Non-uniformly parabolic equations and applications to the random conductance model

We study local regularity properties of linear, non-uniformly parabolic finite-difference operators in divergence form related to the random conductance model on $\mathbb Z^d$. In particular, we provide an oscillation decay assuming only certain summability properties of the conductances and their inverse, thus improving recent results in that direction. As an application, we provide a local limit theorem for the random walk in a random degenerate and unbounded environment.


Introduction
In this contribution, we continue our research [12,13] on regularity and stochastic homogenization of nonuniformly elliptic equations. In [12], we studied local regularity properties of weak solutions of elliptic equations in divergence form (1) ∇ · a∇u = 0 and proved local boundedness and the validity of Harnack inequality under essentially minimal integrability conditions on the ellipticity of the coefficients a. This generalizes the seminal theory of De Giorgi, Nash and Moser [23,31,33] and improves in an optimal way classic results due to Trudinger [35] (see also [32]). In [13], we adapted the regularity theory from [12] to discrete finite-difference equations in divergence form and used this to obtain a quenched invariance principle for random walks among random degenerate conductances (see the next section for details).
In the present contribution, we extend our previous results in two ways (A) (deterministic part) We establish local regularity properties in the sense of an oscillation decay (and thus Hölder-continuity) for solution of discrete version of parabolic equations (2) ∂ t u − ∇ · a∇u = 0 under relaxed ellipticity conditions compared to very recent contributions in the field (see e.g. [3,5,7,21]). (B) (random part) Based on the regularity result in (A), we establish a local limit theorem for random walks among degenerate and unbounded random conductances.
We study local regularity properties of functions u : Q ⊂ R × Z d → R satisfying the parabolic finite-difference equation ∂ t u − L ω u = 0, where L ω is the elliptic operator defined by (4) (L ω u)(x) = y∈Z d ω(x, y)(u(y) − u(x)). 1 We emphasize here that L ω is in fact an elliptic finite-difference operator in divergence form, see (14) below.
Our main deterministic regularity result is the following (see Section 1.3 for notation).
Remark 1. The restrictions on the exponents p and q in Theorem 1 are natural in the sense that they are essentially necessary in order to establish local boundedness for solutions of ∂ t u − L ω u = 0, see Remarks 3 and 4 below. In recent works [3,7], the conclusion of Theorem 1 is contained under the more restrictive relation 1 p + 1 q < 2 d . Note that [3] also contains results for time-depending conductances and [7] allows for more general speed measures. It would be interesting to see to which extend the method of the present paper can also yield improvements in these cases.
Remark 2. In [12], the corresponding statement of Theorem 1 for d ≥ 3 is proven in the continuum setting as a consequence of elliptic Harnack inequality. Note that, in d = 2 we can consider the borderline case p = q = 1 for which we did not proved Harnack inequality in [12]. Previously, elliptic Harnack inequality and thus oscillation decay in the form of Theorem 2 was proven in [5] under the more restrictive relation 1 p + 1 q < 2 d (see also the classic paper [35]). 1.2. Local limit theorem. In what follows we consider random conductances ω that are distributed according to a probability measure P on Ω equipped with the σ-algebra F := B((0, ∞)) ⊗B d and we write E for the expectation with respect to P. We introduce the group of space shifts {τ x : Ω → Ω | x ∈ Z d } defined by (5) τ x ω(·) := ω(· + x) where for any e = {e, e} ∈ B d , e + x := {e + x, e + x} ∈ B d .
For any fixed realization ω, we study the reversible continuous time Markov chain, X = {X t : t ≥ 0}, on Z d with generator L ω given in (4). Following [4], we denote by P ω x the law of the process starting at the vertex x ∈ Z d and by E ω x the corresponding expectation. X is called the variable speed random walk (VSRW) in the literature since it waits at x ∈ Z d an exponential time with mean 1/µ ω (x), where µ ω (x) = y∈Z d ω(x, y) and chooses its next position y with probability p ω (x, y) := ω(x, y)/µ ω (x). Assumption 1. Assume that P satisfies the following conditions: (i) (stationary) P is stationary with respect to shifts, that is P Starting with the seminal contribution [37], a considerable effort has been invested in the derivation of quenched invariance principles under various assumptions on the conductances, see the surveys [16,26] and the discussion below. The following quenched invariance principe is the starting point for the probabilistic aspects of our contribution.
Proof. For d ≥ 3 this is [13, Theorem 2] and for d = 2 this can be found in [16].
In this contribution, we provide a refined convergence statement under slightly stronger moment conditions -namely a local limit theorem. Consider the heat-kernel p ω of X, characterized by (6) p for t ≥ 0 and x, y ∈ Z d .
The local limit theorem is essentially a pointwise convergence result of the (suitably scaled) heat kernel p ω of X towards the Gaussian transition density of the limiting Brownian motion of Theorem 3. Set where Σ as in Theorem 3.
Assumption 2. There exists p ∈ (1, ∞] and q ∈ ( d 2 , ∞] satisfying Assuming the stronger condition 1 p + 1 q < 2 d instead of (8), the conclusion of Theorem 4 was recently proved by Andres and Taylor [7] (for related results in the continuum setting see [21]). Previously, Barlow and Hambly [11] gave general criteria for a local limit theorem to hold. These criteria were applied to uniformly elliptic conductances or supercritical i.i.d. percolation clusters; see [22] for further generalizations. In [20], Boukhadra, Kumagai, and Mathieu identified sharp conditions on the tails of i.i.d. conductances at zero under which the parabolic Harnack inequality and the local limit theorem hold. An inspiring result for the present contribution is [5], where the local limit theorem for the constant speed random walk (CSRW) is proven under Assumption 1 and Assumption 2 with (8) replaced by 1 where the latter turns out be optimal in that case. We conclude this introduction by mentioning other related results: As mentioned above the quenched invariance principle in the form of Theorem 3, for uniformly elliptic conductances (that is p = q = ∞) or on supercritical i.i.d. percolation clusters, was proven by Sidoravicius and Sznitman [37]. In the special case of i.i.d. conductances, that is when P is the product measure, which includes e.g. percolation models, building on the previous works [10,15,18,27,28], Andres, Barlow, Deuschel, and Hambly [1] showed that the quenched invariance principle holds provided that P[ω(e) > 0] > p c with p c = p c (d) being the bond percolation threshold. In particular, due to independence of conductances such situation is very different as they do not require any moment conditions such as (9). In the general ergodic situation, it is known that at least first moments of ω and ω −1 are necessary for a quenched invariance principle to hold (see [9]). Andres, Deuschel, and Slowik [4] obtained the conclusion of Theorem 3 under more restrictive relation 1 p + 1 q < 2 d , see also the very recent extension beyond the nearest-neighbor conductance models [17]. For quenched invariance principles in dynamic environments, see [2,19], for a recent paper on local limit theorem, see [3], as well as [29] for related results. A quantitative quenched invariance principle (under quantified ergodicity assumptions) with degenerate conductances can be found in [6]. Very recently, building on [8], an almost optimal quantitative local limit theorem in the percolation setting was proven by Dario and Gu [24]. Further results in the stationary & ergodic setting under moment conditions include large-scale regularity [14], homogenization in the sense of Γ-convergence [34], or spectral homogenization [25].

