Boundary behaviors for a continuous-state

nonlinear Neveu’s branching process

Linyu Bai and Xu Yang111Corresponding author.

School of Mathematics and Information Science, North Minzu University

Yinchuan 750021, People’s Republic of China

Emails<EMAIL_ADDRESS><EMAIL_ADDRESS>

Abstract. After generalizing the criteria introduced by Chen, we establish the
necessary and sufficient conditions for the extinction, explosion and coming
down from infinity of a continuous-state nonlinear Neveu’s branching process.

Mathematics Subject Classifications (2010): 60G57, 60G17, 60J80.

Key words and phrases: Continuous-state branching process, nonlinear
branching, Neveu’s branching, extinction, explosion, coming down from
infinity.

##  1 Introduction and main result

Branching processes in discrete-state are introduced as probabilistic models
for the stochastic evolution of populations, and their scaling limits are
regarded as continuous-state branching processes (CSBPs for short), which are
nonnegative-valued Markov processes with the additive branching processes. We
refer to Kyprianou [12] and Li [18, 19] and references therein for reviews and
literature on CSBPs.

Recently, generalized versions of CSBPs have been introduced to study the
interactions between individuals and (or) between individuals and the
population. In Li [15], a class of CSBPs with nonlinear branching mechanism is
given by generalized Lamperti transform. A more general version of the
nonlinear CSBP is studied in Ma et al. [20] as the solution to stochastic
differential equation (SDE for short)

$\displaystyle X_{t}$ $\displaystyle\\!\\!=$
$\displaystyle\\!\\!x+\int_{0}^{t}a_{0}(X_{s})\mathrm{d}s+\int_{0}^{t}\int_{0}^{a_{1}(X_{s})}W(\mathrm{d}s,\mathrm{d}u)$
(1.2)
$\displaystyle\\!\\!+\int_{0}^{t}\int_{U}\int_{0}^{a_{2}(X_{s-})}z\tilde{N}(\mathrm{d}s,\mathrm{d}z,\mathrm{d}u)+\int_{0}^{t}\int_{(0,\infty)\setminus
U}\int_{0}^{a_{3}(X_{s-})}zN(\mathrm{d}s,\mathrm{d}z,\mathrm{d}u),$

where $x>0$, $a_{0}$ and $a_{1},a_{2},a_{3}\geq 0$ are Borel function on
$[0,\infty)$, $W(\mathrm{d}s,\mathrm{d}u)$ and
$N(\mathrm{d}s,\mathrm{d}z,\mathrm{d}u)$ denote a Gaussian white noise and an
independent Poisson random measure, respectively. Here $U$ is a Borel set on
$(0,\infty)$ and $\tilde{N}(\mathrm{d}s,\mathrm{d}z,\mathrm{d}u)$ is the
compensated measure of $N(\mathrm{d}s,\mathrm{d}z,\mathrm{d}u)$. The pathwise
unique nonnegative solution to SDE (1.2) is obtained in Dawson and Li [7] and
called a Dawson-Li SDE in Pardoux [23]. SDEs similar to (1.2) are also studied
by Dawson and Li [6] and Fu and Li [11].

There are a series of results on boundary behaviors to SDE (1.2). The
extinction, extinguishing, explosion and coming down from infinity are studied
in Li [15] with identical power rate functions $a_{0},a_{1},a_{2}$ by Lamperti
transform. Using a martingale approach, the conditions for extinction,
extinguishing, explosion, coming down from infinity and staying infinite are
obtained in Li et al. [17]. If $a_{i}(x)=c_{i}a(x)(i=0,1,2)$ for some
constants $c_{i}$ and function $a(x)$, then the asymptotic assertions on the
speeds of coming down from infinity and explosion are studied in Foucart et
al. [9] and in Li and Zhou [13], respectively. The exponential ergodicity is
obtained by Li and Wang [16] by using coupling techniques and long time
behaviour of this SDE with catastrophes is studied by Marguet and Smadi [21].
The exponential ergodicity for this SDE with immigration is also studied in
[14]. The density of the Poisson random measure
$N(\mathrm{d}s,\mathrm{d}z,\mathrm{d}u)$ is
$\mathrm{d}s\mu(\mathrm{d}z)\mathrm{d}u$, $U=(0,\infty)$ and the condition
$\int_{0}^{\infty}(z\wedge z^{2})\mu(\mathrm{d}z)<\infty$ is required in [17,
9, 13, 21].

Using generalized Chen’s criteria techniques, new conditions on non-
extinction, non-explosion, coming down from infinity and staying infinite for
the model of (1.2) are established in Ma et al. [20], which gives an
affirmative answer to boundary behavior of critical cases in Li et al. [17]
and also finds some non-extinction and non-explosion conditions when
$\int_{U}(z\wedge z^{2})\mu(\mathrm{d}z)<\infty$ and
$\int_{(0,\infty)\setminus U}(1\vee\ln(1+z))\mu(\mathrm{d}z)<\infty$. These
techniques are used in Li et al. [17] for the estimation on the first passage
probabilities for SDE (1.2) with $U=(0,\infty)$ and in Ren et al. [24] for the
extinction-extinguishing behaviors for a two-dimensional general CSBPs. These
criteria generalize similar results for Markov jump processes; see Chen [5,
Theorems 2.25 and 2.27].

Inspired by those work, in this paper we study the boundary classification for
a continuous-state nonlinear Neveu’s branching process, which can be regarded
as the solution to the following SDE that is a modification of (1.2):

$\displaystyle
X_{t}=x+\int_{0}^{t}\int_{0}^{r}\int_{0}^{X_{s-}^{\beta}}z\tilde{M}(\mathrm{d}s,\mathrm{d}z,\mathrm{d}u)+\int_{0}^{t}\int^{\infty}_{r}\int_{0}^{X_{s-}^{\theta}}zM(\mathrm{d}s,\mathrm{d}z,\mathrm{d}u),$
(1.3)

where $x,r>0$ and $\beta,\theta\geq 0$,
$M(\mathrm{d}s,\mathrm{d}z,\mathrm{d}u)$ denotes a Poisson random measure on
$(0,\infty)^{3}$ with density $\mathrm{d}sz^{-2}\mathrm{d}z\mathrm{d}u$, and
$\tilde{M}(\mathrm{d}s,\mathrm{d}z,\mathrm{d}u)=M(\mathrm{d}s,\mathrm{d}z,\mathrm{d}u)-\mathrm{d}sz^{-2}\mathrm{d}z\mathrm{d}u$.
When $\beta=\theta=1$, the model in (1.3) is considered by Neveu [22] and the
solution is called continuous-state Neveu’s branching process with branching
mechanism

$\psi(u):=\int_{0}^{\infty}[{\mbox{\rm e}}^{-zu}-1-zu1_{\\{z\leq
r\\}}]z^{-2}\mathrm{d}z=u\ln u+u(\ln r-5{\mbox{\rm e}}^{-1}+1).$

Neveu [22] notes a connection between Ruelle’s generalized random energy
models and the continuous-state Neveu’s branching processes. Bolthausen and
Sznitman [2] explains how the results of replica theory of spin glasses can be
interpreted in terms of a coalescent process (now called the Bolthausen-
Sznitman coalescent). Bertoin and Le Gall [1] gives a precise and complete
form of the relation between the connections between the continuous state
Neveu’s branching processes, the Ruelle’s generalized random energy models,
and the Bolthausen-Sznitman coalescent. Further conclusions are obtained in
Foucart et al. [10].

Our aim is to establish the necessary and sufficient conditions of the
extinction, explosion and coming down from infinity for the continuous-state
nonlinear Neveu’s branching process determined in (1.3) and thus find out the
sharp influence of different forms of big jumps and small jumps on the
boundary behaviors. To this purpose, we adapt the approach in Ma et al. [20]
for the behavior of non-extinction, non-explosion and staying infinite and
establish new criteria for the behavior of extinction, explosion and first
passage probabilities which are of independent interesting and generalize the
criteria in [24, Proposition 2.2] and Chen [5, Theorem 2.27] (see Lemma 2.2 in
the following). We also select power functions and log-logarithm type test
functions which are different from that of Ma et al. [20] (see the
explanations in Remark 3.2).

