# Vortex gap solitons in spin-orbit-coupled Bose-Einstein condensates with
competing nonlinearities

Xiaoxi Xu1 Feiyan Zhao1 Yangui Zhou1 Bin Liu1 Xunda Jiang1<EMAIL_ADDRESS>Boris A. Malomed2,3 Yongyao Li1,4 1 School of Physics and Optoelectronic
Engineering, Foshan University, Foshan 528000, China
2 Department of Physical Electronics, School of Electrical Engineering,
Faculty of Engineering, Tel Aviv University, Tel Aviv 69978, Israel
3 Instituto de Alta Investigación, Universidad de Tarapacá, Casilla 7D, Arica,
Chile
4 Guangdong-Hong Kong-Macao Joint Laboratory for Intelligent Micro-Nano
Optoelectronic Technology, Foshan University, Foshan 528000, China

###### Abstract

The formation and dynamics of full vortex gap solitons (FVGSs) is investigated
in two-component Bose-Einstein condensates with spin-orbit coupling (SOC),
Zeeman splitting (ZS), and competing cubic and quintic nonlinear terms, while
the usual kinetic energy is neglected, assuming that it is much smaller than
the SOC and ZS terms. Unlike previous SOC system with the cubic-only
attractive nonlinearity, in which solely semi-vortices may be stable, with the
vorticity carried by a single component, the present system supports stable
FVGS states, with the vorticity present in both components (such states are
called here “full vortex solitons”, to stress the difference from the half-
vortices). They populate the bandgap in the system’s linear spectrum. In the
case of the cubic self-attraction and quintic repulsion, stable FVGSs with a
positive effective mass exist near the top of the bandgap. On the contrary,
the system with cubic self-repulsion and quintic attraction produces stable
FVGSs with a negative mass near the bottom of the bandgap. Mobility and
collisions of FVGSs with different topological charges are investigated too.
Key words: Full vortex gap solitons, cubic-quintic nonlinearity, soliton
dynamics

##  I Introduction

Since the Bose-Einstein condensates (BECs) have been created in ultracold
gases of alkali metals Anderson1995 ; Bradley1995 ; Davis1995 , they have
become a universal platform to study many phenomena in superfluids, such as
vortices Williams1999 ; Kasamatsu2003 , solitons Burger1999 ; Anderson2001
and many other dynamical effects Heinzen2000 ; Morsch2006 ; Eckardt2009 . In
their straightforward form, one-dimensional (1D) solitons stably exist in BECs
without external traps, while two- and three-dimensional (2D and 3D) solitons
are made unstable by the presence of the collapse, driven by the cubic self-
attraction Berg1998 ; Sulem1999 ; Fibich2015 ; Malomed2005 ; Malomed2016 .

A significant direction in the work with atomic BEC is their use for emulation
of various phenomena which are known in a much more complex form in condensed-
matter physics emulator ; Ahufinger . In particular, synthetic spin-orbit
coupling (SOC), which simulates this fundamental effect well known in physics
of semiconductors Dresselhaus ; Bychkov , has been realized in two-component
BEC by using counterpropagating Raman laser beams to engineer the necessary
coupling between the two atomic states in the binary condensate Lin2011 ;
Liu2014 ; Wu2016 . SOC is a source of many remarkable findings in BECs,
including various extended patterns, vortices, and solitons Wang2010 ;
Kawakami2011 ; Kawakami2012 ; Conduit2012 ; Liu2012 ; Zhou2013 ; Zezyulin2013
; Sakaguchi2013 ; Xu2013 ; Kartashov2013 ; Kartashov2017 . A noteworthy fact
is that SOC affects ground-state properties and enhances stability of the
system, suppressing the critical collapse in the two-dimensional BEC with the
self- and cross-attractive cubic nonlinearity, thus making it possible to
predict absolutely stable 2D solitons Sakaguchi2014 ; Sakaguchi2016 . In 3D,
full suppression of the respective supercritical collapse is not possible, but
SOC was predicted to maintain metastable 3D solitons Zhang2015 . The 2D and 3D
solitons created by SOC may be semi-vortices (SVs), built of zero-vorticity
and vortical components, or mixed modes (MMs), which combine zero-vorticity
and vortex terms in both components Sakaguchi2014 ; Zhang2015 ; Sakaguchi2016
; Malomed2018 . In the context of binary dipolar BECs, SOC may support stable
anisotropic 2D composite solitons Li2017 .

Under the action of strong Zeeman splitting (ZS) added to the SOC system, the
MM states are transformed into stable SV gap solitons Sakaguchi2018 . The same
effect stabilizes 2D solitons, supported by long-range dipole-dipole
interactions between atoms carrying permanent magnetic moments, close to the
edge of the ZS-induced bandgap Liao2017 . However, the dominant component of
the 2D composite gap solitons, reported in Refs. Liao2017  and Sakaguchi2018 ,
is a fundamental mode without any phase structure. The creation of stable
full-vortex gap solitons (FVGSs) in the SOC-affected BEC system, which, unlike
the SVs, would carry vorticity in both components, remains a challenging
problem. In fact, such solitons were considered, in the form of “excited
states” of SVs and MMs, in the system which did not include the ZS terms
Sakaguchi2014 . However, they all were found to be completely unstable states.

To construct stable FVGSs in this work, we use competing cubic and quintic
nonlinearities. This approach is suggested by the prediction of robust 2D
Li2018 ; Zhang2019 ; Lin2021 ; Zheng2021  and 3D Taruell  vortex states in
quantum droplets, which are stabilized by the Lee-Huang-Yang effect, i.e.,
additional quartic self-repulsive nonlinearity, induced by quantum
fluctuations around mean-field states. The stability is provided by the
competition of the quartic self-repulsion with the cubic mean-field attraction
between components in the binary BEC Petrov2015 ; Petrov2016 ; Schmitt2016 ;
Cabrera2018 . A similar possibility is to consider higher-order nonlinearities
of the mean-field origin in BEC Yamaguchi1979 . In that connection, it is
relevant to mention that, in addition to the common two-body (cubic)
interaction, the three-body (quintic) term can be adjusted by means of the
Feshbach resonance Marcelli1999 ; Smith2014 ; Adhikari2017 . In previous
studies, it was demonstrated that competing cubic self-attraction and quintic
repulsion could provide the stability of vortex solitons in BECs (in the
absence of SOC) Mithun2013 ; Abdullaev2001 ; Gao2018 ; PhysicaD ; Xu2021 .

In this work, we consider a possibility to use the interplay of the competing
nonlinearities, SOC, and ZS to support stable FVGSs in the two-component BEC
system. We produce the density distribution and phase structure for families
of stationary vortex solitons with different topological charges, and identify
their stability. Their mobility and collisions are addressed too.

The paper is organized as follows. In Section 2, we introduce the model and
FVGS states in it. In Section 3, we study the dynamics, mobility and
collisions of the FVGSs by means of systematic simulations. The paper is
concluded by Section 4.

##  II The model and stationary solutions

###  II.1 The coupled Gross-Pitaevskii equations

We consider the binary BEC which, in the mean-field approximation, is modelled
by the system of 2D Gross-Pitaevskii equations (GPEs) for wave functions
$\tilde{\psi}_{\pm}(\tilde{x},\tilde{y})$ of two components of the condensate.
The equations include the Rashba-type SOC Rashba ; Sakaguchi2016  with
strength $\lambda$, ZS with chemical-potential difference $2\tilde{\Omega}$
(both $\lambda$ and $\tilde{\Omega}$ are defined to be positive), and a
combination of cubic and quintic terms, which represents two-body and three-
body interactions, respectively Abdullaev2001 ; Braaten2002 ; Bulgac2002 ;
Michinel1 ; Salerno ; Kofane :

$i\hbar{\frac{\partial}{\partial\tilde{t}}}\tilde{\psi}_{\pm}=-{\frac{\hbar^{2}}{2m}}\nabla_{\pm}^{2}\tilde{\psi}\pm\lambda\left({\frac{\partial}{\partial\tilde{x}}}\mp
i{\frac{\partial}{\partial\tilde{y}}}\right)\tilde{\psi}_{\mp}\mp\tilde{\Omega}\tilde{\psi}_{\pm}+g_{1}(|\tilde{\psi}_{\pm}|^{2}+|\tilde{\psi}_{\mp}|^{2})\tilde{\psi}_{\pm}+g_{2}(|\tilde{\psi}_{\pm}|^{2}+|\tilde{\psi}_{\mp}|^{2})_{\pm}^{2}\tilde{\psi}$
(1)

(the tilde designates variables and coefficients which are replaced below by
their scaled counterparts). The strength of the two-body interaction,

$g_{1}=2\sqrt{2\pi}\hbar^{2}a/\left(a_{\perp}m\right),$ (2)

is proportional to the s-wave scattering length $a$ ($a>0$ and $a<0$ represent
the repulsive and attractive interactions, respectively), and inversely
proportional to size $a_{\perp}$ of the confinement in the transverse
direction Dalfovo1999 . The ZS effect can be induced by an artificial magnetic
field in BECs, which breaks the symmetry between the two components of the
binary wave function and produces a bandgap in the energy spectrum of the
system Lin2011 ; Quay2010 ; Goldman2014 .

The nonlinearity in Eq. (1) is written in the Manakov’s form Manakov , i.e.,
with equal coefficients of the self- and cross-interactions, which is a
relevant approximation for mixtures of two different atomic states Manakov-BEC
; Manakov-BEC2 . The magnitude and sign of $a$ can be adjusted to necessary
values in Eq. (2) by means of the Feshbach resonance Feshbach . Further, the
strength of the quintic nonlinearity is

$g_{2}=\frac{4\sqrt{3}\hbar^{2}a^{4}}{ma_{\perp}^{2}}\left[d_{1}+d_{2}\tan\left(s_{0}\ln\frac{|a|}{|a_{0}|}+\frac{1}{2}\right)\right],$
(3)

where $a_{0}$ is the value of $a$ at which a three-body bound state appears,
$d_{1,2}$ and $s_{0}$ being coefficients given in Refs. Bulgac2002 ;
Braaten2002 . It is known that $g_{2}$ can be tuned to arbitrary positive or
negative value. In this work, we focus on the competition between the cubic
and quintic nonlinearities with opposite signs, which implies $g_{1}g_{2}<0$.
As mentioned above, the s-wave scattering length $a$ in Eq. (2) can be tuned
to a desired value by means of the Feshbach-resonance technique, therefore the
value of $g_{1}$ is also adjustable.

Following Refs. Li2017  and Sakaguchi2018 , we focus on the situation when the
SOC energy is much larger than the kinetic energy, which implies the
consideration of configurations with sufficiently large characteristic length
scales, $l\gg\hbar^{2}/\left(m\lambda\right)$. In this case, terms
$\sim\nabla^{2}\psi_{\pm}$ in Eq. (1) may be neglected. By means of scaling,

$\tilde{\psi}_{\pm}=\sqrt{\left|g_{1}/g_{2}\right|}\psi_{\pm},\left(\tilde{x},\tilde{y}\right)=\left(\lambda\left|g_{2}\right|/g_{1}^{2}\right)\left(x,y\right),\tilde{t}=\left(\hbar\left|g_{2}\right|/g_{1}^{2}\right)t,\tilde{\Omega}=\left(g_{1}^{2}/\left|g_{2}\right|\right)\Omega,$
(4)

the so simplified GPE system is cast in the following form, in which the
nonlinearity and SOC coefficients are set to be $1$:

$\displaystyle i{\frac{\partial}{\partial
t}}\psi_{+}=\left({\frac{\partial}{\partial x}}-i{\frac{\partial}{\partial
y}}\right)\psi_{-}-\Omega\psi_{+}-(|\psi_{+}|^{2}+|\psi_{-}|^{2})\psi_{+}+\left(|\psi_{+}|^{2}+|\psi_{-}|^{2}\right)^{2}\psi_{+},$
$\displaystyle i{\frac{\partial}{\partial
t}}\psi_{-}=-\left({\frac{\partial}{\partial x}}+i{\frac{\partial}{\partial
y}}\right)\psi_{+}+\Omega\psi_{-}-(|\psi_{-}|^{2}+|\psi_{+}|^{2})\psi_{-}+\left(|\psi_{-}|^{2}+|\psi_{+}|^{2}\right)^{2}\psi_{-},$
(5)

in the case of the competition between cubic attraction and quintic repulsion,
or

$\displaystyle i{\frac{\partial}{\partial
t}}\psi_{+}=\left({\frac{\partial}{\partial x}}-i{\frac{\partial}{\partial
y}}\right)\psi_{-}-\Omega\psi_{+}+(|\psi_{+}|^{2}+|\psi_{-}|^{2})\psi_{+}-(|\psi_{+}|^{2}+|\psi_{-}|^{2})^{2}\psi_{+},$
$\displaystyle i{\frac{\partial}{\partial
t}}\psi_{-}=-\left({\frac{\partial}{\partial x}}+i{\frac{\partial}{\partial
y}}\right)\psi_{+}+\Omega\psi_{-}+(|\psi_{-}|^{2}+|\psi_{+}|^{2})\psi_{-}-(|\psi_{+}|^{2}+|\psi_{-}|^{2})^{2}\psi_{-},$
(6)

in the opposite case of the cubic repulsion and quintic attraction. The
remaining ZS parameter $\Omega$ may be also fixed by rescaling, but we keep it
in the equations, as it is relevant to display some results for a fixed norm
of the solution, while varying $\Omega$.

