Stochastic homogenization of diffusions in turbulence driven by non-local symmetric Lévy operators

STOCHASTIC HOMOGENIZA TION OF DIFFUSIONS IN TURBULENCE DRIVEN BY NON-LOCAL SYMMETRIC LÉVY OPERA TORS XIN CHEN, JIAN W ANG AND KUN YIN Abstra ct. W e in vestigate the stochastic homogenization of a class of turbulen t diﬀusions generated b y non- lo cal symmetric Lévy op erators with div ergence-free drift ﬁelds in ergo dic random environmen ts, where neither the drift ﬁelds nor their asso ciated stream functions are assumed to b e b ounded. A piv otal step in our pro of is the establishmen t of W 1 ,q loc estimates with q ∈ (1 , 2) for the corresp onding correctors, under mild prior regularity conditions imp osed on the Lévy measure and the stream function. Keyw ords: sto c hastic homogenization; non-lo cal symmetric Lévy op erator; ergo dic random environmen t; div ergence-free drift; jump pro cess. MSC 2020: 60G51; 60G52; 35B27; 82D30 1. Intr oduction and main resul t 1.1. Bac kground. In this pap er, we in v estigate the sto chastic homogenization of the following random op er- ator L ω f ( x ) := L 0 f ( x ) + ⟨ b ( x ; ω ) , ∇ f ( x ) ⟩ , f ∈ C 2 b ( R d ) , (1.1) where b ( · ; ω ) is a divergence-free random ﬁeld. When L 0 corresp onds to the Laplacian op erator, this mo del is closely related to a fundamental paradigm in statistical ﬂuid mechanics, whic h describ es the dynamics of diﬀusive particles conv ected by a random incom- pressible velocity ﬁeld. In particular, the long-time b eha vior of the Mark o v pro cess generated by the random op erator L ω is characterized via the sto chastic homogenization framework (or the inv ariance principle). A core comp onen t of this theory lies in the deriv ation of the eﬀective diﬀusivity , a key parameter that determines the co eﬃcients of the limiting Bro wnian motion induced by the drift ﬁeld b ( x ; ω ) under appropriate regularity conditions. Indep enden tly , Kozlo v [30] and Papanicolaou and V aradhan [39] pioneered the so-called – seen from the particle – metho d to construct the asso ciated corrector in ergo dic random environmen ts. This tec hnique has since emerged as a cornerstone in the researc h of sto c hastic homogenization. Under the assumption that the drift ﬁeld b ( x ; ω ) admits a stream function (see Assumption 1.2 b elow), extensive studies ha v e b een conducted on prop erties of sto chastic homogenization for L ω . F or instance, Osada [37] established the quenched in v ari- ance principle under the condition of a b ounded stream function. Landim, Olla and Y au [32] prov ed the corresp onding result for time-dep endent environmen ts, assuming the existence of a (time-dep endent) b ounded stream. Oelsc hläger [36] relaxed the b oundedness constrain t and analyzed stochastic homogenization under ﬁnite p -moment conditions on the stream function for an y p < ∞ . With ﬁnite second moment assumptions on the stream function, F annjiang and Papanicolaou [20] further demonstrated the L 2 -con v ergence for the densit y of the asso ciated Marko v pro cess. Lev eraging Moser’s iteration argument, F annjiang and Komoro wski [18, 19] prov ed the quenched inv ariance principle for b oth static and time-dep endent ergo dic environmen ts, resp ectiv ely , under strengthened ﬁnite momen t conditions on the stream function. K omoro wski and Olla [29] prop osed a criterion for stochastic homogenization in time-dependent mo dels based on the sp ectral resolution of the drift ﬁeld. Readers are referred to the monograph [28] and the references therein for details on related dev elopmen ts. F ehrman [22] explored sto chastic homogenization in space-time ergo dic settings, where the im- pact of temp oral v ariables was completely analyzed, and the limiting pro cess was shown to deviate from a standard Brownian motion in certain cases. Additionally , F ehrman [21] provided a simple pro of of quenc hed sto c hastic homogenization, and also examined the large-scale Hölder regularit y and the ﬁrst-order Liouville principle of the op erator L ω . Recen tly , signiﬁcant progress has b een made for the critically correlated case, where the drift ﬁeld b ( x ; ω ) do es not p ossess a globally deﬁned stream function. Cannizzaro, Haunschmid-Sibitz and T oninelli [6] derived long- time second-momen t estimates for a Brownian particle in R 2 , sub ject to a random, time-indep endent drift giv en 1 2 XIN CHEN, JIAN W ANG AND KUN YIN b y the curl of the t w o-dimensional Gaussian F ree Field. By com bining stochastic homogenization tec hniques with reﬁned cutoﬀ arguments, Chatzigeorgiou, Morfe, Otto and W ang [7] improv ed these estimates to achiev e the optimal order in the large-time regime. F urthermore, Armstrong, Bou-Rab ee and Kuusi [1] employ ed a renormalization group approac h to analyze the coarse-grained diﬀusivit y across diﬀerent scales, proving both qualitativ e and quantitativ e v ersions of the quenched in v ariance principle with sup er-diﬀusiv e scaling. Notably , the limiting Bro wnian motion in their work is fully determined by the drift ﬁeld. F or additional prop erties of mo dels with critically correlated div ergence-free drift ﬁelds, w e refer the reader to [2, 33, 34, 35, 38] and the references therein. In con trast, results for the case where L 0 is a non-lo cal op erator remain relatively scarce. F or symmetric stable-lik e op erators L 0 , Chen and Yin [12] studied the sto c hastic homogenization of L ω , sho wing that the limiting pro cess is an α -stable Lévy pro cess with no eﬀective diﬀusivit y term. A feature of this work is that the proof do es not require the construction of a corrector. T o the best of our kno wledge, it remains an unkno wn question whether the limiting process in the stochastic homogenization of non-lo cal op erators L 0 can b e a Bro wnian motion with a non-trivial eﬀectiv e diﬀusivity . In this pap er, we fo cus on a class of non- lo cal Lévy operators L 0 whose Lévy measures are L 2 -in tegrable (see Assumption 1.1 for details). In fact, for Mark o v processes generated by symmetric non-lo cal op erators with L 2 -in tegrable jumping k ernels, stochastic homogenization has b een established by Biskup, Chen, Kumagai and W ang [4], Flegel, Heida and Slowik [23], and Piatnitski and Zhizhina [40] via distinct approaches; in all these cases, the limiting pro cess is a Brownian motion with eﬀective diﬀusivity . As discussed in Remark 1.5 below, a primary c hallenge in our w ork is to establish W 1 ,q loc -estimates for the asso ciated corrector, given the weak er regularit y of b oth the non-lo cal Lévy op erator L 0 and the drift ﬁeld b ( x ; ω ) . F or recen t adv ances in the sto chastic homogenization of non-lo cal op erators, w e also refer to [9, 10, 11, 13, 24, 27, 40, 42] and the references therein. 1.2. F ramew ork and main result. Let d ⩾ 2 . Suppose that (Ω , G , P ) is a probability space endo w ed with a measurable group of transformations τ x : Ω → Ω , for all x ∈ R d , suc h that (a) τ 0 = id , where id denotes the identit y map on Ω ; (b) τ x ◦ τ y = τ x + y for ev ery x, y ∈ R d . W e assume that the following stationary , ergo dic and measurable prop erties hold: (i) The stationary condition: P ( τ x A ) = P ( A ) for all A ∈ G and x ∈ R d ; (ii) The ergo dic condition: if A ∈ G and τ x A = A for all x ∈ R d , then P ( A ) ∈ { 0 , 1 } ; (iii) The measurable condition: the function ( x, ω ) 7→ τ x ω is B ( R d ) × G -measurable. Throughout the paper, w e supp ose that L 0 in (1.1) is a symmetric non-lo cal (deterministic) Lévy op erator ha ving the following expression L 0 f ( x ) := p . v . Z R d ( f ( x + z ) − f ( x )) ν ( z ) dz , f ∈ C 2 b ( R d ) with ν ( z ) := 1 | z | d + α 1 {| z | ⩽ 1 } + ν ( z ) 1 {| z | > 1 } , z ∈ R d , (1.2) and the random drift b ( x ; ω ) in (1.1) is deﬁned by b ( x ; ω ) := ˜ b ( τ x ω ) = ( ˜ b 1 ( τ x ω ) , ˜ b 2 ( τ x ω ) , · · · , ˜ b d ( τ x ω )) , x ∈ R d , ω ∈ Ω with ˜ b : Ω → R d . W e alwa ys make the follo wing assumption on the op erator L 0 . Assumption 1.1. The density function ν ( z ) of the jumping me asur e for the op er ator L 0 fulﬁl ls al l the fol lowing c onditions: (i) The c onstant α in (1.2) satisﬁes that α ∈ (1 , 2) . (ii) ν ( z ) = ν ( − z ) for al l z ∈ R d , sup z ∈ R d : | z | > 1 ν ( z ) < ∞ and Z {| z | > 1 } | z | 2 ν ( z ) dz < ∞ . NON-LOCAL SYMMETRIC OPERA TORS WITH DIVERGENCE-FREE DRIFT 3 (iii) Ther e exist c onstants c 0 ⩾ 1 and K 0 ⩾ 2 , and a function γ : R d → R + := (0 , ∞ ) such that for every r ⩾ 1 and x ∈ R d with | x | ⩾ K 0 r , Z B ( x,r ) ν 2 ( z ) dz ⩽ c 0 r d γ 2 ( x ) ν 2 ( x ) (1.3) and Z {| x | ⩾ K 0 } | x | 2 γ 2 ( x ) ν ( x ) dx < ∞ . (1.4) In particular, under Assumption 1.1(ii), Z R d | z | 2 ν ( z ) dz < ∞ ; that is, the Lévy measure ν ( z ) dz corresp onding to the op erator L 0 has ﬁnite second moment. Example 1.6 b elow indicates that Assumption 1.1(iii) can apply to a large class of symmetric non-lo cal Lévy op erators with regular jumping measures. Let H := L 2 (Ω; P ) . Using { τ x } x ∈ R d , we can deﬁne the translation operator T x : H → H and the deriv ativ e op erator D j (if the limit exists) resp ectiv ely as follows: T x ˜ F ( ω ) := ˜ F ( τ − x ω ) , ˜ F ∈ H , x ∈ R d , D j ˜ F ( ω ) := lim ε ↓ 0 T εe j ˜ F ( ω ) − ˜ F ( ω ) ε , ˜ F ∈ H , 1 ⩽ j ⩽ d, where { e j } 1 ⩽ j ⩽ d denotes the canonical orthonormal basis of R d . In particular, it is easy to verify that, if D j ˜ F exists, then D j ˜ F ( τ x ω ) = − ∂ F ( · ; ω ) ∂ x j ( x ) , x ∈ R d , (1.5) where F ( x ; ω ) := ˜ F ( τ x ω ) . Set D := { ˜ F ∈ L ∞ (Ω; P ) : D j ˜ F and D k ( D j ˜ F ) exist and b elong to L ∞ (Ω; P ) for every 1 ⩽ j, k ⩽ d } . A ccording to the stationary prop erty of the transformation { τ x } x ∈ R d and the integration by parts form ula, we ha v e E [ D j ˜ F ˜ G ] = − E [ ˜ F D j ˜ G ] , ˜ F , ˜ G ∈ D . (1.6) Since { T x } x ∈ R d is a strongly contin uous unitary semigroup on H , w e ha v e the following sp ectral expression: T x = Z R d e i ⟨ ξ ,x ⟩ U ( dξ ) , x ∈ R d , (1.7) where U ( dξ ) is the asso ciated pro jection v alued measure on H . By (1.6) and (1.7), we can deﬁne H 1 := D ∥·∥ H 1 as the closed extension of D under the ∥ · ∥ H 1 -norm, where ∥ ˜ F ∥ 2 H 1 := E [ ˜ F 2 ] + d X j =1 E [ | D j ˜ F | 2 ] = Z R d  1 + | ξ | 2  E [ U ( dξ ) ˜ F · ˜ F ] , ˜ F ∈ D . Hence, D j ˜ F is well deﬁned for ev ery ˜ F ∈ H 1 , and the in tegration by parts formula (1.6) still holds for ev ery ˜ F , ˜ G ∈ H 1 . It is also easy to see that F ( · , ω ) ∈ W 1 , 2 loc ( R d ) for every ˜ F ∈ H 1 and a.e. ω ∈ Ω , where F ( x ; ω ) := ˜ F ( τ x ω ) . W e refer the readers to [39, Section 2] for the details of all the con ten ts mentioned ab ov e. W e then put the following assumption on the drift term b ( x ; ω ) = ˜ b ( τ x ω ) . Assumption 1.2. L et ˜ b = ( ˜ b 1 , ˜ b 2 , · · · , ˜ b d ) : Ω → R d b e a r andom ve ctor such that ther e exists a system of str e am functions ˜ H j l : Ω → R , 1 ⩽ j, l ⩽ d , satisfying the fol lowing pr op erties: (i) ˜ H j l ∈ H 1 for every 1 ⩽ j, l ⩽ d ; ˜ H j l = − ˜ H lj for every 1 ⩽ j, l ⩽ d ; and the function x 7→ D l ˜ H j l ( τ x ω ) is c ontinuous for every 1 ⩽ j, l ⩽ d and a.s. ω ∈ Ω . (ii) ˜ b j = P d l =1 D l ˜ H j l for every 1 ⩽ j ⩽ d . It follo ws from Assumption 1.2 that (in the distributional sense) div b ( · ; ω )( x ) = − d X j =1 ∂ b j ( x ; ω ) ∂ x j = d X j =1 D j ˜ b j ( τ x ω ) = d X j,l =1 D j D l ˜ H j l ( τ x ω ) = 0 , x ∈ R d , (1.8) 4 XIN CHEN, JIAN W ANG AND KUN YIN where in the last equalit y we used the anti-symmetry of { ˜ H j l } 1 ⩽ j,l ⩽ d . In particular, the random drift b ( · ; ω ) is div ergence-free. No w, for any ε > 0 , we deﬁne the scaled op erator L ε,ω b y L ε,ω f ( x ) = L ε 0 f ( x ) + ε − 1 D b  x ε ; ω  , ∇ f ( x ) E , f ∈ C 2 b ( R d ) , where L ε 0 f ( x ) := ε − d − 2 p . v . Z R d ( f ( x + z ) − f ( x )) ν  z ε  dz = ε − 2 p . v . Z R d ( f ( x + εz ) − f ( x )) ν ( z ) dz . If ( X ω t ) t ⩾ 0 is a Marko v pro cess asso ciated with the op erator L ω , then ( X ε,ω t ) t ⩾ 0 := ( εX ω ε − 2 t ) t ⩾ 0 is the Marko v pro cess asso ciated with the scaled op erator L ε,ω . F or any λ > 0 and h ∈ C ∞ b ( R d ) , consider the