Local Central Limit Theorem for diffusions in a degenerate and unbounded Random Medium

We study a symmetric diffusion $X$ on $\mathbb{R}^d$ in divergence form in a stationary and ergodic environment, with measurable unbounded and degenerate coefficients. We prove a quenched local central limit theorem for $X$, under some moment conditions on the environment; the key tool is a local parabolic Harnack inequality obtained with Moser iteration technique.


Description of the Main Result
We model the stationary and ergodic random environment by a probability space (Ω, G, µ), on which we define a measure-preserving group of transformations τ x : Ω → Ω, x ∈ R d . One can think about τ x ω as a translation of the environment ω ∈ Ω in direction x ∈ R d . The function (x, ω) → τ x ω is assumed to be B(R d ) ⊗ G-measurable and such that if τ x A = A for all x ∈ R d , then µ(A) ∈ {0, 1}. Given the random environment (Ω, G, µ, {τ x } x∈R d ) we can construct a stationary and ergodic random field simply taking a random variable f : Ω → R and defining f ω (x) := f (τ x ω), x ∈ R d .
Since a ω (x) is modeling a random field, it is not natural to assume its differentiability in x ∈ R d . Therefore the operator defined in (1.1) does not make sense, and the standard techniques from the Stochastic Differential Equations theory or Itô calculus are not helpful nor in the construction of the diffusion process nor in performing the relevant computations. We will exploit Dirichlet Forms theory to construct the diffusion process formally associated with (1.1). Instead of the operator L ω we shall consider the bilinear form obtained by L ω formally integrating by parts, for a proper class of functions u, v ∈ F Λ,ω ⊂ L 2 (R d , Λ ω dx), more precisely F Λ,ω is the closure of C ∞ 0 (R d ) in L 2 (R d , Λ ω dx) with respect to E ω + (·, ·) Λ . It is a classical result of Fukushima [11] that it is possible to associate to (1.2) a diffusion process (X ω , P ω x ) as soon as (λ ω ) −1 and Λ ω are locally integrable. As a drawback, the process cannot in general start from every x ∈ R d but only from almost all, and the set of exceptional points may depend on the realization of the environment.
In [5] it was proved that if λ ω (·) −1 , Λ ω (·) ∈ L ∞ loc (R d ) for µ-almost all ω ∈ Ω then a quenched invariance principle holds for X ω , namely the scaled process X ǫ,ω t := ǫX ω t/ǫ 2 converges in distribution under P ω 0 to a Brownian motion with a non-trivial deterministic covariance structure as ǫ → 0. In that work local boundness was assumed in order to get some regularity for the density of the process X ω and avoid technicalities due to exceptional sets arising from Dirichlet forms theory.
The method. The proof of Theorem 1.1 relies strongly on a priori estimates for solutions to the "formal" parabolic equation It is well known that when x → a ω (x) and x → Λ ω (x) are bounded and bounded away from zero, uniformly in ω ∈ Ω, then a parabolic Harnack's inequality holds for solutions to (1.4), this is a celebrated result due to Moser [16]. He showed that there is a positive constant C P H , which depends only on the uniform bounds on a and Λ, such that for any positive weak solution of (1.4) on (t, t + r 2 ) × B(x, r) we have sup where Q − = (t + 1/4r 2 , t + 1/2r 2 ) × B(x, r/2) and Q + = (t + 3/4r 2 , t + r 2 ) × B(x, r/2). The parabolic Harnack inequality plays a prominent role in the theory of partial differential equations, in particular to prove Hölder continuity for solutions to parabolic equations, as it was observed by Nash [17] and De Giorgi [7], or to prove Gaussian type bounds for the fundamental solution p ω t (x, y) of (1.4) as done by Aronson [2]. It is remarkable that such results do not depend neither on the regularity of a nor of Λ.
In this paper we shall exploit the stability of Moser's method to derive a parabolic Harnack inequality also in the case of degenerate and possibly unbounded coefficients. The technique is quite flexible and can also be applied to discrete space models for which we refer to [1] .
