# Homogenization of stable-like operators with random, ergodic coefficients

Tomasz Klimsiak, Tomasz Komorowski, Lorenzo Marino

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Tomasz Klimsiak, Tomasz Komorowski, Lorenzo Marino. Homogenization of stable-like operators with random, ergodic coefficients. Journal of Differential Equations, 2025, 430, pp.113183. 10.1016/j.jde.2025.02.054. hal-05500063

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# Homogenization of stable-like operators with random, ergodic coefficients

Tomasz Klimsiak 1,2

Tomasz Komorowski 1,3

Lorenzo Marino 3

October 1, 2024

¹ Institute of Mathematics, Polish Academy of Sciences,
Śniadeckich 8, 00-656 Warsaw, Poland, email: tkomorowski@impan.pl, corresponding author

² Faculty of Mathematics and Computer Science, Nicolaus Copernicus University,
Chopina 12/18, 87-100 Toruń, Poland, email: tomas@mat.umk.pl

³ Unité de Mathématiques Appliquées (UMA), ENSTA Paris,
828 Boulevard des Maréchaux, F-91120 Palaiseau, France, email: lorenzo.marino@ensta-paris.fr

## Abstract

We show homogenization for a family of $\mathbb{R}^d$-valued stable-like processes $(X_t^{\varepsilon;\theta})_{t\ge 0}$, $\varepsilon \in (0, 1]$, whose (random) Fourier symbols equal $q_\varepsilon(x, \xi; \theta) = \frac{1}{\varepsilon^\alpha}q(\frac{x}{\varepsilon}, \varepsilon\xi; \theta)$, where

$$q(x, \xi; \theta) = \int_{\mathbb{R}^d} (1 - e^{iy \cdot \xi} + iy \cdot \xi \mathbf{1}_{\{|y| \le 1\}}) \frac{\langle a(x; \theta)y, y \rangle}{|y|^{d+2+\alpha}} dy,$$

for $(x, \xi, \theta) \in \mathbb{R}^{2d} \times \Theta$. Here $\alpha \in (0, 2)$ and the family $(a(x; \theta))_{x \in \mathbb{R}^d}$ of $d \times d$ symmetric, non-negative definite matrices is a stationary ergodic random field over some probability space $(\Theta, \mathcal{H}, m)$. We assume that the random field is deterministically bounded and non-degenerate, i.e. $|a(x; \theta)| \le \Lambda$ and $\mathrm{Tr}(a(x; \theta)) \ge \lambda$ for some $\Lambda, \lambda > 0$ and all $\theta \in \Theta$. In addition, we suppose that the field is regular enough so that for any $\theta \in \Theta$, the operator $-q(\cdot, D; \theta)$, defined on the space of compactly supported $C^2$ functions on $\mathbb{R}^d$, is closable in the space of continuous functions vanishing at infinity and its closure generates a Feller semigroup. We prove the weak convergence of the laws of $(X_t^{\varepsilon;\theta})_{t\ge 0}$, as $\varepsilon \downarrow 0$, in the Skorokhod space, $m$-a.s. in $\theta$, to an $\alpha$-stable process whose Fourier symbol $\bar{q}(\xi)$ is given by $\bar{q}(\xi) = \int_{\Omega} q(0, \xi; \theta) \Phi_*(\theta) m(d\theta)$, where $\Phi_*$ is a strictly positive density w.r.t. measure $m$. Our result has an analytic interpretation in terms of the convergence, as $\varepsilon \downarrow 0$, of the solutions to random integro-differential equations $\partial_t u_\varepsilon(t, x; \theta) = -q_\varepsilon(x, D; \theta)u_\varepsilon(t, x; \theta)$, with the initial condition $u_\varepsilon(0, x; \theta) = f(x)$, where $f$ is a bounded and continuous function on $\mathbb{R}^d$.

**Key words:** Martingale problem, stable-like Feller process, homogenization, stationary and ergodic coefficients.

**2020 Mathematics Subject Classification:** primary 35B27; secondary 60H15, 60H20.
# 1 Introduction

In the present paper, we consider the problem of homogenization for a class of stable-like operators with random coefficients. More precisely, let $(\Theta, \mathcal{H}, m)$ be a probability space. For each $\theta \in \Theta$, we assign an integral operator brown $L^\theta$ on $C_c^2(\mathbb{R}^d)$ - the set of all $C^2$-smooth, compactly supported functions on $\mathbb{R}^d$ - given by

$$L^\theta u(x) := \frac{1}{2} \int_{\mathbb{R}^d} [u(x+z) + u(x-z) - 2u(x)] n(x, dz; \theta), \quad (1.1)$$

where the Lévy kernel $n$ has the form

$$n(x, dz; \theta) := \frac{\langle a(x; \theta)z, z \rangle}{|z|^{d+\alpha+2}} dz, \quad (1.2)$$

for some $\alpha \in (0, 2)$ and $a(x; \theta) = [a_{i,j}(x; \theta)]_{i,j=1}^d$ is a stationary and ergodic random field taking values in $S_d^+$, the family of all non-negative definite, symmetric $d \times d$-matrices.

Throughout the paper, we assume the following:

**Hypothesis 1.1.** The realizations of the random field $(a(x; \theta))_{x \in \mathbb{R}^d}$ are continuous in $x$, deterministically bounded, both in $x$ and $\theta$, and satisfy the non-degeneracy condition, i.e there exist $\lambda, \Lambda > 0$ such that

$$\|a(x; \theta)\| \le \Lambda, \quad \mathrm{Tr}(a(x; \theta)) \ge \lambda, \quad (x, \theta) \in \mathbb{R}^d \times \Theta. \quad (1.3)$$

Here, $\|a\| := \sum_{i,j=1}^d |a_{i,j}|$.

**Hypothesis 1.2.** For any $\theta \in \Theta$, the martingale problem (see Section 2.2 below) associated with the operator $L^\theta$ defined in (1.1) is well-posed on the Skorokhod space $\mathcal{D}$ of all $\mathbb{R}^d$-valued càdlàg paths equipped with the $J_1$-topology (see [7, Section 12] for the definition of the topology).

Hypothesis 1.2 holds e.g. whenever $a(x; \theta)$ is sufficiently smooth in $x$ (see [25, Theorem 4.2]) or it is Lipschitz continuous in $x$ and uniformly elliptic (see [8, Theorem 1.3]), i.e. or some $\nu > 0$

$$\langle a(x; \theta) \xi, \xi \rangle \ge \nu |\xi|^2, \quad (x, \xi, \theta) \in \mathbb{R}^{2d} \times \Theta. \quad (1.4)$$

Assume Hypothesis 1.1. The results [34, Theorem 1.1, Lemma 2.1] and [30, Theorem 4.10.3] show that then Hypothesis 1.2 is equivalent to the following condition:

- for any $\theta \in \Theta$, the operator $L^\theta$ defined in (1.1) is closable in $C_0(\mathbb{R}^d)$ - the set of all continuous functions on $\mathbb{R}^d$ vanishing at infinity - and its closure generates a Feller semigroup.

In particular, it follows from the above that for each $\theta \in \Theta$ there exists a unique $\mathbb{R}^d$-valued, càdlàg, Feller process $(X_t^{x;\theta}; t \ge 0, x \in \mathbb{R}^d)$, defined over some probability space $(\Sigma, \mathcal{A}, \mathbb{P})$, satisfying $X_0^{x;\theta} = x, \mathbb{P}$-a.s. Its random Fourier symbol $q: \mathbb{R}^d \times \mathbb{R}^d \times \Theta \to \mathbb{R}$ equals (due to the symmetry of the measure $n(x, \cdot, \theta)$)

$$q(x, \xi; \theta) = \int_{\mathbb{R}^d} (1 - e^{iz \cdot \xi} + iz \cdot \xi \mathbf{1}_{\{|z| \le 1\}}) n(x, dz; \theta) = \int_{\mathbb{R}^d} (1 - \cos(z \cdot \xi)) n(x, dz; \theta). \quad (1.5)$$
To study the homogenization problem, we introduce the scaled processes $X_t^{\varepsilon,x;\theta}(\zeta) := \varepsilon X_{t\varepsilon^{-\alpha}}^{x/\varepsilon;\theta}(\zeta)$, $t \ge 0$, defined for any $\varepsilon > 0$. The processes are considered over the product probability space $(\Theta \times \Sigma, \mathcal{H} \otimes \mathcal{A}, m \otimes \mathbb{P})$. For each $\theta \in \Theta$, the process $(X_t^{\varepsilon,x;\theta}, t \ge 0, x \in \mathbb{R}^d)$ is Feller and its infinitesimal generator on $C_c^2(\mathbb{R}^d)$ equals

$$L_{\varepsilon}^{\theta} u(x) := \frac{1}{2} \int_{\mathbb{R}^d} [u(x+z) - 2u(x) + u(x-z)] \frac{\langle a(x/\varepsilon; \theta)z, z \rangle}{|z|^{d+\alpha+2}} dz, \quad x \in \mathbb{R}^d.$$

Our main result is the following quenched version of the convergence of the laws of the scaled processes.

**Theorem 1.3.** Under the assumptions spelled out in the foregoing the laws of $(X_t^{\varepsilon,x;\theta})_{t \ge 0}$ converge weakly in $\mathcal{D}$, as $\varepsilon \downarrow 0$, for $m$-a.s. in $\theta$, to the law of an $\alpha$-stable process whose Fourier symbol equals

$$\bar{q}(\xi) = \int_{\Theta} q(0, \xi; \theta) \Phi_{*}(\theta) m(d\theta) = \int_{\mathbb{R}^d} [1 - \cos(z \cdot \xi)] \frac{\langle \bar{a}z, z \rangle}{|z|^{d+\alpha+2}} dz, \quad \xi \in \mathbb{R}^d.$$

Here $\Phi_*$ is a strictly positive density with respect to $m$ and

$$\bar{a} := \int_{\Omega} a(0; \theta) \Phi_{*}(\theta) m(d\theta).$$

Our result has an obvious analytic interpretation in terms of solutions to random integro-differential equations of the form

$$\begin{cases} \partial_t u_{\varepsilon}(t, x; \theta) = L_{\varepsilon}^{\theta} u_{\varepsilon}(t, x; \theta), & t > 0, x \in \mathbb{R}^d, \theta \in \Theta; \\ u_{\varepsilon}(0, x; \theta) = f(x), & \end{cases}$$

where $f$ is a bounded and continuous function in $\mathbb{R}^d$. As its direct consequence one can conclude that for $m$-a.s. $\theta$ the solutions $u_{\varepsilon}(t, x; \theta)$ converge, as $\varepsilon \downarrow 0$, to the solution $\bar{u}(t, x)$ of the following “homogenized” equation:

$$\begin{cases} \partial_t \bar{u}(t, x) = \bar{L} \bar{u}(t, x), & t > 0, x \in \mathbb{R}^d; \\ \bar{u}(0, x) = f(x), & \end{cases}$$

where

$$\bar{L}u(x) := \frac{1}{2} \int_{\mathbb{R}^d} [u(x+z) - 2u(x) + u(x-z)] \frac{\langle \bar{a}z, z \rangle}{|z|^{d+\alpha+2}} dz.$$

## Context

Homogenization of solutions to stochastic differential equations (S.D.E-s) and partial differential equations (P.D.E.-s) with random coefficients is a classical problem in both analysis and the theory of stochastic processes. The pioneering results in the subject have been almost simultaneously obtained in [33] and [37], where the problem of homogenization of solutions to the Dirichlet boundary value problem for elliptic equations in a divergence form, with stationary and ergodic coefficients has been considered. Since then the topic has been developed by many authors and for various types of elliptic
and parabolic differential equations with fast oscillating coefficients. We refer an interested reader
to the monographs [2, 4, 28, 31, 36, 40, 50] and the references therein.

More recently, there has been a growing interest in stochastic homogenization for classes of
integro-differential equations and a related problem of scaling limits of S.D.E.-s with random coeffi-
cients, driven by general Lévy processes. Often such limits require a non-diffusive scaling and the
result of the homogenization is a Lévy process.

