Stochastic homogenisation of high-contrast media

Using a suitable stochastic version of the compactness argument of [V. V. Zhikov, 2000. On an extension of the method of two-scale convergence and its applications. Sb. Math., 191(7--8), 973--1014], we develop a probabilistic framework for the analysis of heterogeneous media with high contrast. We show that an appropriately defined multiscale limit of the field in the original medium satisfies a system of equations corresponding to the coupled"macroscopic"and"microscopic"components of the field, giving rise to an analogue of the"Zhikov function", which represents the effective dispersion of the medium. We demonstrate that, under some lenient conditions within the new framework, the spectra of the original problems converge to the spectrum of their homogenisation limit.


Introduction
Asymptotic analysis of differential equations with rapidly oscillating coefficients has featured prominently among the interests of the applied analysis community during the last half a century. The problem of understanding and quantifying the overall behaviour of heterogeneous media has emerged as a natural step within the general progress of material science, wave propagation and mathematical physics. In this period several frameworks have been developed for the analysis of families of differential operators, functionals and random processes describing multiscale media, all of which have benefitted from the invariably deep insight and mathematical elegance of the work of V. V. Zhikov. In the present paper, we touch upon two subjects in which his contributions have inspired generations of followers: the stochastic approach to homogenisation, in particular through his collaboration with S. M. Kozlov during the 1980s, and the analysis of differential operators describing periodic composites with high contrast, which started with his fundamental contribution [1].
Our present interest in the context of stochastic homogenisation of high-contrast composites stems from the relationships that have recently been indicated between media with negative material properties ('metamaterials'), and more generally time-dispersive media, and 'degenerate' families of differential operators, where e.g. loss of uniform ellipticity of the symbol is known to lead to nonclassical dispersion relations in the limit of vanishing ratio ε of the microscopic (l) and macroscopic (L) lengths: ε = l/L → 0. The work [1] has provided an example, in the periodic context, of what one should expect in the limit as ε → 0 in terms of the two-scale structure of the solution as well as the spectrum of the related differential operator, in the case when the metamaterial is modelled by disjoint 'soft' inclusions with low, order O(ε 2 ) values of the material parameters (say, elastic constants in the context of linearised elasticity), embedded in a connected 'stiff' material with material constants of order O (1). In mathematical terms, the coefficients of the corresponding differential expression alternate between values of different orders in ε, where the contrast increases as ε gets smaller.
In the present article, we introduce a stochastic framework for the analysis of homogenisation problems with soft inclusions and explore the question on what version of the results of [1] can be achieved in this new framework. In particular, we are interested in the equations that describe the stochastic two-scale limit, in an appropriate sense, of the sequence of solutions to the probabilistic version of a Dirichlet problem in a bounded domain of R n . Furthermore, we show that the spectra of such problems converge, in the Hausdorff sense, to the spectrum of the limit problem, which we analyse in a setting that models distributions of soft inclusions whose shapes are taken from a certain finite set and whose sizes vary over an interval. To our knowledge, the present manuscript is the first work containing an analysis of random heterogeneous media with high contrast that results in a 'complete' Hausdorff-type convergence statement for the spectra of the corresponding differential operators. Various aspects of multiscale analysis of high-contrast media in the stochastic context have been looked at in a handful of papers, e.g. [2][3][4].
While in the periodic context norm-resolvent convergence results been obtained for high-contrast media, see [5,6], the stochastic case remains open to developments of a similar nature. It is anticipated that the operator-theoretic approach to problems of the kind we discuss in the present article will provide a general description of the types of spectral behaviour that can occur in the real-world applications where it is difficult to enforce periodicity of the microstructure. On the other hand, as we show in the present work, new wave phenomena should be expected in the stochastic setting (e.g. a non-trivial continuous component of the spectral measure of the homogenised operator for a bounded-domain problem), which makes the related future developments even more exciting.
Next, we outline the structure of the paper. In Section 2 we recall the notion of stochastic twoscale convergence, which we use, in Sections 3 and 4, to pass to the limit, as ε → 0, in a family of homogenisation problems with random soft inclusions. In Section 3 we give a formulation of the high-contrast problem we study and provide some auxiliary statements. In Section 4 we describe the limit problem and prove the strong resolvent convergence of the ε-dependent family to the limit system of equations. In Section 5 we provide a link between the spectra of the Laplacian operator on realisations of the inclusions and of the corresponding stochastic Laplacian. In Section 6 we prove that sequences of normalised eigenfunctions of the ε-dependent problems are compact in the sense of strong stochastic two-scale convergence. Finally, in Section 7 we discuss two examples of the general stochastic setting and describe the structure of the corresponding limit spectrum.
In conclusion of this section, we introduce some notation used throughout the paper. For a Banach space X and its dual X * , we denote by X ·, · X * the corresponding duality. For a Hilbert space H the inner product of a, b ∈ H is also denoted by a, b H and, if H = R n , by a · b. For a set O we denote by χ O its characteristic function, which takes value one on the set O and zero on the complement to O in the appropriate ambient space. For D ⊆ R n we denote by D its closure and by |D| its Lebesgue measure. Further, we use the notation B r (0) for the ball in R n of radius r with the centre at the origin; Y denotes the cube [0, 1) n with torus topology, where the opposite faces are identified; and N l 0 := {0, . . . , l}. For an operator A on some Hilbert space, we denote by SpA its spectrum. Finally, for a Lipschitz open set D ⊂ R n , we denote by − D the (positive) Laplace operator with the Dirichlet boundary condition on ∂D. For x ∈ R n , we denote by [x] the element of Z n which satisfies [x] ≤ x < [x] + (1, . . . , 1). For k = 1, . . . , n, by e k we denote the kth coordinate vectors.