Throughout this paper we always assume that process $(X_{t})_{{t}\geq 0}$ is
defined on filtered probability space
$(\Omega,\mathscr{F},\mathscr{F}_{t},\mathbf{P})$ which satisfies the usual
hypotheses. We use $\mathbf{P}_{x}$ to denote the law of a process started at
$x$, and denote by $\mathbf{E}_{x}$ the associated expectation. For $a,b>0$ we
define the first passage times

$\displaystyle\tau_{a}^{-}:=\inf\\{t>0:X_{t}\leq
a\\},~{}~{}\tau_{b}^{+}:=\inf\\{t>0:X_{t}\geq b\\},$ (1.4)

and the first times of hitting zero $\tau_{0}^{-}$ and reaching infinity
$\tau_{\infty}^{+}$ by

$\displaystyle\tau_{0}^{-}:=\inf\\{t>0:X_{t}=0\\},~{}~{}\tau_{\infty}^{+}:=\lim_{n\rightarrow\infty}\tau_{n}^{+}$

with the convention $\inf\emptyset=\infty$. Then $\lim_{a\to
0+}\tau_{a}^{-}=\tau_{0}^{-}$. Let $C^{2}((0,\infty))$ denote the space of
twice continuously differentiable functions on $(0,\infty)$. We only consider
the solution to (1.3) before $\tau_{0}^{-}\wedge\tau_{\infty}^{+}$, i.e. both
zero and infinity are absorbing states for the solution. By the same argument
as in Li et al. [17, Theorem 3.1] and Ren et al. [24, Lemma A.1], we can show
that SDE (1.2) has a pathwise unique solution. Process $(X_{t})_{t\geq 0}$
becomes extinct iff $\tau_{0}^{-}<\infty$ almost surely; it explodes iff
$\tau_{\infty}^{+}<\infty$ almost surely; it stays infinite iff
$\lim_{x\rightarrow\infty}\mathbf{P}_{x}\\{\tau_{a}^{-}<t\\}=0$ for all $t>0$
and all large $a$; it comes down from infinity iff
$\lim_{a\rightarrow\infty}\lim_{x\rightarrow\infty}\mathbf{P}_{x}\\{\tau_{a}^{-}<t\\}=1$
for all $t>0$. The solution $(X_{t})_{t\geq 0}$ to (1.3) comes down from
infinity if it does not stay infinite by the arguments in [15, Theorem 1.11]
and [8, Lemma 1.2]. The following assertions are the main result of this
paper.

###### Theorem 1.1

For the solution $(X_{t})_{t\geq 0}$ to (1.3) we have

  * (i)

$(X_{t})_{t\geq 0}$ does not extinct iff $\beta\geq 1$;

  * (ii)

$(X_{t})_{t\geq 0}$ does not explode iff either $\beta>\theta+1$ or
$0\leq\theta\leq 1$;

  * (iii)

$(X_{t})_{t\geq 0}$ comes down from infinity iff $\beta>(\theta+1)\vee 2$.

###### Remark 1.2

  * (i)

In the case $\beta=\theta$, the assertions (i) and (ii) of Theorem 1.1 are
given in [15, Theorems 1.8 and 1.10] but the assertion (iii) of Theorem 1.1 is
not presented in [15].

  * (ii)

From the above example we have the following intuition. The extinction
behavior is determined by the small jumps term and the process explodes if the
big jumps term is much larger than the fluctuation of the small jumps term.
The coming down from infinity is due to the fluctuation of the small jumps
much larger than the big jumps.

The rest of the paper is arranged as follows. The results on criteria for
boundary behaviors are stated in Section 2. The proof of Theorem 1.1 is
deferred to Section 3.

##  2 Criteria for boundary behaviors

In this section we state the criteria for non-extinction, extinction, non-
explosion, explosion and staying infinite, which generalize Chen’s criteria
for the uniqueness problem of Markov jump processes. These Chen’s criteria are
first proposed in [3, 4] and can also be found in [5, Theorems 2.25 and 2.27].

Suppose that $g\in C^{2}((0,\infty))$ satisfies

$\displaystyle\sup_{u\geq v,z\geq
1}\Big{[}|g^{\prime}(u)|+|g^{\prime\prime}(u)|+|g(u+z)-g(u)|/z^{\delta}\Big{]}<\infty$
(2.1)

for any $v>0$ and some $0<\delta<1$. For $u>0$, put

$\displaystyle
Lg(u)=u^{\beta}\int_{0}^{r}[g(u+z)-g(u)-zg^{\prime}(u)]z^{-2}\mathrm{d}z+$
$\displaystyle u^{\theta}\int^{\infty}_{r}[g(u+z)-g(u)]z^{-2}\mathrm{d}z.$
(2.2)

By (1.3) and It$\rm\hat{o}$’s formula, we have

$\displaystyle g(X_{t})=$ $\displaystyle
g(x)+\int_{0}^{t}Lg(X_{s})\mathrm{d}s+\int_{0}^{t}\int_{0}^{r}\int_{0}^{X_{s-}^{\beta}}[g(X_{s-}+z)-g(X_{s-})]\tilde{M}(\mathrm{d}s,\mathrm{d}z,\mathrm{d}u)$
$\displaystyle+\int_{0}^{t}\int_{r}^{\infty}\int_{0}^{X_{s-}^{\theta}}[g(X_{s-}+z)-g(X_{s-})]\tilde{M}(\mathrm{d}s,\mathrm{d}z,\mathrm{d}u).$
(2.3)

For $0<a<b<\infty$ recall the definitions of $\tau_{a}^{-}$ and $\tau_{b}^{+}$
in (1.4), and let $\gamma_{a,b}:=\tau_{a}^{-}\wedge\tau_{b}^{+}$ and

$\displaystyle M_{t}^{g}:=g(X_{t})-g(x)-\int_{0}^{t}Lg(X_{s})\mathrm{d}s.$

Then under condition (2.1),

$\displaystyle t\mapsto M_{t\wedge\gamma_{a,b}}^{g}\mbox{ is a martingale}$
(2.4)

for all $0<a<b<\infty$. By Taylor’s formula, for any $u,z>0$ we have

$\displaystyle
g(u+z)-g(u)-zg^{\prime}(u)=z^{2}\int_{0}^{1}g^{\prime\prime}(u+zv)(1-v)\mathrm{d}v.$
(2.5)

The following criteria are on non-extinction, non-explosion and staying
infinite for SDE (1.3) and given in [20, Lemma 3.2].

###### Lemma 2.1

Let $(X_{t})_{t\geq 0}$ be the solution to SDE (1.3).

  * (i)

For any fixed $0<b<\infty$, if there exists a function $g\in
C^{2}((0,\infty))$ strictly positive on $(0,b]$ satisfying (2.1) and
$\lim_{u\rightarrow 0}g(u)=\infty$, and there is a constant $d_{b}>0$ so that
$Lg(u)\leq d_{b}g(u)$ for all $0<u\leq b$, then
$\mathbf{P}_{x}\\{\tau_{0}^{-}\geq\tau_{b}^{+}\\}=1$ for all $0<x<b$.

  * (ii)

For any fixed $0<a<\infty$, if there exists a function $g\in
C^{2}((0,\infty))$ strictly positive on $[a,\infty)$ satisfying (2.1) and
$\lim_{u\rightarrow\infty}g(u)=\infty$, and there is a constant $d_{a}>0$ so
that $Lg(u)\leq d_{a}g(u)$ for all $u>a$, then
$\mathbf{P}_{x}\\{\tau_{\infty}^{+}\geq\tau_{a}^{-}\\}=1$ for all $x>a$.

  * (iii)

If there exists a function $g\in C^{2}((0,\infty))$ strictly positive on
$[u,\infty)$ for all large $u$ satisfying (2.1) and
$\lim_{u\rightarrow\infty}g(u)=0$, and for any large $a>0$, there is a
constant $d_{a}>0$ such that $Lg(u)\leq d_{a}g(u)$ for all $u\geq a$, then
$(X_{t})_{t\geq 0}$ stays infinite.

By the arguments in the proof of [24, Proposition 2.2] one can get the
following criteria which are on the estimation of extinction, explosion and
first passage probabilities for SDE (1.3) as $a=0$, $b=\infty$ and
$0<a<b<\infty$ respectively.

###### Lemma 2.2

Let $(X_{t})_{t\geq 0}$ be the solution to SDE (1.3).

  * (i)

For fixed $0\leq a<b<\infty$, if there exists a non-increasing continuous
function $g$ on $[0,\infty)$ satisfying $g\in C^{2}((0,\infty))$ and (2.1),
and there is a constant $d_{a,b}>0$ so that $Lg(u)\geq d_{a,b}g(u)>0$ for
$a<u\leq b$, then
$\mathbf{P}_{x}\\{\tau_{a}^{-}<\tau_{b}^{+}\\}\geq[g(x)-g(b)]/g(a)$ for each
$a<x<b$.