Actually, Eqs. (5) and (6) can be transformed into each other by simple
substitution

$\psi_{\pm}\rightarrow\psi_{\mp}^{\ast},$ (7)

where $\ast$ stands for the complex conjugate; however, this transformation
does not apply to the full system of Eqs. (1).

The linearization of Eqs. (5) and (6) is carried out by substituting

$\psi_{\pm}=\varphi_{\pm}\exp\left(-i\omega t+ipx+iqy\right)$ (8)

with constant amplitudes $\varphi_{\pm}$, which leads to the dispersion
relation between the frequency $\omega$ and wave numbers $p$ and $q$ Li2017

$\omega^{2}=\Omega^{2}+(p^{2}+q^{2}),$ (9)

which is plotted in Fig. 1. Obviously, Eq. (9) with $\Omega\neq 0$ gives rise
to the bandgap of width $2\Omega$, which may be populated by gap solitons.

Figure 1: (Color online) Dispersion relation given by Eq. (9) with
$\Omega=10$.

###  II.2 Stationary gap solitons existing at bottom and top edges of the
bandgap.

Soliton solutions of Eqs. (5) and (6) with chemical potential $\mu$ are looked
for as

$\psi_{\pm}=e^{-i\mu t}\phi_{\pm}(x,y),$ (10)

where $\phi_{\pm}$ are stationary wave functions. At the bottom and top edges
of the bandgap, the chemical potential of the soliton is defined as

$\mu=\begin{cases}\Omega-\Delta\mu,\quad&\mathrm{top}\text{ }\mathrm{of}\text{
}\mathrm{the}\text{ }\mathrm{bandgap,}\\\
-\Omega+\Delta\mu,\quad&\mathrm{bottom}\text{ }\mathrm{of}\text{
}\mathrm{the}\text{ }\mathrm{bandgap,}\end{cases}$ (11)

with

$0<\Delta\mu<<\Omega.$ (12)

The total norm of the two-component wave function, defined by

$N=N_{+}+N_{-}\equiv\int\int\left[|\psi_{+}(x,y)|^{2}+|\psi_{-}(x,y)|^{2}\right]dxdy,$
(13)

is a dynamical invariant of the system. The value of the norm is a natural
characteristic of the solitons.

Below, we analyze the existence of the gap solitons in the edge regions which
are defined as per Eq. (11). The analysis is presented here for Eq. (6), as,
near the bottom and top edges of the bandgap Eq. (11), the transformation
provided by Eq. (7) automatically produces solutions for the stationary wave
functions $\phi_{\pm}$, defined as per Eq. (10) and Eq. (5):

$\phi_{\mp}^{\mathrm{(top(stable);bottom(unstable))}}\left\\{\mathrm{Eq.~{}}\text{(\ref{1})}\right\\}=\left[\phi_{\pm}^{\mathrm{(bottom(stable);top(unstable))}}\left\\{\mathrm{Eq.~{}}\text{(\ref{2})}\right\\}\right]^{\ast}.$
(14)

The Lagrangian of Eq. (6) is

$\mathcal{L}=\int\int\left(\frac{i}{2}\sum_{+,-}\psi_{\pm}^{\ast}\frac{\partial\psi_{\pm}}{\partial
t}+\mathrm{c.c.}\right)dxdy-H,$ (15)

where $\mathrm{c.c.}$ stands for the complex-conjugate expression, and the
Hamiltonian (another dynamical invariant of the system) is

$\displaystyle H$ $\displaystyle=$
$\displaystyle\int\int\left\\{\left[\frac{1}{2}\left(\psi_{+}^{\ast}\frac{\partial\psi_{-}}{\partial
x}-i\psi_{+}^{\ast}\frac{\partial\psi_{-}}{\partial
y}\right)+\mathrm{c.c.}\right]+\Omega\left(\left|\psi_{-}\right|^{2}-\left|\psi_{+}\right|^{2}\right)\right.$
(16)
$\displaystyle\left.+\frac{1}{2}\left(\left|\psi_{+}\right|^{2}+\left|\psi_{-}\right|^{2}\right)^{2}-\frac{1}{3}\left(\left|\psi_{+}\right|^{2}+\left|\psi_{-}\right|^{2}\right)^{3}\right\\}dxdy.$

The application of the Noether’s theorem Noether  to Eq. (15) produces the
conserved momentum in the same form as in Schrödinger equations, namely,

$\mathbf{P}=i\int\int[\nabla(\psi_{+}^{\ast})\psi_{+}+\nabla(\psi_{-}^{\ast})\psi_{-}]dxdy.$
(17)

On the other hand, the angular momentum, which may be conveniently written in
terms of polar coordinates $\left(r,\theta\right)$ as

$\Lambda=i\int_{0}^{\infty}rdr\int_{0}^{2\pi}d\theta\sum_{+,-}\frac{\partial\psi_{\pm}^{\ast}}{\partial\theta}\psi_{\pm},$
(18)

is not conserved in the framework of the SOC system. Indeed, rewriting Eq. (6)
in terms of $\left(r,\theta\right)$,

$\displaystyle i{\frac{\partial}{\partial
t}}\psi_{+}=e^{-i\theta}\left({\frac{\partial}{\partial
r}}-\frac{i}{r}{\frac{\partial}{\partial\theta}}\right)\psi_{-}-\Omega\psi_{+}+(|\psi_{+}|^{2}+|\psi_{-}|^{2})\psi_{+}-(|\psi_{+}|^{2}+|\psi_{-}|^{2})^{2}\psi_{+},$
$\displaystyle i{\frac{\partial}{\partial
t}}\psi_{-}=-e^{i\theta}\left({\frac{\partial}{\partial
r}}+\frac{i}{r}{\frac{\partial}{\partial\theta}}\right)\psi_{+}+\Omega\psi_{-}+(|\psi_{-}|^{2}+|\psi_{+}|^{2})\psi_{-}-(|\psi_{+}|^{2}+|\psi_{-}|^{2})^{2}\psi_{-},$
(19)

one obtains

$\frac{d\Lambda}{dt}=2\int_{0}^{\infty}rdr\int_{0}^{2\pi}d\theta~{}\mathrm{Im}\left(e^{i\theta}\psi_{+}\frac{\partial\psi_{-}^{\ast}}{\partial
r}\right).$ (20)

Substituting Eq. (10) into Eq. (6) yields the following equations:

$\displaystyle\mu\phi_{+}=\left({\frac{\partial}{\partial
x}}-i{\frac{\partial}{\partial
y}}\right)\phi_{-}-\Omega\phi_{+}+(|\phi_{+}|^{2}+|\phi_{-}|^{2})\phi_{+}-\left(|\phi_{+}|^{2}+|\phi_{-}|^{2}\right)^{2}\phi_{+},$
$\displaystyle\mu\phi_{-}=-\left({\frac{\partial}{\partial
x}}+i{\frac{\partial}{\partial
y}}\right)\phi_{+}+\Omega\phi_{-}+(|\phi_{-}|^{2}+|\phi_{+}|^{2})\phi_{-}-\left(|\phi_{-}|^{2}+|\phi_{+}|^{2}\right)^{2}\phi_{-}.$
(21)

In particular, near the bottom edge of the bandgap, which is defined as per
the bottom line of Eq. (11), the first equation of Eq. (21) amounts, in the
lowest approximation, to a linear relation (cf. Ref. Sakaguchi2016 ),

$\phi_{-}\approx\left(2\Omega\right)^{-1}\left({\frac{\partial}{\partial
x}}+i{\frac{\partial}{\partial y}}\right)\phi_{+}.$ (22)

Substituting this in the first equation of Eq. (21), one obtains

$-\Delta\mu\cdot\phi_{+}=-\left(2\Omega\right)^{-1}\left({\frac{\partial^{2}}{\partial
x^{2}}}+{\frac{\partial^{2}}{\partial
y^{2}}}\right)\phi_{+}-|\phi_{+}|^{2}\phi_{+}+|\phi_{+}|^{4}\phi_{+}.$ (23)

This equation, which combines the effective cubic self-attraction and quintic
repulsion, has a chance to produce stable 2D solitons Michinel2 ; Berntson .
It is well known that such solitons, with all values of the embedded vorticity
(winding number), exist in interval Bulgaria ; Canada ; Berntson

$0<\Delta\mu<3/16,$ (24)

with the respective values of the norm

$N(\Delta\mu\rightarrow 0)\rightarrow 0,N\left(\Delta\mu\rightarrow
3/16\right)\rightarrow\infty.$ (25)

Near the top edge of the bandgap, defined as per the top line of Eq. (11),
approximate relation (22) is replaced by one following from the second
equation of Eq. (21):

$\phi_{+}\approx{\frac{1}{2\Omega}}\left({\frac{\partial}{\partial
x}}-i{\frac{\partial}{\partial y}}\right)\phi_{-}.$ (26)

Substituting it into the second equation of Eq. (21), one obtains

$-\Delta\mu\cdot\phi_{-}=-{\frac{1}{2\Omega}}\left({\frac{\partial^{2}}{\partial
x^{2}}}+{\frac{\partial^{2}}{\partial
y^{2}}}\right)\phi_{-}+|\phi_{-}|^{2}\phi_{-}-|\phi_{-}|^{4}\phi_{-},$ (27)

cf. Eq. (23). It is obvious that this equation, which combines the nonlinear
terms with the signs opposite to those in Eq. (23), i.e., cubic self-repulsion
and quintic attraction, may produce only unstable 2D solitons, which are
subject to the destructive effect of the supercritical collapse.

Figure 2: (Color online) Density patterns, $|\phi_{+}\left(x,y)\right)|^{2}$
and $|\phi_{-}\left(x,y\right)|^{2}$ (the first and second columns,
respectively), and phase distributions in $\phi_{+}\left(x,y\right)$ and
$\phi_{-}\left(x,y\right)$ (the third and fourth columns, respectively) for
the two components of stable solitons located near the bottom edge of the
bandgap, see Eq. (11), as produced by the numerical solution of Eq. (21).
Values of the winding numbers of the components, from top to bottom, are:
$(S_{+},S_{-})=(-2,-1)$, $(-1,0)$, $(0,1)$, $(1,2)$, and $(2,3)$. In all the
panels, we fix the total norm as $N=50$ (see Eq. (13), and the ZS strength is
$\Omega=5$.

##  III Numerical results

###  III.1 Stationary solutions for the vortex gap solitons

The analysis presented in the previous section indicates that stable
stationary solutions for gap solitons may exist near the bottom edge of the
bandgap produced by Eq. (6) or, according to Eqs. (7) and (14), near the top
edge in the case of Eq. (5). The initial guess for solutions of stationary Eq.
(21) was obtained by means of the imaginary-time method, applied to the time-
dependent version of Eq. (23). Then, iterating the initial solution in the
framework of the power-conserved squared-operator method Yang , stationary
solutions were produced in the numerically accurate form. Lastly, stability of
the soliton solutions was tested by real-time simulations of Eq. (6), running
from $t=0$ to $1000$. Actually, this is a very long time, which corresponds,
roughly, to $\sim 1000$ solitons’ relaxation time, thus making conclusions
about the stability of the stationary states fully reliable.

Note that the topological charges (winding numbers) of two components of the
stationary solitons are subject to a simple relation Sakaguchi2014  [the same
relation follows from Eq. (26)],

$S_{+}=S_{-}-1.$ (28)

Typical examples of the numerically constructed stable stationary solution for
the solitons, located close to the bottom edge of the bandgap, with
$(S_{+},S_{-})=(-2,-1),(-1,0),(0,1),(1,2)$ and $(2,3)$, are displayed in Fig.
2. In particular, all the two-component modes in which both $S_{\pm}$ are
different from zero represent FVGSs. Thus, these composite modes may indeed be
stable, unlike their completely unstable counterparts in the form of “excited
states”, in the usual SOC system with the cubic self-attraction Sakaguchi2014
. Further, Fig. 2 shows that component $\phi_{+}$ gives a dominant
contribution to norm Eq.(13), while $\phi_{-}$ plays a subordinate role (i.e.,
$N_{-}\ll N_{+}$, ), being mainly distributed along the inner rim of
$\phi_{+}$ in the case of $|S_{+}|>|S_{-}|$, or along the outer rim in the
opposite case, $|S_{+}|<|S_{-}|$. For the FVGS modes located near the top edge
of the bandgap, Eq. (26) indicates that $\phi_{-}$ plays a dominant role,
while $\phi_{+}$ is a subordinate component $N_{-}\gg N_{+}$, which is mainly
distributed along the inner or outer rim of $\phi_{-}$ for $|S_{+}|<|S_{-}|$
or $|S_{+}|>|S_{-}|$, respectively.

