2.1. Assumptions

Firstly, we list our assumptions on the functions b, c, $\kappa$ (or $\kappa^*$ ), and J.

Assumption 2. The function $(x,z,u,v)\to\kappa^*(x,z,u,v)$ is periodic of period 1 in x and u. For the function $\kappa(x,z,u)\,:\!=\,\kappa^*(x,z,u,u)$ , there exist constants $\kappa_1,\kappa_2,\kappa_3>0$ such that for the same $\beta$ of 1, and all $x,x_1,x_2\in\mathbb R^d$ and $z,u\in\mathbb R^{d}\setminus \{0\}$ ,

(2.1) \begin{align} \kappa_1\le\kappa(x,z,u)\le\kappa_2, \end{align}

(2.2) \begin{align} |\kappa(x_1,z,u)-\kappa(x_2,z,u)|\le \kappa_3|x_1-x_2|^\beta. \end{align}

There exists a function $(x,z,u)\to\kappa_0(x,z,u)$ , periodic of period 1 in x and u and also satisfying (2.1) and (2.2), such that for all $x\in\mathbb R^d$ and $z\in\mathbb R^{d}\setminus \{0\}$ ,

(2.3) \begin{equation} |\kappa^*(x,z,z/\epsilon,z/\epsilon) - \kappa_0(x,z,z/\epsilon)| \to 0, \quad \epsilon\to 0^+. \end{equation}

We assume also that there exists a function $\tilde\kappa\,:\,\mathbb R^d\times\big(\mathbb R^{d}\setminus \{0\}\big)\to [0,\infty)$ such that for all $z\in\mathbb R^{d}\setminus \{0\}$ ,

(2.4) \begin{equation} \sup_{x\in\mathbb R^d} |\kappa(x,\epsilon z,z) - \tilde\kappa(x,z)| \to 0, \quad \epsilon\to 0^+. \end{equation}

In the case $\alpha\in(1,2)$ and $b\ne0$ , we assume further that $\alpha+\beta\ne 2$ and for all $z\in\mathbb R^{d}\setminus \{0\}$ ,

(2.5) \begin{equation} \frac{1}{\epsilon^{\alpha-1}} \sup_{x\in\mathbb R^d} | \kappa(x,\epsilon z,z)- \tilde\kappa(x,z)| \to0, \quad \epsilon\to 0^+. \end{equation}

Assumption 3. Assume that the function J is positive homogeneous of degree $-(d+\alpha)$ for some $\alpha\in(0,2)$ , i.e.

(2.6) \begin{equation} J(r z)=r^{-(d+\alpha)}J(z), \quad r>0,\ z\in\mathbb R^{d}\setminus \{0\}, \end{equation}

and that J is bounded between two positive constants on the unit $(d-1)$ -sphere S, i.e., there exist constants $j_1,j_2>0$ such that for all $\xi\in S$ ,

(2.7) \begin{equation} j_1 \le J(\xi) \le j_2. \end{equation}

In the case $\alpha = 1$ , we assume additionally that for each $x\in\mathbb R^d$ and $r_1,r_2\in(0,\infty)$ ,

(2.8) \begin{equation} \int_{S} \xi \kappa(x,r_1\xi,r_2\xi) J(\xi) d\xi = 0. \end{equation}

We denote by ${\mathcal{C}}\big(\mathbb T^d\big)$ the space of continuous functions on the flat torus $\mathbb T^d\,:\!=\,\mathbb R^d/\mathbb Z^d$ endowed with the supremum norm $\|f\|_\infty \,:\!=\, \sup_{x\in\mathbb T^d}|f(x)|$ . Denote by ${\mathcal{C}}^k\big(\mathbb T^d\big)$ , with integer $k\ge0$ , the space of continuous functions on $\mathbb T^d$ possessing derivatives of orders not greater than k, endowed with the norm $\|f\|_{{\mathcal{C}}^k} \,:\!=\,\|f\|_\infty + \sum_{|\alpha|\le k}\sup_{x\in\mathbb T^d}|\nabla^\alpha f(x)|$ . For a non-integer $\gamma>0$ , the Hölder space ${\mathcal{C}}^\gamma\big(\mathbb T^d\big)$ is defined as the subspace of ${\mathcal{C}}^{\lfloor\gamma\rfloor}$ consisting of functions whose $\lfloor\gamma\rfloor$ th-order partial derivatives are locally Hölder continuous with exponent $\gamma-\lfloor\gamma\rfloor$ . The space ${\mathcal{C}}^\gamma\big(\mathbb T^d\big)$ is a Banach space endowed with the norm

$$\|f\|_{{\mathcal{C}}^\gamma} \,:\!=\,\|f\|_{{\mathcal{C}}^{\lfloor\gamma\rfloor}} + \sup_{x,y\in\mathbb T^d,x\ne y}\frac{|f(x)-f(y)|}{|x-y|^{\gamma-\lfloor\gamma\rfloor}}.$$

The space of infinitely differentiable functions on $\mathbb T^d$ is denoted by ${\mathcal{C}}^\infty\big(\mathbb T^d\big)$ .

2.2. Feller processes

We need some auxiliary operators and processes. In fact, we will rescale the operator $\mathcal A^\epsilon$ and its canonical process in an effective fashion. For this purpose, we define the following non-local operators for $f\in{\mathcal{C}}^\infty\big(\mathbb T^d\big)$ , the space of all smooth functions on the flat torus $\mathbb T^d$ (i.e., smooth periodic functions of period 1):

(2.10) \begin{equation} \begin{split} \tilde{\mathcal A}^\epsilon f(x) =&\ \int_{\mathbb R^{d}\setminus \{0\}} \left[ f( x+z)-f(x) - z\cdot\nabla f(x) \left(\textbf 1_{\{1\}}(\alpha)\textbf 1_B(z) + \textbf 1_{(1,2)}(\alpha)\textbf 1_B(\epsilon z) \right) \right] \\ &\qquad \times \kappa(x,\epsilon z,z)J(z)dz +\big(b(x)+\epsilon^{\alpha-1}c(x)\big) \cdot \nabla f(x) \textbf 1_{(1,2)}(\alpha), \quad \epsilon>0, \end{split}\end{equation}

