# On a skew stable Lévy process

Alexander Iksanov  A.Iksanov Taras Shevchenko National University of Kyiv,
Ukraine; e-mail<EMAIL_ADDRESS>and  Andrey Pilipenko  A. Pilipenko
Institute of Mathematics of Ukrainian National Academy of Sciences; Igor
Sikorsky Kyiv Polytechnic Institute; e-mail<EMAIL_ADDRESS>
###### Abstract.

The skew Brownian motion is a strong Markov process which behaves like a
Brownian motion until hitting zero and exhibits an asymmetry at zero. We
address the following question: what is a natural counterpart of the skew
Brownian motion in the situation that the noise is a stable Lévy process with
finite mean and infinite variance. We define a skew stable Lévy process $X$ as
the limit of a sequence of stable Lévy processes which are perturbed at zero.
We point out a formula for the resolvent of $X$ and show that $X$ is a
solution to a stochastic differential equation with a local time. Also, we
provide a representation of $X$ in terms of Itô‘s excursion theory.

###### Key words and phrases:

Excursion theory; functional limit theorem; resolvent; recurrent extension of
a Markov process; skew Brownian motion; stable Lévy process; stochastic
differential equation with a local time

###### 2020 Mathematics Subject Classification:

Primary: 60F17, 60J35; Secondary: 60J50

##  1\. Introduction and main results

A skew Brownian motion (SBM) with parameter $p\in[0,1]$ appears in the book
[16] as a diffusion that behaves like a Brownian motion until hitting $0$ and
whose excursions select the positive or negative sign with probabilities $p$
and $1-p$, respectively. There are numerous alternative descriptions and
generalizations of the SBM. Portenko constructed an SBM as a solution to a
stochastic differential equation (SDE) with a generalized drift or a diffusion
with a semipermeable membrane [31, 32]. Harrison and Shepp [13] proved that an
SBM is a strong solution to an SDE with a local time drift, which can be
thought of as Dirac’s delta function drift in Portenko’s framework. Harrison
and Shepp also constructed an SBM as an appropriate limit of random walks
perturbed at $0$, see also [14, 23, 24, 28] and references therein. Walsh [35]
described an SBM with the help of a martingale problem, see also [2]. The
detailed review on the SBM is presented in [20].

In this paper we address the following question: what is a natural analogue of
the SBM in the case of a stable noise? For $\alpha\in(1,2)$, denote by
$U_{\alpha}:=(U_{\alpha}(t))_{t\geq 0}$ a symmetric $\alpha$-stable Lévy
process with characteristic function

$\mathrm{E}\exp({\rm i}zU_{\alpha}(t))=\exp(-t|z|^{\alpha}),\quad
z\in{\mathbb{R}},\ t\geq 0.$

This process hits any point with probability 1, see, for instance, [3, p. 63]
or [34, Example 43.42]. We are going to construct a Feller process that
behaves like $U_{\alpha}$ until hitting $0$ and have some “asymmetry” at the
origin. To understand what could be a natural asymmetry at $0$ we attempt to
find analogies with the SBM construction. Note that $U_{\alpha}$ does not hit
$0$ by a single jump and cross the zero level infinitely often before hitting
$0$, see [36, Theorem 6.4]. It does not also exit $0$ by a jump and changes
sign infinitely often on exiting $0$ [34, Theorem 47.1]. Thus, unlike in the
case of SBM, selecting signs of the excursions becomes an issue.

The transition probability density function of the SBM with parameter $p$ at
time $t>0$ is given by

$(x,y)\mapsto\varphi_{t}^{(2)}(x-y)+(2p-1){\rm
sgn}\,(y)\varphi_{t}^{(2)}(|x|+|y|),\quad x,y\in\mathbb{R},$

where $\varphi_{t}^{(2)}$ is the density of a centered normal distribution
with variance $t$. For $t>0$, denote by $\varphi_{t}^{(\alpha)}$ the density
of the random variable $U_{\alpha}(t)$, $\alpha\in(1,2)$. It turns out that,
for $p\in[0,1]$,

$(x,y)\mapsto\varphi^{(\alpha)}_{t}(x-y)+(2p-1){\rm
sgn}\,(y)\varphi^{(\alpha)}_{t}(|x|+|y|),\quad x,y\in\mathbb{R}$

is the transition density of a Markov process, whose martingale
characterization is given in [30]. Now we provide an informal construction of
this process and note that it does not behave like $U_{\alpha}$ outside $0$.
The jumps of the process are governed by the intensity function proportional
to $|x|^{-(1+\alpha)}$, $x\in\mathbb{R}$. These are accumulated until the
process changes sign. At the epoch of sign change a new sign is selected
positive with probability $p$ and negative with probability $1-p$.

Existence and uniqueness of a strong solution to an SDE with a local time were
proved by Harrison and Shepp with the help of Tanaka’s formula and Nakao’s
theorem. This technique fails in the case of an $\alpha$-stable noise, and so
do arguments related to space- and time-change transforms, which are efficient
for one-dimensional diffusions with local times, see [19]. When constructing a
diffusion with a generalized drift Portenko uses a partial differential
equations approach. An essential part of his proof is based on the result
dealing with a jump of the normal derivative of a single layer potential. It
turns out that a similar result, in which the derivative has to be replaced
with some nonlocal operator, holds true for a potential generated by the
process $U_{\alpha}$. Unfortunately, such an approach leads to strongly
continuous semigroups without nonnegativity condition [21, 25].

We also mention here several recent results on strong solutions of SDEs with
singular drifts and additive fractional Brownian motion noise having a small
Hurst parameter, see [1, 7] and references therein. These results are derived
under the assumption that the noise has a local time, which is sufficiently
smooth with respect to the spatial parameter. Observe that the $\alpha$-stable
Lévy process $U_{\alpha}$ does not enjoy such a property.

To define the asymmetry at $0$ in a natural way, we shall use the approach of
Harrison and Shepp. Specifically, we construct perturbations of the
$\alpha$-stable process $U_{\alpha}$ as follows. Let $\zeta_{1}$,
$\zeta_{2},\ldots$ be independent copies of a random variable $\zeta$, which
are independent of $U_{\alpha}$. Assume that $\zeta\neq 0$ almost surely
(a.s.). With these at hand, we define the process
$X_{\zeta}:=(X_{\zeta}(t))_{t\geq 0}$ which satisfies $X_{\zeta}(0)=x\neq 0$
and has the same increments as $U_{\alpha}$ on the time intervals where
$X_{\zeta}$ does not ”touch” $0$. Upon the $k$th touch of $0$ the process
$X_{\zeta}$ has jump $\zeta_{k}$. To make the previous discussion formal, put

$\displaystyle\sigma_{0}=0,\quad\sigma_{k+1}:=\inf\\{t>\sigma_{k}\ :\
\zeta_{k}+U_{\alpha}(t)-U_{\alpha}(\sigma_{k})=0\\},$ (1) $\displaystyle
X_{\zeta}(t):=x+U_{\alpha}(t),\quad t\in[0,\sigma_{1}),$ $\displaystyle
X_{\zeta}(t):=\zeta_{k}+U_{\alpha}(t)-U_{\alpha}(\sigma_{k}),\quad
t\in[\sigma_{k},\sigma_{k+1}),\quad k\geq 1.$

Now we want to successively decrease the perturbations $(\zeta_{k})_{k\geq 1}$
of $U_{\alpha}$. To this end, for each positive integer $n$, replace
$(\zeta_{k})_{k\geq 1}$ with $(\zeta_{k}/n)_{k\geq 1}$, then define the
processes $X_{\zeta/n}$ and send $n\to\infty$. Note that each particular
perturbation tends to $0$. Nevertheless, the smaller the jump from $0$ is, the
smaller the return time to $0$ is. As a consequence, the number of visits to
$0$ increases as $n$ grows.

In this paper we aim at finding a distributional limit (that we denote by $X$)
for the sequence $(X_{\zeta/n})_{n\geq 1}$ and investigating its properties.
In particular, we shall point out the resolvent, the entrance law, the
excursions measure and an SDE that $X$ satisfies. Related to our investigation
are the papers [18, 38, 39], in which some invariance principles are obtained
in terms of convergence of the excursions measures.

Before formulating our results we note that, for $t>0$ and large $n$, the
value $X_{\zeta/n}(t)$ is the sum of $U_{\alpha}(t)$ and a random number
(depending on $t$) of small perturbations from the collection
$(\zeta_{k}/n)_{k\geq 1}$. Intuitively, if the distribution of $\zeta$ is
light-tailed, the contribution of the perturbations should be negligible. We
shall show below that this (trivial) situation occurs whenever
$\mathrm{E}|\zeta|<\infty$. Assume now that the opposite situation prevails,
which particularly means that $\mathrm{E}|\zeta|=\infty$. Then, to make the
sum of perturbations, properly normalized, convergent, it is natural to assume
that the distribution of $\zeta$ belongs to the domain of attraction of a
$\beta$-stable distribution with $\beta\in(0,1)$ (here, $\beta$ could have
been equal to $1$; however, we do not treat this case in the present paper).
This means that the variables $\zeta_{1}+\ldots+\zeta_{n}$, properly
normalized and centered, converge in distribution to a random variable with a
$\beta$-stable distribution. It is known that this happens if, and only if,
the function $x\mapsto\mathrm{P}(|\zeta|>x)$ is regularly varying at $+\infty$
of index $-\beta$ and the limits

$c_{\pm}:=\lim_{x\to+\infty}\frac{\mathrm{P}(\pm\zeta>x)}{\mathrm{P}(|\zeta|>x)}$
(2)

exist and satisfy $c_{-}+c_{+}>0$.

Given next are some notation to be in force throughout the paper. For a
process $Y$, denote by $\sigma$ or $\sigma(Y)$ the first hitting time of $0$,
that is,

$\sigma(Y)=\sigma:=\inf\\{t>0\ :\ Y(t)=0\\}.$

For bounded measurable functions $f:\mathbb{R}\to\mathbb{R}$, put

$V_{\lambda}f(x):=\mathrm{E}^{x}\int_{0}^{\sigma}e^{-\lambda
s}f(U_{\alpha}(s)){\rm d}s,$

so that $V_{\lambda}$ is the resolvent of $U_{\alpha}$ killed at $0$. We write
$D:=D[0,\infty)$ for the Skorokhod space of càdlàg functions defined on
$[0,\infty)$. We always assume that the space $D$ is endowed with the
$J_{1}$-topology. For a measurable function $f:\mathbb{R}\to\mathbb{R}$ and a
measure $\nu$ on $\mathbb{R}$ we write $\langle\nu,f\rangle$ for
$\int_{\mathbb{R}}f(x)\nu({\rm d}x)$ provided that the integral is well-
defined. In particular, if $\nu$ is a probability measure, then
$\langle\nu,f\rangle=\mathrm{E}f(\tau)$, where $\tau$ is a random variable
with distribution $\nu$.

We are ready to formulate our main results.

Theorem A. Assume that either the function $x\mapsto\mathrm{P}(|\zeta|>x)$ is
regularly varying at $+\infty$ of index $-\beta$, $\beta\in(0,1)$ and (2)
holds, or $\mathrm{E}|\zeta|<\infty$. If $X_{\zeta/n}(0)$ converges in
distribution as $n\to\infty$ to some random variable $\xi$, then the processes
$X_{\zeta/n}$ converge in distribution on $D$ to a Feller process $X$ starting
at $\xi$.

(a) If $\beta<\alpha-1$, then the resolvent of $X$ is given by

$R_{\lambda}f(x)=V_{\lambda}f(x)+\mathrm{E}^{x}e^{-\lambda\sigma(U_{\alpha})}\frac{\langle\eta^{\ast},V_{\lambda}f\rangle}{\langle\eta^{\ast},V_{\lambda}1\rangle}$
(3)

for bounded measurable $f$, where $\eta^{\ast}$ is a measure defined by

$\eta^{\ast}({\rm d}x)=(c_{-}1\\!\\!\,{\rm
I}_{(-\infty,0)}(x)+c_{+}1\\!\\!\,{\rm I}_{(0,\infty)}(x))|x|^{-(1+\beta)}{\rm
d}x,\quad x\in\mathbb{R},$ (4)

and the constants $c_{\pm}$ are given in (2).

(b) If $\beta>\alpha-1$ or $\mathrm{E}|\zeta|<\infty$, then
$X(t)=\xi+U_{\alpha}(t)$, $t\geq 0$, where $\xi$ and $U_{\alpha}$ are
independent.

