1. Introduction and main results
Let
$D\subset \mathbb{R}^m$
. For smooth functions
$b\,:\, D\rightarrow \mathbb{R}^m$
and
$\gamma\,:\, D\times \mathbb{R}^n\rightarrow \mathbb{R}^n$
, consider the deterministic dynamical system
\begin{equation}\begin{cases} {\rm d} X_t=b(X_t)\,{\rm d} t,\\[3pt] {\rm d} Y_t=\gamma(X_t,Y_t)\,{\rm d} t.\end{cases}\end{equation}
The feature of this system is that although it consists of two components, the first component is determined completely by itself. The interpretations of the model (1.1) can be explained in at least the following two aspects. The first one arises from the point of view of ecological systems. Regard
$ X_t $
as the quantity of grass and
$Y_t$
as the number of vegetarians at time t. Then, (1.1) demonstrates that the quantity of grass is sufficient and does not change as the number of vegetarians varies; however, the quantity of grass influences the increase in vegetarians. The other aspect comes from the mathematical modelling of infectious diseases. For numerous infections, e.g. measles, many newborn individuals might be immune to the disease since they are protected by maternal antibodies (in particular, via the placenta and the colostrum). This is termed ‘passive immunity’ in the literature. To describe this phenomenon in a mathematical framework, an extra M class is taken into consideration at the beginning of the classical SIR (susceptible–infectious–recovered) dynamics (see, e.g., [Reference Capasso4, Chapter 2]), which leads to the MSIR model (M = maternally derived immunity; see, e.g., [Reference Bichara, Iggidr and Sallet3]):
\begin{equation*} \begin{cases} {\rm d} M_t= \{p\Lambda-(\mu+\eta)M_t\}\,{\rm d} t,\\[3pt] {\rm d} S_t= \{(1-p)\Lambda+\eta M_t-\beta S_tI_t+\gamma I_t-\mu S_t\}\,{\rm d} t,\\[3pt] {\rm d} I_t= \{ \beta S_t I_t-(\mu+\delta+\nu+\gamma) I_t\}\,{\rm d} t,\\[3pt] {\rm d} R_t=\{\nu I_t-\mu R_t\}\,{\rm d} t,\end{cases}\end{equation*}
where
$0\le p <1$
and
$\Lambda, \mu, \eta, \beta, \gamma, \delta, \nu > 0$
. It is apparent that this MSIR model is a typical example of (1.1).
Since the model (1.1) is often subject to environmental noise, it is natural to consider the corresponding stochastic version,
\begin{equation}\begin{cases} {\rm d} X_t=b(X_t)\,{\rm d} t+\sigma^{(1)} (X_t)\, {\rm d} L^{(1)}_t,\\[3pt] {\rm d} Y_t=\gamma(X_t,Y_t)\,{\rm d} t+\sigma^{(2)} (X_t)\, {\rm d} L^{(2)}_t,\end{cases}\end{equation}
where
$\big(L_t^{(1)}\big)_{t\ge0}$
$\big($
resp.
$\big(L_t^{(2)}\big)_{t\ge0}\big)$
is a
$k_1$
-dimensional (resp.
$k_2$
-dimensional) Lévy noise (including the standard Brownian motion),
$\sigma^{(1)}\,:\, D\rightarrow \mathbb{R}^m\otimes\mathbb{R}^{k_1}$
and
$\sigma^{(2)}\,:\, D\rightarrow \mathbb{R}^n\otimes \mathbb{R}^{k_2}$
. Intuitively speaking, (1.2) can be regarded as a class of stochastic differential equation (SDE) with simple interactions.
Before we explain the focus of the current article, let us introduce two motivating examples related to the SDE (1.2).
Example 1.1 (Two-factor type process.) Let
$\big(B_t^{(1)},B_t^{(2)}\big)_{t\ge0}$
be a standard two-dimensional Brownian motion. For
$i=1,2$
and
$\gamma\in \{\alpha, \beta\}\subset (1,2)$
, let
$\big(L^{(i,\gamma)}_t\big)_{t\ge0}$
be a spectrally positive
$\gamma$
-stable process with the Lévy measure
$\nu_\gamma({\rm d} z)=C_\gamma z^{-(1+\gamma)}\textbf{1}_{\{z>0\}}{\rm d} z$
, where
$C_\gamma\,:\!=\,(\gamma \Gamma({-}\gamma))^{-1}$
and
$\Gamma$
is the Gamma function. We suppose that
$\big(B_t^{(1)},B_t^{(2)}\big)_{t\ge0}$
,
$\big(L^{(1,\alpha)}_t\big)_{t\ge0}$
, and
$\big(L_t^{(2,\beta)}\big)_{t\ge0}$
are mutually independent. Set
$\mathbb{R}_+\,:\!=\,[0,\infty)$
. We consider the following process
$(X_t,Y_t)_{t\ge0}$
on
$\mathbb{R}_+\times\mathbb{R}$
solving
\begin{equation}\begin{cases} {\rm d} X_t=b(X_t)\,{\rm d} t+ \theta_1X_{t}^{{1}/{2}} \,{\rm d} B_t^{(1)}+\theta_2X_{t-}^{{1}/{\alpha}}\,{\rm d} L^{(1,\alpha)}_t, & X_0=x\in\mathbb{R}_+,\\[3pt] {\rm d} Y_t=\gamma(X_t, Y_t)\,{\rm d} t+ \delta_1X_{t}^{{1}/{2}} \,{\rm d} B_t^{(2)}+ \delta_2X_{t-}^{1/\beta}\,{\rm d} L^{(2,\beta)}_t, & \, Y_0=y\in \mathbb{R},\end{cases}\end{equation}
where
$\theta_1, \theta_2, \delta_1, \delta_2 \ge 0$
with
$\theta_1+\theta_2>0$
and
$\delta_1+\delta_2>0$
.
In particular, when
$b(x)=a-bx$
and
$\gamma(x,y)=c-dy$
, the corresponding SDE (1.3) has a unique strong solution, and the associated process becomes a two-factor affine model; see, e.g., [Reference Barczy, Döring, Li and Pap2].
Example 1.2 (Gruschin type process) Let
$(X_t, Y_t)_{t\ge0}$
be the process on
$\mathbb{R}^{m+n}\,:\!=\,\mathbb{R}^{m}\times \mathbb{R}^n$
solving the following SDE:
\begin{equation}\begin{cases} {\rm d} X_t=b(X_t)\,{\rm d} t+ {\rm d} B_t^{(1)}, & X_0=x\in\mathbb{R}^m,\\ {\rm d} Y_t=\gamma(X_t, Y_t)\,{\rm d} t+ \sigma(X_t) \,{\rm d} B_t^{(2)}, & \, Y_0=y\in \mathbb{R}^n,\end{cases}\end{equation}
where
$(B_t)_{t\ge0}\,:\!=\,\big(B_t^{(1)}, B_t^{(2)}\big)_{t\ge0}$
is an
$(m+n)$
-dimensional Brownian motion.
In particular, when
$m=n=1$
,
$b(x)=0$
,
$\gamma(x,y)=0$
, and
$\sigma(x)=|x|^l$
for
$l>0$
, the SDE (1.4) is reduced into
\begin{equation*} \begin{cases} {\rm d} X_t={\rm d} B_t^{(1)},\\ {\rm d} Y_t=|X_t|^l\,{\rm d} B_t^{(2)},\end{cases}\end{equation*}
where
$\big(B_t^{(1)},B_t^{(2)}\big)_{t\ge0}$
is a two-dimensional Brownian motion. The generator of the process
$(X_t,Y_t)_{t\ge0}$
solving this SDE is given by
$L=\frac{1}{2}\big(\partial_{11}+|x^{(1)}|^{2l}\partial_{22}\big)$
with
$x=\big(x^{(1)},x^{(2)}\big)$
, where
$\partial_{ii}, i=1,2$
, denotes the second-order gradient operator with respect to the ith component. The associated semigroup is called the classical Gruschin semigroup on
$\mathbb{R}^2$
with order
$l>0$
; see, e.g., [Reference Wang and Xu19].
The objective of this article is to seek verifiable conditions under which SDEs such as (1.3) and (1.4) are exponentially ergodic. Ergodicity is the foundation for a wide class of limit theorems and long-time behavior for Markov processes. In this setting, we will provide a (more direct) strategy via the Markov coupling technique to prove the exponential ergodicity for a class of Markov processes with interactions.
In the following two subsections, we state our main results for the exponential ergodicity for the SDEs (1.3) and (1.4), respectively. For a more general result concerning the exponential ergodicity for a class of Markov processes with interactions, see Section 2.1. For simplicity, we call processes solving (1.3) and (1.4) two-factor type processes and Gruschin type processes, respectively.
