# Radial symmetry and Liouville theorem for master equations

Lingwei Ma  School of Mathematical Sciences, Tianjin Normal University,
Tianjin, 300387, P. R. China  Department of Mathematical Sciences, Yeshiva
University, New York, NY, 10033, USA  Yahong Guo  School of Mathematical
Sciences, Nankai University, Tianjin, 300071, P. R. China  Department of
Mathematical Sciences, Yeshiva University, New York, NY, 10033, USA  Zhenqiu
Zhang Corresponding author. School of Mathematical Sciences and LPMC, Nankai
University, Tianjin, 300071, P. R. China

###### Abstract

This paper has two primary objectives. The first one is to demonstrate that
the solutions of master equation

$(\partial_{t}-\Delta)^{s}u(x,t)=f(u(x,t)),\,\,(x,t)\in
B_{1}(0)\times\mathbb{R},$

subject to the vanishing exterior condition, are radially symmetric and
strictly decreasing with respect to the origin in $B_{1}(0)$ for any
$t\in\mathbb{R}$. Another one is to establish the Liouville theorem for
homogeneous master equation

$(\partial_{t}-\Delta)^{s}u(x,t)=0,\,\,\mbox{in}\,\,\mathbb{R}^{n}\times\mathbb{R},$

which states that all bounded solutions must be constant. We propose a new
methodology for a direct method of moving planes applicable to the fully
fractional heat operator $(\partial_{t}-\Delta)^{s}$, and the proof of our
main results based on this direct method involves the perturbation technique,
limit argument as well as Fourier transform. This study opens up a way to
investigate the geometric behavior of master equations, and provides valuable
insights for establishing qualitative properties of solutions and even for
deriving important Liouville theorems for other types of fractional order
parabolic equations.

Mathematics Subject classification (2020): 35R11; 35K05; 47G30; 35B50; 35B53.

Keywords: master equation; fully fractional heat operator; direct method of
moving planes; radial symmetry; monotonicity; Liouville theorem.

11footnotetext: E-mail<EMAIL_ADDRESS>(L. Ma),
<EMAIL_ADDRESS>(Y. Guo<EMAIL_ADDRESS>(Z. Zhang).

##  1 Introduction

The objective of this paper is to explore the qualitative properties of
solutions to space-time dual nonlocal equations involving the fractional
powers of the heat operator $(\partial_{t}-\Delta)^{s}$. More specifically, we
establish the radial symmetry and monotonicity of solutions for the following
master equation in a unit ball

$\left\\{\begin{array}[]{ll}(\partial_{t}-\Delta)^{s}u(x,t)=f(u(x,t)),&(x,t)\in
B_{1}(0)\times\mathbb{R},\\\ u(x,t)\equiv 0,&(x,t)\in
B^{c}_{1}(0)\times\mathbb{R},\end{array}\right.$ (1.1)

for any $t\in\mathbb{R}$, and prove the Liouville theorem for the homogeneous
master equations

$(\partial_{t}-\Delta)^{s}u(x,t)=0,\,\,\mbox{in}\,\,\mathbb{R}^{n}\times\mathbb{R}$
(1.2)

in the whole space.

The first proposal of such a fully fractional heat operator
$(\partial_{t}-\Delta)^{s}$ is attributed to the mathematician Marcel Riesz,
who introduced it in [26]. This nonlocal operator can be defined in the
following pointwise form

$(\partial_{t}-\Delta)^{s}u(x,t):=C_{n,s}\int_{-\infty}^{t}\int_{\mathbb{R}^{n}}\frac{u(x,t)-u(y,\tau)}{(t-\tau)^{\frac{n}{2}+1+s}}e^{-\frac{|x-y|^{2}}{4(t-\tau)}}\operatorname{d}\\!y\operatorname{d}\\!\tau,$
(1.3)

where $0<s<1$, the integral in $y$ is taken in the Cauchy principal value
sense and the normalization positive constant

$C_{n,s}=\frac{1}{(4\pi)^{\frac{n}{2}}|\Gamma(-s)|},$

with $\Gamma(\cdot)$ denoting the Gamma function. The singular integral in
(1.3) is well defined in $\mathbb{R}^{n}\times\mathbb{R}$ provided

$u(x,t)\in C^{2s+\epsilon,s+\epsilon}_{x,\,t,\,{\rm
loc}}(\mathbb{R}^{n}\times\mathbb{R})\cap\mathcal{L}(\mathbb{R}^{n}\times\mathbb{R})$

for some $\varepsilon>0$, where the slowly increasing function space
$\mathcal{L}(\mathbb{R}^{n}\times\mathbb{R})$ is defined by

$\mathcal{L}(\mathbb{R}^{n}\times\mathbb{R}):=\left\\{u(x,t)\in L^{1}_{\rm
loc}(\mathbb{R}^{n}\times\mathbb{R})\mid\int_{-\infty}^{t}\int_{\mathbb{R}^{n}}\frac{|u(x,\tau)|e^{-\frac{|x|^{2}}{4(t-\tau)}}}{1+(t-\tau)^{\frac{n}{2}+1+s}}\operatorname{d}\\!x\operatorname{d}\\!\tau<\infty,\,\,\forall\,t\in\mathbb{R}\right\\},$

and the definition of the local parabolic Hölder space
$C^{2s+\epsilon,s+\epsilon}_{x,\,t,\,{\rm
loc}}(\mathbb{R}^{n}\times\mathbb{R})$ will be specified in Section 2 . In
particular, if $u$ is bounded, we can ensure the integrability of (1.3) by
assuming only that $u$ is local parabolic Hölder continuous. We notice that
the operator $(\partial_{t}-\Delta)^{s}$ is nonlocal both in space and time,
since the value of $(\partial_{t}-\Delta)^{s}u$ at a given point $(x,t)$
depends on the values of $u$ over the whole $\mathbb{R}^{n}$ and even on all
the past time before $t$. The intriguing aspect of this problem is that
applying the space-time nonlocal operator $(\partial_{t}-\Delta)^{s}$ to a
function that only depends on either space or time, it reduces to a familiar
fractional order operator, as discussed in [27]. More precisely, if $u$ is
only a function of $x$, then

$(\partial_{t}-\Delta)^{s}u(x)=(-\Delta)^{s}u(x),$

where $(-\Delta)^{s}$ is the well-known fractional Laplacian of order $2s$.
While if $u=u(t)$, then

$(\partial_{t}-\Delta)^{s}u(t)=\partial_{t}^{s}u(t),$

where $\partial_{t}^{s}$ is usually denoted by $D_{\rm left}^{s}$,
representing the Marchaud left fractional derivative of order $s$, defined as

$D_{\rm
left}^{s}u(t)=\frac{1}{|\Gamma(-s)|}\int_{-\infty}^{t}\frac{u(t)-u(\tau)}{(t-\tau)^{1+s}}\operatorname{d}\\!\tau.$

Moreover, it should be noted that as $s$ tends to $1$ from the left side, the
fractional power of heat operator $(\partial_{t}-\Delta)^{s}$ converges to the
local heat operator $\partial_{t}-\Delta$ (cf. [17]).

The master equation has a wide range of applications in physical and
biological phenomena, such as anomalous diffusion [20], chaotic dynamics [30],
biological invasions [2], among others. In addition to these areas, it has
also been employed in the financial field [25], where it can model the
correlation between waiting times and price jumps in transactions. From a
probabilistic perspective, the master equation plays a crucial role in the
theory of continuous time random walk, where $u$ represents the distribution
of particles that make random jumps simultaneously with random time lags (cf.
[21]). This is in contrast to the nonlocal parabolic equations

$\partial_{t}u+(-\Delta)^{s}u=f$ (1.4)

or the dual fractional parabolic equation

$\partial_{t}^{\alpha}u+(-\Delta)^{s}u=f,$ (1.5)

where jumps are independent of the waiting times. In other words, the master
equation takes into account the strong correlation between the waiting times
and the particle jumps, whereas the nonlocal parabolic equation (1.4) or the
dual fractional parabolic equation (1.5) do not. It is evident that the master
equation is of great importance in various fields, and continuous research on
it can drive us towards a deeper understanding of complex phenomena.

Nowadays substantial progress has been made in a series of remarkable papers
[1, 12, 27] investigating the existence, uniqueness and regularity of
solutions to master equations. The primary approach used in such studies is
the extension method introduced by Caffarelli and Silvestre [11], which
extends the master equation to a local degenerate parabolic equation in a
higher dimensional space. As far as we know, there is limited understanding of
the geometric behavior of solutions to the master equation. This lack of
results can mainly be attributed to the challenges posed by the non-locality
and strong correlation of the operator $(\partial_{t}-\Delta)^{s}$ within the
framework of master equation. The only related paper we are aware of is [9],
in which Chen and Ma utilized a direct sliding method to establish that the
entire solution of the master equation in (1.1) is monotone increasing and
one-dimensional symmetric in $\mathbb{R}^{n}\times\mathbb{R}$, and thus proved
the Gibbons’ conjecture in this context.

The method of moving planes, which was introduced by Alexandroff in [18], is a
commonly used technique to study the monotonicity and symmetry of solutions to
local elliptic and parabolic equations. However, this approach cannot be
directly applied to pseudo-differential equations involving the fractional
Laplacian, due to the non-locality of this operator. One effective method is
to combine the aforementioned extension method, which enables us to apply the
traditional method of moving planes designed for local equations to the
extended problem, thereby establishing the properties of solutions. Another
useful approach is to convert the given pseudo-differential equations into
their equivalent integral equations. By doing so, one can use the method of
moving planes in integral forms and the regularity lifting to investigate the
properties of solutions (cf. [7, 8]). These two effective methods have been
successfully employed to investigate elliptic equations involving the
fractional Laplacian. However, the above two methods can only be applied to
equations involving the fractional Laplacian, and sometimes one may need to
perform cumbersome calculations and to impose additional restrictions on the
problem, which may not be necessary when dealing with the fractional equations
directly.

A decade later, Chen, Li, and Li [6] made further progress by introducing a
direct method of moving planes. This method removed the restrictions and
greatly simplified the proof process. Since then, this effective direct method
has been widely applied to establish the symmetry, monotonicity, non-
existence, and even to obtain estimates in a boundary layer of solutions for
various elliptic equations and systems involving the fractional Laplacian, the
fully nonlinear nonlocal operators, the fractional $p$-Laplacians as well as
the higher order fractional operators. For more details, please refer to [3,
4, 5, 13, 22, 23, 24, 31] and the references therein. The direct method of
moving planes is noteworthy for its ability not only to explore the symmetry,
monotonicity, and non-existence of positive solutions for the fractional
parabolic equations of type (1.4) (cf. [14, 15, 16, 29]), but also for its
recent generalization to investigate the monotonicity of solutions for the
dual nonlocal parabolic equations of type (1.5) in a half-space (cf. [10]).

In contrast, there has been no progress in investigating the feasibility of a
direct method of moving plane for the master equation, and in exploring how to
utilize this approach to derive qualitative properties of solutions to the
master equation. As we have observed that the kernel of the fully fractional
heat operator $(\partial_{t}-\Delta)^{s}$ possesses a radial decreasing
property, thus it is hopeful to establish the direct method of moving planes,
which is exactly the research objective of this paper. The innovation of this
paper is that it overcomes the difficulties arising from the non-locality and
strong correlation of the operator $(\partial_{t}-\Delta)^{s}$, and avoids the
heavy reliance on extension method as in classical approach when studying the
master equations. By establishing various maximum principles, we have
developed a direct method of moving plane applicable to the master equation,
which has allowed us to obtain the radial symmetry and strict monotonicity of
solution $u(x,t)$ to the master equation (1.1) in a unit ball for any
$t\in\mathbb{R}$. More surprisingly, we have also applied this direct method
to establish the Liouville theorem of the homogeneous master equation in
$\mathbb{R}^{n}\times\mathbb{R}$.

