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Homogenization of Stable-like Feller Processes

We study the homogenization for a class of non-symmetric stable-like Feller processes, and demonstrate its usage in the numerical approximation of first exit time. The jump intensity involves periodic and aperiodic constituents, as well as oscillating and non-oscillating constituents. This means that the noise can come both from the underlying periodic medium and from external environments, and is allowed to have different scales. It turns out that the stable-like process converges in distribution, as the scaling parameter goes to zero, to a L\'evy process. As special cases of our result, some homogenization problems studied in previous works can be recovered. Moreover, we present some numerical experiments to illustrate our result by demonstrating that the first exit time of the original process can be approximated by that of the limit process.


Introduction
As a subclass of Markov processes, Feller processes possess lots of nice properties in both probabilistic and analytic aspects [1,7,14,23]. The generator of a Feller process is in general a nonlocal operator. It looks locally like the generator of a Lévy process, in the sense that it is given by a Lévy-Khintchine type representation with an x-dependent Lévy triplet (b(x), a(x), η(x, ·)). For this reason, Feller processes are sometimes called Lévy-type processes or jump-diffusions, their generators are called Lévy-type operators. Feller processes with no diffusion parts at all, i.e., a ≡ 0, are called pure jump processes. If, in addition, for every x, the jump measure η(x, ·) is a stable Lévy measure [29], the Feller process is called a stable-like process.
Homogenization problems arise from the study of porous media, composite materials and other physical and engineering systems [3,10,11]. Generally speaking, in a periodic structure, such as medium or material, the heterogeneities are relatively small compared to its global dimension. Thus, two scales characterize the motion of particles in the structure: the microscopic one describing the heterogeneities, and the macroscopic one describing the global behavior of particles. The aim of homogenization is precisely to give the macroscopic properties of the particles by taking into account the properties of the microscopic structure.
In this paper, we focus on the homogenization of stable-like processes, periodic in space and locally periodic in noise. Consider a pure jump process in a periodic medium. The drift b(x) and the jump measure η(x, ·) are periodic and have a small scale in the spatial variable x, due to heterogeneities. In mathematical formulation, the small scale is represented by a small parameter > 0. From the realistic point of view, the noise may occur not only from the underlying periodic medium, but also from external environments. So we assume the jump measure η(x, dz) have mixed scales in the noise variable z. These suggest that the generators of stable-like processes in periodic medium will take the following form: (1.1) Here and after, we denote by B the unit open ball in R d , and by S := ∂B the unit sphere. We use Einstein's convention that the repeated indices in a product will be summed automatically. Besides, the drift functions b, c : R d → R d are Borel measurable and periodic, J : R d \ {0} → [0, ∞] is an α-stable Lévy density with α ∈ (0, 2). The jump intensity κ : is a Borel measurable function, periodic in its first variable. The normal scale of κ in z corresponds to the noise constituent coming from external environments. Furthermore, we assume that there exists a function κ * : periodic in its first and third variables, such that κ(x, z, u) = κ * (x, z, u, u). In this case, the jump intensity κ(x/ , z, z/ ) in (1.1) is locally periodic in noise variable z. It means that the small noise scale can be decomposed into two constituents, corresponding to the periodic medium and the external environments respectively.
Under some regularity assumptions (see the next section), each Lévy-type operator A can generate a Feller process on R d , say X . Our aim is to identify the limit of X , when the scaling parameter goes to zero. It turns out that (see Theorem 3.2), the limit process X, in the sense of convergence in distribution, is a Lévy process with the following generator: whereκ is a homogenized jump intensity related to the function κ,c is a homogenized constant.
Homogenization of Feller processes with jumps has been investigated by a number of authors. To name a few, the paper [20] considered the one-dimensional pure-jump case, and [15] studied the homogenization of stable-like processes with variable order. See also [32] for a multi-dimensional generalization with diffusion terms involved. The paper [30] investigated the homogenization problem of a class of pure-jump Lévy processes using a pure analytical approach -Mosco convergence. Recently, the authors studied in [21] the periodic homogenization of SDEs with jump noise and corresponding nonlocal PDEs. Under the setting of the present paper, many homogenization problems in the literature mentioned above can be recovered. See Section 4 for comparisons.
