Ergodicity and Mixing Properties for SDEs with α-Stable Lévy Noises

Abstract

In this paper, we consider a class of stochastic differential equations driven by multiplicative $\alpha$-stable ($0<\alpha <2$) Lévy noises. Firstly, we show that there exists a unique strong solution under a local one-sided Lipschitz condition and a general non-explosion condition. Next, the weak Feller and stationary properties are derived. Furthermore, a concrete sufficient condition for the coefficients is presented, which is different from the conditions for SDEs driven by Brownian motion or general squared-integrable martingales. Finally, some ergodic and mixing properties are obtained by using the Foster–Lyapunov criteria.

1. Introduction

In recent years, there has been increasing interest in studying the $\alpha$-stable Lévy process both regarding stochastic analysis and statistical inference. For example, in the field of stochastic analysis, Bass and Chen [1] considered stochastic differential equations (SDEs) with multiplicative $\alpha$-stable noises ($0<\alpha <2$); the existence and uniqueness of a weak solution were obtained under mild assumptions. Priola [2] studied the pathwise uniqueness for SDEs driven by non-degenerate symmetric $\alpha$-stable processes $(1\le \alpha <2)$ with a bounded and $\beta$-Hölder continuous drift term (≥$1-\alpha /2$). Furthermore, Priola and Zabczyk [3] considered a class of semilinear stochastic evolution equations with additive cylindrical stable Lévy noises and obtained some structural properties for the equations, including Markovianity, irreducibility, stochastic continuity, Feller, and strong Feller properties. In addition, Zhang [4] considered the pathwise uniqueness for SDEs driven by symmetric $\alpha$-stable Lévy processes ($1<\alpha <2$) with time-dependent Sobolev drifts, and Xu [5] considered types of 2D SDEs with additive degenerated $\alpha$-stable noise. Also, types of time-dependent SDEs with $\alpha$-stable-like noises ($0<\alpha <2$, including cylindrical cases) were considered in the study by Chen et al. [6]. For other related topics, we refer the reader to [7,8,9] and the references therein.

On the other hand, from the perspective of statistical inference, Hu and Long [10] considered parameter estimate problems for the drift coefficients of Ornstein–Uhlenbeck processes with additive $\alpha$-stable Lévy noises. Later, least squares estimators for discretely observed stochastic processes driven by small Lévy processes were considered in [11,12]. In addition, minimum distance estimates for general SDEs with small additive $\alpha$-stable Lévy noises were considered in the research by Zhao and Zhang [13].

We note that most of the works mentioned above are concerned with additive $\alpha$-stable Lévy noises. However, due to the diverse demands of practical problems, it is important to consider stochastic differential equations with multiplicative $\alpha$-stable noises. Therefore, in this paper, we consider the following d-dimensional SDE with multiplicative $\alpha$-stable Lévy noises; that is,

$$X_t=X_0+{\int }_0^{t}b\left(X_s\right)ds+{\int }_0^{t}L\left(X_{s-}\right)d{Z}_s,$$

(1)

where $b:{\mathbb{R}}^d\to {\mathbb{R}}^d$ is a continuous function, L is a $d\times d$-matrix-valued function defined on ${\mathbb{R}}^d$ with each $(i,j)$-th element $L_{ij}$ being continuous, and $Z={\left(Z_t\right)}_{t\ge 0}$ is a d-dimensional symmetric $\alpha$-stable ($0<\alpha <2$) Lévy process. Here, the stochastic integral with respect to Z is understood in the sense of [14] (see, e.g., p. 237).

It is widely acknowledged that the existence and uniqueness of solutions to SDEs are foundational for statistical inference. Moreover, properties such as stationarity, ergodicity, and mixing are crucial in statistical inference; see, e.g., Refs. [15,16]. Hence, the main purpose of this paper is to propose some weak sufficient conditions under which

(1)

the equation has a unique strong solution;

(2)

the solution admits an invariant measure;

(3)

the solution has some ergodic or mixing properties.

We note that Equation (1) is a generalization of a stochastic differential equation driven by Brownian motion and a continuous martingale ([17]), and our results in this paper can serve as fundamental tools for further statistical inference. It is worth mentioning that celebrated papers [5,18] considered similar questions for SDEs with jumps. However, the results in [18] were derived under the classical Lipschitz condition, and the equation in [5] is just a 2D stochastic differential equation with additive degenerated noise. Compared to their results, we consider a weaker local one-sided Lipschitz condition (A1) (see Section 2), and our equation has multiplicative noise. In addition, we need to overcome some new difficulties in the local one-sided Lipschitz condition case.

The rest of this paper is organized as follows. In Section 2, under both the local one-sided Lipschitz condition and non-explosion condition, by using the interlacing method, we show that Equation (1) has a unique strong solution, which is the main result of Theorem 1. In our case, the solution does not have a continuity modification with respect to the initial value, which is crucial for the weak Feller property in the Lipschitz case. Although this continuity property is no longer satisfied, we show that the weak Feller property still holds, and the existence and uniqueness of the stationary distribution are derived in Theorem 2. Subsequently, we present a concrete condition to ensure that our local one-sided Lipschitz and non-explosion conditions are satisfied. Finally, under some additional conditions, ergodic, exponentially ergodic, $\beta$-mixing, and exponentially $\beta$-mixing properties are explored in Section 3.

2. Preliminaries

2.1. Basic Assumptions

Let $(\Omega ,\mathcal{F},{\left({\mathcal{F}}_t\right)}_{t\ge 0},\mathbb{P})$ be a complete probability space satisfying the usual conditions. For $d\in \mathbb{N}$ and $0<\alpha <2$, let $Z={\left(Z_t\right)}_{t\ge 0}$ be a d-dimensional normalized symmetric $\alpha$-stable Lévy process; i.e., Z is a process with stationary independent increments and

$$\mathbb{E}exp\{i\langle u,Z_t\rangle \}={exp\{-t|u|}^{\alpha }\},u\in {\mathbb{R}}^d.$$

By using Lévy–Itô decomposition, we have

$$Z_t={\int }_0^t{\int }_{|z|<1}z\tilde{N}(dz,dt)+{\int }_0^t{\int }_{|z|\ge 1}zN(dz,dt),$$

where $N(dz,dt)$ is a Poisson random measure on ${\mathbb{R}}^d\times {\mathbb{R}}_{+}$ with intensity measure $\nu \left(dz\right)dt$ and $\tilde{N}(dz,dt):=N(dz,dt)-\nu \left(dz\right)dt$. Here,

$$\nu \left(dz\right)=\left\{\begin{array}{c}{\displaystyle \frac{c_{\alpha }}{{|z|}^{d+\alpha }}dz,z\ne 0,}\hfill \\ {\displaystyle 0,z=0,}\hfill \end{array}\right.$$

where $c_{\alpha }>0$. Note that we have

$${\int }_{{\mathbb{R}}^d}(1\wedge {|z|}^2)\nu \left(dz\right)<\infty ,$$

(2)

and, for every $0\le q<\alpha$,

$${\int }_{|z|\ge 1}{|z|}^q\nu \left(dz\right)<\infty$$

(3)

holds. In addition, Equation (1) can be rewritten as

$$\begin{array}{ccc}\hfill X_t& =& X_0+{\int }_0^{t}b\left(X_s\right)ds+{\int }_0^t{\int }_{|z|<1}L\left(X_{s-}\right)z\tilde{N}(dz,ds)\hfill \\ & & +{\int }_0^t{\int }_{|z|\ge 1}L\left(X_{s-}\right)zN(dz,dt).\hfill \end{array}$$

(4)

Next, for some suitable function $f:{\mathbb{R}}^d\to \mathbb{R}$, let

$$\mathcal{A}f:=\langle b\left(x\right),\nabla f\rangle +\mathcal{J}f,$$

(5)

where

$$\nabla f=({\partial }_{x_1}f,\cdots ,{\partial }_{x_d}f),$$

$$\mathcal{J}f={\int }_{{\mathbb{R}}^d\setminus \left\{0\right\}}(f(x+L\left(x\right)z)-f\left(x\right)-\langle \nabla f\left(x\right),L\left(x\right)z\rangle 1_{\{|z|<1\}})\nu \left(dz\right),$$

and $\langle ·,·\rangle$ denotes the inner product of two ${\mathbb{R}}^d$-vectors. It is evident that $\mathcal{A}$ is an integro-differential operator, which plays an important role in our basic assumptions. Let $D\left(\mathcal{A}\right)$ be the domain of $\mathcal{A}$. Generally, the domain $D\left(\mathcal{A}\right)$ is too narrow to include the unbounded functions. To overcome this difficulty, we follow the idea of [19] to consider the following truncated operators. That is, for each $m\in \mathbb{N}$, we define

$${\mathcal{A}}_{m}f\left(x\right):=1_{O^m}\left(x\right)\mathcal{A}f\left(x\right),$$

where $O^m:=\{x\in {\mathbb{R}}^d:|x|\le m\}$.

Now, we propose our basic assumptions.

(A1) 

(Local one-sided Lipschitz condition) For each $m\in \mathbb{N}$, suppose that there exists a constant $K_m>0$ such that, for any x, $y\in O^m$, we have

$$\langle b\left(x\right)-b\left(y\right),x-y\rangle +{\displaystyle \frac{a}{2}}{\parallel L\left(x\right)-L\left(y\right)\parallel }^2\le K_m{|x-y|}^2,$$

where $a={\int }_{\{0<|z|<1\}}{|z|}^2\nu \left(dz\right)$ (constant), and $\parallel ·\parallel$ denotes the operator norm of a matrix.

