Periodic Homogenization of a L\'evy-Type Process with Small Jumps

In this article, we consider the problem of periodic homogenization of a Feller process generated by a pseudo-differential operator, the so-called L\'evy-type process. Under the assumptions that the generator has rapidly periodically oscillating coefficients, and that it admits"small jumps"only (that is, the jump kernel has finite second moment), we prove that the appropriately centered and scaled process converges weakly to a Brownian motion with covariance matrix given in terms of the coefficients of the generator. The presented results generalize the classical and well-known results related to periodic homogenization of a diffusion process.


Introduction
The classical reaction-diffusion equation describes the evolution of population density due to random displacement of individuals (diffusion term), movement of individuals within the environment (drift term), and their reproduction (reaction term). In order to characterize long-range effects the diffusion and drift terms are naturally replaced by an integro-differential operator of the following form where ν(x, dy) is a non-negative Borel kernel which describes these effects, that is, it quantifies the property that an individual at x jumps to x + dy. The main goal of this article is to discuss periodic homogenization of the operator L, with kernel ν(x, dy) admitting "small jumps" only (that is, having finite second moment). Our approach is based on probabilistic techniques. More precisely, we discuss periodic homogenization of the stochastic (Markov) process {X t } t≥0 in periodic medium, generated by L. We focus to the case when {X t } t≥0 is a so-called Lévy-type process or, equivalently, when L is a pseudo-differential operator (see below for details). Roughly speaking, we show that the appropriately centered and scaled process {X t } t≥0 : for someb * ∈ R d , converges, as ε → 0, in the path space endowed with the Skorohod J 1 -topology to a d-dimensional zero-drift Brownian motion determined by covariance matrix of the form 1≤i,j≤d , (see Theorem 1.4 for details). Equivalently, according to [13,Theorem 7.1], Let us remark that when b(x) ≡ 0 and ν(x, dy) is symmetric for all x ∈ R d , centralization in eq. (1.2) is not necessary (that is, one can takeb * = 0), and β(x) ≡ 0 in eq. (1.3). Thus, in this case, Σ is reduced to 1≤i,j≤d , (see [66] for more details).
Preliminaries on Lévy-Type Processes. Let (Ω, F , {P x } x∈R d , {F t } t≥0 , {θ t } t≥0 , {X t } t≥0 ), denoted by {X t } t≥0 in the sequel, be a Markov process on state space (R d , B(R d )) (see [12]). Here, d ≥ 1, and B(R d ) denotes the Borel σ-algebra on R d . Due to the Markov property, the associated family of linear operators {P t } t≥0 on B b (R d ) (the space of bounded and Borel measurable functions), defined by forms a semigroup on the Banach space (B b (R d ), · ∞ ), that is, P 0 = Id and P s • P t = P s+t for all s, t ≥ 0. Here, E x stands for the expectation with respect to P x (dω), x ∈ R d , and · ∞ and Id denote the supremum norm and the identity operator, respectively, on the space B b (R d ). Moreover, the semigroup {P t } t≥0 is contractive ( P t f ∞ ≤ f ∞ for all t ≥ 0 and f ∈ B b (R d )) and positivity preserving (P t f ≥ 0 for all t ≥ 0 and f ∈ B b (R d ) satisfying f ≥ 0). The infinitesimal generator (A b , D A b ) of the semigroup {P t } t≥0 (or of the process {X t } t≥0 ) is a linear operator We call (A b , D A b ) the B b -generator for short. A Markov process {X t } t≥0 is said to be a Feller process if its corresponding semigroup {P t } t≥0 forms a Feller semigroup. This means that (i) {P t } t≥0 enjoys the Feller property, that is, P t (C ∞ (R d )) ⊆ C ∞ (R d ) for all t ≥ 0; (ii) {P t } t≥0 is strongly continuous, that is, lim t→0 P t f − f ∞ = 0 for all f ∈ C ∞ (R d ).