following resolv en t equation asso ciated with L ε,ω : λu ε ( x ; ω ) − L ε,ω u ε ( x ; ω ) = h ( x ) , x ∈ R d . (1.9) The existence of a weak solution to the equation (1.9) will b e established in Prop osition 3.1 b elow. Deﬁne a d × d matrix ¯ A = { ¯ a j k } 1 ⩽ j,k ⩽ d b y ¯ a j k = E  Z R d ( z j + ϕ j ( z ; ω )) ( z k + ϕ k ( z ; ω )) ν ( z ) dz  , 1 ⩽ j, k ⩽ d, (1.10) where ϕ : R d × Ω → R d is the corrector constructed in Theorem 2.2. As shown in Lemma 2.3 b elow, ¯ A is a strictly p ositiv e deﬁnite and b ounded matrix, so ¯ Lf ( x ) := 1 2 d X j,k =1 ¯ a j k ∂ 2 f ( x ) ∂ x j ∂ x k , f ∈ C 2 c ( R d ) is a uniformly elliptic second diﬀerential op erator. F or every λ > 0 , set G λ = { h ∈ C ∞ b ( R d ) ∩ W 1 , 2 ( R d ) : ( λ − ¯ L ) − 1 h ∈ C ∞ c ( R d ) } . Then, for an y λ > 0 and h ∈ G λ , there exists a unique solution ¯ u in L 2 ( R d ; dx ) to the following equation λ ¯ u ( x ) − ¯ L ¯ u ( x ) = h ( x ) , x ∈ R d . (1.11) The main theorem of this paper is concerned on the sto c hastic homogenization for the scaled resolven t equation (1.9) to (1.11) as ε → 0 . F or this, we further need the following assumption. Assumption 1.3. Supp ose that d > 4( α − 1) and that ther e exists q ∈ (2 d/ ( d + 2) , 2) such that the fol lowing pr op erties hold. (i) Ther e exists a c onstant r 0 ⩾ 1 such that E  N r 0 ( ω ) max { 4 ,d } 2( α − 1)(2 − q )  < ∞ , (1.12) wher e N r 0 ( ω ) := sup x ∈ B (0 ,r 0 ) , 1 ⩽ j,l ⩽ d    D j ˜ H j l ( τ x ω )    . (ii) F or some r > max n q q − 1 , 2 p ′ 0 p ′ 0 − 2 o with p 0 = dq d − q ∈ (2 , ∞ ) and p ′ 0 = 2 p 0 α 4( α − 1)+ p 0 (2 − α ) ∈ (2 , p ) , E [ | ˜ H j l | r ] < ∞ , 1 ⩽ j, l ⩽ d (1.13) and E        Z {| z | ⩽ 1 } | ˜ H j l ( τ z ω ) − ˜ H j l ( ω ) | 2 | z | d +2 − α dz      p 0 p 0 − 2   < ∞ , 1 ⩽ j, l ⩽ d. (1.14) Theorem 1.4. Supp ose that Assumptions 1 . 1 , 1 . 2 and 1 . 3 ar e satisﬁe d. Then, for any λ > 0 , h ∈ G λ , a.s. ω ∈ Ω and every R ⩾ 1 , p ⩾ 1 , lim ε → 0 Z B (0 ,R ) | u ε ( x ; ω ) − ¯ u ( x ) | p dx = 0 . (1.15) Her e u ε ( x ; ω ) and ¯ u ar e solutions to the e quations (1.9) and (1.11) , r esp e ctively. NON-LOCAL SYMMETRIC OPERA TORS WITH DIVERGENCE-FREE DRIFT 5 W e give some comments on the pro of of Theorem 1.4. Remark 1.5. (i) Compared with the case of symmetric non-lo cal op erators with L 2 -in tegrable jumping k ernels [4, 23, 40] or the case where L 0 is a second-order diﬀeren tial op erator [18, 22, 39], we will employ a diﬀeren t metho d to handle the eﬀect of the drift ﬁeld b ( x ; ω ) in the construction of the associated corrector ϕ ; see (2.1) b elow. In particular, additional analysis based on the sp ectral representation of the drift ﬁeld is carried out; see the pro of of Theorem 2.2 for details. (ii) T o iden tify the limiting operator ¯ L , we m ust address a term inv olving the corrector ϕ in the test function, see the function f ε m ( x ; ω ) deﬁned b y (4.3) in the pro of of Theorem 1.4. F or this reason, the regularit y prop erty ϕ ∈ W 1 ,q loc ( R d ; R d ) for some q ∈ (1 , 2) is essential . In the case L 0 = ∆ , the prop erty ϕ ∈ W 1 , 2 loc ( R d ; R d ) holds automatically for the corrector, thanks to the a priori regularity estimate E  |∇ ϕ ( x ; ω ) | 2  < ∞ , x ∈ R d , (1.16) see, e.g., [18, 22, 39] for further details. Ho w ev er, for the non-lo cal op erator L 0 considered in this pap er, the main diﬃculty is that the a priori estimate (1.16) fails to hold. Instead, we can only obtain w eak er a priori estimates (2.8) and (2.9) below for the corrector corresp onding to L ω . W e develop a no v el pro cedure tailored to non-lo cal op erators for establishing the W 1 ,q loc ( R d ; R d ) regularit y of the corrector ϕ with q ∈ (1 , 2) , whic h is founded on these weak er a priori estimates combined with additional structural conditions imp osed on the stream function { ˜ H j l } 1 ⩽ j,l ⩽ d . F urthermore, owing to the reduced regularit y of the non-lo cal op erator L 0 , compared to the case of second-order diﬀeren tial op erators w e need encounter additional technical c hallenges in eliminating sev eral residual terms throughout the sto c hastic homogenization. F or instance, see step (3) in the pro of of Theorem 1.4. This partly explains wh y the assumption α ∈ (1 , 2) is necessary . (iii) W e note that the approach here diﬀers from those in [8, 14, 43], where Sc hauder estimates for elliptic equations induced b y sup ercritical non-lo cal op erators hav e b een established. This diﬀerence arises partly from the w eak er a priori estimates (2.8) and (2.9), as well as the low er regularit y of the drift ﬁeld b ( x ; ω ) ; for example, we cannot guarantee that the corrector ϕ is globally b ounded on R d . F urthermore, in [8, 14, 43], the Schauder estimates dep end only on the small-jump part of the asso ciated non-lo cal op erator L 0 . By con trast, the situation is diﬀerent for our mo del: in the present setting, the large-jump part of the non-lo cal op erator L 0 also plays a crucial role in the W 1 ,q loc ( R d ; R d ) estimate of the corrector ϕ . Indeed, the regularity condition (1.3) can b e in terpreted as a kind of (large-scale) a v eraged low er b ound for the jumping k ernel asso ciated with large jumps. Belo w we take some esp ecial examples such that Assumption 1.1(iii) is satisﬁed. Example 1.6. The following conclusions hold. (1) Supp ose that ν ( z ) = 1 | z | d + β 1 log β 2 (2+ | z | ) for ev ery | z | > 1 with β 1 > 2 and β 2 ∈ R . Then, (1.3) and (1.4) hold with K 0 = 2 and γ ( x ) ≡ 1 . (2) If ν ( z ) = exp  − a | z | β  for ev ery | z | > 1 with a > 0 and β > 0 , then (1.3) and (1.4) hold with K 0 = 1 1 − 2 − 1 β + 1 and γ ( x ) = exp  a  1 −  1 − 1 K 0  β  | x | β  . (3) If ν ( z ) = 0 for every | z | > 1 , then (1.3) and (1.4) hold with K 0 = 2 and γ ( x ) ≡ 1 . The remainder of this pap er is organized as follows. In the next section, we establish the existence of the corrector ϕ asso ciated with the dual op erator for L ω . In Section 3, we ﬁrst prov e the existence of a weak solution to the scaled resolv en t equation (1.9), and then deriv e the lo cal W 1 ,q ( R d ; R d ) regularity for the corrector ϕ with q ∈ (1 , 2) , as w ell as the relationships b et w een ϕ and the drift ﬁeld b ( x ; ω ) . The ﬁnal section is devoted to the pro of of Theorem 1.4. Notation Throughout the paper, let B ( x, R ) := { z ∈ R d : | z − x | < R } be the op en ball with center x ∈ R d and radius R > 0 . F or ev ery 1 ⩽ p ⩽ ∞ , let L p ( R d ) denote the L p space with resp ect to the Leb esgue measure endo w ed with the norm ∥ f ∥ L p ( R d ) :=  R R d | f ( x ) | p dx  1 /p for 1 ⩽ p < ∞ and ∥ f ∥ L ∞ ( R d ) = sup x ∈ R d | f ( x ) | . F or an y 1 ⩽ p < ∞ , let W 1 ,p ( R d ) denote the Sob olev space as follo ws W 1 ,p ( R d ) := n f ∈ L p ( R d ) : the w eak gradient ∇ f exists 6 XIN CHEN, JIAN W ANG AND KUN YIN and ∥ f ∥ W 1 ,p ( R d ) := ∥ f ∥ L p ( R d ) +  Z R d |∇ f ( x ) | p dx  1 /p < ∞ o . F or every β ∈ (0 , 1) and 1 ⩽ p < ∞ , let W β ,p ( R d ) b e the Sob olev-Slob o deckij space (i.e., the fractional Sob olev space) as follo ws W β ,p ( R d ) := ( f ∈ L p ( R d ) : ∥ f ∥ W β ,p ( R d ) := ∥ f ∥ L p ( R d ) +  Z R d Z R d | f ( x ) − f ( y ) | p | x − y | d + β p dx dy  1 /p < ∞ ) . 2. The corrector and homogenized coefficients Let L ω b e the random op erator given b y (1.1). Since L 0 is a non-lo cal symmetric Lévy op erator and the random drift b ( · ; ω ) is divergence-free suc h that (1.8) is satisﬁed, the dual op erator corresp onding to L ω is formally giv en by L ω ∗ f ( x ) = L 0 f ( x ) − ⟨ b ( x ; ω ) , ∇ f ( x ) ⟩ . The w eak solution to the following equation L ω ∗ ϕ k ( · ; ω )( x ) = L 0 ϕ k ( · ; ω )( x ) − ⟨ b ( x ; ω ) , ∇ ϕ k ( x ; ω ) ⟩ = b k ( x ; ω ) , x ∈ R d , 1 ⩽ k ⩽ d (2.1) pla ys an imp ortan t role in the homogenization theory for L ω , and it is usually called the (dual) corrector in the literature; e.g. see [22]. Here we use the notation ϕ ( x ; ω ) = ( ϕ 1 ( x ; ω ) , · · · , ϕ d ( x ; ω )) with ϕ k ( x ; ω ) denoting the k -th co ordinate of ϕ ( x ; ω ) for all 1 ⩽ k ⩽ d , and the same for b ( x ; ω ) = ( b 1 ( x ; ω ) , · · · , b d ( x ; ω )) . The main purp ose of this section is to verify the existence of the corrector ϕ ( x ; ω ) to the equation (2.1), whic h is inv olv ed in the homogenized matrix co eﬃcien ts { ¯ a j k } 1 ⩽ j,k ⩽ d giv en in (1.10) for the limit op erator ¯ L . 2.1. The existence of the corrector. W e b egin with the following simple lemma. Lemma 2.1. Under Assumption 1 . 1 , ther e exist p ositive c onstants c 1 ⩽ c 2 such that for al l ξ ∈ R d , c 1  | ξ | 2 1 {| ξ | ⩽ 1 } + | ξ | α 1 {| ξ | > 1 }  ⩽ Z R d (1 − e i ⟨ ξ ,z ⟩ ) ν ( z ) dz ⩽ c 2  | ξ | 2 1 {| ξ | ⩽ 1 } + | ξ | α 1 {| ξ | > 1 }  . (2.2) Pr o of. By the change of v ariable, for all ξ ∈ R d , Z {| z | ⩽ 1 } 1 − e − i ⟨ ξ ,z ⟩ | z | d + α dz = | ξ | α Z {| z | ⩽ | ξ |} 1 − e − i D ξ | ξ | ,z E | z | d + α dz = | ξ | α Z {| z | ⩽ | ξ |} 1 − cos D ξ | ξ | , z E | z | d + α dz . Note that there exist p ositiv e constan ts r 0 , c 1 and c 2 suc h that for all ξ , z ∈ R d with | z | ⩽ r 0 , c 1      ξ | ξ | , z      2 ⩽ 1 − cos  ξ | ξ | , z  ⩽ c 2      ξ | ξ | , z      2 , whic h implies immediately that for all ξ ∈ R d with | ξ | ⩽ r 0 , c 3 | ξ | 2 ⩽ Z {| z | ⩽ 1 } 1 − e − i ⟨ ξ ,z ⟩ | z | d + α dz ⩽ c 4 | ξ | 2 . On the other hand, for all ξ ∈ R d with | ξ | > r 0 , c 5 ⩽ Z {| z | ⩽ r 0 } 1 − cos D ξ | ξ | , z E | z | d + α dz ⩽ Z {| z | ⩽ | ξ |} 1 − cos D ξ | ξ | , z E | z | d + α dz ⩽ Z R d 1 − cos D ξ | ξ | , z E | z | d + α dz ⩽ c 6 , whic h yields that for all ξ ∈ R d with | ξ | > r 0 , c 7 | ξ | α ⩽ Z {| z | ⩽ 1 } 1 − e − i ⟨ ξ ,z ⟩ | z | d + α dz ⩽ c 8 | ξ | α . Com bining with b oth estimates ab o v e, we ﬁnd that c 9  | ξ | 2 1 {| ξ | ⩽ 1 } + | ξ | α 1 {| ξ | > 1 }  ⩽ Z {| z | ⩽ 1 } 1 − e − i ⟨ ξ ,z ⟩ | z | d + α dz ⩽ c 10  | ξ | 2 1 {| ξ | ⩽ 1 } + | ξ | α 1 {| ξ | > 1 }  . (2.3) NON-LOCAL SYMMETRIC OPERA TORS WITH DIVERGENCE-FREE DRIFT 7 F urthermore, by using the fact ν ( z ) = ν ( − z ) for all z ∈ R d and T aylor’s expansion, Z {| z | > 1 } (1 − e − i ⟨ ξ ,z ⟩ ) ν ( z ) dz ⩽ c 11 " | ξ | 2 Z {| z | > 1 } | z | 2 ν ( z ) dz ! 1 {| ξ | ⩽ 1 } + Z {| z | > 1 } ν ( z ) dz ! 1 {| ξ | > 1 } # ⩽ c 12  | ξ | 2 1 {| ξ | ⩽ 1 } + | ξ | α 1 {| ξ | > 1 }  . A ccording to this and (2.3), we can prov e (2.2). □ T o mov e further, we need to in tro duce some notations. Deﬁne L 0 ˜ F ( ω ) = p . v . Z R d ( ˜ F ( τ z ω ) − ˜ F ( ω )) ν ( z ) dz , ˜ F ∈ D . Then, L 0 ˜ F ( τ x ω ) = L 0 F ( · ; ω )( x ) , x ∈ R d , ˜ F ∈ D , where F ( x ; ω ) := ˜ F ( τ x ω ) . According to the stationary prop ert y of { τ x } x ∈ R d and the fact ν ( z ) = ν ( − z ) for all z ∈ R d , w e hav e E [ L 0 ˜ F · ˜ G ] = E [ ˜ F · L 0 ˜ G ] = − 1 2 E  Z R d ( ˜ F ( τ z ω ) − ˜ F ( ω ))( ˜ G ( τ z ω ) − ˜ G ( ω )) ν ( z ) dz  , ˜ F , ˜ G ∈ D . (2.4) In particular, for all ˜ F ∈ D , − E [ L 0 ˜ F · ˜ F ] = 1 2 E  Z R d ( ˜ F ( τ z ω ) − ˜ F ( ω )) 2 ν ( z ) dz  . On the other hand, according to (1.7), − E [ L 0 ˜ F · ˜ F ] = − E  Z R d ( T − z ˜ F ( ω ) − ˜ F ( ω )) ν ( z ) dz  · ˜ F ( ω )  = − E  Z R d  Z R d ( e − i ⟨ ξ ,z ⟩ − 1) ν ( z ) dz  [ U ( dξ ) ˜ F · ˜ F ]  , ˜ F ∈ D . (2.5) Next, w e will prov e the existence of the corrector ϕ ( x ; ω ) to the equation (2.1). Theorem 2.2. Supp ose that Assumptions 1 . 1 and 1 . 