One of the standing assumptions of the previous theorem is $\alpha\in(1,2)$.
Put formally $\alpha=2$, so that the noise becomes a Brownian motion $W$, say.
Then, under the assumption $\mathrm{E}|\zeta|<\infty$, a counterpart of the
limit process in Theorem A, still denoted by $X$, is a SBM which solves the
SDE

${\rm d}X(t)={\rm d}W(t)+\gamma{\rm d}L(t).$ (5)

Here, $\gamma=\frac{\mathrm{E}\zeta}{\mathrm{E}|\zeta|}\in[-1,1]$, and $L$ is
a two-sided local time of $X$ at $0$. The claim can be justified with the help
of arguments given in [8] or [14]. Alternatively, this can be shown along the
lines of the proof of Theorem A.

Although a resolvent uniquely determines the corresponding Feller process,
formula (3), being rather implicit, does not shed much light on the properties
of $X$. As a remedy, we characterize a Feller process with resolvent (3) as a
solution to an SDE.

Theorem B. Assume that the function $x\mapsto\mathrm{P}(|\zeta|>x)$ is
regularly varying at $+\infty$ of index $-\beta$, $\beta\in(0,\alpha-1)$ and
(2) holds. Let $X$ be a Feller process with resolvent (3). Then $X$ is a weak
solution to the SDE

$X(t)=X(0)+U_{\alpha}(t)+S_{\beta}(L^{X}_{0}(t)),\quad t\geq 0.$ (6)

Here, $L^{X}_{0}$ is a Blumenthal-Getoor local time of $X$ at $0$, $S_{\beta}$
is a $\beta$-stable Lévy process which is independent of $U_{\alpha}$ and has
the Lévy measure $\eta$ given by

$\eta({\rm d}x)=C(c_{-}1\\!\\!\,{\rm I}_{(-\infty,0)}(x)+c_{+}1\\!\\!\,{\rm
I}_{(0,\infty)}(x))|x|^{-(1+\beta)}{\rm d}x,\quad x\in\mathbb{R},$ (7)

where the constants $c_{\pm}$ are given in (2),

$C:=\Big{(}\int_{0}^{\infty}\mathrm{E}^{x}(1-e^{-\sigma(U_{\alpha})})\eta({\rm
d}x)\Big{)}^{-1}=\frac{\beta\sin\frac{\pi(\beta+1)}{\alpha}}{(c_{-}+c_{+})\Gamma(1-\beta)\cos\frac{\pi\beta}{2}\sin\frac{\pi}{\alpha}}$

and $\Gamma$ is the Euler gamma function.

Furthermore, the process $X$ has a zero sojourn at $0$ with probability $1$.

######  Remark 1.1.

The definition of the Blumenthal-Getoor local time can be found in [5, Theorem
2.3, Chapter V Section 2] or Section 3 below.

######  Remark 1.2.

The weak solution in Theorem A is a triple $(X,U_{\alpha},S_{\beta})$, with
all components being defined on a common probability space, which satisfies
equality (6) a.s. Here, the components are as defined in Theorem A. In
particular, $U_{\alpha}$ and $S_{\beta}$ are independent.

While not discussing a filtration, we only mention that it follows from the
construction that the processes $U_{\alpha}$ and
$(S_{\beta}(L_{0}^{X}(t)))_{t\geq 0}$ are $({\mathcal{F}}^{X}_{t})_{t\geq
0}$-adapted, where $({\mathcal{F}}^{X}_{t})_{t\geq 0}$ is a filtration
generated by $X$ and augmented by events of probability $0$. Observe that the
process $S_{\beta}$ is not $({\mathcal{F}}^{X}_{t})_{t\geq 0}$-adapted.

Comparing equations (5) and (6) we find it reasonable to call the process $X$
with resolvent (3) a skew $\alpha$-stable Lévy process.

Equation (5) has a unique solution if $|\gamma|\leq 1$ and has no solution if
$|\gamma|>1$, see [13]. An interesting problem is to find a counterpart of the
parameter $\gamma$ for equations like (6). Theorems C and D given next provide
a solution to the problem as well as a description of the corresponding
processes with the help of Itô’s excursion theory. We shall recall basic
definitions and results of the theory in Section 3 below.

For $t\geq 0$, put $\eta^{{\rm jump}}_{t}:=\eta P^{0}_{t}$, where the measure
$\eta$ is as defined in Theorem B, and $P^{0}_{t}$ is the semigroup of
$U_{\alpha}$ killed at $0$. For $x\in\mathbb{R}$, denote by $\bar{P}^{x}$ the
semigroup of $U_{\alpha}$ stopped at $0$. It can be checked that
$\bar{P}^{\eta}:=\int_{\mathbb{R}}P^{x}\eta({\rm d}x)$ is the excursion
measure of a skew $\alpha$-stable Lévy process. Denote by $(\eta^{{\rm
c}}_{t})_{t>0}$ and $\hat{P}^{U_{\alpha}}$ the entrance law of $U_{\alpha}$
and the corresponding excursion measure, respectively.

Theorem C. Assume that the function $x\mapsto\mathrm{P}(|\zeta|>x)$ is
regularly varying at $+\infty$ of index $-\beta$, $\beta\in(0,\alpha-1)$ and
(2) holds. Let $p\in[0,1]$ and $X$ be a Feller process having the entrance law
$(p\eta^{{\rm jump}}_{t}+(1-p)\eta^{{\rm c}}_{t})_{t>0}$ and the corresponding
excursion measure $p\bar{P}^{\eta}+(1-p)\hat{P}^{U_{\alpha}}$, where the
measure $\eta$ is defined in (7). Then $X$ is a weak solution to the SDE

$X(t)=X(0)+U_{\alpha}(t)+p^{\frac{1}{\beta}}S_{\beta}(L^{X}_{0}(t)),\quad
t\geq 0,$ (8)

where $L^{X}_{0}$, $S_{\beta}$ are as given in Theorem B.

Furthermore, the process $X$ has a zero sojourn at $0$ with probability $1$.

Theorem D. Assume that the function $x\mapsto\mathrm{P}(|\zeta|>x)$ is
regularly varying at $+\infty$ of index $-\beta$, $\beta\in(0,\alpha-1)$ and
(2) holds. Let $p\in[0,1]$ and $S_{\beta}$ be a $\beta$-stable Lévy process as
defined in Theorem B. Then there exists a unique Feller process $X$ with a
zero sojourn at $0$, which is a weak solution to equation (8), with
$U_{\alpha}$ being an $({\mathcal{F}}_{t}^{X})_{t\geq 0}$-adapted process.

######  Remark 1.3.

Even though we require in Theorem D $({\mathcal{F}}_{t}^{X})_{t\geq
0}$-adaptedness of $U_{\alpha}$, it may follow automatically from (8) and
independence of $U_{\alpha}$ and $S_{\beta}$.

We close the section with the list of open problems which are non-trivial even
for the SBM.

1) Characterize a sticky skew $\alpha$-stable Lévy process and the
corresponding SDE in the way similar to that used for the sticky Brownian
motion, see [10].

2) Consider a time inhomogeneous analogue of a skew stable Lévy process and
describe the corresponding semigroup in terms of partial differential
equations with Feller-Wenzell boundary condition at $0$, see [17] for the
Brownian case.

3) Investigate existence and uniqueness of a strong solution or a path-by-path
solution in the sense of Davie [9] to equation (8). This problem is non-
trivial even in the case where the noise is a Brownian motion and the process
$S_{\beta}$ is a subordinator [27], see also [29, 15].

4) Prove uniqueness of a weak solution to (8) among all (possibly non-Markov)
solutions in the situations that the local time is defined in terms of the
time spent in a neighborhood of $0$, or a number of long excursions, or a
number of a level crossings, etc.

The remainder of the paper is structured as follows. In Section 2 we use a
resolvent technique and prove Theorem A. In Section 3 we recall some basic
facts of Itô’s excursion theory. In Section 4 we prove Theorems B, C and D and
their generalizations.

##  2\. Convergence of resolvents

###  2.1. Discussion and limit theorem

In view of the assumption $X_{\zeta}(0)=x\neq 0$, the process $X_{\zeta}$ does
not visit $0$. It follows from the construction that $X_{\zeta}$ is a strong
Markov process on ${\mathbb{R}}\setminus\\{0\\}$ with càdlàg paths. We shall
investigate distributional convergence of the processes $X_{\zeta/n}$ as
$n\to\infty$. Since we expect that a limit process visits $0$, the machinery
of Markov processes on ${\mathbb{R}}$ rather than
${\mathbb{R}}\setminus\\{0\\}$ has to be exploited. To this end, we introduce
an auxiliary holding and jumping process that spends at $0$ a random period of
time having an exponential distribution (exponential time, in short), then has
jump $\zeta_{k}$ and afterwards behaves like $U_{\alpha}$ until the next visit
to $0$. The evolution just described then iterates, and the successive
exponential times at $0$ are independent and identically distributed.

Here is a formal construction. For $m>0$, denote by $\tau_{1}$,
$\tau_{2},\ldots$ independent random variables having the exponential
distribution of mean $1/m$. Assume that the sequences $(\zeta_{k})_{k\geq 1}$
and $(\tau_{k})_{k\geq 1}$ and the process $U_{\alpha}$ are independent.
Similarly to (1), put

$\tilde{\sigma}_{0}=0,\quad\tilde{\sigma}_{k+1}:=\inf\\{t>\tilde{\sigma}_{k}+\tau_{k}\
:\ \zeta_{k}+U_{\alpha}(t)-U_{\alpha}(\tilde{\sigma}_{k}+\tau_{k})=0\\},\quad
k\geq 0$

and then

$X_{\zeta,m}(t):=0,\quad\text{for}~{}~{}t\in[\tilde{\sigma}_{k},\tilde{\sigma}_{k}+\tau_{k}),\
k\geq 0$

and

$X_{\zeta,m}(t):=\zeta_{k}+U_{\alpha}(t)-U_{\alpha}(\tilde{\sigma}_{k}+\tau_{k}),\quad\text{for}~{}~{}t\in[\tilde{\sigma}_{k}+\tau_{k},\tilde{\sigma}_{k+1}),\
k\geq 0.$

The so defined $X_{\zeta,m}$ is a Feller process on ${\mathbb{R}}$ with càdlàg
paths. Unlike $X_{\zeta}$, the process $X_{\zeta,m}$ visits $0$ and may start
at $0$.

Recall Slutsky’s lemma: if $(X_{n})_{n\geq 1}$ and $(Y_{n})_{n\geq 1}$ are
sequences of random elements in a metric space (with metric ${\rm dist}$)
which satisfy ${\rm
dist}\,(X_{n},Y_{n})\stackrel{{\scriptstyle\mathrm{P}}}{{\to}}0$ as
$n\to\infty$, and the elements $X_{n}$ converge in distribution as
$n\to\infty$ to a random element $X$, then the elements $Y_{n}$ converge in
distribution to $X$, too.

Note that

$d(X_{\zeta},X_{\zeta,m})\stackrel{{\scriptstyle\mathrm{P}}}{{\to}}0,\quad
m\to\infty,$

where $d$ is the $J_{1}$-metric on $D$ and $\overset{\mathrm{P}}{\to}$ denotes
convergence in probability. Hence, distributional convergence of $X_{\zeta/n}$
to $X$ follows if we can show that $X_{\zeta/n,m_{n}}$ converges in
distribution, where $(m_{n})$ is a sequence which diverges to $+\infty$
sufficiently fast as $n\to\infty$.

