# Ergodicity of supercritical SDEs driven by $\alpha$-stable processes and
heavy-tailed sampling

Xiaolong Zhang, Xicheng Zhang  Xiaolong Zhang: School of Mathematics and
Statistics, Wuhan University, Wuhan, Hubei 430072, P.R.China
Email<EMAIL_ADDRESS>Xicheng Zhang: School of Mathematics and
Statistics, Wuhan University, Wuhan, Hubei 430072, P.R.China
Email<EMAIL_ADDRESS>
###### Abstract.

Let $\alpha\in(0,2)$ and $d\in\mathbb{N}$. Consider the following stochastic
differential equation (SDE) driven by $\alpha$-stable process in
$\mathbb{R}^{d}$:

${\mathord{{\rm d}}}X_{t}=b(X_{t}){\mathord{{\rm
d}}}t+\sigma(X_{t-}){\mathord{{\rm d}}}L^{\alpha}_{t},\quad
X_{0}=x\in\mathbb{R}^{d},$

where $b:\mathbb{R}^{d}\to\mathbb{R}^{d}$ and
$\sigma:\mathbb{R}^{d}\to\mathbb{R}^{d}\otimes\mathbb{R}^{d}$ are locally
$\gamma$-Hölder continuous with $\gamma\in(0\vee(1-\alpha)^{+},1]$,
$L^{\alpha}_{t}$ is a $d$-dimensional rotationally invariant $\alpha$-stable
process. Under some dissipative and non-degenerate assumptions on $b,\sigma$,
we show the $V$-uniformly exponential ergodicity for the semigroup $P_{t}$
associated with $\\{X_{t}(x),t\geqslant 0\\}$. Our proofs are mainly based on
the heat kernel estimates recently established in [25] through showing the
strong Feller property and the irreducibility of $P_{t}$. It is interesting
that when $\alpha$ goes to zero, the diffusion coefficient $\sigma$ can grow
faster than drift $b$. As applications, we put forward a new heavy-tailed
sampling scheme.

Keywords: $\alpha$-stable processes, Ergodicity, Strong Feller property,
Irreducibility, Heavy-tailed distribution.

This work is partially supported by NNSFC grants of China (Nos. 12131019,
11731009), and the German Research Foundation (DFG) through the Collaborative
Research Centre(CRC) 1283 “Taming uncertainty and profiting from randomness
and low regularity in analysis, stochastics and their applications”.

##  1\. Introduction

Let $\pi(x)\varpropto\mathrm{e}^{-U(x)}$ be a probability density function,
where $U:{\mathbb{R}}^{d}\to{\mathbb{R}}$ is a potential function. The
classical Langevin Monte Carlo (LMC) method is to use the Langevin diffusion
processes to approximate the target distribution $\mu({\mathord{{\rm
d}}}x)=\pi(x){\mathord{{\rm d}}}x$. More precisely, consider the following
stochastic differential equations (SDEs):

$\displaystyle{\mathord{{\rm d}}}X_{t}=-\nabla U(X_{t}){\mathord{{\rm
d}}}t+\sqrt{2}{\mathord{{\rm d}}}B_{t},\ \ X_{0}=x,$ (1.1)

where $(B_{t})_{t\geqslant 0}$ is a $d$-dimensional standard Brownian motion.
Under some regularity assumptions on $U$, it is well known that the above SDE
admits a unique solution $(X_{t})_{t\geqslant 0}$ and $\mu$ is an invariant
measure of the semigroup $P_{t}$ generated by $X_{t}(x)$. Moreover, under the
following dissipativity assumption: for some $c_{0},c_{1}>0$,

$\langle x,\nabla U(x)\rangle\geqslant c_{0}|x|^{2}-c_{1},$

it is also well known that the law of $X_{t}$ exponentially converges to the
stationary distribution $\mu$ in some sense as $t\to\infty$ (cf. [3, 26]),
i.e., $P_{t}$ is exponentially ergodic. In particular, it provides a way to
generate samples from $\mu$ by the Euler discretization for SDE (1.1) (cf.
[3]). Nowadays, there are numerous works about the analysis of the performance
of LMC algorithms (see [10, 11, 13] and references therein).

On the other hand, in statistics physics, the Langevin diffusion $X_{t}$
represents the position of a particle at time $t$ which is influenced by some
random force, where the random force is usually the sum of many i.i.d. random
pulses with finite variance. By the central limit theorem, the sum of the
pulses converges to a Gaussian distribution. While, if the random pulses have
infinite variance, then the sum of the pulses would converge to an
$\alpha$-stable distribution, a typical class of heavy-tailed distribution.
Moreover, it was already shown in [28] that for $d=1$ and $U(x)=|x|^{\beta}$,
the diffusion process defined by SDE (1.1) is exponentially ergodic if and
only if $\beta\geqslant 1$. Therefore, using continuous Langevin diffusion
(1.1) to simulate the heavy-tailed distribution is not any more a good choice.
Thus, instead of SDE (1.1), it is quite natural to consider the following SDE:

$\displaystyle{\mathord{{\rm d}}}X_{t}=b(X_{t}){\mathord{{\rm
d}}}t+{\mathord{{\rm d}}}L^{\alpha}_{t},\ \ \alpha\in(0,2),$ (1.2)

where $(L^{\alpha}_{t})_{t\geqslant 0}$ is a $d$-dimensional rotationally
invariant $\alpha$-stable process with generator $\Delta^{\frac{\alpha}{2}}$,
the usual fractional Laplacian, and

$b(x)=-\frac{\Gamma(\frac{d-2+\alpha}{2})}{\pi^{\frac{d}{2}}2^{\frac{2-\alpha}{2}}\Gamma(\frac{2-\alpha}{2})}\mathrm{e}^{U(x)}\int_{{\mathbb{R}}^{d}}\frac{\mathrm{e}^{-U(y)}\nabla
U(y)}{|x-y|^{d-(2-\alpha)}}{\mathord{{\rm
d}}}y=\mathrm{e}^{U(x)}\Delta^{\frac{\alpha-2}{2}}(\nabla\mathrm{e}^{-U})(x),$

where $d\geqslant 2$ and $\Delta^{\frac{\alpha-2}{2}}$ is the Riesz potential
(cf. [31, page117]). Formally, one sees that

$\Delta^{\frac{\alpha}{2}}\mathrm{e}^{-U}-\mathord{{\rm
div}}(b\mathrm{e}^{-U})\equiv 0,$

which implies that $\mu$ is an invariant measure of SDE (1.2). Rigorous
theoretical results for the exponential ergodicity of the above SDE (1.2) were
given in [16, Theorem 1.1], and therein, some explicit conditions on $U$ are
provided. This approach called Fractional Langevin Monte Carlo was firstly
introduced in [29, 32]. It has been demonstrated to be useful in modern
machine learning both in optimization and heavy-tailed sampling [15, 30, 36].

Motivated by the above mentioned works, in this paper we consider the
following SDE driven by multiplicative $\alpha$-stable noises:

$\displaystyle{\mathord{{\rm d}}}X_{t}=b(X_{t}){\mathord{{\rm
d}}}t+\sigma(X_{t-}){\mathord{{\rm d}}}L^{\alpha}_{t},\ \ X_{0}=x,$ (1.3)

where $b:\mathbb{R}^{d}\to\mathbb{R}^{d}$ and
$\sigma:\mathbb{R}^{d}\to\mathbb{R}^{d}\otimes\mathbb{R}^{d}$ are two Borel
measurable functions. The main contributions of this paper are two-folds:

  1. (i)

Under some dissipativity, non-degeneracy and locally Hölder regularity
assumptions on $b$ and $\sigma$, we establish the exponential ergodicity of
SDE (1.3) for all $\alpha\in(0,2)$, which, in particular, covers the
supercritical regime $\alpha\in(0,1)$.

  2. (ii)

For a large class of heavy-tailed distributions, we put forward a new
ergodicity SDE driven by multiplicative $\alpha$-stable processes to simulate
it. Compared with the recent work [16], our conditions on the potential
functions are simpler and easier to realize in the computer.

Now we make the following assumptions on $b$ and $\sigma$:

  1. (Hloc)

For any $m\in{\mathbb{N}}$, there is a constant $C_{m}\geqslant 1$ such that
for all $|x|\leqslant m$ and $\xi\in{\mathbb{R}}^{d}$,

$C_{m}^{-1}|\xi|\leqslant|\sigma(x)\xi|\leqslant C_{m}|\xi|.$

Moreover, there are $\gamma\in((1-\alpha)^{+},1]$ and locally bounded
measurable functions $\ell_{1}$ and $\ell_{2}$ on ${\mathbb{R}}^{d}$ such that
for all $x\in{\mathbb{R}}^{d}$ and $|h|\leqslant 1$,

$\displaystyle|b(x+h)-b(x)|\leqslant\ell_{1}(x)|h|^{\gamma},\
|\sigma(x+h)-\sigma(x)|\leqslant\ell_{2}(x)|h|^{\gamma}.$ (1.4)

  1. (H${}^{r,q}_{\rm glo}$)

For given $r>-\alpha$, there are $\varepsilon_{0}\in(0,1]$ small enough and
$c_{0},c_{1}>0$ such that for all $|x|>1$,

$\displaystyle\langle
x,b(x)\rangle+\varepsilon_{0}\ell_{1}(x)|x|+q(\|\sigma(x)\|+\varepsilon_{0}\ell_{2}(x))^{\alpha}|x|^{2-\alpha}\leqslant-
c_{0}|x|^{2+r}+c_{1},$ (1.5)

where $q$ is defined by (3.1) below.

######  Remark 1.1.

Notice that (1.4) is equivalent to the locally $\gamma$-Hölder continuity of
$b$ and $\sigma$. If $b,\sigma$ are locally Lipschitz, i.e. $\gamma=1$ in
(1.4), then one can take $\varepsilon_{0}=0$ in (1.5). When $\gamma<1$, in
order to adopt the stopping time to localize the coefficients, we need to use
the pathwise uniqueness of strong solutions. Thus we must mollify the
coefficients and require $\varepsilon_{0}>0$ in (1.5) to have some uniform
estimates for the approximation coefficients. Examples of $b,\sigma$
satisfying (H${}^{r,q}_{\rm glo}$) shall be provided below. In particular, $b$
and $\sigma$ are allowed to be polynomial growth. It is interesting that when
$\alpha\in(0,1)$, $\sigma$ can grow faster than $b$ (see Example 1.3 below).

Under the above assumptions, it is well known that there is a unique weak
solution to SDE (1.3) (see [7]). The semigroup associated with the Markov
process $X_{t}(x)$ is then defined by

$P_{t}f(x):=\mathbb{E}f(X_{t}(x)),\ \ f\in C_{b}({\mathbb{R}}^{d}).$

By Itô’s formula, one sees that the generator of $P_{t}$ is given by

$\displaystyle\mathscr{L}f(x):=\mathcal{L}_{\sigma}f(x)+\langle b(x),\nabla
f(x)\rangle,$ (1.6)

with

$\displaystyle\mathcal{L}_{\sigma}f(x)=\frac{\alpha
2^{\alpha}\Gamma(\frac{d+\alpha}{2})}{\Gamma(\frac{d}{2})\Gamma(\frac{2-\alpha}{2})}\int_{\mathbb{R}^{d}}[f(x+\sigma(x)z)+f(x-\sigma(x)z)-2f(x)]{\mathord{{\rm
d}}}z/|z|^{d+\alpha}.$ (1.7)

In particular,

$\displaystyle\Delta^{\frac{\alpha}{2}}f(x)=\frac{\alpha
2^{\alpha}\Gamma(\frac{d+\alpha}{2})}{\Gamma(\frac{d}{2})\Gamma(\frac{2-\alpha}{2})}\int_{\mathbb{R}^{d}}[f(x+z)+f(x-z)-2f(x)]{\mathord{{\rm
d}}}z/|z|^{d+\alpha}.$ (1.8)

We call SDE (1.3) being supercritical for $\alpha\in(0,1)$ since in this case,
from the view point of PDE, the drift term plays a dominant role compared with
the diffusion term ${\mathcal{L}}_{\sigma}$ (see [7]).

Our main result of this paper is

######  Theorem 1.2.

Let $r>-\alpha$ and $p\in((-r)\vee 0,\alpha)$. Define $V_{p}(x):=1+|x|^{p}$.
Suppose that (Hloc) and (H${}^{r,q}_{\rm glo}$) hold. Then there is a unique
invariant probability measure $\mu$ associated with $(P_{t})_{t\geqslant 0}$.
Moreover, we have

(i) If $r\geqslant 0$, then there are $\lambda,C_{p}>0$ such that for all
$t>0$ and $x\in{\mathbb{R}}^{d}$,

$\sup_{\|\varphi/V_{p}\|_{\infty}\leqslant
1}|P_{t}\varphi(x)-\mu(\varphi)|\leqslant C_{p}\mathrm{e}^{-\lambda
t}V_{p}(x).$

(ii) If $r>0$, then there are $\lambda,C>0$ such that for all $t>0$,

$\sup_{x\in{\mathbb{R}}^{d}}\|P_{t}(x.\cdot)-\mu\|_{\rm Var}\leqslant
C\mathrm{e}^{-\lambda t}.$

Below we provide two examples to illustrate our main result.

######  Example 1.3.

The following example shows that $b$ and $\sigma$ can be polynomial decay and
growth:

$b(x)=-x(1+|x|^{2})^{\beta/2},\ \
\sigma(x)=(1+|x|^{2})^{\gamma/2}{\mathbb{I}},\ \ \beta\in(-\alpha,\infty),\
\gamma\in(-\infty,1+\tfrac{\beta}{\alpha}).$

For $|x|>1$, by Young’s inequality, we have

$\displaystyle\langle
x,b(x)\rangle+q\|\sigma(x)\|^{\alpha}|x|^{2-\alpha}=-|x|^{2}(1+|x|^{2})^{\frac{\beta}{2}}+q(1+|x|^{2})^{\frac{\alpha\gamma}{2}}|x|^{2-\alpha}$
$\displaystyle\qquad\leqslant-2^{\frac{\beta}{2}\wedge
0}|x|^{2+\beta}+2^{\frac{\alpha(\gamma\vee
0)}{2}}q|x|^{2-\alpha+\alpha(\gamma\vee 0)}\leqslant-2^{(\frac{\beta}{2}\wedge
0)-1}|x|^{2+\beta}+c_{1}.$

In particular, when $\alpha>1$, $\beta$ can be less than $-1$ so that $b$ is
polynomial decay. Moreover, for fixed $\beta>0$, when $\alpha\downarrow 0$,
$\gamma$ can go to infinity. In other words, $\sigma$ can grow faster than
$b$. This is not surprise since one can image that for the SDE driven by
compound Poisson processes (in some sense, corresponding to $\alpha=0$),
$\sigma$ can be arbitrarily growth.