(A2) 

(Non-explosion condition) For each $m\in \mathbb{N}$, suppose that there exists a norm-like function $\phi \in C^2({\mathbb{R}}^d;{\mathbb{R}}_{+})\cap D\left({\mathcal{A}}_m\right)$ and two non-negative constants $c_0$ and $c_1$, independent of m, such that, for all $x\in O^m$, we have

$${\mathcal{A}}_m\phi \left(x\right)\le c_0+c_1\phi \left(x\right)$$

and

$${\int }_{|z|\ge 1}|\phi (x+z)-\phi \left(x\right)|\nu \left(dz\right)\le c_0+c_1\phi \left(x\right).$$

Here, we call a positive function $\phi :{\mathbb{R}}^d\to {\mathbb{R}}_{+}$ with ${lim}_{|x|\to \infty }\phi \left(x\right)=\infty$ a norm-like function.

(A3) 

(Tail symmetry property) Suppose that there exists a constant $K>0$ and an increasing norm-like function $\varphi :{\mathbb{R}}_{+}\to {\mathbb{R}}_{+}$ such that

$$\phi \left(x\right)1_{\{|x|>K\}}=\varphi (|x|)1_{\{|x|>K\}},$$

where $\phi$ is the same function mentioned as in (A2).

Remark 1. 

Before proceeding, we provide some remarks for our basic assumptions.

(1) 

The one-sided Lipschitz condition (A1) is also called a monotonicity condition (see, e.g., Refs. [20,21]) or a dissipative condition (see, e.g., Ref. [22]). It is clear that (A1) is weaker than the classical local Lipschitz condition.

(2) 

The first inequality of non-explosion condition (A2) is borrowed from [19] (cf. condition (CD0) on page 524). If the norm-like function can be chosen as $\phi \left(x\right)={|x|}^2$, then condition (A2) implies that there exists constant $c>0$ such that, for all $x\in O^m$,

$$\langle b\left(x\right),x\rangle +{\int }_{|z|<1}{|z|}^2{\nu \left(dz\right)\parallel L\left(x\right)\parallel }^2\le {c(1+|x|}^2),$$

which is the so-called local one-sided linear growth condition in the case of Brownian motion or squared-integrable martingale driving equations (see, e.g., Ref. [23]). However, for α-stable Lévy noises driving SDEs, the situation is different due to their weak integrable property (see, e.g., (A4) in Section 2.4). From this perspective, (A2) can be viewed as an extension of one-sided linear growth condition.

(3) 

The second inequality of (A2) is used to prove the existence of solution for modified Equation (6) without large jumps.

(4) 

Assumption (A3) implies that the tail of φ is symmetric, which will make our proofs easy to write. That is, (A3) is not an essential condition; it can be replaced by some other conditions.

2.2. Existence and Uniqueness

Now, we present the following existence and uniqueness results for Equation (4).

Theorem 1. 

Under assumptions (A1)–(A3), let μ be the distribution of initial value $X_0$, satisfying ${\int }_{{\mathbb{R}}^d}\phi \left(x\right)\mu \left(dx\right)<\infty$, where φ is the norm-like function in (A2). Then, there exists a unique càdlàg strong solution for Equation (4).

Proof. 

We divide the proof into two steps.

Step 1. In this step, we consider the following modified equation without large jumps:

$$\begin{array}{ccc}\hfill Y_t& =& X_0+{\int }_0^{t}b\left(Y_s\right)ds+{\int }_0^t{\int }_{|z|<1}L\left(Y_{s-}\right)z\tilde{N}(dz,ds).\hfill \end{array}$$

(6)

Firstly, the uniqueness of the solution of Equation (6) follows assumption (A1) by employing a similar procedure as in the proof of Theorem 1 of [20], so we omit it. Next, by applying the method used in [20] (see also [23]), we have that, for any $n\in \mathbb{N}$, under condition (A1), equation

$$\begin{array}{ccc}\hfill X_t& =& X_0+{\int }_0^t{1}_{\{|X_s|\le n\}}b\left(X_s\right)ds+{\int }_0^t{\int }_{|z|<1}{1}_{\{|X_{s-}|\le n\}}L\left(X_{s-}\right)z\tilde{N}(dz,ds)\hfill \end{array}$$

(7)

admits a unique adapted càdlàg solution. We denote the corresponding solutions by ${\left(X_t^n\right)}_{t\ge 0}$ and let ${\tau }_n:=inf\{t\ge 0:|X_t^n|\ge n\}$. In other words, ${\left(X_t^n\right)}_{t\ge 0}$ satisfies the following equation

$$\begin{array}{ccc}\hfill X_t^n& =& X_0+{\int }_0^{t\wedge {\tau }_n}b\left(X_s^n\right)ds+{\int }_0^{t\wedge {\tau }_n}{\int }_{|z|<1}L\left(X_{s-}^n\right)z\tilde{N}(dz,ds).\hfill \end{array}$$

(8)

The uniqueness of Equation (7) implies that $X_t^n=X_t^m$ for $t\in [0,{\tau }_n\wedge {\tau }_m]$. Furthermore, if $n\le m$, then ${\tau }_n\le {\tau }_m$ (a.s.). It follows that there exists a stopping time $\tau$ such that $\tau ={lim}_{n\to \infty }{\tau }_n$ (a.s.). To verify Equation (6) admits a unique solution, we only need to prove $\tau =\infty$ (a.s.). For arbitrary $T>0$ and $0\le t\le T$, applying Itô’s formula to Equation (8), we have

$$\begin{array}{ccc}\hfill \phi \left(X_{t\wedge {\tau }_n}^n\right)& =& \phi \left(X_0\right)+{\int }_0^{t\wedge {\tau }_n}\langle b\left(X_s^n\right),\nabla \phi \left(X_s^n\right)\rangle ds\hfill \\ & & +{\int }_0^{t\wedge {\tau }_n}{\int }_{|z|<1}(\phi (X_{s-}^n+L\left(X_{s-}^n\right)z)-\phi \left(X_{s-}^n\right))\tilde{N}(dz,ds)\hfill \\ & & +{\int }_0^{t\wedge {\tau }_n}{\int }_{|z|<1}(\phi (X_{s-}^n+L\left(X_{s-}^n\right)z)-\phi \left(X_{s-}^n\right)\hfill \\ & & -\langle L\left(X_{s-}^n\right)z,\nabla \phi \left(X_{s-}^n\right)\rangle )\nu \left(dz\right)ds,\hfill \end{array}$$

where $\phi$ is the norm-like function in assumption (A2). By assumption (A2) and taking the expectation, we obtain

$$\begin{array}{ccc}\hfill \mathbb{E}\left(\phi \left(X_{t\wedge {\tau }_n}^n\right)\right)& \le & \mathbb{E}\phi \left(X_0\right)+\mathbb{E}{\int }_0^t[c_0+c_1\phi \left(X_{s\wedge {\tau }_n}^n\right)]ds\hfill \\ & \le & \mathbb{E}\phi \left(X_0\right)+c_{0}T+c_1{\int }_0^t\mathbb{E}\left(\phi \left(X_{s\wedge {\tau }_n}^n\right)\right)ds.\hfill \end{array}$$

By using Gronwall’s inequality, we have

$$\mathbb{E}\left(\phi \left(X_{T\wedge {\tau }_n}^n\right)\right)\le (\mathbb{E}\phi \left(X_0\right)+c_{0}T)e^{c_{1}T}\le C.$$

On the other hand, for a large enough n such that $n>K$ (where K is the constant in assumption (A3)), then

$$\mathbb{P}({\tau }_n\le T)\varphi \left(n\right)\le \mathbb{E}\left(\phi \left(X_{T\wedge {\tau }_n}^n\right)1_{\{{\tau }_n\le T\}}\right)\le \mathbb{E}\left(\phi \left(X_{T\wedge {\tau }_n}^n\right)\right).$$

Therefore,

$$\mathbb{P}({\tau }_n\le T)\varphi \left(n\right)\le C.$$

Let $n\to \infty$ and, by using assumption (A3), we obtain

$$\mathbb{P}(\tau \le T)=0.$$

Since T is arbitrary, we obtain

$$\mathbb{P}(\tau =\infty )=1.$$

Step 2. In this step, we construct the unique solution of Equation (4) by using the interlacing method (see, e.g., Ref. [14] p. 236). Let

$$J_t:={\int }_0^t{\int }_{|z|\ge 1}zN(ds,dz).$$

(9)

For $k\in \mathbb{N}$, let $S_0:=0$,

$$\begin{array}{ccc}\hfill S_1& :=& inf\{t\ge 0:|\Delta J_t|>0\},\hfill \\ & & ⋮\hfill \\ \hfill S_k& :=& inf\{t>S_{k-1}:|\Delta J_t|>0\},\hfill \end{array}$$

and so on denote the k-th jump of ${\left(J_t\right)}_{t\ge 0}$. Note that ${lim}_{k\to \infty }{S}_k=\infty$ almost surely (see, e.g., p. 26 in [24]). Let ${\left(Y_t\right)}_{t\ge 0}$ be the unique solution of Equation (6). For each $t\ge 0$, we construct $X_t$ recursively as follows:

$$X_t=\left\{\begin{array}{cc}{Y}_t,\hfill & \mathrm{for}0\le t<S_1,\hfill \\ X_{t-}+L\left(X_{t-}\right)\Delta J_t,\hfill & \mathrm{for}t=S_1,\hfill \\ X_{S_1}+Y_t-Y_{S_1},\hfill & \mathrm{for}{S}_1\le t<S_2,\hfill \\ X_{t-}+L\left(X_{t-}\right)\Delta J_t,\hfill & \mathrm{for}t=S_2.\hfill \end{array}\right.$$