The sequel of this paper is organized as follows. In the next section, we list our main assumptions, and prove some preliminary results such as the well-posedness of martingale problems, invariance and ergodicity, etc. Some technical results will be left into appendices, with no affect to the smoothness of reading. In Section 3, we prove our main result to identify the homogenization limit. Some examples of resolving the homogenization problems in previous works are presented in Section 4. Section 5 is devoted to some numerical experiments and an application to the first exit time.

General assumptions and preliminary results
In the section, we collect general assumptions and some results we need. The most crucial results are Corollary 2.6 and 2.8. The former allow us to pass the functional convergence in the main theorem in next section, while the latter is the well-posedness for a Poisson equation. Most proofs in this section are quite short. We put other auxiliary but technical results into Appendix.
Firstly, we list our assumptions on the functions b, c, κ (or κ * ) and J.
Assumption H1. The functions b, c are in the Hölder class C β for some β ∈ (0, 1), and they are periodic of period 1.
Assumption H2. The function (x, z, u, v) → κ * (x, z, u, v) is periodic of period 1 in x and u; there exist constants κ 1 , κ 2 , κ 3 > 0, β ∈ (0, 1), such that for all x, x 1 , x 2 ∈ R d and z, u ∈ R d \ {0}, There exists a function κ 0 : We assume also that there exists a function κ 1 : In the case α ∈ (1, 2) and b = 0, we assume further that for all z ∈ R d \ {0}, Assumption H3. Assume that the Lévy density J is positive homogeneous of degree −(d + α) for some α ∈ (0, 2), that is, 6) and is bounded between two positive constants on the unit sphere S, i.e., there exist constants j 1 , j 2 > 0, such that for all ξ ∈ S j 1 ≤ J(ξ) ≤ j 2 . (2.7) In the case α = 1, we assume additionally that for each x ∈ R d and r 1 , r 2 ∈ (0, ∞), We denote by C k (T d ) with integer k ≥ 0 the space of continuous functions on T d possessing derivatives of orders not greater than k, endowed with the norm f k := sup x∈T d |f (x)| + |α|≤k sup x∈T d |∇ α f (x)|. When k = 0, we simply denote by C(T d ) := C 0 (T d ). For a non-integer γ > 0, the Hölder space C γ (T d ) are defined as the subspaces of C γ consisting of functions whose γ -th order partial derivatives are locally Hölder Remark 2.1. (i). We shall always identify a periodic function on R d of period 1 with its restriction on the flat torus T d := R d /Z d . Then Assumption H1 amounts to saying that b, c ∈ C β (T d ).
(ii). The relationship (2.3) ensures that the function κ 0 (x, z, u) is also periodic in x and u and satisfies (2.1). The relationship (2.4) or (2.5) ensures that the function κ 1 (x, z) is periodic in x.
(iii). A typical example for assumptions (2.4) and (2.5) to hold is that κ(x, z, u) can be written as the quotient of two positive homogeneous functions in z. In the case that α ∈ (1, 2) and b = 0, the convergence (2.5) implies (2.4). In this case, there involves singularity in the drift coefficient 1 α−1 b, so that we need more regularities for κ to cancel the singularity in the drift, as we will see in the proof of Theorem 3.2.