To illustrate the main results of this paper, we start by presenting the
notation that will be used throughout the subsequent sections. Let $x_{1}$ be
any given direction in $\mathbb{R}^{n}$,

$T_{\lambda}=\\{x=(x_{1},x^{\prime})\in\mathbb{R}^{n}\mid
x_{1}=\lambda\,\,\mbox{for}\,\,\lambda\in\mathbb{R}\\}$

be the moving planes perpendicular to $x_{1}$-axis,

$\Sigma_{\lambda}=\\{x\in\mathbb{R}^{n}\mid
x_{1}<\lambda\\}\,\,\mbox{and}\,\,\Omega_{\lambda}=\\{x\in B_{1}(0)\mid
x_{1}<\lambda\\}$

be the region to the left of the hyperplane $T_{\lambda}$ in $\mathbb{R}^{n}$
and in $B_{1}(0)$ respectively. We denote the reflection of $x$ with respect
to the hyperplane $T_{\lambda}$ as

$x^{\lambda}=(2\lambda-x_{1},x_{2},\cdots,x_{n}).$

Let $u_{\lambda}(x,t)=u(x^{\lambda},t)$, we define

$w_{\lambda}(x,t)=u_{\lambda}(x,t)-u(x,t),$

which represents the comparison between the values of $u(x,t)$ and
$u(x^{\lambda},t)$. It is obvious that $w_{\lambda}(x,t)$ is an antisymmetric
function of $x$ with respect to the hyperplane $T_{\lambda}$.

We are now ready to present the main results of this paper. Our primary
outcome is the narrow region principle for antisymmetric functions in bounded
domains.

######  Theorem 1.1.

Let $\Omega$ be a bounded narrow domain containing in the narrow slab
$\\{x\in\Sigma_{\lambda}\mid\lambda-l<x_{1}<\lambda\\}$ with some small $l$.
Suppose that

$w(x,t)\in C^{2s+\epsilon,s+\epsilon}_{x,\,t,\,{\rm
loc}}(\Omega\times\mathbb{R})\cap\mathcal{L}(\mathbb{R}^{n}\times\mathbb{R})$

is lower semi-continuous up to the boundary $\partial\Omega$ and bounded from
below in $\Omega\times\mathbb{R}$, and satisfies

$\left\\{\begin{array}[]{ll}(\partial_{t}-\Delta)^{s}w(x,t)=c(x,t)w(x,t),{}&(x,t)\in\Omega\times\mathbb{R},\\\
w(x,t)\geq 0,{}&(x,t)\in(\Sigma_{\lambda}\backslash\Omega)\times\mathbb{R},\\\
w(x,t)=-w(x^{\lambda},t),&(x,t)\in\Sigma_{\lambda}\times\mathbb{R},\end{array}\right.$
(1.6)

where the coefficient function $c(x,t)$ has a uniformly upper bound $C_{0}$.

Then

$w(x,t)\geq 0,\,\,\mbox{in}\,\,\Sigma_{\lambda}\times\mathbb{R}$ (1.7)

for sufficiently small $l$. Furthermore, if $w(x,t)$ attains zero at some
point $(x^{0},t_{0})\in\Omega\times\mathbb{R}$, then

$w(x,t)\equiv 0,\,\,\mbox{in}\,\,\mathbb{R}^{n}\times(-\infty,t_{0}].$ (1.8)

The above narrow region principle is a crucial element in implementing the
method of moving planes, as it provides a starting point to observe some
geometric behavior of solutions. Based on this maximum principle, we construct
a direct method of moving planes that is suitable for the master equations,
and then determine the maximum distance that the plane can be moved while
maintaining the desired property. This allows us to demonstrate the following
radial symmetry and strict monotonicity result.

######  Theorem 1.2.

Let

$u(x,t)\in C^{2s+\epsilon,s+\epsilon}_{x,\,t}(B_{1}(0)\times\mathbb{R})$

be a positive bounded solution of

$\left\\{\begin{array}[]{ll}(\partial_{t}-\Delta)^{s}u(x,t)=f(u(x,t)),&(x,t)\in
B_{1}(0)\times\mathbb{R},\\\ u(x,t)\equiv 0,&(x,t)\in
B^{c}_{1}(0)\times\mathbb{R}.\end{array}\right.$

Suppose that $f\in C^{1}([0,+\infty))$ satisfies $f(0)\geq 0$ and
$f^{\prime}(0)\leq 0$. Then $u(x,t)$ is radially symmetric and strictly
decreasing about the origin in $B_{1}(0)$ for any $t\in\mathbb{R}$.

To establish the Liouville theorem for the homogeneous master equation in
$\mathbb{R}^{n}\times\mathbb{R}$, which has important applications in several
areas of mathematics and physics, we first establish the following maximum
principle for antisymmetric functions in unbounded domains.

######  Theorem 1.3.

Let

$w(x,t)\in C^{2s+\epsilon,s+\epsilon}_{x,\,t,\,{\rm
loc}}(\Sigma_{\lambda}\times\mathbb{R})\cap\mathcal{L}(\mathbb{R}^{n}\times\mathbb{R})$

be upper semi-continuous up to the boundary $T_{\lambda}$ and bounded from
above in $\Sigma_{\lambda}\times\mathbb{R}$. Suppose that

$\left\\{\begin{array}[]{ll}(\partial_{t}-\Delta)^{s}w(x,t)\leq 0,{}&\mbox{at
the points
in}\,\,\Sigma_{\lambda}\times\mathbb{R}\,\,\mbox{where}\,\,w(x,t)>0,\\\
w(x,t)=-w(x^{\lambda},t),&\mbox{in}\,\,\Sigma_{\lambda}\times\mathbb{R},\end{array}\right.$
(1.9)

then there holds that

$w(x,t)\leq 0\,\,\mbox{in}\,\,\Sigma_{\lambda}\times\mathbb{R}.$ (1.10)

######  Remark 1.4.

Compared to the narrow domain principle (Theorem 1.1) and the maximum
principle (Theorem 1.3), although the overall proof methods of both theorems
are based on the contradiction and perturbation technique, the former only has
$t$ belonging to an unbounded interval, while the latter has both $x$ and $t$
belonging to unbounded domains, and $x$ does not belong to a narrow region. In
the latter case, $x$ and $t$ need to be perturbed simultaneously, and it is
more effective to construct a sequence of auxiliary functions in an
antisymmetric form.

By utilizing this maximum principle to show that the solutions are symmetric
with respect to all hyperplanes in $\mathbb{R}^{n}$ for any $t\in\mathbb{R}$,
and combining with Fourier transform, we establish the Liouville theorem for
the homogeneous master equation in $\mathbb{R}^{n}\times\mathbb{R}$.

######  Theorem 1.5.

Assume that

$u(x,t)\in C^{2s+\epsilon,s+\epsilon}_{x,\,t,\,{\rm
loc}}(\mathbb{R}^{n}\times\mathbb{R})$

is a bounded solution of

$(\partial_{t}-\Delta)^{s}u(x,t)=0,\,\,(x,t)\in\mathbb{R}^{n}\times\mathbb{R},$

Then $u(x,t)$ must be a constant.

######  Remark 1.6.

As is well known, the classical Liouville theorem states that any bounded
harmonic function defined on the whole space must be a constant. In fact, this
boundedness condition can be relaxed to one-sided boundedness. However, under
this relaxed condition, the Liouville theorem for the caloric function
satisfying heat equation

$(\partial_{t}-\Delta)u(x,t)=0,\,\,\mbox{in}\,\,\mathbb{R}^{n}\times\mathbb{R}$
(1.11)

is not valid. For instance, the function $u(x,t)=e^{x+t}$ is a positive
nonconstant solution of (1.11) in $\mathbb{R}^{n}\times\mathbb{R}$. Since the
master equations recover the classical parabolic equations in limit cases,
then in this sense Theorem 1.5 can be regarded as a generalization of the
Liouville theorem for the local heat equation to the case of master equations
involving the fractional heat operator $(\partial_{t}-\Delta)^{s}$, where the
bounded condition may not be optimal but is still reasonable.

######  Remark 1.7.

In contrast to the nonlocal elliptic equation $(-\Delta)^{s}u(x)=0$ and the
nonlocal parabolic equations $\partial_{t}u(x,t)+(-\Delta)^{s}u(x,t)=0$, where
the Liouville theorem can be directly proven by the maximum principles in
unbounded domains. But for the space-time nonlocal equation (1.2), where the
operator $(\partial_{t}-\Delta)^{s}$ with respect to the time variable $t$ is
also nonlocal, such maximum principle can only determine that the solution is
solely dependent on the variable $t$. To further prove that the solution must
be a constant, Fourier transform is applied.

To conclude this section, we will provide a brief outline of the structure of
this paper. In Section 2 , we present the definition of parabolic Hölder
space, as well as some auxiliary results that are necessary for proving our
main theorems. Section 3 is devoted to demonstrating two maximum principle:
the narrow domain principle (Theorem 1.1) and the maximum principle in
unbounded domains (Theorem 1.3). By utilizing these maximum principles, we
establish a direct method of moving planes for the master equation, which
enables us to complete the proof of Theorem 1.2 and Theorem 1.5 in the last
section.

##  2 Preliminaries

In this section, we present the definition of the parabolic Hölder space and
collect some useful preliminary estimates, which are necessary for
establishing our main results. Throughout this paper, we use $C$ to denote a
general constant whose value may vary from line to line.

Now we start by stating the definition of parabolic Hölder space
$C^{2\alpha,\alpha}_{x,\,t}(\mathbb{R}^{n}\times\mathbb{R})$ (cf. [19]) as
follows.

  * (i)

When $0<\alpha\leq\frac{1}{2}$, if $u(x,t)\in
C^{2\alpha,\alpha}_{x,\,t}(\mathbb{R}^{n}\times\mathbb{R})$, then there exists
a constant $C>0$ such that

$|u(x,t)-u(y,\tau)|\leq C\left(|x-y|+|t-\tau|^{\frac{1}{2}}\right)^{2\alpha}$

for any $x,\,y\in\mathbb{R}^{n}$ and $t,\,\tau\in\mathbb{R}$.

  * (ii)

When $\frac{1}{2}<\alpha\leq 1$, we say that

$u(x,t)\in
C^{2\alpha,\alpha}_{x,\,t}(\mathbb{R}^{n}\times\mathbb{R}):=C^{1+(2\alpha-1),\alpha}_{x,\,t}(\mathbb{R}^{n}\times\mathbb{R}),$

if $u$ is $\alpha$-Hölder continuous in $t$ uniformly with respect to $x$ and
its gradient $\nabla_{x}u$ is $(2\alpha-1)$-Hölder continuous in $x$ uniformly
with respect to $t$ and $(\alpha-\frac{1}{2})$-Hölder continuous in $t$
uniformly with respect to $x$.

  * (iii)

While for $\alpha>1$, if $u(x,t)\in
C^{2\alpha,\alpha}_{x,\,t}(\mathbb{R}^{n}\times\mathbb{R}),$ then it means
that

$\partial_{t}u,\,D^{2}_{x}u\in
C^{2\alpha-2,\alpha-1}_{x,\,t}(\mathbb{R}^{n}\times\mathbb{R}).$

In addition, we can analogously define the local parabolic Hölder space
$C^{2\alpha,\alpha}_{x,\,t,\,\rm{loc}}(\mathbb{R}^{n}\times\mathbb{R})$.

In the following, we provide the boundedness estimates for the nonlocal
operators $\partial_{t}^{s}$ and $(\partial_{t}-\Delta)^{s}$ acting on smooth
functions, respectively. These estimates are repeatedly employed in
establishing our main results.