(2.11) \begin{equation} \begin{split} \tilde{\mathcal A} f(x) =&\ \int_{\mathbb R^{d}\setminus \{0\}} \left[ f( x+z)-f(x) - z\cdot\nabla f(x) \left(\textbf 1_{\{1\}}(\alpha)\textbf 1_B(z) + \textbf 1_{(1,2)}(\alpha)\right) \right] \\ &\qquad \times\tilde\kappa(x,z)J(z)dz + b(x)\cdot \nabla f(x) \textbf 1_{(1,2)}(\alpha). \end{split}\end{equation}

For notational simplicity, we shall allow the parameter $\epsilon$ to be zero so that $\tilde{\mathcal A}^0\,:\!=\,\tilde{\mathcal A}$ . The periodicity and continuity of the function $x\to\kappa(x,z,u)$ and (2.1), (2.8), and (2.9) imply that $\tilde{\mathcal A}^\epsilon$ , $\epsilon\ge0$ , are all well-defined unbounded operators on $({\mathcal{C}}\big(\mathbb T^d\big),\|\cdot\|_\infty)$ , whose domains contain ${\mathcal{C}}^\infty\big(\mathbb T^d\big)$ . Moreover, it is easy to verify by (2.6) and (2.8) that for each $\epsilon>0$ , the operator $\tilde{\mathcal A}^\epsilon$ is a rescaling of $\mathcal A^\epsilon$ in the sense that

(2.12) \begin{equation} \tilde{\mathcal A}^\epsilon f(x) = \epsilon^{\alpha}(\mathcal A^\epsilon f_\epsilon)(\epsilon x), \quad f\in{\mathcal{C}}^\infty\big(\mathbb T^d\big).\end{equation}

Here and after, we write $f_\epsilon(x)\,:\!=\,f(x/\epsilon)$ .

Denote by $\mathcal D=\mathcal D(\mathbb R_+;\,\mathbb R^d)$ (resp. $\mathcal D_{\textrm{per}}=\mathcal D(\mathbb R_+;\,\mathbb T^d)$ ) the space of all $\mathbb R^d$ -valued (resp. $\mathbb T^d$ -valued) càdlàg functions on $\mathbb R_+\,:\!=\,[0,\infty)$ , equipped with the Skorokhod topology. The following proposition tells us that the operators $\mathcal A^\epsilon$ , $\tilde{\mathcal A}^\epsilon$ , and $\tilde{\mathcal A}$ can generate Feller processes, and the state space of the Feller processes associated to $\tilde{\mathcal A}^\epsilon$ or $\tilde{\mathcal A}$ can be taken as $\mathbb T^d$ , which is a compact space. Meanwhile, the heat kernel estimates for $\tilde{\mathcal A}^\epsilon$ and $\tilde{\mathcal A}$ are crucial in proving the ergodicity of the associated processes. We also find the core for $\tilde{\mathcal A}^\epsilon$ and $\tilde{\mathcal A}$ , which we will use to show the convergence of the associated invariant measures in the sequel.

Proof. All assertions for the case $\alpha\in(0,1)$ follow from [Reference Grzywny and Szczypkowski20, Theorem 1.1, Theorem 1.3, Theorem 1.4, Remark 1.5]; the assertions for the case $\alpha=1$ can be found in [Reference Szczypkowski37, Theorem 2.1, Theorem 2.3, Theorem 2.4]. In particular, for these two cases, the constant C in the estimate (2.13) depends only on $(d,\alpha,\beta,\kappa_1,\kappa_2,\kappa_3,j_1,j_2)$ and hence is independent of $\epsilon\ge0$ , since $\kappa(x,\epsilon z,z)$ and $\tilde\kappa(x,z)$ , the only quantities in $\{\tilde{\mathcal A}^\epsilon\,:\,\epsilon\ge0\}$ that depend on $\epsilon$ when $\alpha\in(0,1]$ , satisfy (2.1) and (2.2) with uniform constants $\kappa_1,\kappa_2$ . For the case $\alpha\in(1,2)$ , the properties of $\tilde p^\epsilon$ can be found in [Reference Chen and Zhang11, Theorem 1.5]; for the reader’s convenience, we have also included the proof in the appendix (see Proposition 5). In particular, by Proposition 5(iii), the constant C of (2.13) depends on $(d,\alpha,\beta,\kappa_1,\kappa_2,\kappa_3,j_1,j_2)$ and the upper bound of the drift of each $\tilde{\mathcal A}^\epsilon$ , $\epsilon\in[0,1]$ . The drift of $\tilde{\mathcal A}^0 = \tilde{\mathcal A}$ is b, while that of $\tilde{\mathcal A}^\epsilon$ with $\epsilon\in(0,1]$ is

\begin{equation*} b(x)+\epsilon^{\alpha-1}c(x) - \int_{1\le |z|<\frac{1}{\epsilon}} z \kappa(x,\epsilon z,z)J(z)dz, \end{equation*}

whose absolute value is bounded by $\|b\|_\infty + \|c\|_\infty + \frac{\kappa_2}{\alpha-1}$ uniformly for $\epsilon\in(0,1]$ . Thus, C in (2.13) can be chosen as independent of $\epsilon\in[0,1]$ . The proofs of the remaining parts are tedious, especially the proof that ${\mathcal{C}}^\infty\big(\mathbb T^d\big)$ is the core of the generators, and we leave them to the appendix; see Proposition 6 and Corollary 5.

Of course, each of the processes $X^\epsilon$ , $\tilde X$ , and $\tilde X^\epsilon$ is defined on its own stochastic basis. However, by taking the product of the probability spaces, it is always possible to assume that

$X^\epsilon$ , $\tilde X$ and $\tilde X^\epsilon$ , $\epsilon>0$ , are all defined on the same probability space $(\Omega,\mathcal F,\mathbb P)$ .