We need some auxiliary operators and processes. In fact, we will rescale the operator A and its canonical process in a effective fashion. For this purpose, we define the following nonlocal operators for f ∈ C ∞ (T d ), the space of all smooth functions on the flat torus T d (i.e., smooth periodic functions of period 1), For notational simplicity, we shall allow the parameter to be zero so thatÃ 0 :=Ã. The periodicity and continuity of the function x → κ(x, z, u) and (2.1), (2.8) and (2.9) imply thatÃ , ≥ 0, are all well-defined unbounded operators on (C(T d ), · 0 ), whose domains contain C ∞ (T d ). Moreover, it is easy to verify by (2.6) that for each > 0, the operatorÃ is a rescaling of A in the sensẽ (2.10) Here and after, we denote f (x) := f (x/ ). Denote by D = D(R + ; R d ) (D per = D(R + ; T d )) the space of all R d -(T d -) valued càdlàg functions on R + , equipped with the Skorokhod topology. The following theorem tells us that the operators A ,Ã andÃ can generate Feller processes, and the state space of the Feller processes associated toÃ orÃ can be taken to be T d , which is a compact space. Meanwhile, the heat kernel estimates forÃ andÃ is crucial in proving the ergodicity of the associated processes. We also find the core forÃ andÃ, which we will use to show the convergence of the associated invariant measures in the sequel.
(i). For every > 0 and x ∈ R d , the martingale problem for (A , δ x ) has a unique solution P x on (D, B(D)). The coordinate process X is an R d -valued Feller process starting from x.
(ii). For every ≥ 0 and x ∈ T d , the martingale problem for (Ã , δ x ) has a unique solutionP x on (D per , B(D per )). The coordinate processX is a T d -valued Feller process starting from x with generator the closure of (Ã , C ∞ (T d )), and has a transition probability densityp (t; x, y), i.e.,P Proof. All the assertions for the case α ∈ (0, 1) follow from [ Therefore, we only need to prove the results for α ∈ (1, 2). The lower bound estimates ofp can be found in [9, Theorem 1.5], we also put them into Appendix for the reader's convenience, see Proposition A.2. The proofs of the rest parts are tedious, especially that C ∞ (T d ) is the core of the generators, and we leave them into Appendix, see Proposition A.4 and Corollary B.1.
Of course, each of the processes X ,X andX is defined on its own stochastic basis. However, by taking product of the probability spaces, it is always possible to assume that: X ,X andX , > 0, are all defined on the same probability space (Ω, F, P).
We also assume for simplicity that If we identify a periodic function on R d of period with its restriction on the -torus In view of this, the canonical process X can also be treated as a processes taking values on T d , via the quotient map R d → T d . We will use this treatment only in the rest of this section, the benefit is the following relation, 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 same finite-dimensional distributions (see, e.g., [7]).

Lemma 2.3. We have the following indentity in law
Proof. We derive the generator for the Feller process 1 X α t t≥0 . For f ∈ C ∞ (T d ), by (2.10), Therefore, the Feller semigroup associated to 1 X α t t≥0 is also generated by the closure of (Ã , C ∞ (T d )).
Denote by {P t } t≥0 (or {P t } t≥0 ) the Feller semigroup ofX (orX). LetX 0 =X andP 0 t =P t . Now utilizing the lower bound of the transition probability densityp (t; x, y), we obtain the following exponential ergodicity.
Proposition 2.4. For each ≥ 0, the Feller processX possesses a unique invariant probability distribution µ on T d . Moreover, there exist positive constants C and ρ such that for each periodic bounded Borel function Proof. The proof follows the lines of [3, Theorem 3.3.2]. Due to the Doeblin-type results in [3, Theorem 3.3.1], it is enough to ensure that for every t > 0, the map (x, y) →p (t; x, y) is bounded from below by a positive constant, which follows immediately from the transition density estimates in Proposition 2.2 and the compactness of the state space T d .
which converges to zero, uniformly in x, as → 0 + , by (2.4) and dominated convergence. Now by the Trotter-Kato approximation theorem (see [13,Theorem III.4 , uniformly for t in compact intervals. Now using Proposition 2.4 and Lemma 2.5, we can obtain a useful functional convergence theorem.
Corollary 2.6. Let f be a bounded Borel function on T d , Then for every t > 0, In particular, for every T > 0, Proof. We follow the lines of [26,Proposition 2.4].
and then by the Markov property, Hence, Using the exponential ergodicity, we can also obtain the well-posedness of the nonlocal Poisson equation.