######  Lemma 2.1.

(cf. [10]) Let

$\eta_{r}(t)\in C_{0}^{\infty}\left((-r^{2}+t_{0},r^{2}+t_{0})\right),$

For some $t_{0}\in\mathbb{R}$ and $r>0$, then there exists a positive constant
$C$ that depends only on $s$ such that

$|\partial_{t}^{s}\eta_{r}(t)|\leq\frac{C}{r^{2s}}\,\,\mbox{in}\,\,(-r^{2}+t_{0},r^{2}+t_{0}).$

######  Lemma 2.2.

(cf. [9]) Let

$\phi_{r}(x,t)\in
C_{0}^{\infty}\left(B_{r}(x^{0})\times(-r^{2}+t_{0},r^{2}+t_{0})\right)$

for some $(x^{0},t_{0})\in\mathbb{R}^{n}\times\mathbb{R}$ and $r>0$, then
there exists a positive constant $C$ that depends only on $s$ and $n$ such
that

$|(\partial_{t}-\Delta)^{s}\phi_{r}(x,t)|\leq\frac{C}{r^{2s}}\,\,\mbox{in}\,\,B_{r}(x^{0})\times(-r^{2}+t_{0},r^{2}+t_{0}).$

##  3 Various maximum principles for antisymmetric functions

In this section, we demonstrate various maximum principles for antisymmetric
functions, namely, Theorem 1.1 and Theorem 1.3 . It will be shown in the later
section that these principles play a crucial role in establishing a direct
method of moving planes for the master equations.

###  3.1 Narrow region principle in bounded domains

Our first objective is to prove Theorem 1.1 , which establishes a narrow
region principle for antisymmetric functions in bounded domains. This
principle is a key ingredient in demonstrating the radial symmetry and
monotonicity of solutions for the master equation (1.1).

###### Proof of Theorem 1.1 ..

We first argue by contradiction to derive (1.7). Due to the narrow region
$\Omega$ is bounded and the function $w$ is lower semi-continuous up to the
boundary $\partial\Omega$, there must exist $x(t)\in\Omega$ such that

$w(x(t),t)=\displaystyle\min_{x\in\Omega}w(x,t)$ (3.1)

for each fixed $t\in\mathbb{R}$. If (1.7) is not valid, we may assume on the
contrary that there exists a positive constant $m$ such that

$\inf_{\Omega\times\mathbb{R}}w(x,t)=\inf_{\mathbb{R}}w(x(t),t)=-m<0.$ (3.2)

Since $t$ belongs to $\mathbb{R}$ that is an unbounded domain, the infimum of
$w(x(t),t)$ with respect to $t$ may not be attainable, but there certainly
exists a sequence $\\{(x(t_{k}),t_{k})\\}\subset\Omega\times\mathbb{R}$ such
that

$w(x(t_{k}),t_{k})=-m_{k}\rightarrow-m\,\,\mbox{as}\,\,k\rightarrow\infty.$

Let $\varepsilon_{k}=m-m_{k}$, then it is evident that $\varepsilon_{k}$ is
nonnegative and $\varepsilon_{k}\rightarrow 0$ as $k\rightarrow 0$. In order
to remedy scenario that the infimum of $w$ with respect to $t$ may not be
attained, we need to perturb $w$ with respect to the variable $t$ such that
the perturbed function can attain its infimum. For this purpose, we introduce
the following auxiliary function

$v_{k}(x,t)=w(x,t)-\varepsilon_{k}\eta_{k}(t),$

where

$\eta_{k}(t)=\eta(t-t_{k})\in C_{0}^{\infty}\left((-1+t_{k},1+t_{k})\right)$

is a smooth cut-off function that satisfies

$\eta(t)=\left\\{\begin{array}[]{ll}1&t\in(-\frac{1}{2},\frac{1}{2}),\\\
0,{}&t\not\in(-1,1),\end{array}\right.$ (3.3)

and $0\leq\eta(t)\leq 1$. One one hand, we have

$v_{k}(x(t_{k}),t_{k})=w(x(t_{k}),t_{k})-\varepsilon_{k}=-m_{k}-m+m_{k}=-m.$

On the other hand, if
$(x,t)\in\Omega\times\left(\mathbb{R}\setminus(-1+t_{k},1+t_{k})\right)$, then
it follows from (3.2) that

$v_{k}(x,t)=w(x,t)\geq-m.$

Based on the above analysis and the exterior condition in (1.6) satisfied by
$w$, there exists

$(\bar{x}^{k},\bar{t}_{k})\in\Omega\times(-1+t_{k},1+t_{k})$

such that

$-m-\varepsilon_{k}\leq
v_{k}(\bar{x}^{k},\bar{t}_{k})=\inf_{\Omega\times\mathbb{R}}v_{k}(x,t)=\inf_{\Sigma_{\lambda}\times\mathbb{R}}v_{k}(x,t)\leq-m.$
(3.4)

From this, it is not difficult to further verify that

$-m\leq w(\bar{x}^{k},\bar{t}_{k})\leq-m+\varepsilon_{k}=-m_{k}<0.$ (3.5)

Next, we derive a contradiction at the minimum point
$(\bar{x}^{k},\bar{t}_{k})$ of $v_{k}$ in $\Sigma_{\lambda}\times\mathbb{R}$.
On the one hand, by performing a direct calculation and combining the anti-
symmetry of $w$ in $x$ with the decreasing property of the kernel of operator
$(\partial_{t}-\Delta)^{s}$ due to
$|\bar{x}^{k}-y|<|\bar{x}^{k}-y^{\lambda}|$, we have

$\displaystyle(\partial_{t}-\Delta)^{s}v_{k}(\bar{x}^{k},\bar{t}_{k})$
$\displaystyle=$ $\displaystyle
C_{n,s}\int_{-\infty}^{\bar{t}_{k}}\int_{\mathbb{R}^{n}}\frac{v_{k}(\bar{x}^{k},\bar{t}_{k})-v_{k}(y,\tau)}{(\bar{t}_{k}-\tau)^{\frac{n}{2}+1+s}}e^{-\frac{|\bar{x}^{k}-y|^{2}}{4(\bar{t}_{k}-\tau)}}\operatorname{d}\\!y\operatorname{d}\\!\tau$
$\displaystyle=$ $\displaystyle
C_{n,s}\int_{-\infty}^{\bar{t}_{k}}\int_{\Sigma_{\lambda}}\frac{v_{k}(\bar{x}^{k},\bar{t}_{k})-v_{k}(y,\tau)}{(\bar{t}_{k}-\tau)^{\frac{n}{2}+1+s}}e^{-\frac{|\bar{x}^{k}-y|^{2}}{4(\bar{t}_{k}-\tau)}}\operatorname{d}\\!y\operatorname{d}\\!\tau$
$\displaystyle+C_{n,s}\int_{-\infty}^{\bar{t}_{k}}\int_{\Sigma_{\lambda}}\frac{v_{k}(\bar{x}^{k},\bar{t}_{k})-v_{k}(y^{\lambda},\tau)}{(\bar{t}_{k}-\tau)^{\frac{n}{2}+1+s}}e^{-\frac{|\bar{x}^{k}-y^{\lambda}|^{2}}{4(\bar{t}_{k}-\tau)}}\operatorname{d}\\!y\operatorname{d}\\!\tau$
$\displaystyle\leq$ $\displaystyle
C_{n,s}\int_{-\infty}^{\bar{t}_{k}}\int_{\Sigma_{\lambda}}\frac{v_{k}(\bar{x}^{k},\bar{t}_{k})-v_{k}(y,\tau)}{(\bar{t}_{k}-\tau)^{\frac{n}{2}+1+s}}e^{-\frac{|\bar{x}^{k}-y^{\lambda}|^{2}}{4(\bar{t}_{k}-\tau)}}\operatorname{d}\\!y\operatorname{d}\\!\tau$
$\displaystyle+C_{n,s}\int_{-\infty}^{\bar{t}_{k}}\int_{\Sigma_{\lambda}}\frac{v_{k}(\bar{x}^{k},\bar{t}_{k})-v_{k}(y^{\lambda},\tau)}{(\bar{t}_{k}-\tau)^{\frac{n}{2}+1+s}}e^{-\frac{|\bar{x}^{k}-y^{\lambda}|^{2}}{4(\bar{t}_{k}-\tau)}}\operatorname{d}\\!y\operatorname{d}\\!\tau$
$\displaystyle=$ $\displaystyle
C_{n,s}\int_{-\infty}^{\bar{t}_{k}}\int_{\Sigma_{\lambda}}\frac{v_{k}(\bar{x}^{k},\bar{t}_{k})-w(y,\tau)+\varepsilon_{k}\eta_{k}(\tau)}{(\bar{t}_{k}-\tau)^{\frac{n}{2}+1+s}}e^{-\frac{|\bar{x}^{k}-y^{\lambda}|^{2}}{4(\bar{t}_{k}-\tau)}}\operatorname{d}\\!y\operatorname{d}\\!\tau$
$\displaystyle+C_{n,s}\int_{-\infty}^{\bar{t}_{k}}\int_{\Sigma_{\lambda}}\frac{v_{k}(\bar{x}^{k},\bar{t}_{k})-w(y^{\lambda},\tau)+\varepsilon_{k}\eta_{k}(\tau)}{(\bar{t}_{k}-\tau)^{\frac{n}{2}+1+s}}e^{-\frac{|\bar{x}^{k}-y^{\lambda}|^{2}}{4(\bar{t}_{k}-\tau)}}\operatorname{d}\\!y\operatorname{d}\\!\tau$
$\displaystyle=$ $\displaystyle
2C_{n,s}v_{k}(\bar{x}^{k},\bar{t}_{k})\int_{-\infty}^{\bar{t}_{k}}\int_{\Sigma_{\lambda}}\frac{e^{-\frac{|\bar{x}^{k}-y^{\lambda}|^{2}}{4(\bar{t}_{k}-\tau)}}}{(\bar{t}_{k}-\tau)^{\frac{n}{2}+1+s}}\operatorname{d}\\!y\operatorname{d}\\!\tau+2C_{n,s}\varepsilon_{k}\int_{-\infty}^{\bar{t}_{k}}\int_{\Sigma_{\lambda}}\frac{\eta_{k}(\tau)e^{-\frac{|\bar{x}^{k}-y^{\lambda}|^{2}}{4(\bar{t}_{k}-\tau)}}}{(\bar{t}_{k}-\tau)^{\frac{n}{2}+1+s}}\operatorname{d}\\!y\operatorname{d}\\!\tau.$

In order to estimate two integrals in the last line of (3.1), we make a change
of variables

$t=\frac{|\bar{x}^{k}-y^{\lambda}|^{2}}{4(\bar{t}_{k}-\tau)}$

and derive

$\displaystyle
C_{n,s}\int_{-\infty}^{\bar{t}_{k}}\int_{\Sigma_{\lambda}}\frac{e^{-\frac{|\bar{x}^{k}-y^{\lambda}|^{2}}{4(\bar{t}_{k}-\tau)}}}{(\bar{t}_{k}-\tau)^{\frac{n}{2}+1+s}}\operatorname{d}\\!y\operatorname{d}\\!\tau$
$\displaystyle=$
$\displaystyle\frac{1}{(4\pi)^{\frac{n}{2}}|\Gamma(-s)|}\int_{\Sigma_{\lambda}}\int_{0}^{+\infty}\frac{e^{-t}}{(\frac{|\bar{x}^{k}-y^{\lambda}|^{2}}{4t})^{\frac{n}{2}+1+s}}\frac{|\bar{x}_{k}-y^{\lambda}|^{2}}{4t^{2}}\operatorname{d}\\!t\operatorname{d}\\!y$
$\displaystyle=$
$\displaystyle\frac{4^{s}}{\pi^{\frac{n}{2}}|\Gamma(-s)|}\int_{\Sigma_{\lambda}}\frac{1}{|\bar{x}^{k}-y^{\lambda}|^{n+2s}}\int_{0}^{+\infty}e^{-t}t^{\frac{n}{2}+s-1}\operatorname{d}\\!t\operatorname{d}\\!y$
$\displaystyle=$
$\displaystyle\frac{4^{s}\Gamma(\frac{n}{2}+s)}{\pi^{\frac{n}{2}}|\Gamma(-s)|}\int_{\Sigma_{\lambda}}\frac{1}{|\bar{x}^{k}-y^{\lambda}|^{n+2s}}\operatorname{d}\\!y.$

Substituting the above equality into (3.1), and applying $\eta_{k}\in[0,1]$
and (3.4) to arrive at