######  Example 1.4.

The following example shows that $b$ and $\sigma$ can even be exponentially
growth:

$b(x)=-x\mathrm{e}^{|x|},\ \ \sigma(x)=\mathrm{e}^{|x|/\beta}{\mathbb{I}},\ \
\beta\in(\alpha,2).$

It is easy to see that for $|h|\leqslant 1$,

$|b(x+h)-b(x)|\leqslant(1+|x|)\mathrm{e}^{|x|+1}|h|=:\ell_{1}(x)|h|,$

and

$|\sigma(x+h)-\sigma(x)|\leqslant\mathrm{e}^{(|x|+1)/\beta}|h|=:\ell_{2}(x)|h|.$

Thus for any $|x|>1$ and $q>1$, by Young’s inequality, one can choose
$\varepsilon_{0}$ small enough so that

$\displaystyle\langle
x,b(x)\rangle+\varepsilon_{0}\ell_{1}(x)|x|+q(\|\sigma(x)\|+\varepsilon_{0}\ell_{2}(x))^{\alpha}|x|^{2-\alpha}$
$\displaystyle\quad=-|x|^{2}\mathrm{e}^{|x|}+\varepsilon_{0}(1+|x|)|x|\mathrm{e}^{|x|+1}+q(\mathrm{e}^{|x|/\beta}+\varepsilon_{0}\mathrm{e}^{(|x|+1)/\beta})^{\alpha}|x|^{2-\alpha}$
$\displaystyle\quad\leqslant-|x|^{2}\mathrm{e}^{|x|}+\tfrac{1}{4}|x|^{2}\mathrm{e}^{|x|+1}+2q\mathrm{e}^{\alpha|x|/\beta}|x|^{2-\alpha}\leqslant-\tfrac{1}{2}|x|^{2}\mathrm{e}^{|x|}+c_{1}.$

Still, when $\alpha\downarrow 0$, the diffusion $\sigma$ can grow faster than
drift $b$.

When the driven noise is Brownian motion, the ergodicity of SDEs (1.3) has
been studied extensively in the literature [2, 6, 12, 33, 35], etc., and the
dissipativity condition (1.5) usually takes the following form (for example,
see [35, (7.1)]):

$2\langle x,b(x)\rangle+\|\sigma(x)\|^{2}\leqslant-c_{0}|x|^{2}+c_{1}.$

In recent years, there also has been a great interest in the study of the
ergodicity for SDEs driven by Lévy processes, see [18, 20, 34, 35] and the
references therein. To the best of the authors’ knowledge, there exist at
least three well developed methods for studying the exponential ergodicity of
a Markov process defined through an SDE: Coupling method (cf. [16, 18, 19, 20,
21, 22]), Functional inequality method (see [2, 5, 33]), Meyn-Tweedie’s
minorization condition (cf. [14, 23, 24, 26]). Each method has its own merits
and may be adapted to different situations. For the coupling method, in order
to construct a successful coupling it is usually assumed that the following
one-sided monotone condition holds (see [21]): for some $R>0$,

$\langle x-y,\nabla U(x)-\nabla
U(y)\rangle\geqslant\begin{cases}-L|x-y|^{2},\quad&|x-y|<R,\\\
K|x-y|^{2},\quad&|x-y|\geqslant R.\end{cases}$ (1.9)

While, for the functional inequality method, it is usually assumed that the
potential function $U$ is at least $C^{2}$-smooth (see [2, 33]). The obvious
advantage of this method is that the quantitative convergence rate can be
calculated from the parameters. Finally, Meyn-Tweedie’s approach is based on
the Lyapunov estimates together with verifying the strong Feller property and
irreducibility for the associated semigroup.

To prove our main theorem, we adopt the Meyn-Tweedie method. As we know, the
strong Feller property reflects the regularization effects of the noise, which
is usually related to the continuity of the transition density of a Markov
process with respect to the starting point. There are many ways to establish
the strong Feller property, e.g., the gradient estimates, Harnack’s
inequalities, etc.. For the irreducibility, when the driving noise is Brownian
motion, there are several ways for choices: by solving a control problem [24],
by Girsanov’s transformation [35], by the classical supported theorem of SDE
[4]. However, when the driven noise is an $\alpha$-stable noise, the above
method for irreducibility does not work any more. In [35], we directly verify
the positivity of the Dirichlet heat kernel by localization method. Therein,
it is restricted to $\alpha\in(1,2)$ since in this case, the heat kernel of
the operator (1.6) is comparable with the one of the fractional Laplacian
$\Delta^{\frac{\alpha}{2}}$. In a recent work [25], when the conditions in
(Hloc) holds globally, i.e., the constant $C_{m}$ does not depend on $m$ and
$\ell_{1}(x),\ell_{2}(x)$ are constants, we obtain the following two-sided
estimates for the density of SDE (1.3): for any $T>0$ and some $C>1$, and for
all $t\in(0,T]$ and $x,y\in{\mathbb{R}}^{d}$,

$\displaystyle
C^{-1}t(t^{\frac{1}{\alpha}}+|\theta_{t}(x)-y|)^{-d-\alpha}\leqslant
p(t,x,y)\leqslant Ct(t^{\frac{1}{\alpha}}+|\theta_{t}(x)-y|)^{-d-\alpha},$
(1.10)

where $\theta_{t}(x)$ solves the following regularized ODE:

$\displaystyle\dot{\theta}_{t}(x)=b*\phi_{t^{1/\alpha}}(\theta_{t}(x)),\ \
\theta_{0}(x)=x,$ (1.11)

where $(\phi_{\varepsilon})_{\varepsilon>0}$ is a family of standard
mollifiers. The above estimates still allow us to show the irreducibility of
$P_{t}$ by localization. For the existence of invariant measures, we use the
standard Krylov-Bogoliubov’s theorem by showing some Lyapunov’s type estimate.

The paper is organized as follows. In Section 2, we recall some basic notions
about the ergodicity of Markov processes. These notions are standard in
literature (see [12, 27]). Moreover, we also prove a general criterion for the
irreducibility of a Markov process in terms of the heat kernel estimates,
which is motivated by the works of [9, 35]. In Section 3, we prove our main
result Theorem 1.2. For the existence of invariant measures, by the classical
Krylov-Bogoliubov’s method, we show the Lyapunov type estimates for
$X_{t}(x)$. For the uniqueness of invariant measure as well as the exponential
ergodicity, we show the strong Feller property and the irreducibility of
$P_{t}$ by the heat kernel estimates and suitable localization and smoothing
arguments. In Section 4, we present an application of our main result to the
heavy-tailed sampling. In Appendix, the proof of weak convergence for
mollifying SDEs is provided for readers’ convenience.

##  2\. Preliminaries

In this section we prepare some notions and criterion about the ergodicity and
irreducibility of general Markov processes in Euclidean spaces. Fix
$N\in{\mathbb{N}}$. Let ${\mathbb{D}}$ be the space of all cádlág functions
from $[0,\infty)$ to ${\mathbb{R}}^{N}$, which is endowed with the Skorokhod
topology so that ${\mathbb{D}}$ becomes a Polish space. Let
$X_{t}(\omega):=\omega_{t}$ be the coordinate process over ${\mathbb{D}}$. For
$s>0$, let $\theta_{s}:{\mathbb{D}}\to{\mathbb{D}}$ be the shift operator:

$\theta_{s}(\omega)(t)=\omega(t+s),\ \ t\geqslant 0.$

Let $({\mathbb{P}}_{x})_{x\in{\mathbb{R}}^{N}}$ be a family of probability
measures over ${\mathbb{D}}$ so that ${\mathscr{M}}:=\\{(X_{t})_{t\geqslant
0},({\mathbb{P}}_{x})_{x\in{\mathbb{R}}^{N}}\\}$ forms a family of Markov
processes with regard to the natural filtration
$({\mathscr{F}}_{t})_{t\geqslant 0}$. More precisely,

(i) For each $x\in{\mathbb{R}}^{N}$, ${\mathbb{P}}_{x}(X_{0}=x)=1$.

(ii) For each $A\in{\mathscr{B}}({\mathbb{D}})$, $x\mapsto{\mathbb{P}}_{x}(A)$
is Borel measurable.

(iii) For any $s<t$ and $A\in{\mathscr{B}}({\mathbb{D}})$, it holds that

$\displaystyle{\mathbb{P}}_{x}(A\circ\theta_{s}|{\mathscr{F}}_{s})={\mathbb{P}}_{X_{s}}(A).$
(2.1)

Let ${\mathcal{B}}_{b}({\mathbb{R}}^{N})$ be the space of all bounded
measurable functions over ${\mathbb{R}}^{N}$. For $t\geqslant 0$ and
$f\in{\mathcal{B}}_{b}({\mathbb{R}}^{N})$, the semigroup associated with
${\mathscr{M}}$ is defined by

$P_{t}f(x):={\mathbb{E}}_{x}f(X_{t}),\ \ x\in{\mathbb{R}}^{N},$

where ${\mathbb{E}}_{x}$ stands for the expectation with respect to
${\mathbb{P}}_{x}$.

The following notions are standard and can be found in [12, 26].

  1. $\bullet$

A probability measure $\mu\in{\mathcal{P}}({\mathbb{R}}^{N})$ is called an
invariant probability measure of $(P_{t})_{t\geqslant 0}$, if

$\mu(P_{t}f)=\mu(f),\ \ \forall t\geqslant 0,\
f\in{\mathcal{B}}_{b}({\mathbb{R}}^{N}).$

  2. $\bullet$

We call $(P_{t})_{t\geqslant 0}$ ergodic, if there exists a unique invariant
probability measure $\mu$ of $(P_{t})_{t\geqslant 0}$, equivalently, for any
$x\in{\mathbb{R}}^{N}$ and $f\in{\mathcal{B}}_{b}({\mathbb{R}}^{N})$,

$\lim_{t\to\infty}\frac{1}{t}\int^{t}_{0}P_{s}f(x){\mathord{{\rm
d}}}s=\mu(f).$

  3. $\bullet$

Let $V:{\mathbb{R}}^{N}\to[1,\infty)$ be a measurable function. We call
$(P_{t})_{t\geqslant 0}$ be $V$-uniformly exponential ergodic if there are
constants $C,\lambda>0$ and an invariant probability measure $\mu$ such that

$\sup_{\|\phi/V\|_{\infty}\leqslant 1}|P_{t}\phi(x)-\mu(\phi)|\leqslant
C\mathrm{e}^{-\lambda t}V(x),\ \ t\geqslant 0,\ x\in{\mathbb{R}}^{N}.$

In particular, if $V\equiv 1$, then we call $(P_{t})_{t\geqslant 0}$ be
uniformly exponentially ergodic.

  4. $\bullet$

$(P_{t})_{t\geqslant 0}$ is said to be strong Feller, if $P_{t}f\in
C_{b}(\mathbb{R}^{N})$ for any $t>0$ and
$f\in\mathcal{B}_{b}(\mathbb{R}^{N})$.

  5. $\bullet$

$(P_{t})_{t\geqslant 0}$ is called irreducible, if for any $t>0$, open ball
$B$ and $x\in\mathbb{R}^{N}$, $P_{t}\mathbf{1}_{B}(x)>0$.

The following general result is taken from [14, Theorem 2.5].

######  Theorem 2.1.

Suppose that $(P_{t})_{t\geqslant 0}$ is strong Feller and irreducible. If
there are $p,\lambda>0$ and $C_{1},C_{2}>0$ such that

${\mathbb{E}}_{x}|X_{t}|^{p}\leqslant C_{1}|x|^{p}\mathrm{e}^{-\lambda
t}+C_{2},\ \ t\geqslant 0,\ x\in{\mathbb{R}}^{N},$

then $(P_{t})_{t\geqslant 0}$ is $V$-uniformly ergodic with $V(x)=1+|x|^{p}$.
If there are $p,T_{0},C_{3}>0$ such that

${\mathbb{E}}_{x}|X_{t}|^{p}\leqslant C_{3},\ \ t\geqslant T_{0},\
x\in{\mathbb{R}}^{N},$

then $(P_{t})_{t\geqslant 0}$ is uniformly exponentially ergodic.

Next we present a general criterion for irreducibility in terms of the heat
kernel estimates. We believe that it could be used to other cases.

Let $\rho:(0,\infty)\times(0,\infty)\to(0,\infty)$ be a continuous function
with the properties that for any $t>0$, $r\mapsto\rho(t,r)$ is decreasing on
$(0,\infty)$, and for any $0<\delta<T<\infty$,

$\displaystyle\sup_{t\in[\delta,T]}\sup_{r>0}\rho(t,r)<\infty.$ (2.2)

In the following we make the following assumptions about the Markov process
${\mathscr{M}}$:

  1. (A1)

Suppose that for each $t>0$, with respect to ${\mathbb{P}}_{x}$, $X_{t}$
admits a family of transition probability density function $p(t,x,y)$ in
${\mathbb{R}}^{N}$ so that

$\displaystyle\mathbb{P}_{x}(X_{t}\in A)=\int_{A}p(t,x,y){\mathord{{\rm
d}}}y,\ \ \forall A\in{\mathscr{B}}({\mathbb{R}}^{N}),\ t>0,\
x\in{\mathbb{R}}^{N}.$ (2.3)

Moreover, $t\mapsto X_{t}$ is stochastically continuous, equivalently,

$\displaystyle{\mathbb{P}}_{x}(X_{t}\not=X_{t-})=0,\ \ \forall
t>0,x\in{\mathbb{R}}^{N}.$ (2.4)

  2. (A2)

We suppose that for any $T>0$, there are constants $\lambda_{0},C_{0}\geqslant
1$ such that for any $t\in(0,T]$ and $x\in{\mathbb{R}}^{N}$, and for Lebesgue
almost all $y\in{\mathbb{R}}^{N}$,

$\displaystyle C_{0}^{-1}\rho(t,\lambda_{0}\Gamma_{t}(x,y))\leqslant
p(t,x,y)\leqslant C_{0}\rho(t,\Gamma_{t}(x,y)/\lambda_{0}),$ (2.5)

where
$\Gamma_{t}(x,y):(0,\infty)\times{\mathbb{R}}^{N}\times{\mathbb{R}}^{N}\to(0,\infty)$
is a measurable function, and for each $y\in{\mathbb{R}}^{N}$,

$(t,x)\mapsto\Gamma_{t}(x,y)$ is continuous.