Probability framework
Let ( , F, P) be a complete probability space. We assume that F is countably generated, which implies that the spaces L p ( ), p ∈ [1, ∞), are separable. For a function u ∈ L 1 ( ), we will sometimes write u for´ u. Definition 2.1: A family (T x ) x∈R n of measurable bijective mappings T x : → on a probability space ( , F, P) is called a dynamical system on with respect to P if: is measurable (for the standard σ -algebra on the product space, where on R n we take the Lebesgue σ -algebra).
We next define the notion of ergodicity for dynamical systems introduced above.

Definition 2.2:
A dynamical system is called ergodic if one of the following equivalent conditions is fulfilled: Henceforth we assume that the dynamical system T x is ergodic.

Remark 2.1:
Note that for the condition (b) the implication P(B) ∈ {0, 1} has to hold, if the symmetric difference between T x B and B is a null set. It can be shown (see, e.g. [7]) that ergodicity is also equivalent to an a priori weaker implication There is a natural unitary action on L 2 ( ) associated with T x : It can be shown that the conditions of Definition 2.1 imply that this is a strongly continuous group (see [8]). It is often necessary that the set of full measure be invariant in the sense that together with the point ω it contains the whole 'trajectory' {T x ω, x ∈ R m }. This requirement can always be met on the basis of the following simple lemma (see [8,Lemma 7.1]).

Lemma 2.1:
Let 0 be a set of full measure in . Then there exists a set of full measure 1 such that 1 ⊆ 0 , and for a given ω ∈ 1 we have T x ω ∈ 0 for almost all x ∈ R m .
For each j = 1, 2, . . . n, we define the infinitesimal generator D j of the unitary group {U(x)} x∈R n by the formula where the limit is taken in L 2 ( ). Notice that iD j , j = 1, . . . , n, are commuting, self-adjoint, closed and densely defined linear operators on the separable Hilbert space L 2 ( ). The domain D j ( ) of such an operator is given by the set of L 2 ( )-functions for which the limit (2) exists. We consider the set and similarly It is shown by the standard semigroup property that W ∞,2 ( ) is dense in L 2 ( ). We also define the space By the smoothening procedure discussed in [8, p.232] (see also the text before Lemma 3.1 below), it is shown that C ∞ ( ) is dense in L p ( ) for all p ∈ [1, ∞) as well as in W k,2 ( ) for all k. Furthermore, it is shown that W 1,2 ( ) is separable. Notice that, due to the infinitesimal generator being closed, D j f can be equivalently defined as the function that satisfies the propertŷ If f ∈ W 1,2 ( ), we may also define D j f (x, ω) := D j f (T x ω) for all x ∈ R n . It can be shown that the following identity holds (see [9]): Moreover, for a.e. ω ∈ the function D i f (·, ω) is the distributional derivative of f (·, ω) : a proof of this fact can be found in [9,Lemma A.7]. Following [10], we denote by · #,k,2 the seminorm on C ∞ ( ) given by By W k,2 ( ) we denote the completion of C ∞ ( ) with respect to the seminorm · #,k,2 . The gradient operator ∇ ω := (D 1 , . . . , D n ) and the operator div ω := ∇ ω · are extended uniquely by continuity to mappings from W 1,2 ( ) to L 2 ( , R n ) and from W 1,2 ( , R n ) to L 2 ( ), respectively. Finally, by a density argument, it is easily seen that W 1,2 ( ) is also the completion of W 1,2 ( ) with respect to · #,1,2 .