(10)

From the structure of our construction, we can see that ${\left(X_t\right)}_{t\ge 0}$ is the uniqueness solution of Equation (4). We finish the proof. □

2.3. Weak Feller Property and Existence of an Invariant Measure

Following Theorem 1, the flow property and homogeneous Markov property of the solution can be proved by taking a similar approach as in Sections 6.4.1 and 6.4.2 of [14]. In the following, we aim to prove the weak Feller property and the existence of an invariant measure for Equation (4). To be more precise, let $P_t(x,·)$ denote the transition probability of the solution of Equation (4). That is,

$$P_t(x,A):=\mathbb{P}(X_t\in A|X_0=x),A\in \mathcal{B}\left({\mathbb{R}}^d\right),$$

where $\mathcal{B}\left({\mathbb{R}}^d\right)$ denotes the Borel $\sigma$-algebra of ${\mathbb{R}}^d$. If there is a probability measure $\pi (·)$ on $({\mathbb{R}}^d,\mathcal{B}\left({\mathbb{R}}^d\right))$ such that

$${\int }_{{\mathbb{R}}^d}{P}_t(x,A)\pi \left(dx\right)=\pi \left(A\right),A\in \mathcal{B}\left({\mathbb{R}}^d\right),$$

then $\pi$ is said to be an invariant measure for Equation (4).

Before proceeding further, we first present several lemmas that will be used in the subsequent discussions.

Lemma 1. 

Under the assumptions of Theorem 1, let ${\left(X_t\right)}_{t\ge 0}$ be the solution of Equation (4). Then, for each $T>0$, there exists a constant $C_{T,\mu }>0$ such that

$$\underset{0\le t\le T}{sup}\mathbb{E}\left(\phi \left(X_t\right)\right)\le C_{T,\mu },$$

where μ is the distribution of initial value $X_0$.

Proof. 

For $0\le t\le T$, applying Itô’s formula to Equation (4), we have

$$\begin{array}{ccc}\hfill \phi \left(X_t\right)& =& \phi \left(X_0\right)+{\int }_0^t\langle b\left(X_s\right),\nabla \phi \left(X_s\right)\rangle ds\hfill \\ & & +{\int }_0^t{\int }_{|z|<1}(\phi (X_{s-}+L\left(X_{s-}\right)z)-\phi \left(X_{s-}\right))\tilde{N}(dz,ds)\hfill \\ & & +{\int }_0^t{\int }_{|z|\ge 1}(\phi (X_{s-}+L\left(X_{s-}\right)z)-\phi \left(X_{s-}\right))\tilde{N}(dz,ds)\hfill \\ & & +{\int }_0^t{\int }_{{\mathbb{R}}^d\setminus \left\{0\right\}}(\phi (X_{s-}+L\left(X_{s-}\right)z)-\phi \left(X_{s-}\right)\hfill \\ & & -\langle L\left(X_{s-}\right)z,\nabla \phi (X_{s-}\rangle 1_{\{|z|<1\}})\nu \left(dz\right)ds\hfill \\ & =& \phi \left(X_0\right)+{\int }_0^t\mathcal{A}\phi \left(X_s\right)ds+M_t,\hfill \end{array}$$

where

$$M_t:={\int }_0^t{\int }_{{\mathbb{R}}^d}(\phi (X_{s-}+L\left(X_{s-}\right)z)-\phi \left(X_{s-}\right))\tilde{N}(dz,ds).$$

For each $m\in \mathbb{N}$, let ${\sigma }_m:=inf\{t\ge 0:|X_t|\ge m\}$. By taking a similar approach as in the proof of Theorem 1, we see that ${lim}_{m\to \infty }{\sigma }_m=\infty$ a.s. Moreover, according to assumption (A2), we have $\phi \in D\left({\mathcal{A}}_m\right)$, and $M={\left(M_t\right)}_{t\ge 0}$ is a local martingale. It follows that

$$\phi \left(X_{t\wedge {\sigma }_m}\right)=\phi \left(X_0\right)+{\int }_0^{t\wedge {\sigma }_m}{\mathcal{A}}_m\phi \left(X_s\right)ds+M_{t\wedge {\sigma }_m}$$

and

$$\begin{array}{ccc}\hfill \mathbb{E}\phi \left(X_{t\wedge {\sigma }_m}\right)& =& \mathbb{E}\phi \left(X_0\right)+\mathbb{E}{\int }_0^t{\mathcal{A}}_m\phi \left(X_{s\wedge {\sigma }_m}\right)ds\hfill \\ & \le & \mathbb{E}\phi \left(X_0\right)+\mathbb{E}{\int }_0^t(c_0+c_1\phi \left(X_{s\wedge {\sigma }_m}\right))ds.\hfill \end{array}$$

By using Gronwall’s inequality again, we obtain

$$\mathbb{E}\left(\phi \left(X_{t\wedge {\sigma }_m}\right)\right)\le (\mathbb{E}\phi \left(X_0\right)+c_{0}T)e^{c_{1}T}\le C_{T,\mu }.$$

Finally, we have

$$\begin{array}{ccc}\hfill \underset{0\le t\le T}{sup}\mathbb{E}\left(\phi \left(X_t\right)\right)& =& \underset{0\le t\le T}{sup}\mathbb{E}\left(\underset{m\to \infty }{lim}\phi \left(X_{t\wedge {\sigma }_m}\right)\right)\hfill \\ & \le & \underset{0\le t\le T}{sup}\underset{m\to \infty }{lim\; inf}\mathbb{E}\left(\phi \left(X_{t\wedge {\sigma }_m}\right)\right)\hfill \\ & \le & C_{T,\mu },\hfill \end{array}$$

where we have used Fatou’s lemma in the second inequality. Hence, we have proved our desired result. □

The next lemma provides a weak continuity property for the solution of our modified Equation (6).

Lemma 2. 

Under the assumptions of Theorem 1, let ${\left(Y_t^x\right)}_{t\ge 0}$ be the solution of the modified Equation (6) with the initial value $x\in {\mathbb{R}}^d$. If ${lim}_{n\to \infty }{x}_n=x$, then, for any $\delta >0$ and $T>0$,

$$\underset{n\to \infty }{lim}\mathbb{P}(\underset{0\le s\le T}{sup}|Y_s^{x_n}-Y_s^x|>\delta )=0.$$

Proof. 

The proof is essentially the same as the proof of Theorem 3 of [20], so we omit the proof. □

Remark 2. 

Under the one-sided Lipschitz condition, we do not have a continuous modification for the solution like Theorem 6.6.3 in [14], which is crucial in their proof of weak Feller property. However, the next lemma shows that the weak Feller property still holds for Equation (4).

Lemma 3. 

Under the assumptions of Theorem 1, let ${\left(X_t^x\right)}_{t\ge 0}$ be the solution of Equation (4) with the initial value $x\in {\mathbb{R}}^d$. Then, ${\left(X_t^x\right)}_{t\ge 0}$ is a weak Feller process. That is,

$$T_t\left(C_b\left({\mathbb{R}}^d\right)\right)\subset C_b\left({\mathbb{R}}^d\right)$$

for each $t\ge 0$, where ${\left(T_t\right)}_{t\ge 0}$ is the semigroup related to ${\left(X_t^x\right)}_{t\ge 0}$ and $C_b\left({\mathbb{R}}^d\right)$ denotes the set of bounded continuous functions on ${\mathbb{R}}^d$.

Proof. 

Recall that, for each $t\ge 0$ and $f\in C_b\left({\mathbb{R}}^d\right)$, the semigroup $T_t$ is defined as

$$T_{t}f\left(x\right):=\mathbb{E}\left(f\left(X_t^x\right)\right).$$

It is clear that $T_{t}f$ is bounded. In the following, we aim to prove the continuity property of $T_{t}f$. For this purpose, let $x_n\to x$ as $n\to \infty$. Recall that ${\left(J_t\right)}_{t\ge 0}$ defined in (9) is a pure jump process and $S_k\to \infty$ almost surely, where ${\left(S_k\right)}_{k\ge 1}$ are its jump times. Hence, for any $\epsilon >0$, there exists large enough $N_1\in \mathbb{N}$ such that

$$\mathbb{P}(S_{N_1}<t)<\epsilon /\parallel f\parallel ,$$

where $\parallel f\parallel$ denotes the bound of function f. It follows that

$$\begin{array}{ccc}\hfill |T_{t}f\left(x_n\right)-T_{t}f\left(x\right)|& =& \mathbb{E}\left(f\left(X_t^{x_n}\right)\right)-\mathbb{E}\left(f\left(X_t^x\right)\right)|\hfill \\ & \le & \mathbb{E}(|f\left(X_t^{x_n}\right)-f\left(X_t^x\right)|1_{\{S_{N_1}<t\}})\hfill \\ & & +\mathbb{E}(|f\left(X_t^{x_n}\right)-f\left(X_t^x\right)|1_{\{t\le S_{N_1}\}})\hfill \\ & \le & 2\epsilon +\mathbb{E}(|f\left(X_t^{x_n}\right)-f\left(X_t^x\right)|1_{\{t\le S_{N_1}\}})\hfill \\ & =& 2\epsilon +\sum _{k=1}^{N_1}\mathbb{E}(|f\left(X_t^{x_n}\right)-f\left(X_t^x\right)|1_{\{t\in (S_{k-1},S_k]\}})\hfill \end{array}$$

