Central limit theorem for random walk in degenerate divergence-free random environment: $\mathcal H_{-1}$ reloaded with relaxed ellipticity

Cen tral limit theorem for random w alk in degenerate div ergence-free random en vironmen t: H − 1 reloaded with relaxed ellipticit y Bálin t Tóth Rényi Institute Budapest F ebruary 19, 2026 Abstract This pap er enhances the result of the w ork Kozma-Tóth (2017) [14]. W e prov e the cen tral limit theorem (in probabilit y w.r.t. the en vironment) for the displacemen t of a random walk er in divergence-free (or, doubly sto c hastic) random en vironment, with substantial ly r elaxe d el lipticity assumptions. Integrabilit y of the recipro cal of the symmetric part of the jump rates is only assumed (rather than their b ound- edness, as in previous works on this type of R WRE). Relaxing ellipticit y inv olv es substan tial change s in the pro of, making it conceptually elemen tary in the sense that it do es not rely on Nash’s inequality in an y disguise. MSC2010: 60F05, 60G99, 60K37 Key words and phrases: random w alk in random en vironment (R WRE), div er- gence-free drift, incompressible flow, cen tral limit theorem 1 In tro duction In the w ork [14] the weak CL T (that is, in probability with resp ect to the environmen t) w as established for random walks in doubly sto chastic (or, divergence-free) random en- vironmen ts, under the conditions of ( ı ) strict el lipticity assumed for the symmetric part of the drift field, and, ( ıı ) H − 1 ( | ∆ | ) assumed for the antisymmetric part of the drift field. The pro of relied on a martingale appro ximation (a la Kipnis-V aradhan) adapted to the non-self-adjoint and non-se ctorial nature of the problem, the tw o substan tial tec hnical comp onen ts b eing: ◦ A functional analytic statemen t about the un b ounded op erator formal ly written as | L + L ∗ | − 1 / 2 ( L − L ∗ ) | L + L ∗ | − 1 / 2 , where L is the infinitesimal generator of the en viron- men t pro cess, as seen from the position of the moving random walk er. 1 ◦ A diagonal heat k ernel upp er b ound whic h follows from Nash’s inequality , v alid only under the assumed strict el lipticity . The assumption of strict ellipticity , how ev er, is conceptually restrictiv e and excludes relev ant applications. In this pap er w e relax the strict ellipticity assumption, replacing it b y an integrabilit y condition on the recipro cals of the conductances. This can b e done only b y "de-Nashifying" the pro of. On the other hand the functional analytic elements are refined. These changes are also of conceptual imp ortance: the present pro of (of a stronger result) do es not in vok e a conceptually higher lev el (than the CL T) element lik e a lo cal heat kernel estimate. Altogether, it is conceptually simpler than that in [14]. (This w as already demonstrated in [30] where the result of [14] w as re-prov ed along a baby v ersion of the arguments in the present pap er.) T o the b est of our kno wledge this is the first time a CL T is prov ed for a random walk in a non-r eversible random en vironment without a strong ellipticit y assumption. The in tegrability conditions imp osed on the random jump rates seem to b e close to optimal. 1.1 Preliminaries Let (Ω , F , π , ( τ z : z ∈ Z d )) b e a probability space with an ergo dic Z d -action. Denote by N := { k ∈ Z d : | k | = 1 } the set of unit elemen ts generating Z d as an additive group. These will serve as the set of elemen tary steps of a contin uous time nearest neigh b our random walk on Z d . Let p : Ω → [0 , ∞ ) N satisfy ( π -a.s.) the follo wing bi-sto chasticity condition X k ∈N p k ( ω ) = X k ∈N p − k ( τ k ω ) , (1) and p : Z d × Ω → [0 , ∞ ) N b e its lifting to a random field ov er Z d , p k ( x, ω ) := p k ( τ x ω ) . (Throughout the pap er, measurable functions f : Ω → R and their lifting to a random field f : Z d × Ω → R , defined as f ( x, ω ) := f ( τ x ω ) , will b e denoted b y the same symbol.) Giv en the random field p : Z d × Ω → [0 , ∞ ) N , define the con tinuous-time random walk in random environmen t (R WRE), t 7→ X ( t ) ∈ Z d as the Marko vian nearest neigh b our random walk with jump rates P ω  X ( t + dt ) = x + k   X ( t ) = x  = p k ( x, ω ) dt + O (( dt ) 2 ) , (2) and initial p osition X (0) = 0 . W e use the notation P ω ( · ) , E ω ( · ) and V ar ω ( · ) for quenche d probabilit y , exp ecta- tion and v ariance. That is: probabilit y , expectation, and v ariance with resp ect to the distribution of the random w alk X ( t ) , c onditione d on fixe d envir onment ω ∈ Ω . The notation P  ·  := R Ω P ω ( · ) d π ( ω ) , E  ·  := R Ω E ω ( · ) d π ( ω ) and V ar  ·  := R Ω V ar ω ( · ) d π ( ω ) + R Ω E ω ( · ) 2 d π ( ω ) − E  ·  2 will b e reserv ed for anne ale d probabilit y , 2 exp ectation and v ariance. That is: probabilit y , exp ectation and v ariance with resp ect to the random w alk tra jectory X ( t ) and the en vironment ω , sampled according to the distribution π . The envir onment pr o c ess (as seen from the p osition of the random walk er) is, t 7→ η t ∈ Ω , defined as η t := τ X t ω . (3) This is a pure jump Mark o v pro cess on the state space Ω . The infinitesimal generator of its Marko vian semigroup is Lf ( ω ) = X k ∈N p k ( ω )( f ( τ k ω ) − f ( ω )) . (4) The linear op erator L is w ell defined for all measurable functions f : Ω → R , not just a formal expression. It is well kno wn (and easy to c hec k, see e.g. [13]) that bi-sto chasticit y (1) of the jump rates p is equiv alent to stationarit y (in time) of the a priori distribution π of the en vironment pro cess t 7→ η t ∈ Ω . Moreo ver, under the conditions (1) and (8) (see b elo w) spatial ergo dicity of (Ω , F , π , ( τ z : z ∈ Z d )) also implies time-wise ergo dicit y of the (time-wise) stationary en vironment pro cess pro cess t 7→ η t ∈ (Ω , F , π ) . See [13] for details. Hence it follows that under these conditions the random w alk t 7→ X ( t ) will hav e stationary and ergo dic anne ale d incremen ts. Though, in the annealed setting the walk is obviously not Mark ovian. It is conv enien t to separate the symmetric and an tisymmetric parts of the jump rates: p k ( ω ) = s k ( ω ) + b k ( ω ) , s k ( ω ) := p k ( ω ) + p − k ( τ k ω ) 2 , b k ( ω ) := p k ( ω ) − p − k ( τ k ω ) 2 , (5) and note that s k ( ω ) = s − k ( τ k ω ) ≥ 0 , b k ( ω ) = − b − k ( τ k ω ) , (6) and (1) is equiv alent to X k ∈N b k ( ω ) = 0 . (7) W e assume for now we ak el lipticity and finiteness of the conductances: π -a.s. for all k ∈ N , 0 < s k ( ω ) < ∞ . (8) Later somewhat stronger integrabilit y conditions will b e imp osed (see (18) further b elow). A ccordingly , we decomp ose the infinitesimal generator in to Hermitian, and an ti-Her- mitian parts (with resp ect to the stationary measure π ) as L = − S + A 3 where S f ( ω ) := − X k ∈N s k ( ω )( f ( τ k ω ) − f ( ω )) , Af ( ω ) := X k ∈N b k ( ω )( f ( τ k ω ) − f ( ω )) . (9) The righ t hand sides of (4) and (9) mak e p erfect s ense for arbitrary