$\displaystyle(\partial_{t}-\Delta)^{s}v_{k}(\bar{x}^{k},\bar{t}_{k})$ (3.7)
$\displaystyle\leq$ $\displaystyle
Cv_{k}(\bar{x}^{k},\bar{t}_{k})\int_{\Sigma_{\lambda}}\frac{1}{|\bar{x}^{k}-y^{\lambda}|^{n+2s}}\operatorname{d}\\!y+C\varepsilon_{k}\int_{\Sigma_{\lambda}}\frac{1}{|\bar{x}^{k}-y^{\lambda}|^{n+2s}}\operatorname{d}\\!y$
$\displaystyle\leq$
$\displaystyle\frac{Cv_{k}(\bar{x}^{k},\bar{t}_{k})}{l^{2s}}+\frac{C\varepsilon_{k}}{l^{2s}}$
$\displaystyle\leq$
$\displaystyle-\frac{Cm}{l^{2s}}+\frac{C\varepsilon_{k}}{l^{2s}}.$

On the other hand, starting from the differential equation in (1.6) and
combining the fact that $C(x,t)$ has a uniformly upper bound $C_{0}$, (3.5)
with Lemma 2.1 , we obtain

$\displaystyle(\partial_{t}-\Delta)^{s}v_{k}(\bar{x}^{k},\bar{t}_{k})$
$\displaystyle=$
$\displaystyle(\partial_{t}-\Delta)^{s}w(\bar{x}^{k},\bar{t}_{k})-\varepsilon_{k}(\partial_{t}-\Delta)^{s}\eta(\bar{t}_{k})$
(3.8) $\displaystyle=$ $\displaystyle
c(\bar{x}^{k},\bar{t}_{k})w(\bar{x}^{k},\bar{t}_{k})-\varepsilon_{k}\partial_{t}^{s}\eta(\bar{t}_{k})$
$\displaystyle\geq$ $\displaystyle-C_{0}m-C\varepsilon_{k}.$

Then a combination of (3.7) and (3.8) yields that

$-C_{0}m\leq-\frac{Cm}{l^{2s}}+\frac{C\varepsilon_{k}}{l^{2s}}+C\varepsilon_{k}\rightarrow-\frac{Cm}{l^{2s}},$

as $k\rightarrow\infty$. Dividing both side of the preceding inequality by
$-m$, we deduce that

$C_{0}\geq\frac{C}{l^{2s}},$

which is a contradiction for sufficiently small $l$. Hence, we conclude that
(1.7) is true.

In the sequel, we remain to demonstrate the validity of (1.8). It follows from
(1.7) that

$w(x^{0},t_{0})=\min_{\Sigma_{\lambda}\times\mathbb{R}}w(x,t)=0.$

The equation in (1.6) obviously implies that

$(\partial_{t}-\Delta)^{s}w(x^{0},t_{0})=0.$ (3.9)

Besides, through a straightforward calculation, we derive

$\displaystyle(\partial_{t}-\Delta)^{s}w(x^{0},t_{0})$ $\displaystyle=$
$\displaystyle-
C_{n,s}\int_{-\infty}^{t_{0}}\int_{\mathbb{R}^{n}}\frac{w(y,\tau)}{(t_{0}-\tau)^{\frac{n}{2}+1+s}}e^{-\frac{|x^{0}-y|^{2}}{4(t_{0}-\tau)}}\operatorname{d}\\!y\operatorname{d}\\!\tau$
$\displaystyle=$ $\displaystyle
C_{n,s}\int_{-\infty}^{t_{0}}\int_{\Sigma_{\lambda}}\frac{w(y,\tau)}{(t_{0}-\tau)^{\frac{n}{2}+1+s}}\left[e^{-\frac{|x^{0}-y^{\lambda}|^{2}}{4(t_{0}-\tau)}}-e^{-\frac{|x^{0}-y|^{2}}{4(t_{0}-\tau)}}\right]\operatorname{d}\\!y\operatorname{d}\\!\tau.$

Since $w(x,t)\geq 0$ in $\Sigma_{\lambda}\times\mathbb{R}$ and

$e^{-\frac{|x^{0}-y^{\lambda}|^{2}}{4(t_{0}-\tau)}}-e^{-\frac{|x^{0}-y|^{2}}{4(t_{0}-\tau)}}<0,$

then it follows from (3.9) that

$w(x,t)\equiv
0\,\,\mbox{for}\,\,(x,t)\in\Sigma_{\lambda}\times(-\infty,t_{0}].$

Finally, the antisymmetry of $w(x,t)$ with respect to $x$ infers that

$w(x,t)\equiv 0\,\,\mbox{in}\,\,\mathbb{R}^{n}\times(-\infty,t_{0}].$

Therefore, the proof of Theorem 1.1 is completed. ∎

###  3.2 Maximum principle in unbounded domains

We proceed to prove Theorem 1.3 , which establishes the maximum principle for
antisymmetric functions in unbounded domains. This principle is a fundamental
ingredient in proving the Liouville theorem for homogeneous master equations.

###### Proof of Theorem 1.3 ..

The proof goes by contradiction, if (1.10) is violated, since $w(x,t)$ has an
upper bound in $\Sigma_{\lambda}\times\mathbb{R}$, then there exists a
positive constant $A$ such that

$\sup_{\Sigma_{\lambda}\times\mathbb{R}}w(x,t)=A>0.$ (3.10)

Note that the set $\Sigma_{\lambda}\times\mathbb{R}$ is unbounded, then the
supremum of $w(x,t)$ may not be attained. Even so, (3.10) implies that there
exists a sequence $\\{(x^{k},t_{k})\\}\subset\Sigma_{\lambda}\times\mathbb{R}$
such that

$0<w(x^{k},t_{k})=A_{k}\rightarrow A,\,\,\mbox{as}\,\,k\rightarrow\infty.$
(3.11)

Let $\varepsilon_{k}=A-A_{k}$, then the sequence $\\{\varepsilon_{k}\\}$ is
nonnegative and tends to zero as $k\rightarrow\infty$.

Given that the supremum of $w(x,t)$ with respect to both $x$ and $t$ may not
be attained, it is necessary to perturb the function $w$ with respect to both
$x$ and $t$ such that the perturbed function not only attains the supremum,
but also preserves the antisymmetry in $x$. With this aim, we need to
introduce an antisymmetric auxiliary function. Let $\phi(x)\in
C_{0}^{\infty}\left(\mathbb{R}^{n}\right)$ satisfy

$\phi(x)=\left\\{\begin{array}[]{r@{\ \ }c@{\ \
}ll}&&e^{1+\frac{1}{|x|^{2}-1}},&\ \ x\in B_{1}(0)\,,\\\\[1.42271pt] &&0,&\ \
x\not\in B_{1}(0)\,.\end{array}\right.$

We denote

$\phi_{\lambda}(x)=\phi\left(x^{\lambda}\right)=\left\\{\begin{array}[]{r@{\ \
}c@{\ \ }ll}&&e^{1+\frac{1}{|x^{\lambda}|^{2}-1}},&\ \ x\in
B_{1}\big{(}0^{\lambda}\big{)}\,,\\\\[1.42271pt] &&0,&\ \ x\not\in
B_{1}\big{(}0^{\lambda}\big{)}\,,\end{array}\right.$

and $r_{k}=\operatorname{dist}(x^{k},T_{\lambda})$, then it is not difficult
to verify that

$\Phi_{k}(x)=\phi\left(\frac{2(x-x^{k})}{r_{k}}\right)-\phi_{\lambda}\left(\frac{2(x^{\lambda}-x^{k})}{r_{k}}\right)\in
C_{0}^{\infty}\left(B_{\frac{r_{k}}{2}}(x^{k})\cup
B_{\frac{r_{k}}{2}}\big{(}(x^{k})^{\lambda}\big{)}\right)$

is an antisymmetric function with respect to the plane $T_{\lambda}$, and

$\max_{x\in\mathbb{R}^{n}}\Phi_{k}(x)=\Phi_{k}(x^{k})=\phi(x^{k})=1.$

We further select a smooth cut-off function of $t$ that

$\eta_{k}(t)=\eta\left(\frac{t-t_{k}}{(\frac{r_{k}}{2})^{2}}\right)\in
C_{0}^{\infty}\left((t_{k}-(\frac{r_{k}}{2})^{2},t_{k}+(\frac{r_{k}}{2})^{2})\right),$

where $\eta$ is defined in (3.3).

Now we choose the antisymmetric auxiliary function as follows

$V_{k}(x,t)=w(x,t)+\varepsilon_{k}\Phi_{k}(x)\eta_{k}(t).$

Let

$Q_{\frac{r_{k}}{2}}(x^{k},t_{k})=B_{\frac{r_{k}}{2}}(x^{k})\times(t_{k}-(\frac{r_{k}}{2})^{2},t_{k}+(\frac{r_{k}}{2})^{2})$

be a parabolic cylinder, then a straightforward calculation implies that

$V_{k}(x^{k},t_{k})=w(x^{k},t_{k})+\varepsilon_{k}=A_{k}+A-A_{k}=A,$

and

$V_{k}(x,t)=w(x,t)\leq
A\,\,\mbox{for}\,\,(x,t)\in(\Sigma_{\lambda}\times\mathbb{R})\setminus
Q_{\frac{r_{k}}{2}}(x^{k},t_{k}).$

Thus, the auxiliary function $V_{k}(x,t)$ must attain its maximum value in
$\Sigma_{\lambda}\times\mathbb{R}$. More precisely, there exists a point
$(\bar{x}^{k},\bar{t}_{k})\in Q_{\frac{r_{k}}{2}}(x^{k},t_{k})$ such that

$A+\varepsilon_{k}\geq
V_{k}(\bar{x}^{k},\bar{t}_{k})=\sup_{\Sigma_{\lambda}\times\mathbb{R}}V_{k}(x,t)\geq
A>0.$ (3.12)

Meanwhile, it follows from the definition of $V_{k}$ that

$A\geq w(\bar{x}^{k},\bar{t}_{k})\geq A-\varepsilon_{k}=A_{k}>0,$

then we can apply the differential inequality in (1.9) to $w$ at point
$(\bar{x}^{k},\bar{t}_{k})$.

Next, we intend to derive a contradiction at the maximum point
$(\bar{x}^{k},\bar{t}_{k})$ of $V_{k}$ in $\Sigma_{\lambda}\times\mathbb{R}$.
On one hand, combining the definition of $V_{k}$ with the differential
inequality in (1.9) and Lemma 2.2 , we obtain

$(\partial_{t}-\Delta)^{s}V_{k}(\bar{x}^{k},\bar{t}_{k})=(\partial_{t}-\Delta)^{s}w(\bar{x}^{k},\bar{t}_{k})+\varepsilon_{k}(\partial_{t}-\Delta)^{s}\left(\Phi_{k}(\bar{x}^{k})\eta_{k}(\bar{t}_{k})\right)\leq\frac{C\varepsilon_{k}}{r_{k}^{2s}}.$
(3.13)

On the other hand, starting from the definition of operator
$(\partial_{t}-\Delta)^{s}$ and utilizing the antisymmetry of auxiliary
function $V_{k}$, the radial decrease of the kernel as well as (3.12), we
compute

