta}_{0}(t)$
for almost all $t\in[0,1]$ and this is enough for our purposes.212121In
particular, there will be two possible choices of the flow line of the vector
field $e^{i(\hat{\bm{h}}/\chi+\theta)}$ started at $\bm{\eta}_{0}(t)$ if
$\bm{\eta}_{0}(t)$ is a double point of $\bm{\eta}_{\theta}$. Finally, we
denote by $\bm{\phi}_{\Theta}^{(t)}$ the union of the flow lines
$\hat{\bm{\phi}}_{\theta}^{(t)}$ and $\hat{\bm{\phi}}_{\theta+\pi}^{(t)}$
(followed in the same direction of $\hat{\bm{\phi}}_{\theta}^{(t)}$). In Fig.
6 we show the various curves that we are considering and we explain what we
mean when we refer to _left_ and _right_ in 4.2.

Figure 6: A chart for the various curves considered in this section. In
particular, the region covered by $\bm{\eta}_{0}([0,t])$ is highlited in green
and the region covered by
$\bm{\eta}_{\theta}([0,\bm{\psi}_{\gamma,\theta}(t)])$ is highlited in dashed
red. By construction, the boundaries of
$\bm{\eta}_{\theta}([0,\bm{\psi}_{\gamma,\theta}(t)])$ are given by
$\hat{\bm{\phi}}_{\theta}^{(t)}$ and $\hat{\bm{\phi}}_{\theta+\pi}^{(t)}$.
_Left_ and _right_ in 4.2 are defined with respect to the flow line
$\bm{\phi}_{\theta}^{(t)}=\hat{\bm{\phi}}_{\theta}^{(t)}\cup\hat{\bm{\phi}}_{\theta+\pi}^{(t)}$
followed in the direction of $\hat{\bm{\phi}}_{\theta}^{(t)}$.

######  Lemma 4.2.

There exists a constant $\overline{q}=\overline{q}_{\gamma}(\theta)\in[0,1]$
such that a.s. for almost all $t\in(0,1)$

$\displaystyle\left\\{x\in[t,1):{\bm{\mathcal{Z}}}_{\rho,\overline{q}}^{(t)}(x)\geq
0\right\\}$ $\displaystyle=\left\\{x\in[t,1):\bm{\eta}_{0}(x)\text{ is weakly
on the left of }{\bm{\phi}}_{\theta}^{(t)}\right\\},$ (77)
$\displaystyle\left\\{x\in[t,1):{\bm{\mathcal{Z}}}_{\rho,\overline{q}}^{(t)}(x)\leq
0\right\\}$ $\displaystyle=\left\\{x\in[t,1):\bm{\eta}_{0}(x)\text{ is weakly
on the right of }{\bm{\phi}}_{\theta}^{(t)}\right\\}.$ (78)

We can now prove 4.1 using 4.2.

###### Proof of 4.1.

Recall that $\bm{\psi}_{\gamma,\theta}(t)\in[0,1]$ denotes the first time when
$\bm{\eta}_{\theta}$ hits the point $\bm{\eta}_{0}(t)$. Fix $t\in(0,1)$ such
that the relations in 4.2 hold (this is true for almost all $t\in(0,1)$). We
want to show that a.s.

$\bm{\psi}_{\gamma,\theta}(t)=\operatorname{Leb}\left(\big{\\{}x\in[0,t)|{\bm{\mathcal{Z}}}_{\rho,q}^{(x)}(t)<0\big{\\}}\cup\big{\\{}x\in[t,1]|{\bm{\mathcal{Z}}}_{\rho,q}^{(t)}(x)\geq
0\big{\\}}\right).$

By definition and time parametrization of $\bm{\eta}_{0}$ and
$\bm{\eta}_{\theta}$, it holds that

$\bm{\psi}_{\gamma,\theta}(t)=\operatorname{Leb}\left(\left\\{x\in[0,1]:\bm{\eta}_{0}(x)\text{
is weakly on the left of }{\bm{\phi}}_{\theta}^{(t)}\right\\}\right).$ (79)

Noting that for $x\in[0,t)$ the point $\bm{\eta}_{0}(x)$ is weakly on the left
of $\bm{\phi}_{\theta}^{(t)}$ if and only if $\bm{\eta}_{0}(t)$ is weakly on
the right of ${\bm{\phi}}_{\theta}^{(x)}$, we obtain that a.s.