We also assume for simplicity that

\begin{equation*} X^\epsilon_0=\tilde X_0=\tilde X^\epsilon_0=0. \end{equation*}

If we identify a periodic function on $\mathbb R^d$ of period $\epsilon$ with its restriction to the $\epsilon$ -torus $\mathbb T^d_\epsilon\,:\!=\,\epsilon\mathbb T^d$ , then each $\mathcal A^\epsilon$ maps ${\mathcal{C}}^\infty\big(\mathbb T^d_\epsilon\big)$ into ${\mathcal{C}}^\infty\big(\mathbb T^d_\epsilon\big)$ by virtue of the periodicity of $\kappa$ in x. In view of this, the canonical process $X^\epsilon$ can also be treated as a process taking values on $\mathbb T^d_\epsilon$ , via the quotient map $\mathbb R^d \to \mathbb T^d_\epsilon$ . We will use this treatment only in the rest of this section; the benefit is the relation below, which follows from the well-known fact that Feller semigroups and Feller processes are in one-to-one correspondence if we identify the processes that have the same finite-dimensional distributions (see, e.g., [Reference Böttcher, Schilling and Wang8]).

Lemma 1. We have the following identity in law:

$$\{\tilde X_t^\epsilon\}_{t\ge0} \stackrel{\mathtt d}{=} \left\{ \textstyle{\frac{{1}}{\epsilon}} X^\epsilon_{\epsilon^{\alpha} t} \right\}_{t\ge0}, \quad{for\ each }\ \epsilon>0.$$

Proof. We derive the generator for the Feller process $\left\{ \textstyle{\frac{{1}}{\epsilon}} X^\epsilon_{\epsilon^{\alpha} t} \right\}_{t\ge0}$ . For $f\in {\mathcal{C}}^\infty\big(\mathbb T^d\big)$ , by (2.12),

\begin{equation*} \lim_{t\downarrow0}\frac{\mathbb E^\epsilon_{\epsilon x} \!\left[ f\!\left( \frac{{1}}{\epsilon} X^\epsilon_{\epsilon^{\alpha} t} \right) \right] -f(x)}{t} = \epsilon^{\alpha} \lim_{t\downarrow0}\frac{\mathbb E^\epsilon_{\epsilon x} \!\left[ f_\epsilon\!\left( X^\epsilon_{\epsilon^{\alpha} t} \right) \right] -f(x)}{\epsilon^{\alpha}t} = \epsilon^{\alpha}(\mathcal A^\epsilon f_\epsilon)(\epsilon x) = \tilde{\mathcal A}^\epsilon f(x). \end{equation*}

Therefore, the Feller semigroup associated to $\left\{ \textstyle{\frac{{1}}{\epsilon}} X^\epsilon_{\epsilon^{\alpha} t} \right\}_{t\ge0}$ is also generated by the closure of $\big(\tilde{\mathcal A}^\epsilon,{\mathcal{C}}^\infty\big(\mathbb T^d\big)\big)$ .

Denote by $\big\{\tilde P^\epsilon_t\big\}_{t\ge0}$ $\big(\text{or}\ \big\{\tilde P_t\big\}_{t\ge0}\big)$ the Feller semigroup of $\tilde X^\epsilon$ (or $\tilde X$ ). Let $\tilde X^{0}=\tilde X$ and $\tilde P^0_t=\tilde P_t$ . Now, utilizing the lower bound of the transition probability density $\tilde p^\epsilon(t;\,x,y)$ , we obtain the following exponential ergodicity.

Proposition 2. For each $\epsilon\in[0,1]$ , the Feller process $\tilde X^\epsilon$ possesses a unique invariant probability distribution $\mu_\epsilon$ on $\mathbb T^d$ . Moreover, there exist positive constants C and $\rho$ which are independent of $\epsilon\in[0,1]$ , such that for each periodic bounded Borel function f on $\mathbb R^d$ (i.e., f is a bounded Borel function on $\mathbb T^d$ ),

\begin{equation*} \sup_{x\in\mathbb T^d} \left| \tilde P^\epsilon_t f(x)-\int_{\mathbb T^d}f(y)\mu_\epsilon(dy) \right| \le C\|f\|_\infty e^{-\rho t} \end{equation*}

for every $t\ge0$ .

Denote by $\mu=\mu_0$ the invariant probability measure of $\tilde X$ .

Proof. By the argument in [Reference Hairer and Pardoux21, Lemma 2.4], we only need to show that $\tilde P_t^\epsilon f\to \tilde P_tf$ in ${\mathcal{C}}\big(\mathbb T^d\big)$ as $\epsilon\to 0^+$ for each $f\in{\mathcal{C}}\big(\mathbb T^d\big)$ and $t\ge0$ .

By Proposition 1, we know that ${\mathcal{C}}^\infty\big(\mathbb T^d\big)$ is a core for each $\tilde{\mathcal A}^\epsilon$ , $\epsilon\ge0$ . Now fix an arbitrary $f\in{\mathcal{C}}^\infty\big(\mathbb T^d\big)$ . If $\alpha\in(0,1]$ , then $\|\tilde{\mathcal A}^\epsilon f-\tilde{\mathcal A} f\|_\infty$ as $\epsilon\to0^+$ by dominated convergence and (2.4). For the case $\alpha\in(1,2)$ , we use the fact that

\begin{equation*} |f( x+z)-f(x) - z\cdot\nabla f(x)| \le \frac{1}{2} \|f\|_{{\mathcal{C}}^2} |z|^2 \textbf 1_{\{|z|\le1\}} + 2\|f\|_{{\mathcal{C}}^1} |z| \textbf 1_{\{|z|>1\}} \le 2\|f\|_{{\mathcal{C}}^2} \big(|z|^2 \wedge |z|\big), \end{equation*}