This yields the uniqueness. Thanks to Corollary B.3, the existence follows from a standard argument of Fredholm alternative ([17, Section 5.3]).
According to the terminology of periodic homogenization, we will refer to equation (2.13) as the cell problem.

Homogenization result
In this section we will prove our homogenization result. Before that, some preparations are needed. Firstly, we need a convergence lemma for locally periodic function.
where p is the conjugate of p, i.e., 1 p + 1 p = 1. Then for every compact set K ⊂ R d , we have In the case that the function φ is separable, that is, φ is of the form φ(x, y) = f (x)g(y) with g periodic, the conclusions of the lemma can be found in [  The limit processX is a Lévy process starting from 0 with Lévy triplet (b, 0,ν) given by
Proof. (i). We first prove the theorem for the case that b ≡ 0 or α ∈ (0, 1]. By [7, Theorem 2.44], we know that the semimartingale characteristics of X relative to the truncation function 1 B are (B , 0, ν ), where By applying the functional central limit theorem in [22,Theorem VIII.2.17], we only need to show that for all t ∈ R + and every bounded continuous function f : R d → R vanishing in a neighbourhood of the origin, the following convergences hold in probability when → 0 + : Clearly, by Corollary 2.6 we have We also have the following convergence in probability when α ∈ (0, 1), where N is a Poisson random measure on R d × [0, ∞) × [0, ∞) with intensity measure J(z)dz × m × m and N is the associated compensated Poisson random measure. The convergence of I 1 is shown in (3.5). For I 2 , we derive in a similar way as (2.11), where the first term at right hand side converges to 0 uniformly in x as → 0 + , by assumption (2.5) and the following fact Using the same argument as in the proof of (3.3), we also have the following locally uniform convergence in t in probability, as → 0 + , For I 3 , we use Itô's isometry to get which goes to zero as → 0 + , where ω d−1 is the surface area of the unit sphere in R d . This implies that I 3 (t) converges to 0 locally uniformly in t in probability. Since the local uniform topology is stronger than the Skorokhod topology in the space D (see, for instance, [22, Proposition VI.1.17]), I 3 converges to 0 in Skorokhod topology in probability and thereby in distribution. Further, it is easy to verify that the semimartingale characteristics of I 4 + I 5 are (0, 0, ν ), whose convergence is proved in (3.4). Combining these convergence together and using the functional central limit theorem again, we get the results.
(ii). The generator of the limit processX, restricted to C ∞ (T d ), is

Examples and comparisons
In this section, we present some examples that cover several results in earlier papers.
Example 4.1 (Pure jump Lévy processes). In the special case that b = c ≡ 0 and κ * (x, z, u, v) ≡ κ * (u) which is a periodic function of period 1 and satisfies κ 1 ≤ κ * (u) ≤ κ 2 for all u, the homogenized constant isκ = T d κ * (u)du andb = 0. This is the case presented in [30,Remark 6]. Note that in that paper, the authors use a pure analytical approach -Mosco convergence -to identify the limit process.
Example 4.2 (SDEs with Lévy noise). Let L α = {L α t } t≥0 be a d-dimensional isotropic α-stable Lévy process on a filtered probability space (Ω, F, P, {F t } t≥0 ) given by where 1 < α < 2, N α is a Poisson random measure on (R d \ {0}) × R + with jump intensity measure ν α (dy) = dy |y| d+α ,Ñ α is the associated compensated Poisson random measure, that is,Ñ α (dy, ds) := N α (dy, ds) − ν α (dy)ds. Consider the following SDE where the functions b, c are all periodic of period 1, the function σ(x, y) is periodic in x of period 1, and odd in y in the sense that σ(x, −y) = −σ(x, y) for all x, y ∈ R d . We assume that σ ∈ C 1,2 (R d × R d ) and there exists constants C 1 > 0, C 2 > 1, such that for all x 1 , x 2 , x, y ∈ R d , Assume in addition that for every x, σ(x, ·) is uniformly continuous and is C 2 -diffeomorphism with inverse τ (x, ·) := σ(x, ·) −1 . Then we know that (4.1) possesses a unique strong solution which is a Feller process, for each > 0, see [21, Theorem 4.2, Corollary 4.3]. Now the generator of the solution processes X x, restricted to C ∞ (T d ) is We can rewrite it, by a change of variables and the oddness of y → σ(x, y), to the form in (1.1) with We assume further that (cf. [21, Assumption H5]), 1 σ(x, y) → ∇ y σ(x, 0) · y, uniformly in x and y, as → 0 + , Then we can prove easily (e.g., by [28,Theorem 7.17]) that for each z, Hence, we conclude that the function κ defined in (4.2) satisfies assumption (2.5) with.