Here, C ∞ (R d ) denotes the space of continuous functions vanishing at infinity. Recall also that a Markov process {X t } t≥0 is said to be a C b -Feller (resp. strong Feller ) process if the corresponding semigroup {P t } t≥0 satisfies P t f ∈ C b (R d ) for all t > 0 and all then, according to [23,Theorem 3.4], is a pseudo-differential operator, that is, it can be written in the form dx denotes the Fourier transform of the function f (x). The function q : R d × R d → C is called the symbol of the pseudodifferential operator. It is measurable and locally bounded in (x, ξ), and is continuous and negative definite as a function of ξ. Hence, by [40,Theorem 3.7.7], the function ξ → q(x, ξ) has for each x ∈ R d the following Lévy-Khintchine representation (or of q(x, ξ)). Let us remark that local boundedness of q(x, ξ) implies local boundedness of the corresponding x-coefficients, and vice versa (see [69, Lemma 2.1 and Remark 2.2]). In the sequel, we assume the following condition on the symbol q(x, ξ): This condition is closely related to the conservativeness property of {X t } t≥0 . Namely, under the assumption that the x-coefficients of q(x, ξ) are uniformly bounded (which is certainly the case in the periodic setting), (LTP2) implies that {X t } t≥0 is conservative, that is, P x (X t ∈ R d ) = 1 for all t ≥ 0 and x ∈ R d (see [68,Theorem 5.2]). Further, note that by combining eqs. (1.4) and (1. is a linear operator of the form eq. (1.1) satisfying the so-called positive maximum principle (Lf (x 0 ) ≤ 0 for for some (or all) λ > 0, then, according to the Hille-Yosida-Ray theorem, L is closable and the closure is the generator of a Feller semigroup. In particular, the corresponding Feller process is a Lévy-type process. In the case when q(x, ξ) does not depend on the variable x ∈ R d , {X t } t≥0 becomes a Lévy process, that is, a stochastic process with stationary and independent increments. Moreover, unlike Feller processes, every Lévy process is uniquely and completely characterized through its corresponding symbol (see [67,Theorems 7.10 and 8.1] and [13,Example 2.26]). According to this, it is not hard to check that every conservative Lévy process satisfies conditions (LTP1) and (LTP2) (see [67,Theorem 31.5]). Thus, the class of processes we consider in this article contains Lévy processes. Throughout this article, the symbol {X t } t≥0 denotes a Feller process satisfying conditions (LTP1) and (LTP2). Such a process is called a Lévy-type process (LTP). If ν(x, dy) ≡ 0, {X t } t≥0 is called a diffusion process. Note that this definition agrees with the standard definition of (Feller-Dynkin) diffusions (see [64, Chapter III.2]). A typical example of a LTP is a solution to the following SDE where Φ : R d → R d×n is locally Lipschitz continuous and bounded (which is not a restriction in the periodic setting), and {Y t } t≥0 is an n-dimensional Lévy process with symbol q Y (ξ). Namely, in [70, Here, for a matrix M, M ′ denotes its transpose. Observe that the following SDE is a special case of eq. (1.6), are locally Lipschitz continuous and bounded, {B t } t≥0 is a p-dimensional Brownian motion, and {Z t } t≥0 is a q-dimensional pure-jump Lévy process (that is, a Lévy process determined by a Lévy triplet of the form (0, 0, ν Z (dy))). Namely, set Φ(x) = Φ 1 (x), Φ 2 (x), Φ 3 (x) for any x ∈ R d , and Y t = (t, B t , Z t ) ′ for t ≥ 0. For more on Lévy-type processes we refer the readers to the monograph [13].
LTPs with Periodic Coefficients. Let τ = (τ 1 , . . . , τ d ) ∈ (0, ∞) d be fixed, and let τ Z d : and for x ∈ R d define In the sequel, we denote Clearly, every τ -periodic function f (x) is completely and uniquely determined by its , and since f | [0,τ ] (x) assumes the same value on opposite faces of [0, τ ], it can be identified by a function f τ : . For notational convenience, we will often omit the subscript τ and simply write x instead of x τ , and f (x) instead of f τ (x).