In addition, let us denote by $|\tilde{X}|=|X_t^x|\vee {sup}_{n\ge 1}|X_t^{x_n}|$. It is clear that there exists large enough $M>0$ such that

$$\mathbb{P}(|\tilde{X}|>M)<\epsilon /\parallel f\parallel .$$

Then, we immediately have

$$\begin{array}{ccc}\hfill |T_{t}f\left(x_n\right)-T_{t}f\left(x\right)|& \le & 2\epsilon +\sum _{k=1}^{N_1}\mathbb{E}\left(|f\left(X_t^{x_n}\right)-f\left(X_t^x\right)|1_{\{t\in (S_{k-1},S_k]\}}\right)\hfill \\ & =& 2\epsilon +\sum _{k=1}^{N_1}\mathbb{E}\left(|f\left(X_t^{x_n}\right)-f\left(X_t^x\right)|1_{\{t\in (S_{k-1},S_k]\}}{1}_{\{|\tilde{X}|>M\}}\right)\hfill \\ & & +\sum _{k=1}^{N_1}\mathbb{E}\left(|f\left(X_t^{x_n}\right)-f\left(X_t^x\right)|1_{\{t\in (S_{k-1},S_k]\}}{1}_{\{|\tilde{X}|\le M\}}\right)\hfill \\ & \le & 4\epsilon +\sum _{k=1}^{N_1}\mathbb{E}\left(|f\left(X_t^{x_n}\right)-f\left(X_t^x\right)|1_{\{t\in (S_{k-1},S_k]\}}{1}_{\{|\tilde{X}|\le M\}}\right).\hfill \end{array}$$

(11)

Furthermore, since f is a bounded continuous function, the uniform continuity property on the interval $[-M,M]$ implies that there exists ${\delta }_0>0$ such that $|f\left(x\right)-f\left(y\right)|<\epsilon /N_1$ for all $x,y\in [-M,M]$ with $|x-y|\le {\delta }_0$. Therefore, for fixed $k\in \{1,2,\cdots ,N_1\}$, we have

$$\begin{array}{ccc}\hfill & & \mathbb{E}(|f\left(X_t^{x_n}\right)-f\left(X_t^x\right)|1_{\{t\in (S_{k-1},S_k]\}}{1}_{\{|\tilde{X}|\le M\}})\hfill \\ & =& \mathbb{E}(|f\left(X_t^{x_n}\right)-f\left(X_t^x\right)|1_{\{t\in (S_{k-1},S_k]\}}{1}_{\{|\tilde{X}|\le M\}}{1}_{\{|X_t^{x_n}-X_t^x|\le {\delta }_0\}})\hfill \\ & & +\mathbb{E}(|f\left(X_t^{x_n}\right)-f\left(X_t^x\right)|1_{\{t\in (S_{k-1},S_k]\}}{1}_{\{|\tilde{X}|\le M\}}{1}_{\{|X_t^{x_n}-X_t^x|>{\delta }_0\}})\hfill \\ & \le & \epsilon /N_1+2\parallel f\parallel \mathbb{E}\left(1_{\{t\in (S_{k-1},S_k]\}}{1}_{\{|\tilde{X}|\le M\}}{1}_{\{|X_t^{x_n}-X_t^x|>{\delta }_0\}}\right)\hfill \\ & \le & \epsilon /N_1+2\parallel f\parallel \mathbb{E}\left(1_{\{t\in (S_{k-1},S_k]\}}{1}_{\{|X_t^{x_n}-X_t^x|>{\delta }_0\}}\right).\hfill \end{array}$$

(12)

Next, we will show that, for all $k\in \{1,2,\cdots ,N_1\}$, there exists a large enough $N_2\in \mathbb{N}$ such that

$$\mathbb{E}(1_{\{t\in (S_{k-1},S_k]\}}{1}_{\{|X_t^{x_n}-X_t^x|>{\delta }_0\}})\le \epsilon /(N_1\parallel f\parallel )$$

(13)

for $n\ge N_2$. Indeed, for $k=1$, due to

$$X_t^x=\left\{\begin{array}{cc}{Y}_t^x,\hfill & \mathrm{for}{S}_0<t<S_1,\hfill \\ X_{t-}^x+L\left(X_{t-}^x\right)\Delta J_t,\hfill & \mathrm{for}t=S_1,\hfill \end{array}\right.$$

where ${\left(Y_t^x\right)}_{t\ge 0}$ is the solution of the modified Equation (6), we have

$$\begin{array}{ccc}& & \mathbb{E}\left(1_{\{t\in (S_0,S_1]\}}{1}_{\{|X_t^{x_n}-X_t^x|>{\delta }_0\}}\right)\hfill \\ & =& \mathbb{E}\left(1_{\{t\in (S_0,S_1)\}}{1}_{\{|X_t^{x_n}-X_t^x|>{\delta }_0\}}\right)+\mathbb{E}\left(1_{\{t=S_1\}}{1}_{\{|X_t^{x_n}-X_t^x|>{\delta }_0\}}\right)\hfill \\ & =& \mathbb{E}\left(1_{\{t\in (S_0,S_1)\}}{1}_{\{|Y_t^{x_n}-Y_t^x|>{\delta }_0\}}\right)+\mathbb{E}\left(1_{\{t=S_1\}}{1}_{\{|X_t^{x_n}-X_t^x|>{\delta }_0\}}\right)\hfill \\ & =:& I_1+I_2.\hfill \end{array}$$

According to Lemma 2, there exists a large enough $n_1\in \mathbb{N}$ such that

$$I_1\le \mathbb{P}(|Y_t^{x_n}-Y_t^x|>{\delta }_0)\le \epsilon /(4N_1\parallel f\parallel ).$$

Furthermore, note that $\nu (|z|\ge 1)<\infty$, so there exists a positive $M_1$ such that, for all $t\ge 0$,

$$\mathbb{P}(1<|\Delta J_t|<M_1)>1-\epsilon /(8N_1\parallel f\parallel ).$$

Then, we have

$$\begin{array}{ccc}\hfill I_2& =& \mathbb{P}(|X_t^{x_n}-X_t^x|>{\delta }_0,t=S_1)\hfill \\ & \le & \mathbb{P}(|Y_{t-}^{x_n}-Y_{t-}^x|>{\displaystyle \frac{{\delta }_0}{2}})+\mathbb{P}(|L\left(Y_{t-}^{x_n}\right)-L\left(Y_{t-}^x\right)||\Delta J_t|>{\displaystyle \frac{{\delta }_0}{2}})\hfill \\ & \le & \epsilon /(8N_1\parallel f\parallel )+\mathbb{P}(|Y_{t-}^{x_n}-Y_{t-}^x|>{\displaystyle \frac{{\delta }_0}{2}})\hfill \\ & & +\mathbb{P}(|L\left(Y_{t-}^{x_n}\right)-L\left(Y_{t-}^x\right)||\Delta J_t|>{\displaystyle \frac{{\delta }_0}{2}},1<|\Delta J_t|<M_1)\hfill \\ & \le & \epsilon /(8N_1\parallel f\parallel )+\mathbb{P}(|Y_{t-}^{x_n}-Y_{t-}^x|>\frac{{\delta }_0}{2})+\mathbb{P}(|L\left(Y_{t-}^{x_n}\right)-L\left(Y_{t-}^x\right)|>{\displaystyle \frac{{\delta }_0}{2M_1}}).\hfill \end{array}$$

By using Lemma 2 and continuous mapping theorem, we establish that there exists large enough $n_2>n_1$ such that

$$I_2\le \epsilon /(2N_1\parallel f\parallel )$$

for $n\ge n_2$. It follows that (13) holds for $k=1$. Note that $N_1$ is a finite number, so we can obtain (13) for all $k\in \{1,2,\cdots ,N_1\}$ by employing similar arguments as above. Finally, combining results (11)–(13), we obtain

$$\begin{array}{c}\hfill |T_{t}f\left(x_n\right)-T_{t}f\left(x\right)|\le 7\epsilon \end{array}$$

for $n\ge N_2$. We finish the proof. □

Next, if we strengthen assumption (A2) to (B2), then Equation (4) admits an invariant measure.

(B2)

Suppose that the constant $c_1$ in (A2) can be changed into some negative constant ${\tilde{c}}_1$.

Theorem 2. 

Under the assumptions of Theorem 1 and supposing that (B2) is satisfied, then there exists an invariant measure for Equation (4).

Proof. 

By using the Krylov–Bogoliubov theorem (see, e.g., Ref. [25], Corollary 3.1.2) and Lemma 3, to show there exists a stationary measure for $P_t$, we only need to prove that, for some $x\in {\mathbb{R}}^d$ and any $\epsilon >0$, there exist $r>0$ and $T>0$ such that, for all $t\ge T$,

$$\mathbb{P}(|X_t^x|>r)\le \epsilon .$$

(14)

In fact, according to the proof of Lemma 1, we have

$$\mathbb{E}\left(\phi \left(X_t^x\right)\right)\le (\phi \left(x\right)+c_{0}t)e^{{\tilde{c}}_{1}t}.$$

By Chebyshev inequality, for $r>K$ (here, K is the constant defined in (A3)), it follows that

$$\begin{array}{ccc}& & \mathbb{P}(|X_t^x|>r)\hfill \\ & =& \mathbb{P}\left(\varphi \right(|X_t^x|)>\varphi \left(r\right))\hfill \\ & \le & {\displaystyle \frac{\mathbb{E}\left(\varphi \right(|X_t^x|))}{\varphi \left(r\right)}}\hfill \\ & \le & {\displaystyle \frac{(\phi \left(x\right)+c_{0}t)e^{{\tilde{c}}_{1}t}}{\varphi \left(r\right)}}.\hfill \end{array}$$

Note that ${\tilde{c}}_1<0$; we immediately obtain (14), from which we obtain our desired result. □

2.4. Verification of Assumptions (A2) and (A3)

In this subsection, we provide a concrete condition for b and L to make sure that assumptions (A2) and (A3) are fulfilled.