Moser's method is based on two steps. One wants first to get a Sobolev inequality to control some L ρ norm in terms of the Dirichlet form and then control the Dirichlet form of any caloric function by a lower moment. This sets up an iteration which leads to bound the L ∞ norm of the caloric function. In the uniform elliptic case this is rather standard and it is possible to control the L 2d/d−2 norm by the L 2 norm. In our case the coefficients are neither bounded from above nor from below and we need to work with a weighted Sobolev inequality, which was already established in [5] by means of Hölder's inequality. Doing so we are able to control locally on balls the L ρ norm by means of the L 2p * norm, In order to start the iteration we need ρ > 2p * which is equivalent to 1/p + 1/q < 2/d. This integrability assumption firstly appeared in [9] in order to extend the results of De Giorgi and Nash to degenerate elliptic equations, although they focus on weights belonging to the Muckenhaupt's class. A similar condition was also recently exploited in [20] to obtain Aronson type estimates for solutions to degenerate parabolic equations.
Following the classic proof of Moser, with some extra care due to the different exponents we get a parabolic Harnack inequality for solution to (1.4) in our setting. In the uniform elliptic and bounded case the constant in front of the Harnack inequality was depending only on uniform bounds on a and Λ. In our setting we cannot expect that to be true for general weights, and the constant will strongly depend on the center and the radius of the ball, in particular we don't have any control for small balls, so that a genuine Hölder's continuity result like the one of Nash is not given. Luckily in the diffusive limit the ergodic theorem helps to control constants and to give Theorem 1.1. Remark 1.2. Given a speed measure θ : Ω → (0, +∞) one can consider also the Dirichlet form (E ω , F θ,ω ) on L 2 (R d , θ ω dx) where E ω is given by (1.2) and F θ,ω is the closure of of C ∞ 0 (R d ) in L 2 (R d , θ ω dx) with respect to E ω + (·, ·) θ . This corresponds to the formal generator One can show along the same lines of the proof for θ = Λ that if where p, q, r ∈ (1, ∞] are such that then the parabolic Harnack inequality still works, in particular a quenched local central limit theorem can still be derived in this situation. Observe that in the case θ = Λ we find back the familiar condition 1/p + 1/q < 2/d. In the case that θ ≡ 1, r = ∞ the condition reads 1/(p − 1) + 1/q < 2/d. Remark 1.3. The condition 1/p + 1/q < 2/d is morally optimal to state Theorem 1.1. Indeed it was shown in [1][See Theorem 5.4] that if 1/p + 1/q > 2/d, then there is an ergodic environment for which the quenched local central limit theorem does not hold. It is not hard construct an example also in the continuous by exploiting the same ideas given in [1].
A summary of the paper is the following. In Section 2 we present a deterministic model obtained by looking at a fixed realization of the environment. We derive Sobolev, Poincaré and Nash inequalities for such a model.
In Section 3 we prove a priori estimates, on-diagonal bounds and Hölder continuity type estimates for caloric functions. The main aim and result of the section is the parabolic Harnack inequality.
In section 4 we prove a local Central Limit Theorem for the deterministic model which we apply to finally get Theorem 1.1.

Deterministic Model and Local inequalities
Since we want to prove a quenched result we will develop a collection of inequalities for a deterministic model. With a slight abuse of notation we will note with a(x), λ(x) and Λ(x) the deterministic versions of a(τ x ω), λ(τ x ω) and Λ(τ x ω).
We are given a symmetric matrix a : R d → R d×d such that Assumption (b.2) plays the role of ergodicity in the random environment model. We are interested in finding a priori estimates for solutions to the formal parabolic equation for t ∈ (0, ∞) and x ∈ R d . Clearly in the way it is stated (2.1) is not well defined since a is only assumed to be measurable. In order to make sense of (2.1) we shall exploit the Dirichlet form framework, see [11] for an exhaustive treatment on the subject.