which follows from Taylor expansion, to derive

(2.14) \begin{equation} \begin{split} &\ \| \tilde{\mathcal A}^\epsilon f-\tilde{\mathcal A} f \|_\infty \\ \le&\ \| \epsilon^{\alpha-1}c \cdot \nabla f \|_\infty + \sup_{x\in\mathbb T^d} \left| \int_{\mathbb R^{d}\setminus \{0\}} z\cdot\nabla f(x) \left(1-\textbf 1_{B}(\epsilon z)\right) \kappa(x,\epsilon z,z)J(z) dz \right| \\ & + \sup_{x\in\mathbb T^d} \left|\int_{\mathbb R^{d}\setminus \{0\}} \left[ f( x+z)-f(x) - z\cdot\nabla f(x) \right] (\kappa(x,\epsilon z,z)-\tilde\kappa(x,z))J(z)dz\right| \\ \le&\ \epsilon^{\alpha-1}\|c\|_\infty \|f\|_{{\mathcal{C}}^1} + \kappa_2 j_2\|f\|_{{\mathcal{C}}^1} \int_{|z|\ge 1/\epsilon} \frac{dz}{|z|^{d+\alpha-1}} \\ & + 2 j_2\|f\|_{{\mathcal{C}}^2} \int_{\mathbb R^{d}\setminus \{0\}} \left( \sup_{x\in\mathbb T^d} |\kappa(x,\epsilon z,z)-\tilde\kappa(x,z)| \right) (|z|^2\wedge|z|) \frac{dz}{|z|^{d+\alpha}}, \end{split} \end{equation}

which converges to zero as $\epsilon\to0^+$ , by (2.4) and dominated convergence. Now by the Trotter–Kato approximation theorem (see [Reference Engel and Nagel15, Theorem III.4.8]), $\tilde P_t^\epsilon f\to \tilde P_tf$ in ${\mathcal{C}}\big(\mathbb T^d\big)$ as $\epsilon\to 0^+$ for all $f\in{\mathcal{C}}\big(\mathbb T^d\big)$ , uniformly for t in compact intervals.

Now, using Proposition 2 and Lemma 2, we can obtain a useful functional convergence theorem.

Corollary 1. Let f be a bounded Borel function on $\mathbb T^d$ . Then for every $t>0$ ,

\begin{equation*} \mathbb E \!\left[ \left| \int_0^t f\!\left( \frac{X_s^\epsilon}{\epsilon} \right) ds - t\int_{\mathbb T^d}f(y)\mu(dy) \right|^2\right] \to 0, \quad {as }\ \epsilon\to 0^+. \end{equation*}

For every $T>0$ ,

(2.15) \begin{equation} \sup_{t\in[0,T]}\left| \int_0^t f\!\left( \frac{X_s^\epsilon}{\epsilon} \right) ds - t\int_{\mathbb T^d}f(y)\mu(dy) \right|\to 0,\quad { in\ probability }\ \mathbb P,\ { as }\ \epsilon\to 0^+. \end{equation}

Proof. We follow the lines of [Reference Pardoux32, Proposition 2.4]. Fix $\epsilon>0$ . Let $\bar f_\epsilon\,:\!=\, f-\int_{\mathbb T^d}f(y)\mu_\epsilon(dy)$ . By virtue of Lemma 1 and Lemma 2, to prove the two limits, it suffices to prove that

(2.16) \begin{equation} \epsilon^\alpha\int_0^{\epsilon^{-\alpha}t} \bar f_\epsilon\big(\tilde X^\epsilon_s\big) ds = \int_0^t \bar f_\epsilon(X^\epsilon_s/\epsilon) ds \to 0, \end{equation}

in $L^2(\Omega,\mathbb P)$ and also in probability uniformly in $t\in[0,T]$ . By Proposition 2, for $0\le s<t$ we have

(2.17) \begin{equation} \mathbb E \!\left( \bar f_\epsilon\big(\tilde X^{\epsilon}_t\big) \Big| \tilde X^{\epsilon}_s\right) = \int_{\mathbb T^d} \bar f_\epsilon(y) \left[\tilde p^\epsilon\big(t-s,\tilde X_s^{\epsilon},y\big)dy-\mu_\epsilon(dy)\right] \le C\|\bar f_\epsilon\|_\infty e^{-\rho(t-s)}, \end{equation}

and then by the Markov property,

(2.18) \begin{equation} \begin{split} \mathbb E\big( \bar f_\epsilon\big(\tilde X^\epsilon_s\big) \bar f_\epsilon\big(\tilde X^\epsilon_t\big) \big) &= \mathbb E\!\left[\bar f_\epsilon\big(\tilde X^\epsilon_s\big) \mathbb E \!\left(\bar f_\epsilon\big(\tilde X^{\epsilon}_t\big)\Big| \tilde X^{\epsilon}_s\right)\right] \\ &\le C\|\bar f_\epsilon\|_\infty^2 e^{-\rho(t-s)} \le 4C\|f\|_\infty^2 e^{-\rho(t-s)}. \end{split} \end{equation}

Hence, if we write $g_\epsilon(s)\,:\!=\,\bar f_\epsilon(X^\epsilon_s/\epsilon)$ , then as $\epsilon\to0^+$ ,

(2.19) \begin{equation} \begin{split} \mathbb E\!\left[\left| \int_0^t g_\epsilon(s) ds \right|^2\right] &= 2\epsilon^{2\alpha}\int_0^{\epsilon^{-\alpha}t} \int_0^{r} \mathbb E \!\left( \bar f_\epsilon\big(\tilde X^\epsilon_s\big) \bar f_\epsilon\big(\tilde X^\epsilon_r\big) \right) ds dr \\ &\le 8C\epsilon^{2\alpha} \|f\|_\infty^2 \int_0^{\epsilon^{-\alpha}t} \int_0^{r} e^{-\rho(r-s)} dsdr \\ &= 8C\epsilon^{2\alpha} \|f\|_\infty^2 \rho^{-2} \left[ -1+\rho\epsilon^{-\alpha}t + e^{-\rho\epsilon^{-\alpha}t} \right] \\ &\to 0. \end{split} \end{equation}