$\displaystyle(\partial_{t}-\Delta)^{s}V_{k}(\bar{x}^{k},\bar{t}_{k})$ (3.14)
$\displaystyle=$ $\displaystyle
C_{n,s}\int_{-\infty}^{\bar{t}_{k}}\int_{\Sigma_{\lambda}}\frac{V_{k}(\bar{x}^{k},\bar{t}_{k})-V_{k}(y,\tau)}{(\bar{t}_{k}-\tau)^{\frac{n}{2}+1+s}}e^{-\frac{|\bar{x}_{k}-y|^{2}}{4(\bar{t}_{k}-\tau)}}\operatorname{d}\\!y\operatorname{d}\\!\tau$
$\displaystyle+C_{n,s}\int_{-\infty}^{\bar{t}_{k}}\int_{\Sigma_{\lambda}}\frac{V_{k}(\bar{x}^{k},\bar{t}_{k})+V_{k}(y,\tau)}{(\bar{t}_{k}-\tau)^{\frac{n}{2}+1+s}}e^{-\frac{|\bar{x}_{k}-y^{\lambda}|^{2}}{4(\bar{t}_{k}-\tau)}}\operatorname{d}\\!y\operatorname{d}\\!\tau$
$\displaystyle\geq$ $\displaystyle
2C_{n,s}\int_{-\infty}^{\bar{t}_{k}}\int_{\Sigma_{\lambda}}\frac{V_{k}(\bar{x}^{k},\bar{t}_{k})}{(\bar{t}_{k}-\tau)^{\frac{n}{2}+1+s}}e^{-\frac{|\bar{x}_{k}-y^{\lambda}|^{2}}{4(\bar{t}_{k}-\tau)}}\operatorname{d}\\!y\operatorname{d}\\!\tau$
$\displaystyle=$ $\displaystyle
2\frac{4^{s}\Gamma(\frac{n}{2}+s)}{\pi^{\frac{n}{2}}|\Gamma(-s)|}V_{k}(\bar{x}^{k},\bar{t}_{k})\int_{\Sigma_{\lambda}}\frac{1}{|\bar{x}_{k}-y^{\lambda}|^{n+2s}}\operatorname{d}\\!y$
$\displaystyle\geq$ $\displaystyle\frac{CA}{r_{k}^{2s}}.$

Finally, a combination of (3.13) and (3.14) yields that

$A\leq C\epsilon_{k},$

which leads to a contradiction for sufficiently large $k$. Hence, we conclude
that (1.10) is valid, and then the proof of Theorem 1.3 is completed. ∎

##  4 The proof of main results

In this section, we will show how the maximum principles established
previously can be used to develop a direct method of moving planes applicable
to the master equations. By means of this direct method, we complete the proof
of our main results (i.e., Theorem 1.2 and Theorem 1.5).

###  4.1 Radial symmetry of solutions in a unit ball

In this subsection, we apply the narrow region principle (Theorem 1.1) to
initiate the moving plane and combine the perturbation technique with the
limit argument to prove that solutions of the master equation subject to the
vanishing exterior condition are radially symmetric and strictly decreasing
with respect to the origin in $B_{1}(0)$ for any $t\in\mathbb{R}$, under
appropriate assumptions on the nonhomogeneous term $f$.

###### Proof of Theorem 1.2 ..

To carrying out the method of moving planes, we choose $x_{1}$ to be any
direction and let $T_{\lambda}$, $\Sigma_{\lambda}$, $\Omega_{\lambda}$,
$x^{\lambda}$, and $w_{\lambda}$ be defined as in Section 1 . By a direct
calculation, we have

$\left\\{\begin{array}[]{ll}(\partial_{t}-\Delta)^{s}w_{\lambda}(x,t)=C_{\lambda}(x,t)w_{\lambda}(x,t),{}&(x,t)\in\Omega_{\lambda}\times\mathbb{R},\\\
w_{\lambda}(x,t)\geq
0,{}&(x,t)\in(\Sigma_{\lambda}\backslash\Omega_{\lambda})\times\mathbb{R},\\\
w_{\lambda}(x,t)=-w_{\lambda}(x^{\lambda},t),{}&(x,t)\in\Sigma_{\lambda}\times\mathbb{R},\\\
\end{array}\right.$ (4.1)

where the coefficient function

$C_{\lambda}(x,t)=\frac{f(u_{\lambda}(x,t))-f(u(x,t))}{u_{\lambda}(x,t)-u(x,t)}$

is bounded ensured by $f\in C^{1}([0,+\infty))$ and the boundedness of $u$.
Now we divide the proof into two steps.

Step 1. Start moving the plane $T_{\lambda}$ from $x_{1}=-1$ to the right
along $x_{1}$-axis. If $\lambda$ is sufficiently close to $-1$, then
$\Omega_{\lambda}$ is a narrow region. Furthermore, the assumptions in Theorem
1.2 guarantee that we can apply the narrow region principle, as established in
Theorem 1.1 , to problem (4.1). This allows us to deduce that

$w_{\lambda}(x,t)\geq 0\,\,\mbox{in}\,\,\Omega_{\lambda}\times\mathbb{R}.$
(4.2)

Note that inequality (4.2) provides a starting point to move the plane
$T_{\lambda}$.

Step 2. In the second step, we continue to move the plane $T_{\lambda}$ to the
right along $x_{1}$-axis as long as (4.2) is valid to its limiting position.
Let

$\lambda_{0}=\sup\left\\{\lambda<0\mid w_{\mu}(x,t)\geq
0,\,\,(x,t)\in\Sigma_{\mu}\times\mathbb{R}\,\,\mbox{for
any}\,\,\mu\leq\lambda\right\\},$

Our purpose is to show that

$\lambda_{0}=0$ (4.3)

by a contradiction argument. Otherwise, if $\lambda_{0}<0$, then the
definition of $\lambda_{0}$ and $w_{\lambda_{k}}(x,t)\geq 0$ in
$(\Sigma_{\lambda_{k}}\setminus\Omega_{\lambda_{k}})\times\mathbb{R}$ implies
that there exists a sequence of negative numbers $\\{\lambda_{k}\\}$ with
$\\{\lambda_{k}\\}\searrow\lambda_{0}$ such that

$\inf_{\Sigma_{\lambda_{k}}\times\mathbb{R}}w_{\lambda_{k}}(x,t)=\inf_{\Omega_{\lambda_{k}}\times\mathbb{R}}w_{\lambda_{k}}(x,t)=-m_{k}<0,$

and

$m_{k}\rightarrow 0,\,\,\mbox{as}\,\,k\rightarrow\infty.$

Since $\mathbb{R}$ is an unbounded interval, the infimum of $w_{\lambda_{k}}$
with respect to $t$ may not be attained, but there must exist a sequence
$\\{(x^{k},t_{k})\\}\subset\Omega_{\lambda_{k}}\times\mathbb{R}$ and a
nonnegative sequence $\\{\varepsilon_{k}\\}\searrow 0$ as
$k\rightarrow\infty$, such that

$\displaystyle
w_{\lambda_{k}}(x^{k},t_{k})=\inf_{x\in\Sigma_{\lambda_{k}}}w_{\lambda_{k}}(x,t_{k})\leq-
m_{k}+\varepsilon_{k}m_{k}<0.$

To address the situation where the infimum of $w_{\lambda_{k}}$ with respect
to $t$ may not be attained, we need to introduce the following auxiliary
function

$\displaystyle
W_{k}(x,t)=w_{\lambda_{k}}(x,t)-\varepsilon_{k}m_{k}\eta_{k}(t),$

where $\eta_{k}(t)=\eta(t-t_{k})$ and $\eta(t)$ is a smooth cut-off function
defined in (3.3). A straightforward computation leads to

$W_{k}(x^{k},t_{k})\leq-m_{k},$

and

$W_{k}(x,t)=w_{\lambda_{k}}(x,t)\geqslant-m_{k}$

for $|t-t_{k}|\geqslant 1$. Hence, the auxiliary function $W_{k}(x,t)$ could
attain its minimum at some point

$(\bar{x}^{k},\bar{t}_{k})\in\Omega_{\lambda_{k}}\times(t_{k}-1,t_{k}+1),$

such that

$\displaystyle-m_{k}-\varepsilon_{k}m_{k}\leq
W_{k}(\bar{x}^{k},\bar{t}_{k})=\inf_{\Sigma_{\lambda_{k}}\times\mathbb{R}}W_{k}(x,t)\leq-
m_{k}.$ (4.4)

It follows that

$-m_{k}\leq w_{\lambda_{k}}(\bar{x}^{k},\bar{t}_{k})\leq-
m_{k}+\varepsilon_{k}m_{k}<0.$ (4.5)

Next, we focus on the estimate of $W_{k}$ at its minimum point
$(\bar{x}^{k},\bar{t}_{k})$. On one hand, analogous to the estimate of (3.7),
applying the antisymmetry of $w_{\lambda_{k}}$ in $x$ and
$|\bar{x}^{k}-y|<|\bar{x}^{k}-y^{\lambda_{k}}|$, we derive

$\displaystyle(\partial_{t}-\Delta)^{s}W_{k}(\bar{x}^{k},\bar{t}_{k})$ (4.6)
$\displaystyle\leq$ $\displaystyle
2C_{n,s}W_{k}(\bar{x}^{k},\bar{t}_{k})\int_{-\infty}^{\bar{t}_{k}}\int_{\Sigma_{\lambda_{k}}}\frac{e^{-\frac{|\bar{x}^{k}-y^{\lambda_{k}}|^{2}}{4(\bar{t}_{k}-\tau)}}}{(\bar{t}_{k}-\tau)^{\frac{n}{2}+1+s}}\operatorname{d}\\!y\operatorname{d}\\!\tau+2C_{n,s}\varepsilon_{k}m_{k}\int_{-\infty}^{\bar{t}_{k}}\int_{\Sigma_{\lambda_{k}}}\frac{e^{-\frac{|\bar{x}^{k}-y^{\lambda_{k}}|^{2}}{4(\bar{t}_{k}-\tau)}}}{(\bar{t}_{k}-\tau)^{\frac{n}{2}+1+s}}\operatorname{d}\\!y\operatorname{d}\\!\tau$
$\displaystyle=$ $\displaystyle
CW_{k}(\bar{x}^{k},\bar{t}_{k})\int_{\Sigma_{\lambda_{k}}}\frac{1}{|\bar{x}^{k}-y^{\lambda_{k}}|^{n+2s}}\operatorname{d}\\!y+C\varepsilon_{k}m_{k}\int_{\Sigma_{\lambda_{k}}}\frac{1}{|\bar{x}^{k}-y^{\lambda_{k}}|^{n+2s}}\operatorname{d}\\!y$
$\displaystyle\leq$
$\displaystyle\frac{-Cm_{k}(C-\varepsilon_{k})}{\operatorname{dist}(\bar{x}^{k},T_{\lambda_{k}})^{2s}}$
$\displaystyle\leq$ $\displaystyle\frac{-Cm_{k}(C-\varepsilon_{k})}{2^{2s}}$

On the other hand, combining the differential equation in (4.1), (4.5) with
the boundedness of $C_{\lambda_{k}}(\bar{x}^{k},\bar{t}_{k})$ and Lemma 2.1 ,
we derive

$\displaystyle(\partial_{t}-\Delta)^{s}W_{k}(\bar{x}^{k},\bar{t}_{k})$
$\displaystyle=$
$\displaystyle(\partial_{t}-\Delta)^{s}w_{\lambda_{k}}(\bar{x}^{k},\bar{t}_{k})-\varepsilon_{k}m_{k}(\partial_{t}-\Delta)^{s}\eta(\bar{t}_{k})$
(4.7) $\displaystyle=$ $\displaystyle
C_{\lambda_{k}}(\bar{x}^{k},\bar{t}_{k})w_{\lambda_{k}}(\bar{x}^{k},\bar{t}_{k})-\varepsilon_{k}m_{k}\partial_{t}^{s}\eta(\bar{t}_{k})$
$\displaystyle\geq$ $\displaystyle-
C_{\lambda_{k}}(\bar{x}^{k},\bar{t}_{k})m_{k}-C\varepsilon_{k}m_{k}.$

Thus, a combination of (4.6) and (4.7) yields that

$C_{\lambda_{k}}(\bar{x}^{k},\bar{t}_{k})\geq-C\varepsilon_{k}+\frac{C(C-\varepsilon_{k})}{2^{2s}}\geq
C(C-\varepsilon_{k}).$

By virtue of $\varepsilon_{k}\rightarrow 0$ as $k\rightarrow\infty$, we deduce
that