measurable functions f : Ω → R . So, disregarding for the time b eing top ological issues in function spaces, the linear op erators L , S , and A are well defined ov er the space of all (classes of equiv alence of ) measurable functions f : Ω → R (w.r.t. the probabili t y measure π ). The symmetric part of the jump rates (lifted to Z d as s k ( x, ω ) := s k ( τ x ω ) ), s − k ( x + k , ω ) = s k ( x, ω ) > 0 (10) defines a collection p ositive r andom c onductanc es on the unoriented edges { x, x + k } . The w eak ellipticity condition (10) means that all edges of the lattice Z d are passable for the random w alker, at least in one direction. Integrabilit y conditions on the conductances s k and their recipro cals s − 1 k will b e imp osed later (see (18) b elo w). A ctually , it will b e con venien t to w ork with the v ariables r k ( ω ) := p s k ( ω ) . Regarding the an tisymmetric parts of the jump rates (lifted to Z d as b k ( x, ω ) := b k ( τ x ω ) ), we ha ve b − k ( x + k , ω ) = − b k ( x, ω ) . (11) That is, they define a r andom flow (or, a lattic e ve ctor field ) on the oriented edges ( x, x + k ) of Z d . In terms of these v ariables the bi-sto chasticit y (1) reads as sour c elessness , or diver genc e-fr e eness of the flo w: π -a.s., for all x ∈ Z d X k ∈N b k ( x, ω ) = 0 (12) Ob viously , the former dominate the latter: | b k ( ω ) | ≤ s k ( ω ) . (13) The extreme case, when π -almost-surely , for all k ∈ N , s k ( ω ) = | b k ( ω ) | , will b e called total ly asymmetric . In this case ev ery edge of the lattice Z d is passable in exactly one of the tw o p ossible directions. Throughout this pap er w e assume that the div ergence-free flo ws/vector fields b k are giv en in diver genc e form . That is, there exists h : Ω → R N ×N , with the follo wing ( π -a.s.) symmetries h k,l ( ω ) = − h − k,l ( τ k ω ) = − h k, − l ( τ l ω ) = − h l,k ( ω ) (14) 4 so that b k ( ω ) = X l ∈N h k,l ( ω ) = 1 2 X l ∈N ( h k,l ( ω ) − h k,l ( τ − l ω )) . (15) It is straightforw ard that (14) and (15) imply (11). The conv erse, how ev er, is not true. It is actually a very subtle issue to determine whether a divergence free vector field (as e.g. b k ( x, ω ) given in (5)) can b e written in this so-called div ergence-form or not. More on this in the App endix. The symmetry conditions (14) mean that the lifted field h : Z d × Ω → R N ×N , h k,l ( x, ω ) := h k,l ( τ x ω ) , is a r andom function of the oriente d plaquettes of Z d , that is, a stationary (with resp ect to spatial translations) str e am tensor . Note that in dimen- sions d = 2 and d = 3 , the stream fields ( x, k , l ) 7→ h k,l ( x, ω ) := h k,l ( τ x ω ) are identified with a sc alar height function on Z 2 ∗ := Z 2 + (1 / 2 , 1 / 2) , resp ectively , a flow/ve ctor field on Z 3 ∗ := Z 3 + (1 / 2 , 1 / 2 , 1 / 2) . In [14] the weak CL T (that is, in probabilit y with resp ect to the en vironment) w as established for the diffusiv ely scaled displacement T − 1 / 2 X ( T ) , as T → ∞ , under the conditions s k + ( s k ) − 1 ∈ L ∞ (16) imp osed on the conductances and h k,l ∈ L 2 (17) imp osed on the stream tensor. Throughout this pap er L p := L p (Ω , π ) , with 1 ≤ p ≤ ∞ . Condition (17) is equiv alen t to b k ∈ H − 1 ( | ∆ | ) . That is, the an tisymmetric part of the jump rates b elong to the H − 1 -space of the lattice Laplacian acting on L 2 . The upp er b ound in (16) could b e relaxed to s k ∈ L 1 without effort, and without altering the pro of in [14]. The low er b ound ( str ong el lipticity ) is, how ever, a serious issue. The main goal of this pap er is to relax it to an integrabilit y condition imp osed on ( s k ) − 1 . 1.2 In tegrabilit y conditions on the jump rates Regarding the symmetric p arts / c onductanc es , w e assume that for all k ∈ N , s k + ( s k ) − 1 ∈ L 1 . (18) Or, equiv alen tly r k ∈ L 2 , (19) ( r k ) − 1 ∈ L 2 . (20) Regarding the antisymmetric p arts / flows , note first that due to (13) and (19) we also ha ve for all k ∈ U b k ∈ L 1 . 5 Beside (19) and (20) we also assume that for all k , l ∈ N , ( r l ) − 1 h k,l ∈ L 2 . (21) Note that assuming strong ellipticit y , ( r k ) − 1 ∈ L ∞ , (21) b ecomes (17). F urthermore, (19) & (21) jointly also imply h k,l ∈ L 1 , (22) Summarizing: W e make the structur al assumption (14)&(15) and the inte gr ability as- sumptions (19) , (20) , (21) . W e will b e explicit ab out exactly whic h integrabilit y assump- tion/condition is used in each step of the forthcoming argumen ts. 1.3 The CL T The lo cal quenche d drift of the random walk is E ω  dX ( t )   X ( s ) : 0 ≤ s ≤ t  = ( ϕ ( η t ) + ψ ( η t )) dt + o ( dt ) . (23) where ϕ, ψ : Ω → R d are ϕ ( ω ) := X k ∈N k s k ( ω ) , ψ ( ω ) := X k ∈N k b k ( ω ) . (24) Due to (10), resp ectiv ely , (11) w e ha ve ϕ i ( ω ) = s e i ( ω ) − s e i ( τ − e i ω ) , ψ i ( ω ) = b e i ( ω ) + b e i ( τ − e i ω ) . (25) In particular, from (19) we hav e ϕ ∈ L 1 and Z Ω ϕ ( ω ) dπ ( ω ) = 0 . (26) This do es not hold a priori for the drift term ψ coming from the divergence-free an tisym- metric part. Ho wev er, from (15) and (22) w e ha ve ψ ∈ L 1 and Z Ω ψ ( ω ) dπ ( ω ) = 0 . (27) F rom ergo dicity under π of the en vironment pro cess t 7→ η t , and from (23), (26) and (27) the strong la w of large num b ers for the displacement of the random walk er readily follo ws. Prop osition 1 (Strong La w of Large Num b ers) . Assuming (19) and (22) , lim t →∞ t − 1 X ( t ) = 0 , a . s . 6 The main result of this pap er is: Theorem 2 (Main theorem: CL T) . Assume the inte gr ability c onditions (19) , (20) , (21) . Then the displac ement t 7→ X ( t ) of the r andom walk c an b e de c omp ose d as X ( t ) = Y ( t ) + Z ( t ) so that the fol lowing limits hold as N → ∞ . (i) F or π -almost al l ω ∈ Ω , t 7→ Y ( t ) is a nonde gener ate squar e inte gr able martingale whose incr ements ar e stationary and er go dic in the anne ale d setting. Thus, due to the Martingale Invarianc e Principle (cf. [17]), for π -almost al l ω ∈ Ω , N − 1 / 2 Y ( N · ) ⇒ W σ ( · ) , in D ([0 , 1]) under the quenche d pr ob ability me asur e P ω ( . . . ) , wher e W σ ( · ) is a Br ow- nian motion with finite and non-de gener ate c ovarianc e σ 2 . (ii) F or any t ≥ 0 and δ > 0 P  N − 1 / 2 | Z ( N t ) | > δ  → 0 . Remark. W e should emphasize that this is muc h stronger than an annealed limit theo- rem. The in v ariance principle for the displacement of the random walk N − 1 / 2 X ( N · ) ⇒ W σ ( · ) , as N → ∞ , (28) holds in pr ob ability (and not av eraged out) with resp ect to the the distribution of the en vironment, (Ω , π ) . In other words, the in v ariance principle (28) holds π -a.s. (that is, in a quenched sense) along s equences N → ∞ increasing sufficien tly fast. 1.4 Examples W e giv e some concrete constructive examples – not cov ered by earlier w orks – where the argumen ts of this pap er w ork and provide a CL T (in the sense of Theorem 2). 