Let now {X t } t≥0 be a LTP with semigroup {P t } t≥0 , symbol q(x, ξ) and Lévy triplet (b(x), c(x), ν(x, dy)), satisfying: Directly from the Lévy-Khintchine formula it follows that (C1) is equivalent to the τ -periodicity of the corresponding Lévy triplet (b(x), c(x), ν(x, dy)), which in turn is equivalent to the τ -periodicity of x → P x (X t − x ∈ dy) (see [66,Section 4]). This immediately implies that {P t } t≥0 preserves the class of all bounded Borel measurable τ -periodic functions, that is, the function x → P t f (x) is τ -periodic for all t ≥ 0 and all τ -periodic f ∈ B b (R d ). Now, together with this, a straightforward adaptation of [48, Proposition 3.8.3] entails that {Π τ (X t )} t≥0 is a Markov process on (T d τ , B(T d τ )) with positivity preserving contraction semigroup {P τ t } t≥0 (on the space (B b (T d τ ), · ∞ )) given by Here, B(T d τ ) stands for the Borel σ-algebra on T d τ (with respect to the standard quotient topology), B b (T d τ ) denotes the class of all bounded Borel measurable functions f : T d τ → R (which can be identified with the class of all τ -periodic bounded Borel measurable functions f : R d → R), and with z x being an arbitrary point in Π −1 τ ({x}). Further, assume that Proposition 1.1. The process {Π τ (X t )} t≥0 admits a unique invariant probability measure π(dx), that is, a measure π(dx) satisfying for some γ, Γ > 0, where · TV denotes the total variation norm on the space of signed measures on B(T d τ Conditions (in terms of the Lévy triplet (b(x), c(x), ν(x, dy))) ensuring (C2) are discussed in Section 3.
The Semimartingale Nature of LTPs. As we have already commented, the problem of homogenization of an operator with the form eq. (1.1) corresponding to a LTP is equivalent to the convergence of the corresponding family of LTPs in the path space endowed with the Skorohod J 1 topology (see [13,Theorem 7.1]). According to [69,Lemma 3.2], {X t } t≥0 is a P x -semimartingale (with respect to the natural filtration) for any x ∈ R d . Therefore, in order to show this convergence, our aim is to employ [42, Theorem VIII.2.17] which states that a sequence of semimartingales converges in the path space endowed with the Skorohod J 1 topology to a process with independent increments if the corresponding semimartingale characteristics converge in probability.
Let us now recall the notion of characteristics of a semimartingale (see [42]). Let where the process {∆S t } t≥0 is defined by ∆S t := S t − S t− and ∆S 0 := S 0 . The process {S(h) t } t≥0 is a special semimartingale, that is, it admits a unique decomposition where {M(h) t } t≥0 is a local martingale, and {B(h) t } t≥0 is a predictable process of bounded variation.
be a semimartingale, and let h : R d −→ R d be a truncation function. Furthermore, let {B(h) t } t≥0 be the predictable process of bounded variation appearing in eq. (1.9), let N(ω, dy, ds) be the compensator of the jump measure µ(ω, dy, ds) := Now, according to [69,Theorem 3.5] and [42,Proposition II.2.17] we see that the (modified) characteristics of a LTP {X t } t≥0 (with respect to a truncation function h(x)) are given by for t ≥ 0 and i, j = 1, . . . , d.
In the sequel, we assume that {X t } t≥0 admits "small jumps" only, that is, As a direct consequence of (C3) and [42, Proposition II.2.29] we see that {X t } t≥0 itself is a special semimartingale, and for the truncation function we can take h(x) = x. In particular, if ν(x, dy) is also symmetric for every x ∈ R d , the first characteristic is a solution to the following stochastic differential equation for any x ∈ R d ,ν(dz) is any given σ-finite non-finite and non-atomic measure on B(R), and k : and ν x, dy =ν {z ∈ R : k(x, z) ∈ dy} . Thus, due to this and (C3) we have that (1.10) From this equation we also read the unique special semimartingale decomposition Main Result. Before stating the main result of this article, we introduce some notation we need. Denote by . . } the space of k times differentiable functions such that all derivatives up to order k are bounded. This space is a Banach space endowed with the norm f k : whereB r (x) stands for the (topologically) closed ball of radius r around x ∈ R d . Also, let According to [11,Theorem 2 This space is called a generalized Hölder space, and it is a normed vector space with the norm (see [5]). Observe that the product of two Hölder exponents is a Hölder exponent, and that if m ψ ∈ (k, k In particular, when ψ(r) = r γ for some γ > 0, C ψ b (R d ) becomes the classical Hölder space of order γ (usually denoted by C γ b (R d )), which is a Banach space together with the above-defined norm (which we denote by · γ ). Since f ↔ f τ gives a one-to-one correspondence between {f : R d → R : f is τ -periodic} and {f τ : T d τ → R}, in an analogous way we define C k (T d τ ) and C ψ (T d τ ). We are now in position to state the main result of this article, the proof of which is given in Section 2.