######  Remark 2.2.

In (A2), for the standard Gaussian case, one usually takes
$\rho(t,r)=t^{-d/2}\mathrm{e}^{-r^{2}/t}$ and $\Gamma_{t}(x,y)=|x-y|$. For the
$\alpha$-stable case, if $\alpha\in[1,2)$, one takes
$\rho(t,r)=t(t^{1/\alpha}+r)^{-d-\alpha}$ and $\Gamma_{t}(x,y)=|x-y|$; and if
$\alpha\in(0,1)$, one takes $\Gamma_{t}(x,y)=|x-\theta_{t}(y)|$, where
$\theta_{t}(y)$ is the ODE flow associated with drift $b$ (see (1.11) above).
We would like to emphasize that we do not make any continuity assumption about
$y\mapsto p(t,x,y)$. Usually, the regularity of the density requires more
regular coefficients in the theory of SDEs.

Let $D$ be an open subset of $\mathbb{R}^{N}$. The exit time of $X_{t}$ from
$D$ is defined by

$\tau_{D}:=\inf\big{\\{}t>0,X_{t}\notin D\big{\\}}.$

We have the following general result.

######  Theorem 2.3.

Let $D_{0}\Subset D$ be two connected domains of ${\mathbb{R}}^{N}$ so that
$D$ has Lebesgue-zero measure boundary. Suppose that (A1) and (A2) hold, and
the above $\Gamma_{t}(x,y)$ and $\rho(t,r)$ satisfy the following conditions:
there are $t_{0},\delta_{0}>0$ small enough and functions $\ell_{0}(t)$,
$\ell_{1}(t)$ defined on $(0,t_{0}]$ such that for all $t\in(0,t_{0}]$,
$y_{0}\in D_{0}$, $x,y\in B_{\delta_{0}}(y_{0})\subset D_{0}$ and
$x^{\prime}\notin D$,

$\displaystyle\Gamma_{t}(x,y)\leqslant\ell_{0}(t),\
\Gamma_{t}(x^{\prime},y)\geqslant\ell_{1}(t),$ (2.6)

and for the $C_{0},\lambda_{0}$ in (2.5),
$t\mapsto\rho(t,\ell_{1}(t)/\lambda_{0})$ is increasing on $(0,t_{0}]$, and

$\displaystyle C_{0}^{-1}\rho\big{(}t,\lambda_{0}\ell_{0}(t)\big{)}\geqslant
2C_{0}\rho\big{(}t,\ell_{1}(t)/\lambda_{0}).$ (2.7)

Then for any $x_{0},y_{0}\in D_{0}$ and $r_{1},r_{2}<{\rm dist}(D_{0},D^{c})$,

$\displaystyle\inf_{x\in B_{r_{1}}(x_{0})}{\mathbb{P}}_{x}\Big{(}X_{t}\in
B_{r_{2}}(y_{0});t<\tau_{D}\Big{)}\geqslant c_{0}>0,$ (2.8)

where $c_{0}$ only depends on $\rho,\lambda_{0},C_{0},t_{0},\delta_{0}$ and
$r_{1},r_{2}$, $x_{0},y_{0},D_{0},D$.

###### Proof.

Let $t_{0}$ and $\delta_{0}$ be as in the assumptions. We divide the proofs
into two steps.

(i) In this step we show that for all $y_{0}\in D_{0}$,
$\delta\in(0,\frac{\delta_{0}}{2}]$ and $t\in(0,t_{0}]$,

$\displaystyle\inf_{x\in B_{2\delta}(y_{0})}{\mathbb{P}}_{x}\Big{(}X_{t}\in
B_{\delta}(y_{0});t<\tau_{D}\Big{)}\geqslant
C_{0}\rho\big{(}t,\ell_{1}(t)/\lambda_{0}\big{)}|B_{\delta}(y_{0})|>0.$ (2.9)

Fix $s\in(0,t)$ and $n\in{\mathbb{N}}$. We define a stopping time $T_{n}$ as
follows:

$T_{n}=\begin{cases}ks2^{-n}\quad&\text{if}\
(k-1)s2^{-n}\leqslant\tau_{D}<ks2^{-n},\\\ \infty\quad&\text{if}\
\tau_{D}\geqslant s.\end{cases}$

Clearly, on $\\{\tau_{D}<s\\}$,

$s\geqslant T_{n}\downarrow\tau_{D}\ \ \mbox{as $n\to\infty$.}$

For any bounded $A\in{\mathscr{B}}(D)$, by the Markov property and (2.3), we
have

$\displaystyle\mathbb{P}_{x}(X_{t}\in A;\tau_{D}<s)$
$\displaystyle=\sum^{2^{n}}_{k=1}\mathbb{P}_{x}(X_{t}\in
A;(k-1)s2^{-n}\leqslant\tau_{D}<ks2^{-n})$
$\displaystyle=\sum^{2^{n}}_{k=1}\mathbb{P}_{x}(X_{t}\in A;T_{n}=ks2^{-n})$
$\displaystyle=\sum^{2^{n}}_{k=1}\mathbb{E}_{x}\Big{(}{\mathbb{P}}_{X_{T_{n}}}(X_{t-T_{n}}\in
A);T_{n}=ks2^{-n}\Big{)}$
$\displaystyle=\sum^{2^{n}}_{k=1}\mathbb{E}_{x}\left(\int_{A}p(t-T_{n},X_{T_{n}},y){\mathord{{\rm
d}}}y;T_{n}=ks2^{-n}\right)$
$\displaystyle=\mathbb{E}_{x}\left(\int_{A}p(t-T_{n},X_{T_{n}},y){\mathord{{\rm
d}}}y;\tau_{D}<s\right)$
$\displaystyle\stackrel{{\scriptstyle\eqref{DD2}}}{{\leqslant}}C_{0}\mathbb{E}_{x}\left(\int_{A}\rho\big{(}t-T_{n},\Gamma_{t-T_{n}}(X_{T_{n}},y)/\lambda_{0}\big{)}{\mathord{{\rm
d}}}y;\tau_{D}<s\right).$

Letting $n\to\infty$ and by (2.2) and the dominated convergence theorem, we
get

$\mathbb{P}_{x}(X_{t}\in A;\tau_{D}<s)\leqslant
C_{0}\mathbb{E}_{x}\left(\int_{A}\rho\big{(}t-\tau_{D},\Gamma_{t-\tau_{D}}(X_{\tau_{D}},y)/\lambda_{0}\big{)}{\mathord{{\rm
d}}}y;\tau_{D}<s\right).$

Below we take $A=B_{\delta}(y_{0})$. For $y\in B_{\delta}(y_{0})$, since
$X_{\tau_{D}}\notin D$ and $r\to\rho(t-\tau_{D},r)$ is decreasing,
$t\mapsto\rho(t,\ell_{1}(t)/\lambda_{0})$ is increasing, by (2.6) we have

$\rho\big{(}t-\tau_{D},\Gamma_{t-\tau_{D}}(X_{\tau_{D}},y)/\lambda_{0}\big{)}\leqslant\rho\big{(}t-\tau_{D},\ell_{1}(t-\tau_{D})/\lambda_{0}\big{)}\leqslant\rho\big{(}t,\ell_{1}(t)/\lambda_{0}\big{)}.$

Hence, for any $s<t$,

$\displaystyle\mathbb{P}_{x}{\Big{(}}X_{t}\in
B_{\delta}(y_{0});\tau_{D}<s{\Big{)}}\leqslant
C_{0}\int_{B_{\delta}(y_{0})}\rho\big{(}t,\ell_{1}(t)/\lambda_{0}\big{)}{\mathord{{\rm
d}}}y.$

On the other hand, by (2.4) and (2.5), we have

$\displaystyle{\mathbb{P}}_{x}(\tau_{D}=t)$
$\displaystyle={\mathbb{P}}_{x}(\tau_{D}=t,X_{t}\in\partial
D)+{\mathbb{P}}_{x}(\tau_{D}=t,X_{t}\notin\partial D,X_{s}\in D,s<\tau_{D})$
$\displaystyle\leqslant{\mathbb{P}}_{x}(X_{t}\in\partial
D)+{\mathbb{P}}_{x}(X_{t}\not=X_{t-})=0.$

Thus, by the lower bound in (2.5) and the above estimates, we arrive at

$\displaystyle\mathbb{P}_{x}{\Big{(}}X_{t}\in
B_{\delta}(y_{0});t<\tau_{D}{\Big{)}}$
$\displaystyle=\mathbb{P}_{x}{\Big{(}}X_{t}\in
B_{\delta}(y_{0}){\Big{)}}-\mathbb{P}_{x}{\Big{(}}X_{t}\in
B_{\delta}(y_{0});\tau_{D}<t{\Big{)}}$
$\displaystyle=\int_{B_{\delta}(y_{0})}p(t,x,y){\mathord{{\rm
d}}}y-\lim_{s\uparrow t}\mathbb{P}_{x}{\Big{(}}X_{t}\in
B_{\delta}(y_{0});\tau_{D}<s{\Big{)}}$
$\displaystyle\geqslant\int_{B_{\delta}(y_{0})}\Big{(}C_{0}^{-1}\rho\big{(}t,\lambda_{0}\Gamma_{t}(x,y)\big{)}-C_{0}\rho\big{(}t,\ell_{1}(t)/\lambda_{0}\big{)}\Big{)}{\mathord{{\rm
d}}}y.$

For $x\in B_{2\delta}(y_{0}),y\in B_{\delta}(y_{0})$, since $r\to\rho(t,r)$ is
decreasing, by (2.6), we obtain

$\displaystyle\mathbb{P}_{x}{\Big{(}}X_{t}\in
B_{\delta}(y_{0});t<\tau_{D}{\Big{)}}$
$\displaystyle\geqslant\Big{(}C_{0}^{-1}\rho\big{(}t,\lambda_{0}\ell_{0}(t)\big{)}-C_{0}\rho\big{(}t,\ell_{1}(t)/\lambda_{0}\big{)}\Big{)}|B_{\delta}(y_{0})|$
$\displaystyle\\!\\!\\!\stackrel{{\scriptstyle\eqref{AA2}}}{{\geqslant}}C_{0}\rho\big{(}t,\ell_{1}(t)/\lambda_{0}\big{)}|B_{\delta}(y_{0})|.$

Thus we get (2.9).

(ii) By (2.3), there is a density $p^{D}(t,x,y)$ so that for each
$A\in{\mathscr{B}}(D)$,

$\int_{A}p^{D}(t,x,y){\mathord{{\rm d}}}y=\mathbb{P}_{x}(X_{t}\in
A;t<\tau_{D}).$

Moreover, by the Markov property, we have for each $0<s<t$,

$\displaystyle\int_{A}p^{D}(t,x,y){\mathord{{\rm
d}}}y=\int_{D}p^{D}(s,x,z){\mathord{{\rm
d}}}z\int_{A}p^{D}(t-s,z,y){\mathord{{\rm d}}}y.$ (2.10)

Indeed, by $s+\tau_{D}\circ\theta_{s}=\tau_{D}$ on $\\{s<\tau_{D}\\}$ and
(2.1),

$\displaystyle\mathbb{P}_{x}(X_{t}\in A;t<\tau_{D})$
$\displaystyle={\mathbb{E}}_{x}\Big{(}\mathbb{P}_{x}\big{(}X_{t}\in
A;t<\tau_{D}|{\mathscr{F}}_{s}\big{)}\Big{)}$
$\displaystyle={\mathbb{E}}_{x}\Big{(}\mathbb{P}_{x}\big{(}X_{t-s}\circ\theta_{s}\in
A;t<s+\tau_{D}\circ\theta_{s},s<\tau_{D}|{\mathscr{F}}_{s}\big{)}\Big{)}$
$\displaystyle={\mathbb{E}}_{x}\Big{(}\mathbb{P}_{X_{s}}\big{(}X_{t-s}\in
A;t-s<\tau_{D}\big{)};s<\tau_{D}\Big{)}$
$\displaystyle=\int_{D}p^{D}(s,x,z)\int_{A}p^{D}(t-s,z,y){\mathord{{\rm
d}}}y{\mathord{{\rm d}}}z.$

Let $x_{0}\in D_{0}$. Since $D_{0}$ is connected, there is a continuous curve
${\mathcal{C}}\subset D_{0}$ connecting $x_{0}$ and $y_{0}$. Let

$\delta:=({\rm dist}(D_{0},\partial D)\wedge\delta_{0})/4.$

It is easy to see that there exists $n\in\mathbb{N}$ large enough and
$\\{x_{i},i=0,1,\cdots,n+1\\}\subset{\mathcal{C}}$ such that

$t\leqslant nt_{0},\ \ x_{i}\in B_{\delta}(x_{i-1}),\ i=1,\cdots,n+1,\
B_{\delta}(x_{n+1})\subset B_{r_{2}}(y_{0}).$

For each $j=1,\cdots,n+1$, since $B_{\delta}(x_{j-1})\subset
B_{2\delta}(x_{j})$, by what we have proved in Step (i),

$\displaystyle\inf_{y_{j-1}\in
B_{\delta}(x_{j-1})}\int_{B_{\delta}(x_{j})}p^{D}(\tfrac{t}{n},y_{j-1},y_{j}){\mathord{{\rm
d}}}y_{j}$ $\displaystyle\geqslant\inf_{y_{j-1}\in
B_{2\delta}(x_{j})}\int_{B_{\delta}(x_{j})}p^{D}(\tfrac{t}{n},y_{j-1},y_{j}){\mathord{{\rm
d}}}y_{j}$ $\displaystyle\geqslant
C_{0}\rho\big{(}\tfrac{t}{n},\ell_{1}(\tfrac{t}{n})/\lambda_{0}\big{)}|B_{\delta}(y_{0})|>0.$