2 hold. Then ther e exist ϕ : R d × Ω → R d and a P -nul l set Λ ⊂ Ω such that the fol lowing pr op erties ar e satisﬁe d. (i) F or every 1 ⩽ k ⩽ d , f ∈ C 1 c ( R d ) and ω / ∈ Λ , 1 2 Z R d Z R d ( ϕ k ( x + z ; ω ) − ϕ k ( x ; ω )) ( f ( x + z ) − f ( x )) ν ( z ) dz dx − d X j =1 Z R d b j ( x ; ω ) ϕ k ( x ; ω ) ∂ f ( x ) ∂ x j dx = − Z R d b k ( x ; ω ) f ( x ) dx. (2.6) (ii) E [ ϕ ( x ; ω )] = 0 for every x ∈ R d , and the c o-cycle pr op erty holds for al l ω / ∈ Λ and x, z ∈ R d , ϕ ( x + z ; ω ) − ϕ ( x ; ω ) = ϕ ( z ; τ x ω ) . (2.7) (iii) ϕ ( · ; ω ) ∈ W α/ 2 , 2 loc ( R d , R d ) for every ω / ∈ Λ , E  Z R d | ϕ ( x ; ω ) | 2 ν ( x ) dx  < ∞ , (2.8) and for every b ounde d subset D ⊂ R d , E  Z D | ϕ ( x ; ω ) | 2 dx  + E  Z D Z R d | ϕ ( x + z ; ω ) − ϕ ( x ; ω ) | 2 ν ( z ) dz dx  < ∞ . (2.9) Pr o of. (1) F or an y R ⩾ 1 , let ρ R : R → R b e a smo oth cut-oﬀ function such that ρ R ( s ) =      s, | s | ⩽ R , ∈ [ − R , R ] , R < | s | < 2 R, 0 , | s | ⩾ 2 R , 8 XIN CHEN, JIAN W ANG AND KUN YIN sup R ⩾ 1 sup s ∈ R | ρ ′ R ( s ) | ⩽ 2 and ρ R ( s ) = − ρ R ( − s ) for ev ery s ∈ R . F or every θ > 0 and R ⩾ 1 , deﬁne L ∗ ,θ,R ˜ F ( ω ) = L 0 ˜ F ( ω ) + θ d X j =1 D j ( D j ˜ F )( ω ) + d X j =1 ˜ b R j ( ω ) D j ˜ F ( ω ) , ˜ F ∈ D , where L 0 is giv en by (1.9), and ˜ b R j ( ω ) := d X l =1 D l ( ρ R ( ˜ H j l ))( ω ) . (2.10) A ccording to (1.5), we can verify that L ∗ ,θ,R ˜ F ( τ x ω ) = L ω ∗ ,θ,R F ( · ; ω )( x ) , where F ( x ; ω ) = ˜ F ( τ x ω ) , b R ( x ; ω ) = ˜ b R ( τ x ω ) and L ω ∗ ,θ,R f ( x ) = L 0 f ( x ) + θ ∆ f ( x ) −  b R ( x ; ω ) , ∇ f ( x )  , f ∈ C 2 b ( R d ) . F urthermore, deﬁne E ∗ ,θ,R ( ˜ F , ˜ G ) = − E [ L ∗ ,θ,R ˜ F · ˜ G ] , ˜ F , ˜ G ∈ D . By (2.2) and (2.5), we obtain that − E [ L 0 ˜ F · ˜ F ] ⩽ c 0 Z R d  1 + | ξ | 2  E [ U ( dξ ) ˜ F · ˜ F ] = c 0 ∥ ˜ F ∥ 2 H 1 , ˜ F ∈ D . On the other hand, note that, for any ˜ F ∈ D , d X j =1 E [ ˜ b R j ˜ F D j ˜ F ] = d X l,j =1 E [ D l ( ρ R ( ˜ H j l )) ˜ F D j ˜ F ] = − d X l,j =1 E [ ρ R ( ˜ H j l ) D l ˜ F D j ˜ F ] − d X l,j =1 E [ ρ R ( ˜ H j l ) ˜ F D l ( D j ˜ F )] = 0 , (2.11) thanks to the anti-symmetry of { ˜ H j l } 1 ⩽ j,l ⩽ d , the fact that ρ R ( s ) = − ρ R ( − s ) for all s ∈ R and the integration b y parts formula (1.6). These, along with the b oundedness of ˜ b R j ( ω ) , yield that E ∗ ,θ,R ( ˜ F , ˜ G ) ⩽ c 1 ( θ , R ) ∥ ˜ F ∥ H 1 ∥ ˜ G ∥ H 1 , E ∗ ,θ,R ( ˜ F , ˜ F ) ⩾ c 2 ( θ , R ) d X j =1 E [ | D j ˜ F | 2 ] , ˜ F , ˜ G ∈ D . Therefore, E ∗ ,θ,R can b e extended to a bilinear closed form on H 1 × H 1 . In particular, according to the Lax- Milgram theorem (cf. see [39, Section 2]), there exists a unique ˜ ϕ θ,R = ( ˜ ϕ 1 ,θ,R , · · · , ˜ ϕ d,θ,R ) with ˜ ϕ k,θ ,R ∈ H 1 , 1 ⩽ k ⩽ d , such that for every 1 ⩽ k ⩽ d and ˜ F ∈ H 1 , θ E [ ˜ ϕ k,θ ,R ˜ F ] + E ∗ ,θ,R ( ˜ ϕ k,θ ,R , ˜ F ) = θ E [ ˜ ϕ k,θ ,R ˜ F ] + θ d X j =1 E [ D j ˜ ϕ k,θ ,R D j ˜ F ] + d X j =1 E [ ˜ b R j ˜ ϕ k,θ ,R D j ˜ F ] + 1 2 E  Z R d ( ˜ ϕ k,θ ,R ( τ z ω ) − ˜ ϕ k,θ ,R ( ω ))( ˜ F ( τ z ω ) − ˜ F ( ω )) ν ( z ) dz  = − E [ ˜ b k ˜ F ] , (2.12) where we also used the anti-symmetry of { ˜ H j l } 1 ⩽ j,l ⩽ d , the fact that ρ R ( s ) = − ρ R ( − s ) for all s ∈ R and the in tegration by parts formula (1.6). A ccording to the fact ˜ ϕ k,θ ,R ∈ H 1 and the standard approximation pro cedure (also due to the fact that ˜ b R j is b ounded), w e get b y (2.11) that d X j =1 E [ ˜ b R j ˜ ϕ k,θ ,R D j ˜ ϕ k,θ ,R ] = 0 . (2.13) NON-LOCAL SYMMETRIC OPERA TORS WITH DIVERGENCE-FREE DRIFT 9 Then, taking ˜ F = ˜ ϕ k,θ ,R in (2.12) (noting that the test function ˜ ϕ k,θ ,R ∈ H 1 ) and using (2.13), we further deriv e that θ E [ | ˜ ϕ k,θ ,R | 2 ] + 1 2 E  Z R d ( ˜ ϕ k,θ ,R ( τ z ω ) − ˜ ϕ k,θ ,R ( ω )) 2 ν ( z ) dz  + θ d X j =1 E [ | D j ˜ ϕ k,θ ,R | 2 ] = − E [ ˜ b k ˜ ϕ k,θ ,R ] . Under Assumption 1.2, we obtain that for every δ ∈ (0 , 1) , | E [ ˜ b k ˜ ϕ k,θ ,R ] | =      d X l =1 E [ D l ˜ H kl ˜ ϕ k,θ ,R ]      =      − d X l =1 E  i Z R d ξ l [ U ( dξ ) ˜ H kl · ˜ ϕ k,θ ,R ]       ⩽ d X l =1 Z R d | ξ l || E [ U ( dξ ) ˜ H kl · ˜ ϕ k,θ ,R ] | ⩽ δ Z R d  | ξ | 2 1 {| ξ | ⩽ 1 } + | ξ | α 1 {| ξ | > 1 }  E [ U ( dξ ) ˜ ϕ k,θ ,R · ˜ ϕ k,θ ,R ] + c 3 ( δ, d ) d X l =1 Z R d  1 + | ξ | 2  E [ U ( dξ ) ˜ H kl · ˜ H kl ] ⩽ c 4 δ E  Z R d ( ˜ ϕ k,θ ,R ( τ z ω ) − ˜ ϕ k,θ ,R ( ω )) 2 ν ( z ) dz  + c 4 c 3 ( δ, d ) d X l =1 ∥ ˜ H kl ∥ 2 H 1 . Here c 3 ( δ, d ) > 0 ma y dep end on δ and d , c 4 > 0 is indep enden t of δ , U ( dξ ) denotes the pro jection v alued measure on H asso ciated with the unitary op erator { T x } x ∈ R d deﬁned by (1.7), in the second inequality w e hav e used Y oung’s inequality , and the last inequality follows from (2.2), (2.4) and (2.5). Putting all the estimates ab o v e together and choosing δ small enough, we arrive at that for 1 ⩽ k ⩽ d , θ E [ | ˜ ϕ k,θ ,R | 2 ] + 1 4 E  Z R d ( ˜ ϕ k,θ ,R ( τ z ω ) − ˜ ϕ k,θ ,R ( ω )) 2 ν ( z ) dz  + θ d X j =1 E [ | D j ˜ ϕ k,θ ,R | 2 ] ⩽ c 5 d X l =1 ∥ ˜ H kl ∥ 2 H 1 . In particular, sup θ ∈ (0 , 1) ,R ⩾ 1 θ E h | ˜ ϕ k,θ ,R | 2 i ⩽ c 5 d X l =1 ∥ ˜ H kl ∥ 2 H 1 , sup θ ∈ (0 , 1) ,R ⩾ 1 θ   d X j =1 E [ | D j ˜ ϕ k,θ ,R | 2 ]   ⩽ c 5 d X l =1 ∥ ˜ H kl ∥ 2 H 1 , sup θ ∈ (0 , 1) ,R ⩾ 1 E  Z R d ( ˜ ϕ k,θ ,R ( τ z ω ) − ˜ ϕ k,θ ,R ( ω )) 2 ν ( z ) dz  ⩽ 4 c 5 d X l =1 ∥ ˜ H kl ∥ 2 H 1 . (2.14) (2) Deﬁne ϕ θ,R = ( ϕ 1 ,θ,R , · · · , ϕ d,θ,R ) b y ϕ k,θ ,R ( x ; ω ) = ˜ ϕ k,θ ,R ( τ x ω ) − ˜ ϕ k,θ ,R ( ω ) , ( x, ω ) ∈ R d × Ω , 1 ⩽ k ⩽ d. (2.15) It is easy to verify that E [ ϕ k,θ ,R ( x ; ω )] = 0 for ev ery x ∈ R d , and ϕ k,θ ,R ( x + z ; ω ) − ϕ k,θ ,R ( x ; ω ) = ϕ k,θ ,R ( z ; τ x ω ) , x, z ∈ R d , ω ∈ Ω . (2.16) This along with (2.14) yields that sup θ ∈ (0 , 1) ,R ⩾ 1 E  Z R d | ϕ k,θ ,R ( z ; ω ) | 2 ν ( z ) dz  = sup θ ∈ (0 , 1) ,R ⩾ 1 E  Z R d    ˜ ϕ k,θ ,R ( τ z ω ) − ˜ ϕ k,θ ,R ( ω )    2 ν ( z ) dz  < ∞ . (2.17) In particular, since ν ( z ) = | z | − d − α ⩾ 1 with | z | ⩽ 1 , it holds that sup θ ∈ (0 , 1) ,R ⩾ 1 E " Z {| z | ⩽ 1 } | ϕ k,θ ,R ( z ; ω ) | 2 dz # ⩽ sup θ ∈ (0 , 1) ,R ⩾ 1 E  Z R d | ϕ k,θ ,R ( z ; ω ) | 2 ν ( z ) dz  < ∞ . F urthermore, by (2.16), we ﬁnd | B (0 , 1) | · sup θ ∈ (0 , 1) ,R ⩾ 1 Z B (0 , 1)+ B (0 , 1) Z Ω | ϕ θ,R ( z ; ω ) | 2 P ( dω ) dz ! = sup θ ∈ (0 , 1) ,R ⩾ 1 Z B (0 , 1) Z B (0 , 1) Z Ω | ϕ θ,R ( z 1 + z 2 ; ω ) | 2 P ( dω ) dz 1 dz 2 10 XIN CHEN, JIAN W ANG AND KUN YIN = sup θ ∈ (0 , 1) ,R ⩾ 1 Z B (0 , 1) Z B (0 , 1) Z Ω | ϕ θ,R ( z 2 ; ω ) + ϕ θ,R ( z 1 ; τ z 2 ω ) | 2 P ( dω ) dz 1 dz 2 ⩽ 2 | B (0 , 1) | sup θ ∈ (0 , 1) ,R ⩾ 1 Z B (0 , 1) Z Ω | ϕ θ,R ( z 2 ; ω ) | 2 P ( dω ) dz 2 + sup θ ∈ (0 , 1) ,R ⩾ 1 Z B (0 , 1) Z Ω | ϕ θ,R ( z 1 ; ω ) | 2 P ( dω ) dz 1 ! < ∞ , where w e used the stationary prop erty of the transformation { τ x } x ∈ R d , and B (0 , 1) + B (0 , 1) := { z ∈ R d : there exist z 1 , z 2 ∈ B (0 , 1) such that z = z 1 + z 2 } . Applying the estimate ab o v e iteratively , w e can obtain that, for ev ery b ounded subset D ⊂ R d , sup θ ∈ (0 , 1) ,R ⩾ 1 Z D Z Ω | ϕ k,θ ,R ( z ; ω ) | 2 P ( dω ) dz < ∞ . (2.18) On the other hand, according to (2.16), it holds that for every b ounded subset D ⊂ R d , sup θ ∈ (0 , 1) ,R ⩾ 1 Z Ω  Z D Z R d | ϕ k,θ ,R ( x + z ; ω ) − ϕ k,θ ,R ( x ; ω ) | 2 ν ( z ) dz dx  P ( dω ) = sup θ ∈ (0 , 1) ,R ⩾ 1 Z D Z Ω Z R d | ϕ k,θ ,R ( z ; τ x ω ) | 2 ν ( z ) dz P ( dω ) dx = | D | · sup θ ∈ (0 , 1) ,R ⩾ 1 Z Ω Z R d | ϕ k,θ ,R ( z ; ω ) | 2 ν ( z ) dz P ( dω ) < ∞ . (2.19) In particular, sup θ ∈ (0 , 1) ,R ⩾ 1 Z Ω Z D Z {| z | ⩽ 1 } | ϕ k,θ ,R ( x + z ; ω ) − ϕ k,θ ,R ( x ; ω ) | 2 1 | z | d + α dz dx ! P ( dω ) ⩽ | D | · sup θ ∈ (0 , 1) ,R ⩾ 1 Z Ω Z R d | ϕ k,θ ,R ( z ; ω ) | 2 ν ( z ) dz P ( dω ) < ∞ . Therefore, { ϕ θ,R } θ ∈ (0 , 1) ,R ⩾ 1 is weakly compact in L 2 (Ω , W α/ 2 , 2 loc ( R d , R d ); P ) , and so there exist a subsequence { ϕ θ m ,R m } m ⩾ 1 (with θ m → 0 and R m → ∞ as m → ∞ ) and ϕ ∈ L 2 (Ω , W α/ 2 , 2 loc ( R d , R d ); P ) such that ϕ θ m ,R m con v erges w eakly to ϕ in L 2 (Ω , W α/ 2 , 2 loc ( R d , R d ); P ) as m → ∞ . F urthermore, by (2.17), without loss of generalit y , we can also conclude that ϕ θ m ,R m con v erges w eakly to ϕ in L 2 ( R d × Ω , R d ; ν ( z ) dz × P ) as m → ∞ . In particular, ϕ ∈ L 2 (Ω , W α/ 2 , 2 loc ( R d , R d ); P ) ∩ L 2 ( R d × Ω , R d ; ν ( z ) dz × P ) . Using (2.16) and follo wing the appro ximation argument in the pro of of [40, Prop osition 4.3], we kno w that the co-cycle prop erty (2.7) still holds for ϕ ( x ; ω ) , and that E [ ϕ ( x ; ω )] = 0 for every x ∈ R d . Thus, from (2.17), (2.18) and (2.19), we can prov e (2.8) and (2.9). (3) In this part, we are going to pro v e (2.6). Let ˜ G ∈ D . T aking ˜ F ( ω ) := ˜ G ( τ − x ω ) in (2.12), we obtain that for ev ery f ∈ C 1 c ( R d ) and 1 ⩽ k ⩽ d , θ m Z Ω  Z R d ˜ ϕ k,θ m ,R m ( ω ) ˜ G ( τ − x ω ) f ( x ) dx  P ( dω ) + θ m d X j =1 Z Ω  Z R d D j ˜ ϕ k,θ m ,R m ( ω ) f ( x ) D j ˜ G ( τ − x ω ) dx  P ( dω ) + 1 2 Z Ω  Z R d Z R d ( ˜ ϕ k,θ m ,R m ( τ z ω ) − ˜ ϕ k,θ m ,R m ( ω ))( ˜ G ( τ − x + z ω ) − ˜ G ( τ − x ω )) ν ( z ) dz f ( x ) dx  P ( dω ) − d X j =1 Z Ω  Z R d ˜ b R m j ( ω ) D j ˜ ϕ k,θ m ,R m ( ω ) f ( x ) ˜ G ( τ − x ω ) dx  P ( dω ) = − Z Ω  Z R d ˜ b k ( ω ) ˜ G ( τ − x ω ) f ( x ) dx  P ( dω ) , (2.20) NON-LOCAL SYMMETRIC OPERA TORS WITH DIVERGENCE-FREE DRIFT 11 where we used the integration by parts form ula (1.6), the anti-symmetry of { ˜ H j l } 1 ⩽ j,l ⩽ d and the fact that ρ R ( s ) = − ρ R ( − s ) for all s ∈ R . A ccording to (2.14), for all 1 ⩽ k , j ⩽ d , we ha v e lim m →∞ θ m Z Ω  Z R d ˜ ϕ k,θ m ,R m ( ω ) f ( x ) ˜ G ( τ − x ω ) dx  P ( dω ) = 0 and lim m →∞ θ m Z Ω  Z R d D j ˜ ϕ k,θ m ,R m ( ω ) f ( x ) D j ˜ G ( τ − x ω ) dx  P ( dω ) = 0 . On the other hand, it holds that Z Ω  Z R d Z R d ( ˜ ϕ k,θ m ,R m ( τ z ω ) − ˜ ϕ k,θ m ,R m ( ω ))( ˜ G ( τ − x + z ω ) − ˜ G ( τ − x ω )) ν ( z ) f ( x ) dz dx  P ( dω ) = lim δ ↓ 0 Z Ω Z R d Z {| z | >δ } ( ˜ ϕ k,θ m ,R m ( τ z ω ) − ˜ ϕ k,θ m ,R m ( ω )) ˜ G ( τ − x + z ω ) ν ( z ) f ( x ) dz dx ! P ( dω ) − Z Ω Z R d Z {| z | >δ } ( ˜ ϕ k,θ m ,R m ( τ z ω ) − ˜ ϕ k,θ m ,R m ( ω )) ˜ G ( τ − x ω ) ν ( z ) f ( x ) dz dx ! P ( dω ) ! =: lim δ ↓ 0 ( I 1 ,δ − I 2 ,δ ) . Using the stationary of the transformation { τ x } x ∈ R d , w e get I 1 ,δ = Z Ω Z R d Z {| z | >δ } ( ˜ ϕ k,θ m ,R m ( τ x ω ) − ˜ ϕ k,θ m ,R m ( τ x − z ω )) ˜ G ( ω ) ν ( z ) f ( x ) dz dx ! P ( dω ) = Z Ω Z R d Z {| z | >δ } ( ˜ ϕ k,θ m ,R m ( τ x + z ω ) − ˜ ϕ k,θ m ,R m ( τ x ω )) ˜ G ( ω ) ν ( z ) f ( x + z ) dz dx ! P ( dω ) , where the second equalit y is due to the c hange of v ariable ˜ x = x − z . By the stationary of the transformation { τ x } x ∈ R d again, it is easy to see that I 2 ,δ = Z Ω Z R d Z {| z | >δ } ( ˜ ϕ k,θ m ,R m ( τ x + z ω ) − ˜ ϕ k,θ m ,R m ( τ x ω )) ˜ G ( ω ) ν ( z ) f ( x ) dz dx ! P ( dω ) . Th us, putting all the estimates ab ov e together yields that Z Ω  Z R d Z R d ( ˜ ϕ k,θ m ,R m ( τ z ω ) − ˜ ϕ k,θ m ,R m ( ω ))( ˜ G ( τ − x + z ω ) − ˜ G ( τ − x ω )) ν ( z ) f ( x ) dz dx  P ( dω ) = Z Ω  Z R d Z R d ( ˜ ϕ k,θ m ,R m ( τ x + z ω ) − ˜ ϕ k,θ m ,R m ( τ x ω ))( f ( x + z ) − f ( x )) ν ( z ) dz dx  ˜ G ( ω ) P ( dω ) = Z Ω  Z R d Z R d ϕ k,θ m ,R m ( z ; τ x ω )( f ( x + z ) − f ( x )) ν ( z ) dz dx  ˜ G ( ω ) P ( dω ) , where in the last equality we used (2.15). Since ϕ θ m ,R m con v erges weakly to ϕ in L 2  R d × Ω , R d ; ν ( z ) dz × P  as m → ∞ , we hav e for ev ery f ∈ C 1 c ( R d ) and x ∈ R d , lim m →∞ Z Ω  Z R d ϕ k,θ m ,R m ( z ; τ x ω )( f ( x + z ) − f ( x )) ν ( z ) dz  ˜ G ( ω ) P ( dω ) = Z Ω  Z R d ϕ k ( z ; τ x ω )( f ( x + z ) − f ( x )) ν ( z ) dz  ˜ G ( ω ) P ( dω ) = Z Ω  Z R d ( ϕ k ( x + z ; ω ) − ϕ k ( x ; ω ))( f ( x + z ) − f ( x )) ν ( z ) dz  ˜ G ( ω ) P ( dω ) , 12 XIN CHEN, JIAN W ANG AND KUN YIN where in the last equality we used the co-cycle prop ert y (2.7). By (2.17) and the stationary of the transformation { τ x } x ∈ R d , it is not diﬃcult to verify that for every b ounded subset D ⊂ R d , sup m ⩾ 1 Z D  Z Ω Z R d ϕ k,θ m ,R m ( z ; τ x ω )( f ( x + z ) − f ( x )) ν ( z ) dz ˜ G ( ω ) P ( dω )  2 dx < ∞ and lim R ↑∞ sup m ⩾ 1 Z {| x | >R }  Z Ω Z R d | ϕ k,θ m ,R m ( z ; τ x ω ) || f ( x + z ) − f ( x ) | ν ( z ) dz ˜ G ( ω ) P ( dω )  dx = 0 . With all the prop erties ab ov e, we can apply the routine limit argumen ts to deduce that for every f ∈ C 1 c ( R d ) , lim m →∞ Z Ω  Z R d Z R d ( ˜ ϕ k,θ m ,R m ( τ z ω ) − ˜ ϕ k,m ( ω ))( ˜ G ( τ − x + z ω ) − ˜ G ( τ − x ω )) ν ( z ) f ( x ) dz dx  P ( dω ) = lim m →∞ Z R d  Z Ω Z R d ϕ k,θ m ,R m ( z ; τ x ω ) ( f ( x + z ) − f ( x )) ν ( z ) dz ˜ G ( ω ) P ( dω )  dx = Z Ω  Z R d Z R d ( ϕ k ( x + z ; ω ) − ϕ k ( x ; ω )) ( f ( x + z ) − f ( x )) ν ( z ) dz dx  ˜ G ( ω ) P ( dω ) . Note that ˜ ϕ θ m ,R m ∈ H 1 , so ϕ θ m ,R m ( · ; ω ) ∈ W 1 , 2 loc ( R d , R d ) and D j ˜ ϕ θ m ,R m ( τ x ω ) = − ∂ ϕ θ m ,R m ( · ; ω ) ∂ x j ( x ) . Then, b y the integration b y parts formula and the an ti-symmetry of { ˜ H j l } 1 ⩽ j,l ⩽ d , w e deduce that for every f ∈ C 1 c ( R d ) and ˜ G ∈ D , Z Ω  Z R d ˜ b R m j ( ω ) D j ˜ ϕ k,θ m ,R