Then, (a) the Poisson equation is the unique solution in the class of continuous and periodic solutions to eq. (1.12) satisfying (b) in any of the following three cases for any initial distribution of {X t } t≥0 , ⇒ denotes the convergence in the space of càdlàg functions endowed with the Skorohod J 1 -topology, and {W t } t≥0 is a d-dimensional zero-drift Brownian motion determined by covariance matrix Σ given in eq. (1.3).
Under (C1), (C2) and the assumption that b * ∈ C b (R d ), in Lemma 2.1 below we show that eq. (1.12) admits a τ -periodic solution β ∈ C b (R d ) (which is also unique in the class of continuous τ -periodic solutions satisfying T d τ β τ (x) π(dx) = 0). However, we require additional smoothness of β(x) in order to apply Itô's formula in the proof of Theorem 1.4 (see Section 2 for details). This additional regularity is given through (C4) (together with (C1) and (C2)). Namely, under these assumptions, we show is symmetric for all x ∈ R d , as already commented, β(x) ≡ 0 and the assertion of the theorem follows without assuming (C4). In the pure-jump case (that is, when c(x) ≡ 0), eq. (1.13) suggests that the Hölder exponent ϕ(r) depends on the behavior of ν(x, dy) on B 1 (0). For example, when for some 0 < κ ≤ κ < ∞, eq. (1.13) trivially holds true. Thus, we only require that β ∈ C ϕψ b (R d ) for some Hölder exponent ψ(r) with m ϕψ > 1. Analogously, if eq. (1.14) holds true, then we only require that β ∈ C 1 b (R d ). Observe that in the pure-jump case we do not require explicitly that β ∈ C 2 b (R d ). Under eq. (1.13) (resp. eq. (1.14)), in Proposition 2.2 we show that when m ϕψ > 1 (resp. β ∈ C 1 b (R d )) we can apply Itô's formula to the process {β(X t )} t≥0 .
Literature Review. Our work relates to the active research on homogenization of integro-differential operators, and Markov processes with jumps. The work is highly motivated by the results in [9,10,66] where, by employing probabilistic techniques, the authors considered periodic homogenization of the operator L with ν(x, dy) ≡ 0 (that is, second-order elliptic operator in non-divergence form), and L with b(x) ≡ 0 and ν(x, dy) being symmetric for all x ∈ R d (that is, integro-differential operator in the balanced form), respectively. In this article, we generalize both results by including the non-local part of the operator L, as well as non-symmetries caused by the drift term b(x) and the Lévy kernel ν(x, dy). In a closely related work [61], by using analytic techniques (the corrector method), the authors discuss periodic homogenization of the operator L with a convolution-type Lévy kernel, that is, L is determined by with λ(x) and µ(x) being measurable, τ -periodic and such that 0 < κ ≤ λ(x), µ(x) ≤ κ < ∞ for all x ∈ R d , and a(y) ≥ 0 being measurable and such that 0 < R d (1 ∨ |y| 2 )a(y) dy < ∞ and a(y) = a(−y) for all y ∈ R d . The homogenized operator is again a second-order elliptic operator with constant coefficients. Observe that this case is not covered by Theorem 1.4 since finiteness of ν(x, dy) excludes regularity properties of the corresponding semigroup assumed in (C4).
There is a vast literature on homogenization of differential operators, mostly based on PDE methods. We refer the interested readers to [9,15,22,43,80] and the references therein. Results related to the problem of periodic homogenization of non-local operators (based on probabilistic techniques) were obtained in [30,31,32,34,36,37,38,59]. In all this works the focus is on the so-called stable-like operators (possibly with variable order), that is, on the case when ν(x, dy) admits "large jumps" of power-type only: for some 0 < κ ≤ κ < ∞ and 0 < α ≤ α < 2. In this case, by using subdiffusive scaling (which depends on the behavior of ν(x, dy) on B c 1 (0)), the homogenized operator is the infinitesimal generator of a stable Lévy process with the index of stability being equal to the power of the scaling factor. The problem of stochastic homogenization (that is, homogenization of operators with random coefficients) of this type of operators has been considered in [63]. PDE and other analytical approaches to the problem of periodic homogenization and stochastic homogenization of stable-like operators can be found in [1,2,3,6,14,28,29,44,72,74,75].