Definition and basic properties
The key property of ergodic systems is the following theorem, due to Birkhoff (for a more general approach, see [11]).
Furthermore, for all f ∈ L p ( ), 1 ≤ p ≤ ∞, and a.e. ω ∈ , the function f ( The elements ω such that (6) holds for every f ∈ L 1 ( ) and bounded open B ⊂ R n are refereed to as typical elements, while the corresponding sets (T x ω) x∈R N are called typical trajectories. Note that the separability of L 1 ( ) implies that almost every ω ∈ is typical, and in what follows we only work with such ω.
For vector spaces V 1 , V 2 , we denote by by V 1 ⊗ V 2 their usual tensor product. We define the following notion of stochastic two-scale convergence, which is a slight variation of the definition given in [12]. In [13], the authors average over the probability space and do not use the Birkhoff Ergodic Theorem. As a consequence, they do not obtain convergence almost everywhere but only in mean, which results in a weaker notion of stochastic two-scale convergence than the one introduced in [12]. In the context of calculus of variations, the first results are obtained in [14,15]. The authors of these papers do not use stochastic two-scale convergence at all, as this was introduced later on, but rely on a formula for non-periodic homogenisation for a.e. ω ∈ as well as on the ergodic theorem. We shall stay in the Hilbert setting (p = 2), as it suffices for our analysis. Finally, we denote by S a bounded open LIpschitz set in R n .

Definition 2.3:
Let (T x ω) x∈R n be a typical trajectory and (u ε ) a bounded sequence in L 2 (S). We say that (u ε ) weakly stochastically two-scale converges to u ∈ L 2 (S × ) and write If additionally u ε L 2 (S) → u L 2 (S× ) , we say that (u ε ) strongly stochastically two-scale converges to u and write u ε 2 − → u.

Remark 2.2:
The convergence of (u ε ) is defined along a fixed typical trajectory and a priori the limit depends on this trajectory. In applications, such as the analysis of the PDE family in Section 4, it often turns out that the limit does not depend on the trajectory chosen. For this reason, and to simplify notation, in what follows we often do not indicate this dependence explicitly. Note also that, by density, the set of admissible test functions g in (7) can be extended to L 2 (S) ⊗ L 2 ( ).
In the next proposition, we collect the properties of stochastic two-scale convergence that we use in the present work. (a) Let (u ε ) be a bounded sequence in L 2 (S). Then there exists a subsequence (not relabelled) and Let (u ε ) be a bounded sequence in W 1,2 (S). Then on a subsequence (not relabelled) u ε u 0 in W 1,2 (S), and there exists u 1 ∈ L 2 (S, W 1,2 ( )) such that (f) Let (u ε ) be a bounded sequence in L 2 (S) such that ε∇u ε is bounded in L 2 (S, R n ). Then there exists u ∈ L 2 (S, W 1,2 ( )) such that on a subsequence Proof: In view of analogies with the periodic case, we just give a sketch of the proof. A proof of (a) can be found in [12,Lemma 5.1]. For the proof of (b), we take an arbitrary g ∈ C ∞ 0 (S) ⊗ C ∞ ( ) and calculate lim inf We obtain the claim by approximating The proof of (c), (d) is straightforward. The proof of (e) goes in the same way as in the periodic case, by the duality argument.
First, one proves that if f ∈ L 2 ( , R n ) is such that then there exists ψ ∈ W 1,2 ( ) such that f = ∇ ω ψ. One then proceeds in the same way as in the periodic case (see [16]). In order to show the claim (f), take the subsequence such that and using integration by parts we conclude from which the claim follows by a density argument, in view of the property (4).

Problem formulation and auxiliary statements
Let S ⊆ R n be a bounded open Lipschitz set. We take O ⊆ such that 0 < P(O) < 1 and for each ω ∈ consider its 'realisation' We assume that the following conditions are satisfied.

