# A Note on the Krein-Rutman Theorem for
Sectorial Operators†111${\dagger}$This work was supported by the National
Natural Science Foundation of China [11871368]

Desheng Li<EMAIL_ADDRESS>Ruijing Wang<EMAIL_ADDRESS>School of
Mathematics, Tianjin University, Tianjin 300350, China  Luyan Zhou
<EMAIL_ADDRESS>School of Mathematical Sciences, Laboratory of Mathematics
and Complex Systems, MOE, Beijing Normal University, Beijing 100875, China

###### Abstract

In this note we present some generalized versions of the Krein-Rutman theorem
for sectorial operators. They are formulated in a fashion that can be easily
applied to elliptic operators. Another feature of these generalized versions
is that they contain some information on the generalized eigenspaces
associated with non-principal eigenvalues, which are helpful in the study of
the dynamics of evolution equations in ordered Banach spaces.

###### keywords:

Sectorial operator, Krein-Rutman theorem, elliptic eigenvalue problem.

###### MSC:

[2020] 35P05, 47B65, 47A75.

††journal: arXiv

##  1 Introduction

It is well known that the Krein-Rutman theorem (see e.g. [14, 19]) for bounded
operators (particularly for compact operators) plays a crucial role in the
discussion of the principal eigenvalue problem of elliptic operators via their
resolvents. But generally only part of the information concerning the
principal eigenvalue and eigenvectors can be obtained. One reason is that
there may be no one-one correspondence between the boundary spectrum of an
elliptic operator $A$ and the peripheral spectrum of its resolvent
$R_{\lambda}(A)$. As a remedy, one has to do some tricky PDE argument when
refined information on the principal eigenvalue and eigenvectors is needed;
see e.g. Du [4, Theorem 1.4], Evans [5, Sec. 6.5, Theorem 3] and Ni [17] etc.
It is therefore of particular interest to give variants of the classical
Krein-Rutman theorem for unbounded operators.

In Greiner et al. [8] it was proved that the spectral bound $s(A)$ of the
generator $A$ of a positive $C_{0}$-semigroup on a Banach space with a normal
reproducing cone is contained in $\sigma(A)$. The monograph [2] contains a far
reaching theory concerning the above question in the framework of generators
of $C_{0}$-semigroups on the functional space $C_{0}(X)$ consisting of
continuous functions vanishing at infinity, where $X$ is a locally compact
space; see [2, Chap. B-III]. These extensions make the Krein-Rutman theorem to
be more efficient in studying the spectral properties of unbounded operators.

When dealing with an elliptic differential operator $L$ on a domain
$\Omega\subset\mathbb{R}^{n}$ as in (4.1), the frequently used spaces in which
$L$ may generate a $C_{0}$-semigroup are the Sobolev ones. However, in many
cases the cone of nonnegative functions can neither have interior nor be
normal in a Sobolev space. (The above two properties were required in most
abstract results in the literature.) Of course, $L$ can be considered as an
operator in the space $C^{k,\alpha}(\overline{\Omega})$, in which the cone
${\mathcal{C}}$ of nonnegative functions have nonempty interior. But as it was
pointed in Kielhofer [12], $L$ can fail to generate a $C_{0}$-semigroup in
this space. To cover this situation, Nussbaum [18] made an effort to extend a
major part of the Krein-Rutman theorem to operators which may not generate a
$C_{0}$-semigroup.

In a recent paper [16], the authors reformulated the classical Perron-
Frobenius theory and Krein-Rutman theorem by using an elementary dynamical
approach. Inspired by this work, in this note we present another
generalization of the Krein-Rutman theorem for sectorial operators in a
formalism that seems to be more natural and suitable for elliptic operators.
It is different in some ways from those mentioned above and can be easily
applied to elliptic operators, which enables us to reduce significantly the
technical PDE argument involved in the investigation of elliptic eigenvalue
problems. Another feature of these generalized versions is that they contain
some information on the generalized eigenspaces associated with non-principal
eigenvalues, which are helpful in the study of the dynamics of evolution
equations in ordered Banach spaces (see e.g. [10]). As an illustrating
example, the principal eigenvalue problem of a general elliptic operator
associated with degenerate mixed boundary condition is discussed.

The remaining part of this paper is organized as follows. In Section 2 we do
some preliminary work, and in Section 3 we prove generalized Krein-Rutman type
theorems for sectorial operators. Section 4 consists of an example
illustrating the theoretical results mentioned above.

##  2 Preliminaries

Let $X$ be a real Banach space with norm $\|\cdot\|$. Given $M\subset X$, the
interior and closure  of $M$ are denoted respectively by
$\stackrel{{\scriptstyle\circ}}{{M}}$ and $\overline{M}$. When we need to
emphasize in which space the interior and closure are taken, we also use the
notations $\mbox{int}_{X}M$ and $\mbox{Cl}_{X}M$ in place of
$\stackrel{{\scriptstyle\circ}}{{M}}$ and $\overline{M}$, respectively. For
$x\in X$, set

$d(x,M)=\inf_{y\in M}\|x-y\|.$

###  2.1 Some basic knowledge in the spectral theory of operators

Let $A$ be a closed densely defined operator in $X$. Denote by $\sigma(A)$ and
$\rho(A)$ the spectrum and resolvent set of $A$, respectively. For
$\lambda\in\rho(A)$, let $R_{\lambda}(A):=(\lambda-A)^{-1}$ be the resolvent
of $A$.

Let $\mathbb{X}=X+{\mathbf{i}}X$ be the complexification of $X$ (see e.g. [16,
Sect. 7] for details), and define the complexification $\mathbb{A}$ of $A$ as

$\mathbb{A}u=Ax+{\mathbf{i}}Ay,\hskip
28.45274pt\forall\,u=x+{\mathbf{i}}y\in\mathbb{X}.$

Given $\mu\in\sigma(A)=\sigma(\mathbb{A})$, let

$\mbox{GE}_{\mu}(\mathbb{A})=\\{\xi\in\mathbb{X}:\,\,(\mathbb{A}-\mu)^{j}\xi=0\mbox{
for some }j\geq 1\\}.$ (2.1)

Then $\mbox{GE}_{\mu}(\mathbb{A})$ is an invariant subspace of $\mathbb{A}$.
For each $\xi\in\mbox{GE}_{\mu}(\mathbb{A})\setminus\\{0\\}$, it is clear that
there is an integer $k\geq 1$ such that

$(\mathbb{A}-\mu)^{j}\xi\neq 0\,\,\,(0\leq j\leq k-1),\hskip
14.22636pt(\mathbb{A}-\mu)^{k}\xi=0.$ (2.2)

For convenience in statement, we call the number $k$ in (2.2) the rank of
$\xi$, denoted by $\mbox{rank}(\xi)$.

For $\mu\in\sigma(A)$, it follows by the invariance of
$\mbox{GE}_{\mu}(\mathbb{A})$ that

$\begin{array}[]{ll}\mbox{GE}_{\mu}(A):=\\{\mbox{Re}\,\xi:\,\,\xi\in\mbox{GE}_{\mu}(\mathbb{A})\\}\,\,\left(=\\{\mbox{Im}\,\xi:\,\,\xi\in\mbox{GE}_{\mu}(\mathbb{A})\\}\right)\end{array}$
(2.3)

is an invariant subspace of $A$ in $X$, which will be referred to as the
generalized eigenspace of $A$ pertaining to $\mu$. (The second equality in
(2.3) is due to the fact that if $\xi\in\mbox{GE}_{\mu}(\mathbb{A})$, then
$\pm\mathbf{i}\xi\in\mbox{GE}_{\mu}(\mathbb{A})$.)

###### Lemma 2.1

Let $\mu\in\sigma(A)$, and $\lambda\in\rho(A)$. Then

$R_{\lambda}(A)\mbox{\em GE}_{\mu}(A)=\mbox{\em GE}_{\mu}(A)=\mbox{\em
GE}_{(\lambda-\mu)^{-1}}(R_{\lambda}(A)).$

Proof. It suffices to check that

$R_{\lambda}(\mathbb{A})\mbox{GE}_{\mu}(\mathbb{A})=\mbox{GE}_{\mu}(\mathbb{A})=\mbox{GE}_{\lambda_{\mu}}(R_{\lambda}(\mathbb{A})),$
(2.4)

where $\lambda_{\mu}:=(\lambda-\mu)^{-1}$.

Let $\xi\in\mbox{GE}_{\mu}(\mathbb{A})$. Then $(\mathbb{A}-\mu)^{k}\xi=0$ for
some $k\geq 1$. Hence

$(\mathbb{A}-\mu)^{k}\left(R_{\lambda}(\mathbb{A})\xi\right)=R_{\lambda}(\mathbb{A})(\mathbb{A}-\mu)^{k}\xi=R_{\lambda}(\mathbb{A})0=0.$

It follows that $R_{\lambda}(\mathbb{A})\xi\in\mbox{GE}_{\mu}(\mathbb{A})$.
Simple calculations also yield

$(\lambda-\mathbb{A})^{k}\left(R_{\lambda}(\mathbb{A})-\lambda_{\mu}\right)^{k}\xi=\lambda_{\mu}^{k}(\mathbb{A}-\mu)^{k}\xi=0.$
(2.5)

Since $\lambda\in\rho(\mathbb{A})$, (2.5) implies that
$\left(R_{\lambda}(\mathbb{A})-\lambda_{\mu}\right)^{k}\xi=0$. Therefore
$\xi\in\mbox{GE}_{\lambda_{\mu}}(R_{\lambda}(\mathbb{A}))$. In conclusion, we
have

$R_{\lambda}(\mathbb{A})\mbox{GE}_{\mu}(\mathbb{A})\subset\mbox{GE}_{\mu}(\mathbb{A})\subset\mbox{GE}_{\lambda_{\mu}}(R_{\lambda}(\mathbb{A})).$

The verification of the inverse inclusions is similar. We omit it.
$\blacksquare$

Denote by $\sigma_{e}(A)$ the essential spectrum of $A$ in the terminology of
Browder [3, pp. 107-108, Def. 11]. Then each
$\mu\in\sigma(A)\setminus\sigma_{e}(A)$ is isolated in $\sigma(A)$ with
$\mbox{GE}_{\mu}(A)$ being a finite-dimensional subspace of $X$; see [3, pp.
108].

The spectral bound $\mbox{spb}(A)$ and essential spectral bound
$\mbox{spb}_{e}(A)$ of $A$ are defined as

$\mbox{spb}(A)=\sup\\{\mbox{Re}\,\mu:\,\,\mu\in\sigma(A)\\},\hskip
14.22636pt\mbox{spb}_{e}(A)=\sup\\{\mbox{Re}\,\mu:\,\,\mu\in\sigma_{e}(A)\\}.$

(We assign $\mbox{spb}_{e}(A)=-\infty$ if $\sigma_{e}(A)=\emptyset$.) Set

$\sigma_{b}(A)=\sigma(A)\cap\\{\mbox{Re}\,z=\mbox{spb}(A)\\}.$

$\sigma_{b}(A)$ is called the boundary spectrum of $A$.