For $x,y\in\mathbb{R}$ and a Markov process $X$, denote by $P_{t}(x,{\rm d}y)$
its transition probability function at time $t>0$. Also, for bounded
measurable functions $f:\mathbb{R}\to\mathbb{R}$, we define the semigroup

$\mathrm{E}^{x}f(X(t)):=\int_{{\mathbb{R}}}f(y)P_{t}(x,{\rm d}y)=\langle
P_{t},f\rangle(x)=P_{t}f(x),\quad t\geq 0,\ x\in\mathbb{R}$

and the resolvent

$R_{\lambda}f(x):=R_{\lambda}^{X}f(x):=\mathrm{E}^{x}\int_{0}^{\infty}e^{-\lambda
t}f(X(t)){\rm d}t=\int_{0}^{\infty}e^{-\lambda t}P_{t}f(x){\rm d}t,\quad
x\in\mathbb{R}.$

Further, for $x,y\in\mathbb{R}$, denote by $P^{0}_{t}(x,{\rm d}y)$ the
transition probability function at time $t>0$ for the process $X$ killed upon
the first visit to $0$, that is,

$P^{0}_{t}(x,A)=\mathrm{P}^{x}(X(t)\in A,t<\sigma),\quad x\in\mathbb{R}$

for Borel sets $A$ on $\mathbb{R}$. Also, we define the semigroup of the
killed process

$\mathrm{E}^{x}f(X(t))1\\!\\!\,{\rm
I}_{\\{t<\sigma\\}}=\int_{{\mathbb{R}}}f(y)P^{0}_{t}(x,{\rm d}y)=\langle
P^{0}_{t},f\rangle(x)=P^{0}_{t}f(x),\quad t\geq 0,\ x\in\mathbb{R}$

and its resolvent

$V_{\lambda}f(x)=V_{\lambda}^{X}f(x):=\mathrm{E}^{x}\int_{0}^{\sigma}e^{-\lambda
t}f(X(t)){\rm d}t=\int_{0}^{\infty}e^{-\lambda t}P^{0}_{t}f(x){\rm d}t,\quad
x\in\mathbb{R}.$

In Section 1 we have used the same notation $V_{\lambda}$ for the resolvent of
the particular killed Markov process $U_{\alpha}$. Hopefully, this does not
lead to a confusion. For later use, we note that if $X$ is a strong Markov
process, then

$R_{\lambda}f(x)=\mathrm{E}^{x}\Big{(}\int_{0}^{\sigma(X)}+\int_{\sigma(X)}^{\infty}\Big{)}e^{-\lambda
t}f(X(t)){\rm d}t\\\
=V_{\lambda}f(x)+\mathrm{E}^{x}e^{-\lambda\sigma(X)}\int_{0}^{\infty}e^{-\lambda
t}f(X(t+\sigma(X))){\rm d}t\\\
=V_{\lambda}f(x)+\mathrm{E}^{x}e^{-\lambda\sigma(X)}R_{\lambda}f(0),\quad
x\in\mathbb{R}.$ (9)

Theorem 2.1 states that the uniform convergence of resolvents entails
distributional convergence of the corresponding Markov processes. We write
$C_{0}({\mathbb{R}})$ for the space of all continuous functions vanishing at
$\pm\infty$, equipped with the supremum norm.

######  Theorem 2.1.

Let $(R^{(n)}_{\lambda})_{n\geq 1}$ be a sequence of resolvents of some Feller
processes $(X_{n})_{n\geq 1}$. Assume that, for each $f\in
C_{0}({\mathbb{R}})$ and each $\lambda>0$,

$\lim_{n\to\infty}\sup_{x\in\mathbb{R}}|R^{(n)}_{\lambda}f(x)-R_{\lambda}f(x)|=0,$

where $R_{\lambda}$ is the resolvent of a Feller process $X$. Assume also that
the variables $X_{n}(0)$ converge in distribution as $n\to\infty$ to $X(0)$.
Then

$X_{n}\Rightarrow X,\quad n\to\infty$

on $D$.

The proof follows from [11, Theorem 2.5, Chapter 4] and [26, Theorem 4.2,
Chapter 3].

We shall write $R^{\zeta,m}_{\lambda}$ for the resolvent of $X_{\zeta,m}$. For
bounded measurable functions $f:\mathbb{R}\to\mathbb{R}$ we intend to
calculate $R^{\zeta,m}_{\lambda}f(0)$. Although this follows a standard
pattern (see, for instance, [5, pp. 136-137]), we provide full details for
completeness. Write

$R^{\zeta,m}_{\lambda}f(0)=\mathrm{E}^{0}\Big{(}\int_{0}^{\tau_{1}}+\int_{\tau_{1}}^{\infty}\Big{)}e^{-\lambda
t}f(X_{\zeta,m}(t)){\rm
d}t=\lambda^{-1}\mathrm{E}(1-e^{-\lambda\tau_{1}})f(0)\\\
+\mathrm{E}e^{-\lambda\tau_{1}}\mathrm{E}^{\zeta}\int_{0}^{\infty}e^{-\lambda
t}f(X_{\zeta,m}(t)){\rm d}t=\frac{1}{m+\lambda}f(0)+\frac{m}{m+\lambda}\langle
P_{\zeta},R^{\zeta,m}_{\lambda}f\rangle.$ (10)

Using (9) we infer

$\langle P_{\zeta},R^{\zeta,m}_{\lambda}f\rangle=\langle
P_{\zeta},V^{U_{\alpha}}_{\lambda}f\rangle+\mathrm{E}^{\zeta}e^{-\lambda\sigma(U_{\alpha})}R_{\lambda}f(0)$

having utilized the equality
$\mathrm{E}^{\zeta}e^{-\lambda\sigma(X_{\zeta,m})}=\mathrm{E}^{\zeta}e^{-\lambda\sigma(U_{\alpha})}$.
Substituting this into (10) and then solving for $R_{\lambda}f(0)$ yields

$\lambda R_{\lambda}^{\zeta,m}f(0)=\frac{\frac{f(0)}{m}+\langle
P_{\zeta},V^{U_{\alpha}}_{\lambda}f\rangle}{\frac{1}{m}+\lambda^{-1}\mathrm{E}^{\zeta}(1-e^{-\lambda\sigma(U_{\alpha})})}=\frac{\frac{f(0)}{m}+\langle
P_{\zeta},V^{U_{\alpha}}_{\lambda}f\rangle}{\frac{1}{m}+\langle
P_{\zeta},V^{U_{\alpha}}_{\lambda}1\rangle}.$ (11)

######  Theorem 2.2.

Assume that either the function $x\mapsto\mathrm{P}(|\zeta|>x)$ is regularly
varying at $+\infty$ of index $-\beta$, $\beta\in(0,1)$ and (2) holds, or
$\mathrm{E}|\zeta|<\infty$. Let $f:\mathbb{R}\to\mathbb{R}$ be any bounded
measurable function.

(a) If $\beta<\alpha-1$, then

$\lim_{n\to\infty}\lambda
R_{\lambda}^{\zeta/n,m_{n}}f(0)=\frac{\langle\eta^{\ast},V^{U_{\alpha}}_{\lambda}f\rangle}{\langle\eta^{\ast},V^{U_{\alpha}}_{\lambda}1\rangle},$
(12)

where $(m_{n})_{n\geq 1}$ is any sequence of positive numbers satisfying
$\lim_{n\to\infty}m_{n}\mathrm{P}(\zeta>n)=\infty$, and $\eta^{\ast}$ is the
measure defined by (4).

(b) If $\beta>\alpha-1$ or $\mathrm{E}|\zeta|<\infty$, then

$\lim_{n\to\infty}R_{\lambda}^{\zeta/n,m_{n}}f(0)=R^{U_{\alpha}}_{\lambda}f(0),$

where $(m_{n})_{n\geq 1}$ is any sequence of positive numbers satisfying
$\lim_{n\to\infty}m_{n}n^{1-\alpha}=\infty$, and $R^{U_{\alpha}}_{\lambda}$ is
the resolvent of $U_{\alpha}$.

We claim that

$\lim_{n\to\infty}\sup_{x\in\mathbb{R}}|R_{\lambda}^{\zeta/n,m_{n}}f(x)-R_{\lambda}f(x)|=0$

for any bounded measurable $f:\mathbb{R}\to\mathbb{R}$ and any $\lambda>0$.
Here, for $x\in\mathbb{R}$,

$R_{\lambda}f(x)=\begin{cases}V^{U_{\alpha}}_{\lambda}f(x)+\lambda^{-1}\mathrm{E}^{x}e^{-\lambda\sigma(U_{\alpha})}\frac{\langle\eta^{\ast},V^{U_{\alpha}}_{\lambda}f\rangle}{\langle\eta^{\ast},V^{U_{\alpha}}_{\lambda}1\rangle},\quad\text{if}\
\beta<\alpha-1,\\\ R^{U_{\alpha}}_{\lambda}f(x),\quad\text{if}\
\beta>\alpha-1~{}\text{or}~{}\mathrm{E}|\zeta|<\infty.\end{cases}$ (13)

To check this, observe that
$V^{U_{\alpha}}_{\lambda}=V^{X_{\zeta/n,m_{n}}}_{\lambda}$,
$\mathrm{E}^{x}e^{-\lambda\sigma(X_{\zeta,m})}=\mathrm{E}^{x}e^{-\lambda\sigma(U_{\alpha})}$
for $x\in\mathbb{R}$. Further,
$R_{\lambda}f(0)=\lambda^{-1}\frac{\langle\eta^{\ast},V^{U_{\alpha}}_{\lambda}f\rangle}{\langle\eta^{\ast},V^{U_{\alpha}}_{\lambda}1\rangle}$
if $\beta<\alpha-1$ and $R_{\lambda}f(0)=R^{U_{\alpha}}_{\lambda}f(0)$ if
$\beta>\alpha-1$ or $\mathrm{E}|\zeta|<\infty$. With these at hand, invoking
(9) yields, for any $x\in\mathbb{R}$,

$|R_{\lambda}^{\zeta/n,m_{n}}f(x)-R_{\lambda}f(x)|=\mathrm{E}^{x}e^{-\lambda\sigma(U_{\alpha})}|R_{\lambda}^{\zeta/n,m_{n}}f(0)-R_{\lambda}f(0)|\\\
\leq|R_{\lambda}^{\zeta/n,m_{n}}f(0)-R_{\lambda}f(0)|~{}\to~{}0,\quad
n\to\infty.$

We have used Theorem 2.2 for the limit relation.

According to [5, Chapter V, §2, Theorem 2.8], $R_{\lambda}$ is the resolvent
of a strongly continuous probability semigroup and, as such, the resolvent of
a Feller process. This observation in combination with Theorem 2.1 leads to
the following.

######  Corollary 2.1.

Assume that the variables $X_{\zeta/n}(0)$ converge in distribution as
$n\to\infty$ to a random variable $\xi$. Then the processes $X_{\zeta/n}$
converge in distribution on $D$ to a Feller process $X$ with the resolvent
$R_{\lambda}$ defined in (13) and $X(0)$ having the same distribution as
$\xi$. In particular, if $\beta>\alpha-1$, then the limit process is
$\xi+U_{\alpha}$.

Theorem A is an immediate consequence of Corollary 2.1.

###  2.2. Auxiliary results

In this section we prove a few preparatory results needed for the proof of
Theorem 2.2. We shall treat the cases $\beta<\alpha-1$ and $\beta>\alpha-1$ or
$\mathrm{E}|\zeta|<\infty$ in slightly different ways. As a consequence, we
provide two collections of auxiliary results designed to deal with these
cases. Throughout this section we write $\sigma$ for $\sigma(U_{\alpha})$,
$R_{\lambda}$ for $R^{U_{\alpha}}_{\lambda}$ and $V_{\lambda}$ for
$V^{U_{\alpha}}_{\lambda}$.

Auxiliary results for the case $\beta<\alpha-1$. For $\lambda>0$, the density
$u_{\lambda}$ of the resolvent kernel of $U_{\alpha}$ satisfies
$u_{\lambda}(x,y)=u_{\lambda}(y-x)$, $x,y\in\mathbb{R}$, where

$u_{\lambda}(x)=\frac{1}{\pi}\int_{0}^{\infty}\frac{\cos(x\theta)}{\lambda+\theta^{\alpha}}{\rm
d}\theta,\quad x\in\mathbb{R}.$

Lemma 2.1 collects a couple of formulae to be used in what follows.

######  Lemma 2.1.

Let $\alpha\in(1,2)$.

(a) For $\gamma\in[0,\alpha-1)$ and $\lambda>0$,

$\int_{0}^{\infty}\frac{\theta^{\gamma}}{\lambda+\theta^{\alpha}}{\rm
d}\theta=\frac{\Gamma(1-\frac{\gamma+1}{\alpha})\Gamma(\frac{\gamma+1}{\alpha})}{\alpha\lambda^{1-\frac{\gamma+1}{\alpha}}}=\frac{\pi}{\alpha\sin\frac{\pi(\gamma+1)}{\alpha}}\frac{1}{\lambda^{1-\frac{\gamma+1}{\alpha}}}.$

In particular,

$u_{\lambda}(0)=\frac{1}{\alpha\sin\frac{\pi}{\alpha}}\frac{1}{\lambda^{1-\frac{\gamma+1}{\alpha}}}.$

(b) For $x\in\mathbb{R}$,

$\int_{0}^{\infty}\frac{1-\cos(xy)}{y^{\alpha}}{\rm
d}y=|x|^{\alpha-1}\frac{\Gamma(2-\alpha)\sin\frac{\pi\alpha}{2}}{\alpha-1}.$

###### Proof.

While the first equality in the first formula of part (a) is a consequence of
[12, formula (3.241)(2)], the second equality follows from Euler’s reflection
formula $\Gamma(1-z)\Gamma(z)=\frac{\pi}{\sin(\pi z)}$ which holds true for
any noninteger $z$. The second equality of part (a) is implied by the first
with $\gamma=0$ and the formula
$u_{\lambda}(0)=\pi^{-1}\int_{0}^{\infty}(\lambda+\theta^{\alpha})^{-1}{\rm
d}\theta$. Part (b) follows from [34, formula (14.18)]. ∎

While formula (14) of Lemma 2.2 will be used in the proof of both parts of
Theorem 2.2, formula (15) will be used in the proof of Theorem 2.2(b).