Caloric Functions
For this section we will follow [3]. Let θ : R d → R be a non-negative function such that θ −1 , θ are locally integrable on R d . Consider the symmetric form E on Then, (E, C ∞ 0 (R d )) is closable in L 2 (R d , θdx) thanks to [18][Ch. II example 3b], since λ −1 , Λ ∈ L 1 loc (R d ) by (b.2). We shall denote by (E, F θ ) such a closure; it is clear that F θ is the completion of C ∞ 0 (R d ) in L 2 (R d , θdx) with respect to E 1 := E + (·, ·) θ . Observe that (E, F θ ) is a strongly local regular Dirichlet form, having C ∞ 0 (R d ) as a core. In the case that θ ≡ 1 we will simply write F. Given an open subset Definition 2.1 (Caloric functions). Let I ⊂ R and G ⊂ R d an open set. We say that a function u : I → F θ is a subcaloric (supercaloric) function in I × G if t → (u(t, ·), φ) θ is differentiable in t ∈ I for any φ ∈ L 2 (G, θdx) and for all non negative φ ∈ F Λ G . We say that a function u : I → F θ is a caloric function in I × G if it is both sub-and supercaloric.
It is clear from the definition that if a function is subcaloric on I × G than it is caloric on I ′ × G ′ whenever I ′ ⊂ I and G ′ ⊂ G.
Moreover, observe that if P G t is the semigroup associated to (E, F θ ) on L 2 (G, θdx) and f ∈ L 2 (G, θdx), for a given open set G ⊂ R d , then the function u(t, ·) = P G t f (·) is a caloric function on (0, ∞) × G. To complete the picture we state the following maximum principle which appeared in [13]. For a real number a denote by a + = a ∨ 0.
As a corollary of this lemma we have the super-mean value inequality for subcaloric functions.

Sobolev inequalities
In this section we will state local inequalities on the flat space L 2 (R d , dx) and on the weighted space L 2 (R d , Λdx). We are interested in Sobolev, Poincaré and Nash type inequalities. The first and the second provide an effective tool for deriving local estimates on solutions to Elliptic and Parabolic degenerate partial differential equation, while the latter will be used to prove the existence of a kernel for the semigroup P t associated to (E, We shall see that the constants appearing in the inequalities are strongly dependent on averages of λ and Λ and in particular on the ball where we focus our analysis.
In the sequel we shall use the symbol to say that the inequality ≤ holds up to a multiplicative constant depending only on the dimension d ≥ 2.
In the next proposition it is enough to assume Λ ∈ L 1 loc (R d ) and λ −1 ∈ L q loc (R d ). The following constant will play an important role in the sequel, observe that ρ is the Sobolev's conjugate of 2q/(q + 1).

Proposition 2.3 (Local Sobolev inequality). Fix a ball
where C B S := λ −1 q,B . Proof. We start proving (2.5) for u ∈ C ∞ 0 (B). Since ρ as defined in (2.4) is the Sobolev conjugate of 2q/(q + 1), by the classical Sobolev's inequality where it is clear that we are integrating over B. By Hölder's inequality and (b.1) we can estimate the right hand side as follows which leads to (2.5) for u ∈ C ∞ 0 (B) after averaging over the ball B. By approximation, the inequality is easily extended to u ∈ F B .
Proof. The proof easily follows from Hölder's inequality and the previous proposition.
Remark 2.5. From these two Sobolev's inequalities it follows that the domains F B and F Λ B coincide for all balls B ⊂ R d . Indeed, from (2.5) and (2.6), since ρ, ρ/p * > 2, we get that (F B , E) and (F Λ B , E) are two Hilbert spaces; therefore F B , F Λ B coincide with their extended Dirichlet space, which by [10, pag 324], is the same, hence Cutoffs. Since assumptions (b.1) and (b.2) only assure local integrability of λ −1 and Λ, we will need to work with functions that are locally in F or F Λ and with cutoff functions.
In view of these notations, for u, v ∈ F θ loc we define the bilinear form Proposition 2.6 (Local Sobolev inequality with cutoff). Fix a ball B ⊂ R d and a cutoff function η ∈ C ∞ 0 (B) as above. Then for all u ∈ F Λ loc ∪ F loc and, for the weighted version Proof. We prove only (2.8), being (2.9) analogous. Take u ∈ F loc ∪F Λ loc , by Lemma A.1 in the appendix, ηu ∈ F B , therefore we can apply (2.5) and get To get (2.8) we compute ∇(ηu) = u∇η + η∇u and we easily estimate Concatenating the two inequalities and averaging over B we get the result.

Nash inequalities
Local Nash inequalities follow as an easy corollary of the Sobolev's inequalities (2.5) and (2.6).