Characteristics of these soliton families are summarized in Fig. 3, in which
we plot the shift of the chemical potential, $\Delta\mu$, defined as per the
bottom line of Eq. (11), and the effective area of the soliton’s dominant
component ($\phi_{\mathrm{d}}$),

$A\equiv\frac{\left(\int\int|\phi_{\mathrm{d}}\left(x,y\right)|^{2}dxdy\right)^{2}}{\int\int|\phi_{\mathrm{d}}\left(x,y\right)|^{4}dxdy},$
(29)

versus the total norm $N$ and ZS strength $\Omega$. Note that $\Delta\mu$ in
panels (a) and (c) of the figure varies precisely in interval Eq. (24), in
agreement with limit relations (25). The figure shows that there are two
threshold values for $N$ and $\Omega$, namely, $N^{\mathrm{c1,c2}}$ and
$\Omega^{\mathrm{c1,c2}}$, with $N^{\mathrm{c1}}<N^{\mathrm{c2}}$ and
$\Omega^{\mathrm{c1}}<\Omega^{\mathrm{c2}}$. When $N<N^{\mathrm{c1}}$ or
$\Omega<\Omega^{\mathrm{c1}}$, no soliton solutions are found, only spatially
uniform states being possible; note that
$N^{\mathrm{c1}}=\Omega^{\mathrm{c1}}=0$ for $S_{+}=0$ [these findings agree
with the known fact that 2D vortex-soliton solutions of Eq. (23) exist if
their norm exceeds a certain threshold value, while there is no existence
threshold for zero-vorticity solutions Michinel2 . In intervals
$N^{\mathrm{c1}}<N<N^{\mathrm{c2}}$ and
$\Omega^{\mathrm{c1}}<\Omega<\Omega^{\mathrm{c2}}$, soliton solutions can be
found, but they are unstable (as shown by dashed and dotted curves in Fig. 3).
Finally, stable solutions exist at $N>N^{\mathrm{c2}}$ and
$\Omega>\Omega^{\mathrm{c2}}$. These conclusions for the 2D cubic-quintic
system are different from the case of its counterpart with the cubic-only
nonlinearity, where the SV solutions with $S_{+}=0$, $S_{-}=1$ [see Eq. (28)]
are stable in the entire bandgap Sakaguchi2016 .

Figure 3: (Color online) The chemical-potential shift, defined as per Eq.
(11), and the effective area of the dominant component, defined as per Eq.
(29), versus the total norm $N$, at a fixed value of the ZS strength $\Omega$
(a,b), and versus $\Omega$ at a fixed value of $N$ (c,d). The results are
produced by the numerical solution of Eq. (21). Blue, red and green curves
represent the dependences for the soliton families with the vorticity in the
dominant component $S_{+}=0$, $\pm 1$ and $\pm 2$, respectively, while the
vorticity in the other component is given by Eq. (28). Solid curves represent
stable branches, which correspond to $N>N^{\mathrm{c2}}$ and
$\Omega>\Omega^{\mathrm{c2}}$ (see the text). These stability boundaries are
designated by color balls on the curves. Dashed and dotted curves represent
unstable branches, at $N<N^{\mathrm{c2}}$ and $\Omega<\Omega^{\mathrm{c2}}$
(the dashed curves pertain to $S_{+}=0$, $-1$ and $-2$, and the dotted ones
correspond to $S_{+}=+1$ and $+2$. The stability of identified by means of
systematic simulations of Eq. (6).

###  III.2 Mobility of the gap solitons initiated by kicking

Construction of moving solitons in the framework of Eqs. (5) and (6) is a
nontrivial problem, as they are not Galilean invariant equations. We have
studied the mobility in direct simulations via Eqs. (5) and (6), respectively,
by applying kick $\eta$ to the stationary solution in the $x$ or $y$
direction. According to the relationship in Eq. (14), the corresponding inputs
are defined as

$\psi_{\pm}(x,y,t=0)=\begin{cases}\phi_{\mp}^{\ast}(x,y)\\{e^{i\eta
x},e^{i\eta y}\\}&\mathrm{for\quad Eq.(\ref{1}),}\\\
\phi_{\pm}(x,y)\\{e^{i\eta x},e^{i\eta y}\\}&\mathrm{for\quad
Eq.(\ref{2}).}\end{cases}$ (30)

The substitution of these expressions in Eq. (17) yields the respective values
of the momentum produced by the kick,

$P_{x,y}=N\eta.$ (31)

For convenience, we name stable solutions of Eq. (5), which are located near
the top edge of the bandgap, as “top-type solitons”, while the stable
solutions of Eq. (6), which are located near the bottom edge of the bandgap,
are named as “bottom-type solitons”.

Figure 4: (Color online) Motion of the soliton at the bottom (a1,a2) and top
(b1,b2) of the bandgap initiated by kick $\eta=0.1$ in the positive $y$
direction, see Eq. (30). The first row is the motion of the soliton located
near the bottom of the bandgap, with parameters
$(N_{+},S_{+},\Omega)=(50,1,10)$. The second row is the same for the soliton
located close to the top of the bandgap, with
$(N_{-},S_{-},\Omega)=(50,1,10)$.  Figure 5: (Color online) (a) The velocity
of the gap soliton, which is kicked along the positive $x$ (red) or $y$ (blue)
direction, according to Eq. (30), versus $\eta$. The velocity is found
numerically as per Eqs. (32) and (33) The soliton is taken near the bottom of
the bandgap, with $(N_{+},S_{+},\Omega)=(50,0,10)$. (b) The effective mass of
the solitons located near the top and bottom of the bandgap (short-dashed and
dashed lines, respectively) versus the ZS strength, $\Omega$. Parameters of
the solitons located near the bottom and top edges of the bandgap types are
selected as $(N_{\pm},S_{\pm},\eta)=(50,0,0.1)$. (c) The effective mass of the
solitons located near the top and bottom of the bandgap (short-dashed and
dashed lines, with $(\Omega,S_{\pm},\eta)=(10,0,0.1)$, respectively) versus
the total norm $N$. The masses are found as per Eq. (34).

Figure 4 displays cross-sections of the two components of the moving solitons
located close to the bottom (a1,a2) and top (b1,b2) of the bandgap, applying
$\eta=0.1$ in the positive $y$ direction, as per Eq. (30). The figure shows
that the solitons in these two cases (for the solitons close to the bottom and
top of bandgap) move steadily along the $y$ directions, which indicates that
they have a negative and positive effective mass, respectively. The signs of
the effective mass at the two edges of the bandgap are consistent with
previous results Li2017 .

To accurately identify the effective mass of the solitons from the numerical
data, as the ratio of momenta given by Eq. (31) [see Eq. (17)] to the
corresponding velocities $V_{x,y}$, the center-of-mass (COM) coordinate of the
soliton’s dominant component is calculated as

$\\{X_{\mathrm{COM}}(t),Y_{\mathrm{COM}}(t)\\}=\frac{\int\int\\{x,y\\}|\psi_{\mathrm{d}}(x,y,t)|^{2}dxdy}{\int\int|\psi_{\mathrm{d}}(x,y,t)|^{2}dxdy}.$
(32)

Then, the velocity is found as

$V_{x}=[X_{\mathrm{COM}}(t)-X_{\mathrm{COM}}(0)]/t,\quad
V_{y}=[Y_{\mathrm{COM}}(t)-Y_{\mathrm{COM}}(0)]/t.$ (33)

Eventually, effective masses are obtained from here as

$M_{x,y}=\frac{P_{x,y}}{V_{x,y}}\equiv\frac{N\eta}{V_{x,y}}.$ (34)

The result, $V_{x}=V_{y}$, as displayed in Fig. 5(a), implies that, naturally,
the effective mass is isotropic in the present system, i.e., $M_{x}=M_{y}$.
Figs. 5(b,c) show that, as mentioned above, the solitons located near the
bottom and top edges of the bandgap have negative and positive effective
masses, respectively. Moreover, the numerical results, displayed in these
figures, demonstrate that the effective masses linearly depend on $\Omega$ and
$N$, and do not depend on winding numbers $S_{\pm}$. The increase of the
masses with the increase of the ZS strength, $\Omega$, is explained by the
fact that gap solitons do not exist at $\Omega=0$. The linear dependence of
the masses on $N$ is a common property of solitons in models of the GPE type.

Figure 6: (Color online) The density distribution in the moving solitons
kicked in the positive $y$ direction with different values of $\eta$ as per
Eq. (30). The first and second rows display the dominant and subordinate
components, respectively. Parameters of the soliton in panels (a1,a2,c1,c2)
are $(N_{+},S_{+},\Omega)=(50,1,10)$, while for the soliton in panels
(b1,b2,d1,d2) they are $(N_{+},S_{+},\Omega)=(50,-1,10)$. The first and second
columns display the solitons kicked with $\eta=0.1$ and measured at $t=800$,
while the third and fourth columns pertain to the solitons kicked with a kick
which is ten times as large, therefore the data are taken at time which is ten
times smaller, $t=80$. Figure 7: (Color online) (a) NOAM of the subordinate
component of the soliton of the bottom-edge type versus $t$ for different
values of the kick, i.e., $\eta=0.1$ and $1$ (the red solid and blue dashed
curves, respectively). Parameters of the soliton are
$(N_{+},S_{-},\Omega)=(50,1,10)$. (b) NOAM of a moving soliton with
$(N_{+},S_{-},\eta)=(50,1,0.1)$ and different values of the ZS strength, i.e.,
$\Omega=5$ and $10$ (the green dashed and red solid curves, respectively). (c)
The oscillation frequency of the NOAM of the subordinate component versus
$\Omega$. Here, we also select the bottom-edge type of the soliton, with
parameters $(N_{+},S_{+},\eta)=(50,0,0.1)$.

Typical examples of the density distribution in the solitons set in motion by
the kick with different values of $\eta$ are shown, for the solitons taken
near the bottom edge of the bandgap, in Fig. 6. The snapshots are taken at the
moment of time when the COM of the soliton kicked in the $y$ direction is
located at $y\approx 8$. It is observed that the dominant component of the
moving solution ($\phi_{+}$, in the present case) maintains its shape, while
the subordinate component, $\phi_{-},$ exhibits a deformation, as a
consequence of the absence of the Galilean invariance in the system.
Accordingly, in the case of a small kick $\eta$, the density distribution of
the subordinate component is still confined within the main body of the moving
soliton, see Figs. 6(a2,b2). In contrast, at large $\eta$, the subordinate
components is lagging behind the main body of the soliton, which gives rise to
its gradual delocalization, see Figs. 6(c2,d2).

In the case of small $\eta$, the subordinate component shows an asymmetric
density distribution and rotates around the COM in the course of the direct
simulations of the motion. This fact explains density oscillations in the
cross-section diagram in Figs. 4(a2,b1). To characterize the motion of the
soliton, we define the normalized orbital angular momentum (NOAM) of the two
components as

$\ \left\langle
L_{\pm}\right\rangle=\frac{\int\psi_{\pm}^{\ast}\hat{L}\psi_{\pm}d\mathbf{r}}{N_{\pm}},$
(35)

where $\hat{L}=-i(x\partial_{y}-y\partial_{x})$ is the angular-momentum
operator. The direct simulations show that the NOAM of the dominant component
stays nearly constant in the course of the motion, while the NOAM of the
subordinate one oscillates as a function of $t$. For small $\eta$, the
variation of $\left\langle L_{-}(t)\right\rangle$ manifests itself as harmonic
oscillations [see the red curve in Fig. 7(a)], while for large $\eta$, the
oscillations of $\left\langle L_{-}(t)\right\rangle$ are anharmonic due to the
above-mentioned trend to delocalization of the density distribution, see the
blue dashed curve in Fig. 7(a). Figure 7(b) displays the harmonic oscillations
of $\left\langle L_{-}(t)\right\rangle$ for different values of $\Omega$.
Further, the numerical results show that the frequency of the harmonic
oscillations linearly depends on the ZS strength $\Omega$ for a fixed value of
the norm of the subordinate component, see Fig. 7(c). On the other hand, the
oscillation frequency does not depend on the winding number. Because the
frequency of the harmonic oscillation is determined by the oscillator’s mass,
the conclusion is the same as produced above on the basis of different
numerical data in Fig. 5(b): the effective mass of the soliton linearly grows
with $\Omega$, but it does not depend on the topological charge of the
soliton.

###  III.3 Collisions between moving gap solitons

Figure 8: (Color online) Collisions of fundamental solitons (ones with
$S_{+}=0$). Panels feature the collisions between the solitons set in motion
as per Eq. (37) with (a) $\eta=0.1$, (b) $\eta=0.5$, (c) $\eta=0.9$, and (d)
$\eta=3$. Here, parameters of the gap solitons are selected as
$(N_{+},\Omega)=(10,10)$. Figure 9: (Color online) Parameters of the colliding
FVGSs in this figure are selected as $(N_{+},\Omega)=(20,10)$. (a,b) The
colliding gap solitons have the same topological charge, with $S_{+}=1$. (a)
An elastic collision between two gap solitons with $\eta=0.1$. (b) An
inelastic collision with $\eta=0.7$. (c,d) Collisions of the solitons with
opposite topological charges, $S_{+}=\pm 1$. The kick corresponding to Eq.
(37) is $\eta=0.1$ in (c), and $\eta=0.7$ in (d).