Hence, for all $x\in B_{\delta}(x_{0})$, by (2.10),

$\displaystyle\int_{B_{r_{2}}(y_{0})}p^{D}(t,x,y){\mathord{{\rm
d}}}y=\int_{D}p^{D}(\tfrac{t}{n},x,y_{1}){\mathord{{\rm
d}}}y_{1}\cdots\int_{D}p^{D}(\tfrac{t}{n},y_{n-1},y_{n}){\mathord{{\rm
d}}}y_{n}\int_{B_{r_{2}}(y_{0})}p^{D}(\tfrac{t}{n},y_{n},y){\mathord{{\rm
d}}}y$
$\displaystyle\qquad\geqslant\int_{B_{\delta}(x_{1})}p^{D}(\tfrac{t}{n},x,y_{1}){\mathord{{\rm
d}}}y_{1}\cdots\int_{B_{\delta}(x_{n})}p^{D}(\tfrac{t}{n},y_{n-1},y_{n}){\mathord{{\rm
d}}}y_{n}\int_{B_{\delta}(x_{n+1})}p^{D}(\tfrac{t}{n},y_{n},y){\mathord{{\rm
d}}}y$
$\displaystyle\qquad\geqslant\Big{(}C_{0}\rho\big{(}\tfrac{t}{n},\ell_{1}(\tfrac{t}{n})/\lambda_{0}\big{)}|B_{\delta}(y_{0})|\Big{)}^{n+1}.$

Thus,

$\inf_{x\in B_{\delta}(x_{0})}{\mathbb{P}}_{x}\Big{(}X_{t}\in
B_{r_{2}}(y_{0});t<\tau_{D}\Big{)}\geqslant\Big{(}C_{0}\rho\big{(}\tfrac{t}{n},\ell_{1}(\tfrac{t}{n})/\lambda_{0}\big{)}|B_{\delta}(y_{0})|\Big{)}^{n+1}>0.$

For general $r_{1}\in(0,{\rm dist}(D_{0},D))$, it follows by the finitely
covering technique. ∎

##  3\. Proof of Theorem 1.2

In this section we prove Theorem 1.2 by verifying the conditions in Theorem
2.1. Below we fix

$r>-\alpha,\ \ p\in((-r)\vee 0,\alpha),$

and define

$\displaystyle V_{p}(x):=(1+|x|^{2})^{p/2},\quad
q:=\frac{2^{4+2\alpha}\pi^{\frac{d}{2}}\Gamma(\frac{d+\alpha}{2})}{\Gamma(\frac{d}{2})^{2}\Gamma(\frac{2-\alpha}{2})}\left(\frac{\alpha}{2-\alpha}+\frac{\alpha}{(\alpha-p)p}\right).$
(3.1)

First of all, we show that $V_{p}$ is a Lyapunov function of the operator
${\mathscr{L}}$.

######  Lemma 3.1.

Suppose that $b,\sigma$ are locally bounded and for some $c_{0},c_{1}>0$,

$\displaystyle q\|\sigma(x)\|^{\alpha}|x|^{2-\alpha}+\langle
x,b(x)\rangle\leqslant-c_{0}|x|^{2+r}+c_{1},\ \ |x|>1.$ (3.2)

Then there are $\kappa_{0},\kappa_{1}>0$ such that for all
$x\in{\mathbb{R}}^{d}$,

$\displaystyle{\mathscr{L}}V_{p}(x)={\mathcal{L}}_{\sigma}V_{p}(x)+\langle
b(x),\nabla
V_{p}(x)\rangle\leqslant-\kappa_{0}V_{p}(x)^{1+\frac{r}{p}}+\kappa_{1}.$ (3.3)

###### Proof.

It suffices to prove (3.3) for $|x|>1$. By (1.7), we make the following
decomposition:

$\displaystyle c^{-1}_{d,\alpha}{\mathcal{L}}_{\sigma}V_{p}(x)$
$\displaystyle=\int_{|z|\leqslant\frac{|x|}{2\|\sigma(x)\|}}\Big{[}V_{p}(x+\sigma(x)z)+V_{p}(x-\sigma(x)z)-2V_{p}(x)\Big{]}\frac{{\mathord{{\rm
d}}}z}{|z|^{d+\alpha}}$
$\displaystyle+\int_{|z|>\frac{|x|}{2\|\sigma(x)\|}}\Big{[}V_{p}(x+\sigma(x)z)+V_{p}(x-\sigma(x)z)-2V_{p}(x)\Big{]}\frac{{\mathord{{\rm
d}}}z}{|z|^{d+\alpha}}=:I_{1}+I_{2},$

where $c_{d,\alpha}:=\frac{\alpha
2^{\alpha}\Gamma(\frac{d+\alpha}{2})}{\Gamma(\frac{d}{2})\Gamma(\frac{2-\alpha}{2})}$.
For $I_{1}$, denoting by $A(x):=(1+|x|^{2}){\mathbb{I}}-(2-p)x\otimes x$, we
have

$\nabla^{2}V_{p}(x)=p(1+|x|^{2})^{\frac{p}{2}-2}A(x).$

Noting that for any $\xi\in\mathbb{R}^{d}$,

$\displaystyle|\langle\xi,A(x)\xi\rangle|\leqslant(1+|x|^{2})|\xi|^{2}+(2-p)|\langle\xi,x\rangle|^{2}\leqslant
3(1+|x|^{2})|\xi|^{2}.$

By Taylor’s expansion, for $|z|\leqslant\frac{|x|}{2\|\sigma(x)\|}$, we have

$\displaystyle|V_{p}(x+\sigma(x)z)+V_{p}(x-\sigma(x)z)-2V_{p}(x)|$
$\displaystyle\quad\leqslant
3p|\sigma(x)z|^{2}\int^{1}_{0}\int^{1}_{-1}(1+|x+s_{1}s_{2}\sigma(x)z|^{2})^{\frac{p}{2}-1}{\mathord{{\rm
d}}}r_{1}{\mathord{{\rm d}}}r_{2}$ $\displaystyle\quad\leqslant
3p\|\sigma(x)\|^{2}|z|^{2}\Big{(}1+\tfrac{|x|^{2}}{4}\Big{)}^{\frac{p}{2}-1}.$

Hence, for $S_{d}:=2\pi^{\frac{d}{2}}/\Gamma(\frac{d}{2})$,

$\displaystyle I_{1}$ $\displaystyle\leqslant
3p\|\sigma(x)\|^{2}\Big{(}1+\tfrac{|x|^{2}}{4}\Big{)}^{\frac{p}{2}-1}\int_{|z|\leqslant\frac{|x|}{2\|\sigma(x)\|}}|z|^{2-d-\alpha}{\mathord{{\rm
d}}}z$
$\displaystyle=3pS_{d}\|\sigma(x)\|^{2}\Big{(}1+\tfrac{|x|^{2}}{4}\Big{)}^{\frac{p}{2}-1}\int_{0}^{\frac{|x|}{2\|\sigma(x)\|}}r^{1-\alpha}{\mathord{{\rm
d}}}r$
$\displaystyle\leqslant\frac{p2^{2+\alpha}S_{d}}{2-\alpha}\|\sigma(x)\|^{\alpha}|x|^{p-\alpha}.$

For $I_{2}$, noting that

$|(1+|x+z|^{2})^{\frac{p}{2}}-(1+|x|^{2})^{\frac{p}{2}}|\leqslant||x+z|^{2}-|x|^{2}|^{\frac{p}{2}}\leqslant
2^{\frac{p}{2}}|x|^{\frac{p}{2}}|z|^{\frac{p}{2}}+|z|^{p},$

we have

$\displaystyle I_{2}\leqslant
2\int_{|z|>\frac{|x|}{2\|\sigma(x)\|}}\Big{[}2^{\frac{p}{2}}|x|^{\frac{p}{2}}|\sigma(x)z|^{\frac{p}{2}}+|\sigma(x)z|^{p}\Big{]}\frac{{\mathord{{\rm
d}}}z}{|z|^{d+\alpha}}\leqslant\frac{2^{3+\alpha}S_{d}}{\alpha-p}\|\sigma(x)\|^{\alpha}|x|^{p-\alpha}.$

Combining the above calculations, we get for $|x|>1$,

$\displaystyle{\mathcal{L}}_{\sigma}V_{p}(x)\leqslant$ $\displaystyle
2^{3+\alpha}pS_{d}c_{d,\alpha}\left(\frac{1}{2-\alpha}+\frac{1}{(\alpha-p)p}\right)\|\sigma(x)\|^{\alpha}|x|^{p-\alpha}$
$\displaystyle\leqslant$ $\displaystyle
p2^{\frac{p}{2}-1}q\|\sigma(x)\|^{\alpha}|x|^{p-\alpha}$

Thus, then by (3.2) and $\nabla V_{p}(x)=px(1+|x|^{2})^{\frac{p}{2}-1}$, we
have

$\displaystyle{\mathcal{L}}_{\sigma}V_{p}(x)+\langle b(x),\nabla
V_{p}(x)\rangle$ $\displaystyle\leqslant
p2^{\frac{p}{2}-1}q\|\sigma(x)\|^{\alpha}|x|^{p-\alpha}+p(1+|x|^{2})^{\frac{p}{2}-1}\langle
x,b(x)\rangle$ $\displaystyle\leqslant
p(1+|x|^{2})^{\frac{p}{2}-1}\Big{(}q\|\sigma(x)\|^{\alpha}|x|^{2-\alpha}+\langle
x,b(x)\rangle\Big{)}$ $\displaystyle\leqslant
p(1+|x|^{2})^{\frac{p}{2}-1}\big{(}-c_{0}|x|^{2+r}+c_{1}\big{)}$
$\displaystyle\leqslant-\kappa_{0}(1+|x|^{2})^{\frac{p+r}{2}}+\kappa_{1}=-\kappa_{0}V_{p}(x)^{1+\frac{r}{p}}+\kappa_{1},$

where in the second inequality we use $(1+|x|^{2})^{1-\frac{p}{2}}\leqslant
2^{1-\frac{p}{2}}|x|^{2-p}$ with $|x|>1$ , and $\kappa_{0}=pc_{0}$. ∎

As a consequence of the above Lyapunov-type estimate, we have the following
result (see [35, Lemma 7.1]).

######  Lemma 3.2.

Let $X_{t}(x)$ be any weak solution of SDE (1.3) starting from $x$. Under the
assumptions of Lemma 3.1, there exists a constant $C>0$ such that for all
$x\in{\mathbb{R}}^{d}$ and $t>0$,

$\displaystyle\left[\mathbb{E}\left(\sup_{s\in[0,t]}V_{p}(X_{s}(x))^{\frac{1}{2}}\right)\right]^{2}+{\mathbb{E}}\left(\int^{t}_{0}V_{p}(X_{s}(x))^{1+\frac{r}{p}}{\mathord{{\rm
d}}}s\right)\lesssim_{C}V_{p}(x)+t,$ (3.4)

and for the $\kappa_{0}$ in (3.3),

$\displaystyle\mathbb{E}V_{p}(X_{t}(x))\leqslant\left\\{\begin{aligned}
&\mathrm{e}^{-\kappa_{0}t}V_{p}(x)+C,\ \ &r=0,\\\ &C(1+t^{-p/r}),\ \
&r>0.\end{aligned}\right.$ (3.5)

###### Proof.

Let $N(t,{\mathord{{\rm d}}}z)$ be the Poisson random measure associated with
$L^{\alpha}_{t}$, i.e.,

$N(t,\Gamma):=\sum_{s\in(0,t]}{\mathbf{1}}_{\Gamma}(L^{\alpha}_{s}-L^{\alpha}_{s-}),\
\ \Gamma\in{\mathscr{B}}({\mathbb{R}}^{d}).$

Let ${\widetilde{N}}(t,{\mathord{{\rm d}}}z):=N(t,{\mathord{{\rm
d}}}z)-t|z|^{-d-\alpha}{\mathord{{\rm d}}}z$ be the compensated Poisson random
measure. By Lévy-Itô’s decomposition, one can write

$\displaystyle L^{\alpha}_{t}=\int_{|z|<1}z{\widetilde{N}}(t,{\mathord{{\rm
d}}}z)+\int_{|z|\geqslant 1}zN(t,{\mathord{{\rm d}}}z).$

Thus SDE (1.3) can be written as

$\displaystyle{\mathord{{\rm d}}}X_{t}=b(X_{t}){\mathord{{\rm
d}}}t+\int_{|z|<1}\sigma(X_{t-})z{\widetilde{N}}({\mathord{{\rm
d}}}t,{\mathord{{\rm d}}}z)+\int_{|z|\geqslant
1}\sigma(X_{t-})zN({\mathord{{\rm d}}}t,{\mathord{{\rm d}}}z).$ (3.6)

By Itô’s formula (see [1, Theorem 4.4.7]) and (3.3), we have

$\displaystyle V_{p}(X_{t})$
$\displaystyle=V_{p}(x)+\int^{t}_{0}\Big{[}\langle\nabla
V_{p}(X_{s}),b(X_{s})\rangle+{\mathcal{L}}_{\sigma}V_{p}(X_{s})\Big{]}{\mathord{{\rm
d}}}s+M_{t}$ (3.7) $\displaystyle\leqslant
V_{p}(x)+\int^{t}_{0}\Big{[}-\kappa_{0}V_{p}(X_{s})^{1+\frac{r}{p}}+\kappa_{1}\Big{]}{\mathord{{\rm
d}}}s+M_{t},$ (3.8)

where $M_{t}$ is a local cádlág martingale. Let $\tau_{n}$ be a sequence of
stopping times localizing $M_{t}$, i.e., $M_{t\wedge\tau_{n}}$ is a martingale
and $\tau_{n}\uparrow\infty$ as $n\to\infty$. Then we have

${\mathbb{E}}V_{p}(X_{t\wedge\tau_{n}})+\kappa_{0}{\mathbb{E}}\int^{t\wedge\tau_{n}}_{0}V_{p}(X_{s})^{1+\frac{r}{p}}{\mathord{{\rm
d}}}s\leqslant V_{p}(x)+\kappa_{1}t.$

In particular, letting $n\to\infty$ and by Fatou’s lemma, we obtain that for
all $t\geqslant 0$,

${\mathbb{E}}V_{p}(X_{t})+\kappa_{0}{\mathbb{E}}\left(\int^{t}_{0}V_{p}(X_{s})^{1+\frac{r}{p}}{\mathord{{\rm
d}}}s\right)\leqslant V_{p}(x)+\kappa_{1}t.$