Proof of Theorem 1.4
Throughout this section we assume that {X t } t≥0 is a d-dimensional LTP with semigroup {P t } t≥0 , symbol q(x, ξ) and Lévy triplet (b(x), c(x), ν(x, dy)), satisfying (C1)-(C4). A crucial step in the proof is an application of Itô's formula. In order to justify this step, we first discuss regularity of a solution to the Poisson equation eq. (1.12).
Solution to the Poisson Equation eq. (1.12). Observe first that for any In particular, Therefore, the zero-resolvent is well defined, and According to [68,Corollary 3.4], is a Feller process. Denote the corresponding Feller generator by (A ∞ τ , D A ∞ τ ). Clearly, for any f τ ∈ D A ∞ τ (which is by definition continuous), and its τ -periodic extension f (x), it holds that f ∈ D According to the argument above, we immediately get the following.
Observe that in Lemma 2.1 we only used the fact that b * τ ∈ C(T d τ ). In the sequel, we discuss additional smoothness of β(x).
and that there is a Hölder exponent φ(r) with m φ > 1 such that Then, it holds that dy). In addition, if eq. (1.14) holds true, then the above relation holds for any f ∈ C 1 b (R d ). Proof. Without loss of generality, assume that m φ ∈ (1, 2]. We follow the proof of [62,Lemma 4.2]. Let f ∈ C φ b (R d ), and let χ ∈ C ∞ c (R d ), 0 ≤ χ ≤ 1, be such that R d χ(x) dx = 1. For n ∈ N define χ n (x) := n d χ(nx), and f n (x) := (χ n * f )(x), where * stands for the standard convolution operator. Clearly, {f n } n∈N ⊂ C ∞ b (R d ), f n φ ≤ f φ for n ∈ N, lim n→∞ f n (x) = f (x) and lim n→∞ ∇f n (x) = ∇f (x) for all x ∈ R d . Next, by employing Itô's formula and eq. (1.10) we have that Now, by letting n → ∞ we see that the left hand-side converges to f (X t ), and the first two terms on the right-hand side (for the second term we employ the dominated convergence theorem) converge to f (X 0 ) and t 0 ∇f (X s− ), b * (X s− ) ds, respectively. Further, Taylor's theorem together with the fact that m φ > 1 implies . Observe that according to eq. (2.2) and (C3), Thus, the dominated convergence theorem implies that the last term converges to Finally, by employing the isometry formula, we have , the dominated convergence theorem implies that, possibly passing to a subsequence, the third term on the left-hand side in eq. (2.3) converges to which proves the desired result. Finally, by following the above arguments, one easily sees that under eq. (1.14) the assertion also holds for any f ∈ C 1 b (R d ), which concludes the proof.
Proof of Theorem 1.4 (b). We now prove the main result of this article. We follow the approach from [30]. Let β ∈ C ϕψ b (R d ) be a τ -periodic solution to eq. (1.12) discussed above. Recall that either if c(x) ≡ 0 and eq. (1.13) (resp. eq. (1.14)) holds true, as assumed in Theorem 1.4 (b) (1), (2) and (3). According to Proposition 2.2, we can apply Itô's formula to the process {β(X t )} t≥0 . Let us consider now the process {X t −b * t − β(X t ) + β(X 0 )} t≥0 . By combining eq. (1.10) and Itô's formula we have that for t ≥ 0 and i, j = 1, . . . , d, and the third modified characteristic reads Consequently, for any x ∈ R d , is a P x -semimartingale (with respect to the natural filtration generated by {X t } t≥0 ) whose (modified) characteristics (relative to h(x) = x) are given by converges in the Skorohod space as ε → 0 if, and only if, {εX ε −2 t − b * ε −1 t} t≥0 converges, and if this is the case the limit is the same. Now, according to [42,Theorem VIII.2.17], in order to prove the desired convergence it suffices to prove that for all t ≥ 0 and i = 1, . . . , d, which is trivially satisfied, for all t ≥ 0 and g ∈ C b (R d ) vanishing in a neighborhood of the origin, and for all t ≥ 0 and i, j = 1, . . . , d, where Σ is given in eq. (1.

3) and
Px − → stands for the convergence in probability.
To prove the convergence in eq. (2.5), first observe that due to τ -periodicity of all components we can replace {X t } t≥0 by {Π(X t )} t≥0 , which is an ergodic Markov process (see Proposition 1.1). The assertion now follows as a direct consequence of the Birkhoff ergodic theorem.