  * (ii)

For fixed $0<a<b\leq\infty$, if there exists a function $g\in
C^{2}((0,\infty))$ satisfying (2.1) and $c:=\sup_{a\leq u<b}g(u)<\infty$, and
there is a constant $d_{a,b}>0$ so that $Lg(u)\geq d_{a,b}g(u)>0$ for $a\leq
u<b$, then $\mathbf{P}_{x}\\{\tau_{b}^{+}<\tau_{a}^{-}\\}\geq[g(x)-g(a)]/c$
for each $a<x<b$.

Proof. Let $0<\tilde{a}<\tilde{b}<\infty$. Recall (1.4) and
$\gamma_{\tilde{a},\tilde{b}}=\tau_{\tilde{a}}^{-}\wedge\tau_{\tilde{b}}^{+}$.
By (2) and (2.4), for all $g\in C^{2}((0,\infty))$ satisfying (2.1),

$\displaystyle\mathbf{E}_{x}[g(X_{t\wedge\gamma_{\tilde{a},\tilde{b}}})]=g(x)+\mathbf{E}_{x}\Big{[}\int_{0}^{t\wedge\gamma_{\tilde{a},\tilde{b}}}Lg(X_{s})\mathrm{d}s\Big{]}=g(x)+\int_{0}^{t}\mathbf{E}_{x}\Big{[}Lg(X_{s})1_{\\{s\leq\gamma_{\tilde{a},\tilde{b}}\\}}\Big{]}\mathrm{d}s$

and then by integration by parts, for any $d>0$,

$\displaystyle\mathbf{E}_{x}[g(X_{t\wedge\gamma_{\tilde{a},\tilde{b}}})]{\mbox{\rm
e}}^{-dt}$ $\displaystyle=$ $\displaystyle
g(x)-d\int_{0}^{t}\mathbf{E}_{x}[g(X_{s\wedge\gamma_{\tilde{a},\tilde{b}}}){\mbox{\rm
e}}^{-ds}]\mathrm{d}s+\int_{0}^{t}{\mbox{\rm
e}}^{-ds}\mathbf{E}_{x}[Lg(X_{s})1_{\\{s\leq\gamma_{\tilde{a},\tilde{b}}\\}}]\mathrm{d}s.$
(2.6)

For part (i), taking $d=d_{a,b}$ in (2) and using the assumption that
$Lg(u)\geq d_{a,b}g(u)>0$ for $a<u\leq b$, we have

$\displaystyle\mathbf{E}_{x}[g(X_{t\wedge\gamma_{\tilde{a},b}})]{\mbox{\rm
e}}^{-d_{a,b}t}$ $\displaystyle\geq$ $\displaystyle
g(x)-d_{a,b}\int_{0}^{t}\mathbf{E}_{x}[g(X_{s\wedge\gamma_{\tilde{a},b}})]{\mbox{\rm
e}}^{-d_{a,b}s}\mathrm{d}s+d_{a,b}\int_{0}^{t}{\mbox{\rm
e}}^{-d_{a,b}s}\mathbf{E}_{x}[g(X_{s})1_{\\{s\leq\gamma_{\tilde{a},b}\\}}]\mathrm{d}s$

for all $a<\tilde{a}<b$, which implies that

$\displaystyle
g(x)\leq\mathbf{E}_{x}[g(X_{t\wedge\gamma_{\tilde{a},b}})]{\mbox{\rm
e}}^{-d_{a,b}t}+d_{a,b}\mathbf{E}_{x}\Big{[}\int_{0}^{t}g(X_{\gamma_{\tilde{a},b}}){\mbox{\rm
e}}^{-d_{a,b}s}1_{\\{s>\gamma_{\tilde{a},b}\\}}\mathrm{d}s\Big{]}.$

Letting $t\rightarrow\infty$ in the above inequality and using the dominated
convergence we obtain

$\displaystyle g(x)\leq
d_{a,b}\mathbf{E}_{x}\Big{[}g(X_{\gamma_{\tilde{a},b}})\int_{\gamma_{\tilde{a},b}}^{\infty}{\mbox{\rm
e}}^{-d_{a,b}s}\mathrm{d}s\Big{]}=\mathbf{E}_{x}[g(X_{\gamma_{\tilde{a},b}}){\mbox{\rm
e}}^{-d_{a,b}\gamma_{\tilde{a},b}}]$

for all $a<\tilde{a}<b$. Letting $\tilde{a}\downarrow a$ we have

$\displaystyle g(x)\leq$
$\displaystyle\mathbf{E}_{x}\Big{[}g(X_{\tau_{a}^{-}\wedge\tau_{b}^{+}}){\mbox{\rm
e}}^{-d_{a,b}(\tau_{a}^{-}\wedge\tau_{b}^{+})}(1_{\\{\tau_{a}^{-}<\tau_{b}^{+}\\}}+1_{\\{\tau_{a}^{-}>\tau_{b}^{+}\\}}+1_{\\{\tau_{a}^{-}=\tau_{b}^{+}=\infty\\}})\Big{]}.$

Since $g(u)\leq g(b)$ for all $u\geq b$, then

$\displaystyle g(x)\leq
g(a)\mathbf{E}_{x}\big{[}1_{\\{\tau_{a}^{-}<\tau_{b}^{+}\\}}\big{]}+\mathbf{E}_{x}\big{[}g(X_{\tau_{b}^{+}})\big{]}\leq
g(a)\mathbf{P}_{x}\\{\tau_{a}^{-}<\tau_{b}^{+}\\}+g(b),$

which gives the assertion (i).

For part (ii), using (2) with $d=d_{a,b}$ and the assumption that $Lg(u)\geq
d_{a,b}g(u)>0$ for $a\leq u<b$ and by the similar argument one can obtain

$\displaystyle
g(x)\leq\mathbf{E}_{x}[g(X_{t\wedge\gamma_{a,\tilde{b}}})]{\mbox{\rm
e}}^{-d_{a,b}t}+d_{a,b}\mathbf{E}_{x}\Big{[}\int_{0}^{t}g(X_{\gamma_{a,\tilde{b}}}){\mbox{\rm
e}}^{-d_{a,b}s}1_{\\{s>\gamma_{a,\tilde{b}}\\}}\mathrm{d}s\Big{]}$

for all $a<\tilde{b}<b$. Letting $t\rightarrow\infty$ and then
$\tilde{b}\uparrow b$ we have

$\displaystyle g(x)\leq$
$\displaystyle\mathbf{E}_{x}\Big{[}g(X_{\tau_{a}^{-}\wedge\tau_{b}^{+}}){\mbox{\rm
e}}^{-d_{a,b}(\tau_{a}^{-}\wedge\tau_{b}^{+})}(1_{\\{\tau_{b}^{+}<\tau_{a}^{-}\\}}+1_{\\{\tau_{b}^{+}>\tau_{a}^{-}\\}})\Big{]},$

which implies

$\displaystyle g(x)\leq c\mathbf{P}_{x}\\{\tau_{b}^{+}<\tau_{a}^{-}\\}+g(a).$

This shows the assertion (ii). $\Box$

##  3 Proof of the main result

In this section we establish the proof for Theorem 1.1 by using the
generalized Chen’s criteria in Lemmas 2.1 and 2.2. For the assertions of non-
extinction and non-explosion we use Lemma 2.1(i)-(ii) and select log-logarithm
type test functions. Applying Lemma 2.2 and choosing power functions we
establish the behaviors of extinction and explosion. We use Lemma 2.1(iii)
choose a power function for proving the behavior of staying infinite. We first
study the following first passage probabilities. For the convenience of
calculation we always assume that $r=1$ in (1.3) in this section.

###### Proposition 3.1

Let $0<a<x<b<\infty$ be fixed. For the solution $(X_{t})_{t\geq 0}$ to (1.3)
we have

  * (i)

$\mathbf{P}_{x}\\{\tau_{a}^{-}<\infty\\}>0$;

  * (ii)

$\mathbf{P}_{x}\\{\tau_{b}^{+}<\infty\\}>0$.