We mention in this context, papers [1, 3, 21, 26, 27, 29, 40, 41] for equations with fast oscillating periodic coefficients and [10, 11, 12, 22, 42, 43, 44] which are concerned with stochastically homogeneous random coefficients. In particular, the paper [22] considers the limit of the martingale problem whose coefficients are driven by some uniquely ergodic Markov process. The work [42] deals with the diffusive limit of non-local operators of convolution type with random ergodic coefficients. The closest case to ours is considered in [43, 44], where one dimensional SDEs, driven by both Brownian and Poisson noises, are examined. The coefficients are stationary and ergodic fields. The principal difference between the present paper and [43, 44] is that the latter look at the situation when the process describing the environment, as viewed from vantage point of the trajectory of the solution of the SDE, has an explicitly given invariant measure. In contrast, the main effort of our article is to construct the invariant measure for the aforementioned process. The present paper is related to the result of [38] where diffusions with no jumps have been considered, i.e. $b \equiv 0$ and $n \equiv 0$. In this case, $q_{\varepsilon}(x, \xi; \theta) = \frac{1}{2}a(x/\varepsilon; \theta)\xi \cdot \xi$ for any $\varepsilon > 0$.

Finally, we mention that in the non-linear framework, homogenization of solutions of integro-
differential equations with an external Dirichlet boundary condition on a bounded domain has been
considered in [45, 46], for a class of fully non-linear, non-local elliptic operators with fast ocillating,
either periodic, or stochastic and ergodic, coefficients, that includes also the operators of the form
(1.1). However, their method of proof is quite different from ours. We should also emphasize that our
result deals with the convergence of the laws of stochastic processes, that constitutes the novelty of
the present paper. In addition [13, 14] consider evolution equations involving non-local $p$-Laplacian
type operators both in periodic and random media.

**About the method of proof. Organization of the paper**

Concerning the organization of the paper, in Sections 2.1–2.2 we introduce the basic notions used
throughout the paper. In Section 2.3 we recall some facts about stable-like processes. To show our
main result, formulated in Theorem 1.3, we embed the space $\Theta$ into $\Omega$ - the set of all continuous and
bounded matrix valued functions, by assigning to each $\theta \in \Theta$ its realization, i.e. $J(\theta)(x) := a(x; \theta)$,
$x \in \mathbb{R}^d$. The space $\Omega$ is Polish, when considered with the standard Fréchet metric. We prove,
see Theorem 2.2, that the conclusion of our main result holds under the additional assumption
that an $\Omega$-valued *environment process*, which describes the random environment $\omega \in \Omega$ from the
vantage point of the trajectory, has an invariant ergodic probability measure $\mu_*$.
This measure is
equivalent with $\mu$ - the push-forward measure of $m$ by $J$. The density $\Phi_* = \frac{d\mu_*}{d\mu} \circ J$ appears in
the statement of Theorem 1.3. Such environment process is rigorously introduced in Section 2.4.
In Section 2.5, we formulate and prove the homogenization result, provided that we know that
the environment process possesses an invariant and ergodic measure equivalent with $\mu$, the law of the random environment. Section 3 is devoted to showing the existence and uniqueness of such a measure, with a strictly positive invariant density $\Phi_*$, see Theorem 3.1. The proof uses a weak form of the Alexandrov-Bakelman-Pucci (ABP) estimates for solutions of the Dirichlet problem for equations involving a non-local operator of the form (1.1). Such estimates follow from the results of [23], see Section A. Using this we can prove that the invariant density exists and is in fact $L^{1+\delta}$ integrable for $0 < \delta < \frac{\alpha}{2d-\alpha}$.

The proof of Theorem 1.3 is presented in Section 3.3 under an additional assumption that the uniform ellipticity condition (1.4) holds. In Section 3.4 we relax this assumption and prove the result in full generality.

Finally in the Appendix, we show some additional facts, which are needed in our paper. As we have already mentioned in the foregoing, Section A formulates a version of the Alexandrov-Bakelman-Pucci estimates for integro-differential operators, see Theorem A.1, that can be inferred from [23, Theorem 1.3]. In Section B we prove a result, see Proposition B.2, about irreducibility of stable-like processes. Section C is devoted to derivation of the formula for the Fourier symbol of an isotropic stable-like process. Section D contains the proof of the fact that the set of $d \times d$, non-negative symmetric matrix valued functions, such that there exists a unique Feller process corresponding to the respective Fourier symbol (1.5), is a Borel subset of $\Omega$.

## Acknowledgments

T. Komorowski acknowledges the support of the NCN grant 2020/37/B/ST1/00426, T. Klimsiak acknowledges the support of the NCN grant 2022/45/B/ST1/01095.

# 2 General setup and notations

## 2.1 Some generalities

We denote by $B(y,r)$ the open ball of radius $r > 0$ centered at $y \in \mathbb{R}^d$, with respect to the Euclidean metric on $\mathbb{R}^d$ and by $\mathcal{L}_d$ the Lebesgue measure on $\mathbb{R}^d$. Let $B_b(\mathbb{R}^d)$ (resp. $C_b(\mathbb{R}^d)$) be the space of all Borel measurable (resp. continuous) and bounded functions on $\mathbb{R}^d$. We denote the supremum norm of any bounded function $f$ on $\mathbb{R}^d$ by $\|f\|_{\infty} := \sup_{x \in \mathbb{R}^d} |f(x)|$. Given $k \in \mathbb{N}$, let $C_b^k(\mathbb{R}^d)$ be the space of all $k$-times differentiable functions with continuous and bounded derivatives. For any $f \in C_b^k(\mathbb{R}^d)$, we denote by $\|f\|_{k,\infty}$ the norm defined as the sum of the supremum norms of the function and all its derivatives up to order $k$ included. For any $f \in C_b^1(\mathbb{R}^d)$, we denote its gradient by $\nabla f$. We also consider the space $C_c^\infty(\mathbb{R}^d)$ of all the compactly supported, smooth functions on $\mathbb{R}^d$. For a given $\lambda > 0$, we denote by $S_d^+(\lambda)$ the set of all matrices $a \in S_d^+$ such that $a - \lambda I_d \in S_d^+$, where $I_d$ is the $d \times d$ identity matrix.
## 2.2 Probability space with a group of measure preserving transformations

Let $\Omega$ be the space of all functions $\omega: \mathbb{R}^d \to S_d^+$, which are continuous and satisfy

$$ \|\omega(x)\| \le \Lambda, \quad \mathrm{Tr}(\omega(x)) \ge \lambda, \quad x \in \mathbb{R}^d, $$

where $\Lambda, \lambda > 0$ are the same constants appearing in (1.3). The space is Polish when equipped with the Fréchet metric

$$ d(\omega_1, \omega_2) := \sum_{K=1}^{+\infty} \frac{1}{2^K} \cdot \frac{\|\omega_1 - \omega_2\|_{\infty, K}}{1 + \|\omega_1 - \omega_2\|_{\infty, K}}, $$

where, for a given $K > 0$,

$$ \|\omega\|_{\infty, K} := \sup_{|x| \le K} |\omega(x)|, \quad \omega \in \Omega. $$

For $r > 0$, we introduce the subspace $\Omega_r$ of $\Omega$ which consists of all functions $\omega: \mathbb{R}^d \to S_d^+(r)$ such that both $\omega$ and $\nabla\omega$ are continuous. The space $\Omega_r$ is Polish when equipped with the Fréchet metric

$$ d_r(\omega_1, \omega_2) := \sum_{K=1}^{+\infty} \frac{1}{2^K} \cdot \frac{\|\omega_1 - \omega_2\|_{1, \infty, K}}{1 + \|\omega_1 - \omega_2\|_{1, \infty, K}}. $$

Here

$$ \|\omega\|_{1, \infty, K} := \sup_{|x| \le K} |\omega(x)| + \sup_{|x| \le K} |\nabla\omega(x)|, \quad \omega \in \Omega_r. $$

We denote by $\mathcal{B}(\Omega_r)$ and $\mathcal{B}(\Omega)$ the Borel $\sigma$-fields of $(\Omega_r, d_r)$ and $(\Omega, d)$, respectively. Moreover, let $B_b(\Omega)$ and $B_b(\Omega_r)$ (resp. $C_b(\Omega)$ and $C_b(\Omega_r)$) be the spaces of all Borel measurable (resp. continuous) and bounded functions on $(\Omega, d)$ and $(\Omega_r, d_r)$, respectively. Note that by [16, Theorem 8.3.7], $\Omega_r$ is a Borel measurable subset of $\Omega$, hence $\mathcal{B}(\Omega_r) \subset \mathcal{B}(\Omega)$.

Consider an additive group of transformations $(\tau_x)_{x \in \mathbb{R}^d}$ acting on $\Omega$ as follows

$$ \tau_x \omega(y) := \omega(x + y), \quad y \in \mathbb{R}^d. $$

Clearly, $\tau_x(\Omega_r) \subset \Omega_r$. Note that for any $f \in C_b(\Omega)$, we have

$$ \lim_{x \to 0} f(\tau_x \omega) = f(\omega), \quad \omega \in \Omega. $$

Given two measure spaces $(\Sigma_i, \mathcal{A}_i, m_i)$, $i = 1, 2$ and a measurable mapping $S: \Sigma_1 \to \Sigma_2$, we denote by $S_{\#}m_1$ the push-forward of $m_1$ through $S$, i.e. the measure on $(\Sigma_2, \mathcal{A}_2)$ given by $S_{\#}m_1(A) = m_1(S^{-1}(A))$ for any $A$ in $\mathcal{A}_2$. We introduce the mapping $J: \Theta \to \Omega$ by letting

$$ J(\theta)(x) := a(x; \theta), \quad x \in \mathbb{R}^d. $$

By the assumptions made in Hypothesis 1.1, the function $J: (\Theta, \mathcal{H}) \to (\Omega, \mathcal{B}(\Omega))$ is measurable. Let $\mu := J_{\#}m$ be a measure on $(\Omega, \mathcal{B}(\Omega))$. By stationarity and ergodicity of $(a(x))_{x \in \mathbb{R}^d}$ the measure $\mu$ is invariant and ergodic under the action of the group, i.e.

$$ (\tau_x)_{\#} \mu = \mu, \quad x \in \mathbb{R}^d \qquad (2.1) $$

and if $A \in \mathcal{B}(\Omega)$ satisfies $\tau_x A = A$ for all $x \in \mathbb{R}^d$, then $\mu(A) = 0$ or 1.
## 2.3 Random stable-like processes

Recall that $\mathcal{D}$ is the space of all càdlàg paths $\zeta: [0, +\infty) \to \mathbb{R}^d$, equipped with the $J_1$-Skorokhod topology. We now introduce the canonical process $(X_t)_{t \ge 0}$ by letting

$$X_t(\zeta) := \zeta(t), \quad \zeta \in \mathcal{D}, \qquad (2.2)$$

and its natural filtration $(\mathcal{F}_t)_{t \ge 0}$ by $\mathcal{F}_t := \sigma(X_s, 0 \le s \le t)$. Then, $\mathcal{F} := \sigma(X_t, t \ge 0)$ is the Borel $\sigma$-algebra on $\mathcal{D}$.

Given the function $\mathbf{a}: \Omega \to S_d^+$ defined by $\mathbf{a}(\omega) := \omega(0)$, consider the associated (random) Fourier symbol

$$\mathbf{q}(\xi; \omega) := \int_{\mathbb{R}^d} (1 - e^{iz \cdot \xi} + iz \cdot \xi \mathbf{1}_{\{|z| \le 1\}}) \frac{\langle \mathbf{a}(\omega)z, z \rangle}{|z|^{d+\alpha+2}} dz = \int_{\mathbb{R}^d} (1 - \cos(z \cdot \xi)) \frac{\langle \mathbf{a}(\omega)z, z \rangle}{|z|^{d+\alpha+2}} dz, \quad (2.3)$$

where $(\xi, \omega) \in \mathbb{R}^d \times \Omega$, and the corresponding operator $\mathbf{q}(D; \omega)$ on $C_c^2(\mathbb{R}^d)$ by

$$\mathbf{q}(D; \omega)u(x) := \int_{\mathbb{R}^d} \mathbf{q}(\xi; \tau_x \omega) \hat{u}(\xi) e^{ix \cdot \xi} d\xi, \quad x \in \mathbb{R}^d. \qquad (2.4)$$

Furthermore, one can conclude, see Appendix C, that

$$\mathbf{q}(\xi; \omega) = C_{d,\alpha} \mathrm{Tr} \left( \mathbf{a}(\omega) \right) |\xi|^\alpha + c_{d,\alpha} \langle \mathbf{a}(\omega) \hat{\xi}, \hat{\xi} \rangle |\xi|^\alpha, \qquad (2.5)$$

where $\hat{\xi} := \xi/|\xi|$ and the constants $C_{d,\alpha}, c_{d,\alpha} > 0$ depend only the dimension $d$ and exponent $\alpha$.