$C_{\lambda_{k}}(\bar{x}^{k},\bar{t}_{k})\geq C_{0}>0$

for sufficiently large $k$. From this, owing to

$C_{\lambda_{k}}(\bar{x}^{k},\bar{t}_{k})=f^{\prime}(\xi)\,\,\mbox{for}\,\,\xi\in(u_{\lambda_{k}}(\bar{x}^{k},\bar{t}_{k}),u(\bar{x}^{k},\bar{t}_{k}))$

and the assumption $f^{\prime}(0)\leq 0$, there must exist a subsequence of
$\\{(\bar{x}^{k},\bar{t}_{k})\\}$ (still denoted by
$\\{(\bar{x}^{k},\bar{t}_{k})\\}$) such that

$u(\bar{x}^{k},\bar{t}_{k})\geq C_{1}>0.$ (4.8)

To proceed, we denote

$\bar{w}_{k}(x,t)=w_{\lambda_{k}}(x,t+\bar{t}_{k})\,\,\mbox{and}\,\,\bar{C}_{k}(x,t)=C_{\lambda_{k}}(x,t+t_{k}).$

It follows from Arzelà-Ascoli theorem that there exists some functions
$\bar{w}(x,t)$ and $\bar{C}(x,t)$ such that

$\lim\limits_{k\rightarrow\infty}\bar{w}_{k}(x,t)=\bar{w}(x,t)\,\,\mbox{and}\,\,\lim\limits_{k\rightarrow\infty}\bar{C}_{k}(x,t)=\bar{C}(x,t).$

Furthermore, on account of the equation

$\displaystyle(\partial_{t}-\Delta)^{s}\bar{w}_{k}(x,t)=\bar{C}_{k}(x,t)\bar{w}_{k}(x,t),\,\,\mbox{in}\,\,\Omega_{\lambda_{k}}\times\mathbb{R},$

and utilizing the regularity theory for master equation established in [27],
we obtain that the limit function $\bar{w}$ satisfies

$\displaystyle(\partial_{t}-\Delta)^{s}\bar{w}(x,t)=\bar{C}(x,t)\bar{w}(x,t),\,\,\mbox{in}\,\,\Omega_{\lambda_{0}}\times\mathbb{R},$
(4.9)

by $\lambda_{k}\rightarrow\lambda_{0}$ as $k\rightarrow\infty$. Due to
$\Omega_{\lambda_{k}}$ is a bounded domain, we may assume that
$\bar{x}^{k}\rightarrow x^{0}$, then applying (4.5) to derive

$\bar{w}(x^{0},0)\leftarrow\bar{w}_{k}(\bar{x}^{k},0)=w_{\lambda_{k}}(\bar{x}^{k},\bar{t}_{k})\rightarrow
0,$

as $k\rightarrow\infty$, that is to say,

$\bar{w}(x^{0},0)=0.$

Combining the limit equation (4.9) with a direct calculation, we obtain

$\displaystyle 0$ $\displaystyle=$
$\displaystyle(\partial_{t}-\Delta)^{s}\bar{w}(x^{0},0)$ $\displaystyle=$
$\displaystyle-
C_{n,s}\int_{-\infty}^{0}\int_{\mathbb{R}^{n}}\frac{\bar{w}(y,\tau)}{(-\tau)^{\frac{n}{2}+1+s}}e^{\frac{|x^{0}-y|^{2}}{4\tau}}\operatorname{d}\\!y\operatorname{d}\\!\tau$
$\displaystyle=$ $\displaystyle
C_{n,s}\int_{-\infty}^{0}\int_{\Sigma_{\lambda_{0}}}\bar{w}(y,\tau)\left[\frac{e^{\frac{|x^{0}-y^{\lambda_{0}}|^{2}}{4\tau}}}{(-\tau)^{\frac{n}{2}+1+s}}-\frac{e^{\frac{|x^{0}-y|^{2}}{4\tau}}}{(-\tau)^{\frac{n}{2}+1+s}}\right]\operatorname{d}\\!y\operatorname{d}\\!\tau.$

Thereby the nonnegativity of $\bar{w}(x,t)$ in
$\Sigma_{\lambda_{0}}\times\mathbb{R}$, the antisymmetry of $\bar{w}(x,t)$
with respect to $x$, and the radial decrease of the kernel ensure that the
following identity

$\bar{w}(x,t)\equiv 0\,\,\mbox{in}\,\,\mathbb{R}^{n}\times(-\infty,0]$ (4.10)

holds.

Furthermore, taking the same translation for $u$ as follows

$u_{k}(x,t)=u(x,t+t_{k}).$

Similarly to the above argument regarding $\bar{w}_{k}$, we also have

$\lim_{k\rightarrow\infty}u_{k}(x,t)=\bar{u}(x,t),$

and $\bar{u}(x,t)$ satisfies the limit equation

$\displaystyle(\partial_{t}-\Delta)^{s}\bar{u}(x,t)=f(\bar{u}(x,t)),\,{(x,t)\in
B_{1}(0)\times\mathbb{R}}.$ (4.11)

Then it follows from (4.8) that

$\bar{u}(x^{0},0)=\lim_{k\rightarrow\infty}u_{k}(\bar{x}^{k},0)=\lim_{k\rightarrow\infty}u(\bar{x}^{k},\bar{t}_{k})>0.$
(4.12)

Now we claim that

$\bar{u}(x,0)>0\,\,\mbox{for any}\,\,x\in B_{1}(0).$ (4.13)

If not, then there exists a point $\bar{x}\in B_{1}(0)$ such that

$\bar{u}(\bar{x},0)=0=\min_{\mathbb{R}^{n}\times\mathbb{R}}\bar{u}(x,0)$

by the exterior condition and the interior positivity of $u$. It follows that

$\displaystyle(\partial_{t}-\Delta)^{s}\bar{u}(\bar{x},0)=-C_{n,s}\int_{-\infty}^{0}\int_{\mathbb{R}^{n}}\frac{\bar{u}(y,\tau)}{(-\tau)^{\frac{n}{2}+1+s}}e^{\frac{|\bar{x}-y|^{2}}{4\tau}}\operatorname{d}\\!y\operatorname{d}\\!\tau\leq
0.$

In contrary, we apply the assumption $f(0)\geqslant 0$ and the limit equation
(4.11) to lead to

$\displaystyle(\partial_{t}-\Delta)^{s}\bar{u}(\bar{x},0)=f(\bar{u}(\bar{x},0))=f(0)\geq
0.$

Thus, we conclude that

$\displaystyle(\partial_{t}-\Delta)^{s}\bar{u}(\bar{x},0)=0.$

Taking into account that $\bar{u}\geq 0$, we arrive at

$\displaystyle\bar{u}(x,t)\equiv
0\,\,\mbox{in}\,\,\mathbb{R}^{n}\times(-\infty,0],$

which contradicts (4.12). From this, we verify that the assertion (4.13) is
valid.

Finally, a combination of $\bar{u}(x,0)\equiv 0$ in $B_{1}^{c}(0)$, (4.13) and
$\lambda_{0}<0$ yields that there must exist $x\in B_{1}^{c}(0)$ such that
$x^{\lambda_{0}}\in B_{1}(0)$ and

$\bar{w}(x,0)=\bar{u}(x^{\lambda_{0}},0)-\bar{u}(x,0)=\bar{u}(x^{\lambda_{0}},0)>0,$

which contradict (4.10). Therefore, we prove that the limiting position must
be $T_{0}$, i.e., $\lambda_{0}=0$. By arbitrarily choosing the direction of
$x_{1}$ and combining with the definition of $\lambda_{0}$, we conclude that
$u(x,t)$ must be radially symmetric and monotone decreasing about the origin
in $x\in B_{1}(0)$ for any $t\in\mathbb{R}$.

We have yet to prove that such decrease is strict, in fact, it is sufficient
to argue that

$w_{\lambda}(x,t)>0\,\,\mbox{in}\,\,\Omega_{\lambda}\times\mathbb{R}$ (4.14)

for any $-1<\lambda<0$. If not, then there exist some $\lambda_{0}\in(-1,0)$
and a point $(x^{0},t_{0})\in\Omega_{\lambda_{0}}\times\mathbb{R}$ such that

$w_{\lambda_{0}}(x^{0},t_{0})=0.$

Combining the differential equation in (4.1) with the definition of nonlocal
operator $(\partial_{t}-\Delta)^{s}$, we deduce that

$\displaystyle 0=(\partial_{t}-\Delta)^{s}w_{\lambda_{0}}(x^{0},t_{0})$
$\displaystyle=$ $\displaystyle
C_{n,s}\int_{-\infty}^{t_{0}}\int_{\mathbb{R}^{n}}\frac{-w_{\lambda_{0}}(y,\tau)}{(t_{0}-\tau)^{\frac{n}{2}+1+s}}e^{-\frac{|x^{0}-y|^{2}}{4(t_{0}-\tau)}}\operatorname{d}\\!y\operatorname{d}\\!\tau$
$\displaystyle=$ $\displaystyle
C_{n,s}\int_{-\infty}^{t_{0}}\int_{\Sigma_{\lambda_{0}}}\frac{w_{\lambda_{0}}(y,\tau)}{(t_{0}-\tau)^{\frac{n}{2}+1+s}}\left[e^{-\frac{|x^{0}-y^{\lambda_{0}}|^{2}}{4(t_{0}-\tau)}}-e^{-\frac{|x^{0}-y|^{2}}{4(t_{0}-\tau)}}\right]\operatorname{d}\\!y\operatorname{d}\\!\tau.$

Since $w_{\lambda_{0}}(x,t)\geq 0$ in $\Sigma_{\lambda_{0}}\times\mathbb{R}$
and

$e^{-\frac{|x^{0}-y^{\lambda_{0}}|^{2}}{4(t_{0}-\tau)}}-e^{-\frac{|x^{0}-y|^{2}}{4(t_{0}-\tau)}}<0,$

then we must have $w_{\lambda_{0}}(x,t)\equiv 0$ in
$\Sigma_{\lambda_{0}}\times(-\infty,t_{0}]$. However, it contradicts the fact
that $w_{\lambda_{0}}(x,t)\not\equiv 0$ in $\Sigma_{\lambda_{0}}$ for any
fixed $t\in(-\infty,t_{0}]$, due to the exterior condition and the interior
positivity of $u(x,t)$. Hence, we verify that the assertion (4.14) is valid,
and thus the proof of Theorem 1.2 is completed. ∎

###  4.2 Liouville type theorem in the whole space

At the end of this paper, we utilize the maximum principle in unbounded
domains (Theorem 1.3) to show that solutions are symmetric with respect to all
hyperplanes in $\mathbb{R}^{n}$ for any $t\in\mathbb{R}$, and combine with
Fourier transform to complete the proof of Liouville theorem (Theorem 1.5) for
homogeneous master equation

$(\partial_{t}-\Delta)^{s}u(x,t)=0,\,\,\mbox{in}\,\,\mathbb{R}^{n}\times\mathbb{R}.$

###### Proof of Theorem 1.5 ..

For any fixed $t\in\mathbb{R}$, we first claim that $u(x,t)$ is symmetric with
respect to any hyperplane in $\mathbb{R}^{n}$. Let $x_{1}$ be any given
direction in $\mathbb{R}^{n}$, where we keep the notations $T_{\lambda}$,
$\Sigma_{\lambda}$, $x^{\lambda}$, $u_{\lambda}$ and $w_{\lambda}$ as defined
above. For any $\lambda\in\mathbb{R}$, according to equation (1.2), we
immediately calculate that $w_{\lambda}$ satisfies

$\left\\{\begin{array}[]{ll}(\partial_{t}-\Delta)^{s}w_{\lambda}(x,t)=0,{}&\mbox{in}\,\,\Sigma_{\lambda}\times\mathbb{R},\\\
w_{\lambda}(x,t)=-w_{\lambda}(x^{\lambda},t),&\mbox{in}\,\,\Sigma_{\lambda}\times\mathbb{R}.\end{array}\right.$

Meanwhile, the boundedness of $u$ implies that $w_{\lambda}$ is also bounded.
Then in terms of Theorem 1.3 , we derive

$w_{\lambda}(x,t)\leq 0\,\,\mbox{in}\,\,\Sigma_{\lambda}\times\mathbb{R}.$

Replacing $w_{\lambda}$ with $-w_{\lambda}$ and following a similar argument
as above, we can obtain

$w_{\lambda}(x,t)\geq 0\,\,\mbox{in}\,\,\Sigma_{\lambda}\times\mathbb{R}.$

Thus, it follows that

$w_{\lambda}(x,t)=0\,\,\mbox{in}\,\,\Sigma_{\lambda}\times\mathbb{R}.$

Thereby the arbitrariness of $\lambda$ implies that $u(x,t)$ is symmetric with
respect to any hyperplane perpendicular to $x_{1}$-axis. Furthermore, since
the choice of $x_{1}$ direction is also arbitrary, we verify that $u(x,t)$ is
symmetric with respect to any hyperplane in $\mathbb{R}^{n}$ for any fixed
$t\in\mathbb{R}$. Therefore, we deduce that $u$ must depend only on $t$, i.e.,

$u(x,t)=u(t)\,\,\mbox{in}\,\,\mathbb{R}^{n}\times\mathbb{R}.$

From this, the proof Theorem 1.5 boils down to showing that the bounded
solution $u(t)$ of

$D_{\rm left}^{s}u(t)=0\,\,\mbox{in}\,\,\mathbb{R}$ (4.15)

must be a constant.