Given a distance-like function
$\rho\,:\,\mathbb{R}^d\times\mathbb{R}^d\rightarrow \mathbb{R}_+$
(i.e.
$\rho(x,y)$
is non-negative, symmetric and lower semicontinuous on
$\mathbb{R}^d\times \mathbb{R}^d$
, and
$\rho(x,y)=0$
if only if
$x=y$
), the Wasserstein-type quasi-metric on
$\mathscr P\big(\mathbb{R}^d\big)$
, the set of probability measures on
$\mathbb{R}^d$
, is defined by
\begin{equation*}\mathbb{W}_\rho(\mu,\nu)=\inf_{\pi\in\mathscr C(\mu,\nu )}\int_{\mathbb{R}^d\times\mathbb{R}^d}\rho(x,y)\,\pi({\rm d} x,{\rm d} y),\qquad \mu,\nu\in\mathscr P\big(\mathbb{R}^d\big),\end{equation*}
where
$\mathscr C(\mu,\nu)$
stands for the couplings of
$\mu$
and
$\nu$
. If the distance-like function
$\rho$
satisfies the triangle inequality in the weak sense, i.e. there exists a constant
$K > 0$
such that
\begin{equation} \rho(x,y)\le K\big(\rho(x,z)+\rho(z,y)\big),\qquad x,y,z\in\mathbb{R}^d,\end{equation}
then
$\big(\mathscr P\big(\mathbb{R}^d\big), \mathbb{W}_\rho\big)$
is a complete space; see the proof of [Reference Chen6, Theorem 5.4] for more details. Let
$P(t, (x,y),\cdot)$
be the transition probability kernel of the process
$(X_t,Y_t)$
with the initial value (x, y).
1.1. Two-factor type processes
We make the following assumptions.
Assumption 1.1. There are constants
$\lambda_1,\lambda_2>0$
and
$r_0\ge0$
such that, for all
$x, \widetilde x\in \mathbb{R}_+$
,
$(b(x)-b(\widetilde x))(x- \widetilde x)\le -\lambda_1(x-\widetilde x)^2\textbf{1}_{\{|x-\widetilde x|\ge r_0\}} + \lambda_2(x-\widetilde x)^2\textbf{1}_{\{|x-\widetilde x|< r_0\}}$
.
Assumption 1.2. There are constants
$\kappa_1, \kappa_2>0$
and
$\kappa_3\ge0$
such that, for all
$x,\widetilde x\in \mathbb{R}_+$
and
$y,\widetilde y\in \mathbb{R}$
,
\begin{align} (\gamma(x,y)-\gamma(\widetilde x,\widetilde y))(y-\widetilde y ) \le - \kappa_1 (y-\widetilde y)^2 +\kappa_2|(x-\widetilde x)(y-\widetilde y)| +\kappa_3(y-\widetilde y)^2\textbf{1}_{\{|y-\widetilde y|\le \kappa_3\}}.\end{align}
Theorem 1.1. Under Assumptions 1.1 and 1.2, the SDE given by (1.3) has a unique strong solution
$(X_t,Y_t)_{t\ge0}$
, and the process
$(X_t,Y_t)_{t\ge0}$
is exponentially ergodic in the sense that there exist a unique invariant probability measure
$\mu$
on
$\mathbb{R}_+\times\mathbb{R}$
and a constant
$\eta>0$
such that, for all
$x\in \mathbb{R}_+$
,
$y\in \mathbb{R}$
, and
$t>0$
,
$\mathbb{W}_{V}\big(P(t, (x,y),\cdot), \mu\big)\le C_0(x,y)\,{\rm e}^{-\eta t}$
, where
$C_0(x,y)>0$
is independent of t, and
\begin{equation*} V(x,\widetilde x, y,\widetilde y)\,:\!=\, \textbf{1}_{\{\kappa_3>0, x\neq \widetilde x\}}+ |x-\widetilde x|+ |y-\widetilde y|\wedge|y-\widetilde y|^2, \qquad x,\widetilde x\in \mathbb{R}_+,\ y,\widetilde y\in\mathbb{R},\end{equation*}
with
$\kappa_3\ge 0$
given in (1.6).
When
$b(x)=a- bx$
with
$a\ge0$
and
$b>0$
, and
$\gamma(x,y)=c-d y$
with
$c\in\mathbb{R}$
and
$d>0$
, the exponential ergodicity in terms of the total variation norm for
$ (X_t,Y_t)_{t\ge0}$
solving (1.3) (in this case the process
$(X_t,Y_t)_{t\ge0}$
is the so-called two-factor affine process) was investigated in [Reference Barczy, Döring, Li and Pap2, Reference Jin, Kremer and Rüdiger11] via the Meyn–Tweedie approach, and in terms of the
$L^1$
-Wasserstein-type distance was studied in [Reference Bao and Wang1] by employing the coupling argument (see [Reference Friesen, Jin and Rüdiger8] for the corresponding results for general affine processes). The novelty of Theorem 1.1 is twofold.
First, (1.3) is universal and incorporates all the frameworks in [Reference Bao and Wang1, Reference Barczy, Döring, Li and Pap2, Reference Jin, Kremer and Rüdiger11]. Under Assumption 1.1, the drift term b(x) satisfies the dissipative condition only for large distances. Meanwhile, under Assumption 1.2, the mapping
$\gamma(x,y)$
may also be dissipative for large distances (when
$\kappa_3>0$
) with respect to the second variable. Hence, all the frameworks in [Reference Bao and Wang1, Reference Barczy, Döring, Li and Pap2, Reference Jin, Kremer and Rüdiger11] satisfy Assumptions 1.1 and 1.2. Therefore, to some extent, Theorem 1.1 improves the corresponding results in [Reference Bao and Wang1, Reference Barczy, Döring, Li and Pap2, Reference Jin, Kremer and Rüdiger11]. In particular, Theorem 1.1 can yield the exponential ergodicity of the process
$(X_t,Y_t)_{t\ge0}$
both in terms of the total variation norm (with respect to the first marginal) and the
$L^1$
-Wasserstein-type distance (when the distance between the second components is large). Note that, according to Theorem 1.1, the process
$(X_t,Y_t)_{t\ge0}$
determined by
\begin{equation}\begin{cases} {\rm d} X_t=(aX_t-bX_t^3)\,{\rm d} t+ \theta_1X_{t}^{{1}/{2}} \,{\rm d} B_t^{(1)}+\theta_2X_{t-}^{{1}/{\alpha}}\,{\rm d} L^{(1,\alpha)}_t,\\[4pt] {\rm d} Y_t=(cX_t+\gamma_0(Y_t)-dY_t)\,{\rm d} t+ \delta_1X_{t}^{{1}/{2}} \,{\rm d} B_t^{(2)}+ \delta_2X_{t-}^{1/\beta}\,{\rm d} L^{(2,\beta)}_t,\end{cases}\end{equation}
with
$X_0=x\in\mathbb{R}_+$
,
$Y_0=y\in \mathbb{R}$
,
$\gamma_0(y)$
being a bounded and Lipschitz continuous function on
$\mathbb{R}$
,
$a,b,c,d>0$
, and
$\theta_1,\theta_2, \delta_1, \delta_2\ge 0$
satisfying
$\theta_1+\theta_2>0$
and
$\delta_1+\delta_2>0$
, is exponentially ergodic.
Second, the proofs of [Reference Bao and Wang1, Reference Friesen, Jin and Rüdiger8] are based on the same coupling idea as the present paper. However, the details for the construction of the coupling are quite different. In detail, reflection or synchronous coupling for the Brownian motion part and synchronous coupling for the pure jump counterpart are employed in [Reference Bao and Wang1, Reference Friesen, Jin and Rüdiger8], but here we apply reflection coupling for the Brownian motion part and exploit the refined basic coupling for the pure jump part. The advantage of our approach lies in that it does not depend on the order-preserving property enjoyed by two-factor affine processes. Note that the arguments in [Reference Bao and Wang1, Reference Friesen, Jin and Rüdiger8] cannot handle exponential ergodicity for the SDE (1.3) (even for (1.7)) since the order-preserving property plays a crucial role in [Reference Bao and Wang1, Reference Friesen, Jin and Rüdiger8]. Besides, the exponential ergodicity of
$ (X_t,Y_t)_{t\ge0}$
is addressed in [Reference Bao and Wang1, Reference Friesen, Jin and Rüdiger8] under the Wasserstein-type distance with the cost function
$((x,y),(\widetilde x,\widetilde y))\mapsto |x-\widetilde x|+ |x-\widetilde x|^\theta+|y-\widetilde y|$
for
$\theta\in(0,1)$
(in particular
$\theta=\frac{1}{2}$
is taken in [Reference Friesen, Jin and Rüdiger8]), while in Theorem 1.1 we take the cost function
$((x,y),(\widetilde x,\widetilde y))\mapsto \textbf{1}_{\{\kappa_3>0, x\neq \widetilde x\}} +|x-\widetilde x| +|y-\widetilde y|^2\wedge |y-\widetilde y|$
. Since the cost functions involved are different, the corresponding results in [Reference Bao and Wang1, Reference Friesen, Jin and Rüdiger8] and Theorem 1.1 cannot be compared.