(A4) 

Suppose that there exist constants $K>0$ and $0<p<1$ such that

$$\underset{|x|>K}{sup}{\displaystyle \frac{\langle b\left(x\right),x\rangle }{{|x|}^2}}<\infty$$

and

$$\underset{|x|>K}{sup}{\displaystyle \frac{\parallel L\left(x\right)\parallel }{{|x|}^{p}ln|x|}}<\infty .$$

We have the following result.

Lemma 4. 

If condition (A4) holds, then conditions (A2) and (A3) are satisfied.

Proof. 

The proof is similar to the proof of Lemma 6; we just provide a sketch. Under (A4), for $f\in Q_{q,K}$ (see Section 3.3), we can prove

$$\mathcal{J}f\left(x\right)=O(|x{|}^{q}ln|x|)$$

and

$$\begin{array}{ccc}\hfill \langle b\left(x\right),\nabla f\left(x\right)\rangle & =& {|x|}^{q-2}(qln|x|+1)\langle b\left(x\right),x\rangle \hfill \\ & \le & {C|x|}^{q}ln|x|\hfill \end{array}$$

for $|x|\to \infty$, which implies our desired result. For the details of the proof, we refer the reader to Lemma 6. □

Upon the above result, we immediately have the following corollary.

Corollary 1. 

Under assumptions (A1) and (A4), suppose that there exists $0<q<\alpha$ such that $\mathbb{E}(|X_0{|}^q)<\infty$, and then Equation (4) admits a unique càdlàg strong solution.

Remark 3. 

We observe that the condition in (A4) for L can be slightly weakened as follows: suppose that there exist constants $K>0$ and $0<p<1$ such that

$$\underset{|x|>K}{sup}{\displaystyle \frac{\parallel L\left(x\right)\parallel }{{|x|}^{p}g(ln|x|)}}<\infty ,$$

where g is a polynomial function.

3. Ergodic and Mixing Properties

3.1. Some Definitions

In this section, we focus on the ergodicity and mixing properties for the solution of Equation (4). The following are some definitions.

Let $P_t(x,·)$ be the transition probability defined as in Section 2.3. $X={\left(X_t^x\right)}_{t\ge 0}$ is called ergodic if there exists a unique stationary distribution $\pi$ such that

$$\parallel P_t{(x,·)-\pi \parallel }_{TV}\to 0,$$

as $t\to \infty$ for any $x\in {\mathbb{R}}^d$, where ${\parallel ·\parallel }_{TV}$ denotes the total variation norm of sign measures. Moreover, X is called exponentially ergodic if there exists a positive constant $\gamma$ and a measurable function $h\left(x\right)$ on $\mathbb{R}$ such that

$$\parallel P_t{(x,·)-\pi (·)\parallel }_{TV}\le h\left(x\right)e^{-\gamma t},$$

for any $x\in {\mathbb{R}}^d$.

On the other hand, the mixing properties, which characterize the weak dependence of the process, are also related to the topic of statistical analysis for stochastic processes. The $\alpha$-mixing and $\beta$-mixing coefficients of the solution of (4) with initial distribution $\mu$ can be defined as (see, e.g., Refs. [18,26])

$${\beta }_X\left(t\right)=\underset{s\ge 0}{sup}\mathbb{E}[\underset{B}{sup}|\mathbb{P}(B|\sigma \left(X_s^{\mu }\right))-\mathbb{P}\left(B\right)|],$$

$${\alpha }_X\left(t\right)=\underset{s\ge 0}{sup}\underset{A,B}{sup}|\mathbb{P}\left(AB\right)-\mathbb{P}\left(A\right)\mathbb{P}\left(B\right)|,$$

where $B\in \sigma \{X_u^{\mu }:u\ge s+t\}$ and $A\in \sigma \{X_u^{\mu },u\le s\}$. It is well known that ${\alpha }_X\left(t\right)\le {\beta }_X\left(t\right)$ for each $t\ge 0$. The process ${\left(X_t\right)}_{t\ge 0}$ is called $\beta$-mixing ($\alpha$-mixing) if ${\beta }_X\left(t\right)=o\left(1\right)$ (${\alpha }_X\left(t\right)=o\left(1\right)$) for $t\to \infty$ and exponentially $\beta$-mixing (exponentially $\alpha$-mixing) if there exists a positive constant $\gamma$ such that ${\beta }_X\left(t\right)=O\left(e^{-\gamma t}\right)$ (${\alpha }_X\left(t\right)=O\left(e^{-\gamma t}\right)$) for $t\to \infty$. Here, $g_1\left(x\right)=O\left(g_2\left(x\right)\right)$ as $x\to \infty$ means that there exists a constant $C>0$ such that $g_1\left(x\right)\le C{g}_2\left(x\right)$ for all x sufficiently large. In this paper, we will just consider the $\beta$-mixing property.

3.2. Additional Conditions

In order to obtain some ergodic and $\beta$-mixing properties, we need the following stronger condition for coefficients b and L compared to (A4).

(A5) 

For some

$$0<q<1\wedge \alpha ,$$

suppose that there exist

$$0<q^{\prime }<1,0<q^{″}\le 2$$

with $q+q^{\prime }<1$, $q+q^{″}>2$ and constants $K,C>0$ such that

$$\underset{|x|>K}{sup}{\displaystyle \frac{\langle b\left(x\right),x\rangle }{{|x|}^{q^{″}}}}\le -C$$

and

$$\underset{|x|>K}{sup}{\displaystyle \frac{\parallel L\left(x\right)\parallel }{{|x|}^{q^{\prime }}ln|x|}}<\infty .$$

To obtain our results, we also need some regular properties. For $0<u<1$, let $Y^u$ be the solution of the following equation

$$\begin{array}{ccc}\hfill Y_t^u& =& X_0+{\int }_0^{t}b\left(Y_s^u\right)ds+{\int }_0^t{\int }_{u<|z|<1}L\left(Y_{s-}^u\right)z\tilde{N}(dz,ds).\hfill \end{array}$$

(15)

(A6) (i)

Let $X^x$ be the solution of Equation (4) with initial value $x\in {\mathbb{R}}^d$. For any $x\in {\mathbb{R}}^d$, suppose that there exists a constant $\Delta >0$ such that $X_{\Delta }^x$ admits a density $p_{\Delta }(x,y)$ with respect to the Lebesgue measure on ${\mathbb{R}}^d$, and ${sup}_{x\in G,y\in {\mathbb{R}}^d}{p}_{\Delta }(x,y)<\infty$ for every compact set $G\subset {\mathbb{R}}^d$.

  (ii)

For $0<u<1$, let $Y^u$ be the solution of Equation (15). Suppose that, for any $x\in {\mathbb{R}}^d$ and constant $\Delta >0$ (appeared in statement (i)), there exist a small enough $u_0$ ($0<u_0<1$) and a positive density $p^{u_0}(x,y)$ such that $Y_{\Delta }^u$ admits a density $p_{\Delta }^u(x,y)$ with respect to the Lebesgue measure on ${\mathbb{R}}^d$ satisfying ${inf}_{u\in (0,u_0]}{p}_{\Delta }^u(x,y)\ge p^{u_0}(x,y)$ for all $y\in {\mathbb{R}}^d$.

Remark 4. 

We note that our condition (A6)-(ii) is a little different from the Assumption 2 provided in [18]. To explore concrete conditions to ensure that these regular properties hold is beyond the study of this article, which will be the subject of future research.

3.3. Main Results

Now, we provide the main results of this section.

Theorem 3. 

Let $0<q<\alpha$ and let $X^{\mu }$ be the solution of Equation (4) with the initial distribution μ, such that ${\int }_{{\mathbb{R}}^d}{|x|}^{q}ln|x|\mu \left(dx\right)<\infty$. Then,

(i) 

if assumptions (A1), (A5), and (A6) hold, then X is ergodic. In particular, if $\mu =\pi$ (invariant measure), then X is also β-mixing;

(ii) 

if assumptions (A1) and (A6) hold and (A5) holds with $q^{″}=2$, then X is both exponentially ergodic and exponentially β-mixing.

First of all, for some $0<q<\alpha$ and $K>1$, let us consider the space

$$Q_{q,K}:=\{f:f\in C^2({\mathbb{R}}^d;{\mathbb{R}}_{+})\mathrm{with}f\left(x\right)1_{\{|x|>K|\}}={|x|}^{q}ln(|x|)1_{\{|x|>K\}}\},$$

(16)

which will serve as a class of test functions for generator $\mathcal{A}$. We have the following result.

Lemma 5. 

For each $m\in \mathbb{N}$, we have

$$Q_{q,K}\subset D\left({\mathcal{A}}_m\right).$$

Proof. 