$\displaystyle\bm{\psi}_{\gamma,\theta}(t)=$
$\displaystyle\operatorname{Leb}\left(\left\\{x\in[t,1]:\bm{\eta}_{0}(x)\text{
is weakly on the left of }{\bm{\phi}}_{\theta}^{(t)}\right\\}\right)$ (80)
$\displaystyle+$
$\displaystyle\operatorname{Leb}\left(\left\\{x\in[0,t):\bm{\eta}_{0}(t)\text{
is weakly on the right of }\bm{\phi}_{\theta}^{(x)}\right\\}\right)$ (81)
$\displaystyle=$
$\displaystyle\operatorname{Leb}\left(\left\\{x\in[t,1]:{\bm{\mathcal{Z}}}_{\rho,\overline{q}}^{(t)}(x)\geq
0\right\\}\right)+\operatorname{Leb}\left(\left\\{x\in[0,t):{\bm{\mathcal{Z}}}_{\rho,\overline{q}}^{(x)}(t)\leq
0\right\\}\right),$ (82)

where in the last equality we used 4.2. Finally, from 3.2 it holds that

$\operatorname{Leb}\left(\left\\{x\in[0,t):{\bm{\mathcal{Z}}}_{\rho,\overline{q}}^{(x)}(t)=0\right\\}\right)=0$

and so we can conclude the proof. ∎

It remains to prove 4.2. We first state and prove a result similar to 4.2 (see
4.3 below) when the pair of space-filling SLEs
$(\bm{\eta}_{0},\bm{\eta}_{\theta})$ is parameterized by an LQG cone instead
of an LQG sphere.

Let $(\mathbb{C},\widetilde{\bm{h}},0,\infty)$ be a $\gamma$-quantum cone
([GHS19, Definition 3.10]) independent of $\hat{\bm{h}}$. We denote by
$(\widetilde{\bm{\eta}}_{0},\widetilde{\bm{\eta}}_{\theta})$ the pair
$(\bm{\eta}_{0},\bm{\eta}_{\theta})$ parametrize by the
$\mu_{\widetilde{\bm{h}}}$-LQG area measure so that
$\widetilde{\bm{\eta}}_{0}(0)=\widetilde{\bm{\eta}}_{\theta}(0)=0$ and
$\mu_{\widetilde{\bm{h}}}(\widetilde{\bm{\eta}}_{0}([s,t]))=\mu_{\widetilde{\bm{h}}}(\widetilde{\bm{\eta}}_{\theta}([s,t]))=t-s$,
for each $s,t\in\mathbb{R}$ with $s<t$.

For $t\in\mathbb{R}_{+}$ (resp. $t\in\mathbb{R}_{-}$), let
$\widetilde{{\bm{\mathcal{X}}}}_{\rho}(t)$ be the net change of the
$\nu_{\widetilde{\bm{h}}}$-LQG length measure of the left outer boundary of
$\widetilde{\bm{\eta}}_{0}([0,t])$ (resp. $\widetilde{\bm{\eta}}_{0}([-t,0])$)
relative to time 0 and $\widetilde{{\bm{\mathcal{Y}}}}_{\rho}(t)$ be the net
change of the $\nu_{\widetilde{\bm{h}}}$-LQG length measure of the right outer
boundary of $\widetilde{\bm{\eta}}_{0}([0,t])$ (resp.
$\widetilde{\bm{\eta}}_{0}([-t,0])$) relative to 0 (see [GHS19, Section
4.2.1]). From [DMS21, Theorems 1.9 and 1.11] (see also [GHS19, Theorem 4.6])
the process
$\widetilde{{\bm{\mathcal{W}}}}_{\rho}(t)=(\widetilde{{\bm{\mathcal{X}}}}_{\rho}(t),\widetilde{{\bm{\mathcal{Y}}}}_{\rho}(t))$
defined above has (up to time reparametrization) the law of a two-dimensional
Brownian motion of correlation $\rho$. In addition, the process
$(\widetilde{{\bm{\mathcal{W}}}}_{\rho}(t))_{t\in\mathbb{R}}$ a.s. determines
$(({\mathcal{C}},\widetilde{\bm{h}},-\infty),\widetilde{\bm{\eta}}_{0})$ as a
curve-decorated quantum surface.

For $u\in\mathbb{R}$, let $\widetilde{\bm{\phi}}_{\theta}^{(u)}$ denote the
union of the two flow lines of the vector fields
$e^{i(\hat{\bm{h}}/\chi+\theta)}$ and $e^{i(\hat{\bm{h}}/\chi+\theta+\pi)}$
started at $\widetilde{\bm{\eta}}_{0}(u)$ (followed in the direction of the
flow line of $e^{i(\hat{\bm{h}}/\chi+\theta)}$). As before these flow lines
are unique for almost all $u\in\mathbb{R}$.

######  Lemma 4.3.

Let
$\widetilde{{\bm{\mathcal{Z}}}}_{\rho,q}=\left\\{\widetilde{{\bm{\mathcal{Z}}}}_{\rho,q}^{(u)}\right\\}_{u\in\mathbb{R}}$
denote the collection of (strong) solutions to the following SDEs indexed by
$u\in\mathbb{R}$ and driven by $\widetilde{{\bm{\mathcal{W}}}}_{\rho}$,