Applying Theorem 3.2, we know that the sequence of solutions X x, converges in distribution to a Lévy processX x starting from x with Lévy triplet (b, 0,ν) given in (3.2). By the symmetry of κ and ν α , the homogenized constantb = T d (I + ∇b(x)) · c(x)µ(dx), where µ is the invariant measure of the Feller process generated bỹ andb is the unique solution to the Poisson equationÃ αb = b. Moreover, the homogenized function is In particular, this also generalizes the result in [16], where the author consider the special case σ(x, y) = σ 0 (x)y.
is periodic of period 1 in the first variable. We refer to the function ρ(x, ξ) as the spectral measure. In general, we do not assume the function ρ to be symmetric in the second variable in the sense that ρ(x, ξ) = ρ(x, −ξ) for all x and ξ, while this is assumed in [15]. Note that for each x ∈ R d , η(x, ·) is a stable Lévy measure with stability index α(x). Consider the following pseudo-differential operator defined for f ∈ C ∞ (T d ), To write the operator A η to the form (1.1), let α 0 := inf x∈R d α(x) (the infimum is attainable since α is continuous and periodic), and define where J 0 is a Lévy density related to α 0 in the manner of Assumption H3. Then for We assume that the function κ * in (4.6) satisfies assumptions (2.1) and (2.2), so that the operator A η generates a Feller process X on T d . This process is called stable-like, since locally it looks like a stable processes (see [15] and the literatures therein). It is easy to check that for each x and z,

Thus, assumption (2.3) holds by taking
while assumptions (2.4) and (2.5) hold trivially with κ 1 = κ. Let η (x, dz) := κ * (x/ , z/ )J 0 (z)dz. Then η is of the form (4.4) with α and ρ replace by Let A η be the operator associated to η as in (4.5), denote by X the Feller process generated by A η . Then Theorem 3.2 implies that the processes X converge in distribution to a Lévy processX with Lévy triplet (b, 0,ν), where for A ∈ B(R d \ {0}), the measure µ is the invariant measure of the Feller process generated bỹ In the special case that ρ(x, ·) is symmetric for each x, we haveÃ η = A η ,b = 0, and the result coincides with that in [15, Theorem 1]. Last but not least, the limit Lévy processX is α 0 -stable, its spectral density isρ Example 4.4 (One-dimensional stable-like Feller processes). Consider the one-dimensional case with α ∈ (1, 2), c ≡ 0 and κ * (x, z, u, v) ≡ κ * (x, v), that is, Here J is the α-stable Lévy density on R \ {0} (see [29,Remark 14.4]), that is, with constants j + , j − > 0, so that assumption (2.7) is fulfilled. Besides assumptions (2.1), (2.2), H1 and H4, we assume further that there exists two functions κ + 0 , κ − 0 : Note that this is the type of assumption in [20]. Then by L'Hôpital's rule, we have Thus, our assumption (2.3) is fulfilled by letting And assumption (2.4) hold trivially. Now using Theorem 3.2, the Feller process generated by A 1d converges in distribution, as → 0 + , to a one-dimensional α-stable Lévy processX with Lévy triplet (b, 0,ν) in (3.2). Let µ be the invariant measure of the Feller process generated bỹ Then the homogenized driftb is whereb is the unique solution to the Poisson equationÃ 1db = b. Define two constantsκ ± : Note that the authors in [20] consider the operators of the formÃ 1d with κ * ( x , z ) and 1 α−1 b( x ) in place of κ * (x, z) and b(x), which is slightly different from A 1d , but the homogenized jump measure therein coincides withν.