1.4.1 A dding stream term to random conductances Let e s k ( x, ω ) = e s k ( τ x ω ) , k ∈ N , x ∈ Z d , b e random conductances as in (10) and assume that the in tegrability conditions (18) hold for them. It has b een kno wn since [4] that the random w alk among these conductances ob eys the CL T as stated in Theorem 2. It is a natural question to ask ho w can one mo dify this random en vironment by adding anti- symmetric, nonr eversible p art to the jump r ates so that the nondegenerate CL T remains v alid. Here is an answer. 7 Let h k,l ( x, ω ) = h k,l ( τ x ω ) , k , l ∈ N , x ∈ Z d , b e a stream tensor, as in (14) (defined on the same probabilit y space, join tly with the conductances) and b k ( x, ω ) = b k ( τ x ω ) , k ∈ N , x ∈ Z d , b e the div ergence-free flo w/vector field given by (15). Let now p k ( ω ) := e s k ( ω ) + 2 b k ( ω ) + . (That is s k ( ω ) := e s k ( ω ) + | b k ( ω ) | .) It will suffice to assume (b eside (18) holding for e s k ) that | h k,l | √ e s k ∈ L 2 . In particular, if  e s k ( x )  x ∈ Z d ,k ∈N at one hand and  h k,l ( x )  x ∈ Z d ,k,l ∈N on the other, are indep endent then it suffices that h k,l ∈ L 2 . If this indep endence do es not hold than, ( e s k ) − 1 ∈ L β and h k,l ∈ L α , with 2 /α + 1 /β = 1 will suffice. α = ∞ , β = 1 means a p erturbation of the random conductance mo del satisfying (18) (without strong ellipticit y assumed!) with a div ergence free drift field whic h arises as the curl of a b ounde d stream tensor. The case α = 2 , β = ∞ means div ergence free drift field arising as the curl of an L 2 stream tensor - that is, in H − 1 ( | ∆ | ) of the lattice-Laplacian, and strongly elliptic, fully cov ered in [14]. 1.4.2 T w o totally asymmetric examples Let h k,l ( x, ω ) = h k,l ( τ x ω ) , k , l ∈ N , x ∈ Z d , b e a stream tensor, as in (14), b k ( x, ω ) = b k ( τ x ω ) , k ∈ N , x ∈ Z d , b e the divergence-free flow/v ector field given by (15) and let the jump rates b e p k ( x, ω ) := 2 b k ( x, ω ) + . (That is s k ( ω ) := | b k ( ω ) | .) Note that eac h edge ( x, x + k ) is passable in exactly one of the t wo directions. It is straightforw ard that the integrabilit y conditions (19) , (20) and (21) follo w from h k,l ∈ L α and | b k | − 1 ∈ L β with 2 /α + 1 /β = 1 . α = ∞ , β = 1 means bounded stream tensor (and, as a consequence, b ounded jump rates) and integrable recipro cal jump rates (no strong ellipticity). The case α = 2 , β = 1 means strong ellipticity and L 2 stream tensor - co vered in [14]. Another totally asymmetric example is the following non-elliptic p erturb ation (though, not a small p erturbation) of elliptic cases cov ered in [14]. Let e h k,l ( x, ω ) = e h k,l ( τ x ω ) , k , l ∈ N , x ∈ Z d , b e a stream tensor, as in (14), e b k ( x, ω ) = e b k ( τ x ω ) , k ∈ N , x ∈ Z d , b e the corresp onding divergence-free flow/v ector field given by (15). Assume e h k,l ∈ L 2 and ( e b k ) − 1 ∈ L ∞ - that is, the flo w/vector field e b k ( x, ω ) is assumed to b e strongly ellip- tic. Without loss of generality , assume inf | e b k | = 1 π -a.s. As concrete examples think ab out either the  2 d d  -mo del on Z d , with d ≥ 3 , or the randomly oriented Manhattan lattice on Z d with d ≥ 4 . F or details see [14], [29]. In addition, let ( ξ k,l ( x )) x ∈ Z d ,k,l , ∈N b e i.i.d. random v ariables mo dulo the tensorial symmetries in (14), and also indep en- den t of the collection ( e h k,l ( x )) x ∈ Z d ,k,l , ∈N , distributed as ξ kl ( x ) ∼ UNI [ − a, + a ] , with the 8 v alue of a to b e sp ecified. Finally , let h k,l ( x ) = e h k,l ( x ) + ξ k.l ( x ) , b k ( x ) given b y (15) and p k ( x ) = 2 b k ( x ) . If 0 ≤ a < 1 / (2( d − 1)) then the mo del is still strongly elliptic with s k = | b k | ≥ 1 − 2( d − 1) a > 0 and as suc h it is cov ered by [14]. If a = 1 / (2( d − 1)) then the mo del is not elliptic and the integrabilit y conditions (19) , (20) , (21) hold. If a > 1 / (2( d − 1)) then conditions (20) , (21) fail just marginally . 1.4.3 R WRE with stationary densit y Let p k ( x, ω ) = p k ( τ x ω ) , x ∈ Z d , k ∈ N , b e the jump rates of a random w alk in random en vironment, as in (2). If the bisto chasticit y condition (1) fails to hold that the envi- ronmen t pro cess t 7→ η t cf (3) is not time-stationary under the a priori measure π . The equation for a stationary Radon-Nikodym deriv ative ϱ ∈ L 1 is, cf [13], X k ∈N ϱ ( τ k ω ) p − k ( τ k ω ) = X k ∈N ϱ ( ω ) p k ( ω ) . (29) It is v ery natural to exp ect that solving (29) is the first step tow ards a CL T for the generic R WRE. This is a v ery hard equation, there seem to b e no general metho ds to handle it. F ew exceptions (when it is solved) are the one-dimens ional cases studied in [23] where a con tinued fraction expansion is prov ed to work, the b alanc e d rwr e case studied first on [15] where a hands-on solution is constructed for that particular mo del, and the rw in Dirichlet-distribute d r andom envir onment studied in [22] where the solution of equation (29) is constructed via deep relations of this particular mo del with the reinforced random w alk problem. The CL T question is settled (p ositively) in [15] and [23], and left widely op en in [22], and any other case when (29) could p ossibly b e solved. Assume that (29) is solved with ϱ ∈ L 1 and define a new R WRE t 7→ e X ( t ) with jump rates e p k ( ω ) := ϱ ( ω ) p k ( ω ) . (30) Ob viously , the jump rates e p k ( x, ω ) are bisto chastic, as in (1). The random w alks X ( t ) and e X ( t ) differ only in the holding times b etw een the successiv e steps: at site x ∈ Z d they wait for EXP ( P k ∈N p k ( τ x ω )) -, resp ectiv ely for EXP ( P k ∈N e p k ( τ x ω )) -distributed holding times and jump to the neighbouring site x + k with the same probabilities ( P k ∈N p k ( x, ω )) − 1 p k ( x, ω ) = ( P k ∈N e p k ( x, ω )) − 1 e p k ( x, ω ) . So, pro ving the CL T for the w alks X ( t ) and e X ( t ) are related through la ws of large num b ers for the sums of the holding times in b oth cases. How ev er, these LLNs hold due to the ergo dic theorem. So, it remains to establish the CL T for the walk e X ( t ) . That is, to c heck the structural conditions (14), (15) and the in tegrability conditions (19) , (20) , (21) for the jump bisto chastic rates e p k ( x, ω ) defined in (30). The truly difficult part is chec king the structural conditions (14), (15). This is w ork for the future. Given (14), (15), the in tegrability conditions could b e controlled by the appropriate choice of the parameters of the Dirichlet-distributed en vironment. 