1.2. Gruschin type processes
Suppose that the coefficients in (1.4) satisfy the following assumptions.
Assumption 1.3. There are constants
$\lambda_1>0$
and
$\lambda_2, \lambda_3\ge 0$
such that, for all
$x,\widetilde x\in \mathbb{R}^m$
,
$\langle b(x)-b(\widetilde x), x- \widetilde x\rangle\le \lambda_2|x-\widetilde x|^2$
and
$\langle b(x),x\rangle\le - \lambda_1 |x|^2+\lambda_3$
.
Assumption 1.4. There are constants
$\kappa_1, \kappa_2>0$
,
$l \ge1$
, and
$\kappa_3\ge0$
such that, for all
$x,\widetilde x\in \mathbb{R}^m$
and
$y,\widetilde y\in \mathbb{R}^n$
,
\begin{multline*} \langle \gamma(x,y)-\gamma(\widetilde x,\widetilde y), y-\widetilde y \rangle \\ \le - \kappa_1 |y-\widetilde y|^2+\kappa_2|y-\widetilde y| (1\wedge|x-\widetilde x|)\big(1+ |x|^{l}+|\widetilde x|^{l}\big) + \kappa_3|y-\widetilde y|^2\textbf{1}_{\{|y-\widetilde y|\le \kappa_3\}} \end{multline*}
and
$\|\sigma(x)-\sigma(\widetilde x)\|^2 \le \kappa_2 (1\wedge|x-\widetilde x|)\big(1+ |x|^{l}+|\widetilde x|^{l}\big) +\kappa_3$
.
Theorem 1.2. Under Assumptions 1.3 and 1.4, the SDE given by (1.4) has a unique strong solution
$(X_t,Y_t)_{t\ge0}$
, and the process
$(X_t,Y_t)_{t\ge0}$
is exponentially ergodic in the sense that there exist a unique invariant probability measure
$\mu$
on
$\mathbb{R}^m\times\mathbb{R}^n$
and a constant
$\eta>0$
such that, for all
$x\in \mathbb{R}^m$
,
$y\in \mathbb{R}^n$
, and
$t>0$
,
$\mathbb{W}_{V}(P(t, (x,y),\cdot), \mu)\le C_0(x,y)\,{\rm e}^{-\eta t}$
, where
$C_0(x,y)>0$
is independent of t, and
\begin{equation*} V(x,\widetilde x, y,\widetilde y)\,:\!=\, \textbf{1}_{\{\kappa_3>0, x\neq \widetilde x\}}+ (1\wedge|x-\widetilde x|)\big(1+ |x|^{l}+|\widetilde x|^{l}\big) +|y-\widetilde y|^2\wedge |y-\widetilde y| ,\end{equation*}
with
$\kappa_3$
as in Assumption 1.4.
Note that the Gruschin semigroup is a typical example of hypoelliptic semigroups. There are a few regularity results concerning the Gruschin semigroups; see, for instance, [Reference Wang18, Reference Wang and Xu19]. However, the long-time behavior of diffusions associated with Gruschin type semigroups is unavailable.
Assumption 1.3 guarantees the existence of the Lyapunov function for the first component
$(X_t)_{t\ge0}$
and ensures that
$(X_t)_{t\ge0}$
is exponentially ergodic, while Assumption 1.4 shows that the function
$\gamma(x,y)$
may be dissipative for large distances with respect to the second variable and of polynomial growth with respect to the first variable, and that the diffusion term
$\sigma(x)$
is allowed to be of polynomial growth. We also note that under Assumption 1.3 the process
$(X_t)_{t\ge0}$
need not preserve the order property as the first component in the two-factor affine processes.
2. Proofs of main results
We present the proofs of Theorems 1.1 and 1.2. For this, we first provide the general framework to handle the exponential ergodicity for a class of Markov processes with interactions, and then provide sufficient conditions for this general result. Some comments on extensions of Theorems 1.1 and 1.2 are given at the end of the section.
2.1. A general result
The setting below is highly motivated by the SDE (1.2), but from the point of view of the generator. Let
$(X_t, Y_t)_{t\ge 0}$
be a strong Markov process on
$D\times \mathbb{R}^n\subset \mathbb{R}^{m+n}\,:\!=\,\mathbb{R}^m\times \mathbb{R}^n$
such that
$(X_t)_{t\ge0}$
itself is a strong Markov process on D, and, given
$(X_t)_{t\ge0}$
,
$(Y_t)_{t\ge0}$
is also a strong Markov process on
$\mathbb{R}^n$
. Let L be the generator of the process
$(X_t,Y_t)_{t\ge0}$
. Then we have
$Lf(x,y)=L_Xf(\cdot,y)(x)+L_{x, Y}f(x,\cdot)(y)$
,
$f\in\mathscr D(L)$
, where
$L_X$
is the infinitesimal generator of the process
$(X_t)_{t\ge0}$
and, for any fixed
$x\in D$
,
$L_{x,Y}$
is the generator of the process
$(Y_t)_{t\ge0}$
for given
$(X_t)_{t\ge0}$
.
A measurable function
$f\,:\, D\times \mathbb{R}^n\rightarrow \mathbb{R} $
belongs to the domain of the extended generator L of the Markov process
$(X_t, Y_t)_{t\ge0}$
if there exists a measurable function
$F\,:\, D\times \mathbb{R}^n \rightarrow\mathbb{R}$
such that the process
$f( X_t, Y_t )-\int^t_0 F( X_s, Y_s) \,{\rm d} s$
,
$t\ge0$
, is well defined, and is an
$\mathscr F^{ X,Y }$
-martingale with respect to
$\mathbb{P}_{(x,y)}$
for any
$(x,y)\in D\times \mathbb{R}^n$
. For any such pair (f, F), we write
$f\in \mathscr D(L)$
and
$Lf=F$
.
Suppose that
$\big(\big(X_t, Y_t\big), \big(\widetilde X_t, \widetilde Y_t\big)\big)_{t\ge0}$
is a strong Markov coupling process of the process
$(X_t, Y_t)_{t\ge0}$
such that
$\big(X_t, \widetilde X_t\big)_{t\ge0}$
itself is a strong Markov process on
$D\times D$
, and, for given
$\big(X_t, \widetilde X_t\big)_{t\ge0}$
,
$\big(Y_t,\widetilde Y_t\big)_{t\ge0}$
is also a strong Markov process on
$\mathbb{R}^n\times \mathbb{R}^n$
. In particular,
$\big(X_t, \widetilde X_t\big)_{t\ge0}$
is a Markov coupling of the process
$(X_t)_{t\ge0}$
. Let
$\mathscr L$
be the generator of the process
$\big(\big(X_t, Y_t\big), \big(\widetilde X_t, \widetilde Y_t\big)\big)_{t\ge0}$
. Then, for any
$f\in\mathscr D(\mathscr L)$
(that is, f belongs to the domain of the extended generator
$ \mathscr L$
, which is defined as
$\mathscr D( L)$
above),
\begin{equation*}\mathscr L\, f(x,\widetilde x, y, \widetilde y)=\mathscr L_{X} f(\cdot, \cdot, y, \widetilde y)(x,\widetilde x)+ \mathscr L_{x,\widetilde x, Y} f(x,\widetilde x, \cdot, \cdot)(y,\widetilde y), \end{equation*}
where
$\mathscr L_{X}$
is the generator of the coupling process
$\big(X_t, \widetilde X_t\big)_{t\ge0}$
and
$\mathscr L_{x,\widetilde x, Y}$
is the generator of the process
$\big(Y_t,\widetilde Y_t\big)_{t\ge0}$
for given
$\big(X_t, \widetilde X_t\big)_{t\ge0}$
. Since
$\big(\big(X_t, Y_t\big), \big(\widetilde X_t, \widetilde Y_t\big)\big)_{t\ge0}$
is a strong Markov coupling process of the process
$(X_t, Y_t)_{t\ge0}$
, and
$\big(X_t, \widetilde X_t\big)_{t\ge0}$
is a Markov coupling of the process
$(X_t)_{t\ge0}$
,
$\mathscr L$
is a coupling operator of L (that is, for any
$h((x,y),(\widetilde x,\widetilde y))=f(x,y)+g(\widetilde x,\widetilde y)$
with
$f,g\in\mathscr D(L)$
,
$\widetilde L h((x,y),(\widetilde x,\widetilde y))=Lf(x,y)+Lg(\widetilde x,\widetilde y)$
) and
$\mathscr L_{X}$
is a coupling operator of
$L_X$
. Hence,
$\mathscr L_{x,\widetilde x, Y}$
is also a coupling operator of the operator
$L_{x,Y}$
and
$L_{\widetilde x,Y}$
.