Proof. We prove only (2.10) being the other completely analogous. By Hölder's inequality with θ ∈ (0, 1) and Now solve for θ, use (2.5) to estimate u ρ,B and the result is obtained.
Note that the condition 1/p + 1/q < 2/d is important to have µ and γ positive, in particular γ ≥ d/2, with the equality holding if p = q = ∞. It is well known that Nash inequality (2.11) for the Dirichlet form (E, F Λ B ) implies the ultracontractivity of the semigroup P B t associated to E on where it is once more worthy to notice that 2/d − 1/γ ≥ 0, with the equality holding for the nondegenerate situation. Furthermore, we have just seen that P t is locally ultracontractive, being P B t ultracontractive for all balls B ⊂ R d . It follows by Theorem 2.12 of [14] that P t admits a symmetric transition kernel p t (x, y)

Poincaré inequalities
Let B ⊂ R d be a ball. Given a weight θ : B → [0, ∞], we denote by We have by Theorem 1.5.2 in [19].
In order to get mean value inequalities for the logarithm of caloric functions and, given that, the parabolic Harnack inequality, we will need a Poincaré inequality with a radial cutoff. The cutoff function η : where Φ is some non-increasing, non-negative càdlàg function non identically zero on (r/2, r]. Proposition 2.9 (Poincaré inequalities with radial cutoff). Let B ⊂ R d be a ball of radius r > 0 and center x 0 and let η be a cutoff as above. If u ∈ F loc , then Proof. We give the proof only for (2.15) being (2.14) similar. We apply Theorem 1 in [8]. Accordingly we define a functional F (u, s) : for u ∈ F Λ , and F (u, s) = ∞ otherwise, being B s the ball of center x 0 and radius s ∈ (r/2, r]. Such functional satisfies F (u + a, s) = F (u, s) for all a ∈ R and u ∈ L 2 (R d , Λdx), moreover for every s ∈ (r/2, r] and u ∈ F Λ by the Poincaré inequality (2.13). It follows from Theorem 1 in [8] that for u ∈ F Λ there exists M > 0, explicitly given by ( Λ 1,B Φ(0))/( Λ 1,B/2 Φ(1/2)), such that Here γ(ds) is a non-zero positive σ-finite Borel measure on (r/2, r] such that as in [8]. Of course such an inequality is local and we can extend it for u ∈ F loc .

Remark on the constants
In this section about inequalities, we have introduced different constants, among them C B,Λ S , C B,Λ P and M B,Λ . Observe that they all strongly depend on the ball B, on the radius and the center as well. Assumption (b.2) helps us to control the behavior of the constants as the radius of the ball increases.
Proof. Since u t > 0 is locally bounded, the power function F : R → R defined by F (x) = |x| 2α with α ≥ 1 satisfies the assumptions of Lemma A.3. Thus, for η ∈ C ∞ 0 (B) as above we have We can estimate by means of Young's inequality 2ab ≤ (ǫa 2 + b 2 /ǫ) with a = E η (u α , u α ) 1/2 and b = ∇η ∞ 1 B u 2α 1/2 1,Λ and for ǫ = 1/2α, we get exploiting that α ≥ 1 Going back to (3.2) we have We now take a smooth cutoff in time ζ : R → [0, 1], ζ ≡ 0 on (−∞, t 1 ], where I = (t 1 , t 2 ). We multiply the inequality above by ζ and integrate in time. This yields after averaging and taking the supremum for t ∈ I we get We use (3.3) together with (2.9) to get (3.1). Observe that ν = 2 − 2p * /ρ is greater than one, since ρ > 2p * by the condition 1/p + 1/q < 2/d. Using Hölder's inequality and some easy manipulation In view of the Sobolev inequality (2.9) we have by (3.3) we can bound each of the two factors. We end up with the following iterative step which is what we wanted to prove.