Assumption 3.1: For a.e. ω ∈ one has
where: are open connected sets with Lipschitz boundary; and C ω > 0 such that for all k ∈ N the following extension property holds: It is easily seen that Assumption 3.1 holds for the examples given in Section 7.1. Denote by the set of typical elements ω ∈ satisfying the conditions listed in Assumption 3.1, and for all ω ∈ , ε > 0 define S ε 0 (ω) as the union of all components εO k ω that are subsets of S and stay sufficiently far from its boundary, in the sense that there exists C = C(ω) > 0 such that We denote the complement of the set S ε 0 (ω) by S ε 1 (ω) := S \ S ε 0 (ω) and the corresponding set indicator functions by χ ε 0 (ω) and χ ε 1 (ω). For each ω ∈ , we consider the following Dirichlet problem in S: for λ < 0 and f ε ∈ L 2 (S), find where For all ω ∈ we also define the Dirichlet operator A ε (ω) in L 2 (S) corresponding to the differential expression −div A ε (·, ω)∇u, e.g. by considering the bilinear form It is well known that the spectrum of A ε (ω) is discrete. The following subspace of W 1,2 ( ) will play an essential role in our analysis: Notice that as a consequence of Ergodic Theorem (Theorem 2.1) one has i.e. W 1,2 0 (O) consists of W 1,2 -functions that vanish on \O. Henceforth we assume that ω ∈ without mentioning it explicitly. The next two lemmas use a standard smoothening (or 'mollification') procedure, which we now describe. We take g ∈ L 2 ( ) and (cf. [8, p.232 (12) from which we infer that Arguing by induction, we show that R δ [g] ∈ W ∞,2 ( ), and if g ∈ L ∞ ( ) then R δ [g] ∈ C ∞ ( ). Before we state and prove the lemmas, we introduce additional notation. We define the space as well as the sets Also, for all m ∈ N we define the set By using the density of Q n in R n it can be seen that for all m ∈ N the set B m is measurable. Notice that for each fixed ω, k, m, where m is large enough, there exist constants C 1 , C 2 > 0 such that In the next lemma we assume that a relaxed version of the right inequality in (13) Proof: Using Ergodic Theorem and the assumption of the lemma, it can be shown that P(O\B m ) → 0 as m → ∞. To prove the density, it suffices to approximate g : , for which we use the above mollification procedure. Notice that for δ > 0 small enough, one has R δ [g] ∈ C ∞ 0 (O). It remains to check R δ [g] → g as δ → 0, but this follows from the strong continuity of the group U(x), see (1): Notice that, by the standard Poincaré inequality, for each D k,m ω there exists C > 0 such that In the following lemma we impose this condition uniformly.

Lemma 3.2:
Assume that for a.e. ω ∈ there exists a constant C > 0 such that and that (14) is satisfied for all k and large enough m. Then the set Proof: We take f ∈ W 1,2 0 (O) and define f M := χ |f |≤M f + χ |f |≥M M. Notice that as a consequence of (5), Notice also that for a.e. ω ∈ there exist C 1 , C 2 > 0 such that for all k, m ∈ N, where m is sufficiently large, we have where we have used (12), (14), (15) and Young's inequality. Using the Ergodic Theorem we conclude that there exists C > 0 such that from which the claim follows.

Limit equations and two-scale resolvent convergence
We define the quadratic form and denote by DW 1,2 ( \O) the completion of The proof of the following lemma is straightforward.
In particular, one has A hom 1 ≤ A 1 .

Remark 4.1:
It follows from the observations in [12, p.265-266] that if the following extension property is satisfied for a.e. ω ∈ : for all u ∈ C ∞ 0 (B 1 (0)) there exists ε 0 ω > 0 and a sequence of functions ( u ε ) such that where C ω is a constant independent of u and ε, then the matrix A hom 1 is positive definite.
Notice that under Assumption 3.1, the extension property in Remark 4.1 is satisfied. We define the space which is clearly a direct sum, naturally embedded in L 2 (S × ). Before stating the next theorem we prove a simple lemma that implies that gives norm bounds for each component of H by the norm in L 2 (S × ).
Proof: By Cauchy-Schwartz inequality we have and hence It remains to bound f 0 L 2 (S× ) by f 0 + f 1 L 2 (S× ) , which is done by the triangle inequality: By P : L 2 (S × ) → H we denote the orthogonal projection. For f ∈ L 2 (S × ) we have For each ε > 0, consider the solution u ε to (11). Then for a.e. ω ∈ one has u ε 2 Remark 4.2: The system (18) and (19) is understood in the weak sense: Noting that W 1,2 0 (O) is a closed subspace of W 1,2 ( ) and bearing in mind Lemma 4.2, it follows by the Lax-Milgram lemma that for all f ∈ L 2 (S × ), x ∈ S the problem (20) has a unique solution in W 1,2 0 (O). Its solutions for the right-hand sides f ∈ L 2 (S × ) and Pf coincide. The solution of the Equation (19) has the form where u ∈ L 2 (S, W 1,2 0 (O)) is the solution of (19) obtained by setting u 0 = 0 and w ∈ W 1,2 0 (O) is the solution of (19) obtained by setting u 0 = 1 and f = 0. Substituting (21) into (18), we obtain an equation on u 0 .