A straightforward possibility is to simulate collisions between solitons which
are set in motion by opposite kicks $\pm\eta$. For the solitons located near
the bottom and top edges of the bandgap, we define the respective initial wave
functions as

$\psi_{\pm}(x,y,t=0)=\phi_{\pm}(x,y+y_{0})e^{-i\eta
y}+\phi_{\pm}(x,y-y_{0})e^{i\eta y},$ (36)

and

$\psi_{\pm}(x,y,t=0)=\phi_{\pm}(x,y+y_{0})e^{i\eta
y}+\phi_{\pm}(x,y-y_{0})e^{-i\eta y},$ (37)

where $2y_{0}$ is the initial separation between the solitons. Due to the
symmetry provided by Eq. (14), we here consider only the collision between the
solitons of the bottom-edge type.

Figures 8 and 9 display several typical examples of collisions between the
fundamental solitons and FVGSs, with different values of $\eta$, which are
displayed by the contour surface of the density of the dominant component. In
Fig. 8(a), a completely inelastic collision is observed for a small value of
$\eta$, with the colliding fundamental solitons merging into a breather. On
the contrary, in Fig. 8(b) a “twisted” quasi-elastic collision is observed for
a moderate value of $\eta$: the colliding fundamental solitons first merge and
then separate into ones moving perpendicular to the initial direction. It is
further shown in Fig. 8(c) that three fragments appear after the collision
when $\eta$ is large, a new fragment being a small-amplitude breather.
Finally, it is shown in Fig. 8(d) that the solitons are completely destroyed
by the collision if $\eta$ is sufficiently large.

The collision between two vortex solitons (FVGSs) shows different dynamics. In
Figs. 9(a,b), the colliding solitons have identical topological charges,
$S_{+}=1$. When $\eta$ is small, the solitons repel each other and maintain
their shapes after the rebound, i.e., they collide elastically. In contrast,
in the case of opposite topological charges, the colliding solitons also repel
each other, but the outcome is different: they split into four fragments, as
shown in Fig. 9(c).

When $\eta$ is large, the collisions of the vortex solitons are always
inelastic. In the case of identical topological charges, collision-generated
fragments are hurled in all directions, see Fig. 9(b). In the case of opposite
topological charges, three fragments are generated by the collision, see Fig.
9(d). If $\eta$ is sufficiently large, the colliding FVGSs are completely
destroyed by the collision. The latter case is not shown here.

Thus, the present system, based on Eq. (6), which is dominated by the SOC
energy demonstrates the behavior completely different from what is common for
usual models with the domination of the kinetic energy: collisions with large
values of velocities tend to be destructive, rather than quasi-elastic.

##  IV Conclusions

In this work, we have investigated the formation of stable FVGSs (full-vortex
gap solitons) and their dynamics in the two-component BEC with SOC (spin-orbit
coupling) and competing cubic and quintic interactions. The difference from
the previously studied 2D solitons in the SOC system with the cubic attractive
interactions is that they might be stable solely in the SV (semi-vortex) form,
with the vorticity carried by a single component. When the SOC and ZS (Zeeman-
splitting) terms are sufficiently strong, the kinetic-energy ones may be
neglected in the system of coupled GPEs (Gross-Pitaevskii equations). The so
simplified system maintains a bandgap in its linear spectrum, populated by
FVGSs. They may be stable near edges of the bandgap. If the absolute values of
the topological charge of the dominant component is larger than that of the
secondary one, the density distribution of the secondary component is
surrounded by that of the dominant component. Otherwise, the dominant
component is surrounded by the secondary one. We represent FVGS families by
displaying dependences of the chemical potential and solitons’ effective area
on the norm and ZS strength. The results reveal stability boundaries for the
families. The mobility of the FVGSs is explored by kicking the stationary
solitons. It is thus found that the FVGSs formed near the top and bottom edges
feature, respectively, positive and negative dynamical masses. For a small
kick, the minor component of solitons oscillates with the frequency
proportional to the ZS strength. In contrast, under the action of a larger
kick the minor suffers strong deformation, eventually leading to
delocalization of the kicked soliton. Collisions between FVGSs with identical
and opposite topological charges were considered too, featuring both elastic
and inelastic outcomes.

A natural direction to extend the present work is to investigate self-trapped
modes of the FVGSs type in a system with long-range dipole-dipole
interactions, as well as in three-dimensional systems.

###### Acknowledgements.

This work was supported by NNSFC (China) through grant Nos. 11905032 &
11874112, Natural Science Foundation of Guangdong province through grant
No.2021A1515010214, the GuangDong Basic and Applied Basic Research Foundation
through grant No. 2021A1515111015, the Key Research Projects of General
Colleges in Guangdong Province through grant No. 2019KZDXM001, the Foundation
for Distinguished Young Talents in Higher Education of Guangdong through
grants Nos. 2018KQNCX279 & 2018KTSCX241, the Special Funds for the Cultivation
of Guangdong College Students Scientific and Technological Innovation through
grant Nos. pdjh2021b0529 & pdjh2022a0538, the Research Fund of Guangdong-Hong
Kong-Macao Joint Laboratory for Intelligent Micro-Nano Optoelectronic
Technology through grant No.2020B1212030010 and the Graduate Innovative
Talents Training Program of Foshan University. The work of B.A.M. is
supported, in part, by the Israel Science Foundation through grants No.
1286/17 and 1695/22.

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$\mathbb{R}^{n}$. Fix $d\in(0,n)$. Let $S\subset\mathbb{R}^{n}$. For $r>0$,
let $B(S,r)=\\{x\in\mathbb{R}^{n}:\operatorname{dist}(x,S)\leq r\\}$, and
$\operatorname{Cont}(S;r)=r^{d-n}\lambda^{n}(B(S,r))$. The upper and lower
($d$-dimensional) Minkowski content of $S$ are respectively defined by

$\overline{\operatorname{Cont}}(S):=\limsup_{r\to
0^{+}}\operatorname{Cont}(S;r),\quad\underline{\operatorname{Cont}}(S):=\liminf_{r\to
0^{+}}\operatorname{Cont}(S;r).$

When $\overline{\operatorname{Cont}}(S)=\underline{\operatorname{Cont}}(S)$,
the common value, denoted by $\operatorname{Cont}(S)$ or
$\operatorname{Cont}_{d}(S)$, is called the ($d$-dimensional) Minkowski
content of $S$.

######  Remark 6.1.

The definition here differs from that in [5, 3.2.37] in two aspects. First, we
allow $d$ to be not an integer; second, we omit a multiplicative constant:
$\frac{\Gamma(\frac{n-d}{2}+1)}{\pi^{\frac{n-d}{2}}}$ for simplicity.

For $S_{1},S_{2}\subset\mathbb{R}^{n}$, from $\lambda^{n}(B(S_{1}\cup
S_{2},r))\leq\lambda^{n}(B(S_{1},r))+\lambda^{n}(B(S_{2},r))$, we get

$\overline{\operatorname{Cont}}(S_{1}\cup
S_{2})\leq\overline{\operatorname{Cont}}(S_{1})+\overline{\operatorname{Cont}}(S_{2}).$
(6.1) $\underline{\operatorname{Cont}}(S_{1}\cup
S_{2})\leq\underline{\operatorname{Cont}}(S_{1})+\overline{\operatorname{Cont}}(S_{2}).$
(6.2)

######  Lemma 6.2.

For any compact sets $S_{1},S_{2}\subset\mathbb{R}^{n}$,

$\underline{\operatorname{Cont}}(S_{1})+\underline{\operatorname{Cont}}(S_{2})\leq\underline{\operatorname{Cont}}(S_{1}\cup
S_{2})+\lim_{\varepsilon\to 0^{+}}\overline{\operatorname{Cont}}(S_{1}\cap
B(S_{1}\cap S_{2},\varepsilon)).$ (6.3)

###### Proof.

Fix $\varepsilon>0$. Since $S_{1}\cap B(S_{2},1/k)\downarrow S_{1}\cap S_{2}$,
there is $k_{0}\in\mathbb{N}$ such that $S_{1}\cap B(S_{2},1/k_{0})\subset
B(S_{1}\cap S_{2},\varepsilon)$. Let $r\in(0,1/(2k_{0}))$. Suppose $x\in
B(S_{1},r)\cap B(S_{2},r)$. Then there exist $y_{j}\in S_{j}$ such that
$|x-y_{j}|\leq r$, $j=1,2$. Then $|y_{1}-y_{2}|\leq 2r\leq 1/k_{0}$. So
$y_{1}\in S_{1}\cap B(S_{2},1/k_{0})\subset S_{1}\cap B(S_{1}\cap
S_{2},\varepsilon)$, which implies that $x\in B(S_{1}\cap B(S_{1}\cap
S_{2},\varepsilon),r)$. So we get $B(S_{1},r)\cap B(S_{2},r)\subset
B(S_{1}\cap B(S_{1}\cap S_{2},\varepsilon),r)$. Thus,

$\operatorname{Cont}(S_{1};r)+\operatorname{Cont}(S_{2};r)\leq\operatorname{Cont}(S_{1}\cup
S_{2};r)+\operatorname{Cont}(S_{1}\cap B(S_{1}\cap S_{2},\varepsilon);r).$

We then get (6.3) by first sending $r$ to $0^{+}$ and then sending
$\varepsilon$ to $0^{+}$. ∎

For $T\subset S\subset\mathbb{R}$, we write $\partial_{S}T$ for the relative
boundary of $T$ in $S$, i.e., $S\cap\overline{T}\cap\overline{S\setminus T}$.

######  Definition 6.3.

Let ${\cal M}$ denote the family of all sets in $\mathbb{R}^{n}$ that can be
expressed as the intersection of an open set with a closed set. Fix $S\in\cal
M$. A $d$-dimensional Minkowski content measure on $S$ is a Borel measure
$\mu$ on $S$ which satisfies that, for any compact set $K\subset S$, (i)
$\mu(K)<\infty$, and (ii) whenever $\mu(\partial_{S}K)=0$,
$\operatorname{Cont}_{d}(K)$ exists and equals $\mu(K)$.

######  Remark 6.4.

A Minkowski content measure $\mu$ on any $S\in\cal M$ is $\sigma$-finite since
$S$ can be expressed as a countable union of compact sets. Given such $\mu$,
for $1\leq k\leq n$, let $E_{\mu,k}$ denote the set of points $c\in\mathbb{R}$
such that $\mu\\{x\in\mathbb{R}^{n}:x_{k}=c\\}>0$. Then $E_{\mu,k}$ is
countable. Let $\cal R_{\mu}$ denote the family of $n$-dimensional rectangles
$R=\prod_{k=1}^{n}I_{k}\subset\mathbb{R}^{n}$ such that each $I_{k}$ is a
compact interval with $\partial I_{k}\cap E_{\mu,k}=\emptyset$. Then
$\mu(\partial R)=0$ for $R\in\cal R_{\mu}$, and every compact $n$-dimensional
rectangle in $\mathbb{R}^{n}$ could be approximated by elements in $\cal
R_{\mu}$.

######  Lemma 6.5.

The Minkowski content measure on any $S\in\cal M$ is unique , when it exists.

###### Proof.

Let $S=F\cap G\in\cal M$, where $F$ is closed and $G$ is open. Let
$\mu_{1},\mu_{2}$ be Minkowski content measures on $S$. Let ${\cal
R}_{\mu_{1},\mu_{2}}^{G}$ be the family of $R\in{\cal R}_{\mu_{1}}\cap{\cal
R}_{\mu_{2}}$ that are contained in $G$. For any $R\in{\cal
R}_{\mu_{1},\mu_{2}}^{G}$, from $\mu_{j}(\partial R)=0$ and
$\partial_{S}(R\cap F)\subset\partial R$, we get $\mu_{j}(R)=\mu_{j}(R\cap
F)=\operatorname{Cont}(R\cap F)$, $j=1,2$, which implies that
$\mu_{1}(R)=\mu_{2}(R)$. Since every compact rectangle in $G$ can be
approximated by elements in ${\cal R}_{\mu_{1},\mu_{2}}^{G}$, we get
$\mu_{1}=\mu_{2}$. ∎

######  Lemma 6.6.

(i) If $\mu$ is the Minkowski content measure on $S\in\cal M$, then for any
compact set $K\subset S$, $\overline{\operatorname{Cont}}(K)\leq\mu(K)$. (ii)
If $S\subset\mathbb{R}^{n}$ is compact, and $\mu$ is a measure on $S$ such
that $\operatorname{Cont}(S)=\mu(S)<\infty$, and for any compact set $K\subset
S$, $\overline{\operatorname{Cont}}(K)\leq\mu(K)$, then $\mu$ is the Minkowski
content measure on $S$.