$\begin{cases}d\widetilde{{\bm{\mathcal{Z}}}}_{\rho,q}^{(u)}(t)=\operatorname{\mathds{1}}_{\\{\widetilde{{\bm{\mathcal{Z}}}}_{\rho,q}^{(u)}(t)>0\\}}d\widetilde{{\bm{\mathcal{Y}}}}_{\rho}(t)-\operatorname{\mathds{1}}_{\\{\widetilde{{\bm{\mathcal{Z}}}}_{\rho,q}^{(u)}(t)\leq
0\\}}d\widetilde{{\bm{\mathcal{X}}}}_{\rho}(t)+(2q-1)\cdot
d{\bm{\mathcal{L}}}^{\widetilde{{\bm{\mathcal{Z}}}}_{\rho,q}^{(u)}}(t),&t\in\mathbb{R}_{>u}\\\
\widetilde{{\bm{\mathcal{Z}}}}_{\rho,q}^{(u)}(u)=0.&\end{cases}$ (83)

There exists a constant $\overline{q}=\overline{q}_{\gamma}(\theta)\in[0,1]$
such that a.s. for almost all $u\in\mathbb{R}$

$\displaystyle\left\\{t\in\mathbb{R}_{\geq
u}:\widetilde{{\bm{\mathcal{Z}}}}_{\rho,\overline{q}}^{(u)}(t)\geq 0\right\\}$
$\displaystyle=\left\\{t\in\mathbb{R}_{\geq
u}:\widetilde{\bm{\eta}}_{0}(t)\text{ is weakly on the left of
}\widetilde{\bm{\phi}}_{\theta}^{(u)}\right\\},$ (84)
$\displaystyle\left\\{t\in\mathbb{R}_{\geq
u}:\widetilde{{\bm{\mathcal{Z}}}}_{\rho,\overline{q}}^{(u)}(t)\leq 0\right\\}$
$\displaystyle=\left\\{t\in\mathbb{R}_{\geq
u}:\widetilde{\bm{\eta}}_{0}(t)\text{ is weakly on the right of
}\widetilde{\bm{\phi}}_{\theta}^{(u)}\right\\}.$ (85)

###### Proof.

We start by recalling that existence and uniqueness of solutions to the SDE in
Eq. 83 are guaranteed by 2.1.

We now fix $u=0$ and we set
$\widetilde{\bm{\phi}}_{\theta}\coloneqq\widetilde{\bm{\phi}}_{\theta}^{(0)}$.
We assume that the two flow lines of the vector fields
$e^{i(\hat{\bm{h}}/\chi+\theta)}$ and $e^{i(\hat{\bm{h}}/\chi+\theta+\pi)}$
started at $\widetilde{\bm{\eta}}_{0}(0)$ are unique. The proof for $u\neq 0$
is similar. For $t\in\mathbb{R}_{\geq 0}$, we consider the event

$E_{t}=\left\\{\widetilde{\bm{\eta}}_{0}(t)\text{ is on the left of
}\widetilde{\bm{\phi}}_{\theta}\right\\},$ (86)

and the random variable

$\displaystyle\widetilde{\bm{\tau}_{t}}$
$\displaystyle=\sup\left\\{s\in\mathbb{R}_{\leq
t}:\widetilde{\bm{\eta}}_{0}\text{ crosses
}\widetilde{\bm{\phi}}_{\theta}\text{ at time }s\right\\}.$ (87)

We also consider the process

${\bm{\mathcal{Q}}}(t)\coloneqq(\widetilde{{\bm{\mathcal{Y}}}}_{\rho}(t)-\widetilde{{\bm{\mathcal{Y}}}}_{\rho}(\widetilde{\bm{\tau}_{t}}))\mathds{1}_{E_{t}}-(\widetilde{{\bm{\mathcal{X}}}}_{\rho}(t)-\widetilde{{\bm{\mathcal{X}}}}_{\rho}(\widetilde{\bm{\tau}_{t}}))\mathds{1}_{E_{t}^{c}},\quad
t\in\mathbb{R}_{\geq 0},$ (88)

where we recall that
$\widetilde{{\bm{\mathcal{W}}}}_{\rho}(t)=(\widetilde{{\bm{\mathcal{X}}}}_{\rho}(t),\widetilde{{\bm{\mathcal{Y}}}}_{\rho}(t))$
is the two-dimensional Brownian motion of correlation $\rho$ encoding
$(({\mathcal{C}},\widetilde{\bm{h}},-\infty),\widetilde{\bm{\eta}}_{0})$. In
[GHS16, Proposition 3.2] it was shown that there exists a constant
$\overline{q}=\overline{q}_{\gamma}(\theta)\in[0,1]$ such that
${\bm{\mathcal{Q}}}(s)$ is a skew Brownian motion of parameter $\overline{q}$
and

$\displaystyle\left\\{t\in\mathbb{R}_{\geq 0}:\bm{Q}(t)\geq 0\right\\}$
$\displaystyle=\left\\{t\in\mathbb{R}_{\geq
0}:\widetilde{\bm{\eta}}_{0}(t)\text{ is weakly on the left of
}\widetilde{\bm{\phi}}_{\theta}\right\\},$ (89)
$\displaystyle\left\\{t\in\mathbb{R}_{\geq 0}:\bm{Q}(t)\leq 0\right\\}$
$\displaystyle=\left\\{t\in\mathbb{R}_{\geq
0}:\widetilde{\bm{\eta}}_{0}(t)\text{ is weakly on the right of
}\widetilde{\bm{\phi}}_{\theta}\right\\}.$ (90)

Therefore in other to complete the proof it is enough to show that
${\bm{\mathcal{Q}}}(s)$ solves the SDE in Eq. 83 for $u=0$. Indeed, thanks to
pathwise uniqueness (2.1), then we have that
${\bm{\mathcal{Q}}}=\widetilde{{\bm{\mathcal{Z}}}}_{\rho,q}^{(0)}$ a.s.