We say that the martingale problem corresponding to $\mathbf{q}(D; \omega)$, $\omega \in \Omega$, is well-posed, cf. [48], if for every Borel probability measure $\nu$ on $\mathbb{R}^d$, there exists a unique Borel probability measure $\mathbb{P}^{\nu;\omega}$ on $\mathcal{D}$, called a solution to the martingale problem for $\mathbf{q}(D; \omega)$ with initial distribution $\nu$, such that

i) $\mathbb{P}^{\nu;\omega}(X_0 \in A) = \nu(A)$ for any Borel measurable $A \subset \mathbb{R}^d$,

ii) for any $f \in C_c^\infty(\mathbb{R}^d)$, the process

$$M_t^\omega[f] := f(X_t) - f(X_0) - \int_0^t [\mathbf{q}(D; \omega)f](X_r) dr, \quad t \ge 0 \qquad (2.6)$$

is a (càdlàg paths) $(\mathcal{F}_t)_{t \ge 0}$-martingale under measure $\mathbb{P}^{\nu;\omega}$.

As usual, we write $\mathbb{P}^{x;\omega} := \mathbb{P}^{\delta_x;\omega}$, $x \in \mathbb{R}^d$. The expectations with respect to $\mathbb{P}^{\nu;\omega}$ and $\mathbb{P}^{x;\omega}$ shall be denoted by $\mathbb{E}^{\nu;\omega}$ and $\mathbb{E}^{x;\omega}$, respectively.

Let $\Omega^{\mathcal{M}}$ be the set of all $\omega \in \Omega$ such that there exists a unique Feller process corresponding to the Fourier symbol $\mathbf{q}(\xi; \omega)$. One can show, see Appendix D, that $\Omega^{\mathcal{M}} \in \mathcal{B}(\Omega)$. Moreover, by the assumptions made in Hypothesis 1.1, the measure $\mu := J_\mu^m$ is supported in $\Omega^{\mathcal{M}}$. By [8, Theorem 1.3], it also follows that $\Omega_\nu \subset \Omega^{\mathcal{M}}$ for any $\nu > 0$.

Theorem 1.1 and Lemma 2.1 in [34] imply that for each $\omega \in \Omega^{\mathcal{M}}$, the operator $\mathbf{q}(D; \omega)$ defined on $C_c^2(\mathbb{R}^d)$ is closable in $C_0(\mathbb{R}^d)$ and its closure generates the Feller transition probability semigroup $(P_t^\omega)_{t \ge 0}$, satisfying

$$P_t^\omega f(x) = \mathbb{E}^{x;\omega} f(X_t), \quad f \in C_0(\mathbb{R}^d), x \in \mathbb{R}^d, t \ge 0. \qquad (2.7)$$
For any $\beta > 0$ and $\omega \in \Omega^{\mathcal{M}}$, we introduce the $\beta$-resolvent operator $R_{\beta}^{\omega}: B_b(\mathbb{R}^d) \to B_b(\mathbb{R}^d)$ as

$$R_{\beta}^{\omega} f(x) := \int_{0}^{\infty} e^{-\beta t} P_{t}^{\omega} f(x) dt, \quad x \in \mathbb{R}^{d}. \qquad (2.8)$$

Since the Fourier symbol of $(\mathbb{P}^{x;\omega})_{x \in \mathbb{R}^d}$ is given by

$$q(x, \xi; \omega) := \mathbf{q}(\xi, \tau_x \omega), \quad (x, \xi, \omega) \in \mathbb{R}^{2d} \times \Omega^{\mathcal{M}},$$

with $\mathbf{q}$ defined in (2.3), one can show that $\tau_x(\Omega^{\mathcal{M}}) \subset \Omega^{\mathcal{M}}$ and

$$\mathbb{P}^{x+y;\omega} = (s_y)_{\sharp} \mathbb{P}^{x;\tau_y\omega}, \quad x, y \in \mathbb{R}^d, \omega \in \Omega^{\mathcal{M}}, \qquad (2.9)$$

where $s_y: \mathcal{D} \to \mathcal{D}$ is given by $s_y(\zeta)(t) := y + \zeta(t)$, $t \ge 0$.

For each $\varepsilon \in (0,1)$ and $\omega \in \Omega^{\mathcal{M}}$, we now consider the scaled process $(\mathbb{P}_{\varepsilon}^{x;\omega})_{x \in \mathbb{R}^d}$ such that $\mathbb{P}_{\varepsilon}^{x;\omega}(X_0 = x) = 1$ and whose Fourier symbol $q_{\varepsilon}: \mathbb{R}^d \times \mathbb{R}^d \to \mathbb{R}^d$ equals

$$q_{\varepsilon}(x, \xi; \omega) := \frac{1}{\varepsilon^{\alpha}} q\left(\frac{x}{\varepsilon}, \varepsilon\xi; \omega\right) = C_{d,\alpha} \mathrm{Tr}\left(\mathbf{a}(\tau_{x/\varepsilon}\omega)\right) |\xi|^{\alpha} + c_{d,\alpha} \langle \mathbf{a}(\tau_{x/\varepsilon}\omega)\hat{\xi}, \hat{\xi} \rangle |\xi|^{\alpha}. \qquad (2.10)$$

Denote by $\mathcal{T}_{\varepsilon}: \mathcal{D} \to \mathcal{D}$ the mapping $\mathcal{T}_{\varepsilon}(\zeta)(t) := \varepsilon\zeta(t/\varepsilon^{\alpha})$. Note that

$$\mathbb{P}_{\varepsilon}^{x;\omega} = (\mathcal{T}_{\varepsilon})_{\sharp} \mathbb{P}^{x/\varepsilon;\omega}. \qquad (2.11)$$

Let $\mathbb{E}_{\varepsilon}^{x;\omega}$ be the expectation with respect to the measure $\mathbb{P}_{\varepsilon}^{x;\omega}$. We denote by $(P_{t,\varepsilon}^{\omega})_{t \ge 0}$ the Feller semigroup associated with the process $(\mathbb{P}_{\varepsilon}^{x;\omega})_{x \in \mathbb{R}^d}$ and by $\mathbf{q}_{\varepsilon}(D; \omega)$ the corresponding generator.

Given a Borel probability measure $\nu$ on $\Omega$ that is supported in $\Omega^{\mathcal{M}}$, we introduce the measure $\mathbb{P}_{\varepsilon}^{x;\nu}$ on $(\Omega \times \mathcal{D}, \mathcal{B}(\Omega) \otimes \mathcal{F})$

$$\mathbb{P}_{\varepsilon}^{x;\nu}(A) = \int_{\Omega} \nu(d\omega) \int_{\mathcal{D}} 1_A(\omega, \zeta) \mathbb{P}_{\varepsilon}^{x;\omega}(d\zeta), \quad A \in \mathcal{B}(\Omega) \otimes \mathcal{F} \qquad (2.12)$$

and let $\mathbb{E}_{\varepsilon}^{x;\nu}$ be the respective expectation. To lighten somewhat the notation, we omit writing the superscript when $x = 0$ and subscript when $\varepsilon = 1$ in the notation of measures and their respective expectations, e.g. we write $\mathbb{E}^{\nu}$ and $\mathbb{P}^{\nu}$ when $x = 0$ and $\varepsilon = 1$.

## 2.4 The random environment as seen from the particle

For each $\omega \in \Omega^{\mathcal{M}}$, we introduce an $\Omega^{\mathcal{M}}$-valued process $(\eta_t)_{t \ge 0}$ over the probability space $(\mathcal{D}, \mathcal{F}, \mathbb{P}^{\omega})$, sometimes referred to as the environment process, defined by

$$\eta_t(\omega) := \tau_{X_t}\omega, \quad t \ge 0. \qquad (2.13)$$

**Proposition 2.1.** For each $\omega \in \Omega^{\mathcal{M}}$, the process $(\eta_t(\omega))_{t \ge 0}$ is $(\mathcal{T}_t)_{t \ge 0}$-Markovian under measure $\mathbb{P}^{\omega}$. Its transition semigroup $(\mathfrak{P}_t)_{t \ge 0}$ is given by

$$\mathfrak{P}_t F(\omega) = \mathbb{E}^{\omega} F(\eta_t(\omega)) = P_t^{\omega} \tilde{F}(\cdot; \omega)(0), \quad F \in B_b(\Omega^{\mathcal{M}}), \omega \in \Omega^{\mathcal{M}}, \qquad (2.14)$$

where $\tilde{F}(y; \omega) := F(\tau_y \omega)$.
The proof of the above result can be obtained following the same arguments as in [31, Proposition 9.7]. We may extend $\mathfrak{P}_t$ to $B_b(\Omega)$ by letting

$$ \mathfrak{P}_t F(\omega) := F(\omega), \quad \omega \in \Omega \setminus \Omega^{\mathcal{M}}. $$

Observe that if $\omega \in \Omega_r$ (resp. $\omega \in \Omega^{\mathcal{M}}$), then $\eta_t(\omega) \in \Omega_r$ (resp. $\eta_t(\omega) \in \Omega^{\mathcal{M}}$) for any $t > 0$. Thus, $\eta_t$ may be regarded as a Markov process on either $\Omega_r$, or $\Omega^{\mathcal{M}}$. Using Markov processes theory nomenclature both $\Omega_r$ and $\Omega^{\mathcal{M}}$ are absorbing sets (see [47, Definition 12.27] and also [47, Theorem 12.30]).

## 2.5 The homogenization result

**Theorem 2.2 (Quenched invariance principle).** Let $x \in \mathbb{R}^d$. Assume that there exists an invariant ergodic probability measure $\mu_*$ for the process $(\eta_t)_{t \ge 0}$, which is equivalent to $\mu$. Then, as $\varepsilon \downarrow 0$, the measures $(\mathbb{P}_\varepsilon^{x;\omega})_{\varepsilon \in (0,1)}$ weakly converge in $\mathcal{D}$, $\mu$-a.s. in $\omega$, to $\bar{\mathbb{Q}}^x$, the law of an $\alpha$-stable process $\{x + \bar{Z}(t)\}_{t \ge 0}$ with the Lévy symbol

$$ \bar{q}(\xi) = \int_{\Omega} \mathbf{q}(\xi; \omega) \mu_*(d\omega) = C_{d,\alpha} \mathrm{Tr}(\bar{\mathbf{a}}) |\xi|^\alpha + c_{d,\alpha} \langle \bar{\mathbf{a}} \hat{\xi}, \hat{\xi} \rangle |\xi|^\alpha, $$

where $\bar{\mathbf{a}} = \int_{\Omega} \mathbf{a}(\omega) \mu_*(d\omega)$. The above means that for $\mu$-a.s. $\omega$

$$ \lim_{\varepsilon \downarrow 0} \mathbb{E}_{\varepsilon}^{x;\omega} f = \bar{\mathbb{E}}^x f $$

holds for any $f$ - a bounded and continuous function on $\mathcal{D}$. Here, $\bar{\mathbb{E}}^x$ is the expectation with respect to $\bar{\mathbb{Q}}^x$. In particular, $\mu$-a.s. $\omega$, we have

$$ \lim_{\varepsilon \downarrow 0} P_{t, \varepsilon}^{\omega} f(x) = \bar{P}_t f(x), \quad f \in C_b(\mathbb{R}^d), \ t > 0, \ x \in \mathbb{R}^d. $$

Here, $(\bar{P}_t)_{t \ge 0}$ is the transition probability semigroup of $(\bar{Z}(t))_{t \ge 0}$.

**Proof.** The family $(\mathbb{P}_\varepsilon^{x;\omega})_{\varepsilon \in (0,1]}$ is tight for any $\omega \in \Omega$. Indeed, there exists a constant $C > 0$ such that for any $u \in C_b^2(\mathbb{R}^d)$, $\varepsilon \in (0,1)$ and $\omega \in \Omega$,

$$ \|q(\cdot, D; \omega)u(\cdot)\|_{\infty} \le C \|u\|_{2,\infty}. \qquad (2.15) $$

Hence, tightness of $(\mathbb{P}_\varepsilon^{x;\omega})_{\varepsilon \in (0,1]}$ follows, see the proof of [30, Theorem 4.9.2], or [48, Theorem A.1].

To finish the proof of the theorem, it suffices therefore to prove the convergence of finite dimensional distributions. In what follows we show that for any $t > 0$, $f \in C_b(\mathbb{R}^d)$,

$$ \lim_{\varepsilon \downarrow 0} \mathbb{E}_{\varepsilon}^{x;\omega} f(X_t) = \bar{\mathbb{E}}^x f(X_t), \quad \mu\text{-a.s. in } \omega. \qquad (2.16) $$

The generalization of (2.16) to the case of a finite number of times is straightforward. Since $\mu$ is invariant under $(\tau_x)_{x \in \mathbb{R}^d}$, it is enough to consider only the case when $x = 0$, thanks to (2.9).