Recall that $\mathcal{A}$ is defined as in (5) and

$${\mathcal{A}}_{m}f\left(x\right)=1_{O^m}\left(x\right)\mathcal{A}f\left(x\right).$$

Since b and L are continuous, we only need to verify that $\mathcal{J}f$ is well defined for $x\in O^m$. For $f\in Q_{q,K}$, let

$${\mathcal{J}}^{*}f\left(x\right):={\int }_{|z|\ge 1}(f(x+L\left(x\right)z)-f\left(x\right))\nu \left(dz\right)$$

(17)

and

$${\mathcal{J}}_{*}f\left(x\right):={\int }_{|z|<1}(f(x+L\left(x\right)z)-f\left(x\right)-\langle \nabla f\left(x\right),L\left(x\right)z\rangle )\nu \left(dz\right).$$

(18)

Then, for ${\mathcal{J}}^{*}$ and $x\in O^m$, we have

$$\begin{array}{ccc}\hfill {\mathcal{J}}^{*}f\left(x\right)& =& {\int }_{|z|\ge 1}f(x+L\left(x\right)z)1_{\{|x+L\left(x\right)z|\le K\}}\nu \left(dz\right)\hfill \\ & & +{\int }_{|z|\ge 1}f(x+L\left(x\right)z)1_{\{|x+L\left(x\right)z|>K\}}\nu \left(dz\right)\hfill \\ & & -{\int }_{|z|\ge 1}f\left(x\right)\nu \left(dz\right)\hfill \\ & \le & 2\underset{|x|\le m\vee K}{sup}|f\left(x\right)|\nu \left(\right\{z:|z|\ge 1\left\}\right)\hfill \\ & & +{\int }_{|z|\ge 1}{|x+L\left(x\right)z|}^{q}ln(|x+L\left(x\right)z|)1_{\{|x+L\left(x\right)z|>K\}}\nu \left(dz\right)\hfill \\ & \le & 2\underset{|x|\le m\vee K}{sup}|f\left(x\right)|\nu \left(\right\{z:|z|\ge 1\left\}\right)\hfill \\ & & +\underset{|x|\le m}{sup}{(|x|\vee \parallel L\left(x\right)\parallel \vee 1)}^2{\int }_{|z|\ge 1}(1+{|z|}^q)\nu \left(dz\right)\hfill \\ & & +\underset{|x|\le m}{sup}{(|x|\vee \parallel L\left(x\right)\parallel \vee 1)}^2{\int }_{|z|\ge 1}(1+{|z|}^q)ln(1+|z|)\nu \left(dz\right)\hfill \\ & <& \infty ,\hfill \end{array}$$

where we have used Equations (2) and (3) in the last inequality. Next, for ${\mathcal{J}}_{*}$, by using Taylor expansion, we have

$$\begin{array}{ccc}\hfill {\mathcal{J}}_{*}f\left(x\right)& =& {\int }_{|z|<1}({\int }_0^1{\int }_0^{1}u\langle {\nabla }^{2}f(x+uvL\left(x\right)z)L\left(x\right)z,L\left(x\right)z\rangle dvdu)\nu \left(dz\right)\hfill \\ & \le & \underset{|x|\le m,|z|<1,0<u,v<1}{sup}\parallel {\nabla }^2{f(x+uvL\left(x\right)z)\parallel \parallel L\left(x\right)\parallel }^2{\int }_{|z|<1}{|z|}^2\nu \left(dz\right)\hfill \\ & <& \infty .\hfill \end{array}$$

We finish the proof. □

3.4. Foster–Lyapunov Criteria

Then, next lemma will play a key role in our proof.

Lemma 6. 

Under assumption (A5), for each $m\in \mathbb{N}$, there exist norm-like functions ${\phi }_1$ and ${\phi }_2$ such that

$${\mathcal{A}}_m{\phi }_1\left(x\right)\le -c_2{\phi }_2\left(x\right)+c_3$$

(19)

for all $x\in {\mathbb{R}}^d$, where $c_2$ and $c_3$ are positive constants.

Proof. 

Suppose that condition (A5) is satisfied; for constants q and K in (A5), we construct the space $Q_{q,K}$ as defined in (16). In the following, we will prove that a norm-like function ${\phi }_1$ can be chosen from $Q_{q,K}$ such that

$$\mathcal{A}{\phi }_1\left(x\right)\le -c_2{\phi }_2\left(x\right)+c_3,$$

which implies that inequality (19) holds. For $f\in Q_{q,K}$ and $|x|>K$, we have

$$\begin{array}{ccc}\hfill \nabla f\left(x\right)& =& {|x|}^{q-2}(qln|x|+1)x,\hfill \\ \hfill {\nabla }^{2}f\left(x\right)& =& {|x|}^{q-4}[q(q-2)ln|x|+2(q-1)]U\hfill \\ & & +{|x|}^{q-2}(qln|x|+1)I_d,\hfill \end{array}$$

where U is the $d\times d$-matrix with the $(i,j)$-th element $U_{i,j}=x_i{x}_j$ and $I_d$ is the $d\times d$ identity matrix.

First, let us recall that

$$\mathcal{J}={\mathcal{J}}^{*}+{\mathcal{J}}_{*},$$

where ${\mathcal{J}}^{*}$ and ${\mathcal{J}}_{*}$ are defined as in (17) and (18), respectively. Now, we aim to establish $|\mathcal{J}f(x)|\to 0$ as $|x|\to \infty$. For ${\mathcal{J}}^{*}$, we have

$$\begin{array}{ccc}\hfill |{\mathcal{J}}^{*}f\left(x\right)|& \le & \left|{\int }_{\left|z\right|\ge 1}{1}_{\left\{\right|x+L\left(x\right)z|\le K\}}\right(f(x+L\left(x\right)z)-f\left(x\right)\left)\nu \left(dz\right)\right|\hfill \\ & & +\left|{\int }_{\left|z\right|\ge 1}{1}_{\left\{\right|x+L\left(x\right)z|>K\}}\right(f(x+L\left(x\right)z)-f\left(x\right)\left)\nu \left(dz\right)\right|\hfill \\ & =:& \left|{\int }_{\left|z\right|\ge 1}{J}_1(x,z)\nu \left(dz\right)\right|+\left|{\int }_{\left|z\right|\ge 1}{J}_2(x,z)\nu \left(dz\right)\right|.\hfill \end{array}$$

According to (A5), we have ${sup}_{|x|>K}{\displaystyle \frac{\parallel L\left(x\right)\parallel }{{|x|}^{q^{\prime }}ln|x|}}<\infty$ and $0<q^{\prime }<1$. Then, it is clear that, for each $|z|\ge 1$, we have

$$1_{\{|x+L\left(x\right)z|\le K\}}\to 0$$

as $|x|\to \infty$. It follows that

$$|J_1(x,z)|\to 0$$

as $|x|\to \infty$. In addition, by using Taylor’s formula, we obtain

$$\begin{array}{ccc}\hfill J_2(x,z)& =& 1_{\{|x+L\left(x\right)z|>K\}}(f(x+L\left(x\right)z)-f\left(x\right))\hfill \\ & =& 1_{\{|x+L\left(x\right)z|>K\}}{\int }_0^1\langle \nabla f(x+uL\left(x\right)z),L\left(x\right)z\rangle du.\hfill \end{array}$$

Then, for each fixed $|z|\ge 1$ and $|x|>K$, we have

$$\begin{array}{ccc}\hfill |J_2(x,z)|& =& 1_{\left\{\right|x+L\left(x\right)z|>K\}}|{\int }_0^1\langle \nabla f(x+uL\left(x\right)z),L\left(x\right)z\rangle du|\hfill \\ & \le & {\int }_0^1{|x+uL\left(x\right)z|}^{q-2}\left|\right(qln|x+uL\left(x\right)z|+1\left)\right||\langle x+uL\left(x\right)z,L\left(x\right)z\rangle |du\hfill \\ & \le & \left|z\right|{\int }_0^1{\parallel L\left(x\right)\parallel |x+uL\left(x\right)z|}^{q-1}\left|\right(qln|x+uL\left(x\right)z|+1\left)\right|du.\hfill \end{array}$$

Again, via (A5), we obtain

$$|J_2{(x,z)|\le C|x|}^{q^{\prime }+q-1}{(ln|x|)}^2$$

for large enough $|x|$. Note that $q^{\prime }+q<1$; it follows that

$$|J_2(x,z)|\to 0$$

as $|x|\to \infty$. Combining the results above and according to the dominated convergence theorem, we immediately have

$${\mathcal{J}}^{*}f\left(x\right)=o\left(1\right)$$

for $|x|\to \infty$. Now, for ${\mathcal{J}}_{*}f\left(x\right)$, according to Taylor’s formula, we have

$$\begin{array}{ccc}\hfill {\mathcal{J}}_{*}f\left(x\right)& =& {\int }_{|z|<1}(f(x+L\left(x\right)z)-f\left(x\right)-\langle \nabla f\left(x\right),L\left(x\right)z\rangle )\nu \left(dz\right)\hfill \\ & =& {\int }_{|z|<1}({\int }_0^1{\int }_0^{1}u\langle {\nabla }^{2}f(x+uvL\left(x\right)z)L\left(x\right)z,L\left(x\right)z\rangle dvdu)\nu \left(dz\right).\hfill \end{array}$$