m ( ω ) ˜ G ( τ − x ω ) f ( x ) dx  P ( dω ) = Z Ω  Z R d ˜ b R m j ( τ x ω ) D j ˜ ϕ k,θ m ,R m ( τ x ω ) f ( x ) dx  ˜ G ( ω ) P ( dω ) = − Z Ω  Z R d ˜ b R m j ( τ x ω ) ∂ ϕ k,θ m ,R m ( x ; ω ) ∂ x j f ( x ) dx  ˜ G ( ω ) P ( dω ) = Z Ω  Z R d b R m j ( x ; ω ) ϕ k,θ m ,R m ( x ; ω ) ∂ f ( x ) ∂ x j dx  ˜ G ( ω ) P ( dω ) , where b R m j ( x ; ω ) = ˜ b R m j ( τ x ω ) . On the other hand, according to the deﬁnition of b R ( x ; ω ) and the fact that ˜ H j l ∈ H 1 for all 1 ⩽ j, l ⩽ d , it is easy to verify that for every b ounded subset D ⊂ R d , lim m →∞ Z Ω  Z D | b ( x ; ω ) − b R m ( x ; ω ) | 2 dx  P ( dω ) = 0 . Therefore, for ev ery f ∈ C 1 c ( R d ) and ˜ G ∈ D , lim m →∞ Z Ω  Z R d ˜ b R m j ( ω ) D j ˜ ϕ k,θ m ,R m ( ω ) f ( x ) ˜ G ( τ − x ω ) dx  P ( dω ) = lim m →∞ Z Ω  Z R d b R m j ( x ; ω ) ϕ k,θ m ,R m ( x ; ω ) ∂ f ( x ) ∂ x j dx  ˜ G ( ω ) P ( dω ) = lim m →∞ Z Ω  Z R d b j ( x ; ω ) ϕ k,θ m ,R m ( x ; ω ) ∂ f ( x ) ∂ x j dx  ˜ G ( ω ) P ( dω ) = Z Ω  Z R d b j ( x ; ω ) ϕ k ( x ; ω ) ∂ f ( x ) ∂ x j dx  ˜ G ( ω ) P ( dω ) , where the last equality is due to the fact that ϕ θ m ,R m con v erges weakly to ϕ in L 2 (Ω , W α/ 2 , 2 loc ( R d , R d ); P ) as m → ∞ . NON-LOCAL SYMMETRIC OPERA TORS WITH DIVERGENCE-FREE DRIFT 13 Putting all the estimates abov e together into (2.20) and letting m → ∞ , we will obtain that for every f ∈ C 1 c ( R d ) and ˜ G ∈ D , 1 2 Z Ω  Z R d Z R d ( ϕ k ( x + z ; ω ) − ϕ k ( x ; ω ))( f ( x + z ) − f ( x )) ν ( z ) dz dx  ˜ G ( ω ) P ( dω ) − d X j =1 Z Ω  Z R d b j ( x ; ω ) ϕ k ( x ; ω ) ∂ f ( x ) ∂ x j dx  ˜ G ( ω ) P ( dω ) = − Z Ω  Z R d b k ( x ; ω ) f ( x ) dx  ˜ G ( ω ) P ( dω ) , where we also used the stationary of the transformation { τ x } x ∈ R d in the right hand side of the equalit y ab ov e. Note that the equation ab ov e holds for every ˜ G ∈ D . By choosing a dense subset of test functions in C 1 c ( R d ) , w e can ﬁnd a common P -null set Λ ⊂ Ω such that (2.6) holds for every f ∈ C 1 c ( R d ) and ω / ∈ Λ . The pro of is complete. □ 2.2. Homogenized co eﬃcients. Lemma 2.3. Supp ose that Assumptions 1 . 1 and 1 . 2 hold. L et ¯ A := { ¯ a j k } 1 ⩽ j,k ⩽ d b e the d × d matrix deﬁne d by (1.10) . Then, ¯ A is strictly p ositive deﬁnite and b ounde d. Pr o of. F or every non-zero ξ := ( ξ 1 , · · · , ξ d ) ∈ R d , d X j,k =1 ¯ a j k ξ j ξ k = 1 2 E   Z R d d X j,k =1 ξ j ξ k ( z j + ϕ j ( z ; ω )) ( z k + ϕ k ( z ; ω )) ν ( z ) dz   = 1 2 E   Z R d  d X j =1 ξ j ( z j + ϕ j ( z ; ω ))  2 ν ( z ) dz   ⩾ 0 . This along with (2.8) implies that ¯ A is p ositiv e deﬁnite and b ounded. Next, w e supp ose that P d j,k =1 ¯ a j k ξ j ξ k = 0 for some non-zero ξ ∈ R d . Then, for a.s. ω ∈ Ω and any z ∈ B (0 , 1) d X j =1 ξ j ( z j + ϕ j ( z ; ω )) = 0 . (2.21) Mean while, by Theorem 2.2, E [ ϕ ( x ; ω )] = 0 and so for all z ∈ B (0 , 1) , E   d X j =1 ξ j ( z j + ϕ j ( z ; ω ))   = ⟨ ξ , z ⟩ , whic h contradicts with (2.21). Therefore, w e conclude that P d j,k =1 ¯ a j k ξ j ξ k > 0 for every non-zero ξ ∈ R d , and so ¯ A is strictly p ositiv e deﬁnite. □ 3. The scaled resol vent equa tion and proper ties of the corrector This section consists tw o parts. In the ﬁrst subsection, we prov e the existence of w eak solution to the scaled resolv en t equation (1.9), where the pro of is partly inspired by that of Theorem 2.2. In the second one, w e will establish some regularity prop erties and estimates for the corrector ϕ to the equation (2.1), which are key ingredien ts to the pro of of Theorem 1.4. 3.1. The existence of solution to the scaled resolven t equation (1.9) . Prop osition 3.1. Supp ose that Assumptions 1 . 1 and 1 . 2 hold. Then, for every h ∈ G λ with λ > 0 and every ε ∈ (0 , 1) , ther e exists a we ak solution u ε,ω := u ε ( · ; ω ) ∈ W α/ 2 , 2 ( R d ) to the e quation (1.9) such that the fol lowing statements hold for a.e. ω ∈ Ω : 14 XIN CHEN, JIAN W ANG AND KUN YIN (i) for every f ∈ C 1 c ( R d ) , ε − d − 2 2 Z R d ( u ε ( x + z ; ω ) − u ε ( x ; ω )) ( f ( x + z ) − f ( x )) ν  z ε  dz + ε − 1 d X j =1 Z R d b j  x ε ; ω  ∂ f ( x ) ∂ x j u ε ( x ; ω ) dx + λ Z R d u ε ( x ; ω ) f ( x ) dx = Z R d h ( x ) f ( x ) dx, (3.1) (ii) sup ε ∈ (0 , 1) ∥ u ε ( · ; ω ) ∥ L ∞ ( R d ) < ∞ , sup ε ∈ (0 , 1) ∥ u ε ( · ; ω ) ∥ L 2 ( R d ) < ∞ , (3.2) and sup ε ∈ (0 , 1) ε − d − 2 Z R d Z R d ( u ε ( x + z ; ω ) − u ε ( x ; ω )) 2 ν  z ε  dz dx < ∞ . (3.3) Pr o of. (1) Let η ( x ) = ( C 0 exp  − 1 1 −| x | 2  , | x | < 1 , 0 , | x | ⩾ 1 , (3.4) where C 0 is a p ositive normalizing constan t so that Z R d η ( x ) dx = 1 . F or ev ery δ ∈ (0 , 1) , let η δ ( x ) := δ − d η  x δ  b e the standard smooth molliﬁer, and denote f ∗ η δ ( x ) := Z R d η δ ( x − y ) f ( y ) dy b y the smo oth con v olution of the function f with resp ect to η δ . F or every θ > 0 and R ⩾ 1 , deﬁne L ε,ω θ,R f ( x ) = L ε 0 f ( x ) + θ ∆ f ( x ) + ε − 1 d X j =1 b R j ∗ η θ  x ε ; ω  ∂ f ( x ) ∂ x j , f ∈ C 2 c ( R d ) , where b R j ( x ; ω ) = ˜ b R j ( τ x ω ) with ˜ b R j b eing deﬁned b y (2.10). Then, by (1.5), the integration b y parts formula and the anti-symmetric prop erty of { ˜ H j l } 1 ⩽ j,l ⩽ d (whic h implies the anti-symmetry of { ρ R ( ˜ H j l ) } 1 ⩽ j,l ⩽ d thanks to the deﬁnition of ρ R ), for ev ery f , g ∈ C 2 c ( R d ) , E ε,ω θ,R ( f , g ) : = − Z R d L ε,ω θ,R f ( x ) g ( x ) dx = ε − d − 2 2 Z R d ( f ( x + z ) − f ( x )) ( g ( x + z ) − g ( x )) ν  z ε  dz + θ Z R d ⟨∇ f ( x ) , ∇ g ( x ) ⟩ dx − d X j,l =1 Z R d ρ R ( H j l ( · ; ω )) ∗ η θ  x ε  ∂ f ( x ) ∂ x j ∂ g ( x ) ∂ x l dx, where ρ R : R → R is the cut-oﬀ function used in (2.10). This implies that for all f , g ∈ C 2 c ( R d ) , | E ε,ω θ,R ( g , h ) | ⩽ c 1 ( ε, θ , R )  ∥ f ∥ W α/ 2 , 2 ( R d ) + ∥ f ∥ W 1 , 2 ( R d )   ∥ g ∥ W α/ 2 , 2 ( R d ) + ∥ g ∥ W 1 , 2 ( R d )  ⩽ c 2 ( ε, θ , R ) ∥ f ∥ W 1 , 2 ( R d ) ∥ g ∥ W 1 , 2 ( R d ) , (3.5) where c 2 ( ε, θ , R ) is a p ositive constan t that ma y dep end on ε , θ and R . In particular, we can extend E ε,ω θ,R to a con tin uous bilinear form on W 1 , 2 ( R d ) × W 1 , 2 ( R d ) . On the other hand, by the an ti-symmetry of { ρ R ( ˜ H j l ) } 1 ⩽ j,l ⩽ d again, w e can show that for all f ∈ W 1 , 2 ( R d ) , E ε,ω θ,R ( f , f ) ⩾ c 3 ( ε, θ , R ) Z R d |∇ f ( x ) | 2 dx. NON-LOCAL SYMMETRIC OPERA TORS WITH DIVERGENCE-FREE DRIFT 15 Therefore, by the Lax-Milgram theorem (e.g., see [39, Section 2]), for any λ > 0 and h ∈ G λ , there exists a unique u ε θ,R ( · ; ω ) ∈ W 1 , 2 ( R d ) suc h that for all f ∈ W 1 , 2 ( R d ) , ε − d − 2 2 Z R d Z R d ( u ε θ,R ( x + z ; ω ) − u ε θ,R ( x ; ω ))( f ( x + z ) − f ( x )) ν  z ε  dz dx + θ Z R d ⟨∇ u ε θ,R ( x ; ω ) , ∇ f ( x ) ⟩ dx + ε − 1 d X j =1 Z R d b R j ∗ η θ  x ε ; ω  ∂ f ( x ) ∂ x j u ε θ,R ( x ; ω ) dx + λ Z R d u ε θ,R ( x ; ω ) f ( x ) dx = Z R d h ( x ) f ( x ) dx. (3.6) T aking f = u ε θ,R ( · ; ω ) in the equation (3.6), and using (2.13) as w ell as its pro of due to the an ti-symmetry of { ρ R ( H j l ) } 1 ⩽ j,l ⩽ d , w e hav e ε − d − 2 2 Z R d Z R d  u ε θ,R ( x + z ; ω ) − u ε θ,R ( x ; ω )  2 ν  z ε  dz dx + θ Z R d   ∇ u ε θ,R ( x ; ω )   2 dx + λ Z R d | u ε θ,R ( x ; ω ) | 2 dx = Z R d h ( x ) u ε θ,R ( x ; ω ) dx ⩽ λ 2 Z R d | u ε θ,R ( x ; ω ) | 2 dx + 1 2 λ Z R d | h ( x ) | 2 dx. In particular, sup ε,θ ∈ (0 , 1) ,R ⩾ 1 ε − d − 2 Z R d Z R d  u ε θ,R ( x + z ; ω ) − u ε θ,R ( x ; ω )  2 ν  z ε  dz dx ⩽ c 4 ( λ ) Z R d | h ( x ) | 2 dx, (3.7) sup ε,θ ∈ (0 , 1) ,R ⩾ 1 Z R d | u ε θ,R ( x ; ω ) | 2 dx ⩽ λ − 2 Z R d | h ( x ) | 2 dx (3.8) and sup ε,θ ∈ (0 , 1) ,R ⩾ 1 θ Z R d   ∇ u ε θ,R ( x ; ω )   2 dx ⩽ c 4 ( λ ) Z R d | h ( x ) | 2 dx. (3.9) Since b R ( · ; ω ) ∗ η θ is a smo oth b ounded vector ﬁeld in R d , there exists a unique Mark o v semigroup { T θ,R ,ε,ω t } t ⩾ 0 asso ciated with the op erator L ε,ω θ,R so that u ε θ,R ( x ; ω ) = Z ∞ 0 e − λt T θ,R ,ε,ω t h ( x ) dt, whic h implies that sup ε,θ ∈ (0 , 1) ,R ⩾ 1 ∥ u ε θ,R ( · ; ω ) ∥ L ∞ ( R d ) ⩽ Z ∞ 0 e − λt ∥ h ∥ L ∞ ( R d ) dt ⩽ λ − 1 ∥ h ∥ L ∞ ( R d ) . (3.10) Hence, according to (3.7) and (3.8) as well as the deﬁnition of ν ( z ) , we kno w that for ev ery ﬁxed ε ∈ (0 , 1) , { u ε θ,R ( · ; ω ) } θ ∈ (0 , 1) ,R ⩾ 1 is weakly compact in W α/ 2 , 2 ( R d ) , so it is compact in L 2 loc ( R d ) , thanks to the compact em b edding of the space W α/ 2 , 2 loc ( R d ) ; see [17, Theorem 7.1]. Then, there exist a subsequence { u ε θ m ,R m ( · ; ω ) } m ⩾ 1 (with θ m → 0 and R m → ∞ as m → ∞ ) and u ε ( · ; ω ) in W α/ 2 , 2 ( R d ) suc h that u ε θ m ,R m ( · ; ω ) con v erges to u ε ( · ; ω ) w eakly in W α/ 2 , 2 ( R d ) and strongly in L 2 loc ( R d ) as m → ∞ . That is, lim m →∞ Z D   u ε θ m ,R m ( x ; ω ) − u ε ( x ; ω )   2 dx = 0 for an y b ounded subset D ⊂ R d , (3.11) and for ev ery f ∈ C ∞ c ( R d ) , lim m →∞ Z R d Z R d (( u ε θ m ,R m ( x + z ; ω ) − u ε ( x + z ; ω )) − ( u ε θ m ,R m ( x ; ω ) − u ε ( x ; ω )))( f ( x + z ) − f ( x )) | z | d + α dz dx = 0 . These tw o estimates, along with the deﬁnition of ν ( z ) again, (3.7) and (3.10), in turn yield that for every f ∈ C ∞ c ( R d ) , lim m →∞ Z R d Z R d (( u ε θ m ,R m ( x + z ; ω ) − u ε ( x + z ; ω )) − ( u ε θ m ,R m ( x ; ω ) − u ε ( x ; ω )))( f ( x + z ) − f ( x )) ν  z ε  dz dx = 0 . (3.12) 16 XIN CHEN, JIAN W ANG AND KUN YIN F urthermore, by (3.9), for all f ∈ C 1 c ( R d ) , lim m →∞ θ m     Z R d  ∇ u ε θ m ,R m ( x ; ω ) , ∇ f ( x )  dx     ⩽ lim m →∞ θ m  Z R d   ∇ u ε θ m ,R m ( x ; ω )   2 dx  1 / 2 ·  Z R d |∇ f ( x ) | 2 dx  1 / 2 ⩽ c 5 lim m →∞ θ 1 / 2 m = 0 . (3.13) F or ev ery f ∈ C 1 c ( R d ) , set the supp ort of f by V ( f ) := supp[ f ] . By the deﬁnition of b R j ( x ; ω ) and the fact that | ρ ′ R ( s ) | ⩽ 2 for all s ∈ R , w e kno w that for every ﬁxed ε ∈ (0 , 1) and ω ∈ Ω , sup R ⩾ 1 ,x ∈ V ( f )    b R j  x ε ; ω     ⩽ c 6 sup x ∈ V ( f ) ε sup 1 ⩽ j,l ⩽ d     H j l ( x ; ω ) ∂ x l     =: c 7 ( ε, f , ω ) < ∞ , where we used the fact that x 7→ D l ˜ H j l ( τ x ω ) is contin uous. Combining this with (3.10) and (3.11), and applying the dominated con v ergence theorem, we get lim m →∞ Z R d b R m j ∗ η θ m  x ε ; ω  ∂ f ( x ) ∂ x j u ε θ m ,R m ( x ; ω ) dx = lim m →∞ Z R d b R m j  x ε ; ω  ∂ f ( x ) ∂ x j u ε ( x ; ω ) dx = Z R d b j  x ε ; ω  ∂ f ( x ) ∂ x j u ε ( x ; ω ) dx, f ∈ C 1 c ( R d ) . Therefore, putting this, (3.12), (3.13) and (3.11) together, we can prov e (3.1) by letting m → ∞ in (3.6). F urthermore, by (3.7), (3.8) and (3.10), we can get the desired conclusions (3.2) and (3.3). □ 3.2. Prop erties of the corrector ϕ . In this part, let ϕ : R d × Ω → R d b e the corrector to the equation (2.1) that w as constructed in Theorem 2 . 2 . W e ﬁrst consider the W 1 ,q loc -regularit y prop ert y (with q ∈ (1 , 2) ) of the corrector ϕ . Lemma 3.2. Supp ose that Assumptions 1 . 1 and 1 . 