To prove the relation in eq. (2.4) we proceed as follows. Fix g ∈ C b (R d ) that vanishes on B δ (0) for some δ > 0. Define now Clearly, for any ε > 0, F ε (x) is bounded, τ -periodic, and satisfies F ε (X t ) = F ε Π τ (X t ) for t ≥ 0, and Now, by the Markov property and exponential ergodicity of {Π τ (X t )} t≥0 , we have Let ε > 0 be such that 2ε β ∞ < δ/2. Then, we have that Consequently, which together with (C3) concludes the proof.

On Structural Properties of LTPs
In this section, we present sufficient conditions for LTPs satisfying strong Feller property, open-set irreducibility, regularity property of the semigroup, and regularity properties of the solution to the Poisson equation eq. (1.11), respectively. Several examples are also included.
(i) Let {X t } t≥0 be a diffusion process (that is, ν(x, dy) ≡ 0). According to [65,Theorem V.24.1], it will be strong Feller if b(x) is measurable, c(x) is continuous and positive definite, and there is a constant Λ > 0 such that Let us also remark that when {X t } t≥0 is a diffusion process generated with a second-order elliptic operator in divergence form with c(x) bounded, measurable and uniformly elliptic, strong Feller property of {X t } t≥0 has been discussed in [4,58,79]. (iii) Recently, there are lots of developments on heat kernel (that is, the transition density function) estimates of Feller processes. The reader is referred to [16,17,18,19,35,45,46] and the references therein for more details. In particular, let where α : R d → (0, 2) is a Hölder continuous function such that for some constants c 1 > 0 and β 1 ∈ (0, 1], and κ : (iv) Let {X t } t≥0 and {X t } t≥0 be LTPs with semigroups {P t } t≥0 and {P t } t≥0 , and Feller generators (A ∞ , D A ∞ ) and (Ã ∞ , DÃ∞), respectively. Suppose that implies that {X t } t≥0 is also strong Feller. Namely, since both {X t } t≥0 and {X t } t≥0 are LTPs, the above relation holds for any f ∈ C ∞ c (R d ). According to [20,Lemma 1.1.1], the boundedness of A ∞ −Ã ∞ and the dominated convergence theorem, it also holds for f ( The claim now follows from Dynkin's monotone class theorem. The assertion above roughly asserts that a bounded perturbation preserves the strong Feller property. Below is an example. LetL where α(x) and κ(x, y) satisfy all the assumptions in (iii), and ν(x, dy) is such that For example, one can take ν(x, dy) = γ(x,y) |y| d+δ 1 B c 1 (0) (y) dy with δ > 0 and γ(x, y) nonnegative, bounded and such that x → γ(x, y) is continuous for almost every y ∈ R d . Further, let L be the operator given in (iii). Then, We remark also that the strong Feller property of LTPs has been discussed in [73]. In the special case when {X t } t≥0 is given through eq. (1.7), the strong Feller property (and the open-set irreducibility) has been discussed in [49] under the assumption that ν Z (R m ) < ∞, and in [54,55] for an arbitrary ν Z (dy), that is, an arbitrary pure-jump Lévy process {Z t } t≥0 . Observe that in both situations non-degeneracy of Φ 2 (x)Φ ′ 2 (x) has been assumed. In the case when Φ 3 (x) ≡ Φ 3 ∈ R d×q the problem has been considered in [2,50,53], and for non-constant (and non-degenerated) Φ 3 (x) in [51].
is uniformly continuous for all i, j, k, l = 1, . . . , d, and c(x) is positive definite, the open-set irreducibility (and the strong Feller property) of the process follows from the support theorem for diffusions, see [33,Lemma 6.1.1] and [39, p. 517]. For support theorem of jump processes one can refer to [77].
(ii) If {X t } t≥0 is a diffusion process generated by a second-order elliptic operator in divergence form eq. (3.2) with uniformly elliptic, bounded and measurable diffusion coefficient, the open-set irreducibility (and the strong Feller property) follows from the corresponding heat kernel estimates (see [4,58,79]). The diffusion processes with jumps or pure jump process considered in [16,17,18,19,35,45,46] are also open-set irreducible, which is a direct consequence of lower bounds of heat kernel obtained in these references.
(iii) Let L andL be the operators from (iv) in the discussion on the strong Feller property. According to [16 For open-set irreducibility of LTPs of the form eq. (1.6) we refer the reader to [2], [49] and [54,55].