Numerical simulations
In this section, we will present a numerical experiment to give some visualization for the homogenization result. Besides, for a jump particle in a periodic structure, a typical question in practical applications is the distribution property of the first exit time for the particle to escape a given domain. We will also give some visualization for the empirical mean of the first exit time. For simplicity, we will use the stable-like processes X with variable order in Example 4.3 for instance.

Numerical scheme
Set the dimension d = 2, and where e 1 = (1, 0), e 2 = (0, 1), e 3 = (−1, 0), e 1 = (0, −1), they form an orthonormal basis for R 2 . Then the minimum of α is α 0 = 1 2 , and ρ is symmetric in ξ. The spectral measure of the process X is ρ (x, dξ) = α(x/ )− 1 2 ρ(dξ), while the spectral measure of the limit processX isρ(dξ) = µ α −1 ( 1 2 ) ρ(dξ). We use the method in [25] to simulate the 'one-step' stable random vectors, and then use the method in [6] to simulate each Feller process X by gluing all 'one-step' stable random vectors together. Note that the distribution of each 'one-step' stable random vector depends on the position of the previous step. As for the limit Lévy processX, the 'one-step' stable random vectors are independent of the previous positions. The following figure (Fig. 1) shows the sample paths on the plane for the processes X with = 10 −1 , 10 −2 , 10 −3 , 10 −4 , 10 −5 and the limit processX. As we can see, when the scaling parameter gets smaller and smaller, the path of X is getting more and more concentrated into some small clusters.

Simulations of first exit time
For x ∈ D and r > 0, define It is easy to see that which is exact the first exit time for the path x to escape the ball of radius r. In order to simplify the notation as before, we denote X 0 :=X. Using [22,Lemma VI.2.10], we know that for all ≥ 0 and ω ∈ Ω, V (X (ω)) is an at most countable subset of R + . It follows that each set U = {r > 0 : P(r ∈ V (X )) = 0}, has full measure in R + , and thus we have Lemma 5.1. The set ∩ >0 U also has full measure in R + . Now for each r ∈ ∩ >0 U , the mapping X → S r (X ) is continuous for all ≥ 0, by virtue of [22,Proposition VI.2.11]. Hence, by the continuous mapping theorem (see, e.g., [14, Corollary 3.1.9]), we have the following corollary, of which the second statement follows from [4,Theorem 25.12].
We choose r = π. The following figure (Fig. 2) shows the empirical mean of the first exit time to escape the ball of radius π for the processes X , = 10 −6 , 10 −12 , 10 −18 , 10 −24 , 10 −30 , and the limit processX. Through this figure, we can see that as gets smaller, the convergence rate of the empirical mean with respect to the number of samples is getting slower.
The next figure (Fig. 3) shows the trend of the empirical mean of S π (X ), = 10 −n , with respect to n. It follows that the difference of empirical mean of S π (X ) with that of S π (X) is almost inversely proportional to n, so as to be proportional to 1 log( −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 is very slow.
Therefore, the way of simulating the first exit time of a particle in a periodic structure by choosing a very small is quite expensive (in computational time) and not precise in general. The advantage of our homogenization result is that we can use directly the limit process we have just identified to study its distribution properties, instead of using approximations.

Appendix A Properties of the semigroups and generators
As we have seen in Section 2, we need the Feller nature of the semigroups and the properties of the generators, in order to study the ergodicity of the canonical Feller processes, see Proposition 2.2 and Proposition 2.4. We devote this section to investigate the semigroups and generators. As corollaries, we can also obtain the solvability of the Possion equations with zeroth-order terms and the generalized Itô's formula, which are used in Corollary 2.8 and the proof of our main result Theorem 3.2.