Since ${\bm{\mathcal{Q}}}(t)$ is a a skew Brownian motion of parameter
$\overline{q}$ then, as shown in [HS81], there exists a standard one-
dimensional Brownian motion $({\bm{\mathcal{B}}}(t))_{t\in\mathbb{R}_{\geq
0}}$ such that

${\bm{\mathcal{Q}}}(t)={\bm{\mathcal{B}}}(t)+(2\overline{q}-1){\bm{\mathcal{L}}}^{{\bm{\mathcal{Q}}}}(t),\quad
t\in\mathbb{R}_{\geq 0}.$ (91)

Setting

${\bm{\mathcal{U}}}(t)\coloneqq(1-\overline{q}){\bm{\mathcal{Q}}}(t)\mathds{1}_{\\{{\bm{\mathcal{Q}}}(t)>0\\}}+\overline{q}{\bm{\mathcal{Q}}}(t)\mathds{1}_{\\{{\bm{\mathcal{Q}}}(t)\leq
0\\}},$ (92)

then from [HS81, Equation (11)] we have that

$d{\bm{\mathcal{U}}}(t)=(1-\overline{q})\mathds{1}_{\\{{\bm{\mathcal{U}}}(t)>0\\}}d{\bm{\mathcal{B}}}(t)-\overline{q}\mathds{1}_{\\{{\bm{\mathcal{U}}}(t)\leq
0\\}}d{\bm{\mathcal{B}}}(t).$ (93)

We also introduce the following stochastic process
$\overline{{\bm{\mathcal{W}}}}_{\rho}(t)=(\overline{{\bm{\mathcal{X}}}}_{\rho}(t),\overline{{\bm{\mathcal{Y}}}}_{\rho}(t))$,
defined for all $t\in\mathbb{R}_{\geq 0}$ by

$\displaystyle\overline{{\bm{\mathcal{X}}}}_{\rho}(t)$
$\displaystyle\coloneqq\int_{0}^{t}\mathds{1}_{\\{{\bm{\mathcal{U}}}(s)>0\\}}d\widetilde{{\bm{\mathcal{X}}}}_{\rho}(s)+\int_{0}^{t}\mathds{1}_{\\{{\bm{\mathcal{U}}}(s)\leq
0\\}}d{\bm{\mathcal{B}}}(s),$ (94)
$\displaystyle\overline{{\bm{\mathcal{Y}}}}_{\rho}(t)$
$\displaystyle\coloneqq\int_{0}^{t}\mathds{1}_{\\{{\bm{\mathcal{U}}}(s)>0\\}}d{\bm{\mathcal{B}}}(s)+\int_{0}^{t}\mathds{1}_{\\{{\bm{\mathcal{U}}}(s)\leq
0\\}}d\widetilde{{\bm{\mathcal{Y}}}}_{\rho}(s).$ (95)

Note that

$(1-\overline{q})\mathds{1}_{\\{{\bm{\mathcal{U}}}(t)>0\\}}d\overline{{\bm{\mathcal{Y}}}}_{\rho}(t)-\overline{q}\mathds{1}_{\\{{\bm{\mathcal{U}}}(t)\leq
0\\}}d\overline{{\bm{\mathcal{X}}}}_{\rho}(t)=(1-\overline{q})\mathds{1}_{\\{{\bm{\mathcal{U}}}(t)>0\\}}d{\bm{\mathcal{B}}}(t)-\overline{q}\mathds{1}_{\\{{\bm{\mathcal{U}}}(t)\leq
0\\}}d{\bm{\mathcal{B}}}(t)=d{\bm{\mathcal{U}}}(t),$ (96)

where in the last equality we used Eq. 93. Setting,
$r(x)=x/(1-\overline{q})\cdot\mathds{1}_{x>0}+x/\overline{q}\cdot\mathds{1}_{x\leq
0}$, and using Eq. 92, we have that
$r({\bm{\mathcal{U}}}(t))={\bm{\mathcal{Q}}}(t)$. Therefore, from 2.2, we
obtain that ${\bm{\mathcal{Q}}}(t)$ satisfies