Proof. For $\lambda,u>0$ define $g_{\lambda}(u)={\mbox{\rm e}}^{-\lambda u}$.
Then

$\displaystyle g^{\prime}_{\lambda}(u)=-\lambda{\mbox{\rm e}}^{-\lambda
u},~{}~{}g^{\prime\prime}_{\lambda}(u)=\lambda^{2}{\mbox{\rm e}}^{-\lambda
u},\qquad u>0.$

By (2.2) and (2.5) we have

$\displaystyle Lg_{\lambda}(u)=$ $\displaystyle
u^{\beta}\lambda^{2}\int^{1}_{0}\mathrm{d}z\int^{1}_{0}{\mbox{\rm
e}}^{-\lambda(u+zv)}(1-v)\mathrm{d}v+u^{\theta}\int^{\infty}_{1}[{\mbox{\rm
e}}^{-\lambda(u+z)}-{\mbox{\rm e}}^{-\lambda u}]z^{-2}\mathrm{d}z$
$\displaystyle\geq$ $\displaystyle
g_{\lambda}(u)\Big{[}2^{-1}u^{\beta}\lambda^{2}\int^{1}_{0}\mathrm{d}z\int^{\frac{1}{2}}_{0}{\mbox{\rm
e}}^{-\lambda
z/2}\mathrm{d}v-u^{\theta}\int^{\infty}_{1}z^{-2}\mathrm{d}z\Big{]}$
$\displaystyle=$ $\displaystyle
g_{\lambda}(u)\big{[}2^{-1}u^{\beta}\lambda(1-{\mbox{\rm
e}}^{-\frac{\lambda}{2}})-u^{\theta}\big{]}.$ (3.1)

Then there are constants $\lambda_{0},d>0$ so that for all $a\leq u\leq b$,

$\displaystyle Lg_{\lambda_{0}}(u)\geq
g_{\lambda_{0}}(u)\big{[}2^{-1}a^{\beta}\lambda_{0}(1-{\mbox{\rm
e}}^{-\frac{\lambda_{0}}{2}})-b^{\theta}\big{]}\geq dg_{\lambda_{0}}(u).$

From Lemma 2.2(i) it follows that

$\displaystyle\mathbf{P}_{x}\\{\tau^{-}_{a}<\infty\\}\geq\mathbf{P}_{x}\\{\tau^{-}_{a}<\tau^{+}_{b}\\}\geq{\mbox{\rm
e}}^{\lambda_{0}a}[{\mbox{\rm e}}^{-\lambda_{0}x}-{\mbox{\rm
e}}^{-\lambda_{0}b}]>0,$ (3.2)

which establishes assertion (i).

Let non-increasing function $g\in C^{2}((0,\infty))$ satisfy $g(u)={\mbox{\rm
e}}^{u}$ for $0<u\leq b+1$ and $g^{\prime}(u)=0$ for all $u\geq b+2$. Then

$\displaystyle Lg(u)=$ $\displaystyle
u^{\beta}\int_{0}^{1}\mathrm{d}z\int^{1}_{0}g^{\prime\prime}(u+zv)(1-v)\mathrm{d}v+u^{\theta}\int_{1}^{\infty}[g(u+z)-g(u)]z^{-2}\mathrm{d}z$
$\displaystyle\geq$ $\displaystyle 2^{-1}u^{\beta}g(u)\geq
2^{-1}a^{\beta}g(u),\qquad a\leq u\leq b.$

By using Lemma 2.2(ii),

$\displaystyle\mathbf{P}_{x}\\{\tau^{+}_{b}<\infty\\}\geq\mathbf{P}_{x}\\{\tau^{+}_{b}<\tau^{-}_{a}\\}\geq({\mbox{\rm
e}}^{x}-{\mbox{\rm e}}^{a})/\sup_{u>0}g(u)>0,$ (3.3)

which completes the proof. $\Box$

Now we are ready to

Proof of Theorem 1.1(i). Suppose that $\beta\geq 1$. For $n\geq 1$ let
$g_{n}\in C^{2}((0,\infty))$ be a strictly positive function so that
$g_{n}(u)=\ln\ln(6nu^{-1})$ for $0<u\leq 2n$ and

$\sup_{u>2n}[g_{n}(u)+|g_{n}^{\prime}(u)|+|g_{n}^{\prime\prime}(u)|]<\infty.$

Then for each $n\geq 1$, there is a constant $c_{n}>0$ so that

$\displaystyle g_{n}(u)\leq c_{n},\qquad u\geq 1.$ (3.4)

Moreover,

$\displaystyle
g_{n}^{\prime}(u)=-\frac{1}{u\ln(6nu^{-1})},~{}~{}g_{n}^{\prime\prime}(u)=\frac{1}{u^{2}\ln(6nu^{-1})}-\frac{1}{u^{2}[\ln(6nu^{-1})]^{2}},\qquad
0<u\leq 2n.$ (3.5)

By (2.2) and a change of variable, we obtain

$\displaystyle Lg_{n}(u)=$ $\displaystyle
u^{\beta-1}\int_{0}^{1}[g_{n}(u+uz)-g_{n}(u)-uzg_{n}^{\prime}(u)]z^{-2}\mathrm{d}z$
$\displaystyle+u^{\beta-1}\int_{1}^{u^{-1}}[g_{n}(u+uz)-g_{n}(u)-uzg_{n}^{\prime}(u)]z^{-2}\mathrm{d}z$
$\displaystyle+u^{\theta-1}\int_{u^{-1}}^{\infty}[g_{n}(u+uz)-g_{n}(u)]z^{-2}\mathrm{d}u$
$\displaystyle=:$ $\displaystyle
u^{\beta-1}[F_{1}(u)+F_{2}(u)]+u^{\theta-1}F_{3}(u),\qquad 0<u\leq 1$ (3.6)

and

$\displaystyle Lg_{n}(u)=$ $\displaystyle
u^{\beta-1}\int_{0}^{u^{-1}}[g_{n}(u+uz)-g_{n}(u)-uzg_{n}^{\prime}(u)]z^{-2}\mathrm{d}z$
$\displaystyle+u^{\theta-1}\int_{u^{-1}}^{\infty}[g_{n}(u+uz)-g_{n}(u)]z^{-2}\mathrm{d}z$
$\displaystyle=:$ $\displaystyle
u^{\beta-1}F_{4}(u)+u^{\theta-1}F_{3}(u),\qquad u>1.$ (3.7)

By (2.5) and (3.5), for $0<u\leq n$ and $0\leq z\leq 1$ we have

$\displaystyle
g_{n}(u+uz)-g_{n}(u)-uzg_{n}^{\prime}(u)=u^{2}z^{2}\int_{0}^{1}g_{n}^{\prime\prime}(u+uzv)(1-v)\mathrm{d}v$
$\displaystyle\quad\leq
u^{2}z^{2}\int_{0}^{1}\frac{1-v}{(u+uzv)^{2}\ln[6n(u+uzv)^{-1}]}\mathrm{d}v\leq\frac{z^{2}}{\ln(3nu^{-1})}\leq
z^{2}/\ln 3,$

which implies

$\displaystyle F_{1}(u_{1})\leq(\ln 3)^{-1},~{}F_{4}(u_{2})\leq(\ln
3)^{-1},\qquad 0<u_{1}\leq 1,~{}1\leq u_{2}\leq n.$ (3.8)

For $0<u\leq 1$ and $1\leq z\leq u^{-1}$, we have $0<u+uz\leq 2n$ and then
$g_{n}(u+uz)\leq g_{n}(u)$, which implies

$\displaystyle g_{n}(u+uz)-g_{n}(u)-uzg_{n}^{\prime}(u)\leq-
uzg_{n}^{\prime}(u)=\frac{z}{\ln(6nu^{-1})},\qquad 0<u\leq 1,~{}1\leq z\leq
u^{-1}$

by (3.5). It thus follows that

$\displaystyle
F_{2}(u)\leq\frac{1}{\ln(6nu^{-1})}\int_{1}^{u^{-1}}z^{-1}\mathrm{d}z=\frac{\ln
u^{-1}}{\ln(6nu^{-1})}\leq 1,\quad 0<u\leq 1.$ (3.9)

In view of (3.4),

$\displaystyle
F_{3}(u)\leq\int_{u^{-1}}^{\infty}g_{n}(u+uz)z^{-2}\mathrm{d}z\leq
c_{n}\int_{u^{-1}}^{\infty}z^{-2}\mathrm{d}z\leq uc_{n},\quad 0<u\leq n.$
(3.10)

Since $\beta\geq 1$, then combining (3.8)–(3.10) with (3)–(3) one obtains

$\displaystyle Lg_{n}(u)\leq n^{\beta-1}(1+(\ln
3)^{-1})+n^{\theta}c_{n},\qquad 0<u\leq n.$ (3.11)

Observe that $g_{n}(u)\geq g_{n}(n)=\ln\ln 6$ for all $0<u\leq n$. It then
follows from (3.11) that

$\displaystyle Lg_{n}(u)\leq[(n^{\beta-1}(1+(\ln
3)^{-1})+n^{\theta}c_{n})/\ln\ln 6]g_{n}(u),\qquad 0<u\leq n.$

Now using Lemma 2.1(i) we obtain
$\mathbf{P}_{x}\\{\tau_{0}^{-}\geq\tau_{n}^{+}\\}=1$ for all $0<x<n$. Letting
$n\rightarrow\infty$ we have
$\mathbf{P}_{x}\\{\tau_{0}^{-}\geq\tau_{\infty}^{+}\\}=1$ for all $x>0$. Since
the process is defined before the first time of hitting zero or explosion,
$\mathbf{P}_{x}\\{\tau_{0}^{-}=\infty$ or $\tau_{\infty}^{+}=\infty\\}=1$.
Then $\mathbf{P}_{x}\\{\tau_{0}^{-}=\infty\\}=1$ for all $x>0$.