Let ${\mathscr{L}}(X)$ be the space of bounded linear operators on $X$. If
$A\in{\mathscr{L}}(X)$, we define the spectral radius $r(A)$ and essential
spectral radius $r_{e}(A)$ as

$r(A)=\sup\\{|\mu|:\,\,\mu\in\sigma(A)\\},\hskip
14.22636ptr_{e}(A)=\sup\\{|\mu|:\,\,\mu\in\sigma_{e}(A)\\}.$

It is a basic knowledge that $r(A)=\lim_{k\rightarrow\infty}\|A^{k}\|^{1/k}$.

###  2.2 Cones and positive operators

Let $X$ be a Banach space. A wedge in $X$ is a closed subset $K\subset X$ with
$K\neq\\{0\\}$ such that $tK\subset K$ for all $t\geq 0$. A convex wedge $K$
with $K\cap(-K)=\\{0\\}$ is called a cone.

A cone $K$ is said to be total (resp., solid), if $\overline{K-K}=X$ (resp.,
$\stackrel{{\scriptstyle\circ}}{{K}}\neq\emptyset$).

From now on we always assume that there has been given a cone $K$ in $X$.

An operator $A\in{\mathscr{L}}(A)$ is called positive (resp. strongly
positive) if $AK\subset K$ (resp.
$A(K\setminus\\{0\\})\subset\stackrel{{\scriptstyle\circ}}{{K}}$).

###### Definition 2.2

[16] In case $K$ is a solid cone, a positive operator $A$ is called weakly
irreducible, if the boundary $\partial K$ of $K$ contains no eigenvectors of
$A$ pertaining to nonnegative real eigenvalues.

###### Remark 2.3

It is almost obvious that strongly positive operators and primitive operators
are weakly irreducible. (Recall that $A$ is said to be primitive, if there is
an integer $m\geq 1$ such that
$A^{m}(K\setminus\\{0\\})\subset\stackrel{{\scriptstyle\circ}}{{K}}$; see [15,
pp. 285].) We also infer from the argument following Definition 7.5 in [16]
that irreducibility (in the terminology of [15]) implies weak irreducibility
defined as above.

We finally recall some generalized versions of the Krein-Rutman Theorem for
bounded operators given in [16] to conclude this section.

###### Theorem 2.4

[16, Theorem 7.3] Let $A\in{\mathscr{L}}(X)$ be a positive operator with

$r_{e}:=r_{e}(A)<r(A):=r.$ (2.6)

If $K$ is total, then the following assertions hold true:

  1. $(1)$

$r$ is an eigenvalue of $A$ with a principal eigenvector $u\in K$.

  2. $(2)$

If $\stackrel{{\scriptstyle\circ}}{{K}}$ contains a principal eigenvector $v$
of $A$, then the algebraic and geometric multiplicities of $r$ coincide.

  3. $(3)$

If $\mu$ is a complex eigenvalue with $|\mu|>r_{e}$, then $\mbox{\em
GE}_{\mu}(A)\cap K=\\{0\\}.$

  4. $(4)$

All eigenvectors of $A$ pertaining to eigenvalues $\mu\neq r$ with
$|\mu|>r_{e}$ are contained in
$X\setminus\stackrel{{\scriptstyle\circ}}{{K}}$.

###### Theorem 2.5

[16, Theorems 7.7, 7.9] Let $A\in{\mathscr{L}}(X)$ be a positive operator
satisfying (2.6). Suppose $K$ is solid, and that $A$ is weakly irreducible.
Then

  1. $(1)$

$r$ is a simple eigenvalue of $A$ with a principal eigenvector
$w\in\stackrel{{\scriptstyle\circ}}{{K}}$;

  2. $(2)$

$\mbox{\em GE}_{\mu}(A)\cap K=\\{0\\}$ for any
$\mu\in\sigma(A)\setminus\\{r\\}$ with $|\mu|>r_{e}$;

  3. $(3)$

if $A$ is strongly positive, then

$|\mu|<r,\hskip 28.45274pt\forall\,\mu\in\sigma(A)\setminus\\{r\\}.$ (2.7)

##  3 Krein-Rutman Type Theorems for Sectorial Operators

Let $X,Y$ be two real Banach spaces with $Y\hookrightarrow X$; moreover, $Y$
is dense in $X$. Denote by $\|\cdot\|$ and $\|\cdot\|_{1}$ the norms of $X$
and $Y$, respectively.

Let $K$ be a cone in $Y$, and $A$ a closed densely defined operator in $X$
with $-A$ being sectorial (see [9, Chap. 1] for definition). We always assume
that the following standing assumptions are fulfilled:

  1. (A1)

$R_{\lambda}(A)K\subset K$ for $\lambda>0$ sufficiently large.

  2. (A2)

$s_{e}:=\mbox{spb}_{e}(A)<\mbox{spb}(A):=s$.

  3. (A3)

$\mbox{GE}_{\mu}(A)\subset Y$ for every $\mu\in\sigma(A)$ with
$\mbox{Re}\,\mu>s_{e}$.

As usual, if $s\in\sigma(A)$, then we call $s$ the principal eigenvalue of
$A$. Consequently eigenvectors pertaining to $s$ are referred to as principal
eigenvectors.

One of our main results is the following general Krein-Rutman type theorem.

###### Theorem 3.1

Assume $K$ is total in $Y$. Then the following assertions hold:

  1. $(1)$

$s$ is an eigenvalue of $A$ admitting a principal eigenvector $u\in K$.

  2. $(2)$

If $\mbox{\em int}_{Y}K\neq\emptyset$ and contains a principal eigenvector of
$A$, then $s$ shares the same algebraic and geometric multiplicities.

  3. $(3)$

All eigenvectors of $A$ corresponding to other eigenvalues $\mu\neq s$ with
$\mbox{\em Re}\,\mu>s_{e}$ are contained in $Y\setminus\mbox{\em int}_{Y}K$.

  4. $(4)$

If $\mu\in\sigma(A)$, $\mbox{\em Re}\,\mu>s_{e}$ and $\mbox{\em Im}\,\mu\neq
0$, then

$\mbox{\em GE}_{\mu}(A)\cap K=\\{0\\}.$ (3.1)

Proof. For $t\in\mathbb{R},\,\,t>s_{e}$, set

$\Sigma_{1}(t):=\\{\mu\in\sigma(A):\,\,\mbox{Re}\,\mu\geq t\\}.$ (3.2)

Since $-A$ is sectorial, $\Sigma_{1}(t)$ is a compact subset of $\mathbb{C}$
for every $t>s_{e}$ (by the definition of sectorial operators). As every
$\mu\in\sigma(A)\setminus\sigma_{e}(A)$ is isolated in $\sigma(A)$, one
concludes that $\Sigma_{1}(t)$ consists of a finite number of elements.

Let $\eta\in(s_{e},s]$, and
$\Sigma_{0}(\eta)=\sigma(A)\setminus\Sigma_{1}(\eta)$. By the finiteness of
$\Sigma_{1}(t)$ ($t>s_{e}$) one trivially checks that for some
$\delta=\delta(\eta)>0$,

$\begin{array}[]{ll}\mbox{Re}\,\mu\leq\eta-\delta,\hskip
28.45274pt\forall\,\mu\in\Sigma_{0}(\eta)\end{array}$ (3.3)

Hence $\Sigma_{0}(\eta)$ and $\Sigma_{1}(\eta)$ form a spectral decomposition
of $\sigma(A)$. Denote by

$X=X_{0}(\eta)\oplus X_{1}(\eta)$ (3.4)

the corresponding decomposition of $X$. Then
$X_{1}(\eta)=\oplus_{\mu\in\Sigma_{1}(\eta)}\mbox{GE}_{\mu}(A)$ is a finite-
dimensional subspace of $X$. By (A3) we have $X_{1}(\eta)\subset Y$.

For notational simplicity, we rewrite $X_{i}(\eta):=X_{i}$ ($i=0,1$). Let us
split the argument below into several steps.

Step 1. We show that

$X_{1}\cap K\neq\\{0\\}.$ (3.5)

Let $P=\mbox{Cl}_{X}K$, the closure of $K$ in $X$. Obviously $P$ is a cone in
$X$. Recalling that $X_{1}\subset Y$, to prove (3.5) it suffices to check that

$X_{1}\cap P\neq\\{0\\}.$

For this purpose, put $\tilde{A}=A-\eta+\delta$, where $\delta$ is the
positive number in (3.3). $\sigma(\tilde{A})$ has a corresponding spectral
decomposition
$\sigma(\tilde{A})=\tilde{\Sigma}_{0}(\eta)\cup\tilde{\Sigma}_{1}(\eta)$ with

$\tilde{\Sigma}_{i}(\eta)=\Sigma_{i}(\eta)-\eta+\delta,\hskip
28.45274pti=0,1.$

We observe that

$\sup\\{\mbox{Re}\,\mu:\,\,\mu\in\tilde{\Sigma}_{0}(\eta)\\}\leq-\delta,\hskip
14.22636pt\inf\\{\mbox{Re}\,\mu:\,\,\mu\in\tilde{\Sigma}_{1}(\eta)\\}\geq\delta.$
(3.6)

The direct sum decomposition of $X$ corresponding to the above spectral
decomposition of $\sigma(\tilde{A})$ remains the same as in (3.4).

We claim that $P\not\subset X_{0}$. Indeed, suppose on the contrary that
$P\subset X_{0}$. Then since $K$ is total in $Y$, one would have

$Y=\mbox{Cl}_{Y}{(K-K)}\subset\mbox{Cl}_{X}{(K-K)}\subset\mbox{Cl}_{X}{(P-P)}\subset
X_{0}.$

(We emphasize that the closures $\mbox{Cl}_{Y}$ and $\mbox{Cl}_{X}$ are taken
with respect to the topologies of $Y$ and $X$, respectively.) Because $Y$ is
dense in $X$, we therefore have
$X=\mbox{Cl}_{X}Y\subset\mbox{Cl}_{X}X_{0}=X_{0}$, a contradiction.

Take a $u_{0}\in P\setminus X_{0}$. Write $u_{0}=x_{0}+x_{1}$, where $x_{i}\in
X_{i}$. Clearly $x_{1}\neq 0$. Let $u(t)=e^{t\tilde{A}}u_{0}$ ($t\geq 0$),
where $e^{t\tilde{A}}$ is the $C_{0}$-semigroup generated by $\tilde{A}$. Then

$u(t)=e^{t\tilde{A}}x_{0}+e^{t\tilde{A}}x_{1}:=x_{0}(t)+x_{1}(t).$

We infer from (3.6) that

$\lim_{t\rightarrow\infty}\|x_{0}(t)\|=0,\hskip
14.22636pt\lim_{t\rightarrow\infty}\|x_{1}(t)\|=\infty.$ (3.7)

By (A1) we have $R_{\lambda}(A)K\subset K\subset P$ for $\lambda>0$
sufficiently large. Therefore

$R_{\lambda}(A)P=R_{\lambda}(A)\,\overline{K}=\overline{R_{\lambda}(A)K}\subset\overline{K}\subset
P,$ (3.8)

where the closures are taken in $X$. (The second equality in (3.8) is due to
the fact that $R_{\lambda}(A)\in{\mathscr{L}}(X)$.) This guarantees that $A$
is semigroup positive, i.e., $e^{tA}P\subset P$ for $t\geq 0$ (see e.g. Kato
[11, Lemma 5.1]). Hence

$e^{t\tilde{A}}P=e^{t(\delta-\eta)}e^{tA}P\subset P,\hskip 28.45274ptt\geq 0.$
(3.9)

In particular, we have $u(t)=e^{t\tilde{A}}u_{0}\in P$ for all $t\geq 0$.