2 hold, and that (1.12) holds with some q ∈ (1 , 2) and d > 4( α − 1) . Then ther e ar e a P -nul l set Λ and ˜ Φ ∈ L q (Ω , R d × R d , P ) , so that ϕ ( · ; ω ) ∈ W 1 ,q loc ( R d , R d ) for every ω / ∈ Λ , and ∇ ϕ ( x ; ω ) = ˜ Φ( τ x ω ) for a.e. x ∈ R d and every ω / ∈ Λ . Pr o of. The pro of is split into tw o parts. (1) Let ϕ θ m ,R m : R d × Ω → R d b e deﬁned b y (2.15) in the pro of of Theorem 2.2, which con v erges w eakly to ϕ in L 2 (Ω , W α/ 2 , 2 loc ( R d , R d ); P ) and L 2 ( R d × Ω , R d ; ν ( z ) dz × P ) as m → ∞ . A ccording to (2.12), (2.20) and the proof of (2.6), w e can verify that there exists a P -null set Λ 0 , so that for ev ery f ∈ C 1 c ( R d ) , ω / ∈ Λ 0 and 1 ⩽ k ⩽ d , θ m Z R d ϕ k,θ m ,R m ( x ; ω ) f ( x ) dx + θ m ˜ ϕ k,θ m ,R m ( ω ) Z R d f ( x ) dx + θ m Z R d ⟨∇ ϕ k,θ m ,R m ( x ; ω ) , ∇ f ( x ) ⟩ dx + 1 2 Z R d Z R d ( ϕ k,θ m ,R m ( x + z ; ω ) − ϕ k,θ m ,R m ( x ; ω )) ( f ( x + z ) − f ( x )) ν ( z ) dz dx + Z R d ⟨ b R m ( x ; ω ) , ∇ ϕ k,θ m ,R m ( x ; ω ) ⟩ f ( x ) dx = − Z R d b k ( x ; ω ) f ( x ) dx, (3.14) where w e used the fact that ϕ k,θ m ,R m ( · ; ω ) ∈ W 1 , 2 ( R d ) for a.s. ω ∈ Ω , due to (2.14). Let r 0 ⩾ 1 b e the constan t in (1.12). Let ψ r 0 : R d → [0 , 1] b e a cut-oﬀ function such that ψ r 0 ∈ C ∞ c ( R d ) , supp[ ψ r 0 ] ⊂ B (0 , r 0 ) , sup x ∈ R d |∇ k ψ r 0 ( x ) | ⩽ c 1 ( k , r 0 ) for k = 1 , 2 , and ψ r 0 ( x ) = 1 for ev ery x ∈ B (0 , r 0 / 2) . Deﬁne ϕ r 0 θ m ,R m ( x ; ω ) = ϕ θ m ,R m ( x ; ω ) ψ r 0 ( x ) . T aking f = f ψ r 0 in the equation (3.14), w e ﬁnd that for every NON-LOCAL SYMMETRIC OPERA TORS WITH DIVERGENCE-FREE DRIFT 17 f ∈ C 1 c ( R d ) , θ m Z R d ϕ r 0 k,θ m ,R m ( x ; ω ) f ( x ) dx + θ m Z R d ⟨∇ ϕ r 0 k,θ m ,R m ( x ; ω ) , ∇ f ( x ) ⟩ dx + 1 2 Z R d Z R d ( ϕ r 0 k,θ m ,R m ( x + z ; ω ) − ϕ r 0 k,θ m ,R m ( x ; ω )) ( f ( x + z ) − f ( x )) ν ( z ) dz dx = Z R d F k,m ( x ; ω ) f ( x ) dx. (3.15) Here, F k,m ( x ; ω ) := 3 X j =0 F k,m,j ( x ; ω ) , where F k,m, 0 ( x ; ω ) = −⟨ b R m ( x ; ω ) , ∇ ϕ r 0 k,θ m ,R m ( x ; ω ) ⟩ , F k,m, 1 ( x ; ω ) = − b k ( x ; ω ) ψ r 0 ( x ) − θ m ˜ ϕ k,θ m ,R m ( ω ) ψ r 0 ( x ) , F k,m, 2 ( x ; ω ) = − Z R d ( ϕ k,θ m ,R m ( x + z ; ω ) − ϕ k,θ m ,R m ( x ; ω )) ( ψ r 0 ( x + z ) − ψ r 0 ( x )) ν ( z ) dz , F k,m, 3 ( x ; ω ) = − ϕ k,θ m ,R m ( x ; ω )  L 0 ψ r 0 ( x ) + θ m ∆ ψ r 0 ( x ) − ⟨ b R m ( x ; ω ) , ∇ ψ r 0 ( x ) ⟩  − 2 θ m ⟨∇ ϕ k,θ m ,R m ( x ; ω ) , ∇ ψ r 0 ( x ) ⟩ . Deﬁne ϕ r 0 ,δ θ m ,R m ( x ; ω ) = ϕ r 0 θ m ,R m ∗ η δ ( x ; ω ) , where η δ : R d → [0 , 1] is the smo oth molliﬁer used in the pro of of Prop osition 3.1. Changing x in to z , taking f ( z ) = η δ ( x − z ) and integrating with resp ect to the v ariable z in (3.15), w e can derive that θ m ϕ r 0 ,δ k,θ m ,R m ( x ; ω ) − θ m ∆ ϕ r 0 ,δ k,θ m ,R m ( x ; ω ) − L 0 ϕ r 0 ,δ k,θ m ,R m ( x ; ω ) = F k,m ∗ η δ ( x ; ω ) , where F k,m ∗ η δ ( x ; ω ) := Z R d F k,m ( z ; ω ) η δ ( x − z ) dz , x ∈ R d . This implies θ m ϕ r 0 ,δ k,θ m ,R m ( x ; ω ) − θ m ∆ ϕ r 0 ,δ k,θ m ,R m ( x ; ω ) − ∆ α/ 2 ϕ r 0 ,δ k,θ m ,R m ( x ; ω ) = G δ k,m ( x ; ω ) , (3.16) where ∆ α/ 2 denotes the fractional Laplacian op erator corresp onding to the Lévy measure | z | − d − α dz , and G δ k,m ( x ; ω ) := F k,m ∗ η δ ( x ; ω ) + Z {| z | > 1 } ( ϕ r 0 ,δ k,θ m ,R m ( x + z ; ω ) − ϕ r 0 ,δ k,θ m ,R m ( x ; ω )) ν ( z ) dz − Z {| z | > 1 } ( ϕ r 0 ,δ k,θ m ,R m ( x + z ; ω ) − ϕ r 0 ,δ k,θ m ,R m ( x ; ω )) 1 | z | d + α dz . A ccording to (3.16), we hav e ϕ r 0 ,δ k,θ m ,R m ( x ; ω ) = Z ∞ 0 e − θ m t T m t G δ k,m ( x ; ω ) dt, where { T m t } t ⩾ 0 : L 2 ( R d ; dx ) → L 2 ( R d ; dx ) is the Marko v semigroup asso ciated with the inﬁnitesimal op erator θ m ∆ + ∆ α/ 2 . Letting δ → 0 yields that ϕ r 0 k,θ m ,R m ( x ; ω ) = Z ∞ 0 e − θ m t T m t G k,m ( x ; ω ) dt, (3.17) where G k,m ( x ; ω ) := F k,m ( x ; ω ) + Z {| z | > 1 } ( ϕ r 0 k,θ m ,R m ( x + z ; ω ) − ϕ r 0 k,θ m ,R m ( x ; ω )) ν ( z ) dz − Z {| z | > 1 } ( ϕ r 0 k,θ m ,R m ( x + z ; ω ) − ϕ r 0 k,θ m ,R m ( x ; ω )) 1 | z | d + α dz 18 XIN CHEN, JIAN W ANG AND KUN YIN Let p t,α ( x ) and q m t ( x ) b e the fundamen tal solutions to the op erators ∆ α/ 2 and θ m ∆ resp ectiv ely . Thus, it holds that T m t f ( x ) = Z R d p m t ( x − y ) f ( y ) dy , f ∈ L 2 ( R d ; dx ) , where p m t ( x ) = Z R d p t,α ( x − y ) q m t ( y ) dy , t > 0 , x ∈ R d . It is w ell known that (see e.g. [15, Lemma 2.2]) |∇ l p t,α ( x ) | ⩽ c 2 t − l/α p t,α ( x ) ⩽ c 3 t − ( d + l ) /α , t > 0 , x ∈ R d , l = 0 , 1 , 2 . This yields that |∇ l p m t ( x ) | ⩽ c 2 t − l/α p m t ( x ) ⩽ c 3 t − ( d + l ) /α , t > 0 , x ∈ R d , l = 0 , 1 , 2 , and here we emphasize that the p ositive constants c 2 and c 3 are indep endent of m . Hence, for ev ery f ∈ L 2 ( R d ; dx ) , t > 0 , x ∈ R d and m ⩾ 1 , |∇ T m t f ( x ) | ⩽ c 2 t − 1 /α T m t | f | ( x ) , (3.18) |∇ T m t f ( x ) | ⩽ c 2 t − 1 /α  Z R d | p m t ( x − z ) | 2 dz  1 / 2  Z R d | f ( z ) | 2 dz  1 / 2 ⩽ c 2 t − 1 /α p m 2 t ( x ) 1 / 2  Z R d | f ( z ) | 2 dz  1 / 2 ⩽ c 4 t − ( d +2) / (2 α )  Z R d | f ( z ) | 2 dz  1 / 2 (3.19) and     ∇ T m t  ∂ f ∂ x j  ( x )     =     Z R d ∂ ∇ p m t ( x − z ) ∂ x j f ( z ) dz     ⩽ c 2 t − 2 /α T m t | f | ( x ) . (3.20) (2) Recall that ϕ r 0 k,θ m ,R m ( x ; ω ) = ϕ k,θ m ,R m ( x ; ω ) ψ r 0 ( x ) , so ϕ r 0 k,θ m ,R m ( · ; ω ) ∈ L 2 ( R d ; dx ) . Recall also that N r 0 ( ω ) := sup x ∈ B (0 ,r 0 ) , 1 ⩽ j,l ⩽ d | D j ˜ H j l ( τ x ω ) | . This implies immediately that sup R ⩾ 1 sup x ∈ B (0 ,r 0 ) | b R ( x ; ω ) | ⩽ ∥ ρ ′ R ∥ ∞ d X j,l =1 sup x ∈ B (0 ,r 0 )    D j ˜ H j l ( τ x ω )    ! ⩽ c 5 N r 0 ( ω ) . (3.21) A ccording to the deﬁnitions of F k,m,j ( ω ) , j = 0 , 1 , 2 , we hav e | F k,m, 0 ( x ; ω ) | ⩽ c 6 N r 0 ( ω ) |∇ ϕ r 0 k,θ m ,R m ( x ; ω ) | , (3.22) | F k,m, 1 ( x ; ω ) | ⩽ c 6 ( N r 0 ( ω ) + θ m | ˜ ϕ k,θ m ,R m ( ω ) | ) 1 B (0 ,r 0 ) ( x ) , and | F k,m, 2 ( x ; ω ) | ⩽ c 7 ∥∇ ψ r 0 ∥ L ∞ ( R d )  Z R d | ϕ k,θ m ,R m ( x + z ; ω ) − ϕ k,θ m ,R m ( x ; ω ) | 2 ν ( z ) dz  1 / 2 1 B (0 ,K 0 r 0 ) ( x ) + c 7 Z B ( x,r 0 ) ν 2 ( z ) dz ! 1 / 2   r d/ 2 0 | ϕ k,θ m ,R m ( x ; ω ) | + Z B (0 ,r 0 ) | ϕ k,θ m ,R m ( z ; ω ) | 2 dz ! 1 / 2   1 B (0 ,K 0 r 0 ) c ( x ) ⩽ c 8 r − 1 0  Z R d | ϕ k,θ m ,R m ( x + z ; ω ) − ϕ k,θ m ,R m ( x ; ω ) | 2 ν ( z ) dz  1 / 2 1 B (0 ,K 0 r 0 ) ( x ) + c 8 r d 0 γ ( x ) ν ( x )   | ϕ k,θ m ,R m ( x ; ω ) | + r − d/ 2 0 Z B (0 ,r 0 ) | ϕ k,θ m ,R m ( z ; ω ) | 2 dz ! 1 / 2   1 B (0 ,K 0 r 0 ) c ( x ) , where K 0 ⩾ 2 and γ ( x ) are given in Assumption 1.1(iii), and we used (1.3) in the last inequality . NON-LOCAL SYMMETRIC OPERA TORS WITH DIVERGENCE-FREE DRIFT 19 Similarly , by (1.3), we also hav e | L 0 ψ ( r ) | ⩽ c 9   ∥∇ 2 ψ r 0 ∥ L ∞ ( R d ) 1 B (0 ,K 0 r 0 ) ( x ) + r d/ 2 0 Z B ( x,r 0 ) ν 2 ( z ) dz ! 1 / 2 1 B (0 ,K 0 r 0 ) c ( x )   ⩽ c 10  r − 2 0 1 B (0 ,K 0 r 0 ) ( x ) + r d 0 γ ( x ) ν ( x ) 1 B (0 ,K 0 r 0 ) c ( x )  , whic h along with (3.21) implies that | F k,m, 3 ( x ; ω ) | ⩽ c 11   N r 0 ( ω ) r − 1 0 + r − 2 0  1 B (0 ,K 0 r 0 ) ( x ) + r d 0 γ ( x ) ν ( x ) 1 B (0 ,K 0 r 0 ) c ( x )  | ϕ k,θ m ,R m ( x ; ω ) | + c 11 r − 1 0 θ m |∇ ϕ k,θ m ,R m ( x ; ω ) | 1 B (0 ,K 0 r 0 ) ( x ) . F urthermore, by the fact supp[ ϕ r 0 θ m ,R m ] ⊂ B (0 , r 0 ) , w e get      Z {| z | > 1 } ( ϕ r 0 k,θ m ,R m ( x + z ; ω ) − ϕ r 0 k,θ m ,R m ( x ; ω )) ν ( z ) dz      ⩽ Z {| z | > 1 } | ϕ r 0 k,θ m ,R m ( x ; ω ) | ν ( z ) dz + Z {| z | > 1: x + z ∈ B (0 ,r 0 ) } | ϕ r 0 k,θ m ,R m ( x + z ; ω ) | ν ( z ) dz ⩽ c 12 | ϕ r 0 k,θ m ,R m ( x ; ω ) | + c 12 Z B ( x,r 0 ) ∩ B (0 , 1) c ν 2 ( z ) dz ! 1 / 2 Z B (0 ,r 0 ) | ϕ r 0 k,θ m ,R m ( z ; ω ) | 2 dz ! 1 / 2 ⩽ c 13 | ϕ r 0 k,θ m ,R m ( x ; ω ) | + c 13 r d/ 2 0  1 B (0 ,K 0 r 0 ) ( x ) + γ ( x ) ν ( x ) 1 B (0 ,K 0 r 0 ) c ( x )  Z B (0 ,r 0 ) | ϕ r 0 k,θ m ,R m ( z ; ω ) | 2 dz ! 1 / 2 and      Z {| z | > 1 } ( ϕ r 0 k,θ m ,R m ( x + z ; ω ) − ϕ r 0 k,θ m ,R m ( x ; ω )) 1 | z | d + α dz      ⩽ c 14 | ϕ r 0 k,θ m ,R m ( x ; ω ) | + c 14 r d/ 2 0 (1 + | x | ) − d − α Z B (0 ,r 0 ) | ϕ r 0 k,θ m ,R m ( z ; ω ) | 2 dz ! 1 / 2 . A ccording to all the estimates ab ov e, we obtain that | G k,m ( x ; ω ) − F k,m, 0 ( x ; ω ) | ⩽ c 15 (1 + N r 0 ( ω )) | ϕ k,θ m ,R m ( x ; ω ) | +  Z R d | ϕ k,θ m ,R m ( x + z ; ω ) − ϕ k,θ m ,R m ( x ; ω ) | 2 ν ( z ) dz  1 / 2  1 B (0 ,K 0 r 0 ) ( x ) + c 15  (1 + N r 0 ( ω )) + θ m | ˜ ϕ k,θ m ,R m ( ω ) | + θ m |∇ ϕ k,θ m ,R m ( x ; ω ) |  1 B (0 ,K 0 r 0 ) ( x ) + c 15 γ ( x ) ν ( x )   | ϕ k,θ m ,R m ( x ; ω ) | + Z B (0 ,r 0 ) | ϕ k,θ m ,R m ( z ; ω ) | 2 dz ! 1 / 2   1 B (0 ,K 0 r 0 ) c ( x ) + c 15 Z B (0 ,r 0 ) | ϕ k,θ m ,R m ( z ; ω ) | 2 dz ! 1 / 2 (1 + | x | ) − d − α , where c 15 > 0 may dep end on r 0 . In particular, Z R d | G k,m ( x ; ω ) − F k,m, 0 ( x ; ω ) | dx ⩽ c 16 q K k,m ( ω ) , Z R d | G k,m ( x ; ω ) − F k,m, 0 ( x ; ω ) | 2 dx ⩽ c 16 K k,m ( ω ) , (3.23) 20 XIN CHEN, JIAN W ANG AND KUN YIN where K k,m ( ω ) := θ 2 m Z B (0 ,K 0 r 0 ) |∇ ϕ k,θ m ,R m ( x ; ω ) | 2 dx +  | N r 0 ( ω ) | 2 + 1  Z B (0 ,K 0 r 0 ) | ϕ k,θ m ,R m ( x ; ω ) | 2 dx + 1 ! + Z R d | ϕ k,θ m ,R m ( x ; ω ) | 2 ν ( x ) dx + θ 2 m | ˜ ϕ k,θ m ,R m ( ω ) | 2 + Z B (0 ,K 0 r 0 )  Z R d | ϕ k,θ m ,R m ( x + z ; ω ) − ϕ k,θ m ,R m ( x ; ω ) | 2 ν ( z ) dz  dx. Here, w e used (1.4), which (along with Cauch y-Sc h wartz inequality) implies that Z {| x | ⩾ K 0 r 0 } | ϕ k,θ m ,R m ( x ; ω ) | γ ( x ) ν ( x ) dx ⩽  Z R d | ϕ k,θ m ,R m ( x ; ω ) | 2 ν ( x ) dx  1 / 2 · Z {| x | ⩾ K 0 r 0 } γ 2 ( x ) ν ( x ) dx ! 1 / 2 ⩽ c 16  Z R d | ϕ k,θ m ,R m ( x ; ω ) | 2 ν ( x ) dx  1 / 2 . Therefore, b y (3.18), (3.22) and (3.23), for any t > 0 , Z R d |∇ T m t G k,m ( x ; ω ) | 2 dx ⩽ c 17 t − 2 /α  | N r 0 ( ω ) | 2 Z R d    ∇ ϕ r 0 k,θ m ,R m ( x ; ω )    2 dx + K k,m ( ω )  . (3.24) A ccording to (3.20), (3.21) and the fact div b R m j ( · ; ω ) = 0 , for ev ery t > 0 and x ∈ R d , |∇ T m t ( F k,m, 0 ( · ; ω )) ( x ) | =       d X j =1 ∇ T m t ∂ ( b R m j ( · ; ω ) ϕ r 0 k,θ m ,R m ( · ; ω )) ∂ x j ! ( x )       ⩽ c 18 t − 2 /α N r 0 ( ω ) T m t | ϕ r 0 k,θ m ,R m ( · ; ω ) | ( x ) , whic h implies immediately that Z R d |∇ T m t ( F k,m, 0 ( · ; ω )) ( x ) | 2 dx ⩽ c 19 t − 4 /α N 2 r 0 ( ω ) Z B (0 ,r 0 ) | ϕ k,θ m ,R m ( x ; ω ) | 2 dx. (3.25) On the other hand, by (3.18), (3.19) and (3.23), we deriv e that for every t > 0 and x ∈ R d , |∇ T m t ( G k,m ( · ; ω ) − F k,m, 0 ( · ; ω )) ( x ) | 2 ⩽ c 20 t − d +4 2 α q K k,m ( ω ) T m t | G k,m ( · ; ω ) − F k,m, 0 ( · ; ω ) | ( x ) , whic h together with (3.23) again gives us that Z R d |∇ T m t ( G k,m ( · ; ω ) − F k,m, 0 ( · ; ω )) ( x ) | 2 dx ⩽ c 20 t − d +4 2 α K k,m ( ω ) . (3.26) Putting (3.17), (3.24), (3.26) and (3.25) together, we know that for ev ery t 0 > 0 , ∥∇ ϕ r 0 k,θ m ,R m ( · ; ω ) ∥ L 2 ( R d ) ⩽ Z t 0 0 e − θ m t ∥∇ T t G k,m ( · ; ω ) ∥ L 2 ( R d ) dt + Z ∞ t 0 e − θ m t ∥∇ T t G k,m ( · ; ω ) ∥ L 2 ( R d ) dt ⩽ c 21  N r 0 ( ω ) ∥∇ ϕ r 0 k,θ m ,R m ( · ; ω ) ∥ L 2 ( R d ) + q K k,m ( ω )   Z t 0 0 t − 1 /α dt  + c 21 N r 0 ( ω ) Z B (0 ,r 0 ) | ϕ k,θ m ,R m ( x ; ω ) | 2 dx ! 1 / 2  Z ∞ t 0 t − 2 /α dt  + c 21 q K k,m ( ω )  Z ∞ t 0 t − ( d +4) / (4 α ) dt  . Since α ∈ (1 , 2) , w e can ﬁnd t 0 ( r 0 ; ω ) = c 22 (1 + N r 0 ( ω )) − α α − 1 with c 22 > 0 small enough so that Z t 0 ( r ; ω ) 0 t − 1 /α dt ⩽ 1 2 c 21 (1 + N r 0 ( ω )) . NON-LOCAL SYMMETRIC OPERA TORS WITH DIVERGENCE-FREE DRIFT 21 Th us, noting that d > 4( α − 1) , with such choice t 0 = t 0 ( r 0 , ω ) , w e can deriv e that ∥∇ ϕ r 0 k,θ m ,R m ( · ; ω ) ∥ L 2 ( R d ) ⩽ c 23  (1 + N r 0 ( ω )) 1 α − 1 + (1 + N r 0 ( ω )) d 4( α − 1)    Z B (0 ,K 0 r 0 ) | ϕ k,θ m ,R m ( x ; ω ) | 2 dx ! 1 / 2 + 1   + (1 + N r 0 ( ω )) d +4 − 4 α 4( α − 1) q K ∗ k,m ( ω ) ! , where K ∗ k,m ( ω ) := θ 2 m Z B (0 ,K 0 r 0 ) |∇ ϕ k,θ m ,R m ( x ; ω ) | 2 dx + Z R d | ϕ k,θ m ,R m ( x ; ω ) | 2 ν ( x ) dx + θ 2 m | ˜ ϕ k,θ m ,R m ( ω ) | 2 + Z B (0 ,K 0 r 0 )  Z R d | ϕ k,θ m ,R m ( x + z ; ω ) − ϕ k,θ m ,R m ( x ; ω ) | 2 ν ( z ) dz  dx. A ccording to (2.17), (2.18), (2.19) and (2.14), sup m ⩾ 1 E  K ∗ k,m ( ω )  + E " Z B (0 ,K 0 r 0 ) | ϕ k,θ m ,R m ( x ; ω ) | 2 dx #! < ∞ . This along with the Hölder inequality and (1.12) yields that for every q ∈ [1 , 2) , sup m ⩾ 1 E " Z B (0 ,r 0 / 2) |∇ ϕ k,θ m ,R m ( x ; ω ) | q dx # ⩽ c 24 sup m ⩾ 1 E h ∥∇ ϕ r 0 k,θ m ,R m ( · ; ω ) ∥ q L 2 ( R d ) i ⩽ c 25  E  (1 + N r 0 ( ω )) max n 2 ( α − 1)(2 − q ) , d 2( α − 1)(2 − q ) o  (2 − q ) / 2 × sup m ⩾ 1 E  K ∗ k,m ( ω )  + E " Z B (0 ,r 0 ) | ϕ k,θ m ,R m ( x ; ω ) | 2 dx #! q / 2 < ∞ , where in the ﬁrst inequality we used the fact that ψ r 0 ( x ) = 1 for x ∈ B (0 , r 0 / 2) . and the second inequalit y is due to the fact d + 4 − 4 α < d . Hence, according to the co-cycle prop ert y (2.16) and the stationary property of { τ x } x ∈ R d , w e can obtain that for all x 0 ∈ R d , sup m ⩾ 1 E " Z B ( x 0 ,r 0 / 2) |∇ ϕ k,θ m ,R m ( x ; ω ) | q dx # = sup m ⩾ 1 E " Z B (0 ,r 0 / 2) |∇ ϕ k,θ m ,R m ( x ; τ x 0 ω ) | q dx # = sup m ⩾ 1 E " Z B (0 ,r 0 / 2) |∇ ϕ k,θ m ,R m ( x ; ω ) | q dx # < ∞ . In particular, by the estimate abov e and (2.18), { ϕ θ m ,R m } m ⩾ 1 is w eakly compact in L 2 (Ω , W 1 ,q loc ( R d , R d ); P ) , and its w eak limit ϕ in L 2 (Ω , W α/ 2 , 2 loc ( R d , R d ); P ) ∩ L 2 ( R d × Ω , R d ; ν ( z ) dz × P ) satisﬁes that for all x 0 ∈ R d , E " Z B ( x 0 ,r 0 / 2) ( | ϕ ( x ; ω ) | q + |∇ ϕ ( x ; ω ) | q ) dx # < ∞ . (3.27) Therefore, there is a P -null set Λ ⊂ Ω suc h that ϕ ( · ; ω ) ∈ W 1 ,q loc ( R d , R d ) for all ω / ∈ Λ . Moreo v er, x 7→ ∇ ϕ θ m ,R m ( x ; ω ) is stationary and there exists ˜ Φ m ∈ L q (Ω , R d × R d ; P ) such that ∇ ϕ θ m ,R m ( x ; ω ) = ˜ Φ m ( τ x ω ) . A ccording to (3.27) and the fact ϕ is the weak limit of ϕ θ m ,R m , w e can ﬁnd ˜ Φ ∈ L q (Ω , R d × R d ; P ) so that ∇ ϕ ( x ; ω ) = ˜ Φ( τ x ω ) for a.e. x ∈ R d and a.s. ω ∈ Ω . The pro of is complete. □ The follo wing lemma shows the sublinear growing prop erty of the scaled corrector ϕ ε ( x ; ω ) := εϕ  x ε ; ω  . Lemma 3.3. Supp ose that Assumptions 1 . 