###### Proof.

(i) Let $S=F\cap G$, where $F$ is closed and $G$ is open. Let $K$ be a compact
subset of $S$. Then for any $k>0$, $K$ can be covered by finitely many
rectangles in $\cal R_{\mu}$ that are contained in $B(K,1/k)\cap G$. Let
$K_{k}$ be the union of these rectangles. Then $K_{k}$ is a compact subset of
$B(K,1/k)\cap G$, and $\mu(\partial K_{k})=0$. Now $K_{k}\cap S=K_{k}\cap F$
is a compact subset of $S$, and $\partial_{S}(K_{k}\cap S)\subset\partial
K_{k}$. So $\mu(\partial_{S}(K_{k}\cap S))=0$. Thus,
$\overline{\operatorname{Cont}}(K)\leq\overline{\operatorname{Cont}}(K_{k}\cap
S)=\mu(K_{k}\cap S)\leq\mu(B(K,1/k))$. Sending $k$ to $\infty$, we get
$\overline{\operatorname{Cont}}(K)\leq\mu(K)$.

(ii) Let $K$ be a compact subset of $S$ such that $\mu(\partial_{S}K)=0$. Let
$L=\overline{S\setminus K}$. Then $L$ is also a compact subset of $S$,
$S=K\cup L$, and $K\cap L=\partial_{S}K$. From $\mu(\partial_{S}K)=0$ we get
$\operatorname{Cont}(S)=\mu(S)=\mu(K)+\mu(L)$. Thus, by the assumption and
(6.2),

$\mu(S)=\operatorname{Cont}(S)\leq\underline{\operatorname{Cont}}(K)+\overline{\operatorname{Cont}}(L)\leq\overline{\operatorname{Cont}}(K)+\overline{\operatorname{Cont}}(L)\leq\mu(K)+\mu(L)=\mu(S).$

So we get $\operatorname{Cont}(K)=\mu(K)$, and conclude that $\mu$ is the
Minkowski content measure on $S$. ∎

######  Corollary 6.7.

Let $S\in\cal M$ and $\mu$ be a measure on $S$. Suppose there exists a family
of compact subsets $\\{F_{\iota}\\}_{\iota\in I}$ of $S$ such that (i) for
every $\iota\in I$, $\operatorname{Cont}(F_{\iota})=\mu(F_{\iota})<\infty$;
(ii) for every compact $K\subset S$, there exists $\iota\in I$ such that
$K\subset F_{\iota}$; and (iii) every compact $K\subset S$ can be expressed as
$\bigcap_{n=1}^{\infty}J_{n}$, where $(J_{n})$ is a decreasing sequence of
sets such that each $J_{n}$ is a finite disjoint union of elements in
$\\{F_{\iota}\\}$. Then $\mu$ is the Minkowski content measure on $S$.

###### Proof.

Let $K_{0}\subset S$ be compact. By (ii) there is $\iota\in I$ such that
$K_{0}\subset F_{\iota}$. So we have $\mu(K_{0})\leq\mu(F_{\iota})<\infty$ by
(i). Suppose further that $\mu(\partial_{S}K_{0})=0$. Let
$\mu_{F_{\iota}}=\mu(\cdot\cap F_{\iota})$ be a measure on $F_{\iota}$. Then
$\operatorname{Cont}(F_{\iota})=\mu_{F_{\iota}}(F_{\iota})<\infty$. Let
$K\subset F_{\iota}$ be compact. By (iii) we express $K=\bigcap J_{n}$, where
$(J_{n})$ is a decreasing sequence of sets each of which is a finite disjoint
union of elements in $\\{F_{\iota}\\}$. By (6.1) and (i), each $J_{n}$
satisfies $\overline{\operatorname{Cont}}(J_{n})\leq\mu(J_{n})$. Since
$\mu(J_{n})\downarrow\mu(K)$, we get
$\overline{\operatorname{Cont}}(K)\leq\mu(K)$. By Lemma 6.6 (ii),
$\mu_{F_{\iota}}$ is the Minkowski content measure on ${F_{\iota}}$. Since
$K_{0}$ is a compact subset of ${F_{\iota}}$, and
$\mu_{F_{\iota}}(\partial_{F_{\iota}}K_{0})\leq\mu(\partial_{S}K_{0})=0$, we
get $\operatorname{Cont}(K_{0})=\mu_{F_{\iota}}(K_{0})=\mu(K_{0})$. Thus,
$\mu$ is the Minkowski content measure on $S$. ∎

######  Example 6.8.

Let $n=1$. Let $B$ be a $1$-dimensional Brownian motion, and let $L_{t}$ be
the usual local time of $B$ at $0$ (cf. [19]). Let $Z$ be the zero set
$\\{t:B_{t}=0\\}$. There is a deterministic constant $C_{0}\in(0,\infty)$ such
that for any $t_{2}\geq t_{1}\geq 0$,
$\operatorname{Cont}_{1/2}(Z\cap[t_{1}{,}t_{2}])=C_{0}(L_{t_{2}}-L_{t_{1}})$
(cf. [6]). By applying Corollary 6.7 with
$\\{F_{\iota}\\}=\\{Z\cap[t_{1},t_{2}]:t_{2}\geq t_{1}\geq 0\\}$, we can
conclude that $C_{0}dL$ is the $1/2$-dimensional Minkowski content measure on
$Z$.

######  Example 6.9.

Let $n=2$. Let $\kappa\in(0,8)$ and $d=1+\frac{\kappa}{8}$. Let $\eta$ be an
SLEκ curve in $\mathbb{H}$ from $0$ to $\infty$. The natural parametrization
of $\eta$ is a random continuously strictly increasing function $\theta$ that
satisfies that for any ${t_{2}\geq t_{1}\geq 0}$,
$\operatorname{Cont}_{d}(\eta[{t_{1},t_{2}}])=\theta({t_{2}})-\theta({t_{1}})$
([10]). By applying Corollary 6.7 with
$\\{F_{\iota}\\}=\\{\eta([t_{1},t_{2}]):t_{2}\geq t_{1}\geq 0\\}$, we can
conclude that $\mu_{\eta}:=\eta_{*}(d\theta)$ is the $d$-dimensional Minkowski
content measure on the image of $\eta$. Here we use the fact that
$\eta^{-1}(K)$ is compact for any compact set $K$, which follows from the
transience of $\eta$, i.e., $\lim_{t\to\infty}\eta(t)=\infty$.

######  Example 6.10.

Let $n\geq 3$. Let $B$ be an $n$-dimensional Brownian motion. There is a
constant $C_{n}\in(0,\infty)$ such that for any $t_{2}\geq t_{1}\geq 0$,
$\operatorname{Cont}_{2}(B[t_{1},t_{2}])=C_{n}(t_{2}-t_{1})$ (cf. [6]). Then
$C_{n}B_{*}{(}\lambda^{1}|_{[0,\infty)}{)}$ is the $2$-dimensional Minkowski
content measure on the image of $B$. The argument is similar to that in the
previous example.

######  Theorem 6.11.

Let $S\subset\mathbb{R}^{n}$ be compact. For $r>0$, let $\mu_{r}$ be the
measure such that $d\mu_{r}=r^{d-n}\mathbbm{1}_{B(S,r)}d\lambda^{n}$. Then
$\mu$ is the Minkowski content measure on $S$ iff $\mu_{r}\to\mu$ weakly as
$r\to 0^{+}$.

###### Proof.

First, suppose $\mu_{r}\to\mu$ weakly as $r\to 0^{+}$. Since $\mu_{r}$ is
supported by $B(S,r)$, $\mu$ is supported by $S$. Since
$|\mu_{r}|=\operatorname{Cont}(S;r)$ by definition, we get
$|\mu|=\lim|\mu_{r}|=\operatorname{Cont}(S)$. Let $K$ be a compact subset of
$S$. Fix $r_{0}>0$. Then for any $r\in(0,r_{0})$,
$\operatorname{Cont}(K;r)\leq\mu_{r}(B(K;r_{0}))$. Sending $r$ to $0^{+}$, we
get $\overline{\operatorname{Cont}}(K)\leq\limsup_{r\to
0^{+}}\mu_{r}(B(K,r_{0}))\leq\mu(B(K,r_{0}))$ since $B(K,r_{0})$ is closed.
Sending $r_{0}$ to $0^{+}$, we get
$\overline{\operatorname{Cont}}(K)\leq\mu(K)$. By Lemma 6.6 (ii), $\mu$ is the
Minkowski content measure on $S$.

Second, suppose $\mu$ is the Minkowski content measure on $S$. Then
$|\mu|=\operatorname{Cont}(S)=\lim_{r\to 0^{+}}|\mu_{r}|$. Thus, the family
$(\mu_{r})_{r\in(0,1]}$ is pre-compact w.r.t. the weak convergence. Suppose
$\mu_{r_{n}}\to\mu^{\prime}$ weakly for some sequence $r_{n}\to 0$. Then
$\mu^{\prime}$ is supported by $S$, and
$|\mu^{\prime}|=\lim|\mu_{r_{n}}|=\lim\operatorname{Cont}(S;r_{n})=\operatorname{Cont}(S)$.
Let $K$ be a compact subset of $S$ such that $\mu(\partial_{S}K)=0$. Then
$\operatorname{Cont}(K)=\mu(K)$. Fix $r_{0}\in(0,1]$. Choose $n$ big enough
such that $r_{n}<r_{0}$. Then
$\mu_{r_{n}}(B(K,r_{0}))\geq\operatorname{Cont}(K,r_{n})$. First sending $n$
to $\infty$ and then sending $r_{0}$ to $0^{+}$, we get
$\mu^{\prime}(K)\geq\operatorname{Cont}(K)=\mu(K)$. Now let $K$ be any compact
subset of $S$. Then we could find a decreasing sequence of compact subsets
$(K_{n})$ of $S$ such that $K=\bigcap_{n}K_{n}$ and $\mu(\partial_{S}K_{n})=0$
for each $n$. Then
$\mu^{\prime}(K)=\lim\mu^{\prime}(K_{n})\geq\lim\mu(K_{n})=\mu(K)$. Since this
holds for any compact $K\subset S$ and $|\mu^{\prime}|=|\mu|$, we get
$\mu^{\prime}=\mu$. This means that any subsequential weak limit of
$(\mu_{r})$ as $r\to 0^{+}$ is $\mu$. Thus, $\mu_{r}\to\mu$ weakly as $r\to
0^{+}$. ∎

######  Theorem 6.12 (Restriction Property).

Let $\mu$ be the Minkowski content measure on $S\in\cal M$. Let
$F\subset\mathbb{R}^{n}$ be closed and $G\subset\mathbb{R}^{n}$ be open.
Suppose $\mu(\partial_{S}(S\cap F))=0$. Let $S^{\prime}=S\cap F\cap G$ and
$\mu^{\prime}=\mu|_{S^{\prime}}$. Then $\mu^{\prime}$ is the Minkowski content
measure on $S^{\prime}$.

###### Proof.

We have $S^{\prime}\in\cal M$ since $\cal M$ is closed under finite
intersections. Let $K\subset S^{\prime}$ be compact. Then $K$ is also a
compact subset of $S$, and so $\mu^{\prime}(K)=\mu(K)<\infty$. Suppose
$\mu^{\prime}(\partial_{S^{\prime}}K)=0$. Then
$\mu(\partial_{S^{\prime}}K)=0$. Let $S_{F}=S\cap F$. Since
$S^{\prime}=S_{F}\cap G$ is relatively open in $S_{F}$, and $K\subset
S^{\prime}$ is compact, we get $\partial_{S^{\prime}}K=\partial_{S_{F}}K$.
From $K\subset S_{F}\subset S$, we get
$\partial_{S}K\subset\partial_{S_{F}}K\cup\partial_{S}S_{F}$. Thus,
$\mu(\partial_{S}K)\leq\mu(\partial_{S^{\prime}}K)+\mu(\partial_{S}S_{F})=0$.
Since $\mu$ is the Minkowski content measure on $S$, we get
$\operatorname{Cont}(K)=\mu(K)=\mu^{\prime}(K)$. So $\mu^{\prime}$ is the
Minkowski content measure on $S^{\prime}$. ∎

The following theorem asserts that the Minkowski content measures are
preserved under covariant transforms [2, Definition 1.1]. For an open set
$G\subset\mathbb{R}^{n}$, we say that a map $\phi:G\to\mathbb{R}^{n}$ is
conformal if it is $C^{1}$ and the Jacobian matrix at every $x\in G$ is some
positive number $s(x)$ times an orthogonal matrix, and the
$|\phi^{\prime}(x)|$ is understood as such (unique) number $s(x)$. When $n=1$,
this means that $\phi$ is $C^{1}$ and $\phi^{\prime}\neq 0$. When $n=2$,
$\phi$ is holomorphic or anti-holomorphic on each component of $G$ with
nonzero complex derivatives. When $n\geq 3$, by Liouville’s theorem, $\phi$ is
a Möbius transformation on each component of $G$.

######  Theorem 6.13 (Conformal Covariance).

Let $\mu$ be the Minkowski content measure on $S\in\cal M$, which is
relatively closed in an open set $G\subset\mathbb{R}^{n}$. Let
$\phi:G\to\mathbb{R}^{n}$ be conformal and injective. Then
$\nu:=\phi_{*}(|\phi^{\prime}|^{d}\cdot\mu)$ is the Minkowski content measure
on $\phi(S)$, where $|\phi^{\prime}|^{d}\cdot\mu$ is the measure $\mu$
weighted by $|\phi^{\prime}|^{d}$, and $\nu$ is the pushforward of
$|\phi^{\prime}|^{d}\cdot\mu$ under $\phi$.