9 1.5 Related earlier w orks Since random walks and/or diffusions in diver genc e-fr e e random drift field are not merely mathematical to ys, metaphoric so-called "mo dels", but are directly motiv ated by the true ph ysical problem of self-diffusion in turbulent and incompressible steady state flow, these problems hav e a notorious and long history . The historic bac kground in the context of strongly elliptic (cf (16)) en vironmen ts was presen ted in sufficient details in section 1.6 of the survey pap er [29]. Here I just mention the most imp ortan t forerunning works - in c hronological order - are [20], [13], [18], [7], [6], [10], [11], [12], [5], [31], [9], [14], [16], [28]. The list is by no means exhaustiv e. Note, ho w ever, that in all these works str ong el lipticity cf (16) (or, an equiv alent condition for the diffusion settings) is assumed. As for the non-elliptic setting, m uch effort has b een put on relaxing the strong ellip- ticit y condition (16) in the context of pure conductance mo dels, where p k = s k , b k = 0 . In this setting the CL T of Theorem 2 under conditions (19) and (20) has b een established already in [4]. The strongest results in the fully quenc h ed setting (that is, CL T for the random w alk in π -almost all environmen t) hav e b een established in [2] ( d = 2 ) and [1] ( d ≥ 3 ) where the quenc hed CL T is pro ved under the dimension dep endent conditions s k ∈ L p , ( s k ) − 1 ∈ L q , 1 p + 1 q < 2 d − 1 . I ha ve no kno wledge of an y such result with relaxed ellipticit y for the non-reversible cases, where b k ≡ 0 . As seen from [14], [28], and also from this pap er, the extension is b y no means straightforw ard. 1.6 Blueprin t of the pro of and structure of the pap er W e follow the usual route. Based on (23) we decompose the displacement of the random w alker as X ( t ) = X ( t ) − Z t 0 ( ϕ ( η s ) + ψ ( η s )) ds | {z } M ( t ) + Z t 0 ϕ ( η s )) ds | {z } I ( t ) + Z t 0 ψ ( η s )) ds | {z } J ( t ) . (31) Here t 7→ M ( t ) is a martingale with respect to the quenc hed measures P ω ( · ) , with stationary and ergo dic annealed (i.e., w.r.t. the measure P  ·  ) increments. In Section 2, after some functional analytic preliminaries (sections 2.1 and 2.2) we giv e a detailed exp osition of the natural H ± -spaces related to the self-adjoin t part of the infinitesimal generator (section 2.3). Section 3 is devoted to pro ving annealed diffusive b ounds on the displacement t 7→ X ( t ) of the random w alk. W e show that under the conditions (19) and (21) imp osed, the drift fields ϕ, ψ : Ω → R d are (component-wise) in H − (section 3.1). Due to V aradhan’s H − -b ound this also implies an annealed diffusive upp er b ound on the last tw o in tegrals on the right hand side of (31) and thus, on the displacemen t (section 3.1). Relying on (20) a 10 diffusiv e low er b ound is also established (section 3.2). This is subtler than corresp onding argumen ts in the random conductance case. Theorem 2 is pro ved in Section 4. An efficien t martingale appro ximation of the in tegral terms t 7→ I ( t ) + J ( t ) in (31) is done by applying the non-reversible (non-self- adjoin t) v ariant of the Kipnis-V aradhan theorem, cf [25], [8], [27]. How ev er, since there is no natural grading of the infinitesimal generator L acting on the Hilb ert space L 2 , the Graded Sector Condition cannot b e applied. Instead, we emplo y a stronger v ersion of the r elaxe d se ctor c ondition introduced in [8] and already used in [14]. (The general theory – in a somewhat abstract form – is summarized in section 4.1.) In this order w e hav e to prov e that the (un b ounded) op erator formal ly defined as B := S − 1 / 2 AS − 1 / 2 (where S := − ( L + L ∗ ) / 2 , A := − ( L − L ∗ ) / 2 ) mak es sense as a skew-self-adjoint op er ator o ver L 2 . This is pro ved in subsection 4.2, concluding the pro of of the main result. This is the no vel part of the pro of. In [14] the argument for proving B ∗ = − B hea vily relied on Nash’s inequalit y which, on its turn assumes strong ellipticit y (16) of the jump rates. Relaxing strong ellipticity to the m uch weak er condition (8) (in particular, (20) ) required a new de-Nashifie d pro of whic h turns out to b e conceptually simpler than the one in [14]. As a b on us we also obtain existence and uniqueness of the so-called harmonic c o or dinates (section 4.3), th us op ening the w a y to a quenc hed CL T - to b e completed in future w ork. In the App endix we state a Helmholtz-t yp e theorem s hedding ligh t on the problem of when and ho w can a div ergence-free flow/v ector field expressed in divergence form, as in (14)&(15). 2 Spaces and op erators 2.1 Basic spaces and op erators W e define v arious function spaces (o v er (Ω , π ) ) and linear op erators acting on them. With usual abuse w e denote classes of e quivalenc e of π -a.s. equal measurable functions simply as functions. Let the space of sc alar -, ve ctor -, r otation-fr e e ve ctor - and diver genc e-fr e e ve ctor - fields b e L := { f : Ω → R : f is F -measurable } V := { u : Ω → R N : u k ∈ L , u k ( ω ) + u − k ( τ k ω ) = 0 , k ∈ N , π -a.s. } U := { u ∈ V : u k ( ω ) + u l ( τ k ω ) = u l ( ω ) + u k ( τ l ω ) , k , l ∈ N , π -a.s. } . These are linear spaces (ov er R ) with no norm or top ology endow ed on them yet. W e call these spaces these names for the obvious reason that their liftings f ( x, ω ) := f ( τ x ω ) ( f ∈ L ) u ( x, ω ) := u ( τ x ω ) ( u ∈ V ) are translation-wise ergo dic scalar- , resp ectiv ely , vector-fields ov er Z d . 11 The linear op erators ∂ k , R k , H k,l : L → L , k , l ∈ N , defined b elow on the whole space L as their domain, will b e the basic primary ob jects used in constructing more complex op erators. T k f ( ω ) := f ( τ k ω ) , ∂ k := T k − I , R k f ( ω ) := r k ( ω ) f ( ω ) , H k,l f ( ω ) := h k,l ( ω ) f ( ω ) (32) Using these basic op erators we further define ∇ : L → U , ( ∇ f ) k := ∂ k f ∇ ∗ : V → L , ∇ ∗ u := X k ∈N u k = − 1 2 X k ∈N ∂ − k u k R : V → V , ( Ru ) k := R k u k (33) H : V → V , ( H u ) k := 1 2 X l ∈N H k,l ( T k + I ) u l (34) = 1 4 X l ∈N ( T − l + I ) H k,l ( T k + I ) u l These op erators are w ell defined on the whole spaces giv en as their resp ective domains. F or the time b eing the sup erscript ∗ is only notation. It will later indicate adjunction with resp ect to the inner pro ducts defined in (38) b elow. On the righ t hand side of (34) the factors ( T − l + I ) / 2 and ( T k + I ) / 2 take care of pro jecting bac k to V . Their necessity and role is a consequence of the spatial discreteness of Z d . One can easily c heck that for any v ∈ V : X k ∈N s k v k = ∇ ∗ R 2 v , (35) for any u ∈ U : X k ∈N b k u k = ∇ ∗ H u. (36) The identit y (35) is straightforw ard. W e chec k (36):  ∇ ∗ H u  ( ω ) = 1 2 X k,l ∈N h k,l ( ω )  u l ( τ k ω ) + u l ( ω )  = 1 4 X k,l ∈N h k,l ( ω )  u l ( τ k ω ) + u l ( ω ) − u k ( τ l ω ) − u k ( ω )  = 1 4 X k,l ∈N h k,l ( ω )  2 u l ( ω ) − 2 u k ( ω )  = X k,l ∈N h k,l ( ω ) u l ( ω ) = X l ∈N b l ( ω ) u l ( ω ) 12 where we hav e used, in turn, the definition (34) of the op erator H , the skew-symmetry (14) of the tensor h k,l ( ω ) , the gr adient pr op erty of u ∈ U , and the skew-symmetry (14) again. Note that the identit y (36) holds only for u ∈ U and not for v ∈ V \ U . T o add to the confusion w e note here that the spaces K := R U = { ( r k u k ) k ∈N : u ∈ U } ⊂ V will also b e relev an t in the forthcoming argumen ts. Using (35) and (36) the Hermitian and