######  Lemma 2.2.

For $\alpha\in(1,2)$ and $\lambda>0$,

$\lambda
V_{\lambda}1(x)=\mathrm{E}^{x}(1-e^{-\lambda\sigma})~{}\sim~{}A_{\lambda,\alpha}|x|^{\alpha-1},\quad
x\to 0,$ (14)

where
$A_{\lambda,\alpha}:=\frac{\alpha\Gamma(2-\alpha)\sin\pi\alpha\sin\frac{\pi\alpha}{2}}{\pi(\alpha-1)}\lambda^{1-\frac{1}{\alpha}}$,
and

$\mathrm{P}^{1}(\sigma>y)\sim B_{\alpha}y^{-1+\frac{1}{\alpha}},\quad
y\to\infty,$ (15)

where
$B_{\alpha}:=\frac{\sin\pi\alpha\,\sin{\frac{\pi\alpha}{2}}\Gamma(1-\alpha)}{\pi\,\Gamma(1-\frac{1}{\alpha})}$

###### Proof.

We start with proving (14). According to [3, Corollary II.5.8],

$\mathrm{E}^{x}e^{-\lambda\sigma}=\frac{u_{\lambda}(-x)}{u_{\lambda}(0)},\quad
x\in\mathbb{R},$

whence

$u_{\lambda}(0)\mathrm{E}^{x}(1-e^{-\lambda\sigma})=\frac{1}{\pi}\int_{0}^{\infty}\frac{1-\cos(x\theta)}{\lambda+|\theta|^{\alpha}}{\rm
d}\theta=\\\
\frac{|x|^{\alpha-1}}{\pi}\int_{0}^{\infty}\frac{1-\cos(y)}{\lambda|x|^{\alpha}+y^{\alpha}}{\rm
d}y\sim\frac{|x|^{\alpha-1}}{\pi}\int_{0}^{\infty}\frac{1-\cos(y)}{y^{\alpha}}{\rm
d}y,\quad x\to 0.$ (16)

Invoking Lemma 2.1 we arrive at (14). Using the first equality in (16) and
Lemma 2.1 we infer

$\mathrm{E}^{1}(1-e^{-\lambda\sigma})=\frac{\alpha\sin{\frac{\pi}{\alpha}}\lambda^{1-\frac{1}{\alpha}}}{\pi}\int_{0}^{\infty}\frac{1-\cos\theta}{\lambda+\theta^{\alpha}}{\rm
d}\theta~{}\sim~{}\frac{\alpha\sin{\frac{\pi}{\alpha}}}{\pi}\int_{0}^{\infty}\frac{1-\cos\theta}{\theta^{\alpha}}{\rm
d}\theta\,\lambda^{1-\frac{1}{\alpha}}\\\
=\frac{\sin\pi\alpha\,\sin{\frac{\pi\alpha}{2}}\Gamma(1-\alpha)}{\pi}\,\lambda^{1-\frac{1}{\alpha}},\quad\lambda\to
0+.$

An application of Corollary 8.1.7 in [4] yields (15). ∎

Lemma (2.3) is the principal ingredient of the proof of Theorem 2.2(a).

######  Lemma 2.3.

Let $\alpha\in(1,2)$ and $g:\mathbb{R}\to\mathbb{R}$ be a bounded measurable
function satisfying $g(x)=O(|x|^{\alpha-1})$ as $x\to 0$. Assume that the
function $x\mapsto\mathrm{P}(|\zeta|>x)$ is regularly varying at $+\infty$ of
index $-\beta$, $\beta\in(0,\alpha-1)$ and (2) holds. Then

$\lim_{n\to\infty}\frac{\mathrm{E}g(\zeta/n)}{\mathrm{P}(\zeta>n)}=\int_{\mathbb{R}}g(x)\eta^{\ast}({\rm
d}x)=\langle\eta^{\ast},g\rangle<\infty,$

where $\eta^{\ast}$ is the measure defined in (4).

###### Proof.

The functions $g^{+}:=\max(g,0)$ and $g^{-}:=\max(-g,0)$ are nonnegative,
bounded and satisfy $g^{\pm}(x)=O(|x|^{\alpha-1})$ as $x\to 0$. Thus, without
loss of generality we can and do assume that $g$ is nonnegative.

Put $G(x):=\mathrm{P}(\zeta\leq x)$ for $x\in\mathbb{R}$. We shall show that

$\lim_{n\to\infty}\frac{\int_{[0,\,\infty)}g(x){\rm
d}G(nx)}{1-G(n)}=c_{+}\beta\int_{0}^{\infty}g(x)x^{-\beta-1}{\rm d}x.$ (17)

Fix any $\varepsilon\in(0,1)$. Then

$\lim_{n\to\infty}\frac{\int_{(\varepsilon,\,\infty)}g(x){\rm
d}G(nx)}{1-G(n)}=c_{+}\beta\int_{\varepsilon}^{\infty}g(x)x^{-\beta-1}{\rm
d}x$

follows from

$\lim_{n\to\infty}\frac{\mathrm{P}(\zeta>nx)}{\mathrm{P}(\zeta>n)}=c_{+}x^{-\beta},\quad
x>0.$

Observe that the usual requirement of continuity of $g$ is not needed here,
for the measure $\eta^{\ast}$ is (absolutely) continuous.

There is a constant $c>0$ such that $g(x)\leq cx^{\alpha-1}$ whenever
$x\in(0,1]$. With this at hand we conclude that

$\int_{[0,\,\varepsilon]}g(x){\rm
d}G(nx)\leq\int_{[0,\,\varepsilon]}cx^{\alpha-1}{\rm
d}G(nx)=\frac{c}{n^{\alpha-1}}\int_{[0,\,n\varepsilon]}x^{\alpha-1}{\rm
d}G(x).$

Further,

$\int_{[0,\,n\varepsilon]}x^{\alpha-1}{\rm
d}G(x)~{}\sim~{}(n\varepsilon)^{\alpha-1}(1-G(n\varepsilon))\frac{\beta}{\alpha-1-\beta}~{}\sim~{}\frac{\beta}{\alpha-1-\beta}\varepsilon^{\alpha-1-\beta}n^{\alpha-1}(1-G(n))$

as $n\to\infty$, where the first asymptotic relation follows from Karamata’s
theorem [4, Theorem 1.6.4]. We infer

$\limsup_{n\to\infty}\int_{[0,\,\varepsilon]}g(x)\frac{{\rm
d}G(nx)}{1-G(n)}\leq
c\frac{\beta}{\alpha-1-\beta}\varepsilon^{\alpha-1-\beta}$

and

$\limsup_{n\to\infty}\int_{[0,\,\infty)}g(x)\frac{{\rm d}G(nx)}{1-G(n)}\leq
c\frac{\beta}{\alpha-1-\beta}\varepsilon^{\alpha-1-\beta}+c_{+}\beta\int_{\varepsilon}^{\infty}g(x)x^{-\beta-1}{\rm
d}x.$

Sending $\varepsilon\to 0+$ we arrive at

$\limsup_{n\to\infty}\int_{[0,\infty)}g(x)\frac{{\rm d}G(nx)}{1-G(n)}\leq
c_{+}\beta\int_{0}^{\infty}g(x)x^{-\beta-1}{\rm d}x.$

For the lower bound, write, for any $\varepsilon>0$,

$\int_{[0,\,\infty)}g(x)\frac{{\rm
d}G(nx)}{1-G(n)}\geq\int_{(\varepsilon,\,\infty)}g(x)\frac{{\rm
d}G(nx)}{1-G(n)}~{}\to~{}c_{+}\beta\int_{\varepsilon}^{\infty}g(x)x^{-\beta-1}{\rm
d}x,\quad n\to\infty.$

Sending $\varepsilon\to 0+$ we obtain

$\liminf_{n\to\infty}\int_{[0,\,\infty)}g(x)\frac{{\rm d}G(nx)}{1-G(n)}\geq
c_{+}\beta\int_{0}^{\infty}g(x)x^{-\beta-1}{\rm d}x,$

and (17) follows.

Starting with

$\lim_{n\to\infty}\frac{\mathrm{P}(-\zeta>nx)}{\mathrm{P}(\zeta>n)}=c_{-}x^{-\beta},\quad
x>0$

and arguing analogously we also conclude that

$\int_{(-\infty,\,0)}g(x)\frac{{\rm
d}G(nx)}{1-G(n)}=c_{-}\beta\int_{-\infty}^{0}g(x)|x|^{-\beta-1}{\rm d}x.$

Combining this with (17) completes the proof of the lemma. ∎

Auxiliary results for the case $\beta>\alpha-1$ or $\mathrm{E}|\zeta|<\infty$.
Lemmas 2.4 and 2.5 will be used for the proof of Theorem 2.2 (b).

######  Lemma 2.4.

For $\lambda>0$ and any bounded and continuous function
$f:\mathbb{R}\to\mathbb{R}$,

$\lim_{x\to 0}\frac{V_{\lambda}f(x)}{V_{\lambda}1(x)}=\lambda
R^{U_{\alpha}}_{\lambda}f(0)=\lambda\int_{\mathbb{R}}u_{\lambda}(y)f(y){\rm
d}y.$ (18)

######  Remark 2.1.

According to this lemma and [5, Theorem 4.2, Section V.4] there exists a
unique recurrent extension of the process $U_{\alpha}$ killed at $0$ which has
a zero sojourn at $0$. As a consequence, this extension coincides with
$U_{\alpha}$. More details on the excursion theory, recurrent extensions of
Markov processes etc. will be given in Section 3.

###### Proof.

For each $x\in\mathbb{R}$, denote by $X_{x}$ the process defined in (1) in
which we formally replace $\zeta$ with $x$. For $k\in{\mathbb{N}}$, denote by
$\sigma_{k}^{(x)}$ the time of the $k$th jump (of size $x$) from $0$ and note
that the random variables $\sigma_{1}^{(x)}$, $\sigma_{2}^{(x)},\ldots$ are
independent and identically distributed. We shall use a representation

$X_{x}(t)=U_{\alpha}(t)+xN_{\alpha}(x,t),\quad t\geq 0,~{}~{}x\in\mathbb{R},$

where $N_{\alpha}(x,t)=\\#\\{k\in{\mathbb{N}}\ :\
\sigma_{1}^{(x)}+\ldots+\sigma_{k}^{(x)}\leq t\\}$.