We impose the following assumptions on
$\mathscr L_{X}$
and
$\mathscr L_{x,\widetilde x, Y}$
, respectively.
Assumption 2.1. There are a constant
$\lambda_1>0$
and a sequence of distance-like functions
$(f_k)_{k\ge1}$
such that, for all
$k\ge1$
,
$f_k\in \mathscr D({\mathscr L}_{X} )$
and
\begin{equation} {\mathscr L}_{X}f_k(x, \widetilde x)\le -\lambda_1 f_k(x, \widetilde x), \qquad x,\widetilde x\in D\ { with }\ |x-\widetilde x|\ge 1/k.\end{equation}
Assumption 2.2. There are a distance-like function
$g\,:\,\mathbb{R}^n\times\mathbb{R}^{n}\rightarrow\mathbb{R}_+$
with
$g\in \mathscr D( {\mathscr L}_{x, \widetilde x,Y})$
for all
$x, \widetilde x\in D$
, and constants
$\lambda_2, \lambda_3>0$
such that, for any
$x,\widetilde x\in D$
with
$|x-\widetilde x|\ge 1/k$
and any
$y,\widetilde y\in\mathbb{R}^{n}$
,
\begin{equation} {\mathscr L}_{x, \widetilde x,Y} g(y,\widetilde y)\le -\lambda_2g(y,\widetilde y)+\lambda_3f_k(x, \widetilde x).\end{equation}
Theorem 2.1. Under Assumptions 2.1 and 2.2, there are constants
$\theta, \eta>0$
such that, for all
$x,\widetilde x\in D$
,
$y, \widetilde y\in \mathbb{R}^n$
, and
$t>0$
,
\begin{equation} \mathbb E^{((x,y), (\widetilde x,\widetilde y))} V\big(X_{t}, \widetilde X_{t},Y_{t}, \widetilde Y_{t}\big) \le \max\{ \theta, \theta^{-1}\}\text{ e}^{-\eta t}V(x, \widetilde x, y, \widetilde y),\end{equation}
where
$\mathbb E^{((x,y), (\widetilde x,\widetilde y))}$
is the expectation of the process
$\big(\big(X_t, Y_t\big), \big(\widetilde X_t, \widetilde Y_t\big)\big)_{t\ge0}$
starting from
$((x,y),(\widetilde x,\widetilde y))$
, and
$V(x,\widetilde x, y,\widetilde y)\,:\!=\,f(x,\widetilde x)+ g(y,\widetilde y)$
,
$x,\widetilde x\in D$
,
$y,\widetilde y\in\mathbb{R}^n$
with
$f(x,\widetilde x)=\liminf\limits_{k\rightarrow\infty} f_k(x,\widetilde x)$
. Moreover, if both
$(x,\widetilde x)\mapsto f(x,\widetilde x)$
and
$(y,\widetilde y)\mapsto g(y,\widetilde y)$
satisfy the triangle inequality in the weak sense (see (1.5) for more details), and, for any
$t>0$
,
$x, x_0\in D$
, and
$y\in \mathbb{R}^n$
,
\begin{equation} \mathbb E^x f(X_t, x_0)+\mathbb E^y g(Y_t,0)<\infty,\end{equation}
then the process
$(X_t,Y_t)_{t\ge0}$
is exponentially ergodic under the quasi-distance
$\mathbb{W}_V$
, i.e. there exist a unique invariant probability measure
$\mu$
on
$D\times\mathbb{R}^{n}$
such that, for all
$x\in D$
,
$y\in \mathbb{R}^{n}$
, and
$t>0$
,
$\mathbb{W}_{V}(P(t, (x,y),\cdot), \mu)\le C_0(x,y)\,{\rm e}^{-\eta t}$
, where
$C_0(x,y)>0$
is independent of t.
Proof. For any
$k\ge1$
and
$\theta>0$
, define
$V_{k,\theta}(x,\widetilde x, y,\widetilde y) = f_k(x,\widetilde x) + \theta g(y,\widetilde y)$
and
$V_{\theta}(x,\widetilde x, y,\widetilde y) = f(x,\widetilde x)+ \theta g(y,\widetilde y)$
for
$x,\widetilde x\in D$
,
$y,\widetilde y\in\mathbb{R}^n$
. We claim that there exist constants
$\theta, \eta>0$
such that, for all
$x,\widetilde x\in D$
,
$y, \widetilde y\in \mathbb{R}^n$
, and
$t>0$
,
\begin{equation} \mathbb E^{((x,y), (\widetilde x,\widetilde y))} V_{\theta}\big(X_{t}, \widetilde X_{t},Y_{t}, \widetilde Y_{t}\big) \le \text{e}^{-\eta t}V_{\theta}(x, \widetilde x, y, \widetilde y).\end{equation}
If (2.3) holds, then the first desired assertion (2.5) follows by the basic fact that
\begin{equation*}(1\wedge\theta)V(x,\widetilde x, y,\widetilde y)\le V_\theta(x,\widetilde x, y,\widetilde y) \le (1\vee\theta)V(x,\widetilde x, y,\widetilde y), \qquad x,\widetilde x\in D, y,\widetilde y\in\mathbb{R}^n.\end{equation*}
Now, since
$(x,\widetilde x)\mapsto f(x,\widetilde x)$
and
$(y,\widetilde y)\mapsto g(y,\widetilde y)$
satisfy the weak form of the triangle inequality in the sense of (1.5),
$((x,y),(\widetilde x, \widetilde y))\mapsto V(x,\widetilde x, y,\widetilde y)$
satisfies the weak form of the triangle inequality, and so
$(\mathscr{P}(D\times \mathbb{R}^n),\mathbb{W}_{V})$
is a complete space. Thus, the second assertion is a consequence of (2.5) by following the argument of [Reference Majka15, Corollary 1.8] and by taking the condition (2.4) into consideration.
We first suppose that
$x=\widetilde x$
. Since
$\big(X_t,\widetilde X_t\big)_{t\ge0}$
is a Markov coupling of the process
$(X_t)_{t\ge0}$
and f is a distance-like function, for any
$t>0$
,
$x\in D$
, and
$y,\widetilde y\in \mathbb{R}^n$
,
\begin{align} \mathbb E^{((x,y), (x,\widetilde y))} V_{\theta}\big(X_{t}, \widetilde X_{t},Y_{t}, \widetilde Y_{t}\big) & = \theta \mathbb E^{((x,y), (x,\widetilde y))} g(Y_{t}, \widetilde Y_{t}) \nonumber \\ & \le \theta \text{e}^{-\lambda_2 t}g(y,\widetilde y)= \text{ e}^{-\lambda_2 t}V_{\theta}(x, x, y, \widetilde y), \end{align}
where the inequality follows from (2.2) with
$x=\widetilde x$
, i.e.
${\mathscr L}_{x, x,Y} g(y,\widetilde y)\le -\lambda_2g(y,\widetilde y)$
.