Remark 4.3:
The following observation was made in [1] in the periodic setting. The formulation (20) can be interpreted from the operator-theoretic point of view. Namely, we define a positive-definite operator A on a dense linear subset of V = W 1,2 0 (S) + L 2 (S, W 1,2 0 (O)) (which is a dense subset of H under the condition of Lemma 3.1), as follows. One takes λ < 0 and defines the domain dom(A) as the set of solutions of (20) obtained for varying f ∈ H. To see that dom(A) is dense in H, take the solutions u 0 + u 1 , w 0 + w 1 ∈ V for f , g ∈ H, respectively. Setting ϕ 0 = u 0 , ϕ 1 = u 1 as the test function in the equation for (w 0 , w 1 ) and ϕ 0 = w 0 , ϕ 1 = w 1 as the test function in the equation for Thus, if g ⊥ u 0 + u 1 then necessarily w 0 + w 1 = 0, which implies g = 0. The operator A : is a bounded linear mapping between Hilbert spaces, where the norm on dom(A) is given by We shall need the following statement for the convergence of spectra of the operators associated with (11). It is proved in the same way as the previous theorem, and we omit the proof.
Then for a.e. ω ∈ one has z ε 2 − z(·, ω), where z ∈ L 2 (S, W 1,2 0 (O)) is the solution of the problem Remark 4.4: Theorem 4.1 and Proposition 4.1 are still valid if, instead of a fixed λ < 0, we take a sequence (λ ε ) ⊆ R such that λ ε → λ ∈ R and lim inf ε→0 dist(λ ε , SpA ε ) > 0, for Theorem 4.1, i.e. lim inf ε→0 dist(λ ε , SpT ε ) > 0 for Proposition 4.1, where T ε := −ε 2 S ε 0 . Notice that SpT ε splits into the spectra of scaled Laplace operators on each inclusion contained in S ε 0 : where K ε ω is defined in (10). Notice that there exists C > 0 such that for all λ ∈ R the solution u ε of (11) satisfies ∇u ε and similarly the solution of (25) satisfies In what follows we denote by − ω the operator generated by the bilinear form As a consequence of Proposition 4.1 and Remark 4.4, we have the following statement.

Corollary 4.1: Assume that Assumption 3.1 holds. Then
As a consequence of Remark 4.4, the sequence of solutions of (25) converges weakly two-scale to the solution of (26), which is a resolvent equation. Moreover, (27) and (29) imply the existence of C > 0 such that , and therefore λ / ∈ Sp(− ω ).

Spectral completeness for inclusions
Next we prove that We shall use the assumptions of Lemma 3.2 as well as assume that for each λ 0 > 0 there exists M λ 0 > 0 such that for a.e. ω ∈ the following implication holds: . (31) Notice that, by regularity theory, the above condition is satisfied for a fixed ω ∈ and k ∈ N, whenever the boundary ∂O k ω is sufficiently regular. In what follows we use a sequence where the constant c 2 is defined in Assumption 3.1.
We will now define a sequence of random variables that is invariant for all ω ∈ O whose realisation is such that the shape that contains the origin is the same. For q = (q 1 , . . . , q n ) ∈ Q n define the set Since for each fixed q ∈ Q n there is a countable set of lines satisfying the property (32), the set O q is measurable.
We define the random variables Notice that F j = +∞ whenever ω / ∈ O, and also, due to the assumption, −c 2 ≤ F j ≤ 0 for a.e. ω ∈ O. We denote by F the random vector For a.e. ω ∈ O, m ∈ N we define the set Furthermore, we introduce the set U ω ⊂ [0, c 2 + 1] n , which is a translation of the set O k 0 ω containing the origin: Finally, we define a characteristic function of the translation of the set F m ω and a measurable function of ω taking values in W 1,2 0 ([0, c 2 + 1] n ): Notice that for a.e. ω ∈ O one has supp ϕ k,m (·, ω) ⊂ U ω .