anti-Hermitian parts of the infinitesimal gen- erator L , defined in (9) are written as S = ∇ ∗ R 2 ∇ = ( ∇ ∗ R )( R ∇ ) A = ∇ ∗ H ∇ = ( ∇ ∗ R )( R − 1 H R − 1 )( R ∇ ) (37) W e will w ork in the follo wing (Riesz-)Leb esgue spaces L p := { f ∈ L : ∥ f ∥ p p := Z Ω | f | p dπ < ∞ , Z Ω f dπ = 0 } V p := { u ∈ V : ∥ u ∥ p p := 1 2 X k ∈N ∥ u k ∥ p p < ∞} U p := { u ∈ U : ∥ u ∥ p p < ∞ , Z Ω u dπ = 0 } with p ∈ { 1 , 2 , ∞} , and the usual interpretation for p = ∞ . Note, that only centred (zero mean) functions are kept in the spaces L p and U p . How ev er, in the space V p this is not the case. F or ( p, q ) ∈ { (1 , ∞ ) , (2 , 2) , ( ∞ , 1) } , f ∈ L p and g ∈ L q , resp ectively , u ∈ V p and v ∈ V q , we define the scalar pro ducts ⟨ f , g ⟩ := Z Ω f ( ω ) g ( ω ) dπ ( ω ) , ⟨ u, v ⟩ := 1 2 X k ∈N ⟨ u k , v k ⟩ . (38) W e don’t in tro duce differen t notation for the norms and scalar pro ducts in L p , resp ec- tiv ely , V p . The precise meaning of ∥·∥ p and ⟨· , ·⟩ will b e alwa ys clear from the context. Finally , let K 2 := ( R U ∞ ) cl 2 (19) ⊂ V 2 (39) where the sup erscript cl 2 denotes closure in ( V 2 , ∥·∥ 2 ) . W e will denote by Π : V 2 → K 2 the orthogonal pro jection from V 2 to K 2 , see (51). The op erators ∂ k : L 2 → L 2 , ∇ : L 2 → V 2 , ∇ ∗ : V 2 → L 2 are b ounded, and their adjoin tness relations (with resp ect to the scalar pro ducts (38)) are ob viously ∂ ∗ k = ∂ − k ( ∇ ) ∗ = ∇ ∗ . The op erators R k , H k,l and R, H defined in (32), (33), and (34), when restricted to L 2 , resp ectively , to V 2 , are unb ounde d with respect to the norms ∥·∥ 2 . Ho wev er, as m ultiplication op erators there is no issue with their prop er definition as densely defined self-adjoin t, resp ectiv ely , skew-self-adjoin t op erators: R k = R ∗ k , H ∗ k,l = H k,l R = R ∗ H ∗ = − H . 13 2.2 The infinitesimal generator - acting on L 2 The Hermitian and an ti-Hermitian parts (w.r.t. the stationary measure π ) of the in- finitesimal generator L = − S + A , given in (9) (or, equiv alen tly , in (37)) are well defined on the whole space L of measurable functions. Ho wev er, their status as (unbounded) linear op erators acting on the Hilb ert space L 2 needs clarification. A ctually , they are prop erly defined as (unbounded) self-adjoin t, resp ectiv ely , sk ew-self-adjoin t op erators, and moreov er, they share a common core of definition. F or K < ∞ , let Ω K := { ω ∈ Ω : max k,l ,m ∈N { r k ( ω ) ± 1 , r k ( τ m ω ) ± 1 , | h k,l ( ω ) | , | h k,l ( τ m ω ) |} ≤ K } (40) and A := { f ∈ L ∞ : ∃ K < ∞ : supp f ⊂ Ω K π -a.s. } . (41) Ob viously , K 7→ Ω K exhausts increasingly Ω , as K → ∞ . Hence, for p ∈ { 1 , 2 } , A ⊂ L ∞ ⊂ L p = A cl p . It is also straightforw ard that S, A : A → L ∞ , and, for f , g ∈ A , ⟨ f , S g ⟩ = ⟨ S f , g ⟩ , ⟨ f , S f ⟩ > 0 , ⟨ f , Ag ⟩ = −⟨ Af , g ⟩ . The pro of of the following statement is routine which we omit. Prop osition 3 (The infinitesimal generator) . Assume (8) (and no mor e). The line ar op er ators S and A in (9) (or, e quivalently, in (37) ) acting on L 2 ar e essential ly self-adjoint, r esp e ctively, essential ly skew-self-adjoint on their c ommon c or e A define d in (41) . Mor e over, S > 0 , as a self-adjoint op er ator. Remark. F or the purp ose of Prop osition 3, defining the common core A in (41), with the simpler and seemingly more natural c hoice Ω K := { ω ∈ Ω : max k,m ∈N { r k ( ω ) , r k ( τ m ω ) } ≤ K } w ould suffice. How ev er, later, in section 4 we will need the more restrictiv e c hoice (40). 2.3 H + , H − , Riesz op erators, isometries W e collect the basic functional analytic facts ab out the H ± spaces and Riesz op erators related to the self-adjoint part S of the infinitesimal generator. In subsection 2.3.1 w e collect textb o ok material ab out the H ± spaces of any p ositive op erator S o ver a Hilb ert space H , while in subsection 2.3.2 we collect those facts whic h are v alid in our sp ecific setting. 14 2.3.1 Abstract setting Let S = S ∗ > 0 (note the strict inequalit y!) b e a (p ossibly unbounded) p ositive op erator acting on an (abstract) separable Hilb ert space ( H , ∥·∥ ) . (In our concrete setting H = L 2 and S = − ( L + L ∗ ) / 2 is the self-adjoint part of the infinitesimal generator L , giv en explicitly in (37).) Since S = S ∗ > 0 , the self-adjoint and p ositive (unbounded) op erators S − 1 , S 1 / 2 and S − 1 / 2 are defined through the Sp ectral Theorem. Note that Dom( S ) = Ran( S − 1 ) , Ran( S ) = Dom( S − 1 ) , Dom( S 1 / 2 ) = Ran( S − 1 / 2 ) , Ran( S 1 / 2 ) = Dom( S − 1 / 2 ) are dense subspaces in ( H , ∥·∥ ) and define the Hilb ert spaces H + :=  f ∈ Dom( S ) : ∥ f ∥ 2 + := ⟨ f , S f ⟩  cl + (42) H − :=  f ∈ Dom( S − 1 ) : ∥ f ∥ 2 − := ⟨ f , S − 1 f ⟩  cl − (43) The sup erscripts cl ± denote closure with resp ect to the Euclidean norm ∥·∥ ± , resp ectively . Th us, ( H + , ∥·∥ + ) and ( H − , ∥·∥ − ) are complete Hilb ert spaces. Obviously , if ∥ S ∥ H→H < ∞ then H − ⊂ H ⊂ H + , and lik ewise, if ∥ S − 1 ∥ H→H < ∞ then H + ⊂ H ⊂ H − . How ever, in most relev ant cases ∥ S ∥ H→H = ∞ = ∥ S − 1 ∥ H→H and neither of the ab o v e subspace- inclusions hold. Prop osition 4 holds true in full generality , in this abstract setting. These are standard facts a v ailable in any textb o ok on functional analysis, prov en with the use of the Sp ectral Theorem. See, e.g., c hapter VI I I in [21], and/or section 2.2. in [9]. The pro of of this prop osition is textb o ok material which we omit. The (unbounded) p ositive op erators S 1 / 2 , S − 1 / 2 are defined in terms of the Sp ectral Theorem. Prop osition 4 ( H ± spaces, general abstract setting) . (i) H + ∩ H = Dom( S 1 / 2 ) and for f ∈ H + ∩ H , ∥ f ∥ + = ∥ S 1 / 2 f ∥ , H − ∩ H = Ran( S 1 / 2 ) and for f ∈ H − ∩ H , ∥ f ∥ − = ∥ S − 1 / 2 f ∥ . (ii) The fol lowing variational formulae hold: for f ∈ H ∥ f ∥ 2 + = sup g ∈H − ∩H  2 ⟨ f , g ⟩ − ∥ g ∥ 2 −  = sup g ∈ Ran( S )  2 ⟨ f , g ⟩ − ⟨ g , S − 1 g ⟩  = sup g ∈ Dom( S ) ⟨ 2 f − g , S g ⟩ ∥ f ∥ 2 − = sup g ∈H + ∩H  2 ⟨ f , g ⟩ − ∥ g ∥ 2 +  = sup g ∈ Dom( S )  2 ⟨ f , g ⟩ − ⟨ g , S g ⟩  = sup g ∈ Ran( S ) ⟨ 2 f − g , S − 1 g ⟩ (44) (iii) On H + ∩ H , r esp e ctively, on H − ∩ H , define the Euclide an norms | ∥·∥ | 2 + := ∥·∥ 2 + + ∥·∥ 2 , | ∥·∥ | 2 − := ∥·∥ 2 − + ∥·∥ 2 . 