Now we show that $X_{1}\cap P\neq\\{0\\}$ and complete the proof of (3.5).
First, by the first equality in (3.7) we see that
$\lim_{t\rightarrow\infty}d\left(u(t),X_{1}\right)=0$. Now suppose on the
contrary that $X_{1}\cap P=\\{0\\}$. Then by [16, Lemma 2.4] one deduces that
$\lim_{t\rightarrow\infty}\|u(t)\|=0$. This contradicts (3.7).

Step 2. The verification of assertions (1) and (2).

Take $\eta=s$. Then $\Sigma_{1}(\eta)=\sigma_{b}(A).$ Let $K_{1}=X_{1}\cap K$,
where $X_{i}=X_{i}(\eta)=X_{i}(s)$ ($i=0,1$) are given as in (3.4). Since
$X_{1}$ is a finite-dimensional subspace of $Y$, (3.5) implies that $K_{1}$ is
a cone in $X_{1}$. As $K$ is total in $Y$, we have

$\begin{array}[]{ll}\mbox{Cl}_{X_{1}}(K_{1}-K_{1})&=\mbox{Cl}_{Y}(K_{1}-K_{1})=\mbox{Cl}_{Y}(X_{1}\cap
K-X_{1}\cap K)\\\\[4.30554pt]
&\subset\mbox{Cl}_{Y}\left(X_{1}\cap(K-K)\right)=X_{1}\cap\mbox{Cl}_{Y}(K-K)\\\\[4.30554pt]
&=X_{1}\cap Y=X_{1}.\end{array}$

That is, $K_{1}$ is total in $X_{1}$. Let $A_{1}=A|_{X_{1}}$. For $\lambda>0$
sufficiently large, we infer from (2.6) that $R_{\lambda}(A_{1})X_{1}=X_{1}$.
Thus by (A1) one easily verifies that

$R_{\lambda}(A_{1})K_{1}\subset K_{1}.$ (3.10)

Note that $\sigma(A_{1})=\Sigma_{1}(\eta)=\sigma_{b}(A)$. Let

$\sigma(A_{1})=\\{\mu_{i}=s+{\mathbf{i}}\beta_{i}:\,\,0\leq i\leq n\\}.$

We may assume that $|\beta_{0}|=\min_{0\leq i\leq n}|\beta_{i}|$. Fix a number
$\lambda>s$ such that (3.10) holds. Then $|\lambda-\mu_{0}|=\min_{0\leq i\leq
n}|\lambda-\mu_{i}|$, and hence

$r(R_{\lambda}(A_{1}))=\sup\\{1/|\lambda-\mu_{i}|:\,\,0\leq i\leq
n\\}=1/|\lambda-\mu_{0}|:=r.$

By Theorem 2.4 one concludes that $r$ is an eigenvalue of $R_{\lambda}(A_{1})$
with an eigenvector $w\in K_{1}$. On the other hand, since
$1/|\lambda-\mu_{i}|<r$ for $\mu_{i}\in\sigma(A_{1})$ with
$\mu_{i}\neq\mu_{0},\overline{\mu}_{0}$, we see that the circle
${\mathbb{S}}_{r}=\\{z\in\mathbb{C}:\,\,|z|=r\\}$ in the complex plane
$\mathbb{C}$ contains at most two eigenvalues of $R_{\lambda}(A_{1})$, i.e.,
$1/(\lambda-\mu_{0})$ and $1/(\lambda-\overline{\mu}_{0})$. Thus one
necessarily has $1/(\lambda-\mu_{0})=1/(\lambda-\overline{\mu}_{0})=r$, which
implies that $\beta_{0}=0$. It follows that $\mu_{0}=s$ is an eigenvalue of
$A$; furthermore, $w$ is an eigenvector of $A$ corresponding to $s$. This
completes the proof of (1).

If $\mbox{int}_{Y}K\neq\emptyset$ and contains a principal eigenvector $v$,
one easily verifies that $\mbox{int}_{X_{1}}K_{1}$ is nonvoid and
$v\in\mbox{int}_{X_{1}}K_{1}$. Thus by Theorem 2.4 we deduce that the
eigenvalue $1/(\lambda-s)$ of $R_{\lambda}(A_{1})$ has the same algebraic and
geometric multiplicities. Consequently by (2.6) the algebraic and geometric
multiplicities of the principal eigenvalue $s$ of $A$ coincide. Hence
assertion (2) holds true.

Step 3. The verification of assertions (3) and (4).

Let $\mu\in\sigma(A)\setminus\\{s\\}$, $\mbox{Re}\,\mu>s_{e}$. Take a real
number $\eta$ with $s_{e}<\eta<s$ such that $\mu\in\Sigma_{1}(\eta)$. Let
$X_{1}=X_{1}(\eta)$, $K_{1}=X_{1}\cap K$, and $A_{1}=A|_{X_{1}}$. Then as in
Step 2 it can be shown that $K_{1}$ is a total cone in $X_{1}$. Furthermore,
(3.10) remains valid for $\lambda>0$ sufficiently large. Take a $\lambda>s$
such that (3.10) holds and consider the resolvent operator
$R_{\lambda}(A_{1})$ of $A_{1}$ on $X_{1}$. Then by Theorem 2.4 (4), we deduce
that $\mbox{int}_{X_{1}}K_{1}$ does not contain eigenvectors of
$R_{\lambda}(A_{1})$ pertaining to the eigenvalue
$\lambda_{\mu}:=1/(\lambda-\mu)$. Now if $A$ has an eigenvector
$v\in\mbox{int}_{Y}K$ corresponding to $\mu$. Then one easily verify that
$v\in\mbox{int}_{X_{1}}K_{1}$ and is an eigenvector of $R_{\lambda}(A_{1})$
corresponding to $\lambda_{\mu}$. This leads to a contradiction and verifies
assertion (3).

If $\mbox{Im}\,\mu\neq 0$, Theorem 2.4 (3) asserts that
$\mbox{GE}_{\lambda_{\mu}}(R_{\lambda}(A_{1}))\cap K_{1}=\\{0\\}$. We also
infer from (2.6) that
$\mbox{GE}_{\mu}(A_{1})=\mbox{GE}_{\lambda_{\mu}}(R_{\lambda}(A_{1}))$.
Therefore $\mbox{GE}_{\mu}(A)\cap K=\mbox{GE}_{\mu}(A_{1})\cap K_{1}=\\{0\\}$.
This completes the proof of assertion (4). $\blacksquare$

###### Remark 3.2

Note that we do not require that $e^{tA}Y\subset Y$ for $t\geq 0$ in the proof
of the above theorem. This allows us to avoid deriving higher regularity
results on the corresponding parabolic equations when applying the theory to
elliptic differential operators.

###### Theorem 3.3

In addition to $(A1)-(A3)$, we also assume that the following hypothesis is
fulfilled:

  1. (A4)

$K$ is a solid cone in $Y$, and
$R_{\lambda}(A)(K\setminus\\{0\\})\subset\mbox{\em int}_{Y}K$.

Then $s$ is a simple eigenvalue with a unique normalized eigenvector
$w\in\mbox{\em int}_{Y}K$. Moreover, for any $\mu\in\sigma(A)\setminus\\{s\\}$
with $\mbox{\em Re}\,\mu>s_{e}$,

$\mbox{\em GE}_{\mu}(A)\cap K=\\{0\\}.$ (3.11)

Proof. The proof follows a fully analogous manner as the one for Theorem 2.5,
and is thus omitted. The interested reader may consult [16, Theorems 7.3, 7.7]
for details. $\blacksquare$

###### Remark 3.4

Under the hypotheses of Theorem 3.3, one may expect that the boundary spectrum
$\sigma_{b}(A)$ consists of exactly one eigenvalue of $A$. Unfortunately the
easy counterexample below indicates that this may fail to be true.

Example 3.1. Let $X=Y=\mathbb{R}^{3}$. For computational convenience, here we
make a convention that $\mathbb{R}^{3}$ consists of column vectors. Denote
$v^{\prime}$ the transpose of a row vector $v=(x,y,z)$. Define a cone $K$ in
$X$ as

$K=\\{(x,y,z)^{\prime}\in X:\,\,z\geq\sqrt{x^{2}+y^{2}}\\}.$

Then $\mbox{int}_{Y}K=\\{(x,y,z)^{\prime}\in X:\,\,z>\sqrt{x^{2}+y^{2}}\\}$.
Let

$A=\left(\begin{matrix}0&-1&0\\\ 1&0&0\\\ 0&0&0\end{matrix}\right).$

For $\lambda>0$, simple computations yield

$R_{\lambda}(A):=(\lambda-A)^{-1}=\left(\begin{matrix}B&O\\\\[4.30554pt]
O&{\lambda^{-1}}\end{matrix}\right),\hskip 14.22636pt\mbox{where
}\,B=\frac{1}{1+\lambda^{2}}\left(\begin{matrix}\lambda&-1\\\
1&\lambda\end{matrix}\right).$

Therefore

$R_{\lambda}(A)u=\frac{1}{1+\lambda^{2}}\left(\lambda x-y,\,x+\lambda
y,\,\frac{1+\lambda^{2}}{\lambda}z\right)^{\prime}:=\frac{1}{1+\lambda^{2}}(\tilde{x},\tilde{y},\tilde{z})^{\prime}.$

Observe that

$\tilde{x}^{2}+\tilde{y}^{2}=(x^{2}+y^{2})+\lambda^{2}(x^{2}+y^{2})=(1+\lambda^{2})(x^{2}+y^{2}).$
(3.12)

Now let $u=(x,y,z)^{\prime}\in K$. Then $x^{2}+y^{2}\leq z^{2}$. Since
$(1+\lambda^{2})/\lambda^{2}>1$, by (3.12) we deduce that

$\tilde{x}^{2}+\tilde{y}^{2}\leq(1+\lambda^{2})z^{2}\leq\left(\frac{1+\lambda^{2}}{\lambda}\right)^{2}z^{2}=\tilde{z}^{2}.$

This implies $(\tilde{x},\tilde{y},\tilde{z})^{\prime}\in K$. Thus we see that
$R_{\lambda}(A)K\subset K$.

If $u\in\partial K$, $u\neq 0$, then $x^{2}+y^{2}=z^{2}\neq 0$. By (3.12) we
find that

$\tilde{x}^{2}+\tilde{y}^{2}=(1+\lambda^{2})z^{2}<\left(\frac{1+\lambda^{2}}{\lambda}\right)^{2}z^{2}=\tilde{z}^{2}.$

Hence $(\tilde{x},\tilde{y},\tilde{z})^{\prime}\in\mbox{int}_{Y}K$. Therefore
$R_{\lambda}(A)u\in\mbox{int}_{Y}K$. This indicates that the operator given by
$A$ satisfies all the requirements in Theorem 3.3. However, all the
eigenvalues of $A$ has the same real part $s=0$. To guarantee the uniqueness
of elements in $\sigma_{b}(A)$, Nussbaum [18] used the notion of
“$u_{0}$-positivity ” due to Krasnosel’skii [13]; see [18, Theorem 1.3]. Here
we remark that if the semigroup $e^{tA}$ has some strong positivity property,
then one can still ensure the uniqueness of elements in $\sigma_{b}(A)$.