1 and 1 . 2 hold. Then lim ε → 0 ε 2 Z B (0 ,r )    ϕ  x ε ; ω     2 dx = 0 , r ⩾ 1 , a . s . ω ∈ Ω . (3.28) 22 XIN CHEN, JIAN W ANG AND KUN YIN Mor e over, if additional ly (1.12) holds with some q ∈ (1 , 2) and d > 4( α − 1) , then lim ε → 0 ε qd d − q Z B (0 ,r )    ϕ  x ε ; ω     qd d − q dx = 0 , r ⩾ 1 , a . s . ω ∈ Ω . (3.29) Pr o of. By applying (2.8) and (2.18), and following the pro of of [40, Prop osition 4.4] (in particular, in the pro of of [40, Proposition 4.4] the symmetric prop ert y of the random op erator L ω is not required), we can establish the desired conclusion (3.28). Deﬁne ϕ ε ( x ; ω ) := εϕ  x ε ; ω  . If additionally (1.12) holds for some q ∈ (1 , 2) and d > 4( α − 1) , then, by Lemma 3.2 and the ergo dic theorem, w e obtain lim ε → 0 Z B (0 ,r ) |∇ ϕ ε ( x ; ω ) | q dx = | B (0 , r ) | · E [ | ˜ Φ | q ] < ∞ , r ⩾ 1 . Recall that d ⩾ 2 and q ∈ (1 , 2) . According to the Sob olev inequality (cf. see [5, Corollary 9.14, p. 284]), it holds that   Z B (0 ,r )      ϕ ε ( x ; ω ) − I B (0 ,r ) ϕ ε ( · ; ω )      qd d − q dx   d − q d ⩽ c 1 ( r ) Z B (0 ,r ) |∇ ϕ ε ( x ; ω ) | q dx, where I B (0 ,r ) ϕ ε ( · ; ω ) := | B (0 , r ) | − 1 Z B (0 ,r ) ϕ ε ( z ; ω ) dz . Hence, for any r ⩾ 1 , ( ϕ ε ( x ; ω ) − I B (0 ,r ) ϕ ε ( · ; ω ) ) ε ∈ (0 , 1) is weakly compact in W 1 ,q loc ( R d , R d ) . Therefore, for an y sequence, there exists its subsequence, which is denoted b y ( ϕ ε m ( x ; ω ) − I B (0 ,r ) ϕ ε m ( · ; ω ) ) m ⩾ 1 , weakly conv erges to some ˆ ϕ ∈ W 1 ,q loc ( R d , R d ) as m → ∞ . On the other hand, according to (3.28), ( ϕ ε m ( x ; ω ) − I B (0 ,r ) ϕ ε m ( · ; ω ) ) m ⩾ 1 con v erges strongly in L 2 ( B (0 , r ); dx ) to 0 . Therefore, ˆ ϕ ( x ; ω ) = 0 for a.e. x ∈ B (0 , r ) and a.s. ω ∈ Ω . F urthermore, due to the compact embedding from L qd d − q loc ( R d , R d ) to W 1 ,q loc ( R d , R d ) (cf. see [5, Theorem 9.16, p. 285]), there also exists one of its subsequence, which is conv ergen t in L qd d − q loc ( R d , R d ) to ˆ ϕ . Since the subsequence is arbitrary , we can ﬁnally deduce that the only p ossible limit ˆ ϕ is zero, and so (3.29) is prov ed. □ W e now establish the relation b etw een the corrector ϕ and the random drift term b ( x ; ω ) . Let ˜ Φ = ∇ ϕ ∈ L q (Ω , R d × R d ; P ) b e that in Lemma 3 . 2 . Lemma 3.4. Supp ose that Assumptions 1 . 1 and 1 . 2 hold. Assume that d > 4( α − 1) , (1.12) and E [ | ˜ H j l | q / ( q − 1) ] < ∞ , 1 ⩽ j, l ⩽ d (3.30) hold with some q ∈ (1 , 2) . Then, d X j =1 E [ ˜ Φ kj ( ω ) ˜ H kj ( ω )] = − 1 2 E  Z R d | ϕ k ( z ; ω ) | 2 ν ( z ) dz  , 1 ⩽ k ⩽ d. (3.31) Pr o of. F or every ε ∈ (0 , 1) , let ϕ ε ( x ; ω ) := εϕ  x ε ; ω  . By (2.6), for every f ∈ C 1 c ( R d ) and 1 ⩽ k ⩽ d , ε − d − 1 2 Z R d Z R d  ϕ k  x + z ε ; ω  − ϕ k  x ε ; ω   ( f ( x + z ) − f ( x )) ν  z ε  dz dx − ε − 1 Z R d D b  x ε ; ω  , ∇ f ( x ) E ϕ ε k ( x ; ω ) dx = − ε − 1 Z R d b k  x ε ; ω  f ( x ) dx. (3.32) F or r ⩾ 1 , c ho ose a function 0 ⩽ ψ r ∈ C ∞ c ( R d ) suc h that supp[ ψ r ] ⊂ B (0 , r ) and Z R d ψ r ( x ) dx = 1 . F or R ⩾ 1 , tak e ˜ ρ R : R → [ − 2 R , 2 R ] to b e a C 1 -function such that ˜ ρ R ( s ) = s for − R ⩽ s ⩽ R , ˜ ρ R ( s ) = 2 R for s ⩾ 2 R , ˜ ρ R ( s ) = − 2 R for s ⩽ − 2 R , and | ˜ ρ R ( s ) | ⩽ 2 | s | and | ˜ ρ ′ R ( s ) | ⩽ 2 for all s ∈ R . Recall that, by Theorem NON-LOCAL SYMMETRIC OPERA TORS WITH DIVERGENCE-FREE DRIFT 23 2.2 and Lemma 3.2, ϕ ( · ; ω ) ∈ W α/ 2 , 2 loc ( R d , R d ) ∩ W 1 ,q loc ( R d , R d ) . By the standard appro ximation pro cedure, we can tak e f ( x ) = ˜ ρ R ( ϕ ε k ( x ; ω )) ψ r ( x ) in (3.32), and obtain that I ε,R 1 ( ω ) + I ε,R 2 ( ω ) = − I ε,R 3 ( ω ) , where, for ﬁxed 1 ⩽ k ⩽ d , I ε,R 1 ( ω ) : = ε − d − 1 2 Z R d Z R d  ϕ k  x + z ε ; ω  − ϕ k  x ε ; ω   × ( ˜ ρ R ( ϕ ε k ( x + z ; ω )) ψ r ( x + z ) − ˜ ρ R ( ϕ ε k ( x ; ω )) ψ r ( x )) ν  z ε  dz dx = ε − d − 1 2 Z R d Z R d  ϕ k  x + z ε ; ω  − ϕ k  x ε ; ω   ( ˜ ρ R ( ϕ ε k ( x + z ; ω )) − ˜ ρ R ( ϕ ε k ( x ; ω ))) ψ r ( x ) ν  z ε  dz dx + ε − d − 1 2 Z R d Z R d  ϕ k  x + z ε ; ω  − ϕ k  x ε ; ω   ( ψ r ( x + z ) − ψ r ( x )) ˜ ρ R ( ϕ ε k ( x ; ω )) ν  z ε  dz dx =: I ε,R 11 ( ω ) + I ε,R 12 ( ω ) and I ε,R 2 ( ω ) : = − ε − 1 Z R d D b  x ε ; ω  , ∇ ( ˜ ρ R ( ϕ ε k ( · ; ω )) ψ r ( · )) ( x ) E ϕ ε k ( x ; ω ) dx = ε − 1 Z R d D b  x ε ; ω  , ∇ ϕ ε k ( x ; ω ) E ˜ ρ R ( ϕ ε k ( x ; ω )) ψ r ( x ) dx, I ε,R 3 ( ω ) : = ε − 1 Z R d b k  x ε ; ω  ˜ ρ R ( ϕ ε k ( x ; ω )) ψ r ( x ) dx. Here in the second equality for the estimate of I ε,R 1 ( ω ) w e c hanged the v ariables x + z 7→ x and z 7→ − z , and used the fact that ν ( z ) = ν ( − z ) for all z ∈ R d ; and in the second equalit y for the estimate of I ε,R 2 ( ω ) we used the an ti-symmetry of { ˜ H j l } 1 ⩽ j,l ⩽ d and the in tegration by parts formula. Belo w, we will estimate I ε,R 1 ( ω ) , I ε,R 2 ( ω ) and I ε,R 3 ( ω ) resp ectively . (1) W e ﬁrst consider I ε,R 3 ( ω ) . Since b k ( x ; ω ) = − P d j =1 ∂ H kj ( x ; ω ) ∂ x j , I ε,R 3 ( ω ) = − d X j =1 Z R d ∂ ∂ x j  H kj  · ε ; ω  ( x ) ˜ ρ R ( ϕ ε k ( x ; ω )) ψ r ( x ) dx = d X j =1 Z R d H kj  x ε ; ω  ∂ ϕ k  x ε ; ω  ∂ x j ˜ ρ ′ R ( ϕ ε k ( x ; ω )) ψ r ( x ) dx + d X j =1 Z R d H kj  x ε ; ω  ˜ ρ R ( ϕ ε k ( x ; ω )) ∂ ψ r ( x ) ∂ x j dx =: I ε,R 31 ( ω ) + I ε,R 32 ( ω ) . Using the fact that | ˜ ρ R ( s ) | ⩽ 2 | s | for all s ∈ R and the ergo dic theorem, for every R ⩾ 1 , lim ε → 0 | I ε,R 32 ( ω ) | ⩽ c 1 d X j =1 lim ε → 0 Z B (0 ,r )    H kj  x ε ; ω     2 dx ! 1 / 2 lim ε → 0 Z B (0 ,r ) | ϕ ε k ( x ; ω ) | 2 dx ! 1 / 2 = c 1 d X j =1 | B (0 , r ) | 1 / 2 E [ | ˜ H kj | 2 ] 1 / 2 lim ε → 0 Z B (0 ,r ) | ϕ ε k ( x ; ω ) | 2 dx ! 1 / 2 = 0 , where the last equality follows from (3.28) and (3.30) (that implies E [ | ˜ H kj | 2 ] < ∞ since q / ( q − 1) > 2 for q ∈ (1 , 2) ). F or every M ⩾ 1 , deﬁne A M ,ε, 1 ( ω ) = { x ∈ B (0 , r ) : | ϕ ε k ( x ; ω ) | ⩾ M } , A M ,ε, 2 ( ω ) = { x ∈ B (0 , r ) : |∇ ϕ ε k ( x ; ω ) | ⩾ M } . 24 XIN CHEN, JIAN W ANG AND KUN YIN Then, b y the fact that ˜ ρ ′ R ( ϕ ε k ( x ; ω )) = 1 for ev ery x ∈ A c R,ε, 1 ( ω ) , I ε,R 31 ( ω ) = d X j =1 Z R d H kj  x ε ; ω  ∂ ϕ k  x ε ; ω  ∂ x j ψ r ( x ) dx − d X j =1 Z A R,ε, 1 ( ω ) H kj  x ε ; ω  ∂ ϕ k  x ε ; ω  ∂ x j (1 − ˜ ρ ′ R ( ϕ ε k ( x ; ω ))) ψ r ( x ) dx. Note that ∇ ϕ ( x ; ω ) = ˜ Φ( τ x ω ) with ˜ Φ ∈ L q (Ω , R d × R d ; P ) , and ˜ H kj ∈ L q q − 1 (Ω; P ) . A ccording to the ergo dic theorem (see e.g. [10, Prop osition 2.1]), we get lim ε → 0 Z R d H kj  x ε ; ω  ∂ ϕ k  x ε ; ω  ∂ x j ψ r ( x ) dx =  Z R d ψ r ( x ) dx  E [ ˜ H kj ( ω ) ˜ Φ kj ( ω )] = E [ ˜ H kj ( ω ) ˜ Φ kj ( ω )] . On the other hand, since | ˜ ρ ′ R ( s ) | ⩽ 2 for all s ∈ R , it holds that for ev ery M ⩾ 1 and 1 ⩽ j ⩽ d ,      Z A R,ε, 1 ( ω ) H kj  x ε ; ω  ∂ ϕ k  x ε ; ω  ∂ x j (1 − ˜ ρ ′ R ( ϕ ε k ( x ; ω ))) ψ r ( x ) dx      ⩽ 3 Z A R,ε, 1 ( ω )    H kj  x ε ; ω          ∂ ϕ k  x ε ; ω  ∂ x j      ψ r ( x ) dx ⩽ c 2 Z A R,ε, 1 ( ω ) ∩ A M ,ε, 2 ( ω ) c ∩ B (0 ,r )    H kj  x ε ; ω         ∂ ϕ ε k ( x ; ω ) ∂ x j     dx + c 2 Z A M ,ε, 2 ( ω ) ∩ B (0 ,r )    H kj  x ε ; ω        ˜ Φ kj  τ x ε ω     dx. By the deﬁnition of A M ,ε, 2 ( ω ) , we hav e Z A R,ε, 1 ( ω ) ∩ A M ,ε, 2 ( ω ) c ∩ B (0 ,r )    H kj  x ε ; ω         ∂ ϕ ε k ( x ; ω ) ∂ x j     dx ⩽ M Z A R,ε, 1 ( ω ) ∩ B (0 ,r )    H kj  x ε ; ω     dx ⩽ M | A R,ε, 1 ( ω ) ∩ B (0 , r ) | 1 / 2 Z B (0 ,r )    H kj  x ε ; ω     2 dx ! 1 / 2 ⩽ M R Z B (0 ,r ) | ϕ ε k ( x ; ω ) | 2 dx ! 1 / 2 Z B (0 ,r )    H kj  x ε ; ω     2 dx ! 1 / 2 , where in the last inequality we used | A R,ε, 1 ( ω ) ∩ B (0 , r ) | ⩽ 1 R 2 Z B (0 ,r ) | ϕ ε k ( x ; ω ) | 2 dx. (3.33) Th us, according to (3.28), (3.30) and the ergo dic theorem, we obtain that for every ﬁxed R ⩾ 1 and M ⩾ 1 , lim ε → 0 Z A R,ε, 1 ( ω ) ∩ A M ,ε, 2 ( ω ) c ∩ B (0 ,r )    H kj  x ε ; ω         ∂ ϕ ε k ( x ; ω ) ∂ x j     dx = 0 . Note again that ∇ ϕ k ( x ; ω ) = ˜ Φ k ( τ x ω ) . Applying the ergo dic theorem, w e derive lim M →∞ lim ε → 0 Z B (0 ,r ) Z A M ,ε, 2 ( ω ) ∩ B (0 ,r )    H kj  x ε ; ω        ˜ Φ kj  τ x ε ω     dx = lim M →∞ lim ε → 0 Z B (0 ,r )    H kj  x ε ; ω        ˜ Φ kj  τ x ε ω     1 n    ˜ Φ k  τ · ε ω     ⩾ M o ( x ) dx = | B (0 , r ) | lim M →∞ E h | ˜ H kj ( ω ) ˜ Φ kj ( ω ) | 1 {| ˜ Φ k | ⩾ M } ( ω ) i = 0 , where the last equality follows from the facts that ˜ H kj ∈ L q q − 1 (Ω; P ) and ˜ Φ kj ∈ L q (Ω; P ) again. NON-LOCAL SYMMETRIC OPERA TORS WITH DIVERGENCE-FREE DRIFT 25 Therefore, combining with all the estimates ab ov e, ﬁrstly letting ε → 0 and then M → ∞ , we ha v e for every ﬁxed R ⩾ 1 , lim ε → 0 I ε,R 3 ( ω ) = d X j =1 E [ ˜ H kj ( ω ) ˜ Φ kj ( ω )] . (2) In this part, we consider I ε,R 1 ( ω ) . By the change of v ariable, we derive I ε,R 11 ( ω ) = ε − 1 2 Z R d Z R d  ϕ k  x ε + z ; ω  − ϕ k  x ε ; ω   ˜ ρ R  εϕ k  x ε + z ; ω  − ˜ ρ R  εϕ k  x ε ; ω  ψ r ( x ) ν ( z ) dz dx = 1 2 Z R d  Z R d    ϕ k  z ; τ x ε ω     2 ν ( z ) dz  ψ r ( x ) dx − ε − 1 2 Z R d Z R d  ϕ k  x ε + z ; ω  − ϕ k  x ε ; ω   χ R  εϕ k  x ε + z ; ω  − χ R  εϕ k  x ε ; ω  ψ r ( x ) ν ( z ) dz dx =: I ε 111 ( ω ) + I ε,R 112 ( ω ) , where χ R ( s ) := s − ˜ ρ R ( s ) and we used the co-cycle property (2.7). Then, using the ergo dic theorem and the fact that Z R d ψ r ( x ) dx = 1 , we hav e lim ε → 0 I ε 111 ( ω ) = 1 2 E  Z R d | ϕ k ( z ; ω ) | 2 ν ( z ) dz  . F or every M ⩾ 1 , deﬁne O M ,ε ( ω ) := n ( x, z ) ∈ R d × R d :    ϕ k  z ; τ x ε ω     ⩾ M o . Note that | χ R ( a ) − χ R ( b ) | ⩽ 2 | b − a | , then for every R ⩾ 1 and δ ∈ (0 , 1) , | I ε,R 112 ( ω ) | ⩽ c 3 Z B (0 ,r ) Z {| z | ⩽ δ }    ϕ k  z ; τ x ε ω     2 ν ( z ) dz dx + c 3 Z Z (( B (0 ,r ) ∩ A R/ 2 ,ε, 1 ( ω )) × B (0 ,δ ) c ) ∩ O R/ 2 ,ε ( ω ) c    ϕ k  z ; τ x ε ω     2 ν ( z ) dz dx + c 3 Z Z ( B (0 ,r ) × R d ) ∩ O R/ 2 ,ε ( ω )    ϕ k  z ; τ x ε ω     2 ν ( z ) dz dx = : c 3  I ε,δ 1121 ( ω ) + I ε,R,δ 1122 ( ω ) + I ε,R 1123 ( ω )  . (3.34) Here w e used the prop ert y that χ R  εϕ k  x ε + z ; ω  − χ R  εϕ k  x ε ; ω  = 0 , ( x, z ) ∈  A R/ 2 ,ε, 1 ( ω ) c × R d  ∩ O R/ 2 ,ε ( ω ) c , whic h follows from the fact that χ R ( a ) − χ R ( b ) = 0 for ev ery a, b ∈ R with | a | ⩽ R/ 2 and | a − b | ⩽ R/ 2 . Using (2.8) and the ergo dic theorem directly , w e get lim δ → 0 lim ε → 0 I ε,δ 1121 ( ω ) = | B (0 , r ) | lim δ → 0 E " Z {| z | ⩽ δ } | ϕ k ( z ; ω ) | 2 ν ( z ) dz # = 0 and lim R →∞ lim ε → 0 I ε,R 1123 ( ω ) = | B (0 , r ) | lim R →∞ E  Z R d | ϕ k ( z ; ω ) | 2 1 {| ϕ k ( · ; ω ) | ⩾ R/ 2 } ( z ) ν ( z ) dz  = 0 . By the deﬁnition of O M ,ε ( ω ) , we obtain that for every δ ∈ (0 , 1) and R ⩾ 1 , lim ε → 0 | I ε,R,δ 1122 ( ω ) | ⩽ R 2 4  lim ε → 0   B (0 , r ) ∩ A R/ 2 ,ε, 1 ( ω )    Z {| z | >δ } ν ( z ) dz ! 26 XIN CHEN, JIAN W ANG AND KUN YIN ⩽ c 4 ( δ, R ) lim ε → 0 Z B (0 ,r ) | ϕ ε k ( x ; ω ) | 2 dx ! 1 / 2 = 0 , where w e hav e used (3.33) and (3.28). Hence, ﬁrstly letting ε → 0 , and then taking R → ∞ and δ → 0 , lim R →∞ lim ε → 0 | I ε,R 112 ( ω ) | = 0 . Therefore, lim R →∞ lim ε → 0 I ε,R 11 ( ω ) = 1 2 E  Z R d | ϕ k ( z ; ω ) | 2 ν ( z ) dz  . By the c hange of v ariable and the mean v alue theorem, w e ha v e | I ε,R 12 ( ω ) | = ε − 1 2     Z R d Z R d  ϕ k  x ε + z ; ω  − ϕ k  x ε ; ω  ( ψ r ( x + εz ) − ψ r ( x )) ˜ ρ R ( ϕ ε k ( x ; ω )) ν ( z ) dz dx     ⩽ ∥∇ ψ r ∥ L ∞ ( R d ) Z B (0 ,K 0 r )  Z R d    ϕ k  z ; τ x ε ω     | z | ν ( z ) dz  · | ϕ ε k ( x ; ω ) | dx + c 5 ε − 1 R Z B (0 ,K 0 r ) c    ϕ k  x ε ; ω     Z B ( x ε , r ε ) ν ( z ) dz ! dx + c 5 ε − 1 R Z B (0 ,K 0 r ) c Z B ( 0 , r ε ) | ϕ k ( z ; ω ) | 2 dz ! 1 / 2 Z B ( x ε , r ε ) ν ( z ) 2 dz ! 