15 Then, ( H + ∩ H , | ∥·∥ | + ) and ( H − ∩ H , | ∥·∥ | − ) ar e (c omplete) Hilb ert sp ac es, and ( H + ∩ H , | ∥·∥ | + ) S 1 / 2 − → ( H − ∩ H , | ∥·∥ | − ) S − 1 / 2 − → ( H + ∩ H , | ∥·∥ | + ) ar e isometries b etwe en them. (iv) The fol lowing cycle of isometries holds. ( H , ∥·∥ ) S − 1 / 2 − → ( H + , ∥·∥ + ) S 1 / 2 − → ( H , ∥·∥ ) S 1 / 2 − → ( H − , ∥·∥ − ) S − 1 / 2 − → ( H , ∥·∥ ) (45) 2.3.2 Concrete setting In Prop osition 5 w e collect those facts ab out the spaces H ± whic h are sp ecific to our particular setting, rely on the concrete realization H = L 2 of the basic Hilb ert space, and on the sp ecial form (37) of the op erator S = S ∗ > 0 in our setting. Prop osition 5 ( H ± spaces, concrete setting) . Assume (19) (and nothing mor e). (i) L ∞ ⊂ H + = L cl + ∞ , and for f ∈ L ∞ , ∥ f ∥ 2 + = ∥ R ∇ f ∥ 2 2 . (46) (ii) H − ⊂ L 1 = H cl 1 − , and for f ∈ L 1 , ∥ f ∥ 2 − = sup g ∈L ∞  2 ⟨ f , g ⟩ − ∥ R ∇ g ∥ 2 2  ≤ ∞ , (47) with the interpr etation of ∥ f ∥ − = ∞ as f ∈ L 1 \ H − . (iii) ∇ ∗ R V 2 = ∇ ∗ R K 2 ⊂ H − = ( ∇ ∗ R K 2 ) cl − , and for v ∈ V 2 , ∥∇ ∗ Rv ∥ 2 − = ∥ Π v ∥ 2 2 . (48) (iv) The fol lowing cycle of Hilb ert sp ac e isometries hold ( L 2 , ∥·∥ 2 ) S − 1 / 2 − → ( H + , ∥·∥ + ) R ∇ − → ( K 2 , ∥·∥ 2 ) ∇ ∗ R − → ( H − , ∥·∥ − ) S − 1 / 2 − → ( L 2 , ∥·∥ 2 ) (49) Remarks: (1) Note the difference b etw een the middle terms in (45) and (49). The main p oin t is that the abstract and non-c omputable op erator S 1 / 2 is substituted b y the explicitly c omputable R ∇ and ∇ ∗ R , and, accordingly , in the middle of the chain the Hilb ert space ( L 2 , ∥·∥ 2 ) is replaced b y ( K 2 , ∥·∥ 2 ) . 16 (2) ∇ ∗ R : ( V 2 , ∥·∥ 2 ) → ( H − , ∥·∥ − ) is actually a p artial isometry with Ker( ∇ ∗ R ) = K ⊥ 2 . W e will denote Λ := R ∇ S − 1 / 2 : L 2 → K 2 , Λ ∗ := S − 1 / 2 ∇ ∗ R : V 2 → L 2 , (50) and refer to them as the Riesz op er ators , due to their apparent analogy to the Riesz op erators ( grad | ∆ | − 1 / 2 , resp ectiv ely , | ∆ | − 1 / 2 div · ) of harmonic analysis. W e ob viously ha ve Λ ∗ Λ = I L 2 , Π := Λ Λ ∗ : V 2 → K 2 , (51) the latter one b eing the orthogonal pro jection from V 2 to K 2 . Pr o of of Pr op osition 5. (i) Due to (19) , for f ∈ L ∞ w e ha v e S f = ∇ ∗ RR ∇ f ∈ L 1 , R ∇ f ∈ K 2 , and the following iden tities are legitimate: ∥ f ∥ 2 + = ⟨ f , ∇ ∗ RR ∇ f ⟩ = ⟨ R ∇ f , R ∇ f ⟩ = ∥ R ∇ f ∥ 2 2 It follows that L ∞ ⊂ H + ∩ L 2 = Dom( S 1 / 2 ) and (46) holds. In order to pro ve H + = L cl + ∞ note that, for f ∈ L ∞ and g ∈ Dom( S 1 / 2 ) , ⟨ f , g ⟩ + = ⟨ S 1 / 2 f , S 1 / 2 g ⟩ . Hence (since L 2 = Ran( S 1 / 2 ) cl 2 ) ⟨ f , g ⟩ + = 0 for all g ∈ Dom( S 1 / 2 ) implies S 1 / 2 f = 0 , and thus f = 0 . (ii) F rom (44), L ∞ ⊂ H + ∩ H and (46) we get, for f ∈ L 1 ∥ f ∥ 2 − ≥ sup g ∈L ∞  2 ⟨ f , g ⟩ − ∥ R ∇ g ∥ 2 2  ≥ sup g ∈L ∞  2 ⟨ f , g ⟩ − 4 X k ∥ r k ∥ 2 2 ∥ g ∥ 2 ∞  =  4 X k ∥ r k ∥ 2 2  − 1 ∥ f ∥ 2 1 and hence, due to (19) , H − ⊂ L 1 . L 1 = H cl 1 − follo ws from L ∞ ⊂ H + and Hahn-Banach. (iii) T o prov e (48) let v ∈ V 2 and apply (47) to ∇ ∗ Rv (19) ∈ L 1 : ∥∇ ∗ Rv ∥ 2 − = sup g ∈L ∞  2 ⟨∇ ∗ Rv , g ⟩ − ∥ R ∇ g ∥ 2 2  = sup g ∈L ∞  2 ⟨ v , R ∇ g ⟩ − ∥ R ∇ g ∥ 2 2  = ∥ Π v ∥ 2 2 . Ab o ve, the middle step is legitimate since g ∈ L ∞ , v ∈ K 2 , ∇ ∗ Rv (19) ∈ L 1 , and R ∇ g (19) ∈ L 2 . The last step holds since K 2 = ( R ∇L ∞ ) cl 2 , by definition (39). T o pro ve H − = ( ∇ ∗ R K 2 ) cl − let g ∈ L ∞ b e such that for all v ∈ K 2 0 = ⟨∇ ∗ Rv , g ⟩ = ⟨ v , R ∇ g ⟩ . Then g = 0 and hence, since H + = L cl + ∞ , it follows that H − = ( ∇ ∗ R K 2 ) cl − (iv) W e only hav e to prov e the middle terms in (49). F rom Prop osition 5 (i) readily follo ws that R ∇ maps isometrically ( H + , ∥·∥ + ) into ( K 2 , ∥·∥ 2 ) . T o prov e that it is also surje ctive assume that v ∈ K 2 is such that for all g ∈ L ∞ 0 = ⟨ v , R ∇ g ⟩ = ⟨∇ ∗ Rv , g ⟩ , and thus, due to ∇ ∗ R K 2 ⊂ H − ⊂ L 1 , we get ∇ ∗ Rv = 0 , and hence, v = 0 due to (48). 17 3 Diffusiv e b ounds In this section w e pro ve annealed diffusiv e b ounds for the displacemen t of the random w alker. 3.1 Upp er Lemma 6 ( H − -b ounds on drifts) . Assume (19) and (21) . Then, ϕ i , ψ i ∈ H − , i = 1 , . . . , d, (52) wher e ϕ i and ψ i ar e the c omp onents of the drift-fields define d in (25) . Pr o of. W e rely on (47) from Prop osition 5 (ii), . ∥ ∂ − k s k ∥ 2 − = sup χ ∈L ∞  2 ⟨ χ, ∂ − k s k ⟩ − ∥ R ∇ χ ∥ 2 2  ≤ sup χ ∈L ∞  2 ⟨ χ, ∂ − k s k ⟩ − ∥ R k ∂ k χ ∥ 2 2  = sup χ ∈L ∞  2 ⟨ R k ∂ k χ, r k ⟩ − ∥ R k ∂ k χ ∥ 2 2  ≤ sup χ ∈L 2  2 ⟨ χ, r k ⟩ − ∥ χ ∥ 2 2  = ∥ r k ∥ 2 2 (19) < ∞ ∥ b k ∥ 2 − = sup χ ∈L ∞  ⟨ χ, X l ∈N ∂ − l h k,l ⟩ − ∥ R ∇ χ ∥ 2 2  = sup χ ∈L ∞ X l ∈N  ⟨ R l ∂ l χ, r − 1 l h k,l ⟩ − ∥ R l ∂ l χ ∥ 2 2  ≤ X l ∈N sup χ ∈L 2  ⟨ χ, r − 1 l h k,l ⟩ − ∥ χ ∥ 2 2  = X l ∈N ∥ r − 1 l h k,l ∥ 2 2 (21) < ∞ . Due to (25), from the previous computations we readily get d X i =1 ∥ ϕ i ∥ 2 − ≤ X k ∈N ∥ r k ∥ 2 2 (19) < ∞ , d X i =1 ∥ ψ i ∥ 2 − ≤ X k,l ∈N ∥ r − 1 l h k,l ∥ 2 2 (21) < ∞ , whic h pro ves the claims in (52). Prop osition 7 (Diffusiv e upp er b ound) . Assume (19) and (21) . Then, lim sup t →∞ t − 1 E  | X ( t ) | 2  < ∞ (53) Pr o of. W e use martingale decomp osition (31) and provide diffusive upp er b ounds sepa- rately for the three terms on the right hand side. The first term t 7→ M ( t ) ∈ R d is a martingale whose conditional co v ariance (brac k et) pro cess is ⟨ M i , M j ⟩ ( t ) = δ i.j ( p e i ( η s ) + p − e i ( η s )) 18 Hence, in the annealed setting, t − 1 E  | M ( t ) | 2  = X k ∈N Z Ω s k ( ω ) dπ ( ω ) = X k ∈N Z Ω r 2 k ( ω ) dπ ( ω ) (19) < ∞ (54) On the other hand, recalling V aradhan’s H − -b ound (cf. [9], Theorem 2.7) from (52) (whic h relies on (19) and (21) ) we conclude sup 0 0 , k ∈ N , define d in (56) b elow. Pr o of. W e m ust start with some notation. Let s k :=  Z Ω s k ( ω ) − 1 dπ ( ω )  − 1 (20) > 0 (56) and note that, due to (10), s k = s − k . F urther, let α, β : Ω → R d , α ( ω ) := X k ∈N p k ( ω ) s k s k ( ω ) − 1 k = X k ∈N s k b k ( ω ) s k ( ω ) − 1 k β ( ω ) := ϕ ( ω ) + ψ ( ω ) − α ( ω ) (see (24) for the definition of ϕ, ψ : Ω → R d ). W e denote θ 0 := 0 , θ i +1 := inf { t > θ i : lim ε ↘ 0 ( X ( t + ε ) − X ( t − ε ))  = 0 } ξ i := lim ε ↘ 0 ( X ( θ i + ε ) − X ( θ i − ε ))  = 0 } ∈ N η − i := η θ − i , η + i := η θ + i . 19 In plain w ords, θ i , i ≥ 1 , is the time of the i -th jump of the contin uous time nearest neigh b our w alk walk t 7→ X ( t ) , ξ i ∈ N is the i -th jump and η − i / η + i is the en vironment seen righ t b efore/after the i -th jump. Finally , we define the c.a.d.l.a.g. pro cesses t 7→ Z ( t ) ∈ R d and t 7→ Y ( t ) ∈ R d Z ( t ) := X i :0 <θ i 0 E  Z ( t 1 ) Y ( t 2 )  = 0 . (62) F urther on, from t 7→ Z ( t ) b eing a forwar d-and-b ackwar d martingale it follows that for an y t 1 , t 2 > 0 E  Z ( t 1 )( I ( t 2 ) + J ( t 2 )  = 0 . (63) 21 Finally from (60), (62), (63) we obtain E  X ( t ) ∧ X ( t )  ≥ E  Z ( t ) ∧ Z ( t )  = t X k ∈N s k k ∧ k where the first inequalit y is meant in the d × d -matrix sense and the last iden tit y follo ws from (61). 