###### Theorem 3.5

In addition to $(A1)-(A4)$, assume that

  1. (A5)

for any $t>0$ and $\mu\in\sigma(A)$ with $\mbox{\em Re}\,\mu>s_{e}$,

$e^{tA}(K_{\mu}\setminus\\{0\\})\in\mbox{\em int}_{Y}K,\hskip
14.22636pt\mbox{where }K_{\mu}=\mbox{\em GE}_{\mu}(A)\cap K.$

Then $\sigma_{b}(A)=\\{s\\}$.

Proof. Let $Y^{\prime}=\oplus_{\mu\in\sigma_{b}(A)}\mbox{GE}_{\mu}(A)$,
$A^{\prime}=A|_{Y^{\prime}}$. Denote $K^{\prime}=Y^{\prime}\cap K$. Then by
Theorem 3.3 we see that $K^{\prime}\neq\\{0\\}$. Hence $K^{\prime}$ is a cone
in $Y^{\prime}$. Let $v\in K^{\prime}$, $v\neq 0$. By (A5) we have
$e^{tA}v\in\mbox{int}_{Y}K$ for $t>0.$ Since $e^{tA}v\in Y^{\prime}$, one
trivially verifies that $e^{tA}v\in\mbox{int}_{Y^{\prime}}{K^{\prime}}$.
Therefore $K^{\prime}$ is a solid cone in $Y^{\prime}$ and

$e^{tA^{\prime}}(K^{\prime}\setminus\\{0\\})\subset\mbox{int}_{Y^{\prime}}{K^{\prime}},\hskip
28.45274ptt>0.$ (3.13)

Now let $\mu:=s+{\mathbf{i}}\beta\in\sigma_{b}(A)$. Then $\lambda:=e^{\mu
t}=e^{st}e^{{\mathbf{i}}\beta t}$ is an eigenvalue of $e^{tA^{\prime}}$ with
$|\lambda|=e^{st}:=r(t)$. But (3.13) implies that $r(t)$ is the unique
eigenvalue of $e^{tA^{\prime}}$ on the circle
${\mathbb{S}}_{r(t)}:=\\{z\in\mathbb{C}|\,\,|z|=r(t)\\}$ for $t>0$. Hence we
necessarily have $e^{{\mathbf{i}}\beta t}=1$, and therefore $\beta
t\in\\{2k\pi:\,\,k\in\mathbb{Z}\\}$ for all $t>0$. But this is impossible
unless $\beta=0$. This proves what we desired. $\blacksquare$

##  4 Principal Eigenvalue Problem of Elliptic Operators Associated with
Degenerate Mixed Boundary Conditions

As an illustrating example, we consider the principal eigenvalue problem of
the elliptic operator $L$ on a smooth bounded domain
$\Omega\subset\mathbb{R}^{n}(n\geq 1)$:

$Lu=-\sum^{n}_{i,j=1}a_{ij}(x)\frac{\partial^{2}u}{\partial x_{i}\partial
x_{j}}+\sum^{n}_{i=1}b_{i}(x)\frac{\partial u}{\partial x_{i}}+c(x)u,$ (4.1)

which is associated with the mixed boundary condition:

$Bu:=\alpha(x)u+\beta(x)\frac{\partial u}{\partial\nu}=0$ (4.2)

on the boundary $\Gamma:=\partial\Omega$ of $\Omega$, where $\nu$ stands for
the unit outward normal vector field on $\Gamma$. The coefficients of $L$ and
$B$ are assumed to be $C^{\infty}$ functions satisfying the hypotheses below:

  1. (H1)

$a_{ij}=a_{ji}$ ($1\leq i,j\leq n$), and there is $\theta>0$ such that

$\sum^{n}_{i,j=1}a_{ij}(x)\varsigma_{i}\varsigma_{j}\geq\theta|\varsigma|^{2},\hskip
28.45274pt\forall\,\varsigma\in\mathbb{R}^{n},\,\,x\in\bar{\Omega};$

  2. (H2)

$c,\alpha,\beta$ are nonnegative functions satisfying that

$\alpha(x)+\beta(x)>0,\hskip 28.45274pt\forall\,x\in\Gamma.$ (4.3)

In the case of the Dirichet boundary condition or the Robin boundary condition
(regular case), this problem has already been well understood; see e.g. [1,
Theorem 12.1], [4, Theorem 1.4] and also [7, 18]. Here we are interested in a
degenerate case where $\beta$ may vanish on a proper subset of $\Gamma$. In
such a situation, if $L$ has a divergence form (hence $L$ enjoys some
symmetric properties), one can find some nice results concerning the principal
eigenvalue problem of $L$ in Taira [21, Theorem 1.2]. As an application of the
abstract results given in Section 3, we deal with the general case and present
a less involved argument on the problem.

$\bullet$ Some fundamental results. First, making use of the classical Hopf’s
lemma, one can easily verify the comparison result below:

###### Lemma 4.1

Let $u\in C^{1}(\overline{\Omega})\cap C^{2}(\Omega)$, $u\not\equiv 0$. Assume
that

$Lu+\lambda u\geq 0\,\,(\mbox{in }\Omega),\hskip 14.22636ptBu\geq
0\,\,(\mbox{on }\Gamma),$

where $\lambda\geq 0$. Then $u(x)>0$ for $x\in\Omega$.

Denote by $W^{s,p}(\Omega)$ ($s\in\mathbb{R}_{+}$, $1\leq p<\infty$) the
Sobolev spaces equipped with the standard norms. We infer from Taira [21,
pp.5, Theorem 1] that the following existence and uniqueness result holds
true.

###### Lemma 4.2

Let $1<p<\infty$, $s>1+1/p$, and let $\lambda\geq 0$. Then for any $f\in
W^{s-2,p}(\Omega)$, the homogeneous boundary value problem

$Lu+\lambda u=f\mbox{ $($in\,\,$\Omega$$)$},\hskip 14.22636pt\mbox{ $Bu=0$
$($on\,\,$\Gamma$$)$}$ (4.4)

has a unique solution $u\in W^{s,p}(\Omega)$. Here the boundary condition is
understood in the sense that $B$ can be viewed as a linear operator from
$W^{s,p}(\Omega)$ to Besov space $B_{*}^{s-1-1/p,\,p}(\Gamma)$ (see [21] pp. 3
for details).

Note that Lemma 4.2 implies that if $f\in C^{1}(\overline{\Omega})$, then the
solution $u$ of (4.4) belongs to $C^{2}(\overline{\Omega})$, and hence it
solves (4.4) in the classical sense. Indeed, if $f\in
C^{1}(\overline{\Omega})$ then $f\in W^{1,p}(\Omega)$ for any $1<p<\infty$.
Lemma 4.2 then asserts that $u\in W^{3,p}(\Omega)$. Taking a number $p>1$
sufficiently large so that $W^{3,p}(\Omega)\hookrightarrow
C^{2}(\overline{\Omega})$, one immediately concludes that $u\in
C^{2}(\overline{\Omega})$.

By virtue of [21, pp.4, Theorem 1] we also deduce that

$\|u\|_{C^{2}(\overline{\Omega})}\leq C\|u\|_{W^{3,p}(\Omega)}\leq
C\|f\|_{W^{1,p}(\Omega)}\leq C\|f\|_{C^{1}(\overline{\Omega})}$ (4.5)

for all $f\in C^{1}(\overline{\Omega})$, where $C$ denotes a general constant
independent of $f$.

$\bullet$ Resolvent strong positivity of $A=-L$. Let $X=L^{2}(\Omega)$, and
set

$Y=\\{u\in C^{1}(\overline{\Omega}):\,\,\mbox{$u$ satisfies \eqref{e:bc}}\\}.$

$Y$ is equipped with the usual norm of $C^{1}(\overline{\Omega})$. Clearly
$Y\hookrightarrow X$.

Let $K$ be the positive cone in $Y$ consisting of nonnegative functions.

Denote by $A$ the operator $-L$ with domain

${\mathcal{D}}(A)=\\{u\in H^{2}(\Omega):\,\,Bu=0\\},$

where the boundary condition $Bu=0$ is understood in the same sense as in
Lemma 4.2. Invoking [21, pp.5, Theorem 2] we deduce that $-A$ is a sectorial
operator in $X$ with compact resolvent. Hence by [Kry, Ex. 2.4, (i)] it is
easy to see that $\mbox{spb}_{e}(A)=-\infty<\mbox{spb}(A)<\infty.$ Thus $A$
fulfills (A2) in Section 3.

We infer from Lemma 4.2 and (4.5) that $R_{\lambda}(A)Y\subset Y$ for
$\lambda\geq 0$; furthermore, $R_{\lambda}(A)|_{Y}$ is compact as an operator
on the space $Y$.

The following result indicates that $A$ fulfills hypotheses (A1) and (A4) in
Section 3. The proof of such a result is somewhat standard. We include the
details in the Appendix part for the readers’ convenience.

###### Lemma 4.3

$R_{\lambda}(A)(K\setminus\\{0\\})\subset\mbox{\em int}_{Y}K$ for each
$\lambda\geq 0$.

$\bullet$ Regularity of the generalized eigenfunctions. Let $\mathbb{A}$ be
the complexification of $A$ with
${\mathcal{D}}(\mathbb{A})={\mathcal{D}}(A)+{\mathbf{i}}{\mathcal{D}}(A)$. We
start with the eigenfunctions of $\mathbb{A}$. Let
$\mu=a+{\mathbf{i}}b\in\sigma(A)$, and let $w=u+{\mathbf{i}}v$ be a
corresponding eigenfunction of $\mathbb{A}$, where $u,v\in D(A)$. Then
$\mathbb{A}w=\mu w$ amounts to say that

$Au=au-bv,\hskip 14.22636ptAv=av+bu.$ (4.6)

Since $u,v\in H^{2}(\Omega)$, by Lemma 4.2 and (4.6) one finds that $u,v\in
H^{4}(\Omega)$. Further by a simple bootstrap argument we finally conclude
that $u,v\in H^{s}(\Omega)$ for all $s\geq 0$. It follows by the Sobolev
embeddings that $u,v\in C^{\infty}(\overline{\Omega})$.

Now let $g\in\mbox{GE}_{\mu}(\mathbb{A})$ and $\mbox{rank}\,(g)\geq 2$. Set
$k=\mbox{rank}\,(g)-1$. Then $(\mathbb{A}-\mu)^{k}g:=w$ is an eigenfunction of
$\mathbb{A}$. Hence $w$ a $C^{\infty}$ function on $\overline{\Omega}$.