In the following we assume that $0\leq\beta<1$ and $0<\rho<1-\beta$. Let
$0<c<1$ be a small enough constant so that

$\displaystyle\tilde{c}:=\rho(1-\rho)2^{\rho-3}c^{\beta+\rho-1}-(1-\rho)^{-1}>0.$
(3.12)

Define the function $g$ on $[0,\infty)$ by $g(u)=c^{\rho}-u^{\rho}$. Then

$\displaystyle g^{\prime}(u)=-\rho
u^{\rho-1},~{}~{}g^{\prime\prime}(u)=\rho(1-\rho)u^{\rho-2},\qquad u>0.$

By (2.2) and (2.5) we have

$\displaystyle Lg(u)=$
$\displaystyle\rho(1-\rho)u^{\beta}\int_{0}^{1}\mathrm{d}z\int_{0}^{1}(u+vz)^{\rho-2}(1-v)\mathrm{d}v-u^{\theta}\int_{1}^{\infty}[(u+z)^{\rho}-u^{\rho}]z^{-2}\mathrm{d}z$
$\displaystyle=:$ $\displaystyle G_{1}(u)-G_{2}(u).$ (3.13)

By a change of variable we have

$\displaystyle G_{1}(u)=$
$\displaystyle\rho(1-\rho)u^{\rho+\beta-1}\int_{0}^{u^{-1}}\mathrm{d}z\int_{0}^{1}(1+zv)^{\rho-2}(1-v)\mathrm{d}v$
$\displaystyle\geq$
$\displaystyle\rho(1-\rho)u^{\rho+\beta-1}\int_{0}^{1}\mathrm{d}z\int_{0}^{1}(1+zv)^{\rho-2}(1-v)\mathrm{d}v$
$\displaystyle\geq$
$\displaystyle\rho(1-\rho)2^{\rho-3}c^{\rho+\beta-1},\qquad 0<u\leq c.$ (3.14)

Since $(u+z)^{\rho}-u^{\rho}\leq z^{\rho}$ for all $u,z\geq 0$, then

$\displaystyle G_{2}(u)\leq
u^{\theta}\int_{1}^{\infty}z^{\rho-2}\mathrm{d}z\leq(1-\rho)^{-1}.$ (3.15)

Combining (3.12)–(3.15) we obtain

$\displaystyle
Lg(u)\geq\tilde{c}=\tilde{c}c^{-\rho}c^{\rho}\geq\tilde{c}c^{-\rho}g(u),\qquad
0<u\leq c.$

By using Lemma 2.2(i) one gets that for $0<x<c$,

$\displaystyle\mathbf{P}_{x}\\{\tau_{0}^{-}<\infty\\}\geq\mathbf{P}_{x}\\{\tau_{0}^{-}<\tau_{c}^{+}\\}\geq[g(x)-g(c)]/g(0)=1-(x/c)^{\rho}.$
(3.16)

By strong Markov property, (3.16) and Proposition 3.1(i), for all $x\geq c$,

$\displaystyle\mathbf{P}_{x}\\{\tau_{0}^{-}<\infty\\}$
$\displaystyle\geq\mathbf{P}_{x}\\{\tau_{0}^{-}\circ\theta_{\tau_{c/2}^{-}}<\infty,\tau_{c/2}^{-}<\infty\\}$
$\displaystyle=\mathbf{E}_{x}\Big{[}\mathbf{E}_{x}\big{[}1_{\\{\tau_{0}^{-}\circ\theta_{\tau_{c/2}^{-}}<\infty\\}}1_{\\{\tau_{c/2}^{-}<\infty\\}}|\mathscr{F}_{\tau_{c/2}^{-}}\big{]}\Big{]}$
$\displaystyle=\mathbf{E}_{x}\Big{[}\mathbf{P}_{c/2}\\{\tau_{0}^{-}<\infty\\}1_{\\{\tau_{c/2}^{-}<\infty\\}}\Big{]}$
$\displaystyle\geq(1-2^{-\rho})\mathbf{P}_{x}\\{\tau_{c/2}^{-}<\infty\\}>0,$

where $\theta_{t}$ denotes the usual shift operator. This concludes the
assertion. $\Box$

Proof of Theorem 1.1(ii). Suppose that $\beta>\theta+1$ or $\theta\leq 1$. For
each $n\geq 1$ let $g_{n}\in C^{2}((0,\infty))$ be a strictly positive
function so that $g_{n}(u)=\ln\ln(6nu)$ for $u\geq n^{-1}$. It then follows
that

$\displaystyle
g_{n}^{\prime}(u)=\frac{1}{u\ln(6nu)},~{}~{}g_{n}^{\prime\prime}(u)=-\frac{1}{u^{2}\ln(6nu)}-\frac{1}{u^{2}(\ln(6nu))^{2}},\qquad
u\geq n^{-1}.$ (3.17)

By (2.2) and a change of variable, we have

$\displaystyle Lg_{n}(u)=$ $\displaystyle
u^{\beta-1}\int_{0}^{\frac{1}{u}}[g_{n}(u+uz)-g_{n}(u)-uzg_{n}^{\prime}(u)]z^{-2}\mathrm{d}z+u^{\theta-1}\int_{\frac{1}{u}}^{\infty}[g_{n}(u+uz)-g_{n}(u)]z^{-2}\mathrm{d}z$
$\displaystyle=:$ $\displaystyle u^{\beta-1}H_{1}(u)+u^{\theta-1}H_{2}(u).$
(3.18)

By (2.5) and (3.17), it is elementary to see that we have

$\displaystyle
g_{n}(u+uz)-g_{n}(u)-uzg_{n}^{\prime}(u)\leq-u^{2}z^{2}\int_{0}^{1}\frac{1-v}{(u+uzv)^{2}\ln(6n(u+uzv))}\mathrm{d}v\leq-\frac{z^{2}}{8\ln(12nu)}$

for all $u\geq n^{-1}$ and $0<z\leq 1$, which implies

$\displaystyle H_{1}(u)\leq-\frac{u^{\beta-2}}{8\ln(12nu)},\quad u\geq
n^{-1}.$ (3.19)

Similarly, for $u\geq n^{-1}$ and $z\geq 0$ we have

$\displaystyle
g_{n}(u+uz)-g_{n}(u)=uz\int_{0}^{1}g_{n}^{\prime}(u+uzv)\mathrm{d}v$
$\displaystyle\quad\leq[\ln(6nu)]^{-1}\int_{0}^{1}\frac{z}{1+zv}\mathrm{d}v=[\ln(6nu)]^{-1}\ln(1+z).$
(3.20)

Observe that

$\displaystyle\lim_{u\to
0}\frac{\int_{u}^{\infty}\frac{\ln(1+z)}{z^{2}}\mathrm{d}z}{1+\ln
u^{-1}}=\lim_{u\to 0}\frac{\frac{\ln(1+u)}{u^{2}}}{u^{-1}}=1,$

which implies

$\displaystyle\int_{u^{-1}}^{\infty}\frac{\ln(1+z)}{z^{2}}\mathrm{d}z\leq
c_{0}(1+\ln u),\qquad u\geq 1$

for some constant $c_{0}>0$. From (3) it thus follows that

$\displaystyle
H_{2}(u)\leq[\ln(6nu)]^{-1}\int_{u^{-1}}^{\infty}\frac{\ln(1+z)}{z^{2}}\mathrm{d}z\leq\tilde{c}_{0},~{}~{}n^{-1}\leq
u\leq 1$ (3.21)

with $\tilde{c}_{0}:=(\ln
6)^{-1}\int_{1}^{\infty}\frac{\ln(1+z)}{z^{2}}\mathrm{d}z<\infty$ and

$\displaystyle
H_{2}(u)\leq[\ln(6nu)]^{-1}\int_{u^{-1}}^{\infty}\frac{\ln(1+z)}{z^{2}}\mathrm{d}z\leq\frac{c_{0}(1+\ln
u)}{\ln(6n)+\ln u}\leq c_{0},\qquad u\geq 1.$ (3.22)

Since $g_{n}(u)\geq\ln\ln 6$ for all $u\geq n^{-1}$, then combining (3) with
(3.19) and (3.21)–(3.22) one obtains