According to [48, Theorem 1.1], for each $\omega \in \Omega^{\mathcal{M}}$, $\xi \in \mathbb{R}$ and $\varepsilon > 0$, the process

$$ \exp \left\{ iX_t \cdot \xi - ix \cdot \xi - \int_0^t q_\varepsilon(X_s, \xi; \omega) ds \right\}, \quad t \ge 0 $$
is a mean one, $(\mathcal{F}_t)$-martingale under the measure $\mathbb{P}_{\varepsilon}^{0;\omega}$. Taking into account the definition of the scaled path measures, cf. (2.11), in order to prove (2.16) it suffices to only show that for any $t > 0$, $\xi \in \mathbb{R}^d$,

$$ \lim_{\varepsilon \downarrow 0} \int_{0}^{t/\varepsilon^{\alpha}} q(X_s, \varepsilon\xi) \, ds = t\bar{q}(\xi), \quad \mathbb{P}^{\mu\text{-a.s.}} \qquad (2.17) $$

Using the environment process $(\eta_t)_{t \ge 0}$ defined in (2.13) and formula (2.5), we see that (2.17) is equivalent with proving that

$$ \lim_{\varepsilon \downarrow 0} \varepsilon^{\alpha} \int_{0}^{t/\varepsilon^{\alpha}} \mathbf{q}(\xi; \eta_s) \, ds = t\bar{q}(\xi), \quad \mathbb{P}^{\mu\text{-a.s.}} $$

By the individual Birkhoff ergodic theorem the above convergence holds $\mathbb{P}^{\mu^*\text{-a.s.}}$. The conclusion of the theorem is then a consequence of the fact that $\mu_*$ and $\mu$ are equivalent. $\square$

In light of the above result, the proof of the main Theorem 1.3 is concluded once we show the existence of a probability measure $\mu_*$ as in the statement of Theorem 2.2. This is going to be the main objective of Section 3.

### 3 On the existence of an ergodic invariant measure for the environment process

The main purpose of the present section is to prove the following result.

**Theorem 3.1.** Let $1 \le p < d/(d - \alpha/2)$. Then, there exists $\Phi_* \in L^p(\mu)$ such that:

(i) $\|\Phi_*\|_{L^1(\mu)} = 1$;

(ii) the measure $\mu_*$ on $(\Omega, \mathcal{B}(\Omega))$ given by $d\mu_* = \Phi_*d\mu$ is invariant under the dynamics of the environment process $(\eta_t)_{t \ge 0}$, i.e.

$$ \int_{\Omega} \mathfrak{P}_t F \, d\mu_* = \int_{\Omega} F \, d\mu_*, \quad F \in B_b(\Omega), t \ge 0; $$

(iii) $\Phi_* > 0$ $\mu$-a.s. on $\Omega$ and in consequence the measure $\mu_*$ is ergodic, i.e. if $F \in L^\infty(\mu_*)$ satisfies $\mathfrak{P}_t F = F$ for any $t > 0$, then $F$ is constant $\mu_*$-a.s. in $\Omega$.

The proof of the theorem is presented in Sections 3.1 – 3.4 below. In Sections 3.1 – 3.3 we prove Theorem 3.1 under some additional assumptions about the non-degeneracy and regularity of the random field. Namely, we suppose that the uniform ellipticity condition (1.4) holds and the realizations of the random field $(a(x; \omega))_{x \in \mathbb{R}^d}$ are deterministically bounded together with their first derivatives. Finally, in Section 3.4 we finish the proof of the theorem by discarding these additional regularity assumptions.
## 3.1 An ergodic theorem

Our first result is a version of the ergodic theorem somewhat analogous to the one that can be found in [39], see also [38, Lemma 3.2]. Before its formulation we need some notation. Let $e_i$, $i = 1, \dots, d$ be the canonical basis of $\mathbb{R}^d$. By the one dimensional unit torus $\mathbb{T}$ we understand the interval $[-1/2, 1/2]$ whose endpoints are identified. Given $M > 0$, we can then denote by $\mathbb{T}_M^d := (MT)^d$ the $d$-dimensional torus of length $M$. Furthermore, we let $Q_M := [-M/2, M/2]^d$. We shall also denote by $\ell_M(dy) := M^{-d}dy$ the normalized Lebesgue measure both on $\mathbb{T}_M^d$ or $Q_M$. Let $B_b(\mathbb{T}_M^d)$ (resp. $C(\mathbb{T}_M^d)$) be the space of all bounded Borel measurable (resp. continuous) functions on $\mathbb{T}_M^d$. Given $k \in \mathbb{N}$, let $C^k(\mathbb{T}_M^d)$ be the space of all $k$-times differentiable functions on $\mathbb{T}_M^d$ with continuous derivatives.

**Lemma 3.2.** There exists $\bar{\omega} \in \Omega_r$ such that the sequence $(\bar{\mu}_M)_{M \in \mathbb{N}}$ of measures on $(\Omega_r, \mathcal{B}(\Omega_r))$, given by

$$ \bar{\mu}_M(A) := \int_{Q_M} 1_A(\tau_y \bar{\omega}) \ell_M(dy), \quad A \in \mathcal{B}(\Omega_r), $$
weakly converges, as $M \to \infty$, to $\mu$.

*Proof.* First, we observe that there exist a metric $\bar{d}$ on $\Omega_r$ which is equivalent with $d_r$ and a countable family $\mathcal{X} := (F_n)_{n \in \mathbb{N}}$ of bounded functions on $\Omega_r$ such that $\mathcal{X}$ is dense in $U_{\bar{d}}(\Omega_r)$ in the supremum norm. Here $U_{\bar{d}}(\Omega_r)$ is the space of all real valued, uniformly continuous in metric $\bar{d}$ functions on $\Omega_r$. This can be seen as follows. Since $\Omega_r$ is Polish, it is well-known that $\Omega_r$ is homeomorphic to a subset of a compact metric space - the Hilbert cube. Therefore, there exists an equivalent metric $\bar{d}$ on $\Omega_r$ such that $(\Omega_r, \bar{d})$ is totally bounded. It then follows that the completion of $\Omega_r$ under $\bar{d}$, denoted by $\bar{\Omega}_r$, is compact. The space $U_{\bar{d}}(\Omega_r)$ is isometrically isomorphic with the space $C(\bar{\Omega}_r)$ of continuous functions on $\bar{\Omega}_r$, both equipped with the topology of uniform convergence. Since it is known (cf. [19, Lemma VI.8.4]) that the latter is separable, so is $U_{\bar{d}}(\Omega_r)$.

By the individual ergodic theorem, we can choose $\tilde{\Omega}_r \subset \Omega_r$ such that $\mu(\tilde{\Omega}_r) = 1$ and for any $\bar{\omega} \in \tilde{\Omega}_r$ and $F \in \mathcal{X}$

$$ \lim_{M \to \infty} \int_{Q_M} F(\tau_y \bar{\omega}) \ell_M(dy) = \lim_{M \to \infty} \frac{1}{M^d} \int_{Q_M} F(\tau_y \bar{\omega}) dy = \int_{\Omega_r} F d\mu. \quad (3.1) $$

A density argument implies that (3.1) holds for any function $F$ in $U_{\bar{d}}(\Omega_r)$ and $\bar{\omega} \in \tilde{\Omega}_r$. The conclusion of the lemma follows then from [49, Theorem 1.1.1]. □

We can now state our version of the ergodic theorem.

**Theorem 3.3.** There exist a sequence $(\omega_n)_{n \in \mathbb{N}}$ in $\Omega_r$ and an increasing sequence of positive numbers $(M_n)_{n \in \mathbb{N}}$ such that $M_n \to \infty$ and

(i) each $\omega_n$ is $M_n$-periodic in each variable, i.e. $\tau_{M_n e_i} \omega_n = \omega_n$, for $n \in \mathbb{N}$ and $i = 1, \dots, d$;

(ii) the following sequence of probability measures on $(\Omega_r, \mathcal{B}(\Omega_r))$

$$ \mu_n(A) := \int_{\mathbb{T}_{M_n}^d} 1_A(\tau_y \omega_n) \ell_n(dy), \quad A \in \mathcal{B}(\Omega_r), \quad (3.2) $$
weakly converges to $\mu$, as $n \to \infty$, i.e.

$$ \lim_{n \to \infty} \int_{\Omega_r} F d\mu_n = \int_{\Omega_r} F d\mu, \quad F \in C_b(\Omega_r). \qquad (3.3) $$

*Proof.* Thanks to Lemma 3.2, there exists $\bar{\omega} \in \Omega_r$ such that

$$ \lim_{M \to \infty} \int_{Q_M} F(\tau_y \bar{\omega}) \ell_M(dy) = \int_{\Omega_r} F d\mu, \quad F \in C_b(\Omega_r). \qquad (3.4) $$

Fix an arbitrary $\rho > 0$ and $F \in U_{d_r}(\Omega_r)$. Let us consider increasing sequences of integers $(M'_n)_{n \in \mathbb{N}}$, $(M_n)_{n \ge 1}$ and $(\omega_n)_{n \in \mathbb{N}} \subset \Omega_r$ satisfying:

- each $\omega_n$ is $M_n$-periodic in every direction of the variable $x$;

- $M'_n < M_n$, $n = 1, 2, \dots$ and $\lim_{n \to \infty} \frac{M'_n}{M_n} = 1$;

- $\lim_{n \to \infty} \sup_{y \in Q_{M'_n}} d_r(\tau_y \omega_n, \tau_y \bar{\omega}) = 0$.

Thanks to (3.4), we have

$$ \lim_{n \to \infty} \left| \int_{Q_{M_n}} F(\tau_y \bar{\omega}) \ell_{M_n}(dy) - \int_{\Omega_r} F d\mu \right| = 0. \qquad (3.5) $$

Since $F \in U_{d_r}(\Omega_r)$, there exists $n_0$ such that

$$ |F(\tau_y \omega_n) - F(\tau_y \bar{\omega})| < \rho, \quad y \in Q_{M'_n}, n \ge n_0. \qquad (3.6) $$

Recalling the definition of the measures $\mu_n$ in (3.2), we infer that

$$ \begin{aligned} \left| \int_{\Omega_r} F d\mu_n - \int_{\Omega_r} F d\mu \right| &= \left| \int_{\mathbb{T}_{M_n}^d} F(\tau_y \omega_n) \ell_{M_n}(dy) - \int_{\Omega_r} F d\mu \right| \\ &\le M_n^{-d} \int_{Q_{M_n}} |F(\tau_y \omega_n) - F(\tau_y \bar{\omega})| dy + \left| \int_{Q_{M_n}} F(\tau_y \bar{\omega}) \ell_{M_n}(dy) - \int_{\Omega_r} F d\mu \right| \end{aligned} \qquad (3.7) $$

The second expression on the utmost right hand-side tends to 0, as $n \to \infty$, thanks to (3.5). The first one on the other hand can be estimated from (3.6) by

$$ \rho + \frac{1}{M_n^d} \int_{Q_{M_n} \setminus Q_{M'_n}} |F(\tau_y \omega_n) - F(\tau_y \bar{\omega})| dy \le \rho + 2 \|F\|_{\infty} \left[ 1 - \left( \frac{M'_n}{M_n} \right)^d \right]. $$

Since $\frac{M'_n}{M_n} \to 1$, as $n \to \infty$, the above estimate together with (3.7) imply that

$$ \limsup_{n \to \infty} \left| \int_{\Omega_r} F d\mu_n - \int_{\Omega_r} F d\mu \right| \le \rho $$

for any arbitrary $\rho > 0$, which in turn implies (3.3) for any $F \in U_d(\Omega_r)$. Finally, another application of [49, Theorem 1.1.1] allows us to conclude the proof of Theorem 3.3. □
## 3.2 Construction of an invariant density for the environment process