Note that it is readily implied in $(\mathbb{A}-\mu)^{k}g=w$ that
$(\mathbb{A}-\mu)^{j}g\in{\mathcal{D}}(\mathbb{A})$ for all $j\leq k$. In
particular,

$(\mathbb{A}-\mu)^{k-1}g:=f_{1}\in{\mathcal{D}}(\mathbb{A})\subset{\mathbb{H}}^{2}(\Omega):=H^{s}(\Omega)+{\mathbf{i}}H^{s}(\Omega).$

Therefore by $(\mathbb{A}-\mu)f_{1}=w$ we find that

$\mathbb{A}f_{1}=w+\mu f_{1}:=\tilde{f}_{1}\in{\mathbb{H}}^{2}(\Omega).$ (4.7)

It follows by Lemma 4.2 that $f_{1}\in{\mathbb{H}}^{4}(\Omega)$. This in turn
implies that $\tilde{f}_{1}\in{\mathbb{H}}^{4}(\Omega)$. (4.7) and Lemma 4.2
then asserts that $f_{1}\in{\mathbb{H}}^{6}(\Omega)$. Once again by a
bootstrap argument we see that $f_{1}\in{\mathbb{H}}^{s}(\Omega)$ for all
$s\geq 0$.

Repeating the above argument with $w$ and $f_{1}$ therein replaced by $f_{1}$
and $(\mathbb{A}-\mu)^{k-2}g:=f_{2}$, respectively, one deduces that
$f_{2}\in{\mathbb{H}}^{s}(\Omega)$ for $s\geq 0$. Continuing this procedure we
finally obtain that $g=f_{k}\in{\mathbb{H}}^{s}(\Omega)$ for all $s\geq 0$.
The Sobolev embeddings then immediately imply that $g$ is a $C^{\infty}$
function.

It follows from the above results that $\mbox{GE}_{\mu}(A)\subset Y$ for any
$\mu\in\sigma(A)$. Hence $A$ fulfills hypothesis (A3). By far we have seen
that the operator $A$ satisfies hypotheses (A1)-(A4).

$\bullet$ The verification of hypothesis (A5). Let $\mu\in\sigma(A)$. Denote
by $\mathbb{A}_{\mu}$ the restriction of $\mathbb{A}$ on
$\mbox{GE}_{\mu}(\mathbb{A})$. Given $g\in\mbox{GE}_{\mu}(\mathbb{A})$, let
$u=u(t)$ be the solution of equation $\dot{u}=\mathbb{A}u$ with $u(0)=g$. Then
$u(t)=e^{t\mathbb{A}}g=e^{t\mathbb{A}_{\mu}}g$. Since $\mathbb{A}_{\mu}$ is a
bounded operator on $\mbox{GE}_{\mu}(\mathbb{A})$, we have

$\begin{array}[]{ll}u(t)&=e^{t\mathbb{A}_{\mu}}g=e^{\mu
t}e^{t(\mathbb{A}_{\mu}-\mu)}g=e^{\mu
t}\sum_{j=0}^{\infty}\frac{t^{j}}{j!}(\mathbb{A}_{\mu}-\mu)^{j}g\\\\[4.30554pt]
&=e^{\mu
t}\left(I+\frac{t}{1!}(\mathbb{A}_{\mu}-\mu)+\cdots+\frac{t^{k-1}}{(k-1)!}(\mathbb{A}_{\mu}-\mu)^{(k-1)}\right)g,\end{array}$
(4.8)

where $k=\mbox{rank}\,(g)$. Noticing that
$(\mathbb{A}_{\mu}-\mu)^{j}g\in\mbox{GE}_{\mu}(\mathbb{A})$ for any integer
$j\geq 0$, by what we have proved above it is clear that
$(\mathbb{A}_{\mu}-\mu)^{j}g$ is a $C^{\infty}$ function. Consequently for
each $t\geq 0$ fixed, $u(t)$ is a $C^{\infty}$ function in the space variable
on $\overline{\Omega}$. We write $u(t,x)=u(t)(x)$ for
$(t,x)\in\mathbb{R}_{+}\times\overline{\Omega}$. Then it can be easily seen
that $u=u(t,x)$ is a complex $C^{\infty}$ function on
$\mathbb{R}_{+}\times\overline{\Omega}$.

Now we come back to the real situation. The above result implies that for each
$v_{0}\in\mbox{GE}_{\mu}(A)$, the function $v(t,x):=v(t)(x)$, where
$v(t)=e^{tA}v_{0}$, is a $C^{\infty}$ function on
$\mathbb{R}_{+}\times\overline{\Omega}$. Therefore $v$ is a classical solution
of the parabolic equation:

$\frac{\partial v}{\partial t}+Lv=0,\hskip 28.45274ptx\in\Omega,\,\,t>0$ (4.9)

associated with the boundary condition $Bv=0$ on $\Gamma$. Thanks to the
Hopf’s lemma for parabolic equations (see e.g. Friedman [6, Chap. 2, Theorem
14] or Smith [20, Chap. 7, Theorem 2.2]), using almost the same argument as in
the proof of Lemma 4.3 (see the Appendix), it can be shown that
$v(t,\cdot)\in\mbox{int}_{Y}K$ if $v_{0}\in K$, $v_{0}\neq 0$. This is
precisely what we desired.

Now that $A$ satisfies hypotheses $(A1)-(A5)$, as a straightforward
application of Theorems 3.3 and 3.5, one immediately obtains the following
result.

###### Theorem 4.4

The following assertions hold true:

  1. $(1)$

The spectral bound $s$ of $A$ is an algebraically simple eigenvalue with a
corresponding eigenvector $w\in\mbox{\em int}_{Y}K$.

  2. $(2)$

$\mbox{\em GE}_{\mu}(A)\cap K=\\{0\\}$ for any
$\mu\in\sigma(A)\setminus\\{s\\}$.

  3. $(3)$

$\sigma_{b}(A)=\\{s\\}$

###### Remark 4.5

We mention that in the case of the Dirichlet (or Robin) boundary condition,
almost all the facts concerning the operator $A$ needed in proving Theorem 3.5
are well known and need not be checked. Therefore the theorem becomes nearly
an immediate consequence of Theorems 3.3 and 3.5.

Appendix: Proof of Lemma 4.3

Proof. Let $f\in K\setminus\\{0\\}$, and $u=R_{\lambda}(A)f$. Then $u\in
C^{2}(\overline{\Omega})$. By Lemma 4.1 we deduce that $u(x)>0$ for
$x\in\Omega$. Set $\Gamma_{0}=\\{x\in\Gamma:\,\,u(x)=0\\}$. Then the classical
Hopf’s lemma asserts that $\frac{\partial u}{\partial\nu}(x)<0$ for
$x\in\Gamma_{0}$. Hence by compactness of $\Gamma_{0}$, there is
$\varepsilon_{0}>0$ such that $\frac{\partial
u}{\partial\nu}(x)\leq-3\varepsilon_{0}$ for $x\in\Gamma_{0}$.

Denote $\|\cdot\|_{1}=\|\cdot\|_{C^{1}(\overline{\Omega})}$. Take a
neighborhood ${\mathcal{W}}$ of $\Gamma_{0}$ in $\Gamma$ such that

$\frac{\partial u}{\partial\nu}(x)\leq-2\varepsilon_{0},\hskip
28.45274ptx\in{\mathcal{W}}.$ (4.10)

Then there exists $\delta>0$ such that for all $h\in Y$ with
$\|h\|_{1}<\delta$,

$\frac{\partial(u+h)}{\partial\nu}(x)\leq-\varepsilon_{0}<0,\hskip
28.45274ptx\in{\mathcal{W}}.$ (4.11)

We claim that $\alpha(x)>0$ for $x\in\Gamma_{0}$. Indeed, if $\alpha(x)=0$
then by (4.3) we have $\beta(x)>0$. Thus by (4.10) one deduces that
$Bu(x)=\beta(x)\frac{\partial u}{\partial\nu}(x)\neq 0$, a contradiction.
Hence the claim holds true. By compactness of $\Gamma_{0}$ we deduce that
$\alpha(x)\geq 2\varepsilon_{1}>0$ for all $x\in\Gamma_{0}$. Therefore by
continuity of $\alpha$ it can be assumed that the neighborhood ${\mathcal{W}}$
of $\Gamma_{0}$ is chosen sufficiently small so that

$\alpha(x)\geq\varepsilon_{1}>0,\hskip 28.45274ptx\in{\mathcal{W}}.$ (4.12)

Now for any $h\in Y$ with $\|h\|_{1}<\delta$, we have at any point
$x\in{\mathcal{W}}$ that

$\alpha(x)(u+h)(x)=-\beta(x)\frac{\partial(u+h)}{\partial\nu}(x)\geq(\mbox{by
}\eqref{e:8.19})\geq 0.$

Hence by (4.12) we see that

$(u+h)(x)\geq 0,\hskip 28.45274ptx\in{\mathcal{W}}.$ (4.13)

Using (4.11) and (4.13) it is not difficult to deduce that there is a
neighborhood $U$ of $\Gamma_{0}$ in $\overline{\Omega}$ such that for any
$h\in Y$ with $\|h\|_{1}<\delta$,

$(u+h)(x)\geq 0,\hskip 28.45274ptx\in U.$ (4.14)

We may assume that $U$ is open relative to $\overline{\Omega}$. Hence
$\Gamma_{1}:=\Gamma\setminus U$ is compact. Because $u$ is positive on
$\Gamma_{1}$, there is $\varepsilon_{2}>0$ such that $u(x)\geq
2\varepsilon_{2}$ for $x\in\Gamma_{1}$. This allows us to pick a neighborhood
$V$ of $\Gamma_{1}$ in $\overline{\Omega}$ such that $u(x)\geq\varepsilon_{2}$
for $x\in V$. Further one can restrict $\delta$ sufficiently small so that

$(u+h)(x)\geq 0,\hskip 28.45274ptx\in V$ (4.15)

for all $h\in Y$, $\|h\|_{1}<\delta$. Note that $G:=U\cup V$ is a neighborhood
of $\Gamma$ in $\overline{\Omega}$.

It can be assumed that both $U,V$ are open relative to $\overline{\Omega}$.
Hence $G$ is open in $\overline{\Omega}$. Consequently
$F:=\overline{\Omega}\setminus G$ is a compact subset of $\overline{\Omega}$.
Since $u$ is positive on $F$, there exists $\varepsilon_{3}>0$ such that
$u(x)\geq 2\varepsilon_{3}$ for $x\in F$. Therefore if $\delta$ is
sufficiently small then $u+h$ is positive on $F$ for all $h\in Y$ with
$\|h\|_{1}<\delta$. Combining this with (4.14) and (4.15) it follows that
$u+h\geq 0$ in $\overline{\Omega}$ for all $h\in Y$ with $\|h\|_{1}<\delta$,
i.e., $u+h\in K$. Hence $u\in\mbox{int}_{Y}K$. $\blacksquare$