The main idea is to use Moser's iteration technique on a sequence of parabolic balls; Proposition 3.1 with suitable choice of the cutoffs and of the parameter α is the iteration step. Fix a parameter τ > 0, let x ∈ R d , and r > 0. Consider also a parameter δ ∈ (0, 1). Then we define the parabolic balls Clearly Q δ ⊂ Q for all δ ∈ (0, 1). Theorem 3.2. Fix τ > 0 and let 1/2 ≤ σ ′ < σ ≤ 1. Assume that 1/p + 1/q < 2/d and let u t be a positive subcaloric function on Q = Q(τ, x, s, r). Then there exists a positive constant C 1 := C 1 (d, p, q) such that Proof. We want to apply (3.1) with a suitable sequence of cutoffs η k and ζ k . Set then σ k − σ k+1 = δ k , then consider a cutoff η k : R d → [0, 1], such that supp η k ⊂ B(σ k r) and η k ≡ 1 on B(σ k+1 r), moreover assume that ∇η ∞ ≤ 2/(rδ k ). Take also a cutoff in time ζ : R → [0, 1], ζ k ≡ 1 on I σ k+1 = (s − σ k+1 τ r 2 , s), ζ k ≡ 0 on (−∞, s − σ k τ r 2 ) and ζ ′ ∞ ≤ 2/(r 2 τ δ k ). Let α k = ν k with ν = 2 − 2p * /ρ as above. Then, an application of (3.1) and using the fact that α k+1 = να k yields where we used the fact that σ k /σ k+1 < 2, and that σ k ∈ [1/2, 1]. This is the starting point for Moser's iteration. Iterating the inequality from i = 0 up to k we get at the price of a constant C 1 > 0 which depends on p, q and the dimension where we exploited the fact that ∞ i=0 1/α i = ν/(ν − 1) and that ∞ i=0 k/α i < ∞. From the inequality above we easily get, taking C 1 larger if needed, and taking the limit as k → ∞ gives the result Corollary 3.3. Fix τ > 0 and let 1/2 ≤ σ ′ < σ ≤ 1. Assume that 1/p + 1/q < 2/d and let u be a subcaloric function in Q = Q(τ, x, s, r). Then there exists a positive constant C 2 := C 2 (q, p, d) which depends only on the dimension and on p, q such that for all α > 0 Proof. To prove (3.5) one can follow the same approach in [19][Theorem 2.2.3] with the only difference that we will consider parabolic balls Q σ instead of balls. Observe that for α > 2 this is just an application of Jensen's inequality.
Observe that (3.5) is not good for the application of Bombieri-Giusti's lemma (B.1) since 2 2 α ν ν−1 is exploding as α approaches zero. To get rid of this problem we develop in the next section the same type of inequalities for supercaloric functions.
Theorem 3.2 can be also applied to obtain a global on-diagonal heat kernel upper bound, as it is done in the next proposition.
Proposition 3.4. Let f ∈ L 2 (R d , Λdx), and assume that (b.1) and (b.2) are satisfied, then there exists a constant C 3 = C 3 (q, p, d, C * ,Λ S ) > 0 such that for all x ∈ R d and t > 0 the following inequality holds where γ was defined in 2.11 and s(x, δ) was defined in Section 2.5.
We chose r = s(0, 1) + 2|z| + √ t where s(0, 1) was defined in Section 2.5. In this way C B,Λ S ≤ 2C * ,Λ S and we can read inequality (3.4) for u(s, z) := P s f (z) as follows with c = c(p, q, d) changing throughout the proof. By definition of r we find τ ∈ (0, 2] such that 3/4τ r 2 = t, and in particular (t, z) ∈ Q 1/2 . This gives and this holds for all z ∈ R d and t > 0.
Now it is left to use the semigroup property and standard techniques to finally get the bound.