Proof: Firstly notice that
is a measurable mapping taking values in the set L 2 ([0, c 2 + 1] n ), with Borel σ -algebra. To check this notice that for each q ∈ Q n the set is measurable: the related proof is similar to that of Lemma 5.1. Further, for ψ ∈ C ∞ 0 (R n ) the norm ψ − χ m ϕ k L 2 (R n ) is written as a limit of Riemann sums, and each Riemann sum can be written in terms of a finite number of χ L q and values of function ϕ k (·). Thus ω → ψ − χ m ϕ k L 2 (R n ) is measurable. Since the topology in L 2 (R n ) is generated by the balls of the form B(ψ, r), where ψ ∈ C ∞ 0 (R n ) and r ∈ Q we have that the mapping given by (35) is measurable. The claim follows by using the fact that the convolution is a continuous (and thus measurable) operator from L 2 to W 1,2 .
Notice that by construction {ϕ k,m (·, ω)} k,m∈N ⊂ C ∞ 0 (U ω ) is a dense subset of W 1,2 0 (U ω ) for a.e. ω ∈ (see also the proof of Lemma 3.2). For 0 ≤ a ≤ b we introduce the following subset of O : For 0 ≤ a ≤ b and a.e. ω ∈ E a,b we also define S a,b,ω ⊂ W 1,2 0 (U ω ) as follows: Finally, for every r ∈ R and k, m ∈ N we define the random variable

Lemma 5.4: For 0 ≤ a ≤ b, the set E a,b is measurable.
Proof: The claim follows by observing that Now we are going to define a measurable mapping from O to the subspace S a,b,ω . We set it to be an L 2 -projection onto S a,b,ω of a specially chosen function of x and ω. We need the following measurability lemma. ϕ(·, ω), S a,b,ω ), ω ∈ E a,b , is a measurable map.

Proof:
The claim follows from the formula For 0 ≤ a ≤ b and ω ∈ E a,b we define a measurable map ω → ϕ a,b (·, ω) as follows: and m 0 (ω) is the minimal value of m in (38) Notice that in this way for a.e. ω ∈ E a,b the L 2projection of ϕ a,b (·, ω) on S a,b,ω is not zero. We also define the random variable R : → [0, +∞) in the following way: By invoking the measurability of O q , q ∈ Q n , see Lemma 5.1, it is easily seen that R is indeed measurable. Next, for 0 ≤ a ≤ b, l > 0 we define the random variable ψ a,b,l : → R by where, for all n ∈ N, Notice that in this way ψ a,b is the value at the origin (taking into account for ω ∈ O the relative position of the origin with respect to the shape) of the (unique) L 2 -projection of ϕ a,b onto S a,b,ω . As a consequence of (31), we have |ψ a,b | ≤ M b . Notice that by construction ψ a,b = 0 if P(E a,b ) > 0. We are ready for the proof of main statement. (b) For all ε > 0 the set E l−ε,l+ε has positive probability.
In the case (a), by the continuity of probability, we conclude that P(E 0 l−ε 0 (l),l+ε 0 (l) ) = 0, where (cf. (36)) By Lemma 2.1 and Corollary 4.1 we infer that In particular, we conclude that l / ∈ Sp(− ω ). In the case (b) we construct a Weyl sequence showing that l ∈ Sp(− ω ). To this end, we define Then, by the above construction and using Ergodic Theorem, one has It follows from the above that Sp(− ω ) consists of exactly those l ∈ R that satisfy the property (b). The set Sp(− ω ) is closed, hence its complement is a countable union of open disjoint intervals. Every element of such an interval (d 1 , d 2 ) satisfies the property (a) with l = (d 1 + d 2 )/2, ε 0 (l) = (d 2 − d 1 )/2, and therefore P(E 0 d 1 ,d 2 ) = 0. Using Lemma 2.1, we obtain The claim follows since there is only countable number of such intervals.