Notation.
• (Sets and L p spaces) For y ∈ Z d , n ≥ 0, we set B(y, n) := y + ([−n, n] ∩ Z) d with the shorthand B(n) = B(0, n). For any S ⊂ Z d we denote by S B d ⊂ B d the set of bonds for which both end-points are contained in S, i.e.
and f L ∞ (S) = sup x∈S |f (x)|. Moreover, normalized versions of · || L p are defined for any finite subset S ⊂ Z d and p ∈ (0, ∞) by where |S| and |S B d | denote the cardinality of S and S B d , respectively. Throughout the paper we drop the subscript in S B d if the context is clear and we set · L ∞ (S) := · L ∞ (S) . Moreover, where |I| denotes the Lebesgue measure of I and |S| the cardinality of S. The discrete divergence is defined for every F : Note that for every f : Z d → R that is non-zero only on finitely many vertices and every F : Finally, we observe that the generator L ω defined in (4) can be written as a second order finitedifference operator in divergence form, in particular (14) ∀u : • (Functions) For a function u : I × V → R with I ⊂ R and V ⊂ Z d , we denote by u t the function u t : V → R given by u t = u(t, ·). We call u : I × V → R caloric (subcaloric or supercaloric) in Moreover, we call u :

Parabolic regularity
2.1. Auxiliary Results. We recall suitable versions of Sobolev inequality, see Proposition 1, and provide an optimization result, formulated in Lemma 1 below, that is central in our proof of Theorem 1.
). Estimate (16) is the discrete analogue of the classical Sobolev inequality on the sphere, since the first and the third term measure f on the boundary of the ball/cube (i.e. sphere) whereas the middle term measures ∇f within this set. The statements of Proposition 1 are standard and the proof can be found e.g. in [13,Theorem 3].
For every δ > 0 it holds where for every m ∈ N Proof of Lemma 1. Inequality (17) was already proven in [13, Step 1 of the proof of Lemma 1]. For convenience for the reader we recall the computations below.

Local boundedness.
In this subsection, we establish local boundedness for non-negative subcaloric functions u. The results of this section, in particular Lemma 2 below, contain the main technical improvements compared to previous related results, e.g. [3,7,21]. In principle, we follow the classical strategy of Moser to obtain the local boundedness. We recall that this strategy is based on (i) Caccioppoli inequalities for (powers of) u (see (27) below), (ii) application of the Sobolev inequality, and (iii) an iteration argument. As in our previous works [12,13] the improvement is mainly obtained by using certain optimized cut-off functions in the Caccioppoli inequality that allow (appealing to Lemma 1) to use Sobolev inequality on "spheres" instead of "balls". Unfortunately, the implementation of this strategy is technically much more involved in the parabolic case compared to the elliptic case treated in [12,13]. Throughout this section, we use the shorthand Let ν ∈ (0, 1), γ > 2 and θ > 1 be given by (22) ν : Then there exist c = c(d, p, q) ∈ [1, ∞) such that the following is true: Let n, m ∈ N with n < m ≤ 2n and 0 < s 1 < s 2 be given and consider The main achievement of Lemma 2 is estimate (24), where at the expense of increasing the domain of integration we control u α in terms of u να with ν < 1. The factor on the right-hand side involving norm of u α (and not u να ) to a small power will be dealt with later. The other two estimates (25) and (26) do not include improvement of integrability, and their proofs are significantly simpler.
Proof of Lemma 2. Throughout the proof we write if ≤ holds up to a positive constant that depends only on d, p, and q. We introduce, Step 1. We claim that there exists c = c(d) ∈ [1, ∞) such that for all η ∈ A(n, m) and α ≥ 1, where we recall the notation f (e) = 1 2 (f (ē)+f (e)). This is a discrete parabolic version of classical Caccioppoli inequality, and is obtained by simply testing the equation with η 2 u with η being a cutoff-function in space, combined with Cauchy-Schwarz inequality and integration in time.