In the following, we show that (2.5) is still valid for the case
$x\neq \widetilde x$
. Define the stopping time
$T_X=\inf\{t>0\,:\, X_t=\widetilde X_t\}$
. By the strong Markov property and (2.6), we have
\begin{align*} \mathbb E^{((x,y), (\widetilde x,\widetilde y))}&\big( V_{\theta}\big(X_{t}, \widetilde X_{t},Y_{t}, \widetilde Y_{t}\big)\textbf{1}_{\{T_X\le t\}}\big) \\ & = \mathbb E^{((x,y), (\widetilde x,\widetilde y))} \Big( \textbf{1}_{\{T_X\le t\}} \mathbb E^{\big(\big(X_{T_X},y\big), \big(X_{T_X},\widetilde y\big)\big)}V_{\theta}\big(X_{t-T_X}, \widetilde X_{t-T_X},Y_{t-T_X}, \widetilde Y_{t-T_X}\big)\Big) \\ & \le \mathbb E^{((x,y), (\widetilde x,\widetilde y))}\Big( \textbf{1}_{\{T_X\le t\}} \text{e}^{-\lambda_2(t-T_X)}V_{\theta}\big(X_{T_X}, \widetilde X_{T_X},Y_{T_X}, \widetilde Y_{T_X}\big)\Big).\end{align*}
This yields that, for any
$t>0$
,
$x,\widetilde x\in D$
with
$x\neq \widetilde x$
, and
$\lambda_0\in (0, \lambda_2]$
,
\begin{align*} \text{e}^{ \lambda_0 t}\mathbb E^{((x,y), (\widetilde x,\widetilde y))}& V_\theta \big(X_{t}, \widetilde X_{t},Y_{t},\widetilde Y_{t}\big) \\ & = \mathbb E^{((x,y), (\widetilde x,\widetilde y))}\Big(\text{e}^{ \lambda_0 (t\wedge T_X)}V_\theta\big(X_{t\wedge T_X}, \widetilde X_{t\wedge T_X},Y_{t\wedge T_X},\widetilde Y_{t\wedge T_X}\big) \textbf{1}_{\{T_X>t\}}\Big) \\ & \quad + \text{e}^{ \lambda_0 t}\mathbb E^{((x,y), (\widetilde x,\widetilde y))}\Big( V_{\theta}\big(X_{t}, \widetilde X_{t},Y_{t}, \widetilde Y_{t}\big)\textbf{1}_{\{T_X\le t\}}\Big) \\ & \le \mathbb E^{((x,y), (\widetilde x,\widetilde y))}\Big(\text{e}^{ \lambda_0 \big(t\wedge T_X\big)}V_\theta\big(X_{t\wedge T_X}, \widetilde X_{t\wedge T_X},Y_{t\wedge T_X},\widetilde Y_{t\wedge T_X}\big) \textbf{1}_{\{T_X>t\}}\Big) \\ & \quad + \mathbb E^{((x,y), (\widetilde x,\widetilde y))}\Big( \textbf{1}_{\{T_X\le t\}} \text{e}^{\lambda_0T_X}V_{\theta}\big(X_{T_X}, \widetilde X_{T_X},Y_{T_X}, \widetilde Y_{T_X}\big)\Big) \\ & = \mathbb E^{((x,y), (\widetilde x,\widetilde y))}\Big(\text{e}^{\lambda_0 (t\wedge T_X)}V_\theta\big(X_{t\wedge T_X}, \widetilde X_{t\wedge T_X},Y_{t\wedge T_X}, \widetilde Y_{t\wedge T_X}\big)\Big).\end{align*}
Thus, (2.5) follows for the case
$x\neq\widetilde x$
if we can verify that there is a constant
$\lambda_0\in (0, \lambda_2]$
such that, for all
$t>0$
and
$y,\widetilde y\in \mathbb{R}^n$
,
\begin{equation} \mathbb E\Big(\text{e}^{\lambda_0 (t\wedge T_X)}V_\theta\big(X_{t\wedge T_X}, \widetilde X_{t\wedge T_X},Y_{t\wedge T_X}, \widetilde Y_{t\wedge T_X} \big)\Big)\le V_\theta(x, \widetilde x, y, \widetilde y ).\end{equation}
Define the stopping time
$T_{X,k}\,:\!=\,\inf\big\{t>0\,:\, |X_t-\widetilde X_t|\le 1/k\big\}$
,
$k\ge1$
. Since
$x\neq\widetilde x$
, there exists a constant
$k_0\ge1$
sufficiently large such that
$|x-\widetilde x|> 1/k_0$
. On the other hand, choosing
$\theta\in(0,1\wedge (\lambda_1/\lambda_3))$
and using (2.1) and (2.2), we have, for all
$k\ge k_0$
,
$x,\widetilde x\in D$
with
$|x-\widetilde x|\ge 1/k$
, and
$y,\widetilde y\in \mathbb{R}^n$
,
\begin{equation*}{\mathscr L}V_{k,\theta}(x,\widetilde x, y, \widetilde y )\le-\big((\lambda_1-\theta\lambda_3)\wedge \theta\lambda_2\big) V_{k,\theta}(x,\widetilde x, y, \widetilde y ).\end{equation*}
Now let
$\lambda_0 =\eta\,:\!=\, (\lambda_1-\theta\lambda_3)\wedge \theta\lambda_2$
. Then, for any
$k\ge k_0$
,
$x,\widetilde x\in D$
with
$|x-\widetilde x|\ge 1/k_0$
,
$y,\widetilde y\in \mathbb{R}^n$
, and
$t>0$
,
\begin{align*} &\mathbb E^{((x,y), (\widetilde x,\widetilde y))}\big(\text{e}^{ \eta (t\wedge T_{X,k})} V_{k,\theta} \big(X_{t\wedge T_{Y,n}},\widetilde X_{t\wedge T_{X,k}},Y_{t\wedge T_{X,k}},\widetilde Y_{t\wedge T_{X,k}}\big)\big)\\ & = V_{k,\theta} ( x, \widetilde x, y, \widetilde y ) + \mathbb E^{((x,y), (\widetilde x,\widetilde y))}\!\!\int_0^{t\wedge T_{X,k}}\!\!\!\!\text{e}^{ \eta s}\big(\eta V_{k,\theta} \big(X_{s},\widetilde X_{s},Y_{s},\widetilde Y_{s}\big) +{\mathscr L}V_{k,\theta}\big(X_{s},\widetilde X_{s}, Y_{s}, \widetilde Y_{s} \big)\big) \,{\rm d} s\\ & \le V_{k,\theta} (x, \widetilde x, y, \widetilde y ).\end{align*}
Consequently, (2.7) holds from the inequality above and Fatou’s lemma. The proof is therefore completed.
2.2. Proofs
The proofs of Theorems 1.1 and 1.2 are based on Theorem 2.1. For this, we need to provide some explicit sufficient conditions to guarantee that Assumption 2.2 holds. First, we assume that, for any
$x\in D$
and
$f\in C_b^2(\mathbb{R}^n)$
,
\begin{align} L_{x,Y}f(y) & = \langle \nabla f(y),\gamma(x,y)\rangle +\tfrac{1}{2}\big\langle \nabla^2f(y),(\sigma\sigma^*)(x)\big\rangle _{\rm HS} \nonumber \\ & \quad + \int_{\mathbb{R}^n}\big(f(y+z)-f(y)-\langle \nabla f(y),z\rangle \textbf{1}_{\{|z|\le 1\}}\big)\,\nu(x,{\rm d} z),\end{align}
where
$\gamma\,:\, D\times\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$
,
$\sigma\,:\, D\rightarrow\mathbb{R}^{n}\otimes\mathbb{R}^{n}$
,
$\nu(x,{\rm d} z)$
is a Lévy kernel on
$D\times \mathscr{B}(\mathbb{R}^n)$
such that
\begin{equation} \int_{\mathbb{R}^{n}}\big(|z|^{\alpha_0}\wedge |z|^2\big)\,\nu(x,{\rm d} z)<\infty, \qquad x\in D ,\end{equation}
for some
$\alpha_0\in(0,1]$
,
$\nabla$
(resp.
$\nabla^2$
) denotes the gradient operator (resp. the Hessian operator) on
$\mathbb{R}^n$
, and
$\langle \cdot,\cdot\rangle $
means the Hilbert–Schmidt inner product on
$\mathbb{R}^n\otimes\mathbb{R}^n$
.
Given the coupling
$\big(X_t,\widetilde X_t\big)_{t\ge 0}$
, we adopt simultaneously the synchronous couplings for the diffusion term and the pure jump term in the second component process
$(Y_t)_{t\ge0}$
. For the synchronous coupling of multidimensional diffusion processes, we refer to, e.g., [Reference Chen and Li5, Reference Eberle, Guillin and Zimmer7, Reference Priola and Wang17] for more details. With regard to the pure jump counterpart, the synchronous coupling is constructed via the following relationship (see, e.g., [Reference Chen6, Theorem 0.21] and [Reference Wang20, Section 3.1]): for given
$x,\widetilde x\in D$
,
\begin{equation*}(y,\widetilde y)\rightarrow \begin{cases}(y+z,\widetilde y+z), \quad & (\nu(x,\cdot)\wedge\nu(\widetilde x,\cdot))({\rm d} z),\\(y+z,\widetilde y), \quad & \nu(x,{\rm d} z) -(\nu(x,\cdot)\wedge\nu(\widetilde x,\cdot))({\rm d} z),\\(z,\widetilde y+z), \quad & \nu(\widetilde x,{\rm d} z) -(\nu(x,\cdot)\wedge\nu(\widetilde x,\cdot))({\rm d} z).\end{cases}\end{equation*}
Thus, the generator of the coupling above is given by
\begin{align} {\mathscr L }_{x,\widetilde x, Y}f(y,\widetilde y) & = \langle \partial_1f(y,\widetilde y),\gamma(x,y)\rangle +\langle \partial_2f(y,\widetilde y), \gamma(\widetilde x,\widetilde y)\rangle \nonumber\\ & \quad + \tfrac{1}{2}\langle \partial_{11}f(y,\widetilde y),(\sigma\sigma^*)(x)\rangle _{\rm HS}+\tfrac{1}{2}\langle \partial_{22}f(y,\widetilde y),(\sigma \sigma^*)(\widetilde x)\rangle _{\rm HS}\nonumber\\ & \quad + \langle \partial_{12}f(y,\widetilde y), (\sigma \sigma^*)(\widetilde x)\rangle _{\rm HS}\nonumber\\ & \quad + \int_{\mathbb{R}^{n}}\big(f(y+z,\widetilde y+z)-f(y,\widetilde y)-\langle \partial_1 f(y,\widetilde y),z\rangle \textbf{1}_{\{|z|\le 1\}}\nonumber\\ & \qquad\qquad - \langle \partial_2 f(y,\widetilde y),z\rangle \textbf{1}_{\{|z|\le 1\}}\big)(\nu(x,\cdot)\wedge\nu(\widetilde x,\cdot))({\rm d} z)\nonumber\\ & \quad + \int_{\mathbb{R}^{n}}\big(f(y+z,\widetilde y)-f(y,\widetilde y)-\langle \partial_1 f(y,\widetilde y),z\rangle \textbf{1}_{\{|z|\le 1\}}\big)\nonumber\\ & \qquad\qquad \times \big(\nu(x,{\rm d} z) -(\nu(x,\cdot)\wedge\nu(\widetilde x,\cdot))({\rm d} z)\big)\nonumber\\ & \quad + \int_{\mathbb{R}^{n}}\big(f(y,\widetilde y+z)-f(y,\widetilde y)-\langle \partial_2 f(y,\widetilde y),z\rangle \textbf{1}_{\{|z|\le 1\}}\big)\nonumber \\ & \qquad\qquad \times \big(\nu(\widetilde x,{\rm d} z) -(\nu(x,\cdot)\wedge\nu(\widetilde x,\cdot))({\rm d} z)\big), \end{align}
where
$\partial_{11}$
(resp.