###### Proof.

Since $S$ is relatively closed in $G$ and $\phi$ is a homeomorphism on $G$,
both $\phi(G)$ and $\phi(G\setminus S)$ are open. So
$\phi(S)=\phi(G)\setminus\phi(G\setminus S)\in\cal M$.

Let $K$ be any compact subset of $G$. Let $m_{K}=\min_{x\in
K}|\phi^{\prime}(x)|$ and $M_{K}=\max_{x\in K}|\phi^{\prime}(x)|$. Let $c>1$.
Then there is $\delta>0$ such that $|\phi^{\prime}(x)|<cM_{K}$ for $x\in
B(K,\delta)$, and $|((\phi)^{-1})^{\prime}(y)|\leq c/m_{K}$ for $y\in
B(\phi(K),\delta)$. Thus, for $r>0$ small enough, we have

$\phi(B(K,\frac{r}{cM_{K}}))\subset
B(\phi(K),r)\subset\phi(B(K,\frac{r}{m_{K}/c})),$

which then implies that

$\frac{(m_{K}/c)^{n}}{(cM_{K})^{n-d}}\operatorname{Cont}(K;\frac{r}{cM_{K}})\leq\operatorname{Cont}(\phi(K);r)\leq\frac{(cM_{K})^{n}}{(m_{K}/c)^{n-d}}\operatorname{Cont}(K;\frac{r}{m_{K}/c}).$

First sending $r$ to $0^{+}$ and then sending $c$ to $1^{+}$, we get

${m_{K}^{n}}{M_{K}^{d-n}}\cdot\underline{\operatorname{Cont}}(K)\leq\underline{\operatorname{Cont}}(\phi(K))\leq\overline{\operatorname{Cont}}(\phi(K))\leq{M_{K}^{n}}{m_{K}^{d-n}}\cdot\overline{\operatorname{Cont}}(K).$
(6.4)

Now assume $K\subset S$ and $\mu(\partial_{S}K)=0$. Since
$\operatorname{Cont}(K)=\mu(K)$ and $m_{K}\leq|\phi^{\prime}|\leq M_{K}$ on
$K$, we get

$\frac{m_{k}^{n}}{M_{K}^{n}}\int_{K}|\phi^{\prime}(x)|^{d}\mu(dx)\leq\underline{\operatorname{Cont}}(\phi(K))\leq\overline{\operatorname{Cont}}(\phi(K))\leq\frac{M_{K}^{n}}{m_{K}^{n}}\int_{K}|\phi^{\prime}(x)|^{d}\mu(dx).$
(6.5)

Let $K_{\phi}\subset\phi(S)$ be compact. Let $K=\phi^{-1}(K_{\phi})$. Then
$K\subset S$ is compact. So $\mu(K)<\infty$. Since $|\phi^{\prime}|$ is
bounded on $K$, $\nu(K_{\phi})<\infty$. Suppose
$\nu(\partial_{\phi(S)}K_{\phi})=0$. We want to show that
$\operatorname{Cont}(K_{\phi})=\nu(K_{\phi})$. Since
$\partial_{S}K=\phi^{-1}(\partial_{\phi(S)}K_{\phi})$, we get
$\mu(\partial_{S}K)=0$.

Fix $c>1$. We may find $R_{1},\dots,R_{N}\in{\cal R}_{\mu}$ which intersect
each other only at their boundaries such that
$K\subset\bigcup_{j=1}^{N}R_{j}\subset G$, and for each $1\leq j\leq N$,
$\max_{x\in R_{k}}|\phi^{\prime}(x)|\leq c\min_{x\in
R_{k}}|\phi^{\prime}(x)|$. Let $K_{j}=K\cap R_{j}$, $1\leq j\leq N$. Then
$\mu(\partial_{S}K_{j})\leq\mu(\partial_{S}K)+\mu(\partial R_{j})=0$. By
(6.5),

$c^{-n}\int_{K_{j}}|\phi^{\prime}(x)|^{d}\mu(dx)\leq\underline{\operatorname{Cont}}(\phi(K_{j}))\leq\overline{\operatorname{Cont}}(\phi(K_{j}))\leq
c^{n}\int_{K_{j}}|\phi^{\prime}(x)|^{d}\mu(dx).$ (6.6)

Summing up the last inequality of (6.6) over $1\leq j\leq N$, and using (6.1)
and that $\mu(K_{j}\cap K_{k})\leq\mu(R_{j}\cap R_{k})=0$ when $j\neq k$, we
get $\overline{\operatorname{Cont}}(K_{\phi})\leq
c^{n}\int_{K}|\phi^{\prime}(x)|^{d}\mu(dx)$.

Let $1\leq k<j\leq N$. Since $\mu(K_{j}\cap K_{k})=0$, we get $\lim_{\delta\to
0^{+}}\overline{\operatorname{Cont}}(K_{j}\cap B(K_{j}\cap K_{k},\delta))=0$
by Lemma 6.6 (i). By (6.4), $\lim_{\varepsilon\to
0^{+}}\overline{\operatorname{Cont}}(\phi(K_{j})\cap
B(\phi(K_{j})\cap\phi(K_{k}),\varepsilon))=0$. By (6.1), we then get
$\lim_{\varepsilon\to 0^{+}}\overline{\operatorname{Cont}}(\phi(K_{j})\cap
B(\phi(K_{j})\cap\bigcup_{k=1}^{j-1}\phi(K_{k}),\varepsilon))=0$ for $2\leq
j\leq n$. By (6.3) and induction, we get
$\underline{\operatorname{Cont}}(K_{\phi})\geq\sum_{j=1}^{N}\underline{\operatorname{Cont}}(\phi(K_{j}))$,
which together with the first inequality of (6.6) implies that
$\underline{\operatorname{Cont}}(K_{\phi})\geq
c^{-n}\int_{K}|\phi^{\prime}(x)|^{d}\mu(dx)$. Combining this inequality with
$\overline{\operatorname{Cont}}(K_{\phi})\leq
c^{n}\int_{K}|\phi^{\prime}(x)|^{d}\mu(dx)$ and sending $c$ to $1^{+}$, we get
$\operatorname{Cont}(K_{\phi})=\int_{K}|\phi^{\prime}(x)|^{d}\mu(dx)=\nu(K_{\phi})$.
Thus, $\nu$ is the Minkowski content measure on $\phi(S)$. ∎

######  Remark 6.14.

It’s easy to extend the notions of Minkowski content and Minkowski content
measure to Riemannian manifolds. The propositions in this subsection will
still hold.

######  Remark 6.15.

Most well-known deterministic fractal sets do not have non-trivial Minkowski
contents. However, we may define the average Minkowski content by

$\operatorname{Cont}_{d}^{\operatorname{a}}(S)=\lim_{t\to\infty}\frac{1}{t}\int_{0}^{t}\operatorname{Cont}_{d}(S;e^{-r})dr,$

and use this to define the average Minkowski content measure. All results in
this subsection hold with the word “average” added before “Minkowski” and
$\operatorname{Cont}$ replaced by $\operatorname{Cont}^{\operatorname{a}}$
except that Theorem 6.11 requires moderate modification. The correct statement
is: for any compact set $S\subset\mathbb{R}^{n}$, $\mu$ is the average
Minkowski content measure on $S$ iff $\mu$ is the weak limit of $\mu_{t}$ as
$t\to\infty$, where $\mu_{t}$ is the measure on $\mathbb{R}^{n}$ that
satisfies
$d\mu_{t}/d\lambda^{n}=\frac{1}{t}\int_{0}^{t}e^{(n-d)s}\mathbbm{1}_{B(S,e^{-s})}ds$.

######  Example 6.16.

Let $S\subset[0,1]$ be the $1/3$-Cantor set. Let $d=\frac{\ln 2}{\ln 3}$. Then
$\operatorname{Cont}_{d}(S,e^{-t})$, $t\geq 0$, has period $\ln(3)$, and is
not constant. So the Minkowski content $\operatorname{Cont}_{d}(S)$ does not
exist, but the average Minkowski content
$\operatorname{Cont}_{d}^{\operatorname{a}}(S)=C_{0}$ for some
$C_{0}\in(0,\infty)$. Then the average Minkowski content measure on $S$ exists
and equals $C_{0}\mu$, where $\mu$ is the Haar probability measure on $S$.

###  6.2 Boundary Minkowski content of SLE${}_{\kappa}({\underline{\rho}})$

Now we come back to the SLE${}_{\kappa}(\underline{\rho})$ curve $\eta$ in
Theorem 4.1 with the additional assumption that
$\rho_{\Sigma}\in(\frac{\kappa}{2}-4,\frac{\kappa}{2}-2)$. This means that
$\eta$ intersects the interval $(-\infty,v_{m})$. Let

$d=1-\alpha=\frac{(\rho_{\Sigma}+4)(\kappa-4-2\rho_{\Sigma})}{2\kappa}.$

It is known that $d$ is the Hausdorff dimension of $\eta\cap(-\infty,v_{m})$
(cf. [16]). We are going to prove the existence of the $d$-dimensional
Minkowski content measure on $\eta\cap(-\infty,v_{m})$.

We write $G_{1}(u)$ for the function $C_{\mathbb{R}}G(w,\underline{v};u)$ in
Theorem 4.1, and $G_{2}(u_{1},u_{2})$ for the function
$G_{\mathbb{R}}(w,\underline{v};u_{1},u_{2})$ in Theorem 5.3. We also define
$G_{2}(u_{1},u_{2})=G_{2}(u_{2},u_{1})$ if $u_{1}<u_{2}\in(-\infty,v_{m})$. We
use $\lambda$ to denote the Lebesgue measure on $\mathbb{R}$.

######  Theorem 6.17.

Almost surely the (random) $d$-dimensional Minkowski content measure
$\mu_{\eta}$ on $\eta\cap(-\infty,v_{m}]$ in $\mathbb{R}$ exists, is atomless,
and satisfies the following properties.

  * (i)

If $\phi$ is a deterministic nonnegative measurable function on
$(-\infty,v_{m}]$, then

$\mathbb{E}\Big{[}\int\phi(u)\mu_{\eta}(du)\Big{]}=\int\phi(u)G_{1}(u)\lambda(du).$
(6.7)

  * (ii)

If $\psi$ is a deterministic nonnegative measurable function on
$(-\infty,v_{m}]^{2}$, then

$\mathbb{E}\Big{[}\int\\!\\!\int\psi(u_{1},u_{2})\mu_{\eta}^{2}(du_{1}\otimes
du_{2})\Big{]}=\int\\!\\!\int\psi(u_{1},u_{2})G_{2}(u_{1},u_{2})\lambda^{2}(du_{1}\otimes
du_{2}).$ (6.8)

  * (iii)

For any deterministic compact set $S\subset(-\infty,v_{m}]$ with
$\operatorname{Cont}(\partial_{{\mathbb{R}}}S)=0$, almost surely

$\operatorname{Cont}(\eta\cap S)=\mu_{\eta}(S).$ (6.9)

  * (iv)

The (minimal) support of $\mu_{{\eta}}$ (i.e., the smallest closed set
$F\subset\mathbb{R}$ such that $\mu_{{\eta}}(F^{c})=0$) is a.s.
$\eta\cap(-\infty,v_{m}]$.

######  Remark 6.18.

We emphasize here that the a.s. existence of the Minkowski content measure
$\mu_{\eta}$ means that, on an event with probability $1$, $\mu_{\eta}$
satisfies properties (i) and (ii) in Definition 6.3 simultaneously for all
compact subsets of $\eta\cap[-\infty,v_{m}]$.

###### Proof.

For $r>0$, define random functions $I_{r},J_{r}$ on $(-\infty,v_{m})$ by
$I_{r}(u)=\mathbbm{1}_{B_{\mathbb{R}}(\eta\cap\mathbb{R},r)}(u)$ and
$J_{r}(u)=r^{-\alpha}I_{r}(u)$. By Theorems 4.1 and 5.3, for
$u\in(-\infty,v_{m})$, $\mathbb{E}[J_{r}(u)]\to G_{1}(u)$ as $r\to 0^{+}$, and
for $u_{1}\neq u_{2}\in(-\infty,v_{m})$,
$\mathbb{E}[J_{r_{1}}(u_{1})J_{r_{2}}(u_{2})]\to G_{2}(u_{1},u_{2})$ as
$r_{1},r_{2}\to 0^{+}$.