Arguing as in (9) and (10) we conclude that

$\frac{V_{\lambda}f(x)}{\lambda
V_{\lambda}1(x)}=\mathrm{E}\int_{0}^{\infty}e^{-\lambda t}f(X_{x}(t)){\rm
d}t,\quad x\in\mathbb{R}.$

In view of this and the Lebesgue dominated convergence theorem, we are left
with showing that

$xN_{\alpha}(x,t)\overset{\mathrm{P}}{\to}0,\quad x\to 0.$ (19)

To prove (19), recall that the process $U_{\alpha}$ is self-similar of index
$1/\alpha$. This implies that, for $k\in{\mathbb{N}}$, $\sigma^{(x)}_{k}$ has
the same distribution as $\sigma^{(1)}_{k}|x|^{\alpha}$ and thereupon
$N_{\alpha}(x,t)$ has the same distribution as $\\#\\{k\in{\mathbb{N}}\ :\
\sigma_{1}^{(1)}+\ldots+\sigma_{k}^{(1)}\leq t|x|^{-\alpha}\\}$. By [22,
Theorem 3.2],

$\mathrm{P}^{1}(\sigma>t|x|^{-\alpha})\,N_{\alpha}(x,t)\overset{{\rm
d}}{\to}S_{1-\frac{1}{\alpha}}^{\leftarrow}(1):=\inf\\{t\geq 0\ :\
S_{1-\frac{1}{\alpha}}(t)>1\\},\quad x\to 0,$

where $\overset{{\rm d}}{\to}$ denotes convergence of one-dimensional
distributions, and $(S_{1-\frac{1}{\alpha}}(t))_{t\geq 0}$ is a drift-free
$(1-\frac{1}{\alpha})$-stable subordinator with

$-\log\mathrm{E}e^{-uS_{1-\frac{1}{\alpha}}(1)}=\Gamma(1/\alpha)u^{1-\frac{1}{\alpha}},\quad
u\geq 0.$

Since, by (15), $P^{1}(\sigma>t|x|^{-\alpha})\sim
B_{\alpha}t^{-1+\frac{1}{\alpha}}|x|^{\alpha-1}$ as $x\to 0$, we infer (19). ∎

######  Lemma 2.5.

Let $a:\mathbb{R}\to[0,\infty)$ and $b:\mathbb{R}\to\mathbb{R}$ be functions
such that $|b(x)|\leq Ca(x)$ for all $x\in{\mathbb{R}}$ and some constant
$C>0$ and that $\lim_{x\to 0}\frac{b(x)}{a(x)}=A\in\mathbb{R}$. Assume that a
sequence $(\mu_{n})_{n\geq 1}$ of probability measures satisfies
$\mu_{n}(\\{0\\})=0$ for large $n$ and, for each $\delta>0$,

$\lim_{n\to\infty}\frac{\int_{|x|\geq\delta}a(x)\mu_{n}({\rm
d}x)}{\int_{|x|<\delta}a(x)\mu_{n}({\rm d}x)}=0.$ (20)

Then

$\lim_{n\to\infty}\frac{\int_{{\mathbb{R}}}b(x)\mu_{n}({\rm
d}x)}{\int_{{\mathbb{R}}}a(x)\mu_{n}({\rm d}x)}=A.$

###### Proof.

Let $\varepsilon>0$ be arbitrary. Select $\delta>0$ such that
$|\frac{b(x)}{a(x)}-A|<\varepsilon$ whenever $|x|\leq\delta$. Then

$\limsup_{n\to\infty}\frac{\int_{{\mathbb{R}}}b(x)\mu_{n}({\rm
d}x)}{\int_{{\mathbb{R}}}a(x)\mu_{n}({\rm
d}x)}=\limsup_{n\to\infty}\frac{\frac{(\int_{|x|\geq\delta}+\int_{|x|<\delta})b(x)\mu_{n}({\rm
d}x)}{\int_{|x|<\delta}a(x)\mu_{n}({\rm
d}x)}}{\frac{(\int_{|x|\geq\delta}+\int_{|x|<\delta})a(x)\mu_{n}({\rm
d}x)}{\int_{|x|<\delta}a(x)\mu_{n}({\rm
d}x)}}=\limsup_{n\to\infty}\frac{\frac{\int_{|x|<\delta}b(x)\mu_{n}({\rm
d}x)}{\int_{|x|<\delta}a(x)\mu_{n}({\rm
d}x)}}{\frac{\int_{|x|<\delta}a(x)\mu_{n}({\rm
d}x)}{\int_{|x|<\delta}a(x)\mu_{n}({\rm d}x)}}=\\\
\limsup_{n\to\infty}\frac{\int_{|x|<\delta}b(x)\mu_{n}({\rm
d}x)}{\int_{|x|<\delta}a(x)\mu_{n}({\rm
d}x)}\leq\limsup_{n\to\infty}\frac{(A+\varepsilon)\int_{|x|<\delta}a(x)\mu_{n}({\rm
d}x)}{\int_{|x|<\delta}a(x)\mu_{n}({\rm d}x)}=A+\varepsilon$

having utilized for the second equality

$\lim_{n\to\infty}\frac{\int_{|x|\geq\delta}b(x)\mu_{n}({\rm
d}x)}{\int_{|x|<\delta}a(x)\mu_{n}({\rm d}x)}=0$

which is a consequence of (20) and the assumption $|b(x)|\leq Ca(x)$ for all
$x\in\mathbb{R}$. An analogous inequality for the lower limit follows
similarly. Sending $\varepsilon\to 0$ completes the proof of the lemma. ∎

###  2.3. Proof of Theorem 2.2

We first prove the theorem in the case $\beta<\alpha-1$ (part (a) of the
theorem). In view of (11) we have to show that

$\lim_{n\to\infty}\frac{f(0)/m_{n}+\langle
P_{\zeta/n},V_{\lambda}f\rangle}{1/m_{n}+\langle
P_{\zeta/n},V_{\lambda}1\rangle}=\frac{\langle\eta^{\ast},V_{\lambda}f\rangle}{\langle\eta^{\ast},V_{\lambda}1\rangle}.$
(21)

The function $V_{\lambda}1$ is nonnegative and bounded (by $\lambda^{-1}$).
According to (14), it satisfies $V_{\lambda}1(x)=O(|x|^{\alpha-1})$ as $x\to
0$. By virtue of

$|V_{\lambda}f(x)|\leq\sup_{y\in\mathbb{R}}|f(y)|V_{\lambda}1(x),\quad
x\in\mathbb{R}$ (22)

and the fact that $f$ is bounded by assumption, we conclude that
$V_{\lambda}f$ is a bounded function satisfying
$V_{\lambda}f(x)=O(|x|^{\alpha-1})$ as $x\to 0$. Hence, an application of
Lemma (2.3) yields

$\langle
P_{\zeta/n},V_{\lambda}f\rangle~{}\sim~{}\mathrm{P}(\zeta>n)\langle\eta^{\ast},V_{\lambda}f\rangle\quad\text{and}\quad\langle
P_{\zeta/n},V_{\lambda}1\rangle~{}\sim~{}\mathrm{P}(\zeta>n)\langle\eta^{\ast},V_{\lambda}1\rangle,\quad
n\to\infty.$

These together with our choice of $m_{n}$ prove (21).

Now we are turning to the proof of part (b). Formula (11) tells us that we
have to prove that

$\lim_{n\to\infty}\frac{f(0)/m_{n}+\langle
P_{\zeta/n},V_{\lambda}f\rangle}{1/m_{n}+\langle
P_{\zeta/n},V_{\lambda}1\rangle}=\lambda R_{\lambda}^{U_{\alpha}}f(0).$ (23)

In the notation of Lemma 2.5, put $a:=V_{\lambda}1$, $b:=V_{\lambda}f$ and
$\mu_{n}:=P_{\zeta/n}$ for $n\geq 1$. Then, in view of (22), we may put
$C:=\sup_{y\in\mathbb{R}}|f(y)|$ and, by Lemma 2.4, $A:=\lambda
R^{U_{\alpha}}_{\lambda}f(0)$. Now relation (23) follows from Lemma 2.5 and
our choice of $m_{n}$ if we can show that

$\lim_{n\to\infty}m_{n}\langle P_{\zeta/n},V_{\lambda}1\rangle=\infty$ (24)

and that, for any $\delta>0$,

$\lim_{n\to\infty}\frac{\int_{|x|\geq\delta}\mathrm{E}^{x}(1-e^{-\lambda\sigma})P_{\zeta/n}({\rm
d}x)}{\int_{|x|<\delta}\mathrm{E}^{x}(1-e^{-\lambda\sigma})P_{\zeta/n}({\rm
d}x)}=0$ (25)

which is condition (20) with
$a(x)=V_{\lambda}1(x)=\lambda^{-1}\mathrm{E}^{x}(1-e^{-\lambda\sigma})$ for
$x\in\mathbb{R}$.

Fix any $\delta>0$. On the one hand,

$\int_{|x|\geq\delta}\mathrm{E}^{x}(1-e^{-\lambda\sigma})P_{\zeta/n}({\rm
d}x)\leq\mathrm{P}(|\zeta|\geq n\delta)=o(n^{1-\alpha}),\quad n\to\infty$

which holds true if $\mathrm{E}|\zeta|<\infty$ or $\beta>\alpha-1$. On the
other hand, appealing to (14) we conclude that, for appropriate constant
$K>0$,

$\int_{|x|<\delta}\mathrm{E}^{x}(1-e^{-\lambda\sigma})P_{\zeta/n}({\rm
d}x)\geq
K\int_{|x|<\delta}|x|^{\alpha-1}P_{\zeta/n}({\rm}x)=Kn^{1-\alpha}\int_{|x|<n\delta}|x|^{\alpha-1}\mathrm{P}_{\zeta}({\rm
d}x)\\\ ~{}\sim~{}K\mathrm{E}(|\zeta|^{\alpha-1})n^{1-\alpha},\quad
n\to\infty.$ (26)

Since $\mathrm{E}(|\zeta|^{\alpha-1})<\infty$ if $\mathrm{E}|\zeta|<\infty$ or
$\beta>\alpha-1$, (25) follows. Finally, (24) is a consequence of (26) and our
assumption $\lim_{n\to\infty}m_{n}n^{1-\alpha}=\infty$.

The proof of Theorem 2.2 is complete.

##  3\. Itô’s excursion theory

According to Corollary 2.1, the processes $X_{\zeta/n}$ converge in
distribution. If $\beta>\alpha-1$ or $\mathrm{E}|\zeta|<\infty$, the limit
process is $U_{\alpha}$ and, as such, well-understood. If $\beta<\alpha-1$, we
only have a resolvent description of the limit process that we call a skew
$\alpha$-stable Lévy process. In this section, we give a probabilistic
representation of this process via Itô’s excursion theory and an SDE, which
include a local time of the process. There are several definitions of a local
time of a Markov process. Usually these produce the same process, up to a
multiplicative constant. To provide an absolutely rigorous formulation of our
results, we should stick to a particular definition that serves our needs. As
far as the skew stable Lévy process is concerned, the exact values of
constants are as important as the parameter of permeability appearing in the
defining equation (5) for the SBM.

We briefly review below some basic facts of Itô’s excursion theory for Markov
processes. We only discuss real-valued processes and leave aside processes
taking values in the other spaces. We follow Blumenthal’s book [5] and cite
specific theorems or point out pages for the most important results. Let
$(X,P^{x})$, $x\in\mathbb{R}$ be a Feller process, that is, a Markov process
whose transition semigroup $(P_{t})_{t\geq 0}$ is strongly continuous on
$C_{0}({\mathbb{R}})$. Without loss of generality we always assume that all
processes to be considered are standard, see the corresponding definition in
[6, Chapter I, §9].Assume that $X$ is recurrent at $0$, that is,
$P^{x}(\sigma<\infty)=1$ for all $x\in{\mathbb{R}}$, where, as before,
$\sigma=\sigma(X)$ denotes the first hitting time of $0$.

Associated with $(X,P^{x})$ are

(i) $\bar{X}(t):=X(t\wedge\sigma)$, $t\geq 0$ the process stopped at $0$ (we
denote by $\bar{P}^{x}$ and $\bar{P}_{t}$ its distribution and semigroup,
respectively), and

(ii) the process killed at $0$ with the semigroup $P^{0}_{t}$ and the
transition probabilities $P^{0}(t,x,A)=P(X(t)\in A,\ t<\sigma)$ for $t\geq 0$
and $x\in\mathbb{R}$.

The killed process and its semigroup will be called minimal process and
minimal semigroup, respectively.

Below we recall elements of Itô’s synthesis theory, which describes all
recurrent extensions of the minimal process.

Assume that

$\text{the function
x}~{}\mapsto\bar{\mathrm{E}}^{x}e^{-\lambda\sigma}~{}\text{is continuous for
each}~{}\lambda>0~{}\text{and}\lim_{|x|\to\infty}\bar{\mathrm{E}}^{x}e^{-\sigma}=0.$

######  Remark 3.1.

All the properties of a minimal semigroup, stated above and below, are
satisfied for the minimal semigroup of a symmetric $\alpha$-stable Lévy
process with $\alpha\in(1,2)$.

To each function $u\in D([0,\infty))$, we associate
$\sigma(u):=\inf\\{t>0:u(t)=0\\}$. Let $\hat{\mathrm{P}}$ be a $\sigma$-finite
measure on $D$ supported by the set of functions $\\{u\ :\
u(t)=0,t\geq\sigma(u)\\}$. The elements of this set will be called excursions
and $\sigma(u)$ will be called the length of excursion $u$. We shall assume
that

$\hat{\mathrm{P}}(1-e^{-\sigma})\leq 1$

and that $\hat{P}(|u(0)|>x)<\infty$ for $x>0$.

Let $N({\rm d}s,{\rm d}u)$ denote a Poisson point measure on $[0,\infty)\times
D$ with intensity ${\rm d}s\times\hat{\mathrm{P}}({\rm d}u)$. Denote by
$(s_{k},u_{k})$ the atoms of $N$, that is, $N=\sum_{k}\delta_{(s_{k},u_{k})}$.
Put $m:=1-\hat{\mathrm{P}}(1-e^{-\sigma})$,

$\tau(s):=ms+\sum_{s_{k}\leq
s}\sigma(u_{k})=ms+\int_{[0,\,s]}\int_{D}\sigma(u)N({\rm d}z,{\rm d}u),\quad
s\geq 0$

and

$\varphi(t):=\inf\\{s\geq 0\ :\ \tau(s)>t\\},\quad t\geq 0.$ (27)

Assume that at least one of the following conditions holds:

$m>0$ or the measure $\hat{\mathrm{P}}$ is infinite.