Moreover, starting from (3.8), by stochastic Gronwall’s inequality ([35, Lemma
3.7]), we also have

$\left[\mathbb{E}\left(\sup_{s\in[0,t]}V_{p}(X_{s}(x))^{\frac{1}{2}}\right)\right]^{2}\lesssim_{C}V_{p}(x)+t.$

On the other hand, starting from (3.7), as above, we have

${\mathbb{E}}V_{p}(X_{t})=V_{p}(x)+\int^{t}_{0}{\mathbb{E}}\Big{[}\langle\nabla
V_{p}(X_{s}),b(X_{s})\rangle+{\mathcal{L}}_{\sigma}V_{p}(X_{s})\Big{]}{\mathord{{\rm
d}}}s.$

For $r\geqslant 0$, by (3.3) and Jensen’s inequality, we have

$\displaystyle\frac{{\mathord{{\rm d}}}}{{\mathord{{\rm
d}}}t}\mathbb{E}V_{p}(X_{t})\leqslant-\kappa_{0}\mathbb{E}\Big{[}V_{p}(X_{t})^{1+\frac{r}{p}}\Big{]}+\kappa_{1}\leqslant-\kappa_{0}\Big{[}\mathbb{E}V_{p}(X_{t})\Big{]}^{1+\frac{r}{p}}+\kappa_{1}.$

Let $f(t):=\mathbb{E}V_{p}(X_{t})$. The above inequality then reads

$f(t)^{\prime}\leqslant-\kappa_{0}f(t)^{1+\frac{r}{p}}+\kappa_{1}.$

If $r=0$, then

$\displaystyle(\mathrm{e}^{\kappa_{0}t}f(t))^{\prime}\leqslant\kappa_{1}\mathrm{e}^{\kappa_{0}t}\Rightarrow
f(t)\leqslant\mathrm{e}^{\kappa_{0}(s-t)}f(s)+\tfrac{\kappa_{1}}{\kappa_{0}}(1-\mathrm{e}^{\kappa_{0}(s-t)}),\
\ t>s\geqslant 0.$ (3.9)

If $r>0$, then by Young’s inequality,

$\displaystyle(t^{\frac{2p}{r}}f(t))^{\prime}$
$\displaystyle\leqslant\tfrac{2p}{r}t^{\frac{2p}{r}-1}f(t)-\kappa_{0}t^{\frac{2p}{r}}f(t)^{1+\frac{r}{p}}+\kappa_{1}t^{\frac{2p}{r}}\leqslant
Ct^{\frac{p}{r}-1}+\kappa_{1}t^{\frac{2p}{r}}.$

Integrating both sides from $0$ to $t$, we obtain

$t^{\frac{2p}{r}}f(t)\leqslant\tfrac{rC}{p}t^{\frac{p}{r}}+\tfrac{r\kappa_{1}}{r+2p}t^{\frac{2p}{r}+1}\Rightarrow
f(t)\leqslant\tfrac{rC}{p}t^{-\frac{p}{r}}+\tfrac{r\kappa_{1}}{r+2p}t,$

which together with (3.9) yields (3.5). ∎

Let $\phi:\mathbb{R}^{d}\to\mathbb{R}_{+}$ be a smooth density function with
support in the unit ball. Define

$\phi_{\varepsilon}(x):=\varepsilon^{-d}\phi(\varepsilon^{-1}x),\ \
x\in{\mathbb{R}}^{d},\ \varepsilon\in(0,1),$

and

$b_{\varepsilon}(x):=b*\phi_{\varepsilon}(x),\quad\sigma_{\varepsilon}(x):=\sigma*\phi_{\varepsilon}(x).$

By (Hloc), it is easy to see that for any $m\in{\mathbb{N}}$ and all
$\varepsilon\in(0,1)$,

$\displaystyle|b_{\varepsilon}(x)-b_{\varepsilon}(y)|+\|\sigma_{\varepsilon}(x)-\sigma_{\varepsilon}(y)\|\leqslant
C_{m+1}|x-y|^{\gamma},\ \ \forall x,y\in B_{m},$ (3.10)

and for some $\varepsilon_{1}\in(0,1)$ depending on $m$, and for all
$\varepsilon\in(0,\varepsilon_{1})$,

$\displaystyle C_{m}^{-1}|\xi|\leqslant|\sigma_{\varepsilon}(x)\xi|\leqslant
C_{m}|\xi|,\ \ \forall x\in B_{m},\ \xi\in{\mathbb{R}}^{d}.$ (3.11)

Moreover, note that by (1.4),

$\displaystyle|b_{\varepsilon}(x)-b(x)|\leqslant\int_{{\mathbb{R}}^{d}}|b(x-y)-b(x)|\rho_{\varepsilon}(y){\mathord{{\rm
d}}}y\leqslant\varepsilon^{\gamma}\ell_{1}(x),$

and

$\displaystyle|\sigma_{\varepsilon}(x)-\sigma(x)|\leqslant\int_{{\mathbb{R}}^{d}}|\sigma(x-y)-\sigma(x)|\rho_{\varepsilon}(y){\mathord{{\rm
d}}}y\leqslant\varepsilon^{\gamma}\ell_{2}(x).$

Hence, by (H${}^{r,q}_{\rm glo}$), for any
$\varepsilon\in(0,\varepsilon^{1/\gamma}_{0})$,

$\displaystyle\langle
x,b_{\varepsilon}(x)\rangle+q|\sigma_{\varepsilon}(x)|^{\alpha}|x|^{2-\alpha}$
$\displaystyle\leqslant\langle
x,b(x)\rangle+\varepsilon^{\gamma}|x|\ell_{1}(x)+q(\|\sigma(x)\|+\ell_{2}(x)\varepsilon^{\gamma})^{\alpha}|x|^{2-\alpha}$
$\displaystyle\leqslant\langle
x,b(x)\rangle+\varepsilon_{0}|x|\ell_{1}(x)+q(\|\sigma(x)\|+\ell_{2}(x)\varepsilon_{0})^{\alpha}|x|^{2-\alpha}$
$\displaystyle\leqslant-c_{0}|x|^{2+r}+c_{1}.$ (3.12)

Now, consider the following approximation SDE:

$\displaystyle{\mathord{{\rm
d}}}X^{\varepsilon}_{t}=b_{\varepsilon}(X^{\varepsilon}_{t}){\mathord{{\rm
d}}}t+\sigma_{\varepsilon}(X^{\varepsilon}_{t-}){\mathord{{\rm
d}}}L^{\alpha}_{t},\quad X^{\varepsilon}_{0}=x.$ (3.13)

Since the coefficients are smooth, but, may be not bounded, by (3.12) and [7,
Theorem 1.1], there is a unique strong solution $X^{\varepsilon}_{t}(x)$ to
the above SDE. Moreover, for any $f\in C_{b}({\mathbb{R}}^{d})$, we have

$\displaystyle\lim_{\varepsilon\to
0}{\mathbb{E}}f(X^{\varepsilon}_{t}(x))={\mathbb{E}}f(X_{t}(x)),$ (3.14)

which is proven in Appendix. Notice that if $b$ and $\sigma$ are locally
Lipschitz, then it is not necessary to mollify $b$ and $\sigma$. It suffices
to consider the truncated $b$ and $\sigma$ as done below.

Let $\chi$ be a cutoff function with

$\chi(x)=1,\ \ |x|\leqslant\tfrac{1}{2},\ \ \chi(x)=0,\ \ |x|>1.$

For $m\in{\mathbb{N}}$, set

$\chi_{m}(x):=\chi(x/m),\ \
b^{\varepsilon}_{m}(x):=b_{\varepsilon}(x)\chi_{m}(x),\quad\sigma^{\varepsilon}_{m}(x):=\sigma_{\varepsilon}(x\chi_{m}(x)).$

By (3.10) and (3.11), it is easy to see that $b^{\varepsilon}_{m}$ and
$\sigma^{\varepsilon}_{m}$ satisfy the following global assumptions:

$\displaystyle|b^{\varepsilon}_{m}(x)-b^{\varepsilon}_{m}(y)|+\|\sigma^{\varepsilon}_{m}(x)-\sigma^{\varepsilon}_{m}(y)\|\leqslant
C_{m}|x-y|^{\gamma},\ \ \forall x,y\in{\mathbb{R}}^{d},$ (3.15)

and

$\displaystyle
C_{m}^{-1}|\xi|\leqslant|\sigma^{\varepsilon}_{m}(x)\xi|\leqslant C_{m}|\xi|,\
\ \forall x,\xi\in{\mathbb{R}}^{d},$ (3.16)

where the constant does not depend on $\varepsilon$. Let $\theta_{t}(x)$ solve
the following ODE:

$\displaystyle\dot{\theta}_{t}(x)=(b^{\varepsilon}_{m}*\phi_{t^{1/\alpha}})(\theta_{t}(x)),\quad\theta_{0}(x)=x.$
(3.17)

We also consider the following SDE with cutoff coefficients:

$\displaystyle{\mathord{{\rm
d}}}X^{\varepsilon,m}_{t}=b^{\varepsilon}_{m}(X^{\varepsilon,m}_{t}){\mathord{{\rm
d}}}t+\sigma^{\varepsilon}_{m}(X^{\varepsilon,m}_{t-}){\mathord{{\rm
d}}}L^{\alpha}_{t},\quad X^{\varepsilon,m}_{0}=x.$ (3.18)

Let $X^{\varepsilon,m}_{t}(x)$ be the unique strong solution of the above SDE.
By [25, Theorem 1.1], $X^{\varepsilon,m}_{t}(x)$ admits a density
$p^{\varepsilon}_{m}(t,x,y)$ with the following estimates: For any $T>0$ and
$m\in{\mathbb{N}}$, there is a constant $C_{0}\geqslant 1$ depending on $m$,
but independent of $\varepsilon\in(0,\varepsilon_{1})$, such that for all
$t\in(0,T]$ and $x,y\in\mathbb{R}^{d}$,

$\displaystyle
p^{\varepsilon}_{m}(t,x,y)\asymp_{C_{0}}t(t^{\frac{1}{\alpha}}+|x-\theta_{t}(y)|)^{-d-\alpha},$
(3.19)

and

$\displaystyle|\nabla\log p^{\varepsilon}_{m}(t,x,\cdot)|(y)\leqslant
C_{0}t^{-\frac{1}{\alpha}}.$ (3.20)

For any $m\in{\mathbb{N}}$ and $x\in B_{m}$, define the exit time of
$X^{\varepsilon,m}_{t}(x)$ from $B_{m}$ by

$\tau^{\varepsilon,x}_{B_{m}}:=\inf\big{\\{}t>0:X^{\varepsilon}_{t}(x)\notin
B_{m}\big{\\}}.$

By the uniqueness of strong solution, we have

$\displaystyle X^{\varepsilon}_{t}(x)=X^{\varepsilon,m}_{t}(x),\quad
t<\tau^{\varepsilon,x}_{B_{m}}.$ (3.21)

######  Lemma 3.3 (Strong Feller property).

For any $t>0$ and bounded measurable function $f$, the function $x\mapsto
P_{t}f(x)$ is bounded continuous.

###### Proof.

By a standard monotonic argument, it suffices to show that for any $t>0$,

$\lim_{x\to y}\sup_{f\in C_{b}({\mathbb{R}}^{d}),\|f\|_{\infty}\leqslant
1}|\mathbb{E}(f(X_{t}(x))-f(X_{t}(y)))|=0.$

Since for any $f\in C_{b}({\mathbb{R}}^{d})$,

$\lim_{\varepsilon\to
0}\mathbb{E}(f(X^{\varepsilon}_{t}(x))-f(X^{\varepsilon}_{t}(y)))\stackrel{{\scriptstyle\eqref{DD9}}}{{=}}\mathbb{E}(f(X_{t}(x))-f(X_{t}(y))),$

furthermore, we only need to prove that

$\displaystyle\lim_{x\to y}\sup_{\varepsilon\in(0,1)}\sup_{f\in
C_{b}({\mathbb{R}}^{d}),\|f\|_{\infty}\leqslant
1}|\mathbb{E}(f(X^{\varepsilon}_{t}(x))-f(X^{\varepsilon}_{t}(y)))|=0.$ (3.22)

Given $x,y\in{\mathbb{R}}^{d}$, let $m>|x|\vee|y|$. For given $f\in
C_{b}({\mathbb{R}}^{d})$ with $\|f\|_{\infty}\leqslant 1$, we have

$\displaystyle|\mathbb{E}(f(X^{\varepsilon}_{t}(x))-f(X^{\varepsilon}_{t}(y)))|$
$\displaystyle\leqslant\big{|}\mathbb{E}\big{[}(f(X^{\varepsilon}_{t}(x))-f(X^{\varepsilon}_{t}(y)))\mathbf{1}_{t<\tau^{\varepsilon,x}_{B_{m}}\wedge\tau^{\varepsilon,y}_{B_{m}}}\big{]}\big{|}+2\mathbb{P}\big{(}t\geqslant\tau^{\varepsilon,x}_{B_{m}}\wedge\tau^{\varepsilon,y}_{B_{m}}\big{)}$
$\displaystyle\\!\\!\\!\\!\\!\stackrel{{\scriptstyle\eqref{equ:loc}}}{{=}}\big{|}\mathbb{E}\big{[}(f(X^{\varepsilon,m}_{t}(x))-f(X^{\varepsilon,m}_{t}(y)))\mathbf{1}_{t<\tau^{\varepsilon,x}_{B_{m}}\wedge\tau^{\varepsilon,y}_{B_{m}}}\big{]}\big{|}+2\mathbb{P}\big{(}t\geqslant\tau^{\varepsilon,x}_{B_{m}}\wedge\tau^{\varepsilon,y}_{B_{m}}\big{)}$
$\displaystyle\leqslant\big{|}\mathbb{E}\big{[}(f(X^{\varepsilon,m}_{t}(x))-f(X^{\varepsilon,m}_{t}(y))\big{]}\big{|}+2\mathbb{P}\big{(}t\geqslant\tau^{\varepsilon,x}_{B_{m}}\wedge\tau^{\varepsilon,y}_{B_{m}}\big{)}.$