Let $(\omega_n)_{n \in \mathbb{N}}$ and $(M_n)_{n \in \mathbb{N}}$ be as in the statement of Theorem 3.3. For each $n \in \mathbb{N}$, we associate with $\omega_n \in \Omega_r$ the operator $\mathbf{q}(D; \omega_n) : C_c^2(\mathbb{R}^d) \to C_0(\mathbb{R}^d)$ as in (2.4) and the corresponding Feller process $(\mathbb{P}^{x;\omega_n})_{x \in \mathbb{R}^d}$. Let $\pi_n : \mathbb{R}^d \to \mathbb{T}_{M_n}^d$ be the canonical projection of $\mathbb{R}^d$ onto $\mathbb{T}_{M_n}^d$. Then, the process $(\tilde{\mathbb{P}}^{x;n})_{x \in \mathbb{T}_{M_n}^d}$ defined by $\tilde{\mathbb{P}}^{x;n} := (\pi_n)_\# \mathbb{P}^{x;\omega_n}$ is strongly Markovian and Feller, with the transition probability densities given by

$$ \tilde{p}_t^n(x, y) := \sum_{m \in \mathbb{Z}^d} p_t^{\omega_n}(x, y + M_n m), \quad t \ge 0, x, y \in \mathbb{T}_{M_n}^d. $$

Here, $p_t^{\omega_n}(x, y)$ are the transition probability densities corresponding to the path measure $\mathbb{P}^{x;\omega_n}$. In addition, for any $\tilde{f} \in C(\mathbb{T}_{M_n}^d)$ we have

$$ \tilde{\mathbb{E}}^{x;n} \tilde{f}(\tilde{X}_t) = \mathbb{E}^{x;\omega_n} f(X_t), \quad x \in \mathbb{T}_{M_n}^d, \qquad (3.8) $$

where $f \in C_b(\mathbb{R}^d)$ is the $M_n$-periodic extension of $\tilde{f}$, $\tilde{X}_t = \pi_n(X_t)$ and $\tilde{\mathbb{E}}^{x;n}$ is the expectation corresponding to $\tilde{\mathbb{P}}^{x;n}$. Moreover, the path measure $\tilde{\mathbb{P}}^{x;n}$ can be characterized as the unique solution to the martingale problem associated with the following integro-differential operator

$$ \tilde{L}^n u(x) = \int_{\mathbb{T}_{M_n}^d} (u(x+z) - u(x) - \nabla u(x) \cdot z \mathbf{1}_{\{|z| \le 1\}}) \tilde{n}_n(x, dz), \quad u \in C^2(\mathbb{T}_{M_n}^d), $$

with a Lévy kernel $\tilde{n}_n(x, dz)$ of the form

$$ \tilde{n}_n(x, dz) := \sum_{m \in \mathbb{Z}^d} \frac{\langle \mathbf{a}(\tau_x \omega_n)(z + M_n m), (z + M_n m) \rangle}{|z + M_n m|^{d+2+\alpha}} dz, \quad x \in \mathbb{R}^d. $$

**Proposition 3.4.** (i) For each $n \in \mathbb{N}$, there exists an invariant density $\phi_n$ for the process $(\tilde{\mathbb{P}}^{x;n})_{x \in \mathbb{T}_{M_n}^d}$, i.e. $\phi_n \ge 0$, $\ell_{M_n}$-a.e. in $\mathbb{T}_{M_n}^d$, $\|\phi_n\|_{L^1(\mathbb{T}_{M_n}^d)} = 1$ and

$$ \int_{\mathbb{T}_{M_n}^d} [\tilde{\mathbb{E}}^{x;n} f(\tilde{X}_t)] \phi_n(x) \ell_{M_n}(dx) = \int_{\mathbb{T}_{M_n}^d} f(x) \phi_n(x) \ell_{M_n}(dx), \quad t \ge 0, f \in B_b(\mathbb{T}_{M_n}^d). \quad (3.9) $$

(ii) Let $(\nu_n)_{n \in \mathbb{N}}$ be a sequence of probability measures on $(\Omega_r, \mathcal{B}(\Omega_r))$ defined as

$$ \nu_n(A) := \int_{\mathbb{T}_{M_n}^d} \mathbf{1}_A(\tau_x \omega_n) \phi_n(x) \ell_{M_n}(dx), \quad A \in \mathcal{B}(\Omega_r), \qquad (3.10) $$

with $\phi_n$ as in (i). Then, there exists $c_* > 0$, depending only on $d, \lambda, \Lambda, \alpha$, such that

$$ \int_{\Omega_r} F \, d\nu_n \le c_* \|F\|_{\infty}^{1-\alpha/(2d)} \left( \int_{\Omega_r} F d\mu_n \right)^{\alpha/(2d)}, \quad F \in B_b(\Omega_r). \qquad (3.11) $$

(iii) The sequence $(\nu_n)_{n \in \mathbb{N}}$ is tight. Any weak limiting point $\nu_*$ of $(\nu_n)_{n \in \mathbb{N}}$ satisfies

$$ \int_{\Omega_r} F \, d\nu_* \le c_* \|F\|_{\infty}^{1-\alpha/(2d)} \left( \int_{\Omega_r} F d\mu \right)^{\alpha/(2d)}, \qquad (3.12) $$

for any non-negative $F \in B_b(\Omega_r)$, with $c_*$ as in (3.11). In consequence, $\nu_*$ is absolutely continuous with respect to $\mu$ and its density $\Phi_* = \frac{d\nu_*}{d\mu}$ belongs to any $L^p(\mu)$, where $1 \le p < \frac{d}{d-\alpha/2}$.
Proof. Let $(\omega_n)_{n \in \mathbb{N}}$, $(M_n)_{n \in \mathbb{N}}$ be the sequences appearing in Theorem 3.3. We define a process $(\hat{\mathbb{P}}^{x;n})_{x \in \mathbb{T}^d}$ by letting $\hat{\mathbb{P}}^{x;n} := (\mathcal{J}_{M_n^{-1}})_{\sharp} \tilde{\mathbb{P}}^{x;M_n; n}$, where $\mathcal{J}_{\varepsilon}$ appeared in (2.11). Such a process corresponds to rescaling of the process $(\hat{\mathbb{P}}^{x;n})$ on $\mathbb{T}^d_{M_n}$, i.e. $\hat{\mathbb{P}}^{x;n} = \pi_{\sharp} \mathbb{P}_{M_n^{-1}}^{x;\omega_n}$, where the family $(\mathbb{P}_{\varepsilon}^{x;\omega})$ has been introduced in (2.10) and $\pi$ is the canonical projection of $\mathbb{R}^d$ onto the unit torus $\mathbb{T}^d$.

Let $\hat{X}_t := \pi(X_t)$ and let $\hat{P}_t^{(n)} f(x) := \hat{\mathbb{E}}^{x;n} f(\hat{X}_t)$, $t \ge 0$, $x \in \mathbb{T}^d$ be the transition probability semigroup on $B_b(\mathbb{T}^d)$ corresponding to $(\hat{\mathbb{P}}^{x;n})_{x \in \mathbb{T}^d}$. It is $C_b$-Feller, thanks to the fact that $(\mathbb{P}^{x;\omega_n})$ is Feller and [9, Theorem 1.9]. Its respective 1-resolvent operator is given by

$$ \hat{R}_1^{(n)} f(x) = \int_0^\infty e^{-t} \hat{P}_t^{(n)} f(x) dt, \quad x \in \mathbb{T}^d, f \in B_b(\mathbb{T}^d) \quad (3.13) $$

**Proof of (i).** Since $\mathbb{T}^d$ is compact and $(\hat{\mathbb{P}}^{x;n})_{x \in \mathbb{T}^d}$ is $C_b$-Feller, the existence of an invariant probability measure $\hat{m}_n$ for the process $(\hat{\mathbb{P}}^{x;n})_{x \in \mathbb{T}^d}$ follows from the Krylov-Bogoliubov theorem (cf. [18, Theorem 3.1.1]). By Theorem A.1, we know that $\hat{m}_n$ is absolutely continuous with respect to $\ell$, the normalized Lebesgue measure on $\mathbb{T}^d$, with a density $\hat{\phi}_n := d\hat{m}_n/d\ell$ such that $\|\hat{\phi}_n\|_{L^1(\mathbb{T}^d)} = 1$. We then write

$$ \int_{\mathbb{T}^d} \hat{\mathbb{E}}^{x;n} [f(\hat{X}_t)] \hat{\phi}_n(x) \ell(dx) = \int_{\mathbb{T}^d} f(x) \hat{\phi}_n(x) \ell(dx), \quad t \ge 0, f \in B_b(\mathbb{T}^d). \quad (3.14) $$

Let us denote $\phi_n := \hat{\phi}_n \circ j_{M_n}$, where $j_{M_n}(x) := x/M_n$. It is a density with respect to the normalized Lebesgue measure on $\mathbb{T}^d_{M_n}$. We claim that $\phi_n(x)$ is invariant for the process $(\tilde{\mathbb{P}}^{x;n})_{x \in \mathbb{T}^d_{M_n}}$. Indeed, for any $f \in B_b(\mathbb{T}^d_{M_n})$, we have by (3.14)

$$ \begin{aligned} \int_{\mathbb{T}^d_{M_n}} f(x) \phi_n(x) \ell_{M_n}(dx) &= \int_{\mathbb{T}^d} f \circ j_{M_n}^{-1}(x) \hat{\phi}_n(x) \ell(dx) \\ &= \int_{\mathbb{T}^d} \hat{\mathbb{E}}^{x;n} [f \circ j_{M_n}^{-1}(\hat{X}_{tM_n^{-\alpha}})] \hat{\phi}_n(x) \ell(dx) = \int_{\mathbb{T}^d_{M_n}} \tilde{\mathbb{E}}^{x;n} [f(\tilde{X}_t)] \phi_n(x) \ell_{M_n}(dx), \end{aligned} $$

and thus, we have concluded the proof of (i).

**Proof of (ii).** Let $F \in B_b(\Omega_r)$ be non-negative and such that $\|F\|_{\infty} \le 1$. Then,

$$ \int_{\Omega_r} F \, d\nu_n = \int_{\mathbb{T}^d_{M_n}} F(\tau_x \omega_n) \phi_n(x) \ell_{M_n}(dx) = \int_{\mathbb{T}^d} F(\tau_{M_n x} \omega_n) \hat{\phi}_n(x) \ell(dx). \quad (3.15) $$

Since $\hat{\phi}_n(x)$ is invariant under the measure $(\hat{\mathbb{P}}^{x;n})_{x \in \mathbb{T}^d}$, we have $(\hat{R}_1^{(n)})^* \hat{\phi}_n = \hat{\phi}_n$. Using this and Theorem A.1, the utmost right-hand side of (3.15) can be rewritten as

$$ \begin{aligned} \int_{\mathbb{T}^d} F(\tau_{M_n x} \omega_n) (\hat{R}_1^{(n)})^* \hat{\phi}_n(x) \ell(dx) &= \int_{\mathbb{T}^d} \hat{R}_1^{(n)} (F(\tau_{M_n} \cdot \omega_n))(x) \hat{\phi}_n(x) \ell(dx) \\ &\le \|\hat{R}_1^{(n)}(F(\tau_{M_n} \cdot \omega_n))\|_{\infty} \le \bar{C} \left( \int_{\mathbb{T}^d} F(\tau_{M_n x} \omega_n) \ell(dx) \right)^{\alpha/(2d)} \\ &= \bar{C} \left( \int_{\mathbb{T}^d_{M_n}} F(\tau_x \omega_n) \ell_{M_n}(dx) \right)^{\alpha/(2d)} = \bar{C} \left( \int_{\Omega_r} F d\mu_n \right)^{\alpha/(2d)}, \end{aligned} $$
where $\bar{C}$ is the same constant as in Theorem A.1. This ends the proof of (3.11) when $0 \le F \le 1$. The estimate follows for a general non-negative $F \in B_b(\Omega_r)$ by considering $F/\|F\|_\infty$.