4 Pro of of Theorem 2 4.1 Kipnis-V aradhan theory - the abstract form The pro of of Theorem 2 is based on the non-reversible (i.e. non-self-adjoin t) and non- graded v ersion of martingale appro ximation a la Kipnis-V aradhan, whose abstract form is summarized concisely in this section. Let (Ω , F , π ) b e a probability space and t 7→ η t ∈ Ω a Mark ov pro cess assumed to b e stationary and ergo dic under the probabilit y measure π , whose infinitesimal generator L and resolven t R λ := ( λI − L ) − 1 act on the Leb esgue spaces L p := { f ∈ L p (Ω , π ) : Z Ω f dπ = 0 } , p ∈ { 1 , 2 , ∞} . W e assume that the infinitesimal generator acting on L 2 decomp oses as L = − S + A, S := − ( L + L ∗ ) / 2 , A := ( L − L ∗ ) / 2 , meaning that the symmetric and antisymmetric parts, S , resp ectiv ely , A , ha ve a common subspace of definition A ⊂ L 2 whic h serves as a core for their closure as (un b ounded) linear op erators acting on L 2 . Moreov er, w e assume that S, A : A → L 2 are essentially self-adjoin t, resp ectively , essen tially sk ew-self-adjoint. W e assume that the self-adjoint part S is ergo dic on its own, that is its eigenv alue 0 is non-degenerate. W e also define the self-adjoin t op erators S 1 / 2 , S − 1 / 2 , in terms of the Sp ectral Theorem, and the Hilb ert spaces ( H + , ∥·∥ + ) , ( H − , ∥·∥ − ) as in (42), (43). W e quote the Kipnis-V aradhan martingale appro ximation in the non-self-adjoint set- ting in the form stated in [25, 8, 27]. See the monograph [9] for historic background. Theorem 9 ([25, 8, 27] Theorem KV) . L et φ ∈ L 1 such that for al l λ > 0 , R λ φ ∈ L 2 . If the fol lowing two c onditions hold lim λ → 0 λ 1 / 2 ∥ R λ φ ∥ 2 = 0 , lim λ → 0 ∥ S 1 / 2 R λ φ − v ∥ 2 = 0 , v ∈ L 2 , (64) then ther e exists a squar e inte gr able martingale t 7→ Z ( t ) , with stationary and er go dic incr ements, adapte d to the natur al filtr ation ( F t ) 0 ≤ t< ∞ of the Markov pr o c ess t 7→ η t , and with varianc e E  | Z ( t ) | 2  = 2 ∥ v ∥ 2 2 t, 22 such that lim t →∞ t − 1 E  | Z t 0 φ ( η s ) ds − Z ( t ) | 2  = 0 . (65) Remarks, comments: (1) In the common traditional form ulations of this or equiv alen t theorems (including, e.g., [25], [19], [9], [8], [27], [29]) it is assumed that φ ∈ L 2 . How ev er, as noted already in [4] for the reversible/self-adjoin t setting, the argumen ts w ork neatly under the weak er assumption φ ∈ L 1 and R λ φ ∈ L 2 , for all λ > 0 . (2) By general abstract arguments, for all λ > 0 ∥ R λ ∥ 2 → 2 ≤ λ − 1 , ∥ S 1 / 2 R λ ∥ 2 → 2 ≤ λ − 1 / 2 , ∥ R λ S 1 / 2 ∥ 2 → 2 ≤ λ − 1 / 2 , ∥ S 1 / 2 R λ S 1 / 2 ∥ 2 → 2 ≤ 1 , and hence, for φ ∈ H − w e ha ve a priori sup λ> 0 λ 1 / 2 ∥ R λ φ ∥ 2 < ∞ , sup λ> 0 ∥ S 1 / 2 R λ φ ∥ 2 < ∞ , still somewhat short of (64). (3) Conditions (64) of Theorem 9 are difficult to chec k directly . (An exception, where this could b e done, is the R WRE problem treated in [25].) Sufficient conditions are known under the names of Strong Sector Condition, resp ectiv ely , Graded Sector Condition. See the monograph [9] for con text and details. Ho wev er, these sector conditions hold only under v ery sp ecial structural assumptions ab out the Marko v pro cess considered: a graded structure of the infinitesimal generator L acting on an accordingly graded Hilb ert space L 2 . This structural assumption simply do esn’t hold in man y cases of in terest, including our current problem. The next theorem, quoted from [8], provides a sufficien t condition whic h do es not assume a graded structure of the infinitesimal generator L acting on the Hilbert space L 2 (Ω , π ) . Let B := { f ∈ H − ∩ L 2 : S − 1 / 2 f ∈ Dom( A ) , AS − 1 / 2 f ∈ H − ∩ L 2 } (66) and B : B → L 2 defined as B f := S − 1 / 2 AS − 1 / 2 f . (67) Note that the op erator B : B → L 2 is un b ounded (except for the cases when the op erator S : L 2 → L 2 is b oundedly inv ertible) and skew symmetric. Indeed, for f , g ∈ B all the straigh tforward steps b elow are legitimate ⟨ f , S − 1 / 2 AS − 1 / 2 g ⟩ = ⟨ S − 1 / 2 f , AS − 1 / 2 g ⟩ = −⟨ AS − 1 / 2 f , S − 1 / 2 g ⟩ = −⟨ S − 1 / 2 AS − 1 / 2 f , g ⟩ Of course, it could happ en that the subspace B is not dense in L 2 , or, even worse, that simply B = { 0 } . Even if B is a dense subspace in L 2 , in principle it could still happ en that the op erator B (whic h in this case is densely defined and sk ew-symmetric) is not essen tially sk ew-self-adjoint. 23 Theorem 10 ([8] Theorem 1) . Assume that ther e exists a subsp ac e C ⊆ B ⊂ L 2 = C cl 2 , and the op er ator B : C → L 2 is essential ly skew-self-adjoint (that is, B = − B ∗ ). Then for any φ ∈ H − ∩ L 1 the c onditions of The or em 9 (and henc e the martingale appr oximation (65) ) hold. Remarks: (1) In [8] the theorem is form ulated in slightly differen t terms. How ev er, it is easy to see that this form follows directly from that of Theorem 1 in [8]. (2) The conditions of Theorem 10 are equiv alent to B ⊂ L 2 = B cl 2 and B : B → L 2 essen tially sk ew-self-adjoint. The formulation of the theorem allows flexibility in choosing the core C ⊆ B . (3) In view of Prop osition 4 (iv) (45), the requirement that B := S − 1 / 2 AS − 1 / 2 b e a w ell- defined sk ew-self-adjoint op erator acting on the Hilb ert space ( L 2 , ∥·∥ 2 ) is equiv alent to A = ( L − L ∗ ) / 2 b e a w ell-defined sk ew-self-adjoint op erator acting on the Hilb ert space ( H + , ∥·∥ + ) . (4) The condition stated in Prop osition 2.1.2 of [19] is precisely a somewhat hidden form of von Neumann’s criterion for (sk ew-)self-adjointness (cf. [21] Theorem VI I I.3) of the op erator B defined ab ov e. The form ulation in Theorem 10 allows for chec king skew-self- adjoin tness in v arious other wa ys, as demonstrated in the forthcoming pro of of our main result. 4.2 Pro of of Theorem 2 W e c heck the conditions of Theorem 10 for the concrete case under consideration, when the op erators L , S = − ( L + L ∗ ) / 2 , and A = ( L − L ∗ ) / 2 given in (4), (9), (37) are the infinitesimal generator of the environmen t pro cess t 7→ η t cf. (3), acting on the Hilb ert space L 2 , and its self-adjoin t and skew-self-adjoin t parts. In this case the subspace B of (66) is B := { f ∈ H − ∩ L 2 : S − 1 / 2 f ∈ Dom( ∇ ∗ H ∇ ) , ∇ ∗ H ∇ S − 1 / 2 f ∈ H − ∩ L 2 } and the op erator B : B → L 2 of (67) acts as B f := S − 1 / 2 ∇ ∗ H ∇ S − 1 / 2 f = Λ ∗ R − 1 H R − 1 Λ f = Λ ∗ Π R − 1 H R − 1 ΠΛ f Recall from (41) the common core A of the op erators S and A , and (noting that A ⊂ Dom( S ) ⊂ Dom( S 1 / 2 ) ) let C := { f = S 1 / 2 g : g ∈ A} The pro of of essen tial sk ew-self-adjointness of the op erator B : C → L 2 relies on v arious parts of Prop osition 5, thus only on the in tegrability condition (19) . Ob viously , C ⊆ H − ∩ L 2 and since A is a core for S and Ker( S ) = Ran( S ) ⊥ = { 0 } , w e also hav e L 2 = C cl 2 . F or f ∈ C , the equation f = S 1 / 2 g determines uniquely g ∈ A ⊂ L ∞ . 