By (3.20), we have

$\displaystyle|\mathbb{E}(f(X^{\varepsilon,m}_{t}(x))-f(X^{\varepsilon,m}_{t}(y)))|$
$\displaystyle=\left|\int_{{\mathbb{R}}^{d}}f(z)(p^{\varepsilon}_{m}(t,x,z)-p^{\varepsilon}_{m}(t,x,z)){\mathord{{\rm
d}}}z\right|$
$\displaystyle\leqslant\|f\|_{\infty}|x-y|\int_{\mathbb{R}^{d}}\int^{1}_{0}|\nabla
p^{\varepsilon}_{m}(t,x+r(y-x),z)|{\mathord{{\rm d}}}z{\mathord{{\rm d}}}r$
$\displaystyle\leqslant
C_{m}t^{-\frac{1}{\alpha}}|x-y|\int^{1}_{0}\int_{\mathbb{R}^{d}}p^{\varepsilon}_{m}(t,x+r(y-x),z){\mathord{{\rm
d}}}z{\mathord{{\rm d}}}r$ $\displaystyle=C_{m}t^{-\frac{1}{\alpha}}|x-y|.$

Moreover, by Chebyshev’s inequality and (3.4),

$\displaystyle\mathbb{P}(t\geqslant\tau^{\varepsilon,x}_{B_{m}})\leqslant\mathbb{P}\left(\sup_{s\leqslant
t}|X^{\varepsilon}_{s}(x)|\geqslant m\right)$
$\displaystyle\leqslant\mathbb{E}\left(\sup_{s\leqslant
t}|X^{\varepsilon}_{s}(x)|^{p/2}\right)/m^{p/2}$ $\displaystyle\leqslant
C(V_{p}(x)+t)^{1/2}/m^{p/2}.$ (3.23)

Combining the above calculations, we obtain that for any $f\in
C_{b}({\mathbb{R}}^{d})$ with $\|f\|_{\infty}\leqslant 1$,

$|\mathbb{E}(f(X^{\varepsilon}_{t}(x))-f(X^{\varepsilon}_{t}(y)))|\leqslant
C_{m}t^{-\frac{1}{\alpha}}|x-y|+C_{0}(V_{p}(x)+V_{p}(y)+t)^{1/2}/m^{p/2},$

where $C_{0}$ does not depend on $m$. Thus we obtain (3.22) by first letting
$x\to y$ and then $m\to\infty$. ∎

######  Lemma 3.4 (Irreducibility).

For any $x_{0},y_{0}\in{\mathbb{R}}^{d}$ and $r,t>0$, we have

$\displaystyle\inf_{x\in B_{r}(x_{0})}\mathbb{P}(X_{t}(x)\in B_{r}(y_{0}))>0.$
(3.24)

###### Proof.

Since $X^{\varepsilon}_{t}(x)$ weakly converges to $X_{t}(x)$ as
$\varepsilon\downarrow 0$, for any open set $A\subset{\mathbb{R}}^{d}$, we
have

$\displaystyle\mathbb{P}(X_{t}(x)\in\overline{A})\geqslant\liminf_{\varepsilon\to
0}\mathbb{P}(X^{\varepsilon}_{t}(x)\in A).$ (3.25)

Below we fix $\varepsilon\in(0,1)$ small enough, and
$x,y_{0}\in{\mathbb{R}}^{d}$, $r>0$. Let $m>2$ be big enough such that
$\\{x\\}\cup B_{r}(y_{0})\subset B_{m-2}$. Then we have

$\displaystyle\mathbb{P}(X^{\varepsilon}_{t}(x)\in B_{r}(y_{0}))$
$\displaystyle\geqslant\mathbb{P}{\Big{(}}X^{\varepsilon}_{t}(x)\in
B_{r}(y_{0});t<\tau^{\varepsilon,x}_{B_{m}}{\Big{)}}\stackrel{{\scriptstyle\eqref{equ:loc}}}{{=}}\mathbb{P}{\Big{(}}X^{\varepsilon,m}_{t}(x)\in
B_{r}(y_{0});t<\tau^{\varepsilon,x}_{B_{m}}{\Big{)}}.$ (3.26)

In order to use Theorem 2.3, we choose

$\rho(t,r)=t(t^{1/\alpha}+r)^{-d-\alpha}.$

Clearly, by (3.19), estimate (2.5) is satisfied for
$\Gamma_{t}(x,y)=|x-\theta_{t}(y)|$. It remains to find $t_{0}$ and
$\delta_{0}$ small enough as well as functions $\ell_{0}(t)$ and $\ell_{1}(t)$
so that the conditions in Theorem 2.3 are satisfied for domains
$D_{0}=B_{m-2}$ and $D=B_{m}$. Note that by (3.17) and the definition of
$b^{\varepsilon}_{m}$, for all $t\in[0,1]$ and $y\in{\mathbb{R}}^{d}$,

$\displaystyle|\theta_{t}(y)-y|\leqslant\int^{t}_{0}|(b^{\varepsilon}_{m}*\phi_{s^{1/\alpha}})(\theta_{s}(y))|{\mathord{{\rm
d}}}s\leqslant\sup_{|x|\leqslant m+1}|b(x)|\cdot t=:C_{1}t.$

Let

$\displaystyle\delta_{0}\leqslant\tfrac{1}{2}\wedge{\rm dist}(y_{0},D_{0}),\ \
t_{0}\leqslant\tfrac{1}{2C_{1}}\wedge\tfrac{\delta_{0}}{C_{1}}.$ (3.27)

For $x^{\prime}\notin D$ and $y\in B_{\delta_{0}}(y_{0})\subset D_{0}$, we
have for $t\in(0,t_{0}]$,

$\displaystyle\Gamma_{t}(x^{\prime},y)=|x^{\prime}-\theta_{t}(y)|$
$\displaystyle\geqslant|x^{\prime}-y_{0}|-|y-y_{0}|-|\theta_{t}(y)-y|\geqslant
2-\delta_{0}-C_{1}t\geqslant 1.$

On the other hand, for $x,y\in B_{\delta_{0}}(y_{0})$, we have

$\displaystyle\Gamma_{t}(x,y)=|x-\theta_{t}(y)|\leqslant|x-y|+|\theta_{t}(y)-y|\leqslant
2\delta_{0}+C_{1}t\leqslant 3\delta_{0}.$ (3.28)

Now, for the $C_{0}$ in (3.19), one can choose $\delta_{0},t_{0}>0$ small
enough such that (3.27) holds and

$t\mapsto t/(t^{1/\alpha}+1)^{d+\alpha}\mbox{ is increasing on $(0,t_{0}]$,}$

and

$\displaystyle\frac{C_{0}^{-1}t}{(t^{1/\alpha}+3\delta_{0})^{d+\alpha}}>\frac{2C_{0}t}{(t^{1/\alpha}+1)^{d+\alpha}},\
\ t\in(0,t_{0}].$ (3.29)

In particular, (2.7) is satisfied. Thus by Theorem 2.3, we conclude

$\mathbb{P}{\Big{(}}X^{\varepsilon,m}_{t}(x)\in
B_{r}(y_{0});t<\tau^{\varepsilon,x}_{B_{m}}{\Big{)}}\geqslant c_{0},$

where $c_{0}$ is independent of $\varepsilon$. This, together with (3.25) and
(3.26), yields

$\mathbb{P}(X_{t}(x)\in\overline{B_{r}(y_{0})})\geqslant c_{0}.$

The proof is thus complete by the strong Feller property that
$x\mapsto\mathbb{P}(X_{t}(x)\in B_{r}(y_{0}))$ is continuous. ∎

Now, we are in a position to give

###### Proof of Theorem 1.2.

First of all, by (3.4) and the standard Krylov-Bogoliubov method (see [12,
Theorem 11.7]), there is an invariant probability measure $\mu$ associated
with $(P_{t})_{t\geqslant 0}$. The uniqueness follows by the strong Feller
property and irreducibility. The exponential convergence (i) and (ii) follow
by (3.5), Theorem 2.1 and Lemmas 3.3 and 3.4. ∎

##  4\. Application to heavy-tailed sampling

In this section we introduce an application of Theorem 1.2 to the heavy-tailed
sampling. Let $\mu({\mathord{{\rm d}}}x)=\mathrm{e}^{-U(x)}{\mathord{{\rm
d}}}x/\int_{{\mathbb{R}}^{d}}\mathrm{e}^{-U(x)}{\mathord{{\rm d}}}x$, where
$U:{\mathbb{R}}^{d}\to{\mathbb{R}}$ is a continuous function. Suppose that
there are $\beta,C_{0}>0$ such that for all $|x|\geqslant 1$,

$\displaystyle U(x)\geqslant(d+\beta)\log|x|-C_{0}.$ (4.1)

The above assumption means that $\mathrm{e}^{-U(x)}$ has a polynomial decay
rate as $|x|\to\infty$, which is usually called the heavy-tailed distribution
compared with the exponential or light tailed distribution. Below we want to
find an ergodic SDE so that the law of the solution $X_{t}$ exponentially
converges to $\mu$ in some sense as $t\to\infty$. Thus, one can sample $\mu$
theoritically from $X_{t}$ when $t$ is large.

Fix $\alpha\in(0,2)$. To construct an ergodic SDE driven by $\alpha$-stable
process, we introduce a vector field $B:{\mathbb{R}}^{d}\to{\mathbb{R}}^{d}$
by

$B(x):=c_{d,\alpha}\int_{{\mathbb{R}}^{d}}y[\varrho_{\alpha}(x+y)-\varrho_{\alpha}(x-y)]|y|^{-d-\alpha}{\mathord{{\rm
d}}}y,$

where
$c_{d,\alpha}=2^{\alpha}\Gamma(\frac{d+\alpha}{2})/\Gamma(\frac{d}{2})\Gamma(\frac{2-\alpha}{2})$
and

$\varrho_{\alpha}(x):=(1+|x|^{2})^{-\frac{d+\alpha}{2}}.$

The vector field $B$ enjoys the following important property.

######  Lemma 4.1.

  1. (i)

$B\in C^{\infty}_{0}$ satisfies that for some
$\kappa_{0}=\kappa_{0}(d,\alpha)>0$,

$\displaystyle|B(x)|\leqslant\kappa_{0}|x|^{1-d-\alpha},\ \ |\nabla
B(x)|\leqslant\kappa_{0}|x|^{-d-\alpha}.$ (4.2)

  2. (ii)

There is a constant $\kappa_{1}>0$ only depending on $d,\alpha$ such that for
all $|x|>1$,

$\displaystyle\langle x,B(x)\rangle\leqslant-\kappa_{1}|x|^{2-d-\alpha}.$
(4.3)

  3. (iii)

$\mathord{{\rm div}}B(x)=\Delta^{\frac{\alpha}{2}}\varrho_{\alpha}(x)$.

###### Proof.

(i) Since $\varrho_{\alpha}\in C^{\infty}_{0}({\mathbb{R}}^{d})$ and for any
$j\in{\mathbb{N}}$,

$\|\nabla^{j}\varrho_{\alpha}/\varrho_{\alpha}\|_{\infty}<\infty,$

we clearly have $B\in C^{\infty}_{0}$. To show the bounds (4.2), let
$\chi:[0,\infty)\to[0,1]$ be a cutoff function with

$\chi(r)=1,\ \ r\in[0,\tfrac{1}{4}],\ \ \chi(r)=0,\ \ r\in[0,\tfrac{1}{2}].$

Define

$\varphi_{|x|}(y):=y|y|^{-d-\alpha}\chi\big{(}\tfrac{|y|}{|x|}\big{)},\ \
\widetilde{\varphi}_{|x|}(y):=y|y|^{-d-\alpha}\big{(}1-\chi\big{(}\tfrac{|y|}{|x|}\big{)}\big{)}.$

For $j=0,1$, by definition we have

$\displaystyle\nabla^{j}B(x)$
$\displaystyle=c_{d,\alpha}\int_{{\mathbb{R}}^{d}}y\Big{(}\nabla^{j}\varrho_{\alpha}(x+y)-\nabla^{j}\varrho_{\alpha}(x-y)\Big{)}|y|^{-d-\alpha}{\mathord{{\rm
d}}}y=c_{d,\alpha}(I_{1}+I_{2}),$

where

$\displaystyle I_{1}$
$\displaystyle:=\int_{{\mathbb{R}}^{d}}\varphi_{|x|}(y)\Big{(}\nabla^{j}\varrho_{\alpha}(x+y)-\nabla^{j}\varrho_{\alpha}(x-y)\Big{)}{\mathord{{\rm
d}}}y,$ $\displaystyle I_{2}$
$\displaystyle:=\int_{{\mathbb{R}}^{d}}\widetilde{\varphi}_{|x|}(y)\Big{(}\nabla^{j}\varrho_{\alpha}(x+y)-\nabla^{j}\varrho_{\alpha}(x-y)\Big{)}{\mathord{{\rm
d}}}y.$

For $I_{1}$, we have

$\displaystyle|I_{1}|$
$\displaystyle\leqslant\int_{{\mathbb{R}}^{d}}\varphi_{|x|}(y)|y|\left(\int^{1}_{-1}|\nabla^{j+1}\varrho_{\alpha}(x+sy)|{\mathord{{\rm
d}}}s\right){\mathord{{\rm d}}}y$
$\displaystyle\lesssim\int_{|y|\leqslant\frac{|x|}{2}}|y|^{2-d-\alpha}\left(\int^{1}_{-1}(1+|x+sy|)^{-(d+\alpha+1+j)}{\mathord{{\rm
d}}}s\right){\mathord{{\rm d}}}y$
$\displaystyle\lesssim(1+|x|)^{-(d+\alpha+1+j)}\int_{|y|\leqslant\frac{|x|}{2}}|y|^{2-d-\alpha}{\mathord{{\rm
d}}}y$
$\displaystyle\lesssim(1+|x|)^{-(d+\alpha+1+j)}|x|^{2-\alpha}\lesssim|x|^{1-j-d-2\alpha}.$