It is now standard to get global on-diagonal estimates for the kernel p t (x, y) of the semigroup P t associated to (E, F Λ ) on L 2 (R d , Λdx). Namely we obtain that for almost all x, y ∈ R d and for all t > 0
Proof. We can always assume that u > ǫ by considering the supersolution u + ǫ and then sending ǫ to zero at the end of the argument. Applying Lemma A.3 with the function F (x) := −|x| −β and β > 0 we get which after some manipulation gives by means of Young's inequality 4ab ≤ 3a 2 + 2b 2 /3 and using the simple fact that (β + 1)/β > 1 we get after averaging d dt We now integrate against a time cutoff ζ : R → [0, 1] to obtain something similar to (3.3). Hence the same approach as in Proposition 3.1 applies and we get Moser's iteration technique with β k = ν k α and α > 0 and the same argument of Theorem 3.2 will finally give We introduce the following parabolic ball. Given x ∈ R d , r, τ > 0 and s ∈ R, δ ∈ (0, 1), we note Theorem 3.6. Fix τ > 0 and let 1/2 ≤ σ ′ < σ ≤ 1. Assume that 1/p + 1/q < 2/d and let u be a positive supercaloric function on Q = Q(τ, x, s, r). Fix 0 < α 0 < ν. Then there exists a positive constant C 5 := C 5 (q, p, d, α 0 ) which depends only on the dimension, on p, q and on α 0 such that for all 0 < α < α 0 ν −1 we have Proof. Assume u is supercaloric on Q = I × B. Applying Lemma A.3 with the function F (x) := |x| β with β ∈ (0, 1) we get d dt which after some manipulation gives this yields after Young's inequality where A is a constant possibly depending on q, p, α 0 and d which will be changing throughout the proof.

Mean value inequalities for log u t
In this section we get mean value inequalities for log u t where u t is a positive supercaloric function on Q = (s − τ r 2 , s) × B(x, r), with τ > 0 fixed. We denote by m Λ := Λdx and by γ Λ := dt × m Λ .
Theorem 3.7. Fix τ > 0 and κ ∈ (0, 1), δ ∈ [1/2, 1). For any s ∈ R and r > 0 and any positive supercaloric function u on Q = (s − τ r 2 , s) × B(x, r), there exist a positive constant C 6 := C 6 (q, p, d, δ) and a constant k := k(u, κ) > 0 such that Proof. We follow closely the strategy adopted in Theorem 5.4.1 of [19]. We can always assume u t ≥ ǫ and then send ǫ to zero in our estimates, since u t + ǫ is still a supercaloric function. We denote as usual B := B(x, r). By Lemma A.3 in the last inequality we exploit Young's inequality 2ab ≤ (1/2a 2 + 2b 2 ). The cutoff function η must be on the form used in (2.15). We take where x, r are the center and the radius of the ball B. We note rewriting (3.12) we get for some constant c > 0 depending only on the dimension and δ. Observe that we fixed δ ∈ [1/2, 1) to stay away from the boundary of B, that for very large radius M B,Λ and C B,Λ P are basically constants and that m Λ (B) is the volume of a ball in L 2 (R d , Λdx); hence what we have above resembles closely what is given in [19]. Let us introduce the following auxiliary functions where s ′ = s − κτ r 2 . We can now rewrite (3.13) as Now set k(u, κ) :=W s ′ and define the two sets Using this in (3.14) we obtain Integrating from s ′ to s yields, for Finally, where in the second but last step we used Markov's inequality and the fact that κ < 1. Working with D − t (ℓ) and K − and using similar arguments proves the second inequality.

Bombieri-Giusti's Lemma B.1 is applicable and we obtain
for all α > 0 and 1/2 ≤ σ ′ < σ ≤ 1. Since by Theorem 3.7 we have then Bombieri-Giusti's lemma is applicable and yields BG which we can assume to be the same as before taking the maximum of the two. Putting the two inequalities together gives the result.  Proof. It follows from the previous theorem for positive supercaloric functions and Corollary 3.3.
We have to remark that the constant appearing in (3.19) it is strongly dependent on the ball B we are considering, in particular depends on its center and its radius. We use here the power of assumption (b.2) to get rid of this dependence for balls which are large enough as it was discussed in Section 2.5.
Indeed for all x ∈ R d we can find s(x, 1) ≥ 1 such that C B,Λ H ≤ 2C * ,Λ H for all B(x, r) with r > s(x, 1).
Proof. Set r k := 2 −k r 0 and let let Q − k and Q + k be accordingly defined as in (3.17) with δ = 1/2 and τ = 1, Notice that Q k+1 ⊂ Q k and actually Q k+1 = Q + k . We set this implies that replacing v k by 1−v k if necessary sup Q − k v k ≥ 1/2. Now for all k such that r k ≥ s(x, 1) we can apply the parabolic Harnack inequality and get We can now iterate the inequality up to k 0 such that r k 0 ≥ r > r k 0 +1 and get Finally since B(x, r) ⊂ B(x, r k 0 ) and t ∈ (t 0 − r 2 k 0 , t 0 ) the claim is proved.