We consider the following operator with η(x, dz) := κ (x, z)J(z)dz. We suppose that the vector field b : R d → R d is in the Hölder class C β with some β ∈ (0, 1) and periodic of period 1, J satisfies (2.9), that is,  with α ∈ (1, 2), κ (x, z) is periodic in x of period 1 and satisfies similar conditions as (2.1) and (2.2), that is, for all x, x 1 , x 2 , z ∈ R d , Note that the operatorsÃ, A andÃ are all of the form (A.1) by choosing appropriate κ . It is easy to verify that L b,η f ∈ C(T d ) for each f ∈ C 1+γ (T d ) with 1 + γ > α. Now we treat L b,η as a perturbation of L η := L 0,η by the gradient operator L b := L b,0 = b · ∇, and follow [5,8,18] to investigate the heat kernel for L b,η .
We introduce the following functions on (0, ∞) × R d for later use: For abbreviation we write c 0 for the set of constants (d, α, β, κ 1 , κ 2 , κ 3 , j 1 , j 2 ). Before investigating the semigroups generated by L b,η , we need some facts for the heat kernels of L η and L b,η . By virtue of the periodicity assumptions on the coefficients, we can choose the underlying space to be is the fundamental solution of L η , then it must be periodic in x due to the periodicity of all coefficients in L η . Now we define q η (t; x, y) := l∈Z d q η (t; x, y + l), then q η is periodic in both x and y. Therefore, we can treat q η as a function from [0, ∞) × T d × T d to R, which is exactly the fundamental solution of L η on the state space T d . The same arguments hold for the operator L b,η . Keeping these in mind, the following facts for the operator L η are adapted from [ Proposition A.1. (i). The fundamental solution q η (t; x, y) : [0, ∞) × T d × T d → R of L η has the following properties: for all (t, y) ∈ (0, ∞) × T d , the function x → q η (t; x, y) is differentiable and the derivative ∇ x q η (t; x, y) is jointly continuous on (0, ∞) × T d × T d ; the integral in L η x q η (t; x, y) is absolutely integrable and the function L η x q η (t; x, y) is jointly continuous on (0, ∞) × T d × T d . For every T > 0, there exist a constant C = C(c 0 , T ) > 0, such that for all t ∈ (0, T ] and x, y ∈ T d , . Define a family of operators by then {T η t } t≥0 forms a Feller semigroup on the Banach space (C(T d ), · 0 ) with generator the closure of (L η , C ∞ (T d )). The domain of the generator contains C 1+γ (T d ) with 1 + γ > α, on which the restriction of the generator is L η .
Note that the joint continuity of ∇ x q η (t; x, y) is not mentioned explicitly in the previous references, but it is a consequence of [18, Lemma 3.1, Lemma 3.5, Theorem 3.7, Lemma 3.10, Eq. (59)]. In addition, it is only shown in the above reference that C 2 (T d ) is contained in the domain of the generator, but we can easily generalize to our case, using the same argument as the proofs of [18, Theorem 1.3.(3a), Proposition 4.9] and the fact that L η f ∈ C(T d ) for each f ∈ C 1+γ (T d ) with 1 + γ > α.
For notational simplicity, the summation over the lattice Z d will be omitted in all coming results. Keep in mind that there will be a summation over Z d when the letter l involved in the expression without ambiguity. The following facts for the heat kernel of L b,η are adapted from [9, Theorem 1.5].
Moreover, q b,η enjoys the following properties.
Define a family of operators T b,η t f = T d q b,η (t; ·, y)f (y)dy, f ∈ C(T d ). (A.9) By Proposition A.2, {T b,η t } t≥0 forms a (one-parameter operator) semigroup which is Markovian (positivity preserving, conservative and sub-Markovian) and Feller (each T b,η t map C(T d ) to C(T d )). We can also prove the strong continuity. Hence, we have This givesL b,η =L b,η and we complete the whole proof.