**Proof of (iii).** According to Theorem 3.3, the sequence $(\mu_n)_{n \in \mathbb{N}}$ weakly converges to $\mu$. It is therefore tight: for any $\varepsilon > 0$, there exists a compact set $K \subset \Omega_r$ such that $\mu_n(K^c) < \varepsilon$ for any $n \in \mathbb{N}$. Hence, by the already proved estimate (3.11), we show that

$$ \nu_n(K^c) \le c_* \mu_n^{\alpha/(2d)}(K^c) \le c_* \varepsilon^{\alpha/(2d)}, $$

which proves tightness of the sequence $(\nu_n)_{n \in \mathbb{N}}$. Letting $n \to +\infty$ in (3.11), we conclude (3.12) for $F \in C_b(\Omega_r)$. Since $\Omega_r$ is Polish, by the Ulam theorem, the measure $\mu + \nu_*$ is Radon and therefore $C_b(\Omega_r)$ is dense in $L^1(\mu + \nu_*)$, see e.g. [20, Proposition 7.9]. Suppose that $F \in B_b(\Omega_r)$ and $(F_n) \subset C_b(\Omega_r)$ is such that $\lim_{n \to +\infty} \|F_n - F\|_{L^1(\mu+\nu_*)} = 0$. By considering $0 \vee (F_n \wedge \|F\|_\infty)$, we can always assume that $0 \le F_n \le \|F\|_\infty$. Each $F_n$ satisfies (3.12). Letting $n \to +\infty$, we conclude the estimate for the limiting $F$. From (3.12), we infer that $\nu_*$ is absolutely continuous with respect to $\mu$. According to the estimate, its density $\Phi_*$ satisfies

$$ \lambda\mu[\Phi_* \ge \lambda] \le \int_{\Omega_r} \mathbf{1}_{[\Phi_* \ge \lambda]} \Phi_* d\mu \le c_* \mu[\Phi_* \ge \lambda]^{\alpha/(2d)}, \quad \lambda \ge 0. $$

Therefore,

$$ \mu[\Phi_* \ge \lambda] \le \frac{C}{\lambda^{2d/(2d-\alpha)}}, $$

for some constant $C > 0$. Thus,

$$ \int_{\Omega} \Phi_{*}^{p} d\mu = p \int_{0}^{+\infty} \lambda^{p-1} \mu[\Phi_{*} \ge \lambda] d\lambda \le p + C \int_{1}^{+\infty} \frac{d\lambda}{\lambda^{1+2d/(2d-\alpha)-p}} < +\infty, \quad (3.16) $$

provided that $1 \le p < 2d/(2d - \alpha)$, which ends the proof of part (iii). □

In order to conclude the proof of Theorem 3.1, we need the following lemma asserting the $C_b$-Feller property for the semigroup generated by the environment process.

**Lemma 3.5.** The semigroup $(\mathfrak{P}_t)_{t \ge 0}$ given by (2.14) is $C_b$-Feller, i.e.

$$ \mathfrak{P}_t(C_b(\Omega_r)) \subseteq C_b(\Omega_r), \quad t > 0. $$

*Proof.* Fix $t > 0$ and $F \in C_b(\Omega_r)$. Let $(\omega_n)_{n \in \mathbb{N}} \subset \Omega_r$ and $\omega \in \Omega_r$ be such that $d_r(\omega_n, \omega) \to 0$, as $n \to \infty$. Then the respective path measures $\mathbb{P}^{0,\omega_n}$ converge to $\mathbb{P}^{0,\omega}$, weakly over $\mathcal{D}$. Tightness of the laws of $(X_t)$ under $\mathbb{P}^{0,\omega_n}$ implies in particular that for any $\varepsilon > 0$ there exists a compact set $K \subset \mathbb{R}^d$ such that

$$ \mathbb{P}^{0,\omega_n}(X_t \notin K) < \varepsilon, n \ge 1 \quad \text{and} \quad \mathbb{P}^{0,\omega}(X_t \notin K) < \varepsilon. $$

Then,

$$ |\mathfrak{P}_t F(\omega_n) - \mathfrak{P}_t F(\omega)| \le |\mathbb{E}^{0;\omega_n}[F(\tau_{X_t}\omega_n), X_t \in K] - \mathbb{E}^{0;\omega}[F(\tau_{X_t}\omega), X_t \in K]| + 2\varepsilon \|F\|_\infty. $$
The function $F$, when restricted to the compact set $\mathcal{H}(K) := \{\tau_y(\omega): y \in K, \omega \in \mathcal{H}\}$, where $\mathcal{H} := \{\omega, \omega_1, \omega_2, \dots\}$, is uniformly continuous. Since $d_r(\omega_n, \omega) \to 0$, we have

$$ \lim_{n \to \infty} |\mathbb{E}^{0;\omega_n}[F(\tau_{X_t}\omega_n), X_t \in K] - \mathbb{E}^{0;\omega_n}[F(\tau_{X_t}\omega), X_t \in K]| = 0. \quad (3.17) $$

The function $y \mapsto F(\tau_y\omega)$ is bounded and continuous on $\mathbb{R}^d$ for each fixed $\omega$. Since $(\mathbb{P}^{0;\omega})$ is a Feller, càdlàg process, its paths are quasi-left-continuous, see e.g. [6, Proposition 1.2.7, p. 21]. Thus, $\mathbb{P}^{0;\omega}[F(\tau_{X_t}\omega) = F(\tau_{X_{t-}}\omega)] = 1$. From the above and the fact that $\mathbb{P}^{0;\omega_n}$ weakly converge to $\mathbb{P}^{0;\omega}$ in $\mathcal{D}$, [7, Theorem 2.7] implies that

$$ \lim_{n \to \infty} [\mathbb{E}^{0;\omega_n}[F(\tau_{X_t}\omega)] - \mathbb{E}^{0;\omega}[F(\tau_{X_t}\omega)]] = 0. \quad (3.18) $$

Hence,

$$ \limsup_{n \to \infty} |\mathbb{E}^{0;\omega_n}[F(\tau_{X_t}\omega), X_t \in K] - \mathbb{E}^{0;\omega}[F(\tau_{X_t}\omega), X_t \in K]| \le 2\varepsilon \|F\|_{\infty}. \quad (3.19) $$

Summarizing, we have shown

$$ \limsup_{n \to \infty} |\mathfrak{P}_t F(\omega_n) - \mathfrak{P}_t F(\omega)| \le 4\varepsilon \|F\|_{\infty}, \quad \varepsilon > 0. $$

Thus, $|\mathfrak{P}_t F(\omega_n) - \mathfrak{P}_t F(\omega)| \to 0$, as $n \to \infty$, and the conclusion of the lemma follows. $\square$

### 3.3 Conclusion of the proof of Theorem 3.1

We show that any weak limiting point $\nu_*$ for the sequence $(\nu_n)_{n \in \mathbb{N}}$, defined in (3.10), is invariant for the environment process $(\eta_t)_{t \ge 0}$ introduced in (2.13). Let us fix $F$ in $C_b(\Omega_r)$. We have

$$ \int_{\Omega_r} \mathfrak{P}_t F(\omega) \nu_n(d\omega) = \int_{\Omega} \mathbb{E}^{0;\omega} F(\eta_t(\omega)) \nu_n(d\omega) = \int_{\mathbb{T}_{M_n}^d} \mathbb{E}^{0;\tau_x\omega_n} F(\tau_{X_t}\tau_x\omega_n) \phi_n(x) \ell_{M_n}(dx), \quad n \in \mathbb{N}. $$

By virtue of (2.9) and (3.8) we can rewrite the utmost right-hand side as

$$ \int_{\mathbb{T}_{M_n}^d} \mathbb{E}^{x;\omega_n} F(\tau_{X_t}\omega_n) \phi_n(x) \ell_{M_n}(dx) = \int_{\mathbb{T}_{M_n}^d} \mathbb{E}^{x;n} F(\tau_{\tilde{X}_t}\omega_n) \phi_n(x) \ell_{M_n}(dx). $$

Using (3.9), we conclude that the right hand side equals

$$ \int_{\mathbb{T}_{M_n}^d} F(\tau_x\omega_n) \phi_n(x) \ell_{M_n}(dx) = \int_{\Omega_r} F(\omega) \nu_n(d\omega). $$

Since $\mathfrak{P}_t F \in C_b(\Omega_r)$, see Lemma 3.5, from the weak convergence of $(\nu_n)$ to $\nu_*$ and the above argument

$$ \int_{\Omega_r} \mathfrak{P}_t F \, d\nu_* = \lim_{n \to \infty} \int_{\Omega_r} \mathfrak{P}_t F \, d\nu_n = \lim_{n \to \infty} \int_{\Omega_r} F \, d\nu_n = \int_{\Omega_r} F \, d\nu_*, $$

which proves the invariance of any weak limiting point $\nu_*$.

Thanks to part (iii) of Proposition 3.4, $\nu_*$ - any weak limiting point $(\nu_n)_{n \ge 1}$ - is absolutely continuous w.r.t. $\mu$. We claim that $\Phi_* := \frac{d\nu_*}{d\mu} > 0$, $\mu$-a.s. in $\Omega_r$. Indeed, let $A := \{\omega \in \Omega_r: \Phi_*(\omega) = 0\}$. Clearly, $\mu(A) < 1$, as $\Phi_*$ is a density w.r.t. $\mu$. Suppose that $\mu(A) > 0$. Let

$$ \phi_A(y; \omega) = \mathbb{1}_A(\tau_y\omega) \quad \text{and} \quad B^\omega := \{y \in \mathbb{R}^d: \tau_y\omega \in A\}. \quad (3.20) $$
We have, see (2.7)-(2.8),

$$
\begin{align*}
0 = \int_0^\infty e^{-t} dt \left( \int_{\Omega_r} \mathbb{1}_A \Phi_* d\mu \right) &= \int_0^\infty e^{-t} dt \left( \int_{\Omega_r} \mathfrak{P}_t \mathbb{1}_A \Phi_* d\mu \right) \\
&= \int_{\Omega_r} \Phi_*(\omega) \mu(d\omega) \int_0^\infty e^{-t} P_t^\omega \phi_A(0; \omega) dt = \int_{A^c} R_1^\omega \phi_A(0; \omega) \Phi_*(\omega) \mu(d\omega).
\end{align*}
$$

Thus,

$$
R_1^\omega \phi_A(0; \omega) = 0 \quad \text{for } \mu\text{-a.e. } \omega \in A^c. \tag{3.21}
$$

Let $(x_n)$ be a dense subset of $\mathbb{R}^d$ and $\nu$ be Borel probability measure on $\mathbb{R}^d$ given by $\nu(dy) := \sum_{n=1}^\infty 2^{-n} \delta_{x_n}(dy)$. We let $\mathfrak{m} := \nu R_1$, i.e.

$$
\int_{\mathbb{R}^d} f \, d\mathfrak{m} = \int_{\mathbb{R}^d} R_1 f \, d\nu, \quad f \in B_b(\mathbb{R}^d). \tag{3.22}
$$

Thanks to Proposition B.2 condition (3.21) implies that for $\mu$-a.e. $\omega \in A^c$, $\mathfrak{m}(B^\omega) = 0$, where $B^\omega$ is defined in (3.20). In other words

$$
\int_{\mathbb{R}^d} \mu(\tau_{-y}(A) \setminus A) \mathfrak{m}(dy) = 0.
$$

Therefore, there exists a Borel subset $Z$ of $\mathbb{R}^d$ such that $\mathfrak{m}(Z) = 1$ and $\mu(\tau_{-y}(A) \setminus A) = 0$ for all $y \in Z$. According to Proposition B.1, the set $Z$ is dense in $\mathbb{R}^d$. Thanks to (2.1) and the continuity of $x \mapsto \mathbb{1}_A \circ \tau_x$ in $L^1(\mu)$, it follows that $\mu(\tau_y(A)\Delta A) = 0$ for all $y \in \mathbb{R}^d$, where $\Delta$ denotes the symmetric difference of sets. This in turn implies that $\mu(A) \in \{0, 1\}$, due to ergodicity of $\mu$ under $\tau_x$, which contradicts the assumption that $0 < \mu(A) < 1$. Thus, we have shown that $\Phi_* > 0$, $\mu$-a.s. in $\Omega_r$. Actually, the above argument proves that the density of any invariant measure, which is absolutely continuous with respect to $\mu$, has to be strictly positive, $\mu$ a.s. in $\Omega_t$. Ergodicity of $\nu_*$ is then an easy consequence of this fact. Indeed, if there had existed $A$ such that $\nu_*(A) \in (0, 1)$ and $\mathbb{1}_A(\eta_t(\omega)) = \mathbb{1}_A(\omega)$, $\nu_*$-a.s. in $\Omega_r$, then both measures $\nu_*(A)^{-1}\mathbb{1}_A\Phi_*d\nu_*$ and $\nu_*(A^c)^{-1}\mathbb{1}_{A^c}\Phi_*d\nu_*$ would have been invariant and of disjoint supports, which leads to a contradiction. This ends the proof of Theorem 3.1. $\square$

3.4 Proof of Theorem 3.1 in the general case

In the present section we shall dispense with the additional regularity assumptions on the coefficients
that has been made in the previous section. Inspecting the proof of Lemma 3.5 we can see that the
argument required only that $(\omega_n)_{n \in \mathbb{N}} \subset \Omega^{\mathcal{M}}$ and $\omega \in \Omega^{\mathcal{M}}$. Therefore, we can conclude the following
variant of the lemma.