By (4.8,4.6), for any $u\in(-\infty,v_{m})$ and $r>0$,

$\mathbb{E}[J_{r}(u)],G_{1}(u)\lesssim|v_{m}-u|^{-\alpha}.$ (6.10)

By (5.4,5.1), for $u_{1}\neq u_{2}\in(-\infty,v_{m})$ and $r_{1},r_{2}>0$,

$\mathbb{E}[J_{r_{1}}(u_{1})J_{r_{2}}(u_{2})],G_{2}(u_{1},u_{2})\lesssim|v_{m}-u_{1}|^{-\alpha}|u_{1}-u_{2}|^{-\alpha}.$
(6.11)

Combining (6.11) with (5.2) and setting
$\zeta:=\zeta_{1}\wedge\zeta_{2}\wedge\frac{1-\alpha}{2}\in(0,1-\alpha)$, we
get

$\displaystyle|\mathbb{E}[J_{r_{1}}(u_{1})J_{r_{2}}(u_{2})]-G_{2}(u_{1},u_{2})|$
$\displaystyle\leq$
$\displaystyle|v_{m}-u_{1}|^{-\alpha}|u_{1}-u_{2}|^{-\alpha}\Big{(}1\wedge\Big{(}\frac{r_{1}\vee
r_{2}}{|v_{m}-u_{1}|\wedge|u_{1}-u_{2}|}\Big{)}^{\zeta}\Big{)}.$ (6.12)

Fix a bounded measurable set $S\subset(-\infty,v_{m}]$. Let
$A=\sup\\{|v_{m}-u|:u\in S\\}$. Define

$Y_{r}=\int_{S}J_{r}(u)\lambda(du)=r^{-\alpha}\lambda(B_{\mathbb{R}}(\eta\cap\mathbb{R},r)\cap
S).$ (6.13)

From (6.11) and that $\alpha\in(0,1)$ we get

$\int_{S}\\!\int_{S}G_{2}(u_{1},u_{2})\lambda(du_{1})\lambda(du_{2})\lesssim\int_{0}^{A}L_{1}^{-\alpha}\int_{0}^{A-L_{1}}L_{2}^{-\alpha}\lambda(dL_{2})\lambda(dL_{1})<\infty.$

The finiteness of this integral together with (6.12) implies that, for any
$r_{1},r_{2}>0$,

$\displaystyle\mathbb{E}[(Y_{r_{1}}-Y_{r_{2}})^{2}]=\mathbb{E}\Big{[}\Big{(}\int_{S}J_{r_{1}}(u)\lambda(du)-\int_{S}J_{r_{2}}(u)\lambda(du)\Big{)}^{2}\Big{]}$
$\displaystyle\leq$ $\displaystyle
2\sum_{j=1}^{2}\sum_{k=1}^{2}\int\\!\\!\int_{\\{(u_{1},u_{2})\in
S^{2}:u_{1}>u_{2}\\}}|\mathbb{E}[J_{r_{j}}(u_{1})J_{r_{k}}(u_{2})]-G(u_{1},u_{2})|\lambda(du_{1})\lambda(du_{2})$
$\displaystyle\lesssim$
$\displaystyle\int_{0}^{A}\\!\\!\int_{0}^{A}L_{1}^{-\alpha}L_{2}^{-\alpha}\Big{(}1\wedge\Big{(}\frac{r_{1}\vee
r_{2}}{L_{1}\wedge
L_{2}}\Big{)}^{\zeta}\Big{)}\lambda(dL_{1})\lambda(dL_{2}),$

where in the last line we used a change of variables: $L_{1}=v_{m}-u_{1}$ and
$L_{2}=u_{1}-u_{2}$. Since $\alpha\in(0,1)$ and $\zeta\in(0,1-\alpha)$, we can
easily bound the last integral and get

$\|Y_{r_{1}}-Y_{r_{2}}\|_{L^{2}}^{2}\lesssim A^{2-2\alpha-\zeta}(r_{1}\vee
r_{2})^{\zeta}.$ (6.14)

This implies that $(Y_{r})$ is Cauchy in $L^{2}$ as $r\to 0^{+}$. Let
$\xi_{S}\in L^{2}$ be the limit.

By (6.14) and Chebyshev’s inequality,

$\sum_{n=1}^{\infty}\mathbb{P}[|Y_{Ae^{-n}}-Y_{Ae^{1-n}}|\geq
A^{1-\alpha}e^{-n\zeta/3}]\lesssim\sum_{n=1}^{\infty}e^{-n\zeta/3}<\infty.$

By Borel-Cantelli lemma, with probability one, $|Y_{Ae^{-n}}-Y_{Ae^{1-n}}|\leq
A^{1-\alpha}e^{-n\zeta/3}$ for sufficiently large $n$, which implies that a.s.
$\lim_{n\to\infty}Y_{e^{-n}}$ converges. The limit is $\xi_{S}$ since
$Y_{Ae^{-n}}\to\xi_{S}$ in $L^{2}$. A similar argument shows that, for any
$k\in\mathbb{N}$, a.s. $Y_{Ae^{-n/2^{k}}}\to\zeta$ as $n\to\infty$.

Let $E_{0}$ be the event that $\lim_{n\to\infty}Y_{e^{-n/2^{k}}}=\zeta$ for
every $k\in\mathbb{N}$. Then $\mathbb{P}[E_{0}]=1$. By the definition of
$I_{r}$ and $J_{r}$, for $r_{1}\geq r_{2}>0$, we have $I_{r_{1}}(u)\geq
I_{r_{2}}(u)$ and $J_{r_{1}}(u)\geq(r_{2}/r_{1})^{\alpha}J_{r_{2}}(u)$, which
implies that $Y_{r_{1}}\geq(r_{2}/r_{1})^{\alpha}Y_{r_{2}}$. Let
$k\in\mathbb{N}$. For $r\in(0,1]$, there is a unique $n_{k}(r)\in\mathbb{N}$
such that $e^{-n_{k}(r)/2^{k}}<r\leq e^{(1-n_{k}(r))/2^{k}}$. Then we get
$e^{-\alpha/2^{k}}Y_{e^{-n_{k}(r)/2^{k}}}\leq Y_{r}\leq
e^{\alpha/2^{k}}Y_{e^{(1-n_{k}(r))/2^{k}}}$, which implies that
$e^{-\alpha/2^{k}}\zeta\leq\liminf_{r\to 0^{+}}Y_{r}\leq\limsup_{r\to
0^{+}}Y_{r}\leq e^{\alpha/2^{k}}\zeta$ on the event $E_{0}$. Since this holds
for every $k\in\mathbb{N}$, and $\mathbb{P}[E_{0}]=1$, we get a.s.
$Y_{r}\to\xi_{S}$ as $r\to 0^{+}$.

So far we have proved that, for any bounded measurable set
$S\subset(-\infty,v_{m}]$, there is a nonnegtive random variable $\xi_{S}\in
L^{2}$ such that, as $r\to 0^{+}$, $\int_{S}J_{r}(u)\lambda(du)\to\xi_{S}$
almost surely and in $L^{2}$. It is clear that a.s. $\xi_{S_{1}\cup
S_{2}}=\xi_{S_{1}}+\xi_{S_{2}}$ if $S_{1}\cap S_{2}=\emptyset.$ By Fubini
Theorem,

$\int_{S}\mathbb{E}[J_{r}(u)]\lambda(du)\to\mathbb{E}[\xi_{S}],\quad\int_{S}\\!\int_{S}\mathbb{E}[J_{r}(u_{1})J_{r}(u_{2})]\lambda(du_{1})\lambda(du_{2})\to\mathbb{E}[\xi_{S}^{2}].$

By polarization, the second limit further implies that for any bounded
measurable sets $S_{1},S_{2}\subset(-\infty,v_{m}]$,

$\int_{S_{1}}\\!\int_{S_{2}}\mathbb{E}[J_{r}(u_{1})J_{r}(u_{2})]\lambda(du_{1})\lambda(du_{2})\to\mathbb{E}[\xi_{S_{1}}\xi_{S_{2}}].$

Since $\mathbb{E}[J_{r}(u)]\to G_{1}(u)$ and
$\mathbb{E}[J_{r}(u_{1})J_{r}(u_{2})]\to G_{2}(u_{1},u_{2})$ as $r\to 0^{+}$,
by (6.10,6.11) and dominated convergence theorem, for any bounded measurable
sets $S,S_{1},S_{2}\subset(-\infty,v_{m}]$,

$\mathbb{E}[\xi_{S}]=\int_{S}G_{1}(u)\lambda(du),\quad\mathbb{E}[\xi_{S_{1}}\xi_{S_{2}}]=\int_{S_{1}}\\!\int_{S_{2}}G_{2}(u_{1},u_{2})\lambda(du_{1})\lambda(du_{2}).$
(6.15)

We now define $\theta(u)=\xi_{[u,v_{m}]}$, $u\in(-\infty,v_{m}]$. Then for
$a<b\in(-\infty,v_{m})$, by (6.15,6.11)

$\mathbb{E}[(\theta(a)-\theta(b))^{2}]=\mathbb{E}[\xi_{[a,b)}^{2}]=\int_{a}^{b}\int_{a}^{b}G_{2}(u_{1},u_{2})du_{1}du_{2}$
$\lesssim\int_{v_{m}-b}^{v_{m}-a}\int_{0}^{b-a}L_{1}^{-\alpha}L_{2}^{-\alpha}\lambda(dL_{2})\lambda(dL_{1})\lesssim|v_{m}-b|^{-\alpha}|b-a|^{2-\alpha}.$

By Kolmogorov continuity theorem, for any $b\in(-\infty,v_{m})\cap\mathbb{Q}$,
$\theta$ has a locally Hölder continuous version on $(-\infty,b]$. Thus,
$\theta$ has a continuous version on $(-\infty,v_{m})$. Such continuous
version is nonnegative and decreasing since for any $a<b\in(-\infty,v_{m}]$,
a.s. $\theta(a)-\theta(b)=\xi_{[a,b)}\geq 0$. By (6.15,6.10),
$\mathbb{E}[\theta(b)]\to 0$ as $b\to v_{m}^{-}$. So $\theta$ has a continuous
decreasing version on $(-\infty,v_{m}]$ with $\theta(v_{m})=0$. Such version
of $\theta$ determines an atomless locally finite measure $\mu_{\eta}$ on
$(-\infty,v_{m}]$. Then for any set $S\subset(-\infty,v_{m}]$, which is a
finite disjoint union of intervals of the form $(a_{j},b_{j}]$, we have a.s.
$\mu_{\eta}(S)=\xi_{S}$. By monotone class theorem, for any bounded measurable
set $S\subset(-\infty,v_{m}]$, a.s. $\mu_{\eta}(S)=\xi_{S}$. By (6.15), we get

$\mathbb{E}[\mu_{\eta}(S)]=\int_{S}G_{1}(u)\lambda(du),$ (6.16)

and if $T=S_{1}\times S_{2}$ for some bounded measurable sets
$S_{1},S_{2}\subset(-\infty,v_{m}]$, then

$\mathbb{E}[\mu_{\eta}^{2}(T)]=\int\\!\\!\int_{T}G_{2}(u_{1},u_{2})\lambda^{2}(du_{1}\otimes
du_{2}).$ (6.17)

By monotone convergence theorem and monotone class theorem, (6.16) holds for
any measurable set $S\subset(-\infty,v_{m}]$, and (6.17) holds for any
measurable set $T\subset(-\infty,v_{m}]^{2}$. By monotone convergence theorem,
these results further imply (6.7,6.8).

Let $S$ be a compact subset of $(-\infty,v_{m}]$ with
$\operatorname{Cont}(\partial S)=0$. For $r>0$, since the symmetric difference
between $B_{\mathbb{R}}(\eta\cap\mathbb{R},r)\cap S$ and
$B_{\mathbb{R}}(\eta\cap S,r)$ is contained in $B_{\mathbb{R}}(\partial S,r)$,
by (6.13) we get $|\int_{S}J_{r}(u)\lambda(du)-\operatorname{Cont}(\eta\cap
S;r)|\leq\operatorname{Cont}(\partial S;r)$. Sending $r$ to $0^{+}$ and using
that a.s. $\int_{S}J_{r}(u)\lambda(du)\to\xi_{S}=\mu_{\eta}(S)$, we get (6.9).

Applying (6.9) to $S$ of the form $[a,b]$, we find that there is an event $E$
with $\mathbb{P}[E]=1$ such that on the event $E$, for any
$a<b\in(-\infty,v_{m}]\cap\mathbb{Q}$,
$\operatorname{Cont}(\eta\cap[a,b])=\mu_{\eta}[a,b]=\theta(a)-\theta(b)$.
Since $\mathbb{P}[E]=1$, by applying Corollary 6.7 with
$\\{F_{\iota}\\}=\\{\eta\cap[a,b]:a<b\in(-\infty,v_{m}]\cap\mathbb{Q}\\}$, we
conclude that $\mu_{\eta}$ is a.s. the Minkowski content measure on
$\eta\cap(-\infty,v_{m}]$.

To prove Part (iv), we need the lemma below.

######  Lemma 6.19.

Let $\tau$ be a finite ${\cal F}$-stopping time. Suppose $\eta^{\tau}$ is a
continuous curve in $\overline{\mathbb{H}}$, which satisfies that a.s.
$f_{\tau}(\eta^{\tau})=\eta(\tau+\cdot)$. Then
$\mu^{\tau}:=(g_{\tau})_{*}((g_{\tau}^{\prime})^{d}\cdot\mu_{\eta}|_{(-\infty,v_{m}\wedge
a_{K_{\tau}})})$ is a.s. the Minkowski content measure on
$\eta^{\tau}\cap(-\infty,V_{m}(\tau)]$. Recall that
$a_{K_{\tau}}=\min(\overline{K_{\tau}}\cap\mathbb{R})$.