Then $(\tau(s))_{s\geq 0}$ is a strictly increasing subordinator. For
$t\in[\tau(s_{k}-),\tau(s_{k})]$, put $X(t):=u_{k}(t-\tau(s_{k}-))$ and, for
$t\notin\cup_{k}[\tau(s_{k}-),\tau(s_{k})]$, put $X(t):=0$.

Further, we assume that the characteristic measure $\hat{P}$ is compatible
with the minimal semigroup. This means that, for all $n\geq 1$, all $0\leq
s_{1}\leq\ldots\leq s_{n}\leq s$ and all bounded measurable functions
$g,g_{1},\ldots,g_{n}$ with $g(0)=0$,

$\hat{P}\Big{(}g(u(t+s))g_{1}(u(s_{1}))\ldots
g_{n}(u(s_{n}));\sigma>s\Big{)}=\hat{P}\Big{(}P^{0}_{t}g(u(s))g_{1}(u(s_{1}))\ldots
g_{n}(u(s_{n}));\sigma>s\Big{)},$

and

$\hat{P}\Big{(}g(u(t));u(0)\in B\Big{)}=\hat{P}\Big{(}P^{0}_{t}g(u(0));u(0)\in
B\Big{)},$

(see [5, Chapter V §1 and V §2 (d)] for more details). Then, according to [5,
Theorem 2.10, Chapter V §2]), $X$ is a Markov extension of the minimal process
with $X(0)=0$. Moreover, each extension (starting at $0$) can be obtained with
the help of merging excursions, under suitable measure $\hat{P}$ satisfying
the compatibility conditions. An extension has a zero sojourn at $0$ if, and
only if, $m=0$.

######  Remark 3.2.

If $X(0)\neq 0$, then the behavior of $X$ until hitting $0$ is determined
uniquely by the minimal semigroup or the semigroup corresponding to the
stopped process.

Note that the measure $\hat{P}$ is finite if, and only if,
$\hat{P}=\bar{P}^{\theta}$ for some finite measure $\theta$ on
${\mathbb{R}}\setminus\\{0\\}$. In this case, $X$ is a holding and jumping
process, with holding time $\|\theta\|$ and jumping distribution
$\frac{\theta}{\|\theta\|}$, where
$\|\theta\|=\theta({\mathbb{R}}\setminus\\{0\\})$.

If the measure $\hat{P}$ is infinite, then $X$ can be obtained as an a.s.
limit of holding and jumping processes. To explain the construction, recall
that [5, Chapter 5.2] a family of $\sigma$-finite measures
$(\theta_{t})_{t>0}$ on the Borel subsets of ${\mathbb{R}}\setminus\\{0\\}$ is
called entrance law for $P^{0}_{t}$, if

$\theta_{s}P_{t}^{0}=\theta_{t+s},t\geq 0,\quad s>0$

and

$\langle\theta_{s},V1\rangle=\mathrm{E}^{\theta_{s}}(1-e^{-\sigma})\leq
1,\quad s>0.$

If $\theta$ is a $\sigma$-finite measure on ${\mathbb{R}}\setminus\\{0\\}$
such that $\mathrm{E}^{\theta}(1-e^{-\sigma})\leq 1$, then $\theta_{t}:=\theta
P_{t}^{0}$ is an entrance law. Another important example of an entrance law is
$\hat{\theta}_{t}(A):=\hat{\mathrm{P}}(X_{t}\in A,\ t<\sigma)$.

Consider a family of holding and jumping processes
$(X_{\varepsilon})_{\varepsilon>0}$ with jumping distribution
$\frac{\hat{\eta}_{\varepsilon}}{\|\hat{\eta}_{\varepsilon}\|}$ and holding
parameter $m+\delta_{\varepsilon}$, where $\delta_{\varepsilon}>0$ is any
function satisfying $\delta_{\varepsilon}=o(\varepsilon)$ as $\varepsilon\to
0$. It follows from the proof of [5, Theorem 2.10, Chapter 5.2] that the
corresponding resolvents converge uniformly. By Theorem 2.1, this entails the
distributional convergence $X_{\varepsilon}\Rightarrow X$ on $D$ as
$\varepsilon\to 0$. The aforementioned proof contains an explicit construction
of $X_{\varepsilon}$, which includes merging of excursions of the length
larger than $\varepsilon>0$, and such that $\lim_{\varepsilon\to
0}X_{\varepsilon}(t)=X(t)\quad\mbox{a.s.}$

Note also that, for any entrance law $(\theta_{t})$, there exists a unique
characteristic measure $\hat{P}$ satisfying $\theta_{t}=\hat{\theta}_{t},t>0,$
see [5, Theorem 4.7, Chapter V]. Moreover, the characteristic measure
$\hat{P}$ can be recovered from the process $X$, see [5, Chapter III], and the
process $(\varphi(t))_{t\geq 0}$ defined in (27) is the Blumenthal-Getoor
local time of $X$ at $0$, see [5, Theorem 2.3, Chapter V §2], that is, a
continuous additive functional which satisfies

$\mathrm{E}^{x}e^{-\sigma}=\mathrm{E}^{x}\int_{0}^{\infty}e^{-t}{\rm
d}\varphi(t),\quad x\in\mathbb{R}.$

Existence, uniqueness and some other properties of the local time are
discussed in [5, p. 91-93]. Therefore, under some natural properties imposed
on the minimal semigroup, there is a one-to-one correspondence between the
recurrent extension of the minimal process, the characteristic measure
$\hat{P}$ and the entrance law $(\theta_{t})$. We stress that the analysis of
the entrance law is simpler than that of the two other objects.

Any entrance law for $P^{0}_{t}$ can be uniquely decomposed as the sum of two
entrance laws

$\theta_{t}=\rho_{t}+\theta P_{t}^{0},\quad t>0,$

where $\theta$ is a $\sigma$-finite measure, and the measure $\rho_{t}$
satisfies

$\lim_{t\to 0+}\rho_{t}({\mathbb{R}}\setminus[-x,x])=0$

for any $x>0,$ see [5, p. 140]. These entrance laws, which are called
continuous entrance and jump entrance, respectively, can also be characterized
as follows:

$\theta_{t}^{c}(\cdot):=\rho_{t}=\hat{P}(X(s)\in\cdot,\ X(0)=0,\ s<\sigma),$
$\theta_{t}^{j}(\cdot):=\hat{P}(X(s)\in\cdot,\ X(0)\neq 0,\ s<\sigma),$

see [5, p. 156]. It follows from Theorems 2.6, 2.8 and the proof of Theorem
2.10 from [5, Chapter V] that if an entrance law is a jump entrance law, that
is, $\theta_{t}=\theta P_{t}^{0},$ where $\theta$ is an infinite measure
satisfying $\mathrm{E}^{\theta}(1-e^{-\sigma})=1$, then the corresponding
recurrent extension $X$ has a zero sojourn at $0$, its resolvent satisfies

$\lambda
R_{\lambda}f(0)=\frac{\langle\theta,V_{\lambda}f\rangle}{\langle\theta,V_{\lambda}1\rangle},$
(28)

the corresponding characteristic measure $\hat{P}$ is equal to
$\bar{P}^{\theta}$.

Here is another conclusion motivated by the aforementioned facts. Assume that
there exists a unique continuous entrance law $\rho_{t}$ with the
characteristic measure $\hat{P}^{c}$ that satisfies
$\hat{P}^{c}(1-e^{-\sigma})=1$. For instance, this is the case for the process
$U_{\alpha}$, see Remark 2.1. Then any entrance law $(\theta_{t})$, which
corresponds to an extension with a zero sojourn at $0$, can be uniquely
represented by

$\theta_{t}=(1-p)\rho_{t}+p\,\theta P_{t}^{0},$ (29)

where $p\in[0,1]$, and the measure $\theta$ satisfies
$\mathrm{E}^{\theta}(1-e^{-\sigma})=1$. The corresponding characteristic
measure is

$\hat{P}=(1-p)\hat{P}^{c}+p\bar{P}^{\theta}.$ (30)

##  4\. SDE for the skew stable Lévy process

In this section all semigroups, resolvents, etc. are related to extensions of
the minimal process corresponding to the process $U_{\alpha}$ killed at 0.

###  4.1. Existence of solutions to SDEs with local times

Let $X$ be a Feller process with the resolvent

$R_{\lambda}f(x)=V_{\lambda}f(x)+\mathrm{E}^{x}e^{-\lambda\sigma}\frac{\langle\theta,V_{\lambda}f\rangle}{\langle\theta,V_{\lambda}1\rangle},\quad\lambda>0,~{}~{}x\in\mathbb{R},$
(31)

where $\theta$ is an infinite measure satisfying
$\mathrm{E}^{\theta}(1-e^{-\sigma})=1$. The right-hand side in (31) is
obtained by a combination of (9) and (28).

######  Theorem 4.1.

The process $X$ is a (weak) solution to the SDE

$X(t)=X(0)+U_{\alpha}(t)+S_{\theta}(L^{X}_{0}(t)),\quad t\geq 0,$ (32)

where $U_{\alpha}$ is a symmetric $\alpha$-stable Lévy process, $S_{\theta}$
is a pure-jump Lévy process with the Lévy measure $\theta$, which is
independent of $U_{\alpha}$, the process $L^{X}_{0}$ is the local time of $X$
at $0$ (see the definition in the previous subsection).

A specialization of Theorem 4.1, with the Lévy measure $\theta=\eta$ given by
(7) and $S_{\theta}=S_{\beta}$, immediately proves Theorem B, except the
explicit formula for the constant $C$. Here is the remaining piece of the
proof. Write

$\frac{1}{C}=\int_{{\mathbb{R}}}\mathrm{E}^{x}(1-e^{-\sigma(U_{\alpha})})\eta({\rm
d}x)=\frac{c_{-}+c_{+}}{\pi
u_{1}(0)}\int_{0}^{\infty}\int_{0}^{\infty}\frac{1-\cos(x\theta)}{1+\theta^{\alpha}}{\rm
d}\theta\frac{{\rm d}x}{{x^{1+\beta}}}\\\ =\frac{c_{-}+c_{+}}{\pi
u_{1}(0)}\frac{\Gamma(1-\beta)\sin\frac{\pi(1+\beta)}{2}}{\beta}\int_{0}^{\infty}\frac{\theta^{\beta}}{1+\theta^{\alpha}}{\rm
d}\theta=\frac{(c_{-}+c_{+})\Gamma(1-\beta)\cos\frac{\pi\beta}{2}\sin\frac{\pi}{\alpha}}{\beta\sin\frac{\pi(\beta+1)}{\alpha}}$

having utilized (7) and (16) for the second equality, Fubini’s theorem and
Lemma 2.1(b) for the third equality and Lemma 2.1(a) for the last equality.

###### Proof of Theorem 4.1.

Since $\mathrm{E}^{\theta}(1-e^{-\sigma})=1$ by assumption, we conclude that
the ‘holding’ parameter $m=1-\mathrm{E}^{\theta}(1-e^{-\sigma})$ is equal to
$0$, and the process $X$ has a zero sojourn at $0$ a.s.

Let $N_{\theta}({\rm d}s,{\rm d}x)$ be a Poisson point measure with intensity
${\rm d}s\times\theta({\rm d}x)$. Let $((s_{k},x_{k}))_{k\geq 1}$ be a
(measurable) enumeration of the atoms of $N_{\theta}$. Let $U_{\alpha,1}$,
$U_{\alpha,2}\ldots$ denote independent copies of $U_{\alpha}$, which are
independent of $N_{\theta}$. Put $N:=\sum_{k\geq
1}\delta_{(s_{k},x_{k}+U_{\alpha,k}(\cdot\wedge\sigma_{k}))}$, where

$\sigma_{k}:=\sigma(x_{k}+U_{\alpha,k}(\cdot)):=\inf\\{t\geq 0\ :\
x_{k}+U_{\alpha,k}(t)=0\\}.$

Then $N$ is a Poisson random measure on $[0,\infty)\times D$ with intensity
${\rm d}s\times\hat{P}={\rm d}s\times\bar{P}^{\theta}$.