$d{\bm{\mathcal{Q}}}(t)=\operatorname{\mathds{1}}_{\\{{\bm{\mathcal{Q}}}(t)>0\\}}d\overline{{\bm{\mathcal{Y}}}}_{\rho}(t)-\operatorname{\mathds{1}}_{\\{{\bm{\mathcal{Q}}}(t)\leq
0\\}}d\overline{{\bm{\mathcal{X}}}}_{\rho}(t)+(2\overline{q}-1)\cdot
d{\bm{\mathcal{L}}}^{{\bm{\mathcal{Q}}}}(t).$ (97)

Note that if we show that
$\overline{{\bm{\mathcal{X}}}}_{\rho}=\widetilde{{\bm{\mathcal{X}}}}_{\rho}$
a.s. and
$\overline{{\bm{\mathcal{Y}}}}_{\rho}=\widetilde{{\bm{\mathcal{Y}}}}_{\rho}$
a.s., i.e.
$\overline{{\bm{\mathcal{W}}}}_{\rho}=\widetilde{{\bm{\mathcal{W}}}}_{\rho}$
a.s., then we can conclude the proof. From Eqs. 94 and 92 we have that

$\overline{{\bm{\mathcal{X}}}}_{\rho}(t)-\overline{{\bm{\mathcal{X}}}}_{\rho}(\widetilde{\bm{\tau}_{t}})=\int_{\widetilde{\bm{\tau}_{t}}}^{t}\mathds{1}_{\\{{\bm{\mathcal{Q}}}(s)>0\\}}d\widetilde{{\bm{\mathcal{X}}}}_{\rho}(s)+\int_{\widetilde{\bm{\tau}_{t}}}^{t}\mathds{1}_{\\{{\bm{\mathcal{Q}}}(s)\leq
0\\}}d{\bm{\mathcal{B}}}(s).$ (98)

Since
$\int_{0}^{t}\mathds{1}_{\\{{\bm{\mathcal{Q}}}(s)=0\\}}d{\bm{\mathcal{B}}}(s)$
is identically zero (this follows for instance using the same arguments as in
Eq. 30) and $d{\bm{\mathcal{L}}}^{{\bm{\mathcal{Q}}}}(s)=0$ when
${\bm{\mathcal{Q}}}(s)<0$, using Eq. 91 we have that

$\int_{\widetilde{\bm{\tau}_{t}}}^{t}\mathds{1}_{\\{{\bm{\mathcal{Q}}}(s)\leq
0\\}}d{\bm{\mathcal{B}}}(s)=\int_{\widetilde{\bm{\tau}_{t}}}^{t}\mathds{1}_{\\{{\bm{\mathcal{Q}}}(s)<0\\}}d{\bm{\mathcal{Q}}}(s).$

In addition, noting that $\\{{\bm{\mathcal{Q}}}(s)<0\\}\subseteq E_{s}^{c}$
(this follows form Eq. 90) and using Eq. 88 we obtain that

$\int_{\widetilde{\bm{\tau}_{t}}}^{t}\mathds{1}_{\\{{\bm{\mathcal{Q}}}(s)<0\\}}d{\bm{\mathcal{Q}}}(s)=\int_{\widetilde{\bm{\tau}_{t}}}^{t}\mathds{1}_{\\{{\bm{\mathcal{Q}}}(s)<0\\}}d\widetilde{{\bm{\mathcal{X}}}}_{\rho}(s).$

Substituting the latter two expressions in Eq. 98 we conclude that for all
$t\in\mathbb{R}_{\geq 0},$

$\overline{{\bm{\mathcal{X}}}}_{\rho}(t)-\overline{{\bm{\mathcal{X}}}}_{\rho}(\widetilde{\bm{\tau}_{t}})=\widetilde{{\bm{\mathcal{X}}}}_{\rho}(t)-\widetilde{{\bm{\mathcal{X}}}}_{\rho}(\widetilde{\bm{\tau}_{t}}).$
(99)

Similarly,
$\overline{{\bm{\mathcal{Y}}}}_{\rho}(t)-\overline{{\bm{\mathcal{Y}}}}_{\rho}(\widetilde{\bm{\tau}_{t}})=\widetilde{{\bm{\mathcal{Y}}}}_{\rho}(t)-\widetilde{{\bm{\mathcal{Y}}}}_{\rho}(\widetilde{\bm{\tau}_{t}})$
for all $t\in\mathbb{R}_{\geq 0}$. From Eq. 94 and Lévy’s characterization
theorem ([KS91, Theorem 3.3.16]), we have that both
$\overline{{\bm{\mathcal{X}}}}_{\rho}(t)$ and
$\overline{{\bm{\mathcal{Y}}}}_{\rho}(t)$ are standard one-dimensional
Brownian motions. Therefore, to conclude that
$\overline{{\bm{\mathcal{W}}}}_{\rho}=\widetilde{{\bm{\mathcal{W}}}}_{\rho}$
a.s., it is enough to show that for $t>s$