$\displaystyle Lg_{n}(u)\leq$
$\displaystyle-u^{\beta-2}[8\ln(12nu)]^{-1}+u^{\theta-1}(c_{0}+\tilde{c}_{0})$
$\displaystyle=$ $\displaystyle
u^{\theta-1}\big{[}c_{0}+\tilde{c}_{0}-u^{\beta-\theta-1}[8\ln(12nu)]^{-1}\big{]}$
$\displaystyle\leq$ $\displaystyle c_{n}\leq c_{n}(\ln\ln
6)^{-1}g_{n}(u),\qquad u\geq n^{-1}.$

for constants $c_{n}>0$ as $\beta>\theta+1$. Similarly, as $0\leq\theta\leq
1$,

$\displaystyle Lg_{n}(u)\leq u^{\theta-1}(c_{0}+\tilde{c}_{0})\leq
n^{1-\theta}(c_{0}+\tilde{c}_{0})(\ln\ln 6)^{-1}g_{n}(u),\qquad u\geq n^{-1}.$

Now using Lemma 2.1(ii) we obtain
$\mathbf{P}_{x}\\{\tau_{\infty}^{+}\geq\tau_{n^{-1}}^{-}\\}=1$ for all
$x>n^{-1}$. Letting $n\rightarrow\infty$ we have
$\mathbf{P}_{x}\\{\tau_{\infty}^{+}\geq\tau_{0}^{-}\\}=1$ for all $x>0$. It
then follows from the fact of the solution to SDE (1.3) defined before
$\tau_{0}^{-}\wedge\tau_{\infty}^{+}$ that
$\mathbf{P}_{x}\\{\tau_{\infty}^{+}=\infty\\}=1$ for all $x>0$.

In the following we assume that $\theta+1\geq\beta$ and $\theta>1$. Let
$\rho=\theta-1>0$. For $u>0$ let $g(u)=1-u^{-\rho}$. Thus

$\displaystyle g^{\prime}(u)=\rho
u^{-\rho-1},~{}g^{\prime\prime}(u)=-\rho(\rho+1)u^{-\rho-2},\qquad u>0.$

It then follows from (2.2) and (2.5) that

$\displaystyle Lg(u)=$
$\displaystyle-\rho(\rho+1)u^{\beta}\int_{0}^{1}\mathrm{d}z\int_{0}^{1}(u+vz)^{-\rho-2}(1-v)\mathrm{d}v+u^{\theta}\int_{1}^{\infty}[u^{-\rho}-(u+z)^{-\rho}]z^{-2}\mathrm{d}z$
$\displaystyle=:$ $\displaystyle I_{1}(u)+I_{2}(u).$ (3.23)

By a change of variable, for all $u\geq 1$,

$\displaystyle I_{1}(u)=$
$\displaystyle-\rho(1+\rho)u^{\beta-\theta}\int_{0}^{u^{-1}}\mathrm{d}z\int_{0}^{1}(1+zv)^{-\rho-2}(1-v)\mathrm{d}v$
$\displaystyle\geq$
$\displaystyle-\rho(1+\rho)u^{\beta-\theta-1}\geq-\rho(1+\rho)$ (3.24)

since $\beta-\theta-1\leq 0$ and

$\displaystyle
I_{2}(u)=u^{\theta-\rho-1}\int_{u^{-1}}^{\infty}[1-(1+z)^{-\rho}]z^{-2}\mathrm{d}z=\int_{u^{-1}}^{\infty}[1-(1+z)^{-\rho}]z^{-2}\mathrm{d}z=:c(u).$
(3.25)

Observe that

$\displaystyle\lim_{u\to
0}\frac{\int_{u}^{\infty}[1-(1+z)^{-\rho}]z^{-2}\mathrm{d}z}{\rho\ln
u^{-1}}=\lim_{u\to 0}\frac{[1-(1+u)^{-\rho}]u^{-2}}{\rho u^{-1}}=1,$ (3.26)

which implies that $c(u_{0})-\rho(1+\rho)>0$ for some large $u_{0}>1$. Then
combining (3) and (3.25) with (3) we obtain

$\displaystyle Lg(u)\geq
c(u_{0})-\rho(1+\rho)\geq[c(u_{0})-\rho(1+\rho)]g(u),\qquad u\geq u_{0},$

which gives

$\displaystyle\mathbf{P}_{x}\\{\tau_{\infty}^{+}<\infty\\}\geq\mathbf{P}_{x}\\{\tau_{\infty}^{+}<\tau_{u_{0}}^{-}\\}\geq
u_{0}^{-\rho}-x^{-\rho}$ (3.27)

for all $x>u_{0}$ by Lemma 2.2(ii). By strong Markov property, (3.27) and
Proposition 3.1(ii), for all $0<x\leq u_{0}$,

$\displaystyle\mathbf{P}_{x}\\{\tau_{\infty}^{+}<\infty\\}$
$\displaystyle\geq\mathbf{P}_{x}\\{\tau_{\infty}^{+}\circ\theta_{\tau_{2u_{0}}^{+}}<\infty,\tau_{2u_{0}}^{+}<\infty\\}$
$\displaystyle=\mathbf{E}_{x}\Big{[}\mathbf{E}_{x}\big{[}1_{\\{\tau_{\infty}^{+}\circ\theta_{\tau_{2u_{0}}^{+}}<\infty\\}}1_{\\{\tau_{2u_{0}}^{+}<\infty\\}}|\mathscr{F}_{\tau_{2u_{0}}^{+}}\big{]}\Big{]}$
$\displaystyle=\mathbf{E}_{x}\Big{[}\mathbf{P}_{2u_{0}}\\{\tau_{\infty}^{+}<\infty\\}1_{\\{\tau_{2u_{0}}^{+}<\infty\\}}\Big{]}$
$\displaystyle\geq
u_{0}^{-\rho}(1-2^{-\rho})\mathbf{P}_{x}\\{\tau_{2u_{0}}^{+}<\infty\\}>0,$

which ends the proof. $\Box$

###### Remark 3.2

At the beginning of the proofs of Theorem 1.1(i)-(ii), if the logarithmic type
test function is selected as in Ma et al. [20], the estimations of (3.9) and
(3.22) fail to hold.

Proof of Theorem 1.1(iii). If $\beta>\theta+1$ and $\beta\geq 2$, then
$(X_{t})_{t\geq 0}$ comes down from infinity by [20, Theorem 2.1(iv)] and
Theorem 1.1(ii). In the following we assume that $\beta\leq(\theta+1)\vee 2$.
For $\rho>0$ and $u>0$ let $g(u)=u^{-\rho}$. Then

$\displaystyle g^{\prime}(u)=-\rho
u^{-\rho-1},~{}g^{\prime\prime}(u)=\rho(\rho+1)u^{-\rho-2},\qquad u>0.$

It then follows from (2.2) and (2.5) that

$\displaystyle Lg(u)=$ $\displaystyle
u^{\beta-1-\rho}\rho(\rho+1)\int_{0}^{u^{-1}}\mathrm{d}z\int_{0}^{1}(1+zv)^{-\rho-2}(1-v)\mathrm{d}v$
$\displaystyle-u^{\theta-1-\rho}\int_{u^{-1}}^{\infty}[1-(1+z)^{-\rho}]z^{-2}\mathrm{d}z,\qquad
u>0.$ (3.28)

By (3.26),

$\displaystyle\int_{u^{-1}}^{\infty}[1-(1+z)^{-\rho}]z^{-2}\mathrm{d}z\geq
2^{-1}\rho\ln u,\qquad u\geq\tilde{c}$ (3.29)

for some constant $\tilde{c}>1$. From (3) and (3.29) it then follows that

$\displaystyle Lg(u)\leq u^{\beta-2-\rho}\rho(\rho+1)-2^{-1}\rho
u^{\theta-1-\rho}\ln u=\rho
u^{\theta-1-\rho}[(\rho+1)u^{\beta-1-\theta}-2^{-1}\ln u]\leq 0,~{}u\geq c$

for some constant $c\geq\tilde{c}$ as $\beta\leq\theta+1$ and

$\displaystyle
Lg(u)\leq\rho(\rho+1)u^{\beta-2-\rho}\leq\rho(\rho+1)g(u),\qquad u\geq 1$

as $\beta\leq 2$. It thus follows that $(X_{t})_{t\geq 0}$ stays infinite by
Lemma 2.1(iii). $\Box$

Acknowledgements. We are grateful to Pei-Sen Li, Zenghu Li and Xiaowen Zhou
for their careful reading and insightful suggestions on the manuscript. This
work was supported by NSFC (No. 12061004), NSF of Ningxia (No. 2021AAC02018)
and Major research project for North Minzu University (No. ZDZX201902).