## References

## References

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Expected liquidated volume on long positions |  2.77e-07 (9.14)*** |  7.573e-09 (3.45)*** |  1.965e-07 (14.98)*** |  5.472e-09 (2.27)**
Unexpected liquidated volume on long positions |  2.539e-09 (5.06)*** |  1.384e-09 (1.42) |  2.455e-09 (4.78)*** |  3.984e-09 (3.70)***
Leverage (average level):
Expected average leverage on short positions |  -0.0004 (-2.21)** |  -0.0001 (-1.40) | - | -
Unexpected average leverage on short positions |  -0.0002 (-3.00)*** |  -0.0002 (-4.37)*** | - | -
Expected average leverage on long positions |  -0.0004 (-2.15)** |  -1.193e-05 (-0.36) | - | -
Unexpected average leverage on long positions |  -0.0003 (-3.65)*** |  -7.237e-05 (-1.93)** | - | -
Leverage (in categories):
Expected: less than $1\times$ | - | - |  -2.112e-08 (-1.21) | -
Unexpected: less than $1\times$ | - | - |  1.688e-08 (3.46)*** | -
Expected: $1\times$ to $5\times$ | - | - |  1.829e-10 (1.08) |  1.684e-08 (0.23)
Unexpected: $1\times$ to $5\times$ | - | - |  9.772e-10 (5.51)*** |  -1.289e-08 (-0.54)
Expected: $5\times$ to $10\times$ | - | - |  1.847e-09 (3.44)*** |  -2.534e-08 (-0.55)
Unexpected: $5\times$ to $10\times$ | - | - |  1.012e-09 (5.27)*** |  1.736e-08 (1.08)
Expected: $10\times$ to $15\times$ | - | - |  -6.697e-10 (-1.50) |  5.39e-08 (0.96)
Unexpected: $10\times$ to $15\times$ | - | - |  8.724e-11 (0.63) |  6.408e-09 (0.44)
Expected: $15\times$ to $20\times$ | - | - |  1.998e-09 (3.75)*** |  9.314e-08 (1.84)*
Unexpected: $15\times$ to $20\times$ | - | - |  2.023e-10 (1.28) |  5.102e-09 (0.33)
Expected: $20\times$ to $25\times$ | - | - |  -1.06e-10 (-0.12) |  2.698e-07 (3.89)***
Unexpected: $20\times$ to $25\times$ | - | - |  2.66e-10 (2.15)** |  6.261e-09 (0.92)
Expected: $25\times$ to $30\times$ | - | - |  -9.034e-11 (-0.76) |  -4.597e-08 (-1.31)
Unexpected: $25\times$ to $30\times$ | - | - |  9.099e-11 (1.70)** |  -7.329e-09 (-0.91)
Expected: $30\times$ to $35\times$ | - | - |  5.761e-10 (1.91)* |  8.934e-08 (1.34)
Unexpected: $30\times$ to $35\times$ | - | - |  -3.618e-12 (-0.02) |  -3.778e-09 (-0.26)
Expected: $35\times$ to $40\times$ | - | - |  2.882e-11 (0.03) |  2.465e-07 (3.10)***
Unexpected: $35\times$ to $40\times$ | - | - |  2.266e-10 (0.93) |  -1.913e-09 (-0.32)
Expected: $40\times$ to $45\times$ | - | - |  1.289e-09 (0.51) |  -2.922e-10 (-0.01)
Unexpected: $40\times$ to $45\times$ | - | - |  3.24e-10 (1.12) |  -2.72e-09 (-0.61)
Expected: $45\times$ to $50\times$ | - | - |  2.626e-11 (0.15) |  1.235e-08 (0.74)
Unexpected: $45\times$ to $50\times$ | - | - |  1.692e-10 (1.73)* |  9.194e-10 (0.30)
Expected: more than $50\times$ | - | - |  -1.529e-10 (-0.29) |  -1.466e-09 (-0.13)
Unexpected: more than $50\times$ | - | - |  -3.74e-11 (-0.20) |  -1.21e-08 (-0.96)
Adjusted $R^{2}$ | 0.398 | 0.339 | 0.427 | 0.260
AIC | -4405 | -3693 | -4424 | -3608
No. of obs. | 747 | 628 | 747 | 628
|  Notes. This table reports regression results on Eq. (31) and Eq. (32). For
the regression with Eq. (32), Leverage (in categories) means trading volumes
in leverage within different ranges, which are grouped into categories. For
GNS, our dataset does not include any transaction with leverage less than
$1\times$, so we drop the variables of trading volumes with the leverage less
than $1\times$. In each cell, the t-statistics are in the parentheses. $*$,
$**$, and $***$ denote significance at 0.1, 0.05, and 0.01 level,
respectively.
---

##  6 Conclusion

In conclusion, this study presents an in-depth Systematic Survey of Knowledge
(SoK) that effectively contrasts the operational dynamics of Centralized
Exchanges (CEXs) and Decentralized Exchanges (DEXs) in the realm of perpetual
futures trading. It significantly enriches the academic discourse by
meticulously delineating the design elements, order matching mechanisms, price
discovery processes, and custody of funds in both CEXs and DEXs, thereby
offering a robust foundation for future research in this field. With the
models in our framework, any exchange could be classified into these predined
models and each model corresponds to a sort of distinctive features. The study
underscores the necessity of developing new theoretical models, moving beyond
conventional frameworks like Kyle’s model[1], to aptly comprehend the
intricacies of DEX models. Through an empirical analysis of transaction data
from August 2020 to September 2023, it reveals a positive correlation between
price volatility and trading volumes in exchanges of CEX Model, in line with
traditional futures markets. However, it also uncovers distinct variations in
DEXs, particularly under the VAMM Pricing Model, where open interest impacts
long and short positions differently. The behavior of traders in GMX and GNS,
who operate under Oracle Pricing and act as price takers, is found to be
consistent with Shalen’s dispersion of beliefs model[2]. Together with Table
2, Table 6 summarizes traders’ distinctive behavior derived by different
design of the models. Furthermore, the study highlights the propensity of
uninformed traders to disproportionately react to positive news than negative
news, thereby deepening our understanding of trader behavior in DEXs and
offering valuable insights for the development of future theoretical models in
this evolving domain.

Table 6: Summary of Empirical Results Exchange Model |  Price Discovering |  Price Volatility & Trading Volume |  Price Volatility & Open Interest |  Symmetry; Explanation |  Exchange Analyzed
---|---|---|---|---|---
LOB Models (i.e., CEX Model Hybrid Model) | Yes |  Positively correlated |  Negatively correlated |  Symmetrical; Kyle’s model [1] |  Binance; Bybit
VAMM Model | Yes |  Positively correlated |  |  Negatively correlated for short positions & positively correlated for long positions
---
(in the long-run); positive (in the short-run)
|  Asymmetrical; Uniswap v3 AMM
---
Perpetual Prorocol V2
Oracle Pricing Model | No |  Positively correlated |  Negatively correlated |  |  Asymmetrical; Shalen’s dispersion of beliefs model [2]
---
GMX; GNS
|  Notes. This table summarizes our empirical results regarding traders’
behavior in exchanges of different exchange models. Since both Hybrid Model
CEX Model adopt limit-order book to match orders, traders’ behavior in
exchanges of these two models should be the same. In this table, we care about
the symmetry of effects between buyers and sellers.
---

In terms of future work, firstly, the observed asymmetry in trading behaviors
between buyers and sellers within the Oracle Price Model DEX warrants a deeper
exploration into the psychological underpinnings of this phenomenon. Future
research could delve into whether uninformed traders disproportionately react
to positive news compared to negative news, and examine the extent to which
cognitive biases or prevailing social norms may influence such behaviors. This
line of inquiry could provide valuable insights into the psychological
mechanisms at play in decentralized exchange environments.

Secondly, the role of liquidity providers, a novel concept in the
decentralized finance (DeFi) ecosystem, presents a rich area for academic
investigation. While this paper primarily analyzes the behaviors of traders in
Perpetual Future Contracts, a comparative study focusing on their
counterparts, the liquidity providers, could yield significant findings.
Future research should consider the impact of variables such as interest rates
and the availability of alternative DeFi products on the behaviors of
liquidity providers, particularly in the context of Perpetual Future
Contracts. Understanding how these factors interact to influence liquidity
provider behaviors could offer a more comprehensive view of market dynamics in
DeFi.

Lastly, the potential of blockchain-based financial instruments to enhance
transparency and accountability in micro financial markets is an important
avenue for future research. The decentralized and transparent nature of
blockchain technology suggests that these instruments could be instrumental in
promoting accountability and transparency in micro financial markets,
potentially encouraging more ethical investment practices. Future studies
should explore the capacity of blockchain-based instruments to achieve these
outcomes, examining the factors that may affect their effectiveness in
fostering responsible financial practices.

##  7 Acknowledgement

ChatGPT has been used in the writing for the purpose of spelling, syntax and
grammar checks. The authors take full responsibility of the output.

## Appendix

Table 7 reports the results of the Augmented Dickey–Fuller (ADF) test, which
is used to check whether the time series we are examining are stationary.
According to Section 5.2, when decomposing the series into expected and
unexpected portions, the series for which we fail to reject the hypotheses of
the existence of a unit root are differenced, i.e., the series with
insignificant statistics in Table 7.

Fig. 14 shows the dynamics of trading volumes, open interests, and
liquidations solely for DEXs. GMX dominated across time with the highest
trading volumes. Although the trading volumes on GNS are far less than those
on GMX, it has open interest comparable to GMX. In addition, liquidations on
long positions are more frequent and higher in value than those on short
positions along the time.

Table 7: Augmented Dickey–Fuller (ADF) test statistics | Perpetual Contracts on Bitcoin
---|---
Variables | GMX | GNS |  Perpetual Protocol V2 | Binance | Bybit
Trading Volume | -4.850*** | -4.178*** | -3.680*** | -3.725*** | -2.609*
OI on Short | -2.896** | -1.868 | -1.621 | -2.930* | -1.428
OI on Long | -3.556*** | -1.559 | -2.179 | - | -
Liquidation on Short | -13.712*** | -16.172*** | -24.228*** | -2.115 | -2.886**
Liquidation on Long | -26.746*** | -8.175*** | -8.540*** | -2.787* | -3.430***
Leverage on Short | -6.769*** | -5.844*** | - | - | -
Leverage on Long | -4.110*** | -4.204*** | - | - | -
$\mathcal{SI}$ | -3.388** | -2.542 | -4.375*** | -3.044** | -3.836***
$\mathcal{LIQ}_{short}$ | -26.480*** | -8.453*** | -23.066*** | -1.799 | -2.982**
$\mathcal{LIQ}_{long}$ | -27.120*** | -2.872** | -15.664*** | -2.764* | -2.260
Leverage (in categories):
less than $1\times$ | -6.106*** | - | - | - | -
$1\times$ to $5\times$ | -5.149*** | -6.247*** | - | - | -
$5\times$ to $10\times$ | -5.194*** | -4.790*** | - | - | -
$10\times$ to $15\times$ | -3.846*** | -8.132*** | - | - | -
$15\times$ to $20\times$ | -9.448*** | -8.472*** | - | - | -
$20\times$ to $25\times$ | -4.339*** | -24.221*** | - | - | -
$25\times$ to $30\times$ | -4.614*** | -7.505*** | - | - | -
$35\times$ to $40\times$ | -2.338 | -15.250*** | - | - | -
$40\times$ to $45\times$ | -3.139** | -3.196*** | - | - | -
$45\times$ to $50\times$ | -2.245 | -14.365*** | - | - | -
More than $50\times$ | -4.361*** | -4.145*** | - | - | -
|  Notes. This table reports the statistics for Augmented Dickey–Fuller (ADF)
test, which is used to test whether a time series is stationary. ADF test
statistics are for the hypothesis that a series contains a unit root. $*$,
$**$, and $***$ denote significance at 0.1, 0.05, and 0.01 level,
respectively.
---
Table 8: Granger Causality Test of Return on Bitcoin and the Net Change in Liquidity in the vBTC-vUSDC Pool on Perpetual Protocol V2 $H_{0}$ | Max-lag | F-statistics |  p-value ($Prob>F$)
---|---|---|---
Return on BTC does not Granger-cause net change in liquidity pool | 15 | 1.624 | 0.079
Net change in liquidity pool does not Granger-cause Return on BTC | 15 | 1.164 | 0.311
|  Notes. This table reports the results of Granger Causality Test in
dectacting the relationship between return on Bitcoin and the net change in
the liquidity contained in vBTC-vUSDC pool on Perpetual Protocol V2.
---

Figure 14: Trading Volumes, Open Interest, and Liquidated Volumes of Perpetual
Futures on the Three DEXs.