24 F urthermore, ∇ ∗ H ∇ S − 1 / 2 f = ∇ ∗ R R − 1 H ∇ g |{z} ∈A | {z } ∈V ∞ ∈ H − ∩ L 2 . The last step is due to Prop osition 5 (iii). Thus, indeed, C ⊂ B ⊂ L 2 = C cl 2 , where the sup erscript cl 2 denotes closure with resp ect to the norm ∥·∥ 2 . Prop osition 11 (Sk ew-self-adjointness of S − 1 / 2 AS − 1 / 2 ) . The line ar op er ator B : C → L 2 is essential ly skew-self-adjoint. Pr o of. Let D := Λ C = { R ∇ g : g ∈ A} ⊂ K 2 = D cl 2 and define the op erator D : D → K 2 as D := Λ B | C Λ ∗ . That is, D := Π R − 1 H R − 1 Π , D u := Π R − 1 H R − 1 u |{z} ∈D | {z } ∈∇A | {z } ∈V ∞ | {z } ∈K 2 , (68) where Λ , Λ ∗ , Π are the Riesz op erators and orthogonal pro jection defined in (50), (51). Since Λ : L 2 → K 2 , Λ ∗ : K 2 → L 2 are isometries , the prop osition will follo w from pro ving that the op erator D : D → K 2 is essentially sk ew-self-adjoint. This is what w e are going to pro ve. The op erator D : D → K 2 is sk ew symmetric. Indeed, from the choice of the core A is follo ws that u ∈ D ⊂ K ∞ readily implies that R − 1 u ∈ ∇A ⊂ U ∞ , H R − 1 u ∈ V ∞ , R − 1 H R − 1 u ∈ V ∞ also hold, and thus all steps of th e following chain are legitimate (not merely formal) ⟨ u, R − 1 H R − 1 v ⟩ = ⟨ R − 1 u, H R − 1 v ⟩ = −⟨ H R − 1 u, R − 1 v ⟩ = −⟨ R − 1 H R − 1 u, v ⟩ Next we define the adjoin t (ov er the Hilb ert space K 2 ) of D : D → K 2 . Its domain is D ∗ := { w ∈ K 2 : ∃ c = c ( w ) < ∞ : ∀ u ∈ D : |⟨ w , R − 1 H R − 1 u ⟩| ≤ c ∥ u ∥ 2 } , and D ∗ : D ∗ → K 2 is defined uniquely b y the Riesz Lemma: for any w ∈ D ∗ , D ∗ w is the unique element of K 2 suc h that for all u ∈ D ⟨ D ∗ w , u ⟩ = ⟨ w , R − 1 H R − 1 u ⟩ . 25 Ob viously , D ⊂ D ∗ , D ∗ | D = − D , and D ≺ D ∗∗ ⪯ − D ∗ In order to conclude D ∗∗ = − D ∗ and th us essential skew-self-adjoin tness of D , as defined in (68), it is sufficient to pro v e that D ∗ is skew-symmetric on D ∗ . F or K < ∞ , let r K k ( ω ) := r k ( ω ) 1 ( K − 1 ≤ r k ( ω ) ≤ K ) , h K k,l ( ω ) := h k,l ( ω ) 1 ( | h k,l ( ω ) | ≤ K ) . These truncated functions inherit the conductance symmetries (6) and stream-tensor (an ti)symmetries (14). W e define the b ounde d op erators R K , ( R K ) − 1 , H K : V 2 → V 2 b y the formulas (32), (33), (34), with the functions r k , h k,l replaced by their truncated v ersions r K k , h K k,l . Lemma 12. F or w ∈ D ∗ D ∗ w = − w lim K →∞ Π ( R K ) − 1 H K ( R K ) − 1 w , (69) wher e wlim denotes we ak limit in the Hilb ert sp ac e ( K 2 , ∥·∥ 2 ) . Pr o of. This is straigh tforward. Let w ∈ D ∗ and u ∈ D . Then − lim K →∞ ⟨ ( R K ) − 1 H K ( R K ) − 1 w , u ⟩ = lim K →∞ ⟨ w , ( R K ) − 1 H K ( R K ) − 1 u ⟩ = ⟨ w , R − 1 H R − 1 u ⟩ = ⟨ D ∗ w , u ⟩ . The first step is legitimate, since for K < ∞ , the op erator ( R K ) − 1 H K ( R K ) − 1 is b ounded and sk ew-self-adjoint. The second step follows from the fact that (due to the choice of the core A ) for an y u = R ∇ g , g ∈ A , there exists a K 0 < ∞ , such that for any K > K 0 , ( R K ) − 1 H K ( R K ) − 1 u = R − 1 H R − 1 u . Finally , since K 2 = D cl 2 , (69) follows. F rom (69) the skew-symmetry of the op erator D ∗ : D ∗ → K 2 drops out: for u, w ∈ D ∗ ⟨ w , D ∗ u ⟩ = lim K →∞ ⟨ w , ( R K ) − 1 H K ( R K ) − 1 u ⟩ = − lim K →∞ ⟨ ( R K ) − 1 H K ( R K ) − 1 w , u ⟩ = −⟨ D ∗ w , u ⟩ . This concludes the pro of of essential skew-self-adjoin tness of the op erator D : D → K 2 . W e also conclude essential sk ew-self-adjointness of the op erator B = Λ ∗ D Λ on the core C = Λ ∗ D . This also concludes chec king all conditions of Theorem 10 in the concrete setting and th us also the pro of of Theorem 2. 26 4.3 Bon us: Harmonic co ordinates - existence and uniqueness Prop osition 13 (Existence and uniqueness of harmonic co ordinates) . Assume (19) and (20) . Given φ ∈ H − ther e exists a unique solution w ∈ R − 1 K 2 (20) ⊂ U 1 (that is: inte gr able gr adient field) of the e quation X k ∈N ( s k ( ω ) + b k ( ω )) | {z } p k ( ω ) w k ( ω ) = φ ( ω ) . (70) Pr o of. W e search for a solution w ∈ R − 1 K 2 ⊂ U 1 . Using (35) and (36) we write the equation (70) as ∇ ∗ ( R 2 + H ) w = φ, and w = R − 1 Λ g , g ∈ L 2 . W riting the equation for g ∈ L 2 w e obtain S 1 / 2 Λ ∗ ( I + R − 1 H R − 1 )Λ g = φ. Since it is assumed that φ ∈ H − w e can multiply the equation from the left with S − 1 / 2 and get ( Λ ∗ Λ |{z} I L 2 + Λ ∗ R − 1 H R − 1 Λ | {z } B ) g = S − 1 / 2 φ Finally , since B = − B ∗ it follows that I + B is inv ertible (ov er L 2 ) with ∥ ( I + B ) − 1 ∥ 2 → 2 ≤ 1 , yielding the solution w = R − 1 Λ ( I + B ) − 1 S − 1 / 2 φ |{z} ∈H − | {z } ∈L 2 | {z } ∈L 2 | {z } ∈K 2 | {z } ∈ R − 1 K 2 ⊂U 1 , and hence ∥ w ∥ 1 ≤ ∥ r − 1 ∥ 2 ∥ φ ∥ − (20) < ∞ . 27 App endix: Helmholtz’s Theorem The conten t of this App endix sheds light on the problem of when and ho w can a div ergence-free flo w/vector field expressed in div ergence form, as in (14)&(15). As w e assume d that this is the case for the an tisymmetric part of the jump rates in (5), in a formal logical sense the mathematical conten t of the paper do es not rely on Prop osition 14 b elow. As noted in section 1.1, given a stationary and ergo dic, div ergence-free vector field/flow b : Z d × Ω → R N , as, e.g. that giv en in (5), it is a difficult and subtle issue to decide whether it can b e written in div ergence form, cf (14)&(15) or not. (Except, of course, if it is a priori given in this form.) The following statemen t sheds light on this problem. It is an ergo dic v arian t of what is usually referred to as Helmholtz’s The or em from classical electro dynamics: "In R 3 , a div ergence-free v ector field (e.g., a time-wise stationary magnetic field) is written as the curl/rotation of a v ector p oten tial." (T o b e taken with a grain of salt!) Prop osition 14 ("Helmholtz’s Theorem") . (i) Assume b ∈ L 1 . Ther e exists a str e am tensor field H : Z d × Ω → R N ×N with H k,l ( x, · ) ∈ L 1w := { f ∈ L : ∥ f ∥ 1w := sup 0 <λ< ∞ λπ ( | f | > λ ) < ∞} , H − k,l ( x + k , ω ) = H k, − l ( x + l , ω ) = H l,k ( x, ω ) = − H k,l ( x, ω ) (71) with stationary incr ements H k,l ( y , ω ) − H k,l ( x, ω ) = H k,l ( y − x, τ y ω ) − H k,l (0 , τ x ω ) , (72) such that Helmholtz’s r elation holds b k ( x, ω ) = X l ∈U H k,l ( x, ω ) = 1 2 X l ∈U ( H k,l ( x, ω ) − H k,l ( x − l , ω )) = X l ∈U H k,l (0 , τ x ω ) (73) (ii) If b ∈ L p , p ∈ (1 , 2] than the same hold with H k,l ( x, · ) ∈ L p . (iii) Assume b ∈ L 2 . Ther e exists a stationary str e am field h : Z d × Ω → R cN ×N with h k,l ( x, ω ) = h k,l ( τ x ω ) , h k,l ∈ L 2 such that H k,l ( x, ω ) = h k,l ( x, ω ) − h k,l (0 , ω ) (74) if and only if b k ∈ H − 1 ( | ∆ | ) := { f ∈ L 2 : lim λ ↘ 0 ⟨ f , ( λI − ∆) − 1 f ⟩ < ∞} . As w e assume (14)&(15)) the mathematical conten t of this pap er formally do es not rely on rop osition 14. Therefore, rather than presenting its full pro of we only give some hin ts. 28 Hints to some elements of the pr o of of Pr op osition 14. The L 2 setting of part (ii) and part (iii) appear as Prop osition 11 in [14], with complete pro ofs. The L 1 setting of part (i) is more subtle. F rom an appropriate adaptation of the Calderón- Zygm und Decomp osition (see e.g. [24]) to the ergo dic con text (Ω , F , π , τ x : Ω → Ω : x ∈ Z d ) , it follows that for f ∈ L 1 , ∥| ∆ | − 1 ∂ l ∂ k f ∥ 1w ≤ C ∥ f ∥ 1 , k , l ∈ N , where the constant C dep ends only on the dimension d . Once this is established, the argumen ts from the pro of of Prop osition 11 in [14] yield the result. A similar (though, not iden tical) adaptation of the Calderón-Zygmund theory to a random conductance setting app ears in [3]. Giv en (i) in the L 1 → L 1w setting, and (ii) in the L 2 → L 2 setting, (ii) in the L p → L p , 1 < p < 2 setting follo ws b y Marcinkiewicz interpolation, cf [24]. 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