For $I_{2}$, by the change of variable, we have

$\displaystyle I_{2}$
$\displaystyle=\int_{{\mathbb{R}}^{d}}\Big{(}\widetilde{\varphi}_{|x|}(x-y)-\widetilde{\varphi}_{|x|}(x+y)\Big{)}(\nabla^{j}\varrho_{\alpha})(y){\mathord{{\rm
d}}}y$
$\displaystyle=|x|^{d}\int_{{\mathbb{R}}^{d}}\Big{(}\widetilde{\varphi}_{|x|}(|x|(\bar{x}-y))-\widetilde{\varphi}_{|x|}(|x|(\bar{x}+y))\Big{)}(\nabla^{j}\varrho_{\alpha})(|x|y){\mathord{{\rm
d}}}y$
$\displaystyle=|x|^{1-\alpha}\int_{{\mathbb{R}}^{d}}\Big{(}\widetilde{\varphi}_{1}(\bar{x}-y)-\widetilde{\varphi}_{1}(\bar{x}+y)\Big{)}(\nabla^{j}\varrho_{\alpha})(|x|y){\mathord{{\rm
d}}}y,$

where $\bar{x}=x/|x|$ and in the last step we have used

$\widetilde{\varphi}_{|x|}(|x|y)=|x|^{1-d-\alpha}y|y|^{-d-\alpha}\big{(}1-\chi\big{(}|y|\big{)}\big{)}=|x|^{1-d-\alpha}\widetilde{\varphi}_{1}(y).$

For $j=0$, since
$|\widetilde{\varphi}_{1}(y)|\leqslant|y|^{1-d-\alpha}{\mathbf{1}}_{|y|>1/4}\leqslant
4^{d+\alpha-1}$, it is easy to see that

$\displaystyle|I_{2}|$ $\displaystyle\leqslant 2\cdot
4^{d+\alpha-1}|x|^{1-\alpha}\int_{{\mathbb{R}}^{d}}|(1+|x||y|)^{-(d+\alpha)}{\mathord{{\rm
d}}}y\lesssim|x|^{1-\alpha-d}.$

For $j=1$, since $\|\mathord{{\rm
div}}_{y}\widetilde{\varphi}_{1}\|_{\infty}<\infty$, by the integration by
parts, we also have

$\displaystyle|I_{2}|$
$\displaystyle=|x|^{-\alpha}\left|\int_{{\mathbb{R}}^{d}}\Big{(}\widetilde{\varphi}_{1}(\bar{x}-y)-\widetilde{\varphi}_{1}(\bar{x}+y)\Big{)}\nabla_{y}\varrho_{\alpha}(|x|y){\mathord{{\rm
d}}}y\right|$
$\displaystyle=|x|^{-\alpha}\left|\int_{{\mathbb{R}}^{d}}\Big{(}\mathord{{\rm
div}}_{y}\widetilde{\varphi}_{1}(\bar{x}-y)-\mathord{{\rm
div}}_{y}\widetilde{\varphi}_{1}(\bar{x}+y)\Big{)}\varrho_{\alpha}(|x|y){\mathord{{\rm
d}}}y\right|$ $\displaystyle\leqslant 2\|\mathord{{\rm
div}}_{y}\widetilde{\varphi}_{1}\|_{\infty}|x|^{-\alpha}\int_{{\mathbb{R}}^{d}}|(1+|x||y|)^{-(d+\alpha)}{\mathord{{\rm
d}}}y\lesssim|x|^{-\alpha-d}.$

Combining the above estimates we obtain (4.2).

(ii) By definition and the symmetry, we have

$\displaystyle-\langle x,B(x)\rangle$
$\displaystyle=c_{d,\alpha}\int_{{\mathbb{R}}^{d}}\langle
x,y\rangle[\varrho_{\alpha}(x-y)-\varrho_{\alpha}(x+y)]|y|^{-d-\alpha}{\mathord{{\rm
d}}}y$ $\displaystyle=2c_{d,\alpha}\int_{\langle x,y\rangle\geqslant 0}\langle
x,y\rangle[\varrho_{\alpha}(x-y)-\varrho_{\alpha}(x+y)]|y|^{-d-\alpha}{\mathord{{\rm
d}}}y.$

For $|x|\geqslant 1$ and $\langle x,y\rangle\geqslant 0$, by the mean-valued
formula, we have

$\displaystyle\varrho_{\alpha}(x-y)-\varrho_{\alpha}(x+y)=(1+|x-y|^{2})^{-\frac{d+\alpha}{2}}-(1+|x+y|^{2})^{-\frac{d+\alpha}{2}}$
$\displaystyle\quad=2(d+\alpha)\langle
x,y\rangle\int^{1}_{0}\big{(}1+s(|x-y|^{2}-|x+y|^{2})+|x+y|^{2}\big{)}^{-\frac{d+\alpha}{2}-1}{\mathord{{\rm
d}}}s$ $\displaystyle\quad=2(d+\alpha)\langle
x,y\rangle\int^{1}_{0}\big{(}1-4s\langle
x,y\rangle+|x+y|^{2}\big{)}^{-\frac{d+\alpha}{2}-1}{\mathord{{\rm d}}}s$
$\displaystyle\quad=2(d+\alpha)\langle
x,y\rangle\int^{1}_{0}\big{(}1+4s\langle
x,y\rangle+|x-y|^{2}\big{)}^{-\frac{d+\alpha}{2}-1}{\mathord{{\rm d}}}s$
$\displaystyle\quad\geqslant 2(d+\alpha)\langle
x,y\rangle\int^{|x|^{-2}}_{0}\big{(}1+4s\langle
x,y\rangle+|x-y|^{2}\big{)}^{-\frac{d+\alpha}{2}-1}{\mathord{{\rm d}}}s$
$\displaystyle\quad\geqslant 2(d+\alpha)\langle
x,y\rangle\big{(}1+4|x|^{-2}\langle
x,y\rangle+|x-y|^{2}\big{)}^{-\frac{d+\alpha}{2}-1}|x|^{-2}.$

Hence,

$\displaystyle-\langle x,B(x)\rangle$
$\displaystyle\geqslant\frac{4(d+\alpha)c_{d,\alpha}}{|x|^{2}}\int_{\langle
x,y\rangle\geqslant 0}\langle x,y\rangle^{2}\big{(}1+4|x|^{-2}\langle
x,y\rangle+|x-y|^{2}\big{)}^{-\frac{d+\alpha}{2}-1}|y|^{-d-\alpha}{\mathord{{\rm
d}}}y$
$\displaystyle\geqslant\frac{4(d+\alpha)c_{d,\alpha}}{|x|^{2}}\int_{\langle
x,y\rangle\geqslant 0,|x-y|\leqslant 1}\langle
x,y\rangle^{2}\big{(}1+4|x|^{-2}\langle
x,y\rangle+|x-y|^{2}\big{)}^{-\frac{d+\alpha}{2}-1}|y|^{-d-\alpha}{\mathord{{\rm
d}}}y.$

Since for $|x|\geqslant 1$,

$|x-y|\leqslant 1\Rightarrow\tfrac{|y|^{2}}{2}\leqslant\langle
x,y\rangle\leqslant|x|^{2}+|x|,$

we further have

$\displaystyle-\langle x,B(x)\rangle$
$\displaystyle\geqslant\frac{4(d+\alpha)c_{d,\alpha}}{|x|^{2}}\int_{|x-y|\leqslant
1}\langle
x,y\rangle^{2}\big{(}2+4(1+|x|^{-1})\big{)}^{-\frac{d+\alpha}{2}-1}|y|^{-d-\alpha}{\mathord{{\rm
d}}}y$
$\displaystyle\geqslant\frac{4(d+\alpha)c_{d,\alpha}}{|x|^{2}10^{\frac{d+\alpha}{2}+1}}\int_{|x-y|\leqslant
1}\langle x,y\rangle^{2}|y|^{-d-\alpha}{\mathord{{\rm
d}}}y\geqslant\kappa_{1}|x|^{2-d-\alpha},$

where $\kappa_{1}>1$ only depends on $d,\alpha$. Thus we obtain (4.3).

(iii) By definition and the integration by parts, we have

$\displaystyle\mathord{{\rm div}}B(x)$
$\displaystyle=c_{d,\alpha}\int_{{\mathbb{R}}^{d}}\langle
y,\nabla_{x}[\varrho_{\alpha}(x+y)-\varrho_{\alpha}(x-y)]\rangle|y|^{-d-\alpha}{\mathord{{\rm
d}}}y$ $\displaystyle=c_{d,\alpha}\int_{{\mathbb{R}}^{d}}\langle
y,\nabla_{y}[\varrho_{\alpha}(x+y)+\varrho_{\alpha}(x-y)-2\varrho_{\alpha}(x)]\rangle|y|^{-d-\alpha}{\mathord{{\rm
d}}}y$ $\displaystyle=-c_{d,\alpha}\int_{{\mathbb{R}}^{d}}\mathord{{\rm
div}}(y|y|^{-d-\alpha})[\varrho_{\alpha}(x+y)+\varrho_{\alpha}(x-y)-2\varrho_{\alpha}(x)]{\mathord{{\rm
d}}}y.$

Since for $y\not=0$,

$\mathord{{\rm div}}(y|y|^{-d-\alpha})=d|y|^{-d-\alpha}+\langle
y,\nabla|y|^{-d-\alpha}\rangle=-\alpha|y|^{-d-\alpha},$

we have by (1.8) and
$c_{d,\alpha}=2^{\alpha}\Gamma(\frac{d+\alpha}{2})/\Gamma(\frac{d}{2})\Gamma(\frac{2-\alpha}{2})$,

$\mathord{{\rm div}}B(x)=\alpha
c_{d,\alpha}\int_{{\mathbb{R}}^{d}}\frac{[\varrho_{\alpha}(x+y)+\varrho_{\alpha}(x-y)-2\varrho_{\alpha}(x)]}{|y|^{d+\alpha}}{\mathord{{\rm
d}}}y=\Delta^{\frac{\alpha}{2}}\varrho_{\alpha}(x).$

The proof is complete. ∎

Figure 1. Pictures of $B(x)$ and $\langle x,B(x)\rangle$ for $d=1$.

Now we take $\sigma$ and $b$ in SDE (1.3) as follows:

$\displaystyle\sigma(x):=(\varrho_{\alpha}(x)\mathrm{e}^{U(x)})^{\frac{1}{\alpha}}{\mathbb{I}},\
\ b(x):=B(x)\mathrm{e}^{U(x)}.$ (4.4)

We have the following main result of this section.

######  Theorem 4.2.

Suppose that $U$ is locally Lipschitz continuous and satisfies (4.1). Then
with the above choices of $\sigma$ and $b$, $\mu({\mathord{{\rm
d}}}x)=\mathrm{e}^{-U(x)}{\mathord{{\rm
d}}}x/\int_{{\mathbb{R}}^{d}}\mathrm{e}^{-U(x)}{\mathord{{\rm d}}}x$ is the
unique invariant probability measure of SDE (1.3) and the conclusions in
Theorem 1.2 hold with $r=\beta-\alpha$.

###### Proof.

Let $\sigma$ and $b$ be defined by (4.4). By (1.7) and the change of variable,
we can write

$\displaystyle\mathcal{L}_{\sigma}f(x)$ $\displaystyle=\frac{\alpha
2^{\alpha}\Gamma(\frac{d+\alpha}{2})}{\Gamma(\frac{d}{2})\Gamma(\frac{2-\alpha}{2})}\int_{\mathbb{R}^{d}}[f(x+\sigma(x)z)+f(x-\sigma(x)z)-2f(x)]{\mathord{{\rm
d}}}z/|z|^{d+\alpha}=\varrho_{\alpha}(x)\mathrm{e}^{U(x)}\Delta^{\frac{\alpha}{2}}f(x).$

Thus,

$\displaystyle{\mathscr{L}}f(x)=\mathcal{L}_{\sigma}f(x)+b\cdot\nabla
f(x)=\varrho_{\alpha}(x)\mathrm{e}^{U(x)}\Delta^{\frac{\alpha}{2}}f(x)+B(x)\mathrm{e}^{U(x)}\cdot\nabla
f(x),$ (4.5)

and by (i) of Lemma 4.1,

$({\mathscr{L}}^{*}\mathrm{e}^{-U})(x)=\Delta^{\frac{\alpha}{2}}\varrho_{\alpha}(x)-\mathord{{\rm
div}}B(x)=0,$

which implies that $\mu({\mathord{{\rm d}}}x)=\mathrm{e}^{-U(x)}{\mathord{{\rm
d}}}x/\int_{{\mathbb{R}}^{d}}\mathrm{e}^{-U(x)}{\mathord{{\rm d}}}x$ is an
invariant probability measure of SDE (1.3).

To check (Hloc) and (H${}^{r,q}_{\rm glo}$), by Remark 1.1, it suffices to
verify (1.5) for $\varepsilon_{0}=0$. By (4.3) and (4.1), for any
$|x|\geqslant(2q/\kappa_{1})^{1/\alpha}\vee 1$, we have

$\displaystyle\langle x,b(x)\rangle+q\|\sigma(x)\|^{\alpha}|x|^{2-\alpha}$
$\displaystyle=\big{[}\langle
x,B(x)\rangle+q\varrho_{\alpha}(x)|x|^{2-\alpha}\big{]}\mathrm{e}^{U(x)})$
$\displaystyle\leqslant\big{[}-\kappa_{1}|x|^{2-d-\alpha}+q|x|^{2-d-2\alpha}\big{]}\mathrm{e}^{U(x)}$
$\displaystyle\leqslant-\tfrac{\kappa_{1}}{2}|x|^{2-d-\alpha}\mathrm{e}^{U(x)}\leqslant-\tfrac{\kappa_{1}}{2\mathrm{e}^{C_{0}}}|x|^{2+\beta-\alpha}.$

So, Theorem 1.2 is applicable. The proof is complete. ∎

######  Remark 4.3.

Suppose that $U(x)=-\ln\varrho_{\alpha}(x)$. Note that by (4.2),

$\|\nabla(B/\varrho_{\alpha})\|_{\infty}<\infty.$

Let $X_{t}$ be the unique solution of the following SDE:

$\displaystyle{\mathord{{\rm d}}}X_{t}=(B/\rho_{\alpha})(X_{t}){\mathord{{\rm
d}}}t+{\mathord{{\rm d}}}L^{\alpha}_{t},\ \ X_{0}=x.$ (4.6)

Then $\mu_{0}({\mathord{{\rm d}}}x)=\varrho_{\alpha}(x){\mathord{{\rm
d}}}x/\int\varrho_{\alpha}(x){\mathord{{\rm d}}}x$ is the unique invariant
probability measure of $X_{t}$. It should be noticed that the distributional
density $p_{\alpha}(x)$ of $L^{\alpha}_{1}$ is comparable with
$\varrho_{\alpha}(x)$ (cf. [35]), i.e., there is a constant
$\kappa=\kappa(d,\alpha)\geqslant 1$ such that for all $x\in{\mathbb{R}}^{d}$,

$\kappa^{-1}\varrho_{\alpha}(x)\leqslant
p_{\alpha}(x)\leqslant\kappa\varrho_{\alpha}(x).$

Let ${\mathscr{L}}_{0}$ be the infinitesimal generator of SDE (4.6), i.e.,

${\mathscr{L}}_{0}=\Delta^{\frac{\alpha}{2}}+B/\varrho_{\alpha}\cdot\nabla.$

Then by (4.5), one sees that

${\mathscr{L}}=(\varrho_{\alpha}\mathrm{e}^{U}){\mathscr{L}}_{0}.$

In particular, the solution of SDE (1.3) with coefficients (4.4) is just a
time change of SDE (4.6) (see [2, Section 1.15.2]).