As conjectured in our analysis on liquidation in Perpetual Protocol V2, we
hypothesis that price rise in Bitcoin attract liquidity providers in offering
liquidity. To detect empirical evidence to our conjecture, we conduct the
Granger Causality Test to see the relationship between the return on Bitcoin
and the net change in liquidity in the vBTC-vUSDC pool on Perpetual Protocol
V2, where the net change is calculated by subtracting the daily added
liquidity by the daily withdrawn liquidity from the pool (measured in
Bitcoin). Return on Bitcoin is calculated in daily frequency with the log-form
return as follows:
$R_{t}=\log\left(\underline{P_{t}}\right)-\log\left(\overline{P_{t}}\right)$,
where $\underline{P_{t}}$ and $\overline{P_{t}}$ denote the close and open
price of Bitcoin on day $t$. We retrieve data from 15 May 2023 to 15 Oct 2023
on Etherscan (https://optimistic.etherscan.io/). The testing results are
reported in Table 8. There exists Granger-causality from the return on Bitcoin
to the net change in vBTC-vUSDC liquidity pool, indicated by the statistically
significant F-statistic with p-value of 0.079. However, the causality in
reversed direction (net change in vBTC-vUSDC liquidity pool Granger-cause
return on Bitcoin) is not evidenced, with p-value of 0.311, supporting our
hypothesis that more liquidity is injected when Bitcoin price jump. The large
lag, i.e., 15 days in this case, may suggests that traders need time to adjust
positions in the liquidity pool in reaction to the rise in Bitcoin price. The
empirical evidence here shows a rough picture of how traders change their
deposit in the liquidity pool while future studies are needed with more
factors controlled.

The ADF test and regression results of Eq. (29) and Eq. (30) for Binance using
data separately before (Period(1)) and after (Period(2)) 1 May 2021 are
reported in Table LABEL:tab_binance. Same with the case in Table
LABEL:tab_eq2, coefficients for both expected and unexpected portion of the
trading volume are estimated to be significantly positive, and those for open
interest are estimated to be negative, substantiating our notion in Section
5.3 that the increase in open interest mitigates the effects of shocks on
price volatility.

Table 9: ADF Test and Regression Results with Truncated Data for Binance Regression Results Eq. (29) and Eq. (30) | ADF Test
---|---
Variables | Period(1) | Period(2) | Variables | Period(1) | Period(2)
Panel a: Regression Results on Eq. (29): | Panel b: Results of ADF test
Intercept |  0.011 (3.23)*** |  0.010 (3.90)*** |  Trading Volume | -2.775* | -3.172**
Lagged volatility |  0.145 (4.58)*** |  0.041 (2.07)** | OI | -1.051 | -3.810***
Trading Activity: |  |  |
Expected trading volume |  5.57e-08 (5.10)*** |  1.84e-08 (8.52)*** |  Liquidation on Short | -1.430 | -4.583***
Unexpected trading volume |  7.208e-08 (8.42)*** |  3.958e-08 (14.24)*** |  Liquidation on Long | -4.977*** | -4.939***
Expected OI |  -1.623e-11 (-6.47)*** |  -1.66e-12 (-2.27)** | $\mathcal{SI}$ | -2.258 | -3.853***
Unexpected OI |  -5.132e-12 (-0.55) |  -1.338e-11 (-5.09)*** | $\mathcal{LIQ}_{\text{short }}$ | -1.822 | -8.152***
Adjusted $R^{2}$ | 0.82 | 0.76 |  | - | -
AIC | -1688 | -5408 |  | - | -
No. of obs. | 269 | 878 |  | - | -
Liquidation: |  |  |
Expected liquidated volume on short |  3.859e-10 (5.57)*** |  8.418e-10 (3.41)*** | $\mathcal{LIQ}_{\text{long}}$ | -5.951*** | -7.284***
Unexpected liquidated volume on short |  1.083e-10 (4.52)*** |  1.367e-09 (8.59)*** |  | - | -
Expected liquidated volume on long |  1.694e-10 (4.9)*** |  1.882e-09 (8.59)*** |  | - | -
Unexpected liquidated volume on long |  9.729e-11 (8.46)*** |  9.686e-10 (9.16)*** |  | - | -
Panel c: Regression Results on Eq. (30): |  |  |
Intercept |  0.007 (2.30)** |  -0.004 (-1.47) |  | - | -
Lagged volatility |  0.176 (5.62)*** |  0.026 (1.34) |  | - | -
Trading Activity: |  |  |
Trading volume |  5.329e-08 (4.27)*** |  2.725e-08 (5.89)*** |  | - | -
OI |  -2.124e-13 (-0.10) |  3.563e-12 (3.61)*** |  | - | -
$\mathcal{SI}$ and $\mathcal{LIQ}$: |  |  |
$\mathcal{SI}$ |  -0.009 (-1.37) |  0.007 (0.59) |  | - | -
$\mathcal{LIQ}_{\text{short }}$ |  0.158 (6.03)*** |  2.188 (11.03)*** |  | - | -
$\mathcal{LIQ}_{\text{long }}$ |  0.156 (12.09)*** |  2.374 (8.15)*** |  | - | -
Adjusted $R^{2}$ | 0.81 | 0.76 |  | - | -
AIC | -1675 | -3649 |  | - | -
No. of obs. | 269 | 878 |  | 269 | 878
|  This table reports the estimation results of Eq. (29) (Panel a) and Eq.
(30) (Panel c), as well as the ADF test (Panel b), for Binance in two periods
separately. Period (1) presents the time from 5 August 2020 to 30 Apirl 2021,
and Period (2) stands for the time from 1 May 2021 to 26 September 2023. ADF
test statistics are for the hypothesis that a series contains a unit root. In
each cell of Panel a and Panel c, the t-statistics are in the parentheses.
$*$, $**$, and $***$ denote significance at 0.1, 0.05, and 0.01 level,
respectively.
---

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# Heavy-tailed probability distributions in social sciences

Lev B. Klebanov, Yulia V. Kuvaeva Department of Probability and Mathematical
Statistics, Charles University, Prague, Czech Republic. e-mail:
<EMAIL_ADDRESS>Department of Finance, Money Circulation and Credit, Ural
State University of Economics, 620144 Yekaterinburg, Russia;
<EMAIL_ADDRESS>
###### Abstract

We present an overview of possible reasons for the appearance of heavy-tailed
distributions in applications to the natural sciences. These distributions
include the laws of Pareto, Lotka, and some new ones. The reasons are
illustrated using suitable toy models.

Key words: heavy-tailed distributions; Pareto law; Lotka law; Zipf law;
probability generating function.

Epigraph

King James Bible:

I returned, and saw under the sun, that the race is not to the swift, nor the
battle to the strong, neither yet bread to the wise, nor yet riches to men of
understanding, nor yet favour to men of skill; but time and chance happeneth
to them all.

Lion Feuchtwanger "Der jüdische Krieg":

Gods give talent to one and luck to another.

##  1 History of the problems

The distributions with heavy (power-like) tails in social sciences were used
more than hundred years.

Relevant studies include the following:

  * 1.

Distribution of big capital. Pareto, 1896 (see [6])). The density is
$p(x)=\frac{\alpha}{x_{o}}\Bigl{(}\frac{x_{o}}{x}\Bigr{)}^{\alpha+1}$, for
$x\geq x_{o}$, $\alpha>0$.

  * 2.

Scientific production. The number of scientists who published one, two and so
on $x$ papers ( the number $x$ published by scientist papers). Lotka (1926)
(see [4]) showed that $n(x)=n_{1}/x^{a}$, where $n_{1}>0$, $a\leq 2$ (in many
cases $a\approx 2$).

  * 3.

Lotka’s law approximately holds for the number of citations of a paper by a
scientist.

  * 4.

For a specific artistic text, the sequence of all words is written in
descending order according to the frequency of their occurrence. Comparing the
frequency of the word and the place in this sequence (rank) leads to $x=B/r$,
$B=const$ (see [7]).

Why do these patterns emerge? Probably, Laws 1 - 3 refer to some individual
human abilities, while Law 4 refers to memory or other functions of the human
brain.

We will not consider the 4th law in the paper and will focus on laws 1 and 3,
more precisely on their qualitative explanation. It is so because Zipf
explained his Law based on the least effort principle. Although there are no
rigorous results on the existence of a mechanism related to this principle in
the human brain, not wasting memory seems natural. However, the application of
the the least effort principle in cases 1-3 does not seem to be related to the
essence of the issues under consideration.

At first glance, everything looks quite simple. The population of a country is
heterogeneous. There are people more capable of business (in case 1) or
scientific work (in cases 2, 3) and people who are not (or less capable) of
such activities.

But how big is the difference in ability, and are all differences in "success"
determined by ability?

Let’s remember the epigraph!

Is there an effect of chance? First, let’s focus on the 1st law. Let’s try to
build a model that explains the reason for its occurrence.

However, the income and the capital distribution are subject to many factors
not fully accounted for. Our interest is not in the whole mechanism of
accumulation and distribution of capital but in the roles of human talent and
chance in this process only. How essential are these roles? Therefore, we have
to use a toy model which assumes all people have identical abilities. If the
role of chance is small, then there will not be many variations in the model
between different investors. In contradiction, if we see a large difference
between investors, this will indicate a significant role of chance.

##  2 Toy model for capital distribution

Let us consider the first toy model of capital distribution leading to the
Pareto law.

Suppose for simplicity that there exists only one business. All possible
investors are equal in their talents and initial capital. Consider the case
when each investor invests one unit of capital in the business. After one
time-unit, the business outcome is $X_{1}$, where $X_{1}$ is a random
variable. Suppose the investor left all this sum into business, and the
conditions on the market remain the same during the following time interval.
Then the outcome after the second-time interval is $X_{1}\cdot X_{2}$, where
$X_{1}$ and $X_{2}$ are independent identically distributed (i.i.d.) random
variables. In the same way, the outcome after the n-th time interval is
$\prod_{i=1}^{n}X_{j}$, and $X_{1},X_{2},\ldots,X_{n}$ are i.i.d. random
variables. Let us suppose that the conditions on the market will change
radically at a random moment $\nu_{p}$ so that investing in that business
becomes not profitable. Therefore, the final outcome is
$\prod_{j=1}^{\nu_{p}}X_{j}$. We are interested in the outcome behavior for
large $\nu_{p}$ values. More precisely, we suppose that

  1. 1.

$\mathbf{X}=\\{X_{1},X_{2},\ldots,X_{n},\ldots\\}$ is a sequence of i.i.d.
positive random variables, $a={\rm I}\kern-1.79993pt{\rm E}\log X_{1}$;

  2. 2.