Since u ≥ 0 is subcaloric, we obtain by the chain rule and estimate (123) 1 2α and thus 1 2 where we use in the last estimate Youngs inequality in the form ab ≤ 1 2 (εa 2 + 1 ε b 2 ) with ε = 2α−1 2α 2 . Combining the previous two displays, we obtain d dt Multiplying (28) with the piecewise smooth function ζ given by and integrating in time, we obtain (27) (using n ≤ m ≤ 2n and thus |B(n)| |B(m)| |B(n)|).
Step 3. We claim that there exists c = c(d, p, q) ∈ [1, ∞) such that (29) min η∈A(n,m) As in the elliptic case, see [13,proof of Theorem 4], the idea is to optimize the cutoff η in (27) via Lemma 1 to get spherical averages of u on the right-hand side. Since we do not have good control of time derivatives, the cutoff should be time-independent, hence providing improved integrability for the averages over spheres and in time. To "move" the time-integral outside we first sacrifice bit of space and time integrability (see Substep 3.1), but which is then dealt with using L 2 control of u, uniform in time (see Substep 3.2) -which then gives rise to the first term on the right-hand side of (29).
Note that the choices for θ and ν (see (22)) yield . The inequalities (34) follow by elementary computations which we provide for the readers convenience in Substep 3.4 below. Appealing to (34) we have the following interpolation inequality and thus by Hölder inequality in time (with exponents θ, θ θ−1 ), we obtain We estimate the second factor on the right-hand side in (35) by Sobolev inequality: Let p * ∈ [1, 2) be defined by Then a combination of Sobolev and Hölder inequality yield Combining (33), (35) and (36) with the observation p * where in the first inequality the factor k −(d−1) θ p from (36) is gone due to averaging in ω L p (S(k)) . To estimate the last factor on the right-hand side in (37), we first split the sum and then use once more Hölder inequality with exponents ( q+1 q , q + 1) to obtain Combining (37) for all s ≥ 1) and definition (21), we obtain and thus (30) follows.
. Then a combination of Sobolev and Hölder inequality yield where the last inequality is valid since 1 Combining (40) with Hölder inequality in the form (38) (with ν replaced by 1), we obtain (32). For d = 2, we argue as above but replace (39) by Substep 3.2. We claim that there exists c 2 = c 2 (d, p, q) ∈ [1, ∞) such that Let Q > 2 be the Sobolev exponent for 2q q+1 in R d given by (31)), we obtain where in the last relation we used and thus (1 + ε)ℓ = δ 2 .
, a combination of Sobolev and Hölder inequality yields (45) and (47), we obtain , and the claimed estimate (41) follows by integration in time.
Let us now verify (34) for d ≥ 3. The last inequality in (34) is trivial, while the first follows directly from 0 < ν < 1 < θ < p since d ≥ 3. Next, we observe that and thus the second and third inequality in (34) are equivalent to θ − δ 2 (p − 1) > 0. In the case d ≥ 3, we have where the last inequality follows from the assumption q > 2 d . Hence, the inequalities (34) are proven.
Step 4. Proof of estimates (24) and (26). Estimate (26) follows directly from (27) and (32). To show (24), we combine (27) and (29) to obtain where c = c(d, p, q) ∈ [1, ∞). The first term on the right-hand side has already the desired form, hence we only need to estimate the second term: Let Q > 2 be the Sobolev exponent for 2q q+1 given by (42), which by (43) satisfies νQ > 2(1 + ε) > 2. Combination of Jensen and Sobolev inequality yield Recall that the H 1 -norm consist of the L 2 norm of the function and its gradient (see (21)), so to get the full H 1 -norm on the left-hand side of (48) we need to add and estimate u α t itself: for that we observe that the assumption 1 ≤ m n ≤ 2 and (49), applied on B(n) instead of B(m), yield Estimate (24) follows from (48)-(50) and 1 ≤ ω L p (B(m)) ω −1 L q (B(m)) .