1 / 2 dx =: I ε,R 121 ( ω ) + I ε,R 122 ( ω ) + I ε,R 123 ( ω ) , where in the inequality ab o v e w e used the facts that | ˜ ρ R ( s ) | ⩽ 2 R and | ˜ ρ R ( s ) | ⩽ 2 s for all s ∈ R , and K 0 > 1 is the constan t in Assumption 1.1(iii). By the Hölder inequality and the ergo dic theorem, lim ε → 0 | I ε,R 121 ( ω ) | ⩽ c 6 lim ε → 0 Z B (0 ,K 0 r ) Z R d    ϕ k  z ; τ x ε ω     2 ν ( z ) dz dx ! 1 / 2 lim ε → 0 Z B (0 ,K 0 r ) | ϕ ε k ( x ; ω ) | 2 dx ! 1 / 2 = c 6 | B (0 , K 0 r ) | 1 / 2 E  Z R d | ϕ k ( z ; ω ) | 2 ν ( z ) dz  1 / 2 lim ε → 0 Z B (0 ,K 0 r ) | ϕ ε k ( x ; ω ) | 2 dx ! 1 / 2 = 0 , where in the inequality we used the fact Z R d | z | 2 ν ( z ) dz < ∞ , and the last equality is due to (2.8) and (3.28). A ccording to (1.3), (1.4), (2.8) and the Hölder inequality , w e derive that for every ﬁxed R ⩾ 1 , lim ε → 0 | I ε,R 122 ( ω ) | ⩽ c 7 r d R lim ε → 0 ε − d − 1 Z B (0 ,K 0 r ) c    ϕ k  x ε ; ω     γ  x ε  ν  x ε  dx ⩽ c 8 ( r , R )  Z R d | ϕ k ( x ; ω ) | 2 ν ( x ) dx  1 / 2 lim ε → 0 ε − 2 Z B  0 , K 0 r ε  c γ 2 ( x ) ν ( x ) dx ! 1 / 2 = 0 where in the last inequality we used Z B  0 , K 0 r ε  c γ 2 ( x ) ν ( x ) dx ⩽ ε 2 ( K 0 r ) 2 Z B  0 , K 0 r ε  c | x | 2 γ 2 ( x ) ν ( x ) dx ! . Using (1.3), (3.28) and (2.8) again, we ﬁnd that lim ε → 0 | I ε,R 123 ( ω ) | ⩽ c 9 Rr d/ 2 lim ε → 0 Z B (0 ,r ) | ϕ ε k ( x ; ω ) | 2 dx ! 1 / 2 ×   lim ε → 0 ε − 2 Z B  0 , K 0 r ε  c γ 2 ( x ) ν ( x ) dx ! 1 / 2 Z B  0 , K 0 r ε  c ν ( x ) dx ! 1 / 2   NON-LOCAL SYMMETRIC OPERA TORS WITH DIVERGENCE-FREE DRIFT 27 ⩽ c 10 ( r , R ) lim ε → 0 Z B (0 ,r ) | ϕ ε k ( x ; ω ) | 2 dx ! 1 / 2 × lim ε → 0   Z B  0 , K 0 r ε  c | x | 2 γ 2 ( x ) ν ( x ) dx ! 1 / 2 Z B  0 , K 0 r ε  c | x | 2 ν ( x ) dx ! 1 / 2   = 0 . Com bining with all the estimates ab ov e yields that for every ﬁxed R ⩾ 1 , lim ε → 0 I ε,R 12 ( ω ) = 0 and so lim R →∞ lim ε → 0 I ε,R 1 ( ω ) = 1 2 E  Z R d | ϕ k ( z ; ω ) | 2 ν ( z ) dz  . (3) No w, we consider I ε,R 2 ( ω ) . A ccording to b k ( x ; ω ) = − P d j =1 ∂ H kj ( x ; ω ) ∂ x j and the in tegration b y parts form ula, w e ﬁnd I ε,R 2 ( ω ) = − d X j,l =1 Z R d ∂ ∂ x j  H lj  · ε ; ω  ( x ) ∂ ϕ ε k ( x ; ω ) ∂ x l ˜ ρ R ( ϕ ε k ( x ; ω )) ψ r ( x ) dx = d X j,l =1 Z R d H lj  x ε ; ω  ˜ ρ R ( ϕ ε k ( x ; ω )) ∂ ϕ ε k ( x ; ω ) ∂ x l ∂ ψ r ( x ) ∂ x j dx, (3.35) where in the second equality we used the anti-symmetry of { H lj } 1 ⩽ l,j ⩽ d . F urthermore, it holds that | I ε,R 2 ( ω ) | ⩽ c 11 d X j,l =1 Z B (0 ,r ) ∩ A M ,ε, 2 ( ω ) c    H lj  x ε ; ω     | ˜ ρ R ( ϕ ε k ( x ; ω )) |     ∂ ϕ ε k ( x ; ω ) ∂ x l     dx + c 11 d X j,l =1 Z B (0 ,r ) ∩ A M ,ε, 2 ( ω )    H lj  x ε ; ω     | ˜ ρ R ( ϕ ε k ( x ; ω )) |     ∂ ϕ ε k ( x ; ω ) ∂ x l     dx =: c 11  I ε,R,M 21 ( ω ) + I ε,R,M 22 ( ω )  . By the deﬁnition of A M ,ε, 2 ( ω ) , we deduce that for every ﬁxed M ⩾ 1 , lim ε → 0 I ε,R,M 21 ( ω ) ⩽ c 12 M d X j,l =1 lim ε → 0 Z B (0 ,r )    H lj  x ε ; ω     2 dx ! 1 / 2 lim ε → 0 Z B (0 ,r ) | ϕ ε k ( x ; ω ) | 2 dx ! 1 / 2 = c 12 M | B (0 , r ) | 1 / 2   d X j,l =1 E [ | ˜ H lj | 2 ] 1 / 2   lim ε → 0 Z B (0 ,r ) | ϕ ε k ( x ; ω ) | 2 dx ! 1 / 2 = 0 , where the last equality follo ws from (3.28) again. Note that | ρ ( s ) | ⩽ 2 R for all s ∈ R and ˜ Φ k ( τ x ω ) = ∇ ϕ k ( x ; ω ) , then lim M →∞ lim ε → 0 I ε,R,M 22 ( ω ) ⩽ c 13 R d X j,l =1 lim M →∞ lim ε → 0 Z B (0 ,r )    H lj  x ε ; ω        ˜ Φ kl ( τ x ε ω )    1 ˜ Φ( τ · ε ω ) ⩾ M ( x ) dx = c 13 R | B (0 , r ) |  lim M →∞ E [ | ˜ H lj ˜ Φ kl ( ω ) | 1 {| ˜ Φ k | ⩾ M } ( ω )]  = 0 , where the last equality follows from (3.30). Putting all the estimates ab ov e together, and ﬁrst letting ε → 0 and then M → ∞ , we can show that for ev ery ﬁxed R ⩾ 1 , lim ε → 0 I ε,R 2 ( ω ) = 0 . A ccording to all the limit prop erties of I ε,R 1 ( ω ) , I ε,R 2 ( ω ) and I ε,R 3 ( ω ) , we get the desired conclusion (3.31). □ 28 XIN CHEN, JIAN W ANG AND KUN YIN Remark 3.5. A ccording to the pro of abov e, w e can obtain that under the assumptions of Lemma 3.4, for ev ery 1 ⩽ k , l ⩽ d , d X j =1 E [( ˜ Φ kj ( ω ) + ˜ Φ lj ( ω ))( ˜ H kj ( ω ) + ˜ H lj ( ω ))] = − 1 2 E  Z R d | ϕ k ( z ; ω ) + ϕ l ( z ; ω ) | 2 ν ( z ) dz  . (3.36) W e emphasize the crucial step to pro v e (3.36) is that one can still make the cancelation in (3.35) with ϕ ε k replaced b y ϕ ε k + ϕ ε l . 4. Pr oof of Theorem 1.4 Pr o of of The or em 1 . 4 . The pro of is split into three steps. (1) F or ev ery r ⩾ 2 and θ ∈ (0 , 1) , deﬁne B θ (0 , r ) = { y ∈ B (0 , r ) : | y − ∂ B (0 , r ) | > θ } . According to (3.2), it holds that Z B (0 ,r ) \ B θ (0 ,r ) | u ε ( x ; ω ) | dx ⩽ ∥ u ε ( · ; ω ) ∥ L ∞ ( R d ) | B (0 , r ) \ B θ (0 , r ) | ⩽ c 1 ( r ) sup ε ∈ (0 , 1) ∥ u ε ( · ; ω ) ∥ L ∞ ( R d ) θ . (4.1) F or every y ∈ R d with | y | ⩽ θ , letting N ε 0 :=  2 θ ε  + 1 , we hav e Z B θ (0 ,r ) | u ε ( x + y ; ω ) − u ε ( x ; ω ) | dx ⩽ N ε 0 − 1 X k =0 Z B θ (0 ,r )     u ε  x + ( k + 1) y N ε 0 ; ω  − u ε  x + k y N ε 0 ; ω      dx =: N ε 0 − 1 X k =0 J ε k ( y ; ω ) . Let y k := ky N ε 0 for 0 ⩽ k ⩽ N ε 0 − 1 . Then, for ev ery 0 ⩽ k ⩽ N ε 0 − 1 , J ε k ( y ; ω ) = | B (0 , ε/ 2) | − d Z B θ (0 ,r ) Z B ( x + y k ,ε/ 2) | u ε ( x + y k +1 ; ω ) − u ε ( x + y k ; ω ) | dz ! dx ⩽ c 2 ε − d Z B θ (0 ,r ) Z B ( x + y k ,ε/ 2) | u ε ( x + y k +1 ; ω ) − u ε ( z ; ω ) | dz dx + Z B θ (0 ,r ) Z B ( x + y k ,ε/ 2) | u ε ( x + y k ; ω ) − u ε ( z ; ω ) | dz dx ! =: c 2 ε − d  J ε k, 1 ( y ; ω ) + J ε k, 2 ( y ; ω )  . Since | y k − y k +1 | ⩽ ε/ 2 and | y k | ⩽ | y | ⩽ θ ⩽ r , it holds that J ε k, 1 ( y ; ω ) ⩽ Z B θ (0 ,r ) Z B (0 ,ε ) | u ε ( x + y k +1 ; ω ) − u ε ( x + y k +1 + z ; ω ) | dz dx ⩽ Z B (0 , 2 r ) Z {| z | ⩽ ε } | u ε ( x + z ; ω ) − u ε ( x ; ω ) | 2 | z | d + α dz ! 1 / 2 Z {| z | ⩽ ε } | z | d + α dz ! 1 / 2 dx ⩽ c 3 ( r ) ε d + α/ 2 Z B (0 , 2 r ) Z {| z | ⩽ ε } | u ε ( x + z ; ω ) − u ε ( x ; ω ) | 2 | z | d + α dz dx ! 1 / 2 . By (3.3) and the deﬁnition of ν ( z ) , sup ε ∈ (0 , 1) ε − (2 − α ) Z B (0 , 2 r ) Z {| z | ⩽ ε } | u ε ( x + z ; ω ) − u ε ( x ; ω ) | 2 | z | d + α dz dx ! < ∞ , NON-LOCAL SYMMETRIC OPERA TORS WITH DIVERGENCE-FREE DRIFT 29 whic h implies immediately that J ε k, 1 ( y ; ω ) ⩽ c 4 ( r ) ε d +1 . F ollo wing the same argumen t, we can ﬁnd that J ε k, 2 ( y ; ω ) ⩽ c 5 ( r ) ε d +1 for ev ery 0 ⩽ k ⩽ N ε 0 − 1 . Hence, w e can obtain that for every y ∈ R d with | y | ⩽ θ , sup ε ∈ (0 , 1) Z B θ (0 ,r ) | u ε ( x + y ; ω ) − u ε ( x ; ω ) | dx ⩽ sup ε ∈ (0 , 1)   N ε 0 − 1 X k =0 J ε k ( y ; ω )   ⩽ c 6 ( r ) θ . By this and (4.1), we can apply [16, Theorem 1.95, p. 37] to conclude that { u ε ( · ; ω ) : ε ∈ (0 , 1] } is precompact as ε → 0 in L 1 ( B (0 , R ); dx ) for all R ⩾ 1 . So, we can ﬁnd a subsequence { u ε m ( · ; ω ) } m ⩾ 1 and u 0 ( · ; ω ) suc h that lim m →∞ Z B (0 ,R ) | u ε m ( x ; ω ) − u 0 ( x ; ω ) | dx = 0 , R ⩾ 1 . Com bining this with (3.2) yields that for every p > 0 , lim m →∞ Z B (0 ,R ) | u ε m ( x ; ω ) − u 0 ( x ; ω ) | p dx = 0 , R ⩾ 1 . (4.2) Th us it remains to prov e that, for ev ery conv ergen t subsequence { u ε m } m ⩾ 1 in L p loc ( R d ) , the limit u 0 ( · ; ω ) is the solution to the equation (1.11). (2) F or every ε ∈ (0 , 1) , let ϕ ε ( x ; ω ) := εϕ  x ε ; ω  . Given any f ∈ C ∞ c ( R d ) , w e take the test function f ε m ( x ; ω ) := f ( x ) + d X j =1 ϕ ε m j ( x ; ω ) ∂ f ( x ) ∂ x j (4.3) (whic h b elongs to W 1 ,q ( R d ) with compact supp orts b y Lemma 3.2) in (3.1), and derive that λ Z R d u ε m ( x ; ω ) f ε m ( x ; ω ) dx − Z R d u ε m ( x ; ω ) L ε m 0 f ( x )( x ) dx + ε − 1 m Z R d  b  x ε m ; ω  , ∇ f ε m ( x ; ω )  u ε m ( x ; ω ) dx + ε − d − 2 m 2 d X j =1 Z R d Z R d  ϕ ε m j ( x + z ; ω ) ∂ f ( x + z ) ∂ x j − ϕ ε m j ( x ; ω ) ∂ f ( x ) ∂ x j  ( u ε m ( x + z ; ω ) − u ε m ( x ; ω )) ν  z ε m  dz dx = Z R d h ( x ) f ε m ( x ; ω ) dx. (4.4) Note that f ∈ C ∞ c ( R d ) . By (3.28) and (4.2), we immediately get that lim m →∞ Z R d u ε m ( x ; ω ) f ε m ( x ; ω ) dx = Z R d u 0 ( x ; ω ) f ( x ) dx and lim m →∞ Z R d h ( x ) f ε m ( x ; ω ) dx = Z R d h ( x ) f ( x ) dx. Belo w we will estimate other terms inv olv ed in the left hand side of (4.4). W e write for every M ⩾ 1 that L ε m 0 f ( x ) = ε − 2 m Z {| z | ⩽ M } + Z {| z | >M } ! ( f ( x + ε m z ) − f ( x ) − ε m ⟨∇ f ( x ) , z ⟩ ) ν ( z ) dz =: 1 2 Z {| z | ⩽ M }  ∇ 2 f ( x ) , z ⊗ z  ν ( z ) dz + G m 1 ,M ( x ) . Belo w without loss of generalit y w e supp ose that supp[ f ] ⊂ B (0 , R 0 ) for some R 0 > 1 . Let K 0 ⩾ 2 b e the constan t in Assumption 1.1(iii). Then, for all x ∈ B (0 , K 0 R 0 ) , b y T aylor’s expansion, | G m 1 ,M ( x ) | ⩽ ε m ∥∇ 3 f ∥ L ∞ ( R d ) 6 Z {| z | ⩽ M } | z | 3 ν ( z ) dz + ∥∇ 2 f ∥ L ∞ ( R d ) 2 Z {| z | >M } | z | 2 ν ( z ) dz ; while for all x ∈ B (0 , K 0 R 0 ) c , b y (1.3), | G m 1 ,M ( x ) | ⩽ ε − 2 m ∥ f ∥ L ∞ ( R d ) Z B  x ε m , R 0 ε m  ν ( z ) dz ⩽ c 7 ( R 0 ) ε − d − 2 m γ  x ε m  ν  x ε m  , 30 XIN CHEN, JIAN W ANG AND KUN YIN whic h along with the Hölder inequality and (1.4) implies that Z {| x | >K 0 R 0 } | G m 1 ,M ( x ) | dx ⩽ c 8 ( R 0 ) Z {| z | > R 0 ε m } | z | 2 γ 2 ( z ) ν ( z ) dz ! 1 / 2 Z {| z | > R 0 ε m } ν ( z ) dz ! 1 / 2 → 0 (4.5) as m → ∞ . Hence, com bining all the estimates ab o v e with (3.2) and (4.2), and ﬁrstly letting m → ∞ and then M → ∞ , we get lim m →∞ − Z R d L ε m 0 f ( x ) u ε ( x ; ω ) dx = − 1 2 Z R d  ∇ 2 f ( x ) , Z R d ( z ⊗ z ) ν ( z ) dz  u 0 ( x ; ω ) dx. (4.6) On the other hand, it holds that ε − d − 2 m 2 Z R d Z R d  ϕ ε m j ( x + z ; ω ) ∂ f ( x + z ) ∂ x j − ϕ ε m j ( x ; ω ) ∂ f ( x ) ∂ x j  ( u ε m ( x + z ; ω ) − u ε m ( x ; ω )) ν  z ε m  dz dx = ε − d − 2 m 2 Z R d Z R d  ϕ ε m j ( x + z ; ω ) − ϕ ε m j ( x ; ω )   u ε m ( x + z ; ω ) ∂ f ( x + z ) ∂ x j − u ε m ( x ; ω ) ∂ f ( x ) ∂ x j  ν  z ε m  dz dx − ε − d − 2 m Z R d Z R d  ∂ f ( x + z ) ∂ x j − ∂ f ( x ) ∂ x j   ϕ ε m j ( x + z ; ω ) − ϕ ε m j ( x ; ω )  ν  z ε m  u ε m ( x ; ω ) dz dx − Z R d L ε m 0  ∂ f ( · ) ∂ x j  ( x ) ϕ ε m j ( x ; ω ) u ε m ( x ; ω ) dx =: I m,j 21 ( ω ) + I m,j 22 ( ω ) + I m,j 23 ( ω ) . By the c hange of v ariable and the deﬁnition of ϕ ε , w e hav e I m,j 21 ( ω ) = ε − 1 m 2 Z R d Z R d  ϕ j  x ε m + z ; ω  − ϕ j  x ε m ; ω  ×  u ε m ( x + ε m z ; ω ) ∂ f ( x + ε m z ) ∂ x j − u ε m ( x ; ω ) ∂ f ( x ) ∂ x j  ν ( z ) dz dx and I m,j 22 ( ω ) = − ε − 1 m Z R d Z R d  ϕ j  x ε m + z ; ω  − ϕ j  x ε m ; ω   ∂ f ( x + ε m z ) ∂ x j − ∂ f ( x ) ∂ x j  ν ( z ) u ε m ( x ; ω ) dz dx. F urthermore, by T aylor’s expansion, we know that for every M ⩾ 1 , ε − 1 m Z R d  ϕ j  x ε m + z ; ω  − ϕ j  x ε m ; ω   ∂ f ( x + ε m z ) ∂ x j − ∂ f ( x ) ∂ x j  ν ( z ) dz = d X k =1 ∂ 2 f ( x ) ∂ x j ∂ x k Z {| z | ⩽ M }  ϕ j  x ε m + z ; ω  − ϕ j  x ε m ; ω  z k ν ( z ) dz ! + G m,j 2 ,M ( x ; ω ) . Here, for ev ery x ∈ B (0 , K 0 R 0 ) , | G m,j 2 ,M ( x ; ω ) | ⩽ ∥∇ 3 f ∥ L ∞ ( R d ) M ε m 2 Z {| z | ⩽ M }     ϕ j  x ε m + z ; ω  − ϕ j  x ε m ; ω      | z | 2 ν ( z ) dz + ∥∇ 2 f ∥ L ∞ ( R d ) Z {| z | >M }     ϕ j  x ε m + z ; ω  − ϕ j  x ε m ; ω      | z | ν ( z ) dz ⩽ c 8 Z R d     ϕ j  x ε m + z ; ω  − ϕ j  x ε m ; ω      2 ν ( z ) dz ! 1 / 2 ×   ε m M Z {| z | ⩽ M } | z | 4 ν ( z ) dz ! 1 / 2 + Z {| z | >M } | z | 2 ν ( z ) dz ! 1 / 2   , NON-LOCAL SYMMETRIC OPERA TORS WITH DIVERGENCE-FREE DRIFT 31 whic h, together with the co-cycle prop erty (2.6) and the stationary prop ert y of the transformation { τ x } x ∈ R d , implies that lim M →∞ lim m →∞ Z B (0 ,K 0 R 0 ) | G m,j 2 ,M ( x ; ω ) | 2 dx ⩽ | B (0 , K 0 R 0 ) | E  Z R d | ϕ j ( z ; ω ) | 2 ν ( z ) dz  lim M →∞ Z {| z | >M } | z | 2 ν ( z ) dz ! = 0; while for ev ery x ∈ B (0 , K 0 R 0 ) c , w e ﬁnd by using (1.3) again that | G m,j 2 ,M ( x ; ω ) | ⩽ ε − 1 m ∥∇ f ∥ L ∞ ( R d ) " Z B  x ε m , R 0 ε m  ν ( z ) dz !     ϕ j  x ε m ; ω      + Z B  0 , R 0 ε m  | ϕ j ( z ; ω ) | 2 dz ! 1 / 2 Z B  x ε m , R 0 ε m  ν ( z ) 2 dz ! 1 / 2 # ⩽ c 9 ε − d − 1 m γ  x ε m  ν  x ε m        ϕ j  x ε m ; ω      + ε d m Z B  0 , R 0 ε m  | ϕ j ( z ; ω ) | 2 dz ! 1 / 2   , whic h, along with the Hölder inequality , giv es us Z {| x | >K 0 R 0 } | G m,j 2 ,M ( x ; ω ) | dx ⩽ c 10 Z {| x | > K 0 R 0 ε m } | ϕ j ( x ; ω ) | 2 ν ( x ) dx ! 1 / 2 Z {| x | > K 0 R 0 ε m } | x | 2 γ 2 ( x ) ν ( x ) dx ! 1 / 2 + c 10 Z B (0 ,R 0 ) ε 2 m     ϕ j  x ε m ; ω      2 dx ! 1 / 2 Z {| x | > K 0 R 0 ε m } | x | 2 ν ( x ) dx ! 1 / 2 Z {| x | > K 0 R 0 ε m } | x | 2 γ 2 ( x ) ν ( x ) dx ! 