Starting from (3.20) and knowing that p t (z, ·) is caloric on the whole R d for almost all z ∈ R d we get the following corollary.
Proof. The proof presented here is a slight variation of the one in [6] due to the fact that we work on R d rather than on graphs. For x ∈ B(o, r) and r 0 > 0 denote where k t := k Σ t from assumption (b.5) is the gaussian kernel with covariance matrix Σ. Then we can split J(t, ǫ) = J 1 (t, ǫ) + J 2 (t, ǫ) + J 3 (t, ǫ) + J 4 (t, ǫ) Fix δ > 0. Thanks to Lemma 3.13 and by the continuity of k t we can chose r 0 ∈ (0, 1) andǭ > 0 small such that for all ǫ <ǭ We can now easily bound sup t∈I |J 4 (t, ǫ)| ≤ δ|B(x, r 0 )|. Furthermore, by assumption (b.4) takinḡ ǫ smaller if needed we get sup t∈I |J 3 (t, ǫ)| ≤ δ|B(x, r 0 )| for all ǫ ≤ǭ. Exploiting (4.1) we have a control on J 1 . Namely sup t∈I |J 1 (t, ǫ)| ≤ δ|B(x, r 0 )|. Finally by assumption (b.5) we have also that sup t∈I |J(t, ǫ)| ≤ δ|B(x, r 0 )|. These estimates can be then used to control |J 2 (t, ǫ)| for ǫ ≤ǭ uniformly in t ∈ I. Namely one gets Next if x ∈ B(o, r) then x ∈ B(z, r 0 ) for some z ∈ X and we can write Since x, z ∈ B(o, r + 1) and |x − z| ≤ r 0 , inequality (4.2) implies that the last addendum is bounded by δ, the second term is also bounded uniformly by δ since z ∈ X . We can finally bound the first term uniformly by δ by means of (4.1). This ends the proof.
The second part of the statement follows easily since, if we assume that λ ω (·) −1 , Λ ω (·) ∈ L ∞ loc (R d ) for µ-almost all ω ∈ Ω, Theorem 4.1 holds for all o ∈ R d , µ-almost surely. Indeed, the density p ω t (x, y) is a continuous function of x and y by classical results in PDE theory [12].

A Dirichlet Forms
Let X be a locally compact metric separable space, and m a positive Radon measure on X such that supp[X] = m. Consider the Hilbert space L 2 (X, m) with scalar product ·, · . We call a symmetric form, a non-negative definite bilinear form E defined on a dense subset D(E) ⊂ L 2 (X, m). Given a symmetric form (E, D(E)) on L 2 (X, m), the form E β := E + β ·, · defines a new symmetric form on L 2 (X, m) for each β > 0. Note that D(E) is a pre-Hilbert space with inner product E β . If D(E) is complete with respect to E β , then E is said to be closed.
A closed symmetric form (E, D(E)) on L 2 (X, m) is called a Dirichlet form if it is Markovian, namely if for any given u ∈ D(E), then v = (0 ∨ u) ∧ 1 belongs to D(E) and E(v, v) ≤ E(u, u).
We say that the Dirichlet form (E, D(E)) on L 2 (X, m) is regular if there is a subset H of D(E) ∩ C 0 (X) dense in D(E) with respect to E 1 and dense in C 0 (X) with respect to the uniform norm. H is called a core for D(E).
We say that the Dirichlet form (E, D(E)) is local if for all u, v ∈ D(E) with disjoint compact support E(u, v) = 0. E is said strongly local if u, v ∈ D(E) with compact support and v constant on a neighborhood of supp u implies E(u, v) = 0.
Hence ηf n is Cauchy in L 2 (B, Λdx) with respect to E + ·, · Λ , which implies that ηu ∈ F Λ B = F B . If u ∈ F loc the proof is similar, and one has only to observe that {f n } is Cauchy in W 2q/(q+1) (B), which by Sobolev's embedding theorem implies that {f n } is Cauchy in L 2 (B, Λdx).