**Lemma 3.6.** Suppose that $F \in C_b(\Omega^{\mathcal{M}})$ and $t \ge 0$. Then $\mathfrak{P}_t F|_{\Omega^{\mathcal{M}}} \in C_b(\Omega^{\mathcal{M}})$.

Let

$$
I_r(\omega)(x) := r I_d + (j_r * \omega)(x), \quad x \in \mathbb{R}^d, \omega \in \Omega,
$$

where $j_r$ is a standard smooth mollifier, i.e. $j_r(x) = r^{-d}j(x/r)$, with $j \in C_c^\infty(\mathbb{R}^d)$, non-negative and $\int_{\mathbb{R}^d} j(x)dx = 1$. We have $I_r : \Omega \to \Omega_r$. Let $\omega_r := I_r(\omega)$ and $\mu^r = (I_r)_\#\mu$. We see that
$\sup_{\omega \in \Omega} d(I_r(\omega), \omega) \to 0$ if $r \downarrow 0$. Recall that by [16, Theorem 8.3.7], $\Omega_r$ is a Borel measurable subset of $\Omega$. Each measure $\mu^r$ can be extended to $\mathcal{B}(\Omega)$ by letting $\mu^r(\Omega \setminus \Omega_r) = 0$. Moreover, the space $\Omega^{\mathcal{M}}$, metrized with the metric obtained by the restriction of $d$, is separable.

By Theorem 3.1, for each $r > 0$, there exists an invariant, ergodic measure $\mu_*^r$ on $\Omega_r$ for the semigroup $(\mathfrak{P}_t)_{t \ge 0}$ on $\Omega_r$. The measure is absolutely continuous w.r.t. $\mu^r$ and $d\mu_*^r = \Phi_*^r d\mu^r$, where $\Phi_*^r$ is positive $\mu^r$-a.s. in $\Omega_r$. We extend $\mu_*^r$ to $\mathcal{B}(\Omega)$ by letting it equal to zero on $\Omega \setminus \Omega_r$. Observe that $\mu^r$ converge weakly to $\mu$, as $r \downarrow 0$. By (3.12), we have for any non-negative $F \in B_b(\Omega)$,

$$ \int_{\Omega} F d\mu_{*}^{r} = \int_{\Omega_{r}} F d\mu_{*}^{r} \le c \|F\|_{\infty}^{1-\frac{\alpha}{2d}} \left[ \int_{\Omega_{r}} F(\omega)\mu^{r}(d\omega) \right]^{\frac{\alpha}{2d}} = c \|F\|_{\infty}^{1-\frac{\alpha}{2d}} \left( \int_{\Omega} F(I_{r}(\omega))\mu(d\omega) \right)^{\frac{\alpha}{2d}}, $$

with $c$ independent of $r \in (0, 1]$. Thus, up to a sub-sequence, $\mu_*^r$ converges weakly, as $r \to 0$, over $\Omega$ to some measure $\mu_*$ which satisfies

$$ \int_{\Omega} F d\mu_* \le c \|F\|_{\infty}^{1-\frac{\alpha}{2d}} \left[ \int_{\Omega} F d\mu \right]^{\frac{\alpha}{2d}}, $$

for all non-negative $F \in B_b(\Omega)$. This, in turn, implies that $\mu_*$ is absolutely continuous with respect to $\mu$ and $\Phi_* := d\mu_*/d\mu$ belongs to $L^p(\mu)$ for any $p \in [1, d/(d - \alpha/2))$ (cf. the proof of Proposition 3.4(iii)). Since $\Omega^{\mathcal{M}}$ is dense in $\Omega$, we infer that measures $\mu_*^r$, restricted to $\Omega^{\mathcal{M}}$, converge weakly, as $r \to 0$, to the restriction of $\mu_*$. Indeed, any $F \in C_b(\Omega^{\mathcal{M}})$ that is uniformly continuous can be uniquely extended to $\tilde{F} \in C_b(\Omega)$. Therefore (with some abuse of notation, we denote by the same symbol for measures and their restrictions)

$$ \lim_{r \downarrow 0} \int_{\Omega^{\mathcal{M}}} F d\mu_*^r = \lim_{r \to 0} \int_{\Omega} \tilde{F} d\mu_*^r = \int_{\Omega} \tilde{F} d\mu_* = \int_{\Omega^{\mathcal{M}}} F d\mu_* $$

By virtue of [7, Theorem 2.1] the above implies the convergence in question. Consequently, by Lemma 3.6 we conclude that

$$ \int_{\Omega} \mathfrak{P}_t F d\mu_* = \int_{\Omega^{\mathcal{M}}} \mathfrak{P}_t F d\mu_* = \lim_{r \downarrow 0} \int_{\Omega^{\mathcal{M}}} \mathfrak{P}_t F d\mu_*^r = \lim_{r \downarrow 0} \int_{\Omega_r} \mathfrak{P}_t F d\mu_*^r. $$

Using the version of Theorem 3.1 already proved on $\Omega_r$ we infer that the utmost right hand side equals

$$ \lim_{r \downarrow 0} \int_{\Omega_r} F d\mu_*^r = \lim_{r \downarrow 0} \int_{\Omega} F d\mu_*^r = \int_{\Omega} F d\mu_* $$

The proof that $\mu_*$ is ergodic can be conducted in the same way as in Section 3.3.

## A Aleksandrov-Bakelman-Pucci estimates

Let $(\mathbb{P}^x)_{x \in \mathbb{R}^d}$ be a Feller process corresponding to the generator $L$ of the form (1.1), where the matrix valued function $x \mapsto a(x) = [a_{i,j}(x)]_{i,j=1}^d$ satisfies the assumptions made in Section 1. In the present section we do not assume that the coefficients $a(x)$ are random.

Let $D \subset \mathbb{R}^d$ be an open set. Define the exit time $\tau_D : \mathcal{D} \to [0, \infty]$ of the canonical process $(X_t)_{t \ge 0}$, see (2.2), from $D$ as

$$ \tau_D := \inf \{t > 0 : X_t \notin D\}. $$
It is a stopping time, i.e. {$\tau_D \le t$} $\in \mathcal{F}_t$ for any $t \ge 0$. The transition semigroup on $D$ with the null exterior condition is defined as

$$P_t^D f(x) := \mathbb{E}^x [f(X_t), t < \tau_D], \quad t \ge 0, x \in D, f \in B_b(D).$$

Furthermore, for any $\beta > 0$, we introduce the $\beta$-resolvent of $L$ on $D$ for any $f \in B_b(D)$ by letting

$$R_{\beta}^{D} f(x) := \int_{0}^{\infty} e^{-\beta t} P_{t}^{D} f(x) dt = \mathbb{E}^{x} \left[ \int_{0}^{\tau_{D}} e^{-\beta t} f(X_{t}) dt \right], \quad x \in D.$$

If we suppose that

$$\mathbb{E}^x \tau_D < \infty, \quad x \in D, \qquad (A.1)$$

then, we can define the resolvent $R^D f$ also for $\beta = 0$. Without the assumption (A.1) the resolvent $R^D f(x)$ can be defined for a non-negative $f$ (not necessarily bounded), but we have to admit the possibility that it equals $\infty$ for some $x$.

Let $\pi : \mathbb{R}^d \to \mathbb{T}^d$ be the canonical projection of $\mathbb{R}^d$ onto $\mathbb{T}^d$. Let $C_{\text{per}}(\mathbb{R}^d)$ be the space of all continuous functions which are 1-periodic in each variable. There is a one-to-one correspondence between $C_{\text{per}}(\mathbb{R}^d)$ and $C(\mathbb{T}^d)$, i.e. for every $\tilde{f} \in C(\mathbb{T}^d)$, there exists a unique $f \in C_{\text{per}}(\mathbb{R}^d)$ such that $f(x) = \tilde{f} \circ \pi(x)$, $x \in \mathbb{R}^d$.

Let us suppose that the function $x \mapsto a(x)$ is 1-periodic, as it is the case discussed in Section 3.2. Thus, the corresponding transition probability semigroup $(P_t)_{t \ge 0}$ has the following property

$$P_t(C_{\text{per}}(\mathbb{R}^d)) \subseteq C_{\text{per}}(\mathbb{R}^d), \quad t > 0.$$

The semigroup $(P_t)_{t \ge 0}$ induces then a strongly continuous semigroup $(\tilde{P}_t)_{t \ge 0}$ on $C(\mathbb{T}^d)$, by virtue of [9, Lemma 1.18]. The respective process $\tilde{\mathbb{P}}^x := (\pi)_x \mathbb{P}^x$, $x \in \mathbb{T}^d$, is strongly Markovian and Feller on $\mathbb{T}^d$, with transition probabilities given by

$$\tilde{P}_t(x, dy) = \sum_{m \in \mathbb{Z}^d} P_t(x, dy + m), \quad t > 0, x, y \in \mathbb{T}^d.$$

Here, $P_t(x, dy)$ are the transition probabilities corresponding to the path measure $\mathbb{P}^x$.

In addition, for any $\tilde{f} \in C(\mathbb{T}^d)$, we have

$$\tilde{\mathbb{E}}^x \tilde{f}(\tilde{X}_t) = \mathbb{E}^x f(X_t), \quad t \ge 0, x \in \mathbb{T}^d,$$

where $f \in C_{\text{per}}(\mathbb{R}^d)$ is the 1-periodic extension of $\tilde{f}$, $\tilde{X}_t = \pi(X_t)$ and $\tilde{\mathbb{E}}^x$ is the expectation corresponding to $\tilde{\mathbb{P}}^x$. The 1-resolvent corresponding to $(\tilde{P}_t)_{t \ge 0}$ is then given by

$$\tilde{R}_1 \tilde{f}(x) := \int_0^\infty e^{-t} \tilde{P}_t \tilde{f}(x) dt, \quad x \in \mathbb{T}^d, \tilde{f} \in B_b(\mathbb{T}^d).$$

**Theorem A.1.** We have

$$\|\tilde{R}_1 \tilde{f}\|_{L^\infty(\mathbb{T}^d)} \le \bar{C} \|\tilde{f}\|_{L^\infty(\mathbb{T}^d)}^{(1-\alpha/2)} \|\tilde{f}\|_{L^d(\mathbb{T}^d)}^{\alpha/2}, \quad \tilde{f} \in L^p(\mathbb{T}^d),$$

where the constant $\bar{C}$ depends only on $\alpha, d, \lambda, \Lambda$.
Proof. Recall that $B(x,r)$ denotes a ball centered at $x$ with radius $r > 0$. Let $D := \{x \in \mathbb{R}^d: \text{dist}(x, Q_1) < 1\}$. Suppose also that $f \in C_{\text{per}}(\mathbb{R}^d)$ is such that $f = \tilde{f} \circ \pi$. Using the strong Markov property, we can then write for any $x \in \mathbb{T}^d$:

$$ \tilde{R}_1 \tilde{f}(x) = \mathbb{E}^x \left[ \int_0^{\tau_D} e^{-t} f(X_t) dt \right] + \mathbb{E}^x \left[ e^{-\tau_D} \int_0^{\infty} e^{-t} \mathbb{E}^{X_{\tau_D}} [f(X_t)] dt \right]. $$

We have then

$$ \sup_{x \in \mathbb{T}^d} |\tilde{R}_1 \tilde{f}(x)| \le \sup_{x \in \mathbb{T}^d} \mathbb{E}^x \left[ \int_0^{\tau_D} e^{-s} |f(X(s))| ds \right] + \sup_{x \in \mathbb{T}^d} \mathbb{E}^x [e^{-\tau_D}] \sup_{x \in \mathbb{T}^d} \left| \mathbb{E}^x \int_0^{\infty} e^{-s} f(X(s)) ds \right|. \quad (A.2) $$

Since

$$ u(x) = \mathbb{E}^x \left[ \int_0^{\tau_D} e^{-s} |f(X(s))| ds \right] $$

is a non-negative solution of the equation

$$ (u - Lu)(x) = |f(x)|, \quad x \in D, \quad u(x) = 0, \quad x \notin D, $$

by [23, Theorem 1.3] there exists a constant $C > 0$ such that the first term on the right hand side of (A.2) can be estimated by

$$ C \|f\|_{L^{\infty}(D)}^{(1-\alpha/2)} \|f\|_{L^d(D)}^{\alpha/2}. $$

In consequence,

$$ \sup_{x \in \mathbb{T}^d} |\tilde{R}_1 \tilde{f}(x)| \le C \|f\|_{L^{\infty}(D)}^{(1-\alpha/2)} \|f\|_{L^d(D)}^{\alpha/2} + \sup_{x \in \mathbb{T}^d} \mathbb{E}^x [e^{-\tau_D}] \sup_{x \in \mathbb{T}^d} |\tilde{R}_1 \tilde{f}(x)|. $$