In the following proposition, which slightly generalizes [36, Lemma 2], we show that a LTP will be open-set irreducible if the corresponding Lévy measure shows enough jump activity. First, recall that a function f : (ii) the function x → R d f (y + x)ν(x, dy) is lower semi-continuous for every non-negative lower semi-continuous function f : R d → R.
Proof. Let x ∈ R d and ρ > 0 be arbitrary, and let f ∈ C ∞ c (R d ) be such that 0 ≤ f ≤ 1 and supp f ⊂ B ρ (x). By assumption, In particular, for any B ⊆ B c ρ (x), Further, let 0 < ε < ρ be arbitrary, and let 0 ≤ f ε ∈ C ∞ c (R d ) be such that f ε (y) = 1, y ∈ B ρ−ε (x) 0, y ∈ B c ρ (x) . Then, for any y ∈ B c ρ (x) we have that Next, take x, y ∈ R d such that r < |x − y| < R, and pick ε, ρ > 0 such that ε < ρ and r + 2ρ < |x − y| < R − 2ρ. Then we have However, since z → 1 B ρ−ε (y) (z) is a lower semi-continuous function, we have that which is in contradiction with the above assumption. Hence, there is t * = t * (x, y, ρ, ε) > 0 such that Fix now ε, ρ > 0 such that ε < ρ and 4ρ < R − r. From the previous discussion it follows that for any x, y ∈ R d with r + 2ρ < |x − y| < R − 2ρ, there is t * * = t * * (x, y, ρ, ε) > 0 such that The assertion now follows by employing the Chapman-Kolmogorov equation.
(i) (Diffusion processes) Let ε ∈ (0, 1), and let {X t } t≥0 be a diffusion process , and c(x) being also positive definite. Then, (C4)(i) with arbitrary t 0 > 0 follows from [ (ii) (Diffusion processes with jumps) Let ε ∈ (0, 1). Assume that b(x) and c(x) are as in (i), and that ν(x, dy) satisfies (a) sup Here, |µ(dz)| stands for the total variation measure of a signed measure µ(dz). Then, (C4)(ii) with ϕ(r) = r 2 follows again from [ Let us give sufficient conditions that {X t } t≥0 also satisfies (C4)(i). Denote by {P t } t≥0 the semigroup of {X t } t≥0 , and let {P t } t≥0 be the semigroup of the diffusion process with coefficients b(x) and c(x). Also, denote by (A ∞ , D A ∞ ) and (Ã ∞ , DÃ∞) the corresponding C ∞ -generators, respectively. Then, Since both processes are LTPs, the above relation holds for any f ∈ C ∞ c (R d ). Assume next that A ∞ −Ã ∞ is a bounded operator on (B b (R d ), · ∞ ). Then, according to [20, Lemma 1.1.1], the boundedness of A ∞ −Ã ∞ and the dominated convergence theorem, the above relation holds for f (x) = 1 O (x) for any open set O ⊆ R d . Thus, it also holds for any f ∈ B b (R d ). Recall also that P t f ∈ C b (R d ) for every f ∈ C b (R d ) and every t ≥ 0 (see [68,Corollary 3.4]). Now, according to (i), there is a measurable function C ε : (0, ∞) → (0, ∞) such that t 0 C ε (s) ds < ∞ and P t f ε ≤ C ε (t) f ∞ for all t > 0 and all τ -periodic f ∈ C b (R d ). Thus, for fixed τ -periodic f ∈ C b (R d ), P t f ∈ C ε b (R d ) and where A ∞ −Ã ∞ stands for the operator norm of A ∞ −Ã ∞ . Thus, {X t } t≥0 satisfies (C4)(i) with ψ(r) = r ε .
(iii) (Pure-jump LTPs) In the pure jump case, sufficient conditions for (C4)(i) are given in [52,Theorem 1.1]. Also, when the underlying process is given as a solution to an SDE of the form eq. (1.7), we refer to [50,51,53] and the references therein.
To construct an example satisfying (C4)(ii), we can again employ a perturbation method. Let {X t } t≥0 and {X t } t≥0 be LTPs with semigroups {P t } t≥0 and {P t } t≥0 , and B b -generators (A b , D A b ) and (Ã b , DÃ b ), respectively. Assume that A b satisfies (C4)(ii) for some Hölder exponents ψ(r) and ϕ(r).