The first result follows. On the other hand, for any $n\in\mathbb N_+$ , since $(\lfloor \frac{nt}{T} \rfloor +1) \frac{T}{n}\ge t$ ,

\begin{equation*} \begin{split} &\ \mathbb E\!\left[ \sup_{t\in[0,T]} \left| \int_0^t g_\epsilon(s) ds \right|^2\right] \\ =&\ \mathbb E\!\left[ \sup_{k=0,\cdots,n} \left| \int_0^{k\frac{T}{n}} g_\epsilon(s) ds \right|^2 + \sup_{t\in[0,T]} \left( \left| \int_0^{\lfloor \frac{nt}{T} \rfloor \frac{T}{n}} g_\epsilon(s) ds \right| + \left| \int_{\lfloor \frac{nt}{T} \rfloor \frac{T}{n}}^t g_\epsilon(s) ds \right| \right)^2 \right] \\ \le &\ 3 \mathbb E\!\left[ \sup_{k=0,\cdots,n} \left| \int_0^{k\frac{T}{n}} g_\epsilon(s) ds \right|^2 \right] + 2 \mathbb E\!\left[ \sup_{t\in[0,T]} \left| \int_{\lfloor \frac{nt}{T} \rfloor \frac{T}{n}}^t g_\epsilon(s) ds \right|^2 \right] \\ \le&\ 3 \sup_{k=0,\cdots,n} \mathbb E\!\left[ \left| \int_0^{k\frac{T}{n}} g_\epsilon(s) ds \right|^2 \right] + 8 \|f\|_\infty^2 \frac{T^2}{n^2}, \end{split} \end{equation*}

which goes to zero, by first letting $\epsilon\to0^+$ and applying (2.19), and then letting $n\to\infty$ . The second result follows by an application of Chebyshev’s inequality.

2.3. Non-local Poisson equation

Using the exponential ergodicity, we can also obtain the well-posedness of the non-local Poisson equation. Denote by ${\mathcal{C}}^\gamma_\mu\big(\mathbb T^d\big)$ , $\gamma>0$ , the class of all $f\in{\mathcal{C}}^\gamma\big(\mathbb T^d\big)$ which are centered with respect to the invariant measure $\mu$ , in the sense that $\int_{\mathbb T^d}f(x)\mu(dx)=0$ . It is easy to check that ${\mathcal{C}}_\mu\big(\mathbb T^d\big)$ is a closed subset, and hence a sub-Banach space, of ${\mathcal{C}}\big(\mathbb T^d\big)$ under the norm $\|\cdot\|_\infty$ .

Proof. Since $\mu$ is invariant with respect to $\{\tilde P_t\}_{t\ge0}$ , it is easy to see that ${\mathcal{C}}_\mu\big(\mathbb T^d\big)$ is $\{\tilde P_t\}_{t\ge0}$ -invariant, in the sense that $\tilde P_t\big({\mathcal{C}}_\mu\big(\mathbb T^d\big)\big)\subset{\mathcal{C}}_\mu\big(\mathbb T^d\big)$ for all $t\ge0$ . The first part of the lemma then follows from the corollary in [Reference Engel and Nagel15, Subsection II.2.3]. By the exponential ergodicity result in Proposition 2, we have

(2.21) \begin{equation} \|\tilde P_t^\mu f\|_\infty\le C\|f\|_\infty e^{-\rho t} \end{equation}

for all $f\in{\mathcal{C}}_\mu\big(\mathbb T^d\big)$ and $t\ge0$ . This yields the second part of the lemma, using [Reference Engel and Nagel15, Theorem II.1.10(ii)].

Corollary 2. Let $\alpha\in(1,2)$ . For every $f\in{\mathcal{C}}^\beta_\mu\big(\mathbb T^d\big)$ , there exists a unique solution in ${\mathcal{C}}^{\alpha+\beta}_\mu\big(\mathbb T^d\big)$ to the Poisson equation

(2.22) \begin{equation} \tilde{\mathcal A} u+f = 0. \end{equation}

Proof. If $u\in {\mathcal{C}}^{\alpha+\beta}_\mu\big(\mathbb T^d\big)$ is a solution, then by (2.21),

\begin{equation*} \int_0^\infty \tilde P^\mu_t f dt = - \int_0^\infty \tilde P^\mu_t \tilde{\mathcal A}_\mu u\, dt = - \int_0^\infty \frac{d}{dt} \tilde P^\mu_t u\, dt = u - \lim_{t\to\infty}\tilde P^\mu_t u = u. \end{equation*}

This yields the uniqueness. Thanks to Corollary 6, the existence follows from a standard Fredholm alternative argument ([Reference Gilbarg and Trudinger19, Section 5.3]).

In accordance with the terminology of periodic homogenization, we will refer to equation (2.22) as the cell problem.

5.1. Homogenization result

For there to exist a jump process with jump kernel (5.1), we need some assumptions [Reference Knopova, Kulik and Schilling27]:

• The function $\pmb\alpha\,:\,\mathbb R^d\to(0,2)$ is of class ${\mathcal{C}}^1$ , periodic of period 1, and satisfies that for all $x\in\mathbb R^d$ ,

  (5.2) \begin{equation} 0< \alpha \,:\!=\, \min_{x\in\mathbb R^d}\pmb\alpha(x) \le \bar\alpha \,:\!=\, \max_{x\in\mathbb R^d}\pmb\alpha(x)<2, \end{equation}
  where the minimum and maximum of $\pmb\alpha$ are attainable since $\pmb\alpha$ is continuous and periodic.

• The function $\rho\,:\, \mathbb R^d\to \mathcal M(S)$ is periodic of period 1 and symmetric, i.e., $\rho(x,\xi)=\rho(x,-\xi)$ for all $x\in\mathbb R^d$ and $\xi\in S$ ; it has a density, again denoted by $\rho$ , i.e, $\rho(x,d\xi) = \rho(x,\xi)d\xi$ ; and it satisfies the following conditions:

• – $\inf_{x\in\mathbb R^d} \left(\rho(x,S) \wedge \inf_{\theta\in S} \int_S (\theta\cdot\xi)^2 \rho(x,d\xi) \right) >0$ ;

• – $\rho$ is Lipschitz in the sense that there exists $C>0$ such that for all $x,y \in\mathbb R^d$ ,

  \begin{equation*} \left| \rho(x,S) - \rho(y,S) \right| + \mathcal W_1(\hat\rho(x,\cdot), \hat\rho(y,\cdot)) \le C|x-y|, \end{equation*}
  where $\hat\rho = (\rho(\cdot,S))^{-1}\rho$ is the normalized probability measure of $\rho$ and $\mathcal W_1$ is the Wasserstein-1 distance of probability measures (e.g., [Reference Villani39]);

• – $\rho$ is dominated by a probability function $\rho_0$ on S; that is, there exists a constant $C>0$ such that $\rho(x,\xi) \le C\rho_0(\xi)$ for all $x\in\mathbb R^d$ , $\xi\in S$ .