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# Indoor Localization for IoT Using Adaptive Feature Selection: A Cascaded
Machine Learning Approach

Mohamed I. AlHajri∗,‡, Nazar T. Ali†, Raed M. Shubair‡
∗Electrical Engineering and Computer Science, Massachusetts Institute of
Technology, USA
†Electrical and Computer Engineering, Khalifa University, UAE
‡Research Laboratory of Electronics, Massachusetts Institute of Technology,
USA
Emails: $\\{$malhajri<EMAIL_ADDRESS><EMAIL_ADDRESS>
# Indoor Localization for IoT Using Adaptive Feature Selection:
A Cascaded Machine Learning Approach

Mohamed I. AlHajri∗,‡, Nazar T. Ali†, Raed M. Shubair‡
∗Electrical Engineering and Computer Science, Massachusetts Institute of
Technology, USA
†Electrical and Computer Engineering, Khalifa University, UAE
‡Research Laboratory of Electronics, Massachusetts Institute of Technology,
USA
Emails: $\\{$malhajri<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Evolving Internet-of-Things (IoT) applications often require the use of
sensor-based indoor tracking and positioning, for which the performance is
significantly improved by identifying the type of the surrounding indoor
environment. This identification is of high importance since it leads to
higher localization accuracy. This paper presents a novel method based on a
cascaded two-stage machine learning approach for highly-accurate and robust
localization in indoor environments using adaptive selection and combination
of RF features. In the proposed method, machine learning is first used to
identify the type of the surrounding indoor environment. Then, in the second
stage, machine learning is employed to identify the most appropriate selection
and combination of RF features that yield the highest localization accuracy.
Analysis is based on $k$-Nearest Neighbor ($k$-NN) machine learning algorithm
applied on a real dataset generated from practical measurements of the RF
signal in realistic indoor environments. Received Signal Strength, Channel
Transfer Function, and Frequency Coherence Function are the primary RF
features being explored and combined. Numerical investigations demonstrate
that prediction based on the concatenation of primary RF features enhanced
significantly as the localization accuracy improved by at least 50% to more
than 70%.

###### Index Terms:

Indoor Localization, IoT, Machine Learning, $k$-Nearest Neighbor, Received
Signal Strength, Channel Transfer Function, Frequency Coherence Function.

##  I Introduction

The advent of Internet-of-Things (IoT) has revolutionized the field of
telecommunications opening the door for unprecedented applications such as [1,
2, 3, 4, 5, 6, 7, 8, 9]. These applications are enabled through smart sensors
that cooperate efficiently to provide the desired service. The efficient
deployment of IoT-based sensors primarily depends on accurate localization and
adjustment of sensor power consumption according to the radio frequency (RF)
propagation channel which is dictated by the type of the surrounding indoor
environment.

The complexity of the indoor wireless channel, due to the severe multipath
effect and absence of Line-of-Sight (LOS) path, emphasizes the need for a
sophisticated yet efficient approach for indoor localization. Location
fingerprinting based on various machine learning algorithms has been reported
in the literature using different RF features [10, 11, 12, 13, 14, 15, 16, 17,
18, 19, 20, 21, 22, 23]. These have included Received Signal Strength (RSS),
Channel Transfer Function (CTF), and Frequency Coherence Function (FCF).
Developing a machine learning approach that is based on a hybrid combination
of such primary RF features can greatly enhance localization accuracy, since
more information will be available to the algorithm. Such an approach does not
exist in the literature.

This paper presents a novel method based on a cascaded two-stage machine
learning approach for highly-accurate and robust localization in indoor
environments using adaptive selection and combination of RF features. In the
proposed method, and based on the authors prior work [24, 25], machine
learning is first used to identify the type of the indoor environment based on
real data measurements of the RF signal in different indoor scenarios. Then,
in the second stage, machine learning is employed to identify the most
appropriate selection and combination of RF features that yield the highest
localization accuracy. The most appropriate selection and combination of RF
features of the second stage is dependent on the type of indoor environment,
which is being identified from the first stage. Analysis is based on
$k$-Nearest Neighbor ($k$-NN) machine learning algorithm applied on a real
dataset generated from practical measurements of the RF signal in realistic
indoor environments. Received Signal Strength, Channel Transfer Function, and
Frequency Coherence Function are the primary RF features being explored and
combined.

##  II Signatures of Indoor Environment

###  II-A Primary RF Features

In a multipath rich indoor environment, the received signal $Y(f)$ contains
replicas of the RF transmitted signal $X(f)$. The ratio of $Y(f)$ to $X(f)$
defines the CTF, $H(f)$, which contains the multipath effects of the wireless
channel. Hence, $H(f)$ can be represented as the superposition of the gains
associated with the multipath components as follows [26]:

$H(f)=\sum\limits^{L}_{l=1}a_{l}\exp{[-j(2\pi f\tau_{l}-\theta_{l})]}$ (1)

where $a_{l}$, $\tau_{l}$, and $\theta_{l}$ are the amplitude, delay, and
phase of the $l^{th}$ multipath component; $L$ is the total number of
multipath components; and $f$ is a frequency within bandwidth of operation of
the channel. The analysis in this paper considers an indoor scenario with a
high Signal-to-Noise Ratio (SNR)[27, 28]. The CTF, $H(f)$, is considered as an
RF feature that would be distinctively unique for every spatial position
within the indoor environment. Under frequency selective fading, this RF
signature becomes more sensitive to channel variations due to the rapid
fluctuations of the gains associated with its multipath components.

The complex autocorrelation of CTF, $H(f)$, defines another channel metric
known as FCF, $R(f)$, given by [18]:

$R(f)=\int\limits^{\infty}_{-\infty}H(\hat{f})H^{*}(\hat{f}+f)~{}d\hat{f}$ (2)

FCF given in (2) represents the frequency domain coherence of the radio
channel. It is interpreted as the autocorrelation of the CTF, $H(f)$, due to
different frequency shifts. FCF is known for its slow varying nature in the
spatial domain[18], which makes it a strong candidate as an RF signature.

###  II-B Hybrid RF Features

Characterizing the type of an indoor environment by a machine learning
algorithm becomes a more accurate process when the algorithm makes decisions
based on concatenation of primary RF features of the same environment. In such
a way, the accuracy of prediction increases due to the wealth of information
available to the algorithm from the multi-dimensional dataset constructed from
the concatenation of the combined primary RF features of the same environment.
Hence, the approach is based on hybrid features. The investigations were
carried out using multi-dimensional information generated from various
possible combinations of all or subset of RSS, CTF, and FCF which form the
hybrid features. Results, presented later in section 5, based on these hybrid
features show indeed a substantial improvement in the algorithm prediction
resulting in a significant enhancement in environment separability and
localization accuracy.

Uniform Manifold Approximation and Projection (UMAP) is a non-linear
dimensionality reduction technique that is well-suited for embedding high-
dimensional data for visualization in a low-dimensional space [29]. In this
paper, we mapped the 5-D hybrid RF feature vector (RSS + CTF + FCF) into 2-D,
with results visualized in Fig. 1. It should be noted that the hybrid RF
feature vector (RSS + CTF + FCF) is 5-D since it is the concatenation of the
real-valued RSS followed by the real and imaginary parts of CTF and FCF,
respectively. Fig. 1 indicates that the resulting 2-D mapped hybrid RF
features vector of the Sports Hall (open space) is closest to that of the Main
Lobby (low cluttered). Fig. 1 also indicates that the 2-D mapped hybrid RF
features vector of the Narrow Corridor (medium cluttered) is surrounded by
that from the Main lobby (low cluttered) and Lab (highly cluttered). The 2-D
embedding of the mapped RF features in Fig. 1 shows that further insights can
be gained from the UMAP visualization to confirm the advantage of using the
concept of hybrid RF features.

Figure 1: Using hybrid RF features in UMAP algorithm allowed different
environments to be classified into distinctive clusters.

##  III Dataset Generation from Real Measurements

A dataset has been generated through a measurement campaign in which readings
of the RF signal were taken in realistic indoor environments of four different
types within Khalifa University campus, Sharjah, UAE. The frequency domain
CTF, $H(f)$, was obtained for four indoor environments: highly cluttered
(Laboratory), medium cluttered (Narrow Corridor), low cluttered (Lobby), and
open space (Sports Hall).

The measurement setup was similar to that used in [25] has been utilized. The
dataset generated for each environment consists of 196 grid points $\times$ 10
iterations/(grid point). The dataset has been divided into 75% training
dataset and 25% testing dataset. The dataset has been made available as open-
access online so that it can be utilized for further explorations. It can be
accessed online as per details in [30, 17].