Without loss of generality we can and do assume that $X(0)=0$ and that the
process $X$ is built upon the Poisson point measure $N$ with the help of Itô’s
procedure. Here are details of the construction. Put

$\tau(s):=\sum_{s_{k}\leq s}\sigma(x_{k}+U_{\alpha,k}(\cdot)),\quad s\geq 0,$

and

$\varphi(t):=\inf\\{s\geq 0\ :\ \tau(s)>t\\},\quad t\geq 0.$

For $t\in[\tau(s_{k}-),\tau(s_{k}))$, put
$X(t):=x_{k}+U_{\alpha,k}(t-\tau(s_{k}-))$, and, for
$t\notin\bigcup_{k}[\tau(s_{k}-),\tau(s_{k}))$, put $X(t):=0$.

######  Lemma 4.1.

For almost all $t>0$ with respect to the Lebesgue measure,

$\mathrm{P}\Big{(}\exists k\ :\ \ t\in(\tau(s_{k}-),\tau(s_{k}))\ \
\mbox{and}\ \ \tau(s_{k}-)=\varphi(t)\Big{)}=1.$

The proof follows from the fact that $X$ has a zero sojourn at $0$ and
Fubini’s theorem.

Put

$N_{\theta}^{(\varepsilon)}:=\sum_{k}1\\!\\!\,{\rm
I}_{\\{|x_{k}|>\varepsilon\\}}\delta_{(s_{k},x_{k})},\ \ \
N^{(\varepsilon)}:=\sum_{k\geq 1}1\\!\\!\,{\rm
I}_{\\{|x_{k}|>\varepsilon\\}}\delta_{(s_{k},x_{k}+U_{\alpha,k}(\cdot\wedge\sigma_{k}))}.$
(33)

Then $N_{\theta}^{(\varepsilon)}$ is a Poisson point measure on
$[0,\infty)\times({\mathbb{R}}\setminus\\{0\\})$ with intensity ${\rm
d}s\times(1\\!\\!\,{\rm I}_{\\{|x|>\varepsilon\\}}\theta({\rm d}x))$, and
$N^{(\varepsilon)}$ is a Poisson point measure on $[0,\infty)\times D$ with
intensity ${\rm d}s\times\bar{P}^{1\\!\\!\,{\rm
I}_{\\{|x|>\varepsilon\\}}\theta({\rm d}x)}$.

Assuming that $m=0$ we now construct a process $X^{(\varepsilon)}$ with the
help of the Poisson random measure $N^{(\varepsilon)}$, along the lines of
construction of $X$ with the help of $N$. Put

$\tau^{(\varepsilon)}(s):=\sum_{s_{k}\leq s,\
|x_{k}|>\varepsilon}\sigma(x_{k}+U_{\alpha,k}(\cdot)),\quad s\geq 0$

and

$\varphi^{(\varepsilon)}(t):=\inf\\{s\geq 0\ :\ \tau^{(\varepsilon)}(s)\geq
t\\},\quad t\geq 0.$

For $t\in[\tau^{(\varepsilon)}(s_{k}-),\tau^{(\varepsilon)}(s_{k}))$, put
$X^{(\varepsilon)}(t):=x_{k}+U_{\alpha,k}(t-\tau^{(\varepsilon)}(s_{k}-))$
provided that $|x_{k}|>\varepsilon$. Otherwise, we put
$X^{(\varepsilon)}(t):=0$.

######  Remark 4.1.

In contrast to the process $X$, a.s. there does not exist a $t$ such that
$X^{(\varepsilon)}(t)=0$. The process $X^{(\varepsilon)}$ jumps upon
‘touching’ $0$.

In view of $\int_{\mathbb{R}}\int_{0}^{T}1\\!\\!\,{\rm
I}_{\\{|x|>\varepsilon\\}}{\rm d}s\eta({\rm d}x)<\infty$ for any $T>0$, the
Poisson point process $N_{\eta}^{(\varepsilon)}$ can be represented by
$\sum_{k\geq 1}\delta_{(s^{(\varepsilon)}_{k},x^{(\varepsilon)}_{k})}$, where
$0<s^{(\varepsilon)}_{1}<s^{(\varepsilon)}_{2}<s^{(\varepsilon)}_{3}<\ldots$,
and each interval $[0,T]$ contains finitely many $s_{k}^{(\varepsilon)}$ a.s.
Put
$\sigma_{k}^{(\varepsilon)}:=\sigma(x^{(\varepsilon)}_{k}+U^{(\varepsilon)}_{\alpha,k}(\cdot))$,
where $U^{(\varepsilon)}_{\alpha,k}$ is a process from the collection
$(U_{\alpha,j})$, which corresponds to $s_{k}^{(\varepsilon)}$ in the
representation of $N$. According to the construction of $X^{(\varepsilon)}$,

$X^{(\varepsilon)}(0)=x_{1}^{(\varepsilon)},$
$X^{(\varepsilon)}(\sigma_{1}^{(\varepsilon)}+\ldots+\sigma_{k}^{(\varepsilon)})=x^{(\varepsilon)}_{k+1},$
$X^{(\varepsilon)}(t)=x^{(\varepsilon)}_{k}+U^{(\varepsilon)}_{\alpha,k}(t),t\in[\sigma_{1}^{(\varepsilon)}+\ldots+\sigma_{k}^{(\varepsilon)},\sigma_{1}^{(\varepsilon)}+\ldots+\sigma_{k}^{(\varepsilon)}+\sigma_{k+1}^{(\varepsilon)}),$

and

$\sigma_{k+1}^{(\varepsilon)}=\inf\\{t\geq\sigma_{1}^{(\varepsilon)}+\ldots+\sigma_{k}^{(\varepsilon)}\
\ :\ \
x_{k}^{(\varepsilon)}+U_{\alpha,k}^{(\varepsilon)}(t-(\sigma_{1}^{(\varepsilon)}+\ldots+\sigma_{k}^{(\varepsilon)}))=0\\}.$

Plainly, the process $U^{(\varepsilon)}_{\alpha}$ defined by
$U^{(\varepsilon)}_{\alpha}(0):=0,$ and

$U^{(\varepsilon)}_{\alpha}(t):=U^{(\varepsilon)}_{\alpha}(\sigma_{1}^{(\varepsilon)}+\ldots+\sigma_{k}^{(\varepsilon)})+U_{\alpha,k}^{(\varepsilon)}(t-(\sigma_{1}^{(\varepsilon)}+\ldots+\sigma_{k}^{(\varepsilon)})),$

for
$t\in[\sigma_{1}^{(\varepsilon)}+\ldots+\sigma_{k}^{(\varepsilon)},\sigma_{1}^{(\varepsilon)}+\ldots+\sigma_{k}^{(\varepsilon)}+\sigma_{k+1}^{(\varepsilon)})$,
is a symmetric $\alpha$-stable Lévy process, which is independent of
$N_{\theta}^{(\varepsilon)}$. In particular, $U^{(\varepsilon)}_{\alpha}$ is
independent of

$S_{\theta}^{(\varepsilon)}(s):=\int_{[0,\,s]}\int_{|x|>\varepsilon}xN_{\theta}^{(\varepsilon)}({\rm
d}z,{\rm d}x)=\int_{[0,\,s]}\int_{{\mathbb{R}}\setminus\\{0\\}}x1\\!\\!\,{\rm
I}_{\\{|x|>\varepsilon\\}}N_{\theta}({\rm d}z,{\rm d}x).$ (34)

Note also that

$X^{(\varepsilon)}(t)=U^{(\varepsilon)}_{\alpha}(t)+S_{\theta}^{(\varepsilon)}(\varphi^{(\varepsilon)}(t)),\quad
t\geq 0.$

######  Remark 4.2.

Let $\zeta_{\varepsilon}$ be a random variable with distribution
$\mathrm{P}(\zeta_{\varepsilon}\in
A)=\frac{\int_{|x|>\varepsilon}1\\!\\!\,{\rm I}_{A}(x)\theta({\rm
d}x)}{\theta({|x|>\varepsilon})}$. Then the distribution of $X_{\varepsilon}$
coincides with that of $X_{\zeta_{\varepsilon}}$, see (1) for the definition.

######  Lemma 4.2.

For each $T>0$, almost surely

$\lim_{\varepsilon\to
0}\sup_{s\in[0,\,T]}|\tau^{(\varepsilon)}(s)-\tau(s)|=0,$
$\lim_{\varepsilon\to
0}\sup_{t\in[0,\,T]}|\varphi^{(\varepsilon)}(t)-\varphi(t)|=0$

and

$\lim_{\varepsilon\to
0}\sup_{s\in[0,\,T]}|S^{(\varepsilon)}_{\theta}(s)-S_{\theta}(s)|=0,$

where
$S_{\theta}(s)=\int_{0}^{s}\int_{{\mathbb{R}}\setminus\\{0\\}}xN_{\theta}({\rm
d}z,{\rm d}x)$ for $s\geq 0$.

The proof follows from the construction and the fact that the processes
$(\tau(t))$ and $S_{\theta}$ are pure-jump Lévy processes of (locally) finite
variation.

######  Lemma 4.3.

For almost all $t>0$ with respect to the Lebesgue measure,

$\mathrm{P}(\exists\varepsilon_{0}>0\
\forall\varepsilon\in(0,\varepsilon_{0}):\ \
\varphi^{(\varepsilon)}(t)=\varphi(t))=1.$ (35)

###### Proof.

It follows from Lemmas 4.1 and 4.2 that for a.e. $t>0$ with probability $1$
there exists $k$ such that

$\ t\in(\tau(s_{k}-),\tau(s_{k}))\ \ \mbox{and}\ \ \tau(s_{k}-)=\varphi(t)$

and

$\lim_{\varepsilon\to 0}\tau^{(\varepsilon)}(s_{k}-)=\tau(s_{k}-),\ \
\lim_{\varepsilon\to 0}\tau^{(\varepsilon)}(s_{k})=\tau(s_{k}).$

These entail $t\in(\tau^{(\varepsilon)}(s_{k}-),\tau^{(\varepsilon)}(s_{k}))$
for small $\varepsilon>0$ and thereupon (35). ∎

######  Corollary 4.1.

For each $t>0$,

$\lim_{\varepsilon\to
0}S_{\theta}^{(\varepsilon)}(\varphi^{(\varepsilon)}(t))=S_{\theta}(\varphi(t))\quad\text{a.s.}$

A fixed time $t_{0}$ is not a jump-time of a Lévy process a.s. Hence, by
Fubini’s theorem, for each $k\geq 1$ and almost all $t_{0}>0$ with respect to
the Lebesgue measure,

$\mathrm{P}\left(\lim_{t\to
t_{0}}U_{\alpha,k}(t)=U_{\alpha,k}(t_{0})\right)=1\quad\text{a.s.}$

This together with Lemmas 4.1 and 4.2 enables us to conclude that

$\mathrm{P}\left(\lim_{t\to t_{0}}X^{(\varepsilon)}(t)=X(t_{0})\right)=1$

for almost all $t_{0}>0$. Invoking Corollary 4.1 we infer

$\lim_{t\to t_{0}}U^{(\varepsilon)}_{\alpha}(t)=\lim_{t\to
t_{0}}\left(X^{(\varepsilon)}(t)-S_{\theta}^{(\varepsilon)}(\varphi^{(\varepsilon)}(t))\right)=X(t_{0})-S_{\theta}(\varphi(t_{0}))\quad\text{a.s.}$
(36)

for almost all $t_{0}>0$.

Denote the process $(X(t)-S_{\theta}(\varphi(t)))$ by $U_{\alpha}$. The paths
of this process are càdlàg, for so are the paths of $X$ and
$(S_{\theta}(\varphi(t)))$. The process $U_{\alpha}$ is a symmetric
$\alpha$-stable Lévy process as a limit of symmetric $\alpha$-stable Lévy
processes. Hence, (36) entails

$X(t)=U_{\alpha}(t)+S_{\theta}(\varphi(t)),\quad t\geq 0\quad\text{a.s.}$

Since $(U^{(\varepsilon)}_{\alpha}(t))$ is independent of $S_{\theta}$, so is
$U_{\alpha}$.

∎

######  Remark 4.3.

We have not proved that $\lim_{\varepsilon\to 0}X^{(\varepsilon)}=X$ a.s. on
$D$. However, this has never been claimed.