We still consider the operator $\mathcal A^\epsilon$ in (1.1), with coefficients as follows:

• $b=c\equiv0$ ; $J(z) = |z|^{-(d+\alpha)}$ is the density of a rotation-invariant $\alpha$ -stable Lévy measure with $\alpha$ given in (5.2);

• $\kappa^*$ is given by

  \begin{equation*} \kappa^*(x,z,u,v)\equiv \kappa^*(x,v) \,:\!=\, \rho(x,v/|v|) |v|^{\alpha-\pmb\alpha(x)}. \end{equation*}

The resulting function $\kappa(x,z,u)\equiv\kappa^*(x,u)$ does not satisfy either (2.1) or (2.2) in general. But (2.3) still holds with

$$\kappa_0(x,z,u)\equiv\kappa_0(x,z)\,:\!=\, \rho(x,z/|z|) \textbf 1_{\{\pmb\alpha(x)=\alpha\}},$$

and (2.4) holds trivially with $\tilde\kappa = \kappa = \kappa^*$ .

Note that because of the symmetry of $\rho$ , the indicator function $\textbf 1_{[1,2)}(\alpha)$ in (1.1) has no effect. The jump measure of $\mathcal A^\epsilon$ is of the form (5.1) with $\pmb\alpha$ and $\rho$ replaced by

(5.3) \begin{equation} \pmb\alpha_\epsilon(x) \,:\!=\, \pmb\alpha(x/\epsilon), \quad \rho^\epsilon(x,\xi) \,:\!=\, \epsilon^{\pmb\alpha(x/\epsilon)-\alpha} \rho(x/\epsilon,\xi). \end{equation}

Since $\kappa$ and $\tilde\kappa$ coincide and the jump kernel $\rho$ is symmetric, we see that $\tilde{\mathcal A}^\epsilon \equiv \tilde{\mathcal A}$ for all $\epsilon$ (cf. (2.10) and (2.11)). Their jump measure is given by (5.1) with $\rho(x,d\xi) = \rho(x,\xi)d\xi$ .

The counterpart of Proposition 1 is the following, where the well-posedness is taken from [Reference Knopova, Kulik and Schilling27, Theorem 3.1] and the heat kernel estimate is adapted from [Reference Kolokoltsov28, Proposition 3.1, Theorem 5.1].

Proposition 3. Under the conditions listed above, for every $x\in\mathbb R^d$ , the martingale problems for $(\mathcal A^\epsilon,\delta_x)$ , $\epsilon>0$ , and $(\tilde{\mathcal A},\delta_x)$ have unique solutions $\mathbb P^\epsilon_x$ on $(\mathcal D,\mathcal B(\mathcal D))$ and $\tilde{\mathbb P}_x$ on $(\mathcal D_{\textrm{per}},\mathcal B(\mathcal D_{\textrm{per}}))$ , respectively. The coordinate processes $X^\epsilon$ and $\tilde X$ are respectively $\mathbb R^d$ - and $\mathbb T^d$ -valued Feller processes, starting from x. Moreover, $\tilde X$ has a jointly continuous transition probability density $\tilde p(t;\,x,y)$ satisfying that for every $T>0$ , there exist constants $0<C_1<1$ , $C_2, C_3>0$ , and $\delta\in(0,1)$ such that for all $t\in (0,T]$ and $x,y\in \mathbb T^d$ ,

(5.4) \begin{equation} \begin{split} \tilde p(t;\,x,y) &\ge \sum_{l\in\mathbb Z^d} \bigg\{ C_1 \left[t^{-d/\pmb\alpha(x)}\wedge \left(t |x-y+l|^{-(d+\pmb\alpha(x))}\right)\right] \left( 1- C_2 t^\gamma \right) \\ &\qquad\qquad - C_3 t^\delta \left[ 1 \wedge |x-y+l|^{-(d+\pmb\alpha(x))} \right] \bigg\} \vee 0, \end{split} \end{equation}

with any $0< \gamma < 1/(d+\pmb\alpha(x))$ and $0<\delta<1- \beta(d+\pmb\alpha(x))$ .

By (5.3), we have for all $\epsilon>0$ and ‘good’ test functions $f\,:\,\mathbb R^d\to\mathbb R$ that

(5.5) \begin{equation} \mathcal A^\epsilon f(x) = \epsilon^{-\alpha} \big(\tilde{\mathcal A} f_{1/\epsilon}\big)(x/\epsilon),\end{equation}

so that Lemma 1 holds with $\{X^\epsilon_t\}_{t\ge0} \stackrel{\mathtt d}{=} \big\{ \epsilon \tilde X_{t/\epsilon^{\alpha}} \big\}_{t\ge0}$ in this case. Proposition 2 and Lemma 2 hold trivially with $\tilde X^\epsilon \equiv \tilde X$ , $\tilde P^\epsilon_t \equiv \tilde P_t$ , and $\mu_\epsilon \equiv \mu$ . In particular, to prove the counterpart of Proposition 2, as indicated in its own proof, it suffices to show that there exists a $t_0>0$ such that $\tilde p(t_0;\,x,y)$ is bounded from below by a positive constant independent of $x, y\in\mathbb T^d$ . To this end, we choose $\gamma_0>0$ and $t_0 <1$ such that

\begin{equation*}\left\{ \begin{aligned} &\gamma_0< 1/(d+\bar\alpha), \quad t_0^{1/\alpha} \ge 1/\sqrt{2}, \\ &1- C_2 t_0^{\gamma_0} >0, \quad C_1 t_0^{-d/\bar\alpha} \big(1- C_2 t_0^{\gamma_0}\big) - C_3 >0. \end{aligned}\right.\end{equation*}