$\partial_{22}$
) denotes the second-order gradient operator with respect to the first (resp. second) variable, and
$\partial_{12}$
stands for the second-order gradient operator by taking gradient with respect to the first component followed by doing so with respect to the second component.
Let
\begin{equation} g_0(r)=\eta(r)r^2+(1-\eta(r))r^{\alpha_0},\quad\quad r\ge0,\end{equation}
where
$\alpha_0\in (0,1]$
is introduced in (2.9) and
$\eta\,:\,[0,\infty)\rightarrow[0,1]$
is defined as
\begin{equation*}\eta(r)=\begin{cases}1,\qquad\qquad \qquad\qquad \qquad\qquad \qquad\qquad \qquad\qquad& r\in\big[0,\frac{5}{8}\big],\\[5pt]-30\cdot 4^5\cdot\big[\frac{1}{5}\big(r-\frac{7}{8}\big)^5+\frac{1}{8}\big(r-\frac{7}{8}\big)^4+\frac{1}{48}\big(r-\frac{7}{8}\big)^3\big],\quad & \frac{5}{8}<r<\frac{7}{8}, \\[5pt]0,\qquad\qquad \qquad\qquad \qquad\qquad \qquad\qquad \qquad\qquad& r\ge\frac{7}{8}.\end{cases}\end{equation*}
A direct calculation shows that
$\eta\in C^2(\mathbb{R}_+)$
and is decreasing. Moreover, it is easy to see that
$g_0\in C^2(\mathbb{R}_+)$
is an increasing function satisfying
$g_0(r)\le r^{\alpha_0}$
for all
$r\ge0$
.
Proposition 2.1. Assume that Assumption 2.1 holds. Suppose further that there are constants
$c^*_1>0$
and
$c^*_2, c_3^*\ge0$
such that, for all
$k\ge1$
,
$ x, \widetilde x\in D$
with
$|x-\widetilde x|\ge 1/k$
, and
$y,\widetilde y\in\mathbb{R}^{n}$
,
\begin{equation} \langle y-\widetilde y, \gamma(x,y)-\gamma(\widetilde x,\widetilde y)\rangle \le - c_1^* |y-\widetilde y|^2+c_2^*|y-\widetilde y|f_k(x,\widetilde x) \end{equation}
and
\begin{equation} \|\sigma(x)-\sigma(\widetilde x)\|^2+\int_{\mathbb{R}^{n}}\big(|z|^2\wedge|z|^{\alpha_0}\big)|\nu(x,\cdot)-\nu(\widetilde x,\cdot)|({\rm d} z)\le c_3^* f_k(x,\widetilde x),\end{equation}
where
$\{f_k\}_{k\ge1}$
is a sequence of distance-like functions given in Assumption 2.1. Then, Assupmtion 2.2 holds, i.e. there are constants
$c_1,c_2>0$
such that, for any
$k\ge1$
,
$x,\widetilde x\in D$
with
$|x-\widetilde x|\ge 1/k$
, and
$y,\widetilde y\in \mathbb{R}^n$
,
${\mathscr L}_{x, \widetilde x,Y} g(y,\widetilde y)\le -c_1g(y,\widetilde y)+c_2f_k(x, \widetilde x)$
, where
$g(y,\widetilde y)\,:\!=$
$g_0(|y-\widetilde y|)$
with
$g_0$
given by (2.11).
Proof. According to the definition of the function
$g_0$
, there exist constants
$c_3>0$
and
$c_4\ge1$
such that
\begin{equation} \|g^{\prime}_0\|_\infty+\|g^{\prime\prime}_0\|_\infty\le c_3,\quad c_4^{-1} g^{\prime}_0(r)r\le g_0(r)\le c_4 g^{\prime}_0(r)r, \quad g^{\prime}_0(r)/r\le c_4, \quad r\ge0.\end{equation}
From (2.10), we deduce that, for any
$k\ge1$
,
$x,\widetilde x\in D$
with
$|x-\widetilde x|\ge 1/k$
, and
$y,\widetilde y\in \mathbb{R}^n$
,
\begin{align*} {\mathscr L}_{x,\widetilde x, Y} g_0(|y-\widetilde y|) & = \frac{g^{\prime}_0(|y-\widetilde y|)}{|y-\widetilde y|}\langle y-\widetilde y,\gamma(x,y)-\gamma(\widetilde x,\widetilde y)\rangle \\ & \quad + \bigg[\frac{g^{\prime\prime}_0(|y-\widetilde y|)}{2|y-\widetilde y|^2}\langle (y-\widetilde y)\otimes (y-\widetilde y),\Gamma(x,\widetilde x)\rangle _{\rm HS}\\ & \qquad \quad + \frac{g^{\prime}_0(|y-\widetilde y|)}{2|y-\widetilde y|}\big(\|\sigma(x)-\sigma(\widetilde x)\|^2\\ & \qquad\qquad\qquad\qquad\quad\,\,{-}\langle (y-\widetilde y)/|y-\widetilde y|,\Gamma(x,\widetilde x)(y-\widetilde y)/|y-\widetilde y|\rangle _{\rm HS}\big)\bigg]\\ & \quad + \int_{\mathbb{R}^{n}}\bigg(g_0(|y+z-\widetilde y|)-g_0(|y-\widetilde y|)-\frac{g^{\prime}_0(|y-\widetilde y|)}{|y-\widetilde y|}\langle y-\widetilde y,z\rangle \textbf{1}_{\{|z|\le 1\}}\bigg)\\ & \qquad\qquad \times \big(\nu(x,{\rm d} z) -(\nu(x,\cdot)\wedge\nu(\widetilde x,\cdot))({\rm d} z)\big)\\ & \quad + \int_{\mathbb{R}^{n}}\bigg(g_0(|y-\widetilde y-z|)-g_0(|y-\widetilde y|)-\frac{g^{\prime}_0(|y-\widetilde y|)}{|y-\widetilde y|}\langle \widetilde y-y,z\rangle \textbf{1}_{\{|z|\le 1\}}\bigg) \\ & \qquad\qquad \times \big(\nu(\widetilde x,{\rm d} z) -(\nu(x,\cdot)\wedge\nu(\widetilde x,\cdot))({\rm d} z)\big)\\ & =\!:\, \Lambda_1+\Lambda_2+\Lambda_3+\Lambda_4,\end{align*}
where
$\Gamma(x, \widetilde x )\,:\!=\,(\sigma(x)-\sigma(\widetilde x))(\sigma(x)-\sigma(\widetilde x))^*\in \mathbb{R}^{n}\otimes\mathbb{R}^{n}$
.