Remark 3.
In recent works [3,7] the statement of Theorem 5 is proven under the more restrictive relation The restrictions on p and q in Theorem 5 are essentially optimal: Counterexamples to elliptic regularity in the form [17,Theorem 2.6] show that local boundedness in the form (51) with (52) fails already for L ω -harmonic functions if 1 p + 1 q > 2 d−1 , see [13, Remark 2 and 4] for a more detailed discussion. The additional restriction on q, namely q > d 2 , is not present in the corresponding elliptic version of Theorem 5 (see [13,Theorem 2] and Theorem 7 below) and can be related to trapping phenomena for random walks in random environments. In Remark 4 below we discuss this in more detail and show that local boundedness in the form of Corollary 1 below (which is a direct consequence of Theorem 5) is not valid for q < d 2 . Proof of Theorem 5. Without loss of generality we consider t 0 = 0 and x 0 = 0, and we use the shorthand Q σ (τ, n) = Q σ (0, 0, τ, n). Throughout the proof we write if ≤ holds up to a positive constant that depends only on d, p and q. The proof is divided in three steps: (i) using Lemma 2 and an iteration argument, we obtain a one-step improvement; (ii) the one-step improvement and a Moser iteration-type argument yield local boundedness in the form (51) where the L 2 -norm on the right-hand side is replaced by a slightly stronger norm of u; (iii) finally a well-known interpolation argument yield the claimed estimate.