Since, for any $x,y\in\mathbb T^d = [0,1]^d$ , there is always an $l\in\mathbb Z^d$ such that $|x-y+l|\le 1/\sqrt{2} \le t_0^{1/\alpha} \le t_0^{1/\pmb\alpha(x)} <1$ , we obtain from (5.4) that

\begin{equation*} \tilde p(t;\,x,y) \ge C_1 t_0^{-d/\pmb\alpha(x)} \left( 1- C_2 t^{\gamma_0} \right) - C_3 t_0^\delta \ge C_1 t_0^{-d/\bar\alpha} \big(1- C_2 t_0^{\gamma_0}\big) - C_3,\end{equation*}

where the last quantity is positive and independent of x, y. This proves that Proposition 2 holds true for the case here. As a consequence, Corollary 1 also holds. Therefore, Part (i) of the proof of Theorem 1 can still proceed with no obstacles. In conclusion, we get the following homogenization result for stable-like processes, which recovers the result of [Reference Franke17, Theorem 1].

Theorem 2. Under the same assumptions as Proposition 3 , we have the following weak convergence on the space $\mathcal D(\mathbb R_+;\,\mathbb R^d)$ :

$$X^\epsilon \ \Rightarrow \bar X, \quad {as }\ \epsilon\to 0^+,$$

where the limit process $\bar X$ is a Lévy process with Lévy triplet $(0,0,\bar \nu)$ given by

\begin{equation*} \bar\nu(A) = \int_0^\infty \int_S \textbf 1_A(r\xi) \left( \int_{\mathbb T^d} \rho(x,\xi) \textbf 1_{\{\pmb\alpha(x)=\alpha\}} \mu(dx) \right) d\xi \frac{dr}{r^{1+\alpha}}, \quad A\in\mathcal B\big(\mathbb R^{d}\setminus \{0\}\big), \end{equation*}

with $\mu$ being the invariant probability measure of $\tilde X$ with jump measure (5.1).

Remark 4. In the special case that $\mu(\{x\in\mathbb T^d\,:\, \pmb\alpha(x)=\alpha\}) =0$ , the homogenized measure $\bar{\nu}$ is trivially zero, i.e., $X^\epsilon$ converges weakly to the constant zero process. In this sense the scaling (5.5) is strong. One might expect that it would give a non-trivial limit if we changed the scaling (5.5) to

\begin{equation*} \mathcal A^\epsilon f(x) = \big[ \epsilon^{-\pmb\alpha} \big(\tilde{\mathcal A} f_{1/\epsilon}\big) \big](x/\epsilon). \end{equation*}

But the latter cannot yield a scaling for the generated processes $X^\epsilon$ and $\tilde X$ like Lemma 1. Therefore, the functional convergence in Corollary 1 and thus the final homogenization result are unknown in this case.

5.2. Numerical simulations

In this subsection, we will present a numerical experiment to help visualize the homogenization result. Furthermore, for a jump particle in a periodic structure, a typical topic of interest in practical applications is the distribution of the first exit time at which the particle escapes a given domain. We will also give some visualizations for the empirical mean of the first exit time.

5.2.1. Numerical scheme.

Set the dimension $d=2$ , let

\begin{equation*} \begin{split} \pmb\alpha(x) &= 1+ \frac{1}{4}\bigg[ \textbf 1_{\big[0,\frac{3}{8}\big)}(x_1)\cos\left( \textstyle{\frac{8\pi}{3}} x_1 \right) + \textbf 1_{\big(\frac{5}{8},1\big]}(x_1)\cos\left( \textstyle{\frac{8\pi}{3}} (1-x_1) \right) -\textbf 1_{\big[\frac{3}{8},\frac{5}{8}\big]}(x_1) \\ &\qquad\qquad + \textbf 1_{\big[0,\frac{3}{8}\big)}(x_2)\cos\left( \textstyle{\frac{8\pi}{3}} x_2 \right) + \textbf 1_{\big(\frac{5}{8},1\big]}(x_2)\cos\left( \textstyle{\frac{8\pi}{3}} (1-x_2) \right) -\textbf 1_{\big[\frac{3}{8},\frac{5}{8}\big]}(x_2) \bigg] \end{split}\end{equation*}

for $x=(x_1,x_2)\in [0,1]^2$ , and let

\begin{equation*} \rho(x,d\xi) \equiv \rho(d\xi) \,:\!=\, \sum_{i=1}^4 \delta_{e_i}(d\xi), \quad \xi\in\mathbb S^1,\end{equation*}

where $e_1=(1,0)$ , $e_2=(0,1)$ , $e_3=({-}1,0)$ , $e_4=(0,-1)$ form canonical orthonormal bases for $\mathbb R^2$ . It is easy to verify that such $\pmb\alpha$ and $\rho$ satisfy all conditions listed at the beginning of the last subsection. The minimum of $\pmb\alpha$ is $\alpha = \frac{1}{2}$ . The spectral measure of the process $X^\epsilon$ is $\rho^\epsilon(x,d\xi) = \epsilon^{\alpha(x/\epsilon)-\frac{1}{2}} \rho(d\xi)$ , while the spectral measure of the limit process $\bar X$ is $\bar\rho(d\xi) = \mu\!\left(\pmb\alpha^{-1}\big(\frac{1}{2}\big)\right) \rho(d\xi)$ .

We use the method in [Reference Modarres and Nolan31] to simulate the ‘one-step’ stable random vectors, and then use the scheme developed in [Reference Böttcher6] to simulate the stable-like processes $X^\epsilon$ and $\tilde X$ by gluing all one-step stable random vectors together. The convergence of the latter scheme is proved in [Reference Böttcher and Schilling7]. Note that the distribution of each one-step stable random vector depends on the position of the previous step. As for the limit Lévy process $\bar X$ , the one-step vectors are independent of the previous positions.