Note that the boundedness of $u$ implies that $u$ belongs to a one-side
distributional space ${\mathcal{L}}^{-}_{s}(\mathbb{R})$. More precisely,

$u(t)\in{\mathcal{L}}^{-}_{s}(\mathbb{R})=\\{u\in L^{1}_{\rm
loc}(\mathbb{R})\mid\int_{-\infty}^{t}\frac{|u(\tau)|}{1+|\tau|^{1+s}}\operatorname{d}\\!\tau<+\infty\,\,\mbox{for
any}\,\,t\in\mathbb{R}\\},$

in which one can define $D_{\rm left}^{s}u$ as a distribution

$\int_{-\infty}^{+\infty}\left(D_{\rm
left}^{s}u(t)\right)\psi(t)\operatorname{d}\\!t=\int_{-\infty}^{+\infty}u(t)D_{\rm
right}^{s}\psi(t)\operatorname{d}\\!t$

for any $\psi\in\mathcal{S}$ (cf. [28]). Here $D_{\rm right}^{s}$ is the
Marchaud right fractional derivative, defined as

$D_{\rm
right}^{s}\psi(t)=\frac{1}{|\Gamma(-s)|}\int_{t}^{+\infty}\frac{\psi(t)-\psi(\tau)}{(\tau-t)^{1+s}}\operatorname{d}\\!\tau,$

which only takes into account the values of $\psi$ that occur after time $t$
in the future. Moreover, in such a setting $u$ is a tempered distribution,
then we can define its Fourier transform and the inverse Fourier transform in
the sense of distributions. Applying the fact presented in [28] that

$\mathcal{F}(D_{\rm right}^{s}\psi)(\rho)=(-i\rho)^{s}\mathcal{F}(\psi)(\rho)$

for any $\psi\in\mathcal{S}$, then it follows from (4.15) that

$\displaystyle 0=\int_{-\infty}^{+\infty}\left(D_{\rm
left}^{s}u(t)\right)\psi(t)\operatorname{d}\\!t$ $\displaystyle=$
$\displaystyle\int_{-\infty}^{+\infty}u(t)D_{\rm
right}^{s}\psi(t)\operatorname{d}\\!t$ (4.16) $\displaystyle=$
$\displaystyle\int_{-\infty}^{+\infty}u(t)\mathcal{F}^{-1}\left((-i\rho)^{s}\mathcal{F}(\psi)(\rho)\right)(t)\operatorname{d}\\!t$

for any $\psi\in\mathcal{S}$.

In the sequel, we show that

$\langle\mathcal{F}u,\phi\rangle=0\,\,\mbox{for any}\,\,\phi\in
C_{0}^{\infty}(\mathbb{R}\setminus\\{0\\}).$ (4.17)

Let $\phi\in C_{0}^{\infty}(\mathbb{R}\setminus\\{0\\})$, then the function
$\frac{\phi(\rho)}{(-i\rho)^{s}}$ also belongs to
$C_{0}^{\infty}(\mathbb{R}\setminus\\{0\\})\subset\mathcal{S}$. There must
exist a function $\psi\in\mathcal{S}$ such that

$\mathcal{F}(\psi)(\rho)=\frac{\phi(\rho)}{(-i\rho)^{s}}.$

It follows from (4.16) that

$\displaystyle\langle\mathcal{F}u,\overline{\phi}\rangle$ $\displaystyle=$
$\displaystyle\langle\mathcal{F}u,\overline{(-i\rho)^{s}\mathcal{F}(\psi)(\rho)}\rangle$
$\displaystyle=$ $\displaystyle\langle
u,\overline{\mathcal{F}^{-1}\left((-i\rho)^{s}\mathcal{F}(\psi)(\rho)\right)}\rangle$
$\displaystyle=$
$\displaystyle\int_{-\infty}^{+\infty}u(t)\overline{\mathcal{F}^{-1}\left((-i\rho)^{s}\mathcal{F}(\psi)(\rho)\right)(t)}\operatorname{d}\\!t=0.$

Hence, the assertion (4.17) is valid, which implies that $\mathcal{F}(u)$ is
supported at the origin. From this, we conclude that $u(t)$ is a polynomial of
$t$. While the boundedness of $u$ indicates that

$u(t)\equiv C.$

In conclusion, we complete the proof of Theorem 1.5 . ∎

## Acknowledgments

The authors are grateful for the fruitful discussions with professor Wenxiong
Chen (Yeshiva University) during the preparation of this paper. The work of
the first author is partially supported by the National Natural Science
Foundation of China (NSFC Grant No. 12101452), and the work of the third
author is partially supported by the National Natural Science Foundation of
China (NSFC Grant No. 12071229).

## Conflict of interest

The authors declare that they have no conflict of interest.

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# Bertlmann’s socks from a Gödelian perspective.

Marcello Poletti<EMAIL_ADDRESS>San Giovanni Bianco, Italy

###### Abstract

Quantum mechanics is a theory that is as effective as it is counterintuitive.
While quantum practices operate impeccably, they compel us to embrace
enigmatic phenomena like the collapse of the state vector and non-locality,
thereby pushing us towards untenable ”hypotheses non fingo” stances. However,
a century after its inception, we are presented with a promising interpretive
key, intimated by Wheeler as early as 1974[1]. The interpretative paradoxes of
this theory might be resolved if we discern the relationship between logical
undecidability and quantum undecidability. It will be demonstrated how both
are intricately linked to an observer/observed relational issue, and how the
idiosyncratic behaviours of quantum physics can be reconciled with the
normative, following this path.

##  I A physics problem

> What, exactly, is bizarre, paradoxical, and incomprehensible in quantum
> mechanics?

Almost everything. QM is replete with bizarre effects. The wave-particle
duality, the tunnel effect, the superposition principle, non-commutativity,
the use and necessity of complex numbers, the uncertainty principle, the
probabilistic nature of the theory, non-locality, the measurement problem, and
the list could go on. QM is a physical theory that prevents us from
constructing a sensible physical picture of the world, despite the fact that,
in the practice of ”shut up and calculate”[2], it works splendidly.

The debate around this topic has been heated since the inception of the
theory[3]. A hundred years later, a considerable number of thinkers have
thrown in the towel, resigning themselves to an ”it is what it is” attitude.
However, an equally large group perseveres; how can we contemplate going
beyond QM if we do not understand QM at all?

One of the milestones in this debate is the work of John Bell. In ’64[4], Bell
managed to define in a concrete, clear, and pragmatic way a boundary between
quantum strangeness and classical physics. QM is strange insofar as it
violates certain inequalities. Bell’s work is extraordinary (and finally
recognised with the 2022 Nobel Prize) as it allows us to ignore the technical
details of QM, shifting the focus from the quirks of the theory to the
oddities of reality: it does not matter if QM is correct or not, if it is
incomplete or not, if there is a better theory or not; if concrete experiments
violate Bell’s inequalities, it is reality itself that shows bizarre and
nonsensical aspects. And reality, as has now been widely verified,
particularly since Aspect’s[5] work, violates Bell’s inequalities.

In ’81[6], in an iconic paper, Bell reduced his work to the essential by
asking what difference there is between Professor Bertlmann’s unmatched socks
and two entangled electrons in a Stern-Gerlach type experiment. The answer is
that experiments with socks yield results that adhere to the rules of
classical probability, whereas experiments with electrons yield results that
adhere to the rules of quantum probability, and the two results differ in that
the electrons violate Bell’s inequalities, producing outcomes that are
impossible in classical probability and, therefore, in the usual macroscopic
world.

Both in classical probability and in quantum mechanics, ”probability” is a
real number in the range [0,1]. In general, therefore, the ”probability of
event $p$” can coincide in the two cases. The two paths diverge when posing
more complex questions, involving more measurements, such as determining the
probability of $p\land q$. In the ’81 paper, Bell retraces his reasoning using
an extraordinarily elementary form of inequalities, the Wigner-D’Espagnat (WE)
inequality:

$(A,B)\cup(\neg B,C)\supseteq(A,C)$ (1)

Things that are A and B along with things that are not B and C include things
that are A and C.

The demonstration of 1 is trivial. If an element $x\in(A,C)$, then either
$x\in B$ in which case $x\in(A,B)$, or $x\in\neg B$, in which case $x\in(\neg
B,C)$. That’s all there is to it. And yet, QM violates this banal inequality
in a Stern-Gerlach type experiment, how is that possible?

What is bewildering is the fact that 1 is not about QM, it’s not even about
physics, but is a fundamental property of set theory, the most fundamental
layer of mathematics itself, that mathematics in direct contact with formal
logic. Does QM violate logic? Does reality violate logic? The very existence
of a term such as ”quantum logic”[7] seems to suggest something of the sort,
but this would trigger an explosive paradox. If classical logic were to be
rejected, then the entire mathematical apparatus upon which QM is built would
also need to be rejected, thus leading to the rejection of QM itself. The
proposition ”QM derives an alternative logic” is very similar to the liar’s
paradox.

Bell, from his inequalities, draws philosophical conclusions, summarising them
to the extreme: If quantum objects possessed well-defined properties, like the
colour of socks, then they could be categorised into classical sets and
therefore would respect classical probability theory. Hence, it is NOT
possible to assign well-defined properties a priori to quantum objects.
Consequently, the correlated behaviour of entangled objects, even at great
distances, necessarily implies a form of distant communication. QM, or any
theory compatible with Aspect’s results, is therefore non-local.

The necessity of non-locality is a bewildering paradox within the paradox. The
mechanism by which entangled particles should ”communicate at a distance” is
entirely absent in QM, it is a magical phenomenon on which it is not even
possible to imagine making concrete hypotheses.

However, Bell’s philosophical conclusions are certainly weaker than his
technical conclusions. There is a continuous oscillation, a lexical ambiguity,
between what a quantum object is and what a sentient agent can know about such
an object. This ambiguity constantly pervades the entire debate on the
foundations[8], from its origins to current textbooks.

Does a quantum object not possess well-defined properties, or is it impossible
to assign well-defined properties to a quantum object? Does the uncertainty
principle specify the limits of what we can measure or the limits of the
properties themselves of a quantum object?

The first seems to refer to something that ontologically pertains to the
object itself, the second to something that concerns the relationship between
an object and an observer.[9, 10]

The transition from inequalities to non-locality can be schematised:

  1. 1.

QM produces probability values impossible in classical probability.

  2. 2.

Therefore, it is not possible to assign well-defined properties to quantum
objects.

     * •

Alternatively: quantum objects do not possess well-defined properties.

  3. 3.

Therefore, the correlation of entangled objects is mediated by the
instantaneous exchange of information, in a magical process, at the moment of
state vector collapse.