$\mathbb{E}\left[\overline{{\bm{\mathcal{W}}}}_{\rho}(t)-\widetilde{{\bm{\mathcal{W}}}}_{\rho}(t)\middle|\mathcal{F}_{s}\right]=\overline{{\bm{\mathcal{W}}}}_{\rho}(s)-\widetilde{{\bm{\mathcal{W}}}}_{\rho}(s),$
(100)

where
$\mathcal{F}_{s}\coloneqq\sigma((\overline{{\bm{\mathcal{W}}}}_{\rho},\widetilde{{\bm{\mathcal{W}}}}_{\rho})|_{[0,s]})$.
Indeed, the latter equation implies that
$\overline{{\bm{\mathcal{W}}}}_{\rho}-\widetilde{{\bm{\mathcal{W}}}}_{\rho}$
is a $\mathcal{F}_{t}$-martingale and then in [GHS16, Lemma 3.17] it is shown
that if Eq. 99 holds then the quadratic varion of
$\overline{{\bm{\mathcal{W}}}}_{\rho}-\widetilde{{\bm{\mathcal{W}}}}_{\rho}$
must be zero and so
$\overline{{\bm{\mathcal{W}}}}_{\rho}=\widetilde{{\bm{\mathcal{W}}}}_{\rho}$
a.s.

We proceed with the proof of Eq. 100 by showing that the law of
$(\overline{{\bm{\mathcal{W}}}}_{\rho}(\cdot)-\overline{{\bm{\mathcal{W}}}}_{\rho}(s))|_{[s,\infty)}$
(resp.
$(\widetilde{{\bm{\mathcal{W}}}}_{\rho}(\cdot)-\widetilde{{\bm{\mathcal{W}}}}_{\rho}(s))|_{[s,\infty)}$)
given $\mathcal{F}_{s}$ is the unconditional law of
$\overline{{\bm{\mathcal{W}}}}_{\rho}$ (resp.
$\widetilde{{\bm{\mathcal{W}}}}_{\rho}$).

We set
$\widetilde{\mathfrak{W}}_{\rho}(t)\coloneqq\widetilde{{\bm{\mathcal{W}}}}_{\rho}(t)-\widetilde{{\bm{\mathcal{W}}}}_{\rho}(\widetilde{\bm{\tau}_{t}})$.
From [GHS16, Proposition 3.4 and Lemma 3.16], we have that:

  * •

the processes $\widetilde{\mathfrak{W}}_{\rho}|_{[0,s]}$ and
$\widetilde{{\bm{\mathcal{W}}}}_{\rho}|_{[0,s]}$ (resp.
$(\widetilde{\mathfrak{W}}_{\rho}(\cdot)-\widetilde{\mathfrak{W}}_{\rho}(s))|_{[s,\infty)}$
and
$(\widetilde{{\bm{\mathcal{W}}}}_{\rho}(\cdot)-\widetilde{{\bm{\mathcal{W}}}}_{\rho}(s))|_{[s,\infty)}$)
determine each other. In particular,
$(\widetilde{\mathfrak{W}}_{\rho}(\cdot)-\widetilde{\mathfrak{W}}_{\rho}(s))|_{[s,\infty)}$
is independent of $\widetilde{\mathfrak{W}}_{\rho}|_{[0,s]}$.

  * •

$\widetilde{\mathfrak{W}}_{\rho}|_{[0,s]}$ (resp.
$(\widetilde{\mathfrak{W}}_{\rho}(\cdot)-\widetilde{\mathfrak{W}}_{\rho}(s))|_{[s,\infty)}$)
determines ${\bm{\mathcal{Q}}}|_{[0,s]}$ (resp.
$({\bm{\mathcal{Q}}}(\cdot)-{\bm{\mathcal{Q}}}(s))|_{[s,\infty)}$).

Now from the definition of $\overline{{\bm{\mathcal{W}}}}_{\rho}$ in Eq. 94
and the relations in Eqs. 91 and 92, we also have that:

  * •

the process $\overline{{\bm{\mathcal{W}}}}_{\rho}|_{[0,s]}$ (resp.
$(\overline{{\bm{\mathcal{W}}}}_{\rho}(\cdot)-\overline{{\bm{\mathcal{W}}}}_{\rho}(s))|_{[s,\infty)}$)
is a.s. determined by
$(\widetilde{{\bm{\mathcal{W}}}}_{\rho},{\bm{\mathcal{Q}}})|_{[0,s]}$ (resp.
$(\widetilde{{\bm{\mathcal{W}}}}_{\rho}(\cdot)-\widetilde{{\bm{\mathcal{W}}}}_{\rho}(s),{\bm{\mathcal{Q}}}(\cdot)-{\bm{\mathcal{Q}}}(s))|_{[s,\infty)}$).
Therefore, from the items above,
$\overline{{\bm{\mathcal{W}}}}_{\rho}|_{[0,s]}$ (resp.
$(\overline{{\bm{\mathcal{W}}}}_{\rho}(\cdot)-\overline{{\bm{\mathcal{W}}}}_{\rho}(s))|_{[s,\infty)}$)
is a.s. determined by $\widetilde{\mathfrak{W}}_{\rho}|_{[0,s]}$ (resp.
$(\widetilde{\mathfrak{W}}_{\rho}(\cdot)-\widetilde{\mathfrak{W}}_{\rho}(s))|_{[s,\infty)}$).