###### Proof.

By DMP, conditionally on ${\cal F}_{\tau}$, $\eta^{\tau}$ is an
SLE${}_{\kappa}(\underline{\rho})$ curve started from $W(\tau)$ with force
points $\underline{V}(\tau)$. Thus, the Minkowski content measure
$\mu_{\eta^{\tau}}$ on $\eta^{\tau}\cap(-\infty,V_{m}(\tau)]$ a.s. exists. By
Theorem 6.12, $\mu_{\eta}|_{(-\infty,v_{m}\wedge a_{K_{\tau}})}$ and
$\mu_{\eta^{\tau}}|_{(-\infty,V_{m}(\tau))}$ are respectively the Minkowski
content measure on $\eta\cap(-\infty,v_{m}\wedge a_{K_{\tau}})$ and
$\eta^{\tau}\cap(-\infty,V_{m}(\tau))$. Since $g_{\tau}$ is injective and
$C^{1}$ on $(-\infty,v_{m}\wedge a_{K_{\tau}})$, and sends
$\eta\cap(-\infty,v_{m}\wedge a_{K_{\tau}})$ to
$\eta^{\tau}\cap(-\infty,V_{m}(\tau))$, by Theorem 6.13,
$\mu_{\eta^{\tau}}|_{(-\infty,V_{m}(\tau))}=(g_{\tau})_{*}((g_{\tau}^{\prime})^{d}\cdot\mu_{\eta}|_{(-\infty,v_{m}\wedge
a_{K_{\tau}})})=\mu^{\tau}$. Since $\mu_{\eta^{\tau}}$ is supported by
$(-\infty,V_{m}(\tau)]$, and is atomless, we get
$\mu^{\tau}=\mu_{\eta^{\tau}}$. So we get the conclusion. ∎

Now we prove Part (iv). Let $\operatorname{supp}(\nu)$ denote the (minimal)
support of a measure $\nu$. First consider the case that $w=0$ and
$v_{m}=0^{-}$. Then $\eta$ is an SLE${}_{\kappa}(\rho_{\Sigma})$ curve started
from $0$ with the force point $0^{-}$. It satisfies the scaling property: for
any $a>0$, $a\eta$ has the same law as $\eta$ up to a linear time-change. For
$L\in(-\infty,0)$, let $p(L)=\mathbb{P}[\mu_{\eta}[L,0]=0]$. Then $p(L)$ is
increasing with $\lim_{L\to-\infty}p(L)=\mathbb{P}[\mu_{\eta}=0]$ and
$\lim_{L\to 0^{-}}p(L)=\mathbb{P}[0\not\in\operatorname{supp}(\mu_{\eta})]$.
By Theorem 6.13, $\mu_{\eta}$ has the same law as $a^{d}\mu_{\eta}(\cdot/a)$,
which implies that $p(L)=p(aL)$ for any $a>0$. So $p$ is constant on
$(-\infty,0)$. Denote the constant value by $p_{0}\in[0,1]$. Then
$\mathbb{P}[\mu_{\eta}=0]=\mathbb{P}[0\not\in\operatorname{supp}(\mu_{\eta})]=p_{0}$.

Let $\tau=\tau^{*}_{-1}$. Then $\tau$ is a finite ${\cal F}$-stopping time,
and $\eta(\tau)\in(-\infty,-1)$. By DMP,
$g_{\tau}\circ\eta(\tau+\cdot)-W(\tau)$ has the same law as $\eta$, and is
independent of ${\cal F}_{\tau}$. By Lemma 6.19, we have
$\mathbb{P}[\eta|_{(-\infty,\eta(\tau)]}=0|{\cal F}_{\tau}]=p_{0}$. Since
$\mu_{\eta}[\eta(\tau),0]=\operatorname{Cont}(\eta[0,\tau]\cap(-\infty,0])$ is
${\cal F}_{\tau}$-measurable, we have

$p_{0}=\mathbb{P}[\mu_{\eta}=0]=\mathbb{P}[\\{\mu_{\eta}|_{[\eta(\tau),0]}=0\\}\cap\\{\mu|_{(-\infty,\eta(\tau)]}=0\\}]=p_{0}\mathbb{P}[\mu_{\eta}|_{[\eta(\tau),0]}=0]\leq
p_{0}^{2}.$

If $p_{0}=1$, then by $\mathbb{P}[\mu_{\eta}=0]=p_{0}$ and (6.7), we get
$0=\mathbb{E}[\mu_{\eta}(-\infty,0]]=\int_{-\infty}^{0}G_{1}(u)\lambda(du)>0$,
which is a contradiction. So $p_{0}=0$, and we get
$\mathbb{P}[0\in\operatorname{supp}(\mu_{\eta})]=1$.

Now we consider the general case. To prove that a.s.
$\operatorname{supp}(\mu_{\eta})=\eta\cap(-\infty,v_{m}]$, it suffices to
prove that, for any $a<b\in(-\infty,v_{m}]\cap\mathbb{Q}$, on the event
$\eta\cap(a,b)\neq\emptyset$, a.s. $\mu_{\eta}(a,b)>0$. Fix
$a<b\in(-\infty,v_{m}]\cap\mathbb{Q}$. Let $\tau=\tau^{*}_{b}$. Then $\tau$ is
a finite ${\cal F}$-stopping time, and $\eta(\tau)\in(-\infty,b]$. By DMP,
$g_{\tau}(\eta(\tau+\cdot))-W(\tau)$, denoted by $\eta^{\tau}$, has the law of
an SLE${}_{\kappa}(\rho_{\Sigma})$ curve started from $0$ with the force point
$0^{-}$. Let $\mu^{\tau}$ be the Minkowski content measure on
$\eta^{\tau}\cap(-\infty,0]$. By the last paragraph, a.s.
$0\in\operatorname{supp}(\mu^{\tau})$. By Lemma 6.19, a.s.
$\eta(\tau)\in\operatorname{supp}(\mu_{\eta}|_{(-\infty,\eta(\tau)]})$. On the
event $\eta\cap(a,b)\neq\emptyset$, we have $\eta(\tau)\in(a,b]$, which
together with a.s.
$\eta(\tau)\in\operatorname{supp}(\mu_{\eta}|_{(-\infty,\eta(\tau)]})$ implies
that a.s. $\mu_{\eta}(a,b)>0$ on this event. The proof is now complete. ∎

######  Remark 6.20.

Let $S=[v_{m}-A,v_{m}]$ for some $A>0$. By (6.8) and (6.11), for any
$\varepsilon\in(0,d)$,

$\mathbb{E}\Big{[}\int\\!\\!\int_{S^{2}}\frac{\mu_{\eta}^{2}(du_{1}\otimes
du_{2})}{|u_{1}-u_{2}|^{d-\varepsilon}}\Big{]}=\int\\!\\!\int_{S^{2}}|u_{1}-u_{2}|^{-c}G_{2}(u_{1},u_{2})\lambda^{2}(du_{1}\otimes
du_{2})$
$\lesssim\int_{0}^{A}\\!\\!\int_{0}^{A}L_{1}^{-\alpha}L_{2}^{-\alpha-(d-\varepsilon)}\lambda(dL_{1})\lambda(dL_{2})\leq\frac{A^{1-\alpha}}{1-\alpha}\cdot\frac{A^{\varepsilon}}{\varepsilon}<\infty,$

which implies that a.s.
$\int\\!\\!\int_{S^{2}}\frac{\mu_{\eta}^{2}(du_{1}\otimes
du_{2})}{|u_{1}-u_{2}|^{d-\varepsilon}}<\infty$. This shows that $\mu_{\eta}$
has finite $(d-\varepsilon)$-energy. So, on the event $\\{\eta\cap
S\neq\emptyset\\}$, $\mu_{\eta}|_{S}$ could serve as a Frostman measure on
$\eta\cap S$.

######  Theorem 6.21.

For $u\in(-\infty,v_{m})$, let

$M_{u}(t)=\mathbbm{1}_{\\{t<\tau^{*}_{u}\\}}g_{t}^{\prime}(u)^{\alpha}C_{\mathbb{R}}G(W(t),\underline{V}(t);g_{t}(u)),\quad
t\geq 0.$

Let $S\subset(-\infty,v_{m})$ be compact, and define the supermartingale
$M_{S}(t)=\int_{S}M_{u}(t)\lambda(du)$. Then we have the Doob-Meyer
decomposition:

$M_{S}(t)=\mathbb{E}[\mu_{\eta}(S)|{\cal F}_{t}]-\mu_{\eta}(\eta[0,t]\cap
S),\quad t\geq 0.$

###### Proof.

Since $\mu_{\eta}(S)-\mu_{\eta}(\eta[0,t]\cap S)=\mu_{\eta}(\eta(t,\infty)\cap
S)$ and $\mu_{\eta}(\eta[0,t]\cap S)=\operatorname{Cont}(\eta[0,t]\cap
S)\in{\cal F}_{t}$, it suffices to show
$\mathbb{E}[\mu_{\eta}(\eta(t,\infty)\cap S)|{\cal F}_{t}]=M_{S}(t)$. Since
$\eta(t,\infty)\subset(-\infty,a_{K_{t}})$ and $\mu_{\eta}$ is supported by
$(-\infty,v_{m})$, $\mu_{\eta}(\eta(t,\infty)\cap
S)=\mu_{\eta}((-\infty,a_{K_{t}}\wedge v_{m})\cap S)$.

Let $\eta^{t}$ be the SLE${}_{\kappa}(\underline{\rho})$ curve conditionally
on ${\cal F}_{t}$ started from $W(t)$ with force points $\underline{V}(t)$
such that $\eta(t+\cdot)=f_{t}\circ\eta^{t}$. Let $\mu^{t}$ be the the
Minkowski content measure on $\eta^{t}\cap(-\infty,V_{m}(t)]$. By (6.7), if
$\phi$ is an ${\cal F}_{t}$-measurable random nonnegative measurable function
on $(-\infty,V_{m}(t))$111More precisely, $\phi$ is an ${\cal F}\times{\cal
B}(\mathbb{R})$-measurable function defined on
$\\{(\omega,x)\in\Omega\times\mathbb{R}:x<V_{m}(\omega,t)\\}$. Here we use
$V_{m}(\omega,t)$ to emphasize the dependence of $V_{m}(t)$ on
$\omega\in\Omega$., then

$\mathbb{E}\Big{[}\int\phi(x)\mu^{t}(du)\Big{|}{\cal
F}_{t}\Big{]}=\int\phi(x)C_{\mathbb{R}}G(W(t),\underline{V}(t);x)\lambda(dx).$
(6.18)

By Lemma 6.19,
$\mu^{t}=(g_{t})_{*}((g_{t}^{\prime})^{d}\cdot\mu_{\eta}|_{(-\infty,a_{K_{t}}\wedge
v_{m})})$. Thus,

$\displaystyle\mathbb{E}[\mu_{\eta}((-\infty,a_{K_{t}}\wedge v_{m})\cap
S)|{\cal F}_{t_{0}}]=\mathbb{E}\Big{[}\int_{(-\infty,a_{K_{t}}\wedge
v_{m})\cap S}g_{t}^{\prime}(g_{t}^{-1}(x))^{-d}\mu^{t}(dx)\Big{|}{\cal
F}_{t}\Big{]}$ $\displaystyle=$ $\displaystyle\int_{(-\infty,V_{m}(t))\cap
g_{t}(S)}g_{t}^{\prime}(g_{t}^{-1}(x))^{-d}C_{\mathbb{R}}G(W(t),\underline{V}(t);x)\lambda(dx)\qquad(\mbox{by
}(\ref{eta-S*}))$ $\displaystyle=$
$\displaystyle\int_{(-\infty,a_{K_{t}}\wedge v_{m})\cap
S}g_{t}^{\prime}(u)^{\alpha}C_{\mathbb{R}}G(W(t),\underline{V}(t);g_{t}(u))\lambda(du)\qquad(x=g_{t}(u))$
$\displaystyle=$ $\displaystyle\int_{(-\infty,a_{K_{t}}\wedge v_{m})\cap
S}M_{u}(t)\lambda(du)=\int_{S}M_{u}(t)\lambda(du)=M_{S}(t),$

where the second last “$=$” holds because $S\subset(-\infty,v_{m}]$ and
$M_{u}(t)=0$ for $u\geq a_{K_{t}}$. So we get
$\mathbb{E}[\mu_{\eta}(\eta(t,\infty)\cap S)|{\cal F}_{t}]=M_{S}(t)$, as
desired. ∎

######  Remark 6.22.

Note that the above proof does not use the Doob-Meyer decomposition theorem.
When all $\rho_{j}$ equal $0$, the theorem shows that, if $\eta$ is a chordal
SLEκ curve ($\kappa\in(4,8)$) with no force point, the measure $\mu_{\eta}$
agrees with the covariant measure derived in [2] up to a multiplicative
constant. So we proved the conjecture in [2, Section 7.3].

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