In all path sampling, we always use time-step size $\Delta t=0.01$ . There are two ways to approximately compute $\mu\!\left(\pmb\alpha^{-1}\big(\frac{1}{2}\big)\right)$ , as mentioned in Remark 3(iii). We first generate 1000 sample paths of the process $\tilde X$ with 100 steps by the above-mentioned scheme, and count the number of samples at the final time step inside the set $\pmb\alpha^{-1}\big(\frac{1}{2}\big) = \big[\frac{3}{8},\frac{5}{8}\big]^2$ . Then we use a sample path with time length $T=100$ , and calculate the time-average $\frac{1}{t} \int_0^t \textbf 1_{\pmb\alpha^{-1}\big(\frac{1}{2}\big)} \big(\tilde X_s\big) ds$ for varying t. Figure 1 shows the results from using these two methods. In particular, it shows that when t is large, the two results are very close.

Figure 1.

Computations of $\mu\!\left(\pmb\alpha^{-1}\big(\frac{1}{2}\big)\right)$ . The horizontal coordinate indicates the number of steps $t/\Delta t$ , and the vertical coordinate indicates the time-average of times of a single path inside $\pmb\alpha^{-1}\big(\frac{1}{2}\big)$ , with time-step size $\Delta t=0.01$ . The dashed line shows the results of the Monte Carlo method, implemented by generating 1000 samples with 100 steps.

Figure 2 shows the sample paths on the plane for the processes $X^\epsilon$ with $\epsilon=10^{-1},10^{-2},10^{-3},10^{-4},10^{-5}$ and the limit process $\bar X$ . As we can see, when the scaling parameter $\epsilon$ gets smaller and smaller, the path of $X^\epsilon$ becomes more and more concentrated into small clusters.

Figure 2.

Sample paths for $X^\epsilon$ and $\bar X$ on the time interval [0, 10] with time-step size $0.01$ . The coordinates represent the particle positions in $\mathbb R^2$ .

5.2.2. Simulations of first exit time.

For $x\in\mathcal D$ and $r>0$ , define

\begin{align*} S_r(x) &\,:\!=\, \inf\{t\ge0\,:\, |x(t)|\ge r \text{ or } |x(t{-})|\ge r\}, \\ S_{r+}(x) &\,:\!=\, \inf\{t\ge0\,:\, |x(t)|> r \text{ or } |x(t{-})|> r\}, \\ V(x) &\,:\!=\, \{r>0\,:\, S_r(x)<S_{r+}(x)\}.\end{align*}

It is easy to see that

\begin{equation*} S_r(x) = \inf\Big\{t\ge0\,:\, \sup_{0\le s\le t}|x(s)|\ge r \Big\},\end{equation*}

which is exactly the first exit time for the path x to escape the ball of radius r. In order to simplify the notation as before, we define $X^0 \,:\!=\, \bar X$ . Using [Reference Jacod and Shiryaev25, Lemma VI.2.10], we know that for all $\epsilon\ge0$ and $\omega\in\Omega$ , $V(X^\epsilon(\omega))$ is an at most countable subset of $\mathbb R_+$ . It follows that each set

\begin{equation*} U^\epsilon = \{ r>0\,:\, \mathbb P(r\in V(X^\epsilon) ) = 0 \}\end{equation*}

has full measure in $\mathbb R_+$ , and thus we have the following result.

Lemma 5. The set $\cap_{\epsilon>0} U^\epsilon$ also has full measure in $\mathbb R_+$ .

Now for each $r\in \cap_{\epsilon>0} U^\epsilon$ , the mapping $X^\epsilon \mapsto S_r(X^\epsilon)$ is continuous for all $\epsilon\ge0$ , by virtue of [Reference Jacod and Shiryaev25, Proposition VI.2.11]. Hence, by the continuous mapping theorem (see, e.g., [Reference Ethier and Kurtz16, Corollary 3.1.9]), we have the following corollary, of which the second statement follows from [Reference Billingsley4, Theorem 25.12].

Corollary 3. For each $r\in \cap_{\epsilon>0} U^\epsilon$ , $S_r(X^\epsilon)\Rightarrow S_r(\bar X)$ as $\epsilon\to0^+$ . If in addition the family $\{S_r(X^\epsilon)\}_{\epsilon>0}$ is uniformly integrable, then $\mathbb E(S_r(X^\epsilon))\to \mathbb E(S_r(\bar X))$ .

We choose $r=\pi$ . Figure 3 shows the empirical mean of the first exit time to escape the ball of radius $\pi$ for the processes $X^\epsilon$ , $\epsilon=10^{-6},10^{-12},10^{-18},10^{-24},10^{-30}$ , and the limit process $\bar X$ . From this figure, we can see that as $\epsilon$ gets smaller, the convergence rate of the empirical mean with respect to the number of samples decreases.

Figure 3.

Empirical mean of the first exit time for $X^\epsilon$ and $\bar X$ . The horizontal and vertical coordinates indicate the number of test samples and the average to the present, respectively. The labeled points give the values of the empirical mean of 200 samples with time-step size 0.01.

Figure 4 shows the trend of the empirical mean of $S_\pi(X^{\epsilon})$ , $\epsilon=10^{-n}$ , with respect to n. It follows that the difference between the empirical mean of $S_\pi(X^{\epsilon})$ and that of $S_\pi(\bar X)$ is almost inversely proportional to n, so as to be proportional to $\frac{1}{\log(\epsilon^{-1})}$ . Even though this rate is not strict, we can still conclude from the figure that the convergence rate of the mean first exit time with respect to $\epsilon$ is very slow.

Figure 4.

Empirical mean of the first exit time for $X^{\epsilon}$ with $\epsilon = 10^{-n}$ , $n=1,2,\cdots,30$ . The horizontal coordinate indicates the parameter n, and the vertical coordinate indicates the empirical mean of $S_\pi(X^{\epsilon})$ . The dashed line, provided for reference, gives the empirical mean of $S_\pi(\bar X)$ . All empirical means are simulated with 200 samples with time-step size 0.01.

Therefore, the method of simulating the first exit time of a particle in a periodic structure by choosing a very small $\epsilon$ is quite expensive (in computational time) and not precise in general. The advantage of our homogenization result is that we can directly use the limit process we have just identified to study its distribution properties, instead of using approximations.