Similarly, for large enough $|x|$ (at least $|x|>K$), we have

$$\begin{array}{ccc}\hfill |{\mathcal{J}}_{*}f\left(x\right)|& \le & {\int }_{|z|<1}{|z|}^2\nu \left(dz\right)\underset{|z|<1,0\le u,v\le 1}{sup}{\parallel L\left(x\right)\parallel }^2\parallel {\nabla }^{2}f(x+uvL\left(x\right)z)\parallel \hfill \\ & \le & C\underset{|z|<1,0\le u,v\le 1}{sup}{\parallel L\left(x\right)\parallel }^2\parallel {\nabla }^{2}f(x+uvL\left(x\right)z)\parallel \hfill \\ & \le & C\underset{|z|<1,0\le u,v\le 1}{sup}{|x|}^{2q^{\prime }}{(ln|x|)}^2{|x+uvL\left(x\right)z|}^{q-4}[q(q-2)ln|x\hfill \\ & & +uvL\left(x\right)z|+2(q-1\left)\right]\parallel U\parallel \hfill \\ & & +C\underset{|z|<1,0\le u,v\le 1}{sup}{|x|}^{2q^{\prime }}{(ln|x|)}^2{|x+uvL\left(x\right)z|}^{q-2}\hfill \\ & & (qln|x+uvL\left(x\right)z|+1)\hfill \\ & \le & {C|x|}^{2q^{\prime }+q-2}{(ln|x|)}^3.\hfill \end{array}$$

By (A5), it is easy to check that $2q^{\prime }+q-2<0$. It follows that

$$|{\mathcal{J}}_{*}f\left(x\right)|\to 0$$

as $|x|\to \infty$. Hence, we have proved

$$|\mathcal{J}f(x)|\to 0$$

as $|x|\to \infty$.

After that, according to (A5), for $|x|>K$, we have

$$\begin{array}{ccc}\hfill \langle b\left(x\right),\nabla f\left(x\right)\rangle & =& {|x|}^{q-2}(qln|x|+1)\langle b\left(x\right),x\rangle \hfill \\ & \le & -{C|x|}^{q+q^{″}-2}ln|x|.\hfill \end{array}$$

Recall that

$$\begin{array}{ccc}\hfill \mathcal{A}f\left(x\right)& =& \langle b\left(x\right),\nabla f\left(x\right)\rangle +\mathcal{J}f\left(x\right).\hfill \end{array}$$

Then, via (A5) again, we have $0<q+q^{″}-2<1\wedge \alpha$. Hence, there exist positive constants $c_2$ and $c_3$, such that

$$\begin{array}{ccc}\hfill \mathcal{A}f\left(x\right)& \le & -c_2{|x|}^{q+q^{″}-2}ln|x|+c_3\hfill \end{array}$$

for all $x\in {\mathbb{R}}^d$, which implies that the first inequality in the lemma holds with ${\phi }_1=f$ and ${\phi }_2={|x|}^{q+q^{″}-2}ln|x|$. The proof is completed. □

According to the proof of the above lemma, we immediately have the following corollary.

Corollary 2. 

Suppose that assumption (A5) holds with $q^{″}=2$. Then, for each $m\in \mathbb{N}$, there exists a norm-like function φ, such that

$${\mathcal{A}}_m\phi \left(x\right)\le -c_2\phi \left(x\right)+c_3$$

for all $x\in {\mathbb{R}}^d$, where $c_2$ and $c_3$ are positive constants.

3.5. Irreducibility for Some $\Delta$-Skeleton Chain

For some $\Delta >0$ and $n=0,1,\cdots$, let us define $X_n^{\Delta }:=X_{n\Delta }$. We refer to $X^{\Delta }:={\left(X_n^{\Delta }\right)}_{n\ge 0}$ as the $\Delta$-skeleton chain of X. The main result of this subsection is the following proposition.

Proposition 1. 

Suppose that (A1), (A4), and (A6) are satisfied and let $X={\left(X_t\right)}_{t\ge 0}$ be the solution of Equation (4). Then, there exists some $\Delta >0$ such that $X^{\Delta }$ is Lebesgue-irreducible. That is, for any $x\in {\mathbb{R}}^d$ and nonempty open set $A\in \mathcal{B}\left({\mathbb{R}}^d\right)$, we have $P_{\Delta }(x,A)>0$, where $P_{\Delta }(x,A)=\mathbb{P}(X_{\Delta }\in A|X_0=x)$.

Before approaching Proposition 1, we need the following result, which is a special case of Theorem 3 in [20]. For the sake of completeness, we provide its proof below. Recall that, for $0<u<1$, $Y^u$ is the solution of Equation (15) and Y is the solution of the modified Equation (6) for the case $u=0$. Moreover, for $k\in \mathbb{N}$, let $u_k={\displaystyle \frac{1}{k+1}}$.

Lemma 7. 

Under assumptions (A1) and (A4), for any $t>0$ and $\epsilon \in (0,1)$, we have

$$\underset{k\to \infty }{lim}\mathbb{P}(\underset{0\le s\le t}{sup}|Y_s^{u_k}-Y_s|>\epsilon )=0.$$

Proof. 

For any $m\in \mathbb{N}$, $\epsilon \in (0,1)$ and $i\in \mathbb{N}$, let

$$\begin{array}{ccc}\hfill {\tau }_m& :=& inf\{s\ge 0:|Y_s|\ge m-1\},\hfill \\ \hfill {\tau }_{k\epsilon }& :=& inf\{s\ge 0:|Y_s^{u_k}-Y_s|\ge \epsilon \},\hfill \\ \hfill {\sigma }_{ki}& :=& inf\{s\ge 0:|Y_s^{u_k}|\ge i\},\hfill \end{array}$$

and let ${\tau }^k:={\tau }_{k\epsilon }\wedge {\tau }_m$. First, for any $t>0$, we aim to prove

$$\mathbb{E}(|Y_{t\wedge {\tau }^k}^{u_k}-Y_{t\wedge {\tau }^k}{|}^2)\le {\displaystyle \frac{C_m}{K_m}}{e}^{K_{m}t}(1-e^{-K_{m}t}){\int }_{0<|z|<u_k}{|z|}^2\nu \left(dz\right),$$

(20)

where $C_m:={max}_{|x|\le m}{\parallel L\left(x\right)\parallel }^2$. Observe that

$$\begin{array}{ccc}\hfill Y_t^{u_k}-Y_t& =& {\int }_0^t(b\left(Y_s^{u_k}\right)-b\left(Y_s\right))ds\hfill \\ & & +{\int }_0^t{\int }_{0<|z|<1}(L\left(Y_{s-}^{u_k}\right)1_{(u_k,1)}\left(z\right)-L\left(Y_{s-}\right))z\tilde{N}(dz,ds).\hfill \end{array}$$

Then, by applying Itô’s formula to $|Y_t^{u_k}-Y_t{|}^2{e}^{-K_{m}t}$, we obtain

$$\begin{array}{ccc}\hfill |Y_t^{u_k}-Y_t{|}^2{e}^{-K_{m}t}& =& 2{\int }_0^t{e}^{-K_{m}s}\langle b\left(Y_s^{u_k}\right)-b\left(Y_s\right),Y_s^{u_k}-Y_s\rangle ds\hfill \\ & & +{\int }_0^t{\int }_{0<|z|<1}{e}^{-K_{m}s}{|(L\left(Y_{s-}^{u_k}\right)1_{(u_k,1)}\left(z\right)-L\left(Y_{s-}\right))z|}^2\nu \left(dz\right)ds\hfill \\ & & -{\int }_0^t{K}_m{e}^{-K_{m}s}{|Y_s^{u_k}-Y_s|}^{2}ds+M_t^k\hfill \\ & =& 2{\int }_0^t{e}^{-K_{m}s}(\langle b\left(Y_s^{u_k}\right)-b\left(Y_s\right),Y_s^{u_k}-Y_s\rangle \hfill \\ & & +{\displaystyle \frac{1}{2}}{\int }_{u_k<|z|<1}|(L\left(Y_{s-}^{u_k}\right)-L\left(Y_{s-}\right){)z|}^2\nu \left(dz\right)-K_m{|Y_s^{u_k}-Y_s|}^2)ds\hfill \\ & & +{\int }_0^t{\int }_{0<|z|<u_k}{e}^{-K_{m}s}{|L\left(Y_{s-}\right)z|}^2\nu \left(dz\right)ds+M_t^k,\hfill \end{array}$$

where ${\left(M_t^k\right)}_{t\ge 0}$ is a local martingale. Next, for fixed k, according to the proof of Theorem 1, we have ${\sigma }_{ki}\uparrow \infty$ a.s. as $i\to \infty$. Let ${\tau }_i^k:={\tau }^k\wedge {\sigma }_{ki}$. By (A1), we have

$$\begin{array}{ccc}& & \mathbb{E}(|Y_{t\wedge {\tau }_i^k}^{u_k}-Y_{t\wedge {\tau }_i^k}{|}^2{e}^{-K_m(t\wedge {\tau }_i^k)})\hfill \\ & \le & \mathbb{E}{\int }_0^{t\wedge {\tau }_i^k}{\int }_{0<|z|<u_k}{e}^{-K_{m}s}\parallel L\left(Y_{s-}\right){\parallel }^2{|z|}^2\nu \left(dz\right)ds.\hfill \end{array}$$