Convergence of spectrum
In our analysis we keep in mind the examples set in Section 7, for which it is shown that Sp(− ω ) ⊆ Sp(A). In the present section we assume that this holds, as well as the conclusion of Theorem 5.1, i.e.
We are interested in approximating the spectra SpA ε (ω) of the operators A ε (ω) (see Section 3) by the spectrum SpA of the limit operator. We claim that SpA ε (ω) → SpA for a.e. ω ∈ , where the convergence is understood in the Hausdorff sense: (a) For all λ ∈ SpA there are λ ε ∈ SpA ε (ω) such that λ ε → λ.
We prove this claim by adapting the argument of [17]. First, we introduce the notion of strong resolvent convergence. Definition 6.1: Let A ε (ω) and A be the operators acting on L 2 (S) and on H ⊂ L 2 (S × ), respectively. We say that A ε (ω) strongly two-scale resolvent converge to A and write It can be shown that the property (a) is satisfied if we have strong two-scale resolvent convergence (see the proof of [17, Proposition 2.2]). Theorem 4.1 shows that the following implication holds: It can be shown that this is equivalent to strong two-scale resolvent convergence (see [17,Proposition 2.8]) and thus the property (a) is satisfied. In order to prove (b), we start from the eigenvalue problem for the operator A ε (ω) (it has a compact resolvent and its spectrum is discrete), i.e.
If we have that s ε → s and u ε 2 − u, then we would also have Au = su. However, the problem would be if u = 0, because then s / ∈ SpA. The next lemma tells us if s / ∈ Sp(− ω ) then necessarily the sequence of eigenvalues is compact with respect to the strong two-scale convergence and thus s belongs to the point spectrum of the operator A, since then necessarily u = 0. Theorem 6.1: Suppose that (39) holds and that for each ε > 0, (s ε , u ε ) satisfy (40). If s ε → s / ∈ Sp(− ω ), then for a.e. ω ∈ the sequence (u ε ) is compact in the sense of strong two-scale convergence.
We use Assumption 3.1 and for each ε extend u ε | S ε 0 , denoting the extensions by u ε . Notice that there exists C > 0 such that The difference z ε := u ε − u ε satisfies: From the estimate (41) we see that ( u ε ) is weakly compact in W 1,2 0 (S) and thus there exists . Furthermore, as a consequence of (27), (29) and (39), the following estimate holds for some C > 0 : Therefore, from Proposition 4.1 and Remark 4.4 we conclude that z ε 2 − z ∈ L 2 (S, W 1,2 0 (O)), where the limit z satisfieŝ We also consider the problem In the same way as before we conclude that for some C > 0: By testing (42) with m ε and (44) with z ε we conclude Finally, by testing (43) with m(x, ·) and (45) with z(x, ·) and integrating over S we concludê which completes the proof.

Spectrum of the limit operator: examples
This section is devoted to the description of the spectrum of the limit operator. Since it crucially depends on the intrinsic properties of the microscopic part of the operator and the properties of the probability space, it does not seem feasible (at least at the current stage of research in this area) to provide a characterisation of the spectrum in a general setting. We shall consider several interesting, from the point of view of applications, examples of probability spaces and configurations of soft inclusions. The general example of a finite number of shapes of randomly varying size is described in Section 7.1. Then we consider the case of a single shape of fixed size in Section 7.2, and the case of a single shape of randomly varying size in Section 7.3, for which we provide the full description of the spectrum of the limit operator with the proofs. The characterisation of the spectrum in the general case of Section 7.1 is analogous to the case of a single shape considered in Section 7.3.

The setting of finite number of shapes of varying size
Let ( ω j ) j∈Z n be a sequence of independent and identically distributed random vectors taking values in N l 0 × [r 1 , r 2 ], where 0 < r 1 ≤ r 2 ≤ 1 and ( , F , P) is an appropriate probability space. We also assume that we have a finite number of shapes Y k ⊂ Y := [0, 1) n , k ∈ N l 0 , that represent the inclusions, where the first and the second components of ω j = (k j , r j ) model the shape and the size, respectively. We also set Y 0 = ∅. On there is a natural shift T z ( ω j ) = ( ω j−z ), which is ergodic. We next state the discrete analogue of Lemma 2.1. Lemma 7.1: Assume that 0 ⊆ is a set of full measure. Then there exists a subset 1 ⊆ 0 of full measure such that for each ω ∈ 1 , z ∈ Z n we have T z ω ∈ 0 .
We treat Y as a probability space with Lebesgue measure dy and the standard algebra L of Lebesgue measurable sets, and define It is easily seen that O is measurable. For a fixed ω = ( ω, y), the realisation O ω consists of the inclusions r j Y k j + j − y, j ∈ Z n . Next, we describe the generators D j , j = 1, 2, . . . , n, in the present example. Taking f ∈ W 1,2 ( ) and using the above lemma, note that there exists a subset of full measure 1 ⊆ such that for all ω ∈ 1 and z ∈ Z n we have Using this fact and the statement following (5), we infer that and