2.3.
Proof of Theorem 1. With the local boundedness statement Theorem 5, the oscillation decay can be proven by already established methods. The following argument is essentially the parabolic version (in the form of [36, Section 5.2]) of Moser's proof, see [30], of the De Giorgi theorem in the elliptic case. In recent works [3,7] this strategy is already adapted to the discrete and degenerate situation that we consider here but under more restrictive summability assumption on ω and ω −1 . However, in order to keep the presentation self-contained we provide a detailed proof below. First, we introduce a suitable regularization of the map z → (− log(z)) + , defined by is the smallest solution of 2c log( 1 c ) = 1 − c. Notice that g ∈ C 1 ((0, ∞)) is non-negative, convex and non-increasing. (see (11) for the definition of m(·)). Then, for any (73) σ 1 ∈ (0, λ) and σ 2 ∈ (λ, 1) satisfying there exists h = h(d, λ, ω L 1 (B(n)) , σ 2 ) ∈ (0, 1) such that Moreover, there exists c = c(d) < ∞ such that (74) holds with Proof of Lemma 3. The proof follows the argument of [36, Lemma 5.2.3] (and discrete variants [3,7]).
Proof of Theorem 6. Without loss of generality, we assume ε = 1. Consider the function (t, x) → W t (x) := G(u t (x)), where G(s) := g( s+γ h ) with s ∈ R and suitable constants 0 < γ < h which are specified later.
Step 1. W is a subcaloric function, i.e. d dt W t (x) − L ω W t (x) ≤ 0 for all (t, x) ∈ Q(n). Indeed, this is a consequence of the convexity of G in the form combined with the fact that u is supercaloric and G ′ ≤ 0, and thus Step 2.
Squaring the above expression and integrating in time from −σ 1 n 2 to 0, we obtain (87) using (89).
Finally, we recall the computations that yield (91): For every s ∈ [1, d), we have (with the notation of Estimate (91) follows from (92) with s = 2d d+2 (and thus s * d = 2) and Hölder inequality in the form Step 3. Conclusion.
Next, we combine Proposition 2 with Theorem 1 to obtain large-scale (Hölder-) continuity of the heat kernel provided, we control ω L p (B) and ω −1 L q (B) in the limit |B| → ∞.
From Proposition 2, we directly deduce the corresponding heat kernel estimate for the random conductance model. This extends [29,Proposition 3.6] to the case of unbounded conductances.
Corollary 2. Suppose that Assumption 1 is satisfied and that there exists p ∈ (1, ∞), q ∈ ( d 2 , ∞) satisfying 1 p + 1 q < 2 d−1 such that (9) is valid. Let ν = ν(d, p, q) ∈ (0, 1) be as in (22). Then there exists a random variable X ≥ 0 such that Proof of Corollary 2. By Proposition 2 and the definition of C (see (52)) there exists c = c(d, p, q) ∈ [1, ∞) such that for every t ≥ 1 with C max given by where M denotes the maximal operator given by for all r ≥ 1 and f : Z d → R.

3.2.
Proof of Theorem 4. By now it is well-established that quenched invariance principles (see Theorem 3) combined with additional regularity properties of the heat kernel yield local limit theorems, see [5,11]. Hence we only provide sketch of the proof.
Proof of Theorem 4. We only show that for every x ∈ R d and t > 0 (109) lim n→∞ |n d p ω n 2 t (0, ⌊nx⌋) − k t (x)| = 0 P-a.s., where the Gaussian heat kernel k t is defined in (7). From the pointwise result (109) the desired claim (10)  where the right-hand side tends to zero as δ → 0.
In the proof of Theorem 4 we used the following consequence of the spatial ergodic theorem:

Elliptic regularity: Proof of Theorem 2
We adapt the classical strategy of Moser [30] to the non-uniformly elliptic and discrete setting. As in the parabolic case, a key ingredient is local boundedness for non-negative subharmonic functions.
Proof. In [13, Corollary 1, Proposition 4] the corresponding estimates with max u replaced by max |u| are proven for harmonic functions (i.e. u satisfying L ω u = 0) without any sign condition on u. The proofs apply almost verbatim to non-negative subharmonic functions and thus are omitted here.
Step 1. W is subharmonic, i.e. −L ω W ≤ 0 in B(4n). This follows from Step 1 of the proof of Theorem 6.
Step 4. The case d = 2. Assumption (115) and a suitable version of Sobolev inequality (see (91)) yield Proof of Theorem 2. Appealing to the weak Harnack inequality Theorem 8 the proof follows by the same argument as in the parabolic case, see Theorem 1.