Now, by taking $i\to \infty$, we obtain

$$\begin{array}{ccc}& & \mathbb{E}(|Y_{t\wedge {\tau }^k}^{u_k}-Y_{t\wedge {\tau }^k}{|}^2{e}^{-K_m(t\wedge {\tau }^k)})\hfill \\ & \le & \mathbb{E}{\int }_0^{t\wedge {\tau }^k}{\int }_{0<|z|<u_k}{e}^{-K_{m}s}\parallel L\left(Y_{s-}\right){\parallel }^2{|z|}^2\nu \left(dz\right)ds\hfill \\ & \le & C_m{\int }_0^t{e}^{-K_{m}s}ds{\int }_{0<|z|<u_k}{|z|}^2\nu \left(dz\right)\hfill \\ & =& {\displaystyle \frac{C_m}{K_m}}(1-e^{-K_{m}t}){\int }_{0<|z|<u_k}{|z|}^2\nu \left(dz\right),\hfill \end{array}$$

where $C_m:={max}_{|x|\le m}{\parallel L\left(x\right)\parallel }^2$, which implies estimate (20). By (20), we immediately have

$$\underset{k\to \infty }{lim}\mathbb{E}\left({|Y_{t\wedge {\tau }^k}^{u_k}-Y_{t\wedge {\tau }^k}|}^2\right)\to 0.$$

Moreover,

$$\begin{array}{ccc}\hfill \mathbb{E}\left(|,Y_{t\wedge {\tau }^k}^{u_k},-,Y_{t\wedge {\tau }^k},{|}^2\right)& \ge & \mathbb{E}\left(|,Y_{t\wedge {\tau }^k}^{u_k},-,Y_{t\wedge {\tau }^k},{|}^2,1_{\{{\tau }_{k\epsilon }<t\wedge {\tau }_m\}}\right)\hfill \\ & \ge & {\epsilon }^2\mathbb{P}({\tau }_{k\epsilon }<t\wedge {\tau }_m),\hfill \end{array}$$

which implies

$$\underset{k\to \infty }{lim}\mathbb{P}({\tau }_{k\epsilon }<t\wedge {\tau }_m)=0.$$

Due to

$$\mathbb{P}(\underset{s\le t\wedge {\tau }_m}{sup}|Y_s^{u_k}-Y_s|\ge \epsilon )\le \mathbb{P}({\tau }_{k\epsilon }<t\wedge {\tau }_m),$$

therefore,

$$\underset{k\to \infty }{lim}\mathbb{P}(\underset{s\le t\wedge {\tau }_m}{sup}|Y_s^{u_k}-Y_s|\ge \epsilon )=0.$$

Finally, note that ${\tau }_m\uparrow \infty$ a.s. and let $m\to \infty$; we finish the proof. □

Now, we are in a position to provide the proof of Proposition 1.

Proof of Proposition 1. 

By using the Markov property, to obtain our result, it suffices to verify that, for any $y\in {\mathbb{R}}^d$ and $\epsilon >0$, there exists $\Delta >0$ such that

$$\mathbb{P}(X_{\Delta }\in B(y;\epsilon ))>0,$$

(21)

where $B(y;\epsilon ):=\{\tilde{y}\in {\mathbb{R}}^d:|\tilde{y}-y|<\epsilon \}$. In the following, let $\Delta$ be the constant in assumption (A6). Let $T_1:=inf\{s\ge 0:|Z_s-Z_{s-}|>1\}$. Since

$$\mathbb{P}(T_1>\Delta )\ge \mathbb{P}(N([0,\Delta ]\times [1,\infty ))=0)=exp\{-\Delta \nu \left([1,\infty )\right)\}>0,$$

we only need to prove that there exists $\Delta >0$ such that

$$\mathbb{P}(Y_{\Delta }^0\in B(y;\epsilon ))>0.$$

By assumption (A6)-(ii), there exists a small $u_0>0$ such that, for any $u\in (0,u_0]$, we have

$$\mathbb{P}(Y_{\Delta }^u\in B(y;\epsilon /2))\ge {\int }_{B(y;\epsilon /2)}{p}^{u_0}(x,\tilde{y})d\tilde{y}>0.$$

Next, by Lemma 7, we can choose large enough $k_0\in \mathbb{N}$ such that $u_{k_0}\le u_0$ and

$$\mathbb{P}(|Y_{\Delta }^{u_{k_0}}-Y_{\Delta }^0|>\epsilon /2)<{\int }_{B(y;\epsilon /2)}{p}^{u_0}(x,\tilde{y})d\tilde{y}.$$

It follows that

$$\begin{array}{ccc}& & \mathbb{P}(Y_{\Delta }^0\in B(y;\epsilon ))\hfill \\ & \ge & \mathbb{P}(Y_{\Delta }^{u_{k_0}}\in B(y;\epsilon /2),|Y_{\Delta }^{u_{k_0}}-Y_{\Delta }^0|\le \epsilon /2)\hfill \\ & \ge & \mathbb{P}(Y_{\Delta }^{u_{k_0}}\in B(y;\epsilon /2))-\mathbb{P}(|Y_{\Delta }^{u_{k_0}}-Y_{\Delta }^0|>\epsilon /2)\hfill \\ & \ge & {\int }_{B(y;\epsilon /2)}{p}^{u_0}(x,\tilde{y})d\tilde{y}-\mathbb{P}(|Y_{\Delta }^{u_{k_0}}-Y_{\Delta }^0|>\epsilon /2)\hfill \\ & >& 0,\hfill \end{array}$$

from which we obtain our result. □

3.6. Proof of the Main Result

In this subsection, we will provide the proof of Theorem 3.

Proof of Theorem 3. 

We will prove the ergodic properties by using the Foster–Lyapunov criteria proposed in [19]. According to Lemma 3, Proposition 1, and condition (A6)-(i), by applying similar arguments as in the proof of Proposition 3.1 of [18], we can obtain that there exists a constant $\Delta >0$ such that every compact set of ${\mathbb{R}}^d$ is petite for the $\Delta$-skeleton chain of X. (For concepts such as petite set and $\Delta$-skeleton chain, we refer the reader to [18] and references therein.) Furthermore, upon Lemma 6 and Corollary 2, the conditions of Theorems 5.1 and 6.1 of [19] are fulfilled. It follows that X is ergodic under assumption (A5) and is exponenitally ergodic under assumption (A5) with $q^{″}=2$.

In the following, under (A5), we will prove that X is $\beta$-mixing in the case $\mu =\pi$, that is, in the case that X is stationary. By using the Markov property of X, it is clear that

$$\begin{array}{ccc}\hfill {\beta }_X\left(t\right)& =& \underset{s\ge 0}{sup}\mathbb{E}[\underset{B\in \sigma (X_u^{\mu }:u\ge s+t)}{sup}|\mathbb{P}(B|\sigma \left(X_s^{\mu }\right))-\mathbb{P}\left(B\right)|]\hfill \\ & =& \underset{s\ge 0}{sup}\int {\parallel P_t(x,·)-\mu P_{t+s}(·)\parallel }_{TV}\mu P_s\left(dx\right)\hfill \\ & =& \int \parallel P_t{(x,·)-\pi (·)\parallel }_{TV}\pi \left(dx\right),\hfill \end{array}$$

where $\mu P_s\left(dx\right)$ denotes the distribution of $X_s^{\mu }$. Then, the $\beta$-mixing property is clear by using the ergodic result and the dominated convergence theorem.

Next, we aim to show the exponentially $\beta$-mixing property under (A5) with $q^{″}=2$. Note that

$$\begin{array}{ccc}\hfill {\beta }_X\left(t\right)& =& \underset{s\ge 0}{sup}\int {\parallel P_t(x,·)-\mu P_{t+s}(·)\parallel }_{TV}\mu P_s\left(dx\right)\hfill \\ & \le & \underset{s\ge 0}{sup}\int {\parallel P_t(x,·)-\pi (·)\parallel }_{TV}\mu P_s\left(dx\right)\hfill \\ & & +\underset{s\ge 0}{sup}\int {\parallel \mu P_{t+s}(·)-\pi (·)\parallel }_{TV}\mu P_s\left(dx\right)\hfill \\ & \le & \underset{s\ge 0}{sup}\int {\parallel P_t(x,·)-\pi (·)\parallel }_{TV}\mu P_s\left(dx\right)\hfill \\ & & +\underset{s\ge 0}{sup}\int {\parallel P_{t+s}(x,·)-\pi (·)\parallel }_{TV}\mu P_s\left(dx\right).\hfill \end{array}$$

Upon the exponential ergodicity property, we immediately obtain

$$\begin{array}{ccc}\hfill {\beta }_X\left(t\right)& \le & 2\underset{s\ge 0}{sup}\int h\left(x\right)\mu P_s\left(dx\right)e^{-\gamma t}\hfill \\ & =& 2\underset{s\ge 0}{sup}\mathbb{E}\left(h\left(X_s^{\mu }\right)\right)e^{-\gamma t}.\hfill \end{array}$$

Indeed, according to Theorem 6.1 of [19], function h can be chosen as $f+1$, where $f\in Q_{q,K}$. Finally, by Lemma 5 and Lemma 1, we have that ${sup}_{s\ge 0}\mathbb{E}\left(h\left(X_s^{\mu }\right)\right)<\infty .$ We finish the proof. □

4. Conclusions

In this paper, we consider a class of multidimensional stochastic differential equations driven by multiplicative $\alpha$-stable Lévy noise, where $0<\alpha <2$. Under the local one-sided Lipschitz condition and a general non-explosion condition, we establish the existence and uniqueness of the solution for our equation. After that, the weak Feller property and the stationary property are proved. Finally, we provide some ergodic and mixing properties for the solution by using the Foster–Lyapunov criteria. The results of this paper can serve as fundamental tools for further stochastic analysis and statistical inference.