From the items above, we can conclude that that the law of
$(\overline{{\bm{\mathcal{W}}}}_{\rho}(\cdot)-\overline{{\bm{\mathcal{W}}}}_{\rho}(s))|_{[s,\infty)}$
(resp.
$(\widetilde{{\bm{\mathcal{W}}}}_{\rho}(\cdot)-\widetilde{{\bm{\mathcal{W}}}}_{\rho}(s))|_{[s,\infty)}$)
given $\mathcal{F}_{s}$ is the unconditional law of
$\overline{{\bm{\mathcal{W}}}}_{\rho}$ (resp.
$\widetilde{{\bm{\mathcal{W}}}}_{\rho}$). This ends the proof. ∎

It remains to deduce 4.2 from 4.3.

###### Proof of 4.3.

Note that the only difference between the pairs
$(\widetilde{\bm{\eta}}_{0},\widetilde{\bm{\eta}}_{\theta})$ and
$(\bm{\eta}_{0},\bm{\eta}_{\theta})$ is in their time parametrization. More
precisely,

  * •

$(\widetilde{\bm{\eta}}_{0},\widetilde{\bm{\eta}}_{\theta})$ are parametrized
using the $\gamma$-quantum cone $(\mathbb{C},\widetilde{\bm{h}},0,\infty)$ and
therefore the left and right boundary measures of $\widetilde{\bm{\eta}}_{0}$
are encoded by the Brownian motion
$\widetilde{{\bm{\mathcal{W}}}}_{\rho}(t)=(\widetilde{{\bm{\mathcal{X}}}}_{\rho}(t),\widetilde{{\bm{\mathcal{Y}}}}_{\rho}(t))$
of correlation $\rho$;

  * •

$(\bm{\eta}_{0},\bm{\eta}_{\theta})$ are parametrized using the $\gamma$-LQG
sphere $({\mathbb{C}}\cup\\{\infty\\},\bm{h},\infty)$ and therefore the left
and right boundary measures of $\bm{\eta}_{0}$ are encoded by the Brownian
excursion
${\bm{\mathcal{E}}}_{\rho}(t)=({\bm{\mathcal{X}}}_{\rho}(t),{\bm{\mathcal{Y}}}_{\rho}(t))$
of correlation $\rho$.

For any $\varepsilon>0$, we consider the following curve-decorated quantum
surfaces:

  * •

the curve-decorated quantum surface
$(\widetilde{\mathcal{S}}_{\varepsilon},\widetilde{\bm{\eta}}_{0}|_{[\varepsilon,1-\varepsilon]})$,
where $\widetilde{\mathcal{S}}_{\varepsilon}$ denotes the quantum surface
obtained by restricting the quantum cone field $\widetilde{\bm{h}}$ to
$\widetilde{\bm{\eta}}_{0}([\varepsilon,1-\varepsilon])$.

  * •

the curve-decorated quantum surface
$(\mathcal{S}_{\varepsilon},\bm{\eta}_{0}|_{[\varepsilon,1-\varepsilon]})$,
where $\mathcal{S}_{\varepsilon}$ denotes the quantum surface obtained by
restricting the quantum sphere field $\bm{h}$ to
$\bm{\eta}_{0}([\varepsilon,1-\varepsilon])$.

From [GHS16, Lemma 3.12] the curve-decorated quantum surface
$(\widetilde{\mathcal{S}}_{\varepsilon},\widetilde{\bm{\eta}}_{0}|_{[\varepsilon,1-\varepsilon]})$
is a.s. determined by
$(\widetilde{{\bm{\mathcal{W}}}}_{\rho}(\varepsilon+t)-\widetilde{{\bm{\mathcal{W}}}}_{\rho}(\varepsilon))_{0\leq
t\leq 1-2\varepsilon}$, while the curve-decorated quantum surface
$(\mathcal{S}_{\varepsilon},\bm{\eta}_{0}|_{[\varepsilon,1-\varepsilon]})$ is
a.s. determined by
$({\bm{\mathcal{E}}}_{\rho}(\varepsilon+t)-{\bm{\mathcal{E}}}_{\rho}(\varepsilon))_{0\leq
t\leq 1-2\varepsilon}$.

The law of
$({\bm{\mathcal{E}}}_{\rho}(\varepsilon+t)-{\bm{\mathcal{E}}}_{\rho}(\varepsilon))_{0\leq
t\leq 1-2\varepsilon}$ is absolutely continuous w.r.t. the law of
$(\widetilde{{\bm{\mathcal{W}}}}_{\rho}(\varepsilon+t)-\widetilde{{\bm{\mathcal{W}}}}_{\rho}(\varepsilon))_{0\leq
t\leq 1-2\varepsilon}$ (see A.1). This implies that curve-decorated quantum
surface
$(\mathcal{S}_{\varepsilon},\bm{\eta}_{0}|_{[\varepsilon,1-\varepsilon]})$ is
absolutely continuous w.r.t. the curve-decorated quantum surface
$(\widetilde{\mathcal{S}_{\varepsilon}},\widetilde{\bm{\eta}}_{0}|_{[\varepsilon,1-\varepsilon]})$.