######  Remark 4.4.

The locally Lipschitz assumption on $U$ can be relaxed as locally
$\gamma$-Hölder continuity with $\gamma\in((1-\alpha)^{+},1)$. If it is so, we
need to check (1.5) for some $\varepsilon_{0}>0$.

##  5\. Appendix

In this appendix we sketch the proof of weak convergence (3.14). First of all,
by (3.12) and (3.4), there is a constant $C>0$ such that for all
$\varepsilon\in(0,1)$ and any $x\in{\mathbb{R}}^{d}$ and $T>0$,

$\displaystyle\left[\mathbb{E}\left(\sup_{s\in[0,T]}V_{p}(X^{\varepsilon}_{s}(x))^{\frac{1}{2}}\right)\right]^{2}\lesssim_{C}V_{p}(x)+T.$
(5.1)

For $\theta\in(0,T)$, let $\eta,\eta^{\prime}$ be two stopping times with
$0\leqslant\eta\leqslant\eta^{\prime}+\theta\leqslant T$. For any
$m\in{\mathbb{N}}$, we have

$\displaystyle{\mathbb{P}}\left(|X^{\varepsilon}_{\eta}-X^{\varepsilon}_{\eta^{\prime}}|\geqslant\delta\right)$
$\displaystyle\leqslant{\mathbb{P}}\left(|X^{\varepsilon}_{\eta}-X^{\varepsilon}_{\eta^{\prime}}|\geqslant\delta;T<\tau^{\varepsilon,x}_{B_{m}}\right)+{\mathbb{P}}\left(T\geqslant\tau^{\varepsilon,x}_{B_{m}}\right)$
$\displaystyle={\mathbb{P}}\left(|X^{\varepsilon,m}_{\eta}-X^{\varepsilon,m}_{\eta^{\prime}}|\geqslant\delta;T<\tau^{\varepsilon,x}_{B_{m}}\right)+{\mathbb{P}}\left(T\geqslant\tau^{\varepsilon,x}_{B_{m}}\right)$
$\displaystyle\leqslant{\mathbb{P}}\left(|X^{\varepsilon,m}_{\eta}-X^{\varepsilon,m}_{\eta^{\prime}}|\geqslant\delta\right)+2{\mathbb{P}}\left(T\geqslant\tau^{\varepsilon,x}_{B_{m}}\right).$

By SDE (3.18) and (3.15), (3.16), it is by now standard to derive that for
fixed $m\in{\mathbb{N}}$,

$\displaystyle\lim_{\theta\downarrow
0}\sup_{\varepsilon\in(0,1)}\sup_{0\leqslant\eta\leqslant\eta^{\prime}+\theta\leqslant
T}{\mathbb{P}}\left(|X^{\varepsilon,m}_{\eta}-X^{\varepsilon,m}_{\eta^{\prime}}|\geqslant\delta\right)=0,$

which together with (3.23) yields that for any $T,\delta>0$,

$\lim_{\theta\downarrow
0}\sup_{\varepsilon\in(0,1)}\sup_{0\leqslant\eta\leqslant\eta^{\prime}+\theta\leqslant
T}{\mathbb{P}}\left(|X^{\varepsilon}_{\eta}-X^{\varepsilon}_{\eta^{\prime}}|\geqslant\delta\right)=0.$

Thus, by Aldous’ criterion (see [17, p.356, Theorem 4.5]), the law
${\mathbb{Q}}^{\varepsilon}$ of
$(X^{\varepsilon}_{\cdot},L^{\alpha}_{\cdot})$, $\varepsilon\in(0,1)$ in
${\mathbb{D}}\times{\mathbb{D}}$ is tight. Let ${\mathbb{Q}}$ be any
accumulation point of $({\mathbb{Q}}^{\varepsilon})_{\varepsilon\in(0,1)}$.
Without loss of generality, we assume that for some subsequence
$\varepsilon_{k}\to 0$,
$({\mathbb{Q}}^{k})_{k\in{\mathbb{N}}}:=({\mathbb{Q}}^{\varepsilon_{k}})_{k\in{\mathbb{N}}}$
weakly converges to ${\mathbb{Q}}$ as $k\to\infty$. By Skorokhod’s
representation theorem, there is a probability space
$(\widetilde{\Omega},\widetilde{\mathscr{F}},\widetilde{\mathbb{P}})$ and
${\mathbb{D}}\times{\mathbb{D}}$-valued processes
$(\widetilde{X}^{k},\widetilde{L}^{k})$ and $(\widetilde{X},\widetilde{L})$
such that

$(\widetilde{X}^{k},\widetilde{L}^{k})\to(\widetilde{X},\widetilde{L})\ \mbox{
in ${\mathbb{D}}\times{\mathbb{D}}$},\ \ \widetilde{\mathbb{P}}-a.s.,$

and

$\widetilde{\mathbb{P}}\circ(\widetilde{X}^{k},\widetilde{L}^{k})^{-1}={\mathbb{Q}}^{k},\
\ \widetilde{\mathbb{P}}\circ(\widetilde{X},\widetilde{L})^{-1}={\mathbb{Q}}.$

Moreover, $\widetilde{L}^{k}$ and $\widetilde{L}$ are still $\alpha$-stable
Lévy processes, and

$\widetilde{X}_{t}^{k}=x+\int^{t}_{0}b_{\varepsilon_{k}}(\widetilde{X}^{k}_{s}){\mathord{{\rm
d}}}s+\int^{t}_{0}\sigma_{\varepsilon_{k}}(\widetilde{X}^{k}_{s}){\mathord{{\rm
d}}}\widetilde{L}^{k}_{s}.$

By [17, Theorem 6.22, p.383] and taking limits, one sees that

$\widetilde{X}_{t}=x+\int^{t}_{0}b(\widetilde{X}_{s}){\mathord{{\rm
d}}}s+\int^{t}_{0}\sigma(\widetilde{X}_{s}){\mathord{{\rm
d}}}\widetilde{L}_{s},$

and $(\widetilde{X},\widetilde{L})$ is a weak solution of SDE (1.3). Finally,
by the weak uniqueness of [7] we obtain the weak convergence (3.14).

Acknowledgement: The authors would like to thank Jian Wang for useful
conversation about the heavy-tailed sampling.

## References

  * [1] Applebaum D.: Lévy processes and stochastic calculus. Cambridge Studies in Advanced Mathematics. Vol. 116, Cambridge University Press, Cambridge, 2009.
  * [2] Bakry D, Gentil I. and Ledoux M.: Analysis and geometry of Markov diffusion operators. Grundlehren der mathematischen Wissenschaften. Vol. 348, Springer, Cham, 2014.
  * [3] Bubeck S., Eldan R. and Lehec J.: Sampling from a log-concave distribution with projected Langevin Monte Carlo. Discrete Comput. Geom. Vol. 59, No. 4, pp. 757-783 (2018).
  * [4] Carmona P.: Existence and uniqueness of an invariant measure for a chain of oscillators in contact with two heat baths. Stochastic Process. Appl. Vol. 117, No. 8, pp. 1076-1092 (2007).
  * [5] Chen M.: Eigenvalues, inequalities, and ergodic theory. Probability and its Applications (New York). Springer-Verlag London, Ltd., London, 2005.
  * [6] Cerrai S.: Second order PDE’s in finite and infinite dimension. A probabilistic approach. Vol. 1762, Springer-Verlag, Berlin, 2001.
  * [7] Chen Z. Q., Zhang X. and Zhao G.: Supercritical SDEs driven by multiplicative stable-like Lévy processes. Trans. Amer. Math. Soc, Volume 374, Number 11, November 2021, Pages 7621-7655.
  * [8] Cheng X., Chatterji N. S., Abbasi-Yadkori Y., Bartlett P. L. and Jordan M. I.: Sharp convergence rates for Langevin dynamics in the nonconvex setting. arXiv: 1805.01648 (2020).
  * [9] Chung K. and Zhao Z.: From Brownian motion to Schrödinger’s equation. Grundlehren der mathematischen Wissenschaften. Vol. 312, Springer-Verlag, Berlin, 1995.
  * [10] Dalalyan A. S. and Karagulyan A.: User-friendly guarantees for the Langevin Monte Carlo with inaccurate gradient. Stochastic Process. Appl. Vol. 129, No. 12, pp. 5278-5311 (2019).
  * [11] Dalalyan A. S. and Tsybakov A. B.: Sparse regression learning by aggregation and Langevin Monte-Carlo. J. Comput. System Sci. Vol. 78, No. 5, pp. 1423-1443 (2012).
  * [12] Da Prato G. and Zabczyk J.: Stochastic equations in infinite dimensions. Encyclopedia of Mathematics and its Applications. Vol. 44, Cambridge University Press, Cambridge, 1992.
  * [13] Girolami M. and Calderhead B.: Riemann manifold Langevin and Hamiltonian Monte Carlo methods. J. R. Stat. Soc. Ser. B Stat. Methodol. Vol. 73, No. 2, pp. 123-214 (2011).
  * [14] Goldys B. and Maslowski B.: Exponential ergodicity for stochastic reaction-diffusion equations. Stochastic partial differential equations and applications-VII. Vol. 245, Chapman & Hall/CRC, Boca Raton, FL, 2006.
  * [15] Gürbüzbalaban M., Simsekli U. and Zhu L.: The heavy-tail phenomenon in SGD. International Conference on Machine Learning. PMLR. pp. 3964-3975 (2021).
  * [16] Huang L., Majka M. and Wang J.: Approximation of heavy-tailed distributions via stable-driven SDEs. Bernoulli. Vol. 27, No. 3, pp. 2040-2068 (2021).
  * [17] Jacod J. and Shiryaev A. N.: Limit theorems for stochastic processes. Grundlehren der mathematischen Wissenschaften. Vol. 288, Springer-Verlag, Berlin, Second edition, 2003.
  * [18] Kulik A. M.: Exponential ergodicity of the solutions to SDE’s with a jump noise. Stochastic Process. Appl. Vol. 119, No. 2, pp. 602-632 (2009).
  * [19] Liang M. and Wang J.: Gradient estimates and ergodicity for SDEs driven by multiplicative Lévy noises via coupling. Stochastic Process. Appl. Vol. 130, No. 5, pp. 3053-3094 (2020).
  * [20] Liang M., Majka M. and Wang J.: Exponential ergodicity for SDEs and McKean-Vlasov processes with Lévy noise. Ann. Inst. Henri Poincaré Probab. Stat. Vol. 57, No. 3, pp. 1665-1701 (2021).
  * [21] Luo D. and Wang J.: Refined basic couplings and Wasserstein-type distances for SDEs with Lévy noises. Stochastic Process. Appl. Vol. 129, No. 9, pp. 3129-3173 (2019).
  * [22] Majka M. B.: Coupling and exponential ergodicity for stochastic differential equations driven by Lévy processes. Stochastic Process. Appl. Vol. 127, No. 12, pp. 4083-4125 (2017).
  * [23] Masuda H.: Ergodicity and exponential $\beta$-mixing bounds for multidimensional diffusions with jumps. Stochastic Process. Appl. Vol. 117, No. 1, pp. 35-56 (2007).
  * [24] Mattingly J.C., Stuart A.M. and Higman D.J.: Ergodicity for SDEs and approximations: locally Lipschitz vector fields and degenerate noise. Stoch. Proc. Appl. Vol. 101, pp. 185-232 (2002) .
  * [25] Menozzi S. and Zhang X.: Heat kernel of supercritical SDEs with unbounded drifts. to appear in Journal de l’École polytechnique-Mathématiques, arXiv: 2012.14775 (2020).
  * [26] Meyn S. and Tweedie R. L.: Markov chains and stochastic stability. Cambridge University Press, Cambridge, Second edition, 2009.
  * [27] Revuz D. and Yor M.: Continuous martingales and Brownian motion. Vol. 293, Springer-Verlag, Berlin, 1999.
  * [28] Roberts G. O. and Tweedie R. L.: Exponential convergence of Langevin distributions and their discrete approximations. Bernoulli. Vol. 2, No. 4, pp. 341-363 (1996).
  * [29] Simsekli U.: Fractional langevin monte carlo: Exploring lévy driven stochastic differential equations for markov chain monte carlo. International Conference on Machine Learning. PMLR. pp. 3200-3209 (2017).
  * [30] Simsekli U., Zhu L, Teh Y.W. and Gürbüzbalaban M.: Fractional underdamped langevin dynamics: Retargeting sgd with momentum under heavy-tailed gradient noise. International Conference on Machine Learning. PMLR. pp. 8970-8980 (2020).
  * [31] Stein E.M.: Singular integrals and differentiability properties of functions. Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970.
  * [32] Nguyen T., Simsekli U. and Richard, D.: Non-asymptotic analysis of Fractional Langevin Monte Carlo for non-convex optimization. International Conference on Machine Learning. PMLR. pp. 4810-4819 (2019).
  * [33] Wang F.: Functional inequalities Markov semigroups and spectral theory. Elsevier, 2006.
  * [34] Wang J.: $L^{p}$-Wasserstein distance for stochastic differential equations driven by Lévy processes. Bernoulli. Vol. 22, No. 3, pp. 1598-1616 (2016).
  * [35] Xie L. and Zhang X.: Ergodicity of stochastic differential equations with jumps and singular coefficients. Ann. Inst. Henri Poincaré Probab. Stat. Vol. 56, No. 1, pp. 175-229 (2020).
  * [36] Ye N. and Zhu Z.: Stochastic Fractional Hamiltonian Monte Carlo. Proceedings of the 27th International Joint Conference on Artificial Intelligence. pp. 3019-3025 (2018).