Due to (2.12) and (2.14), for any
$k\ge1$
,
$x,\widetilde x\in D$
with
$|x-\widetilde x|\ge 1/k$
, and
$y,\widetilde y\in \mathbb{R}^n$
,
\begin{equation*} \Lambda_1 \le g^{\prime}_0(|y-\widetilde y|)\big({-} c_1^* |y-\widetilde y|+c_2^*f_k(x,\widetilde x)\big)\le-c_5g_0(|y-\widetilde y|)+ c_6f_k(x,\widetilde x)\end{equation*}
holds with some constants
$c_5,c_6>0$
. Using the fact that the matrix
$\Gamma$
is non-negative definite, and (2.13) as well as (2.14), we find that there is a constant
$c_7>0$
such that, for any
$k\ge1$
,
$x,\widetilde x\in D$
with
$|x-\widetilde x|\ge 1/k$
, and
$y,\widetilde y\in \mathbb{R}^n$
,
$\Lambda_2 \le c_7 f_k(x,\widetilde x)$
. Furthermore, we can derive that, for any
$k\ge1$
,
$x,\widetilde x\in D$
with
$|x-\widetilde x|\ge 1/k$
, and
$y,\widetilde y\in \mathbb{R}^n$
,
\begin{equation} \Lambda_3 \le c_8 f_k(x,\widetilde x)\end{equation}
holds with some constant
$c_8>0$
. Indeed, by using
$g_0(r)\le r^{\alpha_0}$
for
$r\ge0$
,
$g_0(r)=r^{\alpha_0}$
for
$r\ge1$
, and the fact that
$(a+b)^{\alpha_0}\le a^{\alpha_0}+b^{\alpha_0}$
for all
$a,b\ge0$
, we find that, for any
$y,\widetilde y\in \mathbb{R}^n$
with
$|y-\widetilde y|\ge2$
and any
$z\in \mathbb{R}^n$
,
\begin{equation*}g_0(|y+z-\widetilde y|)-g_0(|y-\widetilde y|)\le |y+z-\widetilde y|^{\alpha_0}-|y-\widetilde y|^{\alpha_0}\le |z|^{\alpha_0}.\end{equation*}
Then, for any
$k\ge1$
,
$x,\widetilde x\in D$
with
$|x-\widetilde x|\ge 1/k$
, and any
$y,\widetilde y\in \mathbb{R}^n$
with
$|y-\widetilde y|\ge 2$
, according to the mean value theorem, we have
\begin{align*} \Lambda_3 & = \int_{\{|z|\le 1\}}\bigg(g_0(|y+z-\widetilde y|)-g_0(|y-\widetilde y|)-\frac{g^{\prime}_0(|y-\widetilde y|)}{|y-\widetilde y|}\langle y-\widetilde y,z\rangle \bigg)\\ & \qquad\qquad \times \big(\nu(x,{\rm d} z) -(\nu(x,\cdot)\wedge\nu(\widetilde x,\cdot))({\rm d} z)\big)\\ & \quad + \int_{\{|z|> 1\}} \big(g_0(|y+z-\widetilde y|)-g_0(|y-\widetilde y|)\big) \big(\nu(x,{\rm d} z) -(\nu(x,\cdot)\wedge\nu(\widetilde x,\cdot))({\rm d} z)\big)\\ & \le \frac{\|g^{\prime\prime}_0\|_\infty}{2}\int_{\{|z|\le 1\}} |z|^2\big(\nu(x,{\rm d} z) -(\nu(x,\cdot)\wedge\nu(\widetilde x,\cdot))({\rm d} z)\big)\\ & \quad + \int_{\{|z|> 1\}} |z|^{\alpha_0}\big(\nu(x,{\rm d} z) -(\nu(x,\cdot)\wedge\nu(\widetilde x,\cdot))({\rm d} z)\big)\\ & \le c_9 \int_{\mathbb{R}^n} \big(|z|^2 \wedge |z|^{\alpha_0}\big)\, \big(\nu(x,{\rm d} z) -(\nu(x,\cdot)\wedge\nu(\widetilde x,\cdot))({\rm d} z)\big)\end{align*}
for some constant
$c_9>0$
. On the other hand, for any
$k\ge1$
,
$x,\widetilde x\in D$
with
$|x-\widetilde x|\ge 1/k$
, and any
$y,\widetilde y\in \mathbb{R}^n$
with
$|y-\widetilde y|\le 2$
, by the mean value theorem again, we have
\begin{align*} \Lambda_3 & = \int_{\{|z|\le 1\}}\bigg( g_0(|y+z-\widetilde y|)-g_0(|y-\widetilde y|)-\frac{g^{\prime}_0(|y-\widetilde y|)}{|y-\widetilde y|}\langle y-\widetilde y,z\rangle \bigg)\\ & \qquad\qquad \times \big(\nu(x,{\rm d} z) -(\nu(x,\cdot)\wedge\nu(\widetilde x,\cdot))({\rm d} z)\big)\\ & \quad + \int_{\{|z|> 1\}}\big(g_0(|y+z-\widetilde y|)-g_0(|y-\widetilde y|)) \big(\nu(x,{\rm d} z) -(\nu(x,\cdot)\wedge\nu(\widetilde x,\cdot))({\rm d} z)\big)\\ & \le \frac{\|g^{\prime\prime}_0\|_\infty}{2}\int_{\{|z|\le 1\} } |z|^2\big(\nu(x,{\rm d} z) -(\nu(x,\cdot)\wedge\nu(\widetilde x,\cdot))({\rm d} z)\big)\\ & \quad + \int_{\{|z|> 1\} } (2+|z|)^{\alpha_0}\,\big(\nu(x,{\rm d} z) -(\nu(x,\cdot)\wedge\nu(\widetilde x,\cdot))({\rm d} z)\big)\\ & \le c_{10} \int_{\mathbb{R}^n} \big(|z|^2 \wedge |z|^{\alpha_0}\big)\,\big(\nu(x,{\rm d} z) -(\nu(x,\cdot)\wedge\nu(\widetilde x,\cdot))({\rm d} z) \big)\end{align*}
for some constant
$c_{10}>0$
. Combining both estimates above with (2.13) yields (2.15). Similarly, there exists a constant
$c_{11}>0 $
such that
$\Lambda_4\le c_{11} f_k(x,\widetilde x)$
.
Therefore, the desired assertion follows from all the estimates for
$\Lambda_i$
$(1\le i\le 4)$
.
Now, we are in a position to present the proofs of Theorems 1.1 and 1.2.
Proof of Theorem
1.1.According to [Reference Fu and Li9, Theorem 5.6], under Assumption 1.1 the first SDE in (1.3) has a unique strong solution
$(X_t)_{t\ge0}$
taking values in
$\mathbb{R}_+$
. Once
$(X_t)_{t\ge0}$
is available, the second SDE in (1.3) also has a unique strong solution
$(Y_t)_{t\ge0}$
since Assumption 1.2 yields that the coefficient
$\gamma(x,\cdot)$
satisfies the one-sided Lipschitz condition; see, e.g., [Reference Gyöngy and Krylov10, Theorem 1] or [Reference Majka16, Theorem 1.1]. Therefore, the SDE (1.3) has a unique strong solution
$(X_t,Y_t)_{t\ge0}$
. We can also follow the proof of [Reference Barczy, Döring, Li and Pap2, Theorem 2.2] (which only considers the affine two-factor models but works for general cases) to directly construct the strong solution to the SDE (1.3) with the aid of the Itô formula.
We split the proof into two cases.
Case (i):
$\kappa_3=0$
. For the first component process
$(X_t)_{t\ge0}$
, we consider the Markov coupling
$\big(X_t, \widetilde X_t\big)_{t\ge0}$
, where
$(\widetilde X_t)_{t\ge0}$
is constructed by applying the coupling by reflection for the Brownian motion and the refined basic coupling (see, e.g., [Reference Luo and Wang14]) for the spectrally positive stable process. Roughly speaking, the refined basic coupling is a revised version of the classical maximal coupling which will couple two marginal processes to meet together with half of the biggest jump intensity but fully adopt for the characterization of Lévy jumps. Then, it is seen from the proof of [Reference Li and Wang12, Theorem 3.1] that, under Assumption 1.1, Assumption 2.1 holds for
$f_k(x,\widetilde x) = f_0(|x-\widetilde x|)$
for all
$x,\widetilde x\in \mathbb{R}_+$
with
$|x-\widetilde x|\ge 1/k$
, where the explicit expression for
$f_0$
, satisfying
$c_1 r\le f_0(r)\le c_2 r$
for some
$c_2\ge c_1>0$
, is given in [Reference Li and Wang12, (4.2)]. (That is, in this case we can choose
$f_k(x,\widetilde x) = f_0(|x-\widetilde x|)$
independent of k.) For the second component process
$(Y_t)_{t\ge0}$
, we adopt simultaneously the synchronous couplings for the diffusion term and the pure jump term. Then, we can see that the coupling process
$\big((X_t,Y_t),\big(\widetilde X_t, \widetilde Y_t\big)\big)_{t\ge0}$
has the strong Markov property. Moreover, in this case,
$\sigma(x)=\delta_1 x^{1/2}$
and
$\nu(x,{\rm d} z)=\delta_2^\beta x\nu_\beta({\rm d} z)$
in (2.8), and (2.9) is satisfied with
$\alpha_0=1$
. In particular, by Assumption 1.2, (2.12) and (2.13) hold with
$f_k(x,\widetilde x) = f_0(|x-\widetilde x|)$
as mentioned above. Indeed, when
$\kappa_3=0$
, by (1.6) we know that, for all
$x,\widetilde x\in \mathbb{R}_+$
and
$y,\widetilde y\in \mathbb{R}$
,
\begin{align*} (\gamma(x,y)-\gamma(\widetilde x,\widetilde y))(y-\widetilde y ) & \le - \kappa_1 (y-\widetilde y)^2 +\kappa_2|(x-\widetilde x)(y-\widetilde y)|\\ & \le - \kappa_1 (y-\widetilde y)^2+\kappa_2c_1^{-1}|y-\widetilde y|f_0(|x-\widetilde x|)\\ & = - \kappa_1 (y-\widetilde y)^2+\kappa_2c_1^{-1}|y-\widetilde y| f_k(x,\widetilde x).\end{align*}
Therefore, according to Proposition 2.1, Assumption 2.2 is satisfied with
$f_k(x,\widetilde x) = f_0(|x-\widetilde x|)$
and
$g(y,\widetilde y) = g_0(|y-\widetilde y|)$
, with
$g_0$
given by (2.11).