######  Remark 4.4.

For each $T>0$, the process $(U^{(\varepsilon)}_{\alpha}(t))_{t\in[0,T]}$ was
obtained by merging finitely (but randomly) many fragments of paths of
independent $\alpha$-stable Lévy processes. However, the process
$(U_{\alpha}(t))_{t\in[0,T]}$ is built upon a countable number of paths. We
cannot offer a good formula for $U_{\alpha}(t)$, other than
$U_{\alpha}(t)=X(t)-S_{\theta}(\varphi(t))$ and attract the reader attention
to the fact that the set $\\{t\geq 0:X(t)=0\\}$ is uncountable (the set does
not coincide with a countable set of the endpoints of excursion intervals). A
similar noise representation, in terms of the difference of a Markov process
and a generalized drift, is typical in the framework of diffusions with
semipermeable membrane, see [31, 33]. In this context the Markov process $Y$,
say is first constructed via a semigroup technique. Then the generalized drift
is identified with an additive functional of $Y$. Finally, it has to be
checked that the difference of $Y$ and the generalized drift is a stochastic
integral.

The Lévy measure $\theta$ of the process $S_{\theta}$ appearing in (32)
satisfies $\mathrm{E}^{\theta}(1-e^{-\sigma})=1$. Below we construct a
process, similar to $S_{\theta}$, which satisfies (32) when the latter
equality does not necessarily holds. Denote by $(\rho_{t})$,
$\hat{P}_{\alpha}$ and $N_{\alpha}$ the entrance law, the intensity of
excursion measure and the corresponding Poisson point measure of a symmetric
$\alpha$-stable Lévy process. Recall (see Lemma 2.4 and Remark 2.1) that there
exists a unique continuous entrance law with a zero sojourn at $0$, and that
this entrance law corresponds to a symmetric $\alpha$-stable Lévy process. Let
$\theta$ be a sigma-finite measure on ${\mathbb{R}}\setminus\\{0\\}$
satisfying $\mathrm{E}^{\theta}(1-e^{-\sigma})=1$. Define the entrance law
$\theta_{t}:=\theta P^{0}_{t}$, the characteristic measure $\bar{P}^{\theta}$
and the corresponding Poisson point measure $N$, which is independent of
$N_{\alpha}$. For $p\in[0,1]$, define the entrance law
$p\theta_{t}+(1-p)\rho_{t}$, the intensity of excursions measure
$p\bar{P}^{\theta}+(1-p)\hat{P}_{\alpha}$ and the Poisson point measure
$N(p\,{\rm d}t,{\rm d}u)+N_{\alpha}((1-p)\,{\rm d}t,{\rm d}u)$, see (29) and
(30). Using these objects we now construct a Feller process $X$ with the help
of Itô’s procedure. Since
$(p\bar{P}^{\theta}+(1-p)\hat{P}_{\alpha})(1-e^{-\sigma})=1$, the process $X$
has a zero sojourn at $0$.

######  Theorem 4.2.

For each $p\in[0,1]$, the process $X$ is a (weak) solution to the SDE

$X(t)=U_{\alpha}(t)+S_{\theta}(pL^{X}_{0}(t)),\quad t\geq 0,$ (37)

where $U_{\alpha}$ is a symmetric $\alpha$-stable Lévy process, $S_{\theta}$
is a pure-jump Lévy process with the Lévy measure $\theta$, which is
independent of $U_{\alpha}$, and the process $L^{X}_{0}$ is the local time of
$X$ at $0$.

######  Remark 4.5.

The process $U_{\alpha}$ appearing in (37) is different from the
$\alpha$-stable Lévy process which is built upon $N_{\alpha}$ alone.

######  Remark 4.6.

The process $S_{\theta}(p\cdot)$ has the same distribution as
$S_{p\theta}(\cdot)$. As a consequence, $X$ is also a solution to

$X(t)=U_{\alpha}(t)+S_{p\theta}(L^{X}_{0}(t)),\quad t\geq 0.$

Theorem C follows from the last remark specialized to $\theta=\eta^{\ast}$ (so
that $S_{\theta}=S_{\beta}$), where $\theta^{\ast}$ is the measure defined in
(7).

###### Proof of Theorem 4.2.

Define the Poisson random measures $N^{(\varepsilon)}_{\eta}$ and
$N^{(\varepsilon)}$ via (33).

Put $\theta^{(\varepsilon)}({\rm d}x):=1\\!\\!\,{\rm
I}_{\\{|x|>\varepsilon\\}}\theta({\rm d}x)$,
$\theta^{(\varepsilon)}_{t}:=\theta^{(\varepsilon)}P^{0}_{t}$ and

$q_{\varepsilon}:=\frac{1-p\int_{|x|>\varepsilon}\mathrm{E}^{x}(1-e^{-\sigma})\theta({\rm
d}x)}{\hat{P}_{\alpha}(1-e^{-\sigma})}.$

Use now Itô’s procedure to build the process $X^{(\varepsilon)}$ upon the
Poisson point measure $N^{(\varepsilon)}(p\,{\rm d}t,{\rm
d}u)+N_{\alpha}(q_{\varepsilon}\,{\rm d}t,{\rm d}u)$, which corresponds to the
entrance law $p\theta^{(\varepsilon)}_{t}+q_{\varepsilon}\rho_{t}$ and the
characteristic measure $p\bar{P}^{1\\!\\!\,{\rm
I}_{\\{|x|>\varepsilon\\}}\theta({\rm d}x)}+q_{\varepsilon}\hat{P}_{\alpha}$.
It follows from the construction that $X^{(\varepsilon)}$ has a zero sojourn
at $0$, and that

$X^{(\varepsilon)}(t)=U^{(\varepsilon)}_{\alpha}(t)+S_{\theta}^{(\varepsilon)}(p\varphi^{(\varepsilon)}(t)),\quad
t\geq 0,$

where the process $S_{\theta}^{(\varepsilon)}$ is defined in (34). The
remainder of the proof mimics that of Theorem 4.1.

We proceed with a comment.

Assume that we have constructed the process $\tilde{X}^{(\varepsilon)}$ with
the help of the Poisson point measure $N^{(\varepsilon)}(p_{\varepsilon}\,{\rm
d}t,{\rm d}u)+N_{\alpha}(q{\rm d}t,{\rm d}u)$, where

$p_{\varepsilon}=\frac{p\int_{{\mathbb{R}}}\mathrm{E}^{x}(1-e^{-\sigma})\theta({\rm
d}x)}{\int_{|x|>\varepsilon}\mathrm{E}^{x}(1-e^{-\sigma})\theta({\rm d}x)}.$

Then

$\tilde{X}^{(\varepsilon)}(t)=\tilde{U}^{(\varepsilon)}_{\alpha}(t)+S_{\theta}^{(\varepsilon)}(p_{\varepsilon}\tilde{\varphi}^{(\varepsilon)}(t)),\quad
t\geq 0.$

It is likely that the limit relation $\lim_{\varepsilon\to
0}S_{\theta}^{(\varepsilon)}(p_{\varepsilon}\tilde{\varphi}^{(\varepsilon)}(t))=S_{\theta}(p\varphi(t))$
a.s. also holds. However, this fact does not follow directly from Lemmas 4.2
and 4.3. Indeed, the composition functional is not continuous on $D\times
C_{\uparrow}([0,\infty))$, where $C_{\uparrow}([0,\infty))$ is the set of
nondecreasing continuous functions, see [37, Section 13.2]. The composition
$(S_{\theta},\varphi)\mapsto S_{\theta}\circ\varphi$ would be a.s. continuous,
if the function $\varphi$ were a.s. strictly increasing, which is not the
case. The local time $\varphi$ has many intervals of constancy, which are
directly connected to the points of growth of $S_{\theta}$. ∎

###  4.2. Uniqueness for SDEs with local times for the skew Levy process

######  Theorem 4.3.

Assume that there exists a filtered probability space
$(\Omega,({\mathcal{F}}_{t}),{\mathcal{F}},\mathrm{P}),$ a
$({\mathcal{F}}_{t})$-adapted symmetric $\alpha$-stable process $U_{\alpha}$
with $\alpha\in(1,2)$ and a homogeneous Feller process
$(X,({\mathcal{F}}_{t}))$ with a zero sojourn at $0$ which satisfy

$X(t)=U_{\alpha}(t)+H(L^{X}_{0}(t)),\quad t\geq 0,$

where $H$ is a Lévy process of bounded variation, which is independent of
$U_{\alpha}$. Then there exist $p\in[0,1]$ and a sigma-finite measure $\theta$
on ${\mathbb{R}}\setminus\\{0\\}$ satisfying
$\mathrm{E}^{\theta}(1-e^{-\sigma})=1$, for which the processes $(X,H)$ has
the same distribution as the processes $(X(\cdot),S_{\theta}(p\cdot))$
appearing in Theorem 4.2.

Theorem D is a specialization of Theorem 4.3, with the Lévy measure
$\theta=\eta$ given by (7) in which case $S_{\theta}=S_{\beta}$.

######  Remark 4.7.

Under the assumptions of Theorem 4.3 imposed on $(X,U_{\alpha})$ there exists
neither a solution to the equation

${\rm d}X(t)={\rm d}U_{\alpha}(t)+\gamma{\rm d}L^{X}_{0}(t),\quad t\geq 0$

for $\gamma\neq 0$, nor a solution to (37) for $p>1$.

###### Proof.

Since the local time only grows on the set $\\{t\geq 0\ :\ X(t)=0\\}$, the
process $X$ is a recurrent extension of a symmetric $\alpha$-stable Lévy
process killed at $0$. Hence, there exists a unique value of $p\in[0,1]$ and a
unique measure $\theta$ satisfying $\mathrm{E}^{\theta}(1-e^{-\sigma})=1$, for
which the entrance law of $X$ is given by $(1-p)\rho_{t}+p\,\theta P^{0}_{t}$,
see (29) and (30). It follows from Theorem 4.2 that there exists a version
$\widetilde{X}$ of $X$ satisfying

$\widetilde{X}(t)=\widetilde{U}_{\alpha}(t)+\widetilde{S}_{\theta}(pL^{\widetilde{X}}_{0}(t)),\quad
t\geq 0.$

Observe that the processes $X$ and $\widetilde{X}$ are semimartingales. Since
the local times $(L^{X}_{0}(t))$ and $(L^{\widetilde{X}}_{0}(t))$ are a.s.
continuous processes which do not increase a.s. on the sets $\\{s\geq 0\ :\
X(s)\neq 0\\}$ and $\\{s\geq 0\ :\ \widetilde{X}(s)\neq 0\\}$, respectively,
we conclude that, for any $\delta>0$, with probability $1$,

$\int_{[0,\,t]}1\\!\\!\,{\rm I}_{\\{|\widetilde{X}(s-)|\geq\delta\\}}{\rm
d}(\widetilde{S}_{\theta}(pL^{\widetilde{X}}_{0}(s)))=0,\quad t>0$

and

$\int_{[0,\,t]}1\\!\\!\,{\rm I}_{\\{|X(s-)|\geq\delta\\}}{\rm
d}(H(L^{X}_{0}(s)))=0,\quad t\geq 0.$

This entails

$\int_{[0,\,t]}1\\!\\!\,{\rm I}_{\\{|X(s-)|\geq\delta\\}}{\rm
d}X(s)=\int_{[0,\,t]}1\\!\\!\,{\rm I}_{\\{|X(s-)|\geq\delta\\}}{\rm
d}U_{\alpha}(s)$

and

$\int_{[0,\,t]}1\\!\\!\,{\rm I}_{\\{|\widetilde{X}(s-)|\geq\delta\\}}{\rm
d}\widetilde{X}(s)=\int_{[0,\,t]}1\\!\\!\,{\rm
I}_{\\{|X(s-)|\geq\delta\\}}{\rm d}\widetilde{U}_{\alpha}(s).$

Sending $\delta\to 0+$ we conclude that there exists a measurable function
$F:D\to D$ satisfying $\widetilde{U}_{\alpha}=F(\widetilde{X})$ and
$U_{\alpha}=F(X)$ a.s. Since the process $X$ has the same distribution as
$\widetilde{X}$, there exists a measurable function $G:D\to D$ satisfying
$\widetilde{L}^{\widetilde{X}}_{0}=G(\widetilde{X})$ and $L^{X}_{0}=G(X)$, see
[5, Section III.3]. As a consequence, the distributions of the pairs
$(H(L^{\widetilde{X}}_{0}),L^{\widetilde{X}}_{0})$ and
$(S_{\theta}(pL^{X}_{0}),L^{X}_{0})$ are the same. This, in its turn, ensures
that the distributions of the Lévy processes $(H(t))$ and $(S_{\theta}(pt))$
are the same. ∎

Acknowledgement. The present work was supported by the National Research
Foundation of Ukraine (project 2020.02/0014 ‘Asymptotic regimes of perturbed
random walks: on the edge of modern and classical probability’). The authors
are very grateful to M. I. Portenko for numerous helpful discussions.

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