1 / 2 → 0 as m → ∞ . Here w e used (1.4) and the argument for (4.5). Hence, com bining all the estimates ab ov e together with (3.2) and (4.2) yields that lim m →∞ I m,j 22 ( ω ) = − lim M →∞ lim m →∞ d X k =1 Z R d ∂ 2 f ( x ) ∂ x j ∂ x k Z {| z | ⩽ M }  ϕ j  x ε m + z ; ω  − ϕ j  x ε m ; ω  z k ν ( z ) dz ! u 0 ( x ; ω ) dx = − lim M →∞ lim m →∞ d X k =1 Z R d ∂ 2 f ( x ) ∂ x j ∂ x k Z {| z | ⩽ M } ϕ j  z ; τ x ε m ω  z k ν ( z ) dz ! u 0 ( x ; ω ) dx = − d X k =1 E  Z R d ϕ j ( z ; ω ) z k ν ( z ) dz  Z R d ∂ 2 f ( x ) ∂ x j ∂ x k u 0 ( x ; ω ) dx, where in the second equality we used the co-cycle prop erty (2.7). Since supp h ∂ f ∂ x j i ⊂ B (0 , R 0 ) , w e can easily use the mean v alue theorem and (1.3) to get that     L ε m 0  ∂ f ∂ x j  ( x )     ⩽ c 11  1 B (0 ,K 0 R 0 ) ( x ) + 1 B (0 ,K 0 R 0 ) c ( x ) ε − d − 2 m γ  x ε m  ν  x ε m  . This along with the second inequality in (4.5) yields that    I m,j 23 ( ω )    ⩽ c 12 sup ε ∈ (0 , 1) ∥ u ε ( · ; ω ) ∥ L ∞ ( R d ) " ε m Z B (0 ,K 0 R 0 )     ϕ j  x ε m ; ω      dx + Z {| x | > K 0 R 0 ε m } | ϕ j ( x ; ω ) | 2 ν ( x ) dx ! 1 / 2 Z {| x | > K 0 R 0 ε m } | x | 2 γ 2 ( x ) ν ( x ) dx ! 1 / 2 # . 32 XIN CHEN, JIAN W ANG AND KUN YIN Then, according to (2.8), (3.28) and (1.4), we can obtain immediately that lim m →∞ | I m,j 23 ( ω ) | = 0 . Therefore, putting b oth estimates for I m,j 22 ( ω ) and I m,j 23 ( ω ) together, we ﬁnd that for every 1 ⩽ j ⩽ d , lim m →∞ " ε − d − 2 m 2 Z R d Z R d  ϕ ε m j ( x + z ; ω ) ∂ f ( x + z ) ∂ x j − ϕ ε m j ( x ; ω ) ∂ f ( x ) ∂ x j  ( u ε m ( x + z ; ω ) − u ε m ( x ; ω )) ν  z ε  dz dx − ε − 1 m 2 Z R d Z R d  ϕ j  x ε m + z ; ω  − ϕ j  x ε m ; ω   u ε m ( x + ε m z ; ω ) ∂ f ( x + ε m z ) ∂ x j − u ε m ( x ; ω ) ∂ f ( x ) ∂ x j  ν ( z ) dz dx # = − d X k =1 E  Z R d ϕ j ( z ; ω ) z k ν ( z ) dz  Z R d ∂ 2 f ( x ) ∂ x j ∂ x k u 0 ( x ; ω ) dx. (4.7) (3) In this part, we deal with the drift term in the right hand side of (4.4). It holds that Z R d ε − 1 m  b  x ε m ; ω  , ∇ f ε m ( x ; ω )  u ε m ( x ; ω ) dx = d X j =1 Z R d ε − 1 m b j  x ε m ; ω  ∂ f ( x ) ∂ x j u ε m ( x ; ω ) dx + d X k,j =1 Z R d ε − 1 m b k  x ε m ; ω  ∂ ϕ j  x ε m ; ω  ∂ x k ∂ f ( x ) ∂ x j u ε m ( x ; ω ) dx + I m 3 ( ω ) , where I m 3 ( ω ) := d X k,j =1 Z R d ε − 1 m b k  x ε m ; ω  ϕ ε m j ( x ; ω ) ∂ 2 f ( x ) ∂ x j ∂ x k u ε m ( x ; ω ) dx. Note that ε − 1 m b k  x ε m ; ω  = − P d l =1 ∂ ∂ x l  H kl  · ε m  ( x ) . By using the in tegration by parts form ul a, w e deriv e I m 3 ( ω ) = d X j,k,l =1 Z R d H kl  x ε m ; ω  ∂ ϕ j  x ε m ; ω  ∂ x l ∂ 2 f ( x ) ∂ x j ∂ x k u ε m ( x ; ω ) dx + d X j,k,l =1 Z R d H kl  x ε m ; ω  ϕ ε m j ( x ; ω ) ∂ ∂ x l  ∂ 2 f ( · ) ∂ x j ∂ x k u ε m ( · ; ω )  ( x ) dx =: I m 31 ( ω ) + d X j,k,l =1 I m,j k l 32 ( ω ) . A ccording to Lemma 3.2, ∇ ϕ ( x ; ω ) = ˜ Φ( τ x ω ) with ˜ Φ ∈ L q (Ω; P ) for some q ∈ (1 , 2) . Since, by (1.13), ˜ H kl ∈ L r (Ω; P ) for some r > q / ( q − 1) , it follows from (4.2) and the ergo dic theorem that lim m →∞ I m 31 ( ω ) = lim m →∞ d X j,k,l =1 Z R d H kl  x ε m ; ω  ∂ ϕ j  x ε m ; ω  ∂ x l ∂ 2 f ( x ) ∂ x j ∂ x k u 0 ( x ; ω ) dx = 1 2 lim m →∞ d X j,k,l =1 Z R d  H kl  x ε m ; ω  + H j l  x ε m ; ω    ∂ ϕ k  x ε m ; ω  ∂ x l + ∂ ϕ j  x ε m ; ω  ∂ x l   ∂ 2 f ( x ) ∂ x j ∂ x k u 0 ( x ; ω ) dx − lim m →∞ d X j,k,l =1 Z R d H kl  x ε m ; ω  ∂ ϕ k  x ε m ; ω  ∂ x l ∂ 2 f ( x ) ∂ x j ∂ x k u 0 ( x ; ω ) dx NON-LOCAL SYMMETRIC OPERA TORS WITH DIVERGENCE-FREE DRIFT 33 = d X j,k =1 1 2 d X l =1  E h ( ˜ Φ kl ( ω ) + ˜ Φ j l ( ω ))( ˜ H kl ( ω ) + ˜ H j l ( ω )) i − 2 E h ˜ Φ kl ( ω ) ˜ H kl ( ω ) i ! Z R d ∂ 2 f ( x ) ∂ x j ∂ x k u 0 ( x ; ω ) dx = − 1 2 d X j,k =1 Z R d  1 2 E  Z R d ( ϕ j ( z ; ω ) + ϕ k ( z ; ω )) 2 ν ( z ) dz  − E  Z R d | ϕ k ( z ; ω ) | 2 ν ( z ) dz  ∂ 2 f ( x ) ∂ x j ∂ x k u 0 ( x ; ω ) dx = − 1 2 d X j,k =1 Z R d E  Z R d ϕ j ( z ; ω ) ϕ k ( z ; ω ) ν ( z ) dz  ∂ 2 f ( x ) ∂ x j ∂ x k u 0 ( x ; ω ) dx, where in the fourth equality we used (3.31) and (3.36). Let ψ ∈ C ∞ c ( R d ) be such that ψ ( x ) = 1 for ev ery x ∈ B (0 , R 0 ) and supp[ ψ ] ⊂ B (0 , R 0 + 1) . A ccording to the P arsev al equality , | I m,j k l 32 ( ω ) | =     Z R d H kl  x ε m ; ω  ϕ ε m j ( x ; ω ) ψ ( x ) ∂ ∂ x l  ∂ 2 f ( · ) ∂ x j ∂ x k u ε m ( · ; ω )  ( x ) dx     =     i Z R d F  H kl  · ε m ; ω  ϕ ε m j ( · ; ω ) ψ ( · )  ( ξ ) F  ∂ 2 f ( · ) ∂ x j ∂ x k u ε m ( · ; ω )  ( ξ ) ξ l dξ     ⩽ Z R d     F  H kl  · ε m ; ω  ϕ ε m j ( · ; ω ) ψ ( · )  ( ξ )     2  1 {| ξ | ⩽ ε − 1 m } + ε 2 − α m | ξ | 2 − α 1 {| ξ | >ε − 1 m }  dξ ! 1 / 2 × Z R d     F  ∂ 2 f ( · ) ∂ x j ∂ x k u ε m ( · ; ω )( ξ )      2  | ξ | 2 1 {| ξ | ⩽ ε − 1 m } + ε − (2 − α ) m | ξ | α 1 {| ξ | >ε − 1 m }  dξ ! 1 / 2 =:  I m,j k l 321 ( ω )  1 / 2 ×  I m,j k l 322 ( ω )  1 / 2 . Here F ( f )( ξ ) denotes the F ourier transform of f ∈ L 1 ( R d ; dx ) . A ccording to (2.3) and its pro of, we can verify directly that ε − (2 − α ) m Z {| z | ⩽ ε m } 1 − e − i ⟨ ξ ,z ⟩ | z | d + α dz ⩾ c 11  | ξ | 2 1 {| ξ | ⩽ ε − 1 m } + ε − (2 − α ) m | ξ | α 1 {| ξ | >ε − 1 m }  , ε − (2 − α ) m Z {| z | ⩽ ε m } 1 − e − i ⟨ ξ ,z ⟩ | z | d + α dz ⩽ c 12  | ξ | 2 1 {| ξ | ⩽ ε − 1 m } + ε − (2 − α ) m | ξ | α 1 {| ξ | >ε − 1 m }  . (4.8) So, b y (4.8), we get I m,j k l 322 ( ω ) ⩽ c 13 ε − (2 − α ) m Z R d Z {| z | ⩽ ε m }    ∂ 2 f ( x + z ) ∂ x j ∂ x k u ε m ( x + z ; ω ) − ∂ 2 f ( x ) ∂ x j ∂ x k u ε m ( x )    2 | z | d + α dz dx ⩽ c 14 ∥∇ 2 f ∥ L ∞ ( R d ) ε − (2 − α ) m Z R d Z {| z | ⩽ ε m } | u ε m ( x + z ; ω ) − u ε m ( x ) | 2 | z | d + α dz dx + c 14 ∥ u ε m ( · ; ω ) ∥ L ∞ ( R d ) ε − (2 − α ) m Z R d Z {| z | ⩽ ε m }    ∂ 2 f ( x + z ) ∂ x j ∂ x k − ∂ 2 f ( x ) ∂ x j ∂ x k    2 | z | d + α dz dx. A ccording to (3.2) and (3.3), sup m ⩾ 1 I m,j k l 322 ( ω ) < ∞ . On the other hand, using (4.8) again, w e obtain I m,j k l 321 ( ω ) ⩽ c 15 Z R d     H kl  x ε m ; ω  ϕ ε m j ( x ; ω ) ψ ( x )     2 dx + c 15 ε 2 − α m Z R d Z {| z | ⩽ ε m }    H kl  x + z ε m ; ω  ϕ ε m j ( x + z ; ω ) ψ ( x + z ) − H kl  x ε m ; ω  ϕ ε m j ( x ; ω ) ψ ( x )    2 | z | d +2 − α dz dx 34 XIN CHEN, JIAN W ANG AND KUN YIN ⩽ c 16 Z B (0 ,R 0 +2)     H kl  x ε m ; ω  ϕ ε m j ( x ; ω )     2 1 + ε 2 − α m sup x ∈ R d Z {| z | ⩽ ε m } ( ψ ( x + z ) − ψ ( x )) 2 | z | d +2 − α dz ! dx + c 16 ε 2 − α m Z B (0 ,R 0 +2)     H kl  x ε m ; ω      2    Z {| z | ⩽ ε m }    ϕ ε m j ( x + z ; ω ) − ϕ ε m j ( x ; ω )    2 | z | d +2 − α dz    dx + c 16 ε 2 − α m Z B (0 ,R 0 +2) | ϕ ε m j ( x ; ω ) | 2    Z {| z | ⩽ ε m }  H kl  x + z ε m ; ω  − H kl  x ε m ; ω  2 | z | d +2 − α dz    dx =: I m,j k l 3211 ( ω ) + I m,j k l 3212 ( ω ) + I m,j k l 3213 ( ω ) . Recall that p 0 = dq / ( d − q ) , so 2 p 0 p 0 − 2 = 2 q d q ( d +2) − 2 d ; on the other hand, the condition q > 2 d d +2 implies that p 0 > 2 . According to (1.13) and the fact p ′ 0 < p 0 , we ha v e ˜ H kl ∈ L 2 qd q ( d +2) − 2 d (Ω; P ) . By the Hölder inequalit y and (3.29), w e can show immediately that lim m →∞ I m,j k l 3211 ( ω ) ⩽ c 17 lim m →∞ Z B (0 ,R 0 +2)     H kl  x ε m ; ω      2 qd q ( d +2) − 2 d dx ! q ( d +2) − 2 d qd Z B (0 ,R 0 +2)    ϕ ε m j ( x ; ω )    qd d − q dx ! 2( d − q ) qd = 0 . Similarly , also by (1.14), (3.29) and the Hölder inequality , w e derive that lim m →∞ I m,j k l 3213 ( ω ) = 0 . No w we turn to the estimate for I m,j k l 3212 ( ω ) . F or every p > 2 , by the Hölder inequality , I m,j k l 3212 ( ω ) ⩽ c 16 ε 2 − α m Z B (0 ,R 0 +2)     H kl  x ε m ; ω      2 p p − 2 dx ! p − 2 p ×     Z B (0 ,R 0 +2)    Z {| z | ⩽ ε m }    ϕ ε m j ( x + z ; ω ) − ϕ ε m j ( x ; ω )    2 | z | d +2 − α dz    p 2 dx     2 p . (4.9) F urthermore, for every δ > 1 ,     Z B (0 ,R 0 +2)    Z {| z | ⩽ ε m }    ϕ ε m j ( x + z ; ω ) − ϕ ε m j ( x ; ω )    2 | z | d +2 − α dz    p 2 dx     2 p ⩽ ε α m    Z B (0 ,R 0 +2)   Z {| z | ⩽ 1 }    ϕ j  x ε m + z ; ω  − ϕ j  x ε m ; ω     p | z | d + (2 − α ) pδ 2 dz     Z {| z | ⩽ 1 } 1 | z | d − ( δ − 1)(2 − α ) p p − 2 dz   p − 2 2 dx    2 p ⩽ c 18 ε α m   Z B (0 ,R 0 +2) Z {| z | ⩽ 1 }    ϕ j  x ε m + z ; ω  − ϕ j  x ε m ; ω     p | z | d + (2 − α ) pδ 2 dz dx   2 p = c 18 ε α + 2 d p m Z B  0 , R 0 +2 ε m  Z {| z | ⩽ 1 } | ϕ j ( x + z ; ω ) − ϕ j ( x ; ω ) | p | z | d + (2 − α ) pδ 2 dz dx ! 2 p . NON-LOCAL SYMMETRIC OPERA TORS WITH DIVERGENCE-FREE DRIFT 35 Let χ m ∈ C ∞ c ( R d ) b e such that χ m ( x ) = 1 for every x ∈ B  0 , R 0 +2 ε m + 1  , χ m ( x ) = 0 for every x ∈ B  0 , R 0 +2 ε m + 2  c and ∥∇ χ m ∥ L ∞ ( R d ) ⩽ 2 . Then, Z B  0 , R 0 +2 ε m  Z {| z | ⩽ 1 } | ϕ j ( x + z ; ω ) − ϕ j ( x ; ω ) | p | z | d + (2 − α ) pδ 2 dz dx ! 2 p = Z B  0 , R 0 +2 ε m  Z {| z | ⩽ 1 } | ϕ j ( x + z ; ω ) χ m ( x + z ) − ϕ j ( x ; ω ) χ m ( x ) | p | z | d + (2 − α ) pδ 2 dz dx ! 2 p ⩽ ∥ ϕ j χ m ∥ 2 W (2 − α ) δ 2 ,p ( R d ) . Thanks to α ∈ (1 , 2) , w e can tak e δ > 1 suc h that θ 0 := (2 − α ) δ α ∈ (0 , 1) , and p 0 > p 1 > 2 so that 1 p 1 = 1 − θ 0 p 0 + θ 0 2 , where p 0 := dq d − q > 2 . In particular, (2 − α ) δ 2 = (1 − θ 0 ) · 0 + θ 0 α 2 . Below, we take p = p 1 . By the interpolation of Beso v spaces, see e.g. [3, Theorem 6.4.5] or [44, Section 2.4.1, Theorem (a)], ∥ ϕ j χ m ∥ 2 W (2 − α ) δ 2 ,p 1 ( R d ) ⩽ ∥ ϕ j χ m ∥ 2(1 − θ 0 ) L p 0 ( R d ) ∥ ϕ j χ m ∥ 2 θ 0 W α 2 , 2 ( R d ) ⩽ c 16 Z B  0 , R 0 +3 ε m  | ϕ j ( x ; ω ) | p 0 dx ! 2(1 − θ 0 ) p 0 × Z B  0 , R 0 +3 ε m  Z {| z | ⩽ 1 } | ϕ j ( x + z ; ω ) − ϕ j ( x ; ω ) | 2 | z | d + α dz dx + Z B  0 , R 0 +3 ε m  | ϕ j ( x ; ω ) | 2 dx ! θ 0 . Here in the last inequality we used ∥∇ χ m ∥ L ∞ ( R d ) ⩽ 2 . Hence, according to all the estimates ab o v e, we get ε 2 − α m     Z B (0 ,R 0 +3)    Z {| z | ⩽ ε m }    ϕ ε m j ( x + z ; ω ) − ϕ ε m j ( x ; ω )    2 | z | d +2 − α dz    p 1 2 dx     2 p 1 ⩽ c 17 ε 2 m Z B (0 ,R 0 +3)     ϕ j  x ε m ; ω      p 0 dx ! 2(1 − θ 0 ) p 0 ×    Z B (0 ,R 0 +3) Z {| z | ⩽ 1 }    ϕ j  z ; τ x ε m ω     2 | z | d + α dz dx + Z B (0 ,R 0 +3)     ϕ j  x ε m ; ω      2 dx    θ 0 . Putting this into (4.9) with p = p 1 , and applying (3.29) and (2.8), as w ell as 2 < p 1 < p 0 with δ > 1 close to 1 (and then p 1 is close to p ′ 0 = 2 p 0 α 4( α − 1)+ p 0 (2 − α ) ) so that 2 p 1 / ( p 1 − 2) < r , we can prov e that lim m →∞ I m,j k l 3212 ( ω ) = 0 . Therefore, com bining the estimate for I m 3 ( ω ) with (4.6) and (4.7) yields that − Z R d u ε m ( x ; ω ) L ε m 0 f ( x )( x ) dx + ε − 1 m Z R d  b  x ε m ; ω  , ∇ f ε m ( x ; ω )  u ε m ( x ; ω ) dx + ε − d − 2 m 2 d X j =1 Z R d Z R d  ϕ ε m j ( x + z ; ω ) ∂ f ( x + z ) ∂ x j − ϕ ε m j ( x ; ω ) ∂ f ( x ) ∂ x j  ( u ε m ( x + z ; ω ) − u ε m ( x ; ω )) ν  z ε  dz dx 36 XIN CHEN, JIAN W ANG AND KUN YIN = J m ( ω ) − 1 2 d X j,k =1 E  Z R d ( z j + ϕ j ( z ; ω )) ( z k + ϕ k ( z ; ω )) ν ( z ) dz  Z R d ∂ 2 f ( x ) ∂ x j ∂ x k u 0 ( x ; ω ) dx + ε − 1 m 2 d X j =1 Z R d Z R d  ϕ j  x ε m + z ; ω  − ϕ j  x ε m ; ω   u ε m ( x + ε m z ; ω ) ∂ f ( x + ε m z ) ∂ x j − u ε m ( x ; ω ) ∂ f ( x ) ∂ x j  ν ( z ) dz dx + d X k,j =1 Z R d ε − 1 m b k  x ε m ; ω  ∂ ϕ j  x ε m ; ω  ∂ x k ∂ f ( x ) ∂ x j u ε m ( x ; ω ) dx + d X j =1 Z R d ε − 1 m b j  x ε m ; ω  ∂ f ( x ) ∂ x j u ε m ( x ; ω ) dx = J m ( ω ) − 1 2 d X j,k =1 ¯ a j k Z R d ∂ 2 f ( x ) ∂ x j ∂ x k u 0 ( x ; ω ) dx, where J m ( ω ) satisﬁes that lim m →∞ J m ( ω ) = 0 , in the second equality w e hav e used the anti-symmetry of { ˜ H lj } 1 ⩽ l,j ⩽ d , (2.6) and the deﬁnition of ¯ A = { ¯ a j k } 1 ⩽ j,k ⩽ d . 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Xin Chen: Sc ho ol of Mathematical Sciences, Shanghai Jiao T ong Univ ersit y , 200240 Shanghai, P .R. China. Email: chenxin217@sjtu.edu.cn Kun Yin: Sc ho ol of Mathematical Sciences, Shanghai Jiao T ong Univ ersit y , 200240 Shanghai, P .R. China. Email: epsilonyk@sjtu.edu.cn Jian W ang: Sc hool of Mathematics and Statistics & Key Lab oratory of Analytical Mathematics and Applications (Ministry of Education) & F ujian Pro vincial Key Lab oratory of Statistics and Artiﬁcial In telligence, F ujian Normal Univ ersit y , 350007 F uzhou, P .R. China. Email: jianwang@fjnu.edu.cn

Stochastic homogenization of diffusions in turbulence driven by non-local symmetric Lévy operators

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