Simple example
In this section we set l = 0, r 1 = r 2 = 1, so that N l 0 × [r 1 , r 2 ] = {0, 1}, and, by a standard procedure, see e.g. [18], identify the elements of the probability space with sequences ω = ( ω z ) z∈Z n whose components ω z take values in the two-element set {0, 1}. Let Y 1 be an open subset of Y whose closure is contained in Y ('soft inclusion'). The value 0 or 1 of ω z , z ∈ Z n , corresponds to the absence or the presence of the inclusion in the 'shifted cell' Y + z, respectively. We also set Then, for a given ω = ( ω, y) ∈ , the realisation O ω = {x : T x ( ω, y) ∈ O} is the union of the sets ('inclusions') Y 1 + z − y over all z ∈ Z n such that ω z = 1. For this example the space consists of all functions of the form It is also important to understand how one applies the stochastic gradient. For a function v(ω) ∈ W 1,2 0 (O) ⊆ L 2 ( ) we have (see (46)) Consider formally the spectral problem for the limit operator: We write the solution to the 'microscopic' Equation (50) in the form In other words, v is given by (47) with v ω (y) satisfying whenever ω such that ω 0 = 1 and v ω = 0 otherwise. We label the eigenvalues of the operator in (52) in the increasing order, where we repeat multiple eigenvalues, so that ν j , j ∈ N, and ν j , j ∈ N, are, respectively, the eigenvalues whose eigenfunctions ϕ j have non-zero integral over Y 1 and the eigenvalues whose eigenfunctions ϕ j have zero integral over Y 1 . Following [1], we write the solution to (52) via the spectral decomposition and Substituting the obtained representation for u 1 into the 'macroscopic' equation (49) yields where is a stochastic version of the 'Zhikov function' β in [1]. Assume for the moment that S = R n . Then the intervals where β(λ) ≥ 0 are the 'spectral bands' of A, and additionally a Bloch-type spectrum is given by {ν j : j ∈ N}. The set {λ : β(λ) < 0} \ {ν j : j ∈ N} corresponds to the gaps in the spectrum of A.
In the setting of this paper, namely, for a bounded domain S ⊂ R n , instead of each spectral band β(λ) ≥ 0 lying to the left of ν j we have a 'band' of discrete spectrum: a countable set of eigenvalues with the accumulation point at the right end ν j of each band, where μ k are the eigenvalues of the operator −div A hom 1 ∇ defined by the form The Bloch-type spectrum of A consists of eigenvalues ν j of infinite multiplicity with eigenfunctions of the form Summarising, the spectrum of A is given by

More advanced example
Here we allow the inclusions to randomly change size, so that l = 0, 0 < r 1 < r 2 < 1. By analogy with the previous section, we assume that consists of sequences ω = ( ω z ) z∈Z n such that ω z ∈ {0} ∪ [r 1 , r 2 ], z ∈ Z n . We also assume that the restriction to to [r 1 , r 2 ] of the probability measure on is absolutely continuous with respect to Lebesgue measure. As before, consider Y 1 ⊂ Y, and denote by Y 1,r := r(Y 1 − y c ) + y c , where y c is the centre of Y, the 'scaled inclusion', requiring that Y 1,r 2 ⊂ Y, in order for the extension property in Assumption 3.1 to hold. The values 0 or r ∈ [r 1 , r 2 ] of ω z correspond to the absence of an inclusion or the presence of the inclusion Y 1,r in the cell Y + z, respectively. Furthermore, define O := {ω = ( ω, y) : y ∈ Y 1, ω 0 } ⊆ . Then a realisation O ω = {x : Consider the spectral problem for u 1 , namelŷ and separate the variables, as in Section 7.2: The stochastic gradient is given by and therefore the problem (62) is equivalent tô For each r ∈ [r 1 , r 2 ], the eigenvalues ν j,r , ν j,r and (orthonormal) eigenfunctions ϕ j,r , ϕ j,r of the operator − y acting in W 1,2 0 (Y 1,r ) are obtained by scaling the eigenvalues and eigenfunctions of − y acting in W 1,2 0 (Y 1 ), in particular, ν j,r = r −2 ν j , ν j,r = r −2 ν j . Therefore, the formula (53) with ν j , ϕ j replaced by ν j,r , ϕ j,r gives the solution to If 0 < r 1 ≤ r 2 and the set {ν j,r : j ∈ N, r ∈ [r 1 , r 2 ]} has gaps, then for λ ∈ R \ {ν j,r : j ∈ N, r ∈ [r 1 , r 2 ]} the solution to (63) is given by (61), where the functions v ω (y) solve (64) with r = ω 0 . Substituting it into the spectral problem for (49) yields the problem (55) with the Zhikov-type function β given by (cf. (56)) The integral in (65) is well defined for λ ∈ R \ {ν j,r : j ∈ N, r ∈ [r 1 , r 2 ]}, and the description of the spectrum on the intervals where β(λ) > 0 follows Section 7.2. It is clear that if the set j∈N, r∈[r 1 ,r 2 ] {ν j,r , ν j,r } has gaps, then σ (A) also has gaps. We are going to prove the theorem in several steps formulated in the following lemmas. We begin by studying the spectrum of the 'microscopic' part of the limit operator. Proof: Let λ = ν j 0 ,r 0 for some j 0 ∈ N, r 0 ∈ [r 1 , r 2 ], and assume that v