$\nu=\\{\nu_{p},p\in\Delta\subset(0,1)\\}$ is a family of positive integer-
valued random variables independent with the sequence $\mathbf{X}$, ${\rm
I}\kern-1.79993pt{\rm E}\nu_{p}=1/p$.

Generally, the information on the $\nu$–family is unavailable. We shall
consider a few cases starting with a simple one.

  3. 3.

${\rm I}\kern-1.79993pt{\rm P}\\{\nu_{p}=k\\}=p\cdot(1-p)^{k-1}$,
$k=1,2,\ldots$ i.e. $\nu_{p}$ has geometric distribution.

Define $Z_{p}=\prod_{j=1}^{\nu_{p}}X_{j}^{p}$.

######  Theorem 2.1.

Suppose that the items 1. – 3. hold. Let $a\neq 0$. Then

$\lim_{p\to 0}{\rm I}\kern-1.79993pt{\rm
P}\\{Z_{p}<x\\}=1-x^{-1/a},\;\text{for}\;x\geq 1,\;a>0$

and

$\lim_{p\to 0}{\rm I}\kern-1.79993pt{\rm
P}\\{Z_{p}<x\\}=x^{1/a},\;\text{for}\;x\leq 1,\;a<0.$

In the case of $a>0$ (profitable business), we have a Pareto distribution
which Wilfredo Pareto had proposed on the basis of empirical study (see [6]).
For the proof of Theorem 2.1 see [2]. In the paper, [2] is obtained the result
for $a=0$. For this case $Z_{p}$ must be changed by
$Z_{p}^{\prime}=\prod_{j=1}^{\nu_{p}}X_{j}^{\sqrt{p}}$. Under the condition of
the existence of logarithmic second moment of $X_{1}$ the product
$Z_{p}^{\prime}$ converges in distribution to the mixture of the given in
Theorem 2.1 distributions. It is well-known that Pareto distribution has heavy
tails. This fact implies that capital belongs to a relatively small number of
people. Now we see Pareto distribution appears in a very natural way described
as a limit distribution for a product of a random number $\nu_{p}$ of random
variables $X_{j}$. The value of $\nu_{p}$, $p\in(0,1)$ in 3. had a geometric
distribution. What will happen with others ("natural") distributions? Below we
consider two additional cases:

  1. 4.

$\nu_{p}$ has a probability generating function

$\mathcal{P}(z,p,m)=\frac{p^{1/m}z}{(1-(1-p)z^{m}))^{1/m}},\quad
p\in(0,1),\quad m\in\mathbb{N}.$

  2. 5.

$\nu_{p}$ has a probability generating function

$\mathcal{P}(z,n)=\frac{1}{T_{n}(1/z)},$

where $T_{n}(u)$ is Chebyshev polynomial of the first kind and $n=1/\sqrt{p}$
is its degree. ${\rm I}\kern-1.79993pt{\rm E}\nu_{p}=1/p$.

Let us consider case 4. The following result holds.

######  Theorem 2.2.

Suppose that the items 1., 2. and 4. hold. Let $a\neq 0$. Then

$\lim_{p\to 0}{\rm I}\kern-1.79993pt{\rm
P}\\{Z_{p}<x\\}=\int_{1}^{x}\frac{1}{b^{1/m}\Gamma(1/m)u^{1+1/m}\log^{1-1/m}(u)}du,\;\text{for}\;x\geq
1,$

where $b>0$ is a parameter.

###### Proof.

Consider $\log Z_{p}=p\sum_{j=1}^{\nu_{p}}Y_{j}$, where $Y_{j}=\log X_{j}$.
From the result of the paper [5] it follows that the limit distribution of
$\log Z_{p}$ as $p\to 0$ has the density
$\exp\\{-u/b\\}/\bigl{(}u^{1-1/m}b^{1/m}\Gamma(1/m)\bigr{)}$, $u>0$. Now it is
sufficient to pass the limit distribution of $Z_{p}$ from its logarithm
density. ∎

######  Theorem 2.3.

Suppose that the items 1., 2. and 5. hold. Let $a=0$ and the second
logarithmic moment of $X_{1}$ exists. Then

$\lim_{p\to 0}{\rm I}\kern-1.79993pt{\rm
P}\\{Z_{p}^{\prime}<x\\}=\frac{2}{\pi}\arctan(x^{b}),\quad\text{for}\quad
x>0,$

where $b>0$ is a parameter.

###### Proof.

Similarly to the proof of the previous Theorem, we have to pass from
$Z_{p}^{\prime}$ to its logarithm, apply the corresponding result from [3],
and went back to the limit distribution for initial random variables. ∎

All three models constructed above do not take into account any abilities of
the investing in the given enterprise people but lead to heavy-tailed
distributions. The difference between investors is only in the occurrence of
some unfavorable event for them (moment $\nu_{p}$). Objection that this moment
is the same for the whole store, i.e. it is insolvent for all investors at
once because the investors invested in the business at different times.
Therefore, the period for which the investment was made is different for each
investor. So we see that the dependence on the moment and the case are really
very high. I do not deny that the dependence on the talent of the investor is
indeed significant, but it would be very difficult to separate this component
from random.

##  3 Citations distribution

A similar situation occurs when studying the distribution of citations of
scientific publications. Let us make some Assumptions.

Assumption 1.

All scientists under consideration are equal in their scientific and literary
abilities.

Assumption 2.

Paper citations occur independently.

Assumption 3. The probability that an article will be repeatedly cited depends
on the number of previous citations. It is growing with the growing citations
number. More precisely,

Assuming the probability that an article having $\mathit{k-1(k\geq 1)}$
citations will have no further citations is

$p_{k}=\frac{1}{(a\,k+b)},$

where$a>0$ a $b\geq 0$ are real numbers satisfying $a+b>1$.

Let $Y$ be a random variable describing the number of citations during the
considered period. Assumption 1 implicitly de-facto implies that $Y$ has the
same distribution for different papers because the scientific abilities of the
authors are supposed to be the same.

In view of citation independence, the probability that a paper is cited
exactly $n$ times is

$\mathrm{I}\kern-1.79993pt\mathrm{P}\\{Y=n\\}=p_{n}\prod_{k=1}^{n-1}(1-p_{k})=\frac{\Bigl{(}\frac{a+b-1}{a}\Bigr{)}_{n-1}}{(a\,n+b)\Bigl{(}\frac{a+b}{a}\Bigr{)}_{n-1}},$

where $(a)_{n}=a(a+1)\ldots(a+n-1)$ is Pochhammer symbol.

It is not difficult to calculate the probability

$\mathrm{I}\kern-1.79993pt\mathrm{P}\\{Y\geq
m\\}=\frac{\Bigl{(}\frac{a+b-1}{a}\Bigr{)}_{m-1}}{\Bigl{(}\frac{a+b}{a}\Bigr{)}_{m-1}}\stackrel{{\scriptstyle[}}{{m}}\to\infty]{}{\thicksim}\frac{\Gamma((a+b)/a)}{\Gamma((a+b-1)/a)}\frac{1}{m^{1/a}}\;.$
(3.1)

Distribution of citations The relation (3.1) shows that the distribution of
citations has a heavy tail, the severity of which depends on the value of the
parameter $a$ responsible for the degree of influence of previous citations.
Therefore, a larger value of $a$ corresponds to a heavier tail. In any case,
the presence of such a tail makes it possible to conclude that the citation
intensity of almost identical scientists can differ significantly, which leads
to a significant stratification of the scientific community through various
random circumstances that have nothing to do with research abilities. Thus,
the citation number seems meaningless as an indicator of scientific value.

Make now some remarks on the Impact Factor distribution.

Let us now consider the possibility of using the impact factor of a journal as
an indicator of the scientific significance of a paper published in it. The
impact factor of a journal is calculated as the ratio of the number of
citations of papers published over a certain period to the number of these
papers themselves. The idea of considering such an average value is connected
with the idea that, according to the law of large numbers, the influence of
chance will be leveled. However, we shall show, this is not true.

Mention that there exists a rather large literature stating the scientific
journals’ impact factor has essential value. Based on the observed data, the
presence of asymmetry in the distribution of the impact factor and the
presence of a heavy tail is noted. However, these circumstances are not
analyzed from a theoretical point of view, and only comments are made on the
advisability of replacing the arithmetic mean with some other statistics for
the purpose of statistical data analysis. We note one of the typical works of
this kind [1]. True, the author notes the similarity of the distribution of
some data with the Pareto distribution, but the mathematical analysis of the
reasons for its occurrence is not carried out. In addition, not a
mathematically strictly defined distribution is considered, but only its
"naive" form. Below we will try to clarify the appearance of heavy tails of
the impact factor distribution.

We assume that the number of papers submitted to the journal has a Poisson
distribution. For simplicity, let us assume that the number of citations for
each of the submitted papers has a Sibuya distribution. Then the citation
distribution of all papers has a probability generating function that is a
superposition of the generating functions of the Sibuya and Poisson laws.
Probability generating function of this superposition is
$\mathcal{P}(z)=e^{-\lambda(1-z)^{p}}$ for fixed $\lambda>0$ and $p\in(0,1)$.
Clearly, this distribution has a heavy tail with index $p$. In view of the
fact that $p<1$ the law of large numbers is inapplicable in this situation.
Moreover, in this case, the impact factor increases with the number of
publications without increasing their scientific significance. The observed
increase (over time) in the impact factors of leading journals confirms this
circumstance.

Now we can conclude the impact factor distribution has a heavy tail again and
cannot be used as an indicator of scientific significance.

##  4 Conclusions

  * I.

It is shown that distributions with heavy tails can arise in some
manifestations of social inequality (the distribution of capital, the number
of citations, the impact factor) due to purely random reasons. In this case,
the spread in the magnitude of inequality is significant.

  * II.

The circumstance specified in 1. makes it impossible to use such indices as
the number of citations and/or the impact factor of a journal as an indicator
of the scientific significance (scientific quality) of a published work.

## Acknowledgment

The work by Lev B. Klebanov was partially supported by GA ČR Grant 19-28231X
EXPRO.

## References

  * [1] Blanford C. F. (2016) Impact factors, citation distributions and journal stratification. Journal of Materials Science volume 51, 10319–10322.
  * [2] Klebanov L.B., Melamed J.A., Rachev S.T. (1987) On the products of a random number of random variables in connection with a problem from mathematical economics. In: Stability Problems for Stochastic Models, Lecture Notes in Mathematics, 1412, 103–109.
  * [3] Klebanov L.B., Kakosyan A.V., Rachev S.T., Temnov G. (2012) On a class of distributions stable under random summations. Journal of Applied Probability, 49, 303–318.
  * [4] Lotka, Alfred J. (1926). "The frequency distribution of scientific productivity". Journal of the Washington Academy of Sciences. 16 (12): 317–324.
  * [5] Melamed J.A. (1989). Limit theorems in the set-up of summation of a random number of independent and identically distributed random variables. In: Stability Problems for Stochastic Models, Lecture Notes in Mathematics, 1412, 194–228.
  * [6] Pareto, Vilfredo (1964) Cours d’Économie Politique: Nouvelle édition par G.-H. Bousquet et G. Busino, Librairie Droz, Geneva, 299–345.
  * [7] Zipf G.K. (1949) Human Behavior and the Principle of Least Effort. Cambridge. Addison–Westley.