Since $D \subseteq [-3, 3]^d$, we can now use periodicity of $f$ to conclude that

$$ \gamma \sup_{x \in \mathbb{T}^d} |\tilde{R}_1 \tilde{f}(x)| \le 3^d C \|\tilde{f}\|_{L^{\infty}(\mathbb{T}^d)}^{(1-\alpha/2)} \|\tilde{f}\|_{L^d(\mathbb{T}^d)}^{\alpha/2}, \quad (A.3) $$

where

$$ \gamma := 1 - \sup_{x \in \mathbb{T}^d} \mathbb{E}^x [e^{-\tau_D}]. \quad (A.4) $$

Since $B(x, 1) \subseteq D$ for any $x \in \mathbb{T}^d$, by [35, Proposition 3.8] we infer that for each $t \in (0, 1]$

$$ \begin{aligned} 1 - \mathbb{E}^x [e^{-\tau_D}] &\ge 1 - \mathbb{E}^x [e^{-\tau_{B(x,1)}}] \ge (1 - e^{-t}) \mathbb{P}^x(\tau_{B(x,1)} > t) \\ &\ge (1 - e^{-t}) (1 - C(d) t q_*) . \end{aligned} $$

Here a positive constant $C(d)$ depends only on the dimension $d$ and

$$ q_* := \sup_{x \in \mathbb{T}^d} \sup_{y \in B(x, 1)} \sup_{|\xi| \le 1} |q(y, \xi)| \le \Lambda(dC_{d,\alpha} + c_{d,\alpha}), $$

see (2.10)

Finally, if one chooses $t := (2C(d)\Lambda(dC_{d,\alpha} + c_{d,\alpha}))^{-1}$, then

$$ \gamma > \frac{1}{2} (1 - e^{-t}) $$

and the conclusion of the theorem follows.

<figure><img src="image_2.png" /></figure>
## B Some property of the resolvent

As in the previous section, let $(\mathbb{P}^x)_{x \in \mathbb{R}^d}$ be a Feller process corresponding to the generator $L$ of the form (1.1), where the matrix valued function $x \mapsto a(x) = [a_{i,j}(x)]_{i,j=1}^d$ satisfies the assumptions made in Section 1. By [23, Theorem 1.3], $(\mathbb{P}_x)_{x \in \mathbb{R}^d}$ is in fact a strongly Feller process. The set $N_x := \{y \in \mathbb{R}^d : \langle a(x)y, y \rangle = 0\}$ is a linear space and by (1.3) it cannot be the entire $\mathbb{R}^d$. As a result $\text{supp } n(x, \cdot) = \mathbb{R}^d$ for each $x \in \mathbb{R}^d$. Recall that measure $\mathfrak{m}$ has been introduced in (3.22).

**Proposition B.1.** Let $B$ be a Borel subset of $\mathbb{R}^d$ such that $\mathfrak{m}(B) = 0$. Then $B^c$ is a dense subset of $\mathbb{R}^d$.

*Proof.* Suppose by contradiction that there exists an open ball $B(x_0, r_0) \subseteq B$. Then,

$$R_1 \mathbb{1}_{B(x_0, r_0)}(x_n) = 0, \quad n \ge 1.$$

By the strong Feller property of $(\mathbb{P}_x)_{x \in \mathbb{R}^d}$, the function $x \mapsto R_1 \mathbb{1}_{B(x_0, r_0)}(x)$ is continuous and thus, $R_1 \mathbb{1}_{B(x_0, r_0)}(x) = 0, x \in \mathbb{R}^d$. As a result,

$$0 = R_1 \mathbb{1}_{B(x_0, r_0)}(x_0) = \mathbb{E}^{x_0} \int_0^\infty e^{-t} \mathbb{1}_{B(x_0, r_0)}(X_t) dt \ge \mathbb{E}^{x_0} \int_0^{\tau_{B(x_0, r_0)}} e^{-t} dt,$$

which leads to a contradiction. □

**Proposition B.2.** Let $B$ be a Borel subset of $\mathbb{R}^d$ such that $\mathfrak{m}(B) > 0$. Then,

$$R_1 \mathbb{1}_B(x) > 0, \quad \text{for all } x \in \mathbb{R}^d.$$

*Proof.* Let $g \in C_b(\mathbb{R}^d)$ be non-negative. By [17, Lemma 3.5] (see also [5, Proposition 1]) $u := R_1 g$ is a viscosity solution to

$$-Lu + u = g \quad \text{in } \mathbb{R}^d. \qquad (B.1)$$

Let $f \in L^d(\mathbb{R}^d)$, and let $(f_n)_{n \in \mathbb{N}} \subset C_b(\mathbb{R}^d)$ be such that $0 \le f_n \le 1$, $n \ge 1$ and $\|f_n - f\|_{L^d} \le 1/n$, $n \ge 1$. By [23, Theorem 1.3], we have for any $r \ge 0$

$$\sup_{|x|<r} |R_1 f_n(x) - R_1 f(x)| \to 0, \quad n \to \infty.$$

Consequently, by [5, Theorem 1] $R_1 f$ is a viscosity supersolution to (B.1) with $g$ replaced by 0. By [15, Theorem 2] if $R_1 f(x)$ for some $x \in \mathbb{R}^d$, then $R_1 f \equiv 0$. By [32, Theorem 4.10] once we prove that $e^{-t} \mathfrak{m} P_t \le \mathfrak{m}$, $t > 0$, then the asserted implication follows. The last inequality, however, is a direct consequence of the definition of $\mathfrak{m}$. □

## C Calculation on Fourier symbol

Using the spherical change of coordinates $y = \ell \boldsymbol{\sigma}$, $\ell > 0$, $\boldsymbol{\sigma} \in \mathbb{S}^{d-1}$, in the integral, we can write

$$\mathbf{q}(\xi; \omega) = |\xi|^\alpha \int_0^{+\infty} \frac{d\ell}{\ell^{\alpha+1}} \int_{\mathbb{S}^{d-1}} F(\ell \boldsymbol{\sigma} \cdot \hat{\xi}) \langle \mathbf{a}(\omega) \boldsymbol{\sigma}, \boldsymbol{\sigma} \rangle S(d\boldsymbol{\sigma}), \quad (\xi, \omega) \in \mathbb{R}^d \times \Omega,$$
where $F(t) = 1 - \cos(t)$, $t \in \mathbb{R}$, and $\hat{\xi} = \xi/|\xi|$. Observe that

$$ \int_{\mathbb{S}^{d-1}} F(\ell \boldsymbol{\sigma} \cdot \hat{\xi}) \langle \mathbf{a}(\omega) \boldsymbol{\sigma}, \boldsymbol{\sigma} \rangle S(d\boldsymbol{\sigma}) = \sum_{i \neq j} \int_{\mathbb{S}^{d-1}} F(\ell \boldsymbol{\sigma} \cdot \hat{\xi}) \mathbf{a}_{ij}(\omega) \boldsymbol{\sigma}_i \boldsymbol{\sigma}_j S(d\boldsymbol{\sigma}) \\ + \sum_{i=1}^{d} \int_{\mathbb{S}^{d-1}} F(\ell \boldsymbol{\sigma} \cdot \hat{\xi}) \mathbf{a}_{ii}(\omega) (\boldsymbol{\sigma}_i^2 - \frac{1}{d} |\boldsymbol{\sigma}|^2) S(d\boldsymbol{\sigma}) \\ + \frac{1}{d} \mathrm{Tr}(\mathbf{a}(\omega)) \int_{\mathbb{S}^{d-1}} F(\ell \boldsymbol{\sigma} \cdot \hat{\xi}) S(d\boldsymbol{\sigma}) $$

Note that $\boldsymbol{\sigma}_i \boldsymbol{\sigma}_j$ (for $i \neq j$), $\boldsymbol{\sigma}_i^2 - \frac{1}{d}|\boldsymbol{\sigma}|^2$ (for $i = 1, \dots, d$) and the constants are harmonic polynomials in $\boldsymbol{\sigma} = (\boldsymbol{\sigma}_1, \dots, \boldsymbol{\sigma}_d) \in \mathbb{R}^d$. By the Hecke-Funck theorem, see e.g. [24, p. 181], we can then conclude that

$$ \mathbf{q}(\xi; \omega) = c_d |\xi|^\alpha \langle \mathbf{a}(\omega) \hat{\xi}, \hat{\xi} \rangle + C_d \mathrm{Tr}(\mathbf{a}(\omega)) |\xi|^\alpha, $$

where

$$ c_d = |\mathbb{S}^{d-1}| \int_0^{+\infty} \frac{d\ell}{\ell^{\alpha+1}} \int_{-1}^1 (1 - \cos(\ell t)) P_2(t) (1 - t^2)^{\frac{d-3}{2}} dt, $$

$$ C_d = |\mathbb{B}^d| \int_0^{+\infty} \frac{d\ell}{\ell^{\alpha+1}} \int_{-1}^1 (1 - \cos(\ell t)) (P_0(t) - P_2(t)) (1 - t^2)^{\frac{d-3}{2}} dt $$

and $P_n(t) = \cos(n \arccos t)$ is the $n$-th degree Tchebyshev polynomial, while $|\mathbb{S}^{d-1}|$, $|\mathbb{B}^d|$ denote the area of the unit sphere and the volume of the unit ball in $\mathbb{R}^d$, respectively.

# D Borel measurability of $\Omega^{\mathcal{M}}$

We shall show that $\Omega^{\mathcal{M}}$, introduced in Section 2.3, is a Borel measurable subset of $\Omega$. For any $x \in \mathbb{R}^d$, $\omega \in \Omega$, we denote by $\Pi_{x;\omega}$ the set of all solutions of the martingale problem for $L^\omega$ with the initial measure $\delta_x$. By [34, Corollary 3.2 and Lemma 2.1], the definition of $\Omega$ ensures that the set $\Pi_{x;\omega}$ is non-empty for any $(x, \omega) \in \mathbb{R}^d \times \Omega$. For any $f \in C_0(\mathbb{R}^d)$ and $\lambda > 0$ we introduce

$$ \pi_{\lambda}^{*}(x, f; \omega) := \sup_{\mathbb{P} \in \Pi_{x;\omega}} \mathbb{E}^{\mathbb{P}} \int_{0}^{\infty} e^{-\lambda t} f(X_t) dt, \quad \pi_{\star}^{\lambda}(x, f; \omega) := \inf_{\mathbb{P} \in \Pi_{x;\omega}} \mathbb{E}^{\mathbb{P}} \int_{0}^{\infty} e^{-\lambda t} f(X_t) dt. $$

By [35, Theorem 4.1] for any $f \in C_0(\mathbb{R}^d)$, $\omega \in \Omega$ and $\lambda > 0$ there exists a strong Markov process $(\mathbb{Q}_{f,\lambda}^{x;\omega}, x \in \mathbb{R}^d)$ such that each $\mathbb{Q}_{f,\lambda}^{x;\omega} \in \Pi_{x;\omega}$ and

$$ \pi_{\lambda}^{*}(x, f; \omega) = \mathbb{E}^{\mathbb{Q}_{f,\lambda}^{x;\omega}} \int_{0}^{\infty} e^{-\lambda t} f(X_t) dt, \quad \pi_{\star}^{\lambda}(x, f; \omega) = \mathbb{E}^{\mathbb{Q}_{f,\lambda}^{x;\omega}} \int_{0}^{\infty} e^{-\lambda t} f(X_t) dt. $$

From the above formulas, one infers that $\pi_{\lambda}^{*}(x, f; \cdot)$ (resp. $\pi_{\star}^{\lambda}(x, f; \cdot)$) is upper (resp. lower) semi-continuous. Hence, both of these functions are Borel measurable in $\omega \in \Omega$. Let $S$ be a countable and dense subset of $(0, \infty) \times \mathbb{R}^d \times C_0(\mathbb{R}^d)$. Observe that

$$ \Omega^{\mathcal{M}} = \{ \omega \in \Omega : |\pi_{\lambda}^{*}(x, f; \omega) - \pi_{\star}^{\lambda}(x, f; \omega)| = 0, \ (\lambda, x, f) \in S \}. $$

This implies that $\Omega^{\mathcal{M}}$ is a Borel measurable subset of $\Omega$.
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