1 is solid, it is the central result of Bell’s work. 2 is weak or at least
ambiguous. 3 seems reasonable, given 2, but requires a leap of faith in a
magical phenomenon. Probably 3 is widely accepted because the collapse itself
is a magical phenomenon. Therefore, 3 does nothing more than add a bit of
harmless nonsense to an already broadly irrational scheme.

2 and 3 have dominated and still dominate the debate on the foundations of QM.
2 is directly related to the ontology of quantum objects and 3 is
psychologically striking due to the apparent violation of relativity and its
magical nature.

However, 2 and 3 are informal deductions, and not very robust, derived from 1.
The answer to the opening question ”What exactly is bizarre about QM?” is all
in 1. QM is bizarre in that it violates classical probability, QM violates WE.

Let’s take a step back. Forget about non-locality and quantum ontology. The
new question to investigate is ”How is it possible to violate WE? And what can
legitimately be deduced from it?”.[11]

##  II A Logic Problem.

> How is it possible to violate the Wigner-D’Espagnat inequality?

A leap has been made, from physics to logic. In this new domain, there are no
complex ontological evaluations, nor are there complicated structures such as
space-time in which to define a sophisticated concept like ”locality”.

As we have seen, the Wigner-D’Espagnat inequality (WE) is a basic inequality
in set theory, the demonstration of which is equally elementary. We repeat it:

If an element $x\in(A,C)$, then either $x\in B$, in which case $x\in(A,B)$, or
$x\in\neg B$, in which case $x\in(\neg B,C)$.

The proof of WE relies on the law of the excluded middle, and the strongest
temptation to violate WE is to violate this fundamental principle. It makes
sense, the narrative of quantum mechanics (QM) is replete with propositions in
this direction: ”the electron is both here and there”, ”the electron is both a
wave and a particle”, and so on. But violating the law of the excluded middle
comes at a tremendous cost, as almost every known theorem depends on it either
directly or indirectly.

Moreover, logic does not deal with the things of the world, but defines the
rules of the language with which we describe the world; it is not in itself,
”violable”.

Imagine that an alien lands on earth and tells us: ”On our planet, all aliens
are mortal, but one of these aliens, Socrates, is immortal”. How would we
react? Would we have discovered that reality violates syllogisms? Clearly not.
Syllogisms are not a constraint on reality but on language; we would probably
simply ask the alien: ”What do you mean by ’all’?”

This is what logic does: it defines the correct use of certain abstract terms
such as ”is”, ”exists”, ”all”, ”none”. A real event that contradicts logic
cannot exist by construction.

In the specific case, $\neg p$ indicates the only possible alternative to $p$.
Attempting to violate these principles, or trying slightly more subtle
approaches such as violating double negation, appears to be a contrivance in
any case. Attempts in this direction, from Peirce[12] to fuzzy logic[13], are
not viable.

To get out of the mire, the only promising route seems to be observing that in
logic too, there is a subtle problem of observer and object of observation.
Consider a proposition $p$. The proposition $p$ is not true or false in
itself, but is true or false depending on the formal system of axioms chosen.
In a system of axioms that includes, for example, $p\land q$, $p$ will be
true, in a system of axioms that includes $\neg p\land q$, it will be false.
The truth or falsity of $p$ depends on the context.

Not only that, but in the context where $p$ assumes a definite truth value, a
stronger logical condition applies: not only is $p$ true (or false), but it is
also decidable.

We identify a system of axioms with an observer $O$, an abstract entity that
has the information of the axioms with which some theorems can be inferred.
Relative to $O$, the proposition $p$ could be:

  * •

Manifestly true: that is, $p$ is provable.

  * •

Manifestly false: that is, $\neg p$ is provable.

  * •

Undecidable: that is, neither $p$ nor $\neg p$ is provable.

The subtle difference between truth and provability, the core of Gödel’s
theorems[14], will be the key to ”violating” WE, the key to making sense of
quantum superpositions of the type $\ket{true}+\ket{false}$. As mentioned, WE
cannot really be violated without altering classical logic. Consider its
stronger version

$\overline{(A,B)}\cup\overline{(\neg B,C)}\supseteq\overline{(A,C)}$

The things that are manifestly A and B, united with the things that are
manifestly not B and C, include the things that are manifestly A and C.

In this strong form, WE is immediately violated by an element $x$, manifestly
$A$ and $C$ and for which, relative to $O$, the property $B$ is undecidable.

What exactly is happening? It happens that WE is true; it’s a valid theorem of
set theory. However, concretely, a real observer $O$ (a system of axioms)
”measuring” this theorem would invariably be in the stronger condition of
having to verify not only its truth but also its provability in $O$ and thus
would face a deadlock in the presence of any undecidable properties.

More generally, a set $A$ can be thought of, in absolute terms, as well-
defined by entities $x$ that belong to the set $A$. However, relative to a
concrete observer $O$, $A$ is only defined for those elements whose membership
in $A$ is true and provable, leaving a margin of unknowability on a residue of
possible undecidable elements.

Observe that there is no violation of classical logic. The ternary system
(manifestly true, manifestly false, undecidable) is not an alternative to the
binary system (true, false), but is simply the inclusion of the concept of
provability in the discussion.

Classical probability is defined as the ratio of the number of favourable
events to the number of possible events. In set terms:

$|p|=\frac{||P||}{||P||+||\neg P||}=\frac{||P||}{||P||+||P^{c}||}$

But relative to $O$, the measure of a set can be concretely undecidable.

Probability straddles the mathematical formalism and the pragmatics of
measurement. However, the classical definition has a metaphysical flavour, and
relative to a concrete, real observer $O$, it may not be applicable.

A very similar situation arises with Boolean algebra. It is certainly correct.
Assigning the value 1 to true propositions and the value 0 to false ones, the
logical combination of propositions can be expressed as products, sums, and
differences of those numbers. But there’s a problem: if $p$ is undecidable,
Boolean algebra ceases to be concretely applicable by a certain observer for
whom $p$ is undecidable.

The logic problem leads to a probability problem: how to define probabilities
in the case of undecidable properties?

##  III A Probability Problem

> How to construct a theory of probability that incorporates undecidable
> propositions?

This question has been formally treated in [15], here only a brief summary
will be provided.

The most natural definition of probability in relation to an observer $O$ is

$[p]=\frac{||\overline{P}||}{||\overline{P}||+||\overline{\neg P}||}$ (2)

That is, the ratio between the number of manifestly valid events and the
number of manifest events.

We will call ”quantum probability” the theory of probability that derives from
2.

Joint probability in classical probability is dominated by Bayes’ theorem, so:

$|p||q|_{p}=|q||p|_{q}$

Where $|q|_{p}$ and $|p|_{q}$ are ”the probability of $q$, given $p$” and ”the
probability of $p$, given $q$”.

Conversely, from 2 it is easy to obtain:

$[p][q]_{p}\neq[q][p]_{q}$

Quantum probability is intrinsically non-commutative. This makes sense,
measuring $p$ implies an upheaval on the possibility of measuring $q$ as the
possibility of verifying $q$ while keeping $p$ undecided decays. To construct
an adequate algebra for quantum probability, it is appropriate to associate to
$p$ and $q$ Hermitian linear operators (projectors) P and Q that act on a
certain state vector $\ket{\psi}$. In a few steps, the Born rule is obtained:

$[p]=\expectationvalue{\bm{P}}{\psi}$

Moreover, the necessity of complex numbers in the algebra is directly
connected to the necessity to respect the law of excluded middle while
maintaining an additional degree of freedom to define undecidable states. The
quantum probability defined by 2 eventually coincides with the first
postulates of QM.

QM has nothing illogical but abides, without exceptions, by classical logic.
WE, in particular, is not exactly violated. What is violated is a strong
version of WE which is the only version practically feasible by a concrete
observer $O$.

In this sense, QM is a relational theory of measurement[16]. QM does not
dictate how a system $S$ is in itself, but how an observer $O$ interacts with
$S$. The state vector $\ket{\psi}$ does not define an ontology of reality but
at most an ontology of information relative to $S$ available in $O$.

Does this mean that it is possible, at least in principle, to elaborate a
complete theory, albeit not practically feasible, a hidden variable theory
beyond QM?

And this again shifts the focus to the philosophy of science.

##  IV A Problem in Philosophy of Science.

> Is it possible to complete QM?

It is certainly possible to imagine constructing such a theory, but it cannot
be a physical theory.

Physics is rigidly subjected to the scientific method and is, above all, a
theory of measurement and only secondarily, potentially, a theory of reality.
What every rational person implicitly accepts is that the Galilean method is
the guide towards reality. That is, what science does is to assume that a good
theory of reality should produce good experimental predictions.

This is sensible but there is a problem, the theory of measurement is always
by construction relative to the observer who carries out that measurement.
Even admitting that the ”true” theory of probability is classical probability,
physics cannot make use of it as it is forced by the scientific method to
predict the concrete measures of observer $O$ and not theoretical metaphysical
values. Even if we remove from reality any form of ”quantum ontology”, physics
remains quantum physics, a theory of measurement in $O$.

So, has physics come to an end? Or is it time to abandon the scientific
method? Certainly not!

It is time to deeply assimilate the fact that the greatest strength of
physics, the scientific method, is an extraordinary wealth but also a
constraint that prevents physics from being directly a theory of reality
(metaphysics[17]), and always, first and foremost, a theory of measurement.

This is what quantum mechanics is, an extraordinary theory of measurement, a
theory that illustrates how $O$ describes and interacts with a system $S$.

From Bell’s work, it is not permissible to deduce non-locality, but something
much weaker and significant. In a certain system $O$, there are propositions
relative to reality, to the universe, that are undecidable. That is, it is not
true that in every small arbitrary spatiotemporal vicinity, the entire
information of the whole universe and its history is available.

This fact can also be considered obvious. However, it is notable how classical
physics violates this basic principle.

A Newtonian particle of mass $m$ communicates its presence everywhere in the
universe through a holomorphic function, its gravitational field, which allows
every other body in the universe to deduce exactly its position and velocity.
This is how Neptune and Pluto were discovered.

This is what QM teaches us, the boundary conditions are not always sufficient
to deduce the entire state of the system. The information is not available
everywhere. Every observer experiences some undecidable properties.

##  V Conclusions

The interpretation of QM discussed here is, in some respects, disappointing,
perhaps frustrating, as it wipes away all the magic of quantum mechanics at
one stroke, without replacing it with anything equally striking. In this
perspective, it is entirely inappropriate to say that ”an electron is both
here and there” or that ”an electron takes all possible paths simultaneously.”
Reality is simply reduced to an electron that will be somewhere and to our
objective impossibility of assigning it a well-defined position.

Likewise, it is inappropriate to say that ”a quantum computer performs many
calculations in parallel”, leaving the listener astounded, but, much more
banally, quantum computing exploits the peculiarities of quantum probability
to produce new efficient algorithms.

And similarly, the many worlds[18, 19] and the many minds[20] are entirely
superfluous, as is any role of consciousness or non-locality. Finally,
”quantum logic” does not indicate anything well-defined, as QM adheres to the
rules of the usual classical logic. Are we ready to accept that QM might be a
theory, after all, ”normal”?

Einstein was right, QM is incomplete, it does not describe all things in the
world. The state vector is not even remotely a thing of the world, no more
than any probability distribution in any statistical problem is.

But Bohr was also right, QM is not completable because it does not deal with
the reality of the world, but only with what we can measure about the world.

Physics naturally aspires to say something about the world, something
profoundly true about the world. But physics is subject by construction to the
scientific method and cannot, properly, produce a theory of the world, but at
most a theory of measurement, and the two do not coincide.

QM shows to all of us this limit, the limit where a really powerful theory in
predicting measurements says little about what the world is. Physics is
always, by construction, relational. It always deals with what an observer $O$
(ourselves in the first place) observes around him. And these observations,
these measurements are subject to the apparent logical paradoxes of quantum
mechanics for the sole fact that we forget to constantly remind ourselves that
physics does not deal with the truth of generic propositions $p$, but with
constrained propositions: ”$p$ is demonstrable in $O$”.

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