From [GHS16, Lemma 3.10] it follows that the flow lines
$\left\\{\widetilde{\bm{\phi}}^{(t)}_{\theta}\right\\}_{t\in[\varepsilon,1-\varepsilon]}$
(resp.
$\left\\{{\bm{\phi}}^{(t)}_{\theta}\right\\}_{t\in[\varepsilon,1-\varepsilon]}$)
run until they exit $\widetilde{\bm{\eta}}_{0}([\varepsilon,1-\varepsilon])$
(resp. $\bm{\eta}_{0}([\varepsilon,1-\varepsilon])$) are a.s. determined by
$\widetilde{\mathcal{S}}_{\varepsilon}$ (resp. $\mathcal{S}_{\varepsilon}$).

Finally, since by [GHS16, Lemma 3.6], $\widetilde{\bm{\eta}}_{0}$ (resp.
$\bm{\eta}_{0}$) hits points on $\widetilde{\bm{\phi}}^{(t)}_{\theta}$ (resp.
$\bm{\phi}^{(t)}_{\theta}$) in chronological order,
$\widetilde{\bm{\phi}}^{(t)}_{\theta}$ (resp. $\bm{\phi}^{(t)}_{\theta}$)
cannot revisit $\widetilde{\bm{\eta}}_{0}([\varepsilon,1-\varepsilon])$ (resp.
$\bm{\eta}_{0}([\varepsilon,1-\varepsilon])$) after exiting this region. Hence
it is possible to determine from $\widetilde{\mathcal{S}}_{\varepsilon}$
(resp. $\mathcal{S}_{\varepsilon}$) what points of $\widetilde{\bm{\eta}}_{0}$
(resp. $\bm{\eta}_{0}$) are to the left or right of
$\widetilde{\bm{\eta}}_{\theta}$ (resp. $\bm{\eta}_{\theta}$).

On the other hand, thanks to 1.7 (resp. 2.1) the processes
$\left\\{{\bm{\mathcal{Z}}}_{\rho,\overline{q}}^{(t)}|_{[t,1-\varepsilon]}\right\\}_{t\in[\varepsilon,1-\varepsilon]}$
in the statement of 4.2 (resp.
$\left\\{\widetilde{{\bm{\mathcal{Z}}}}_{\rho,\overline{q}}^{(t)}|_{[t,1-\varepsilon]}\right\\}_{t\in[\varepsilon,1-\varepsilon]}$
in the statement of 4.3) are a.s. determined – through the same solution map
$F_{1-\varepsilon}$ – by
$({\bm{\mathcal{E}}}_{\rho}(\varepsilon+t)-{\bm{\mathcal{E}}}_{\rho}(\varepsilon))_{0\leq
t\leq 1-2\varepsilon}$ (resp.
$(\widetilde{{\bm{\mathcal{W}}}}_{\rho}(\varepsilon+t)-\widetilde{{\bm{\mathcal{W}}}}_{\rho}(\varepsilon))_{0\leq
t\leq 1-2\varepsilon}$).

Therefore, by absolute continuity, we can deduce from 4.3 that a.s. for almost
all $t\in[\varepsilon,1-\varepsilon]$

$\displaystyle\left\\{x\in[t,1-\varepsilon]:{\bm{\mathcal{Z}}}_{\rho,\overline{q}}^{(t)}(x)\geq
0\right\\}$
$\displaystyle=\left\\{x\in[t,1-\varepsilon]:\bm{\eta}_{0}(x)\text{ is weakly
on the left of }\bm{\phi}_{\theta}^{(t)}\right\\},$ (101)
$\displaystyle\left\\{x\in[t,1-\varepsilon]:{\bm{\mathcal{Z}}}_{\rho,\overline{q}}^{(t)}(x)\leq
0\right\\}$
$\displaystyle=\left\\{x\in[t,1-\varepsilon]:\bm{\eta}_{0}(x)\text{ is weakly
on the right of }\bm{\phi}_{\theta}^{(t)}\right\\}.$ (102)

Since $\varepsilon>0$ can be made arbitrarily small, this proves 4.2. ∎

##  Appendix A Absolute continuity between correlated Brownian excursions in
cones and correlated Brownian motions

Let ${\bm{\mathcal{W}}}_{\rho}$ be a two-dimensional Brownian motion of
correlation $\rho\in(-1,1)$ and ${\bm{\mathcal{E}}}_{\rho}$ a two-dimensional
Brownian excursion of correlation $\rho$.

######  Proposition A.1.

For every $\varepsilon>0$, the distribution of
$({\bm{\mathcal{E}}}_{\rho}(\varepsilon+t)-{\bm{\mathcal{E}}}_{\rho}(\varepsilon))_{t\in[0,1-2\varepsilon]}$
is absolutely continuous w.r.t. the distribution of