##  IV Machine Learning Algorithm

The machine learning algorithm adopted in this paper is $k$-Nearest Neighbor
($k$-NN) which is a non-parametric method used for classification [31, 32]. In
this algorithm, the $k$ neighbors of each hybrid RF features vector will
determine the type of environment. The $k$-neighbors are associated with the
shortest $k$ Euclidean distances, which are defined as follows [10]:

$d(\mathbf{s}_{m},\mathbf{s}_{n})=\|\mathbf{s}_{m}-\mathbf{s}_{n}\|_{2}=\sqrt{\sum_{i=1}^{K}\left[s_{m}(i)-s_{n}(i)\right]^{2}}$
(3)

where $K$ is the length of the RF features vector, $\mathbf{s}_{m}$ and
$\mathbf{s}_{n}$ are the vectors of hybrid RF features at positions $m$ and
$n$, respectively.

Figure 2: Confusion matrix using $k$-NN algorithm ($k$ = 1) based on hybrid RF
features CTF + FCF.

##  V Cascaded Machine Learning Approach

###  V-A Machine Learning for Indoor Environment Identification

This section is based on prior work by the authors [25, 24] where a machine
learning approach was used for indoor environment classification based on real
measurements of the RF signal. Extensive numerical investigations [25, 24]
showed that a machine learning algorithm using $k$-NN method, utilizing a
hybrid combination of CTF and FCF, outperforms other methods such as Decision
Tree [33] and Support Vector Machine (SVM) [34] in identifying the type of the
indoor environment with a classification accuracy of 99.3%. The required time
was found to be less than 10$\mu$s, which verifies that the adopted $k$-NN
machine learning algorithm is a successful candidate for real-time IoT
deployment scenarios. The accuracy of the adopted technique is represented in
terms of the confusion matrix, shown in Fig. 2, for which the diagonal
elements represent the percentage of accurate prediction for each type of
environment. The off-diagonal elements of the confusion matrix represent the
percentage of misclassification. The results in Fig. 2 have been obtained for
490 test cases per environment.

TABLE I: Comparison of localization distance error, RMSE, for different types
of indoor environments using $k$-NN algorithm ($k$ = 1) based on various
primary and hybrid RF features.

| Primary RF Features | Hybrid RF Features |
---|---|---|---
| RSS | CTF | FCF | RSS + CTF | RSS + FCF | CTF + FCF | RSS + CTF + FCF | $\alpha$
A: Lab | 109.19 cm | 39.75 cm | 49.56 cm | 39.99 cm | 100.25 cm | 39.68 cm | 39.9 cm | 63.7%
B: Narrow Corridor | 105.3 cm | 30.53 cm | 45.89 cm | 31.16 cm | 97.94 cm | 30.49 cm | 30.71 cm | 71.0%
C: Lobby | 106.25 cm | 27.52 cm | 38.9 cm | 27.84 cm | 94.98 cm | 27.18 cm | 27.82 cm | 74.4%
D: Sports Hall | 111.3 cm | 63.33 cm | 55.2 cm | 63.91 cm | 103.52 cm | 63.32 cm | 63.9 cm | 50.4%

###  V-B Machine Learning for Localization Position Estimation

Studies in the literature indicate that a substantial gain in accuracy is
achieved by improving the construction of the RF signature, rather than the
use of more complex algorithms [13, 35, 36, 37]. As a result, in the second
stage and after the type of indoor environment has been identified, a specific
selection and combination of RF feature is utilized based on the identified
indoor environment. We have adopted the $k$-NN algorithm again in the second
stage in order to estimate the sensor position by comparing the testing RF
feature to the training database of RF features, with the purpose of finding
the best matching entry.

Table I shows the localization distance error, RMSE, using various primary and
hybrid RF features. It can be seen from that the choice of the best RF
signature is dependent on the type of the indoor environment. In the case of
the Lab, Narrow Corridor, and Lobby, a hybrid RF feature combining CTF + FCF
produced the least error RMSE, whereas for the case of the Sports Hall, a
primary RF feature based on FCF has the best performance. These findings
clearly indicate the importance of incorporating information on the type of
the indoor environment for improving location estimation.

In order to quantitatively assess the improvement in localization estimation
when the proposed method is used, we define $\alpha$ (last column in Table I)
to represent the percentage reduction in localization distance error, RMSE,
when the proposed method is used compared to the baseline method which uses
RSS as the RF feature:

$\alpha=\left(\frac{\text{RMSE}_{\text{RSS}}-\text{RMSE}_{\beta}}{\text{RMSE}_{\text{RSS}}}\right)\times
100\%$ (4)

where $\text{RMSE}_{\text{RSS}}$ is the distance error due to RSS considered
to be the baseline method, and $\text{RMSE}_{\beta}$ is the distance error due
to the RF feature that produces the lowest RMSE. It is evident from the
computed values of $\alpha$ as given in the last column in Table I, that the
proposed method significantly reduces the percentage error in localization by
at least 50% to more than 70%.

We have also studied the effect of varying the number of neighbors, $k$, on
the performance of different RF signatures in order to decide, for every type
of environment, the best combination of primary features that yield the
highest localization accuracy. The results in Fig. 3 \- 5 indicate that
increasing $k$ reduces the accuracy since the localization distance error,
RMSE, increases. This is attributed to the fact that increasing the number of
neighbors, $k$, means that the algorithm attempts to find the precise location
of the sensor node within a neighboring area that is large due to having many
neighbors, $k$. In such case, the local information available to the algorithm
reduces since prediction is made based on a larger number of neighbors,
causing a deterioration in the performance of the algorithm.

An interesting observation is that all RF features, both primary and hybrid,
produce a steady performance when $k$ increases beyond the range $40\leq k\leq
50$. Another interesting observation is that primary features produce the best
performance when the indoor environment has a fewer number of multipath
components, as the case for indoor open space of the Sports Hall in Fig. 6,
where the best performance is obtained using only FCF. On the other hand, and
as evident from Figs. 3 \- 5 which correspond to indoor environments having a
large number of multipath components, the best performance is obtained using
hybrid features CTF + FCF. Such interesting finding reveals the fact that
using the proposed hybrid RF features concept becomes more necessary in a
cluttered environment having a large number of multipath components, a
scenario that is encountered in many indoor environments that involve narrow
corridors or closer boundaries.

The reduction in the localization distance error, RMSE, using the proposed
method is verified by comparing it to other methods in the literature. It is
evident from Table II that the the proposed method yields the lowest
localization distance error, RMSE, compared to the other existing methods in
the literature such as those by Bahl et al. [10], Li et al. [38], Alsindi et
al. [28],[18].

TABLE II: Comparison of localization distance error, RMSE, using the proposed
method and other methods in the literature.

| [10] | [38] | [28] | [18] |  | Proposed
---
A: Lab | 109.19 cm | 108.76 cm | 49.56 cm | 39.75 cm | 39.68 cm
B: Narrow Corridor | 105.3 cm | 104.3 cm | 45.89 cm | 30.53 cm | 30.49 cm
C: Lobby | 106.25 cm | 105.1 cm | 38.9 cm | 27.52 cm | 27.18 cm
D: Sports Hall | 111.3 cm | 110.9 cm | 55.2 cm | 63.33 cm | 55.2 cm
Average Performance | 108.0 cm | 107.3 cm | 47.4 cm | 40.3 cm | 38.1 cm

Figure 3: Comparison of RMSE using $k$-NN algorithm ($k$=1) based on various
primary and hybrid RF features for Lab (highly cluttered). Figure 4:
Comparison of RMSE using $k$-NN algorithm ($k$=1) based on various primary and
hybrid RF features for Narrow Corridor (medium cluttered). Figure 5:
Comparison of RMSE using $k$-NN algorithm ($k$=1) based on various primary and
hybrid RF features for Lobby (low cluttered). Figure 6: Comparison of RMSE
using $k$-NN algorithm ($k$=1) based on various primary and hybrid RF features
for Sports Hall (open space).

##  VI Conclusion

This paper developed a cascaded two-stage machine learning approach for
highly-accurate and robust indoor localization for IoT using the concept of
hybrid RF features. In the first stage, $k$-NN algorithm was used to identify
the type of indoor environment. In the second stage, $k$-NN algorithm was
employed to identify the most appropriate selection and combination of RF
features that yield the highest localization accuracy, based on the identified
indoor environment. Investigations were carried out using a real dataset
generated from practical measurements of the RF signal in realistic indoor
environments. Results show that the prediction of the algorithm based on the
concatenation of primary RF features enhanced significantly as the
localization accuracy improved by at least 50% to more than 70%. The
significant improvement in localization accuracy attained using the proposed
two-stage machine learning method is attributed to the cascaded performance
gains due to the identification of the type of environment and the
concatenation of primary RF features.

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