Now let
$V(x,\widetilde x, y,\widetilde y)=|x-\widetilde x| + |y-\widetilde y|\wedge|y-\widetilde y|^2$
,
$x,\widetilde x\in \mathbb{R}_+$
,
$y,\widetilde y\in\mathbb{R}$
. It is clear that there are constants
$0<c_3\le c_4$
such that, for all
$x,\widetilde x\in \mathbb{R}_+$
and
$y,\widetilde y\in \mathbb{R}$
,
\begin{equation*}c_3(f_0(|x-\widetilde x|)+g_0(|y-\widetilde y|))\le V(x,\widetilde x, y,\widetilde y)\le c_4(f_0(|x-\widetilde x|)+g_0(|y-\widetilde y|)),\end{equation*}
and
$(y,\widetilde y) \mapsto |y-\widetilde y|\wedge|y-\widetilde y|^2$
satisfies the weak form of the triangle inequality. On the other hand, due to
$\alpha,\beta\in (1,2)$
, we can see from the Itô formula that, for all
$t>0$
and
$x,y\in \mathbb{R}_+$
,
$\mathbb E^{(x,y)}(|X_t|+|Y_t|)<\infty$
; see, e.g., the proof of [Reference Fu and Li9, Proposition 2.3]. Therefore, the desired assertion follows from Theorem 2.1.
Case (ii):
$\kappa_3>0$
. For the first component process
$(X_t)_{t\ge0}$
, we still apply the coupling by reflection for the Brownian motion and the refined basic coupling for the spectrally positive stable process. Suppose that Assumption 1.1 holds. According to the proof of [Reference Li and Wang12, Theorem 3.2], there are a constant
$\lambda_1>0$
and a bounded, nondecreasing, and concave continuous function
$f^*_0$
such that Assumption 2.1 holds for
$f_k(x,\widetilde x)$
, where
$f_k(x,x) = 0$
and
$f_k(x,\widetilde x)= 1+ f_0^*(|x-\widetilde x|)+f_0(|x-\widetilde x|)$
for all
$x,\widetilde x\in \mathbb{R}_+$
with
$|x-\widetilde x|\ge 1/k$
. Here,
$f_0$
is the same function taken in the case
$\kappa_3=0$
(i.e. from [Reference Li and Wang12, (4.2)]). We consider the same coupling for the second component process
$(Y_t)_{t\ge0}$
as in the case
$\kappa_3=0$
. Recall that
$\sigma(x)=\delta_1 x^{1/2}$
and
$\nu(x,{\rm d} z)=\delta_2^\beta x\nu_\beta({\rm d} z)$
in (2.8), and that (2.9) is satisfied with
$\alpha_0=1$
. Then, by Assumption 1.2, for
$x,\widetilde x\in \mathbb{R}_+$
with
$|x-\widetilde x|\ge 1/k$
and
$y,\widetilde y\in \mathbb{R}$
,
\begin{align*} (\gamma(x,y)-\gamma(\widetilde x,\widetilde y))(y-\widetilde y ) & \le - \kappa_1 (y-\widetilde y)^2 +\kappa_2|(x-\widetilde x)(y-\widetilde y)|+\kappa_3|y-\widetilde y|^2\textbf{1}_{\{|y-\widetilde y|\le \kappa_3\}}\\ & \le - \kappa_1 (y-\widetilde y)^2+ c_5 |y-\widetilde y|f_k(x,\widetilde x),\end{align*}
and (2.13) holds as well since
$f_k(x,\widetilde x)\ge f_0(|x-\widetilde x|)$
for all
$x,\widetilde x\in \mathbb{R}_+$
with
$|x-\widetilde x|\ge 1/k$
. Hence, by Proposition 2.1, Assumption 2.2 is satisfied with
$f_k(x,\widetilde x)$
and
$g(y,\widetilde y)=g_0(|y-\widetilde y|)$
, with
$g_0$
given by (2.11).
Note that there is a constant
$0<c_6\le c_7$
such that, for all
$x,\widetilde x\in \mathbb{R}_+$
,
\begin{equation*}c_6\textbf{1}_{\{x\neq \widetilde x\}} (1+|x-\widetilde x|)\le \lim_{k\rightarrow \infty} f_k(x,\widetilde x)\le c_7\textbf{1}_{\{x\neq \widetilde x\}} (1+|x-\widetilde x|).\end{equation*}
With this at hand, the desired assertion follows by similar arguments to Case (i).
Proof of Theorem
1.2.Following the argument at the beginning of the proof of Theorem 1.1, we can see that the SDE (1.3) has a unique strong solution
$(X_t, Y_t)_{t\ge0}$
under Assumptions 1.3 and 1.4.
In the first place, we consider the case
$\kappa_3=0$
. For the first component process
$(X_t)_{t\ge0}$
, we will directly apply the reflection coupling. Then, according to Assumption 1.3 and [Reference Eberle, Guillin and Zimmer7, Theorem 2.2], we know that Assumption 2.1 holds for
$f_k(x,\widetilde x) = f(x,\widetilde x)$
for all
$x,\widetilde x\in \mathbb{R}_+$
with
$|x-\widetilde x|\ge 1/k$
, where
$f(x,\widetilde x)$
satisfies
\begin{equation*}c_1(1\wedge|x-\widetilde x|)\big(1+|x|^l+|\widetilde x|^l\big)\le f(x,\widetilde x)\le c_2 (1\wedge|x-\widetilde x|)\big(1+|x|^l+|\widetilde x|^l\big)\end{equation*}
for any
$l>0$
with
$c_1, c_2>0$
; see [Reference Eberle, Guillin and Zimmer7, Remark 2.1 and Assumption 2.4] for more details. For the second component process
$(Y_t)_{t\ge0}$
, we adopt the synchronous coupling. It is clear that the coupling process
$((X_t,Y_t),\big(\widetilde X_t, \widetilde Y_t\big))_{t\ge0}$
enjoys the strong Markov property, and that (2.12) and (2.13) hold under Assumption 1.4. Note that
$(x,\widetilde x)\mapsto (1\wedge|x-\widetilde x|)\big(1+|x|^l+|\widetilde x|^l\big)$
satisfies the weak form of the triangle inequality. Then, following the argument at the end of the proof of Theorem 1.1, we can obtain the desired assertion by invoking Proposition 2.1 and Theorem 2.1.
Next, we turn to the case
$\kappa_3>0$
. Then, we can follow the arguments in Case (ii) of the proof of Theorem 1.1 and apply [Reference Eberle, Guillin and Zimmer7, Theorem 2.1] (instead of [Reference Eberle, Guillin and Zimmer7, Theorem 2.2]). Since the details are similar to those for the proof of Theorem 1.1, we omit them here.
Finally, we give a remark on the extension of Theorems 1.1 and 1.2.
Remark 2.1. According to [Reference Li and Wang12, Theorem 3.1] and the proof of Theorem 1.1, we can consider more general two-factor type processes by replacing the first component process in the SDE (1.3) by a class of continuous-state nonlinear branching processes as studied in [Reference Li and Wang12]. Similarly, by [Reference Liang, Majka and Wang13, Sections 2.2 and 2.3] and the proof of Theorem 1.1, we can consider more general Gruschin type processes by replacing the Brownian motion with general Lévy noise.
Acknowledgements
The authors thank two referees for their helpful comments and corrections.
Funding information
The research of Jianhai Bao is supported by the National Natural Science Foundation of China (Nos. 11831014 and 12071340). The research of Jian Wang is supported by the National Natural Science Foundation of China (Nos. 11831014, 12071076, 12225104, and 2022YFA1000033) and the Education and Research Support Program for Fujian Provincial Agencies.
Competing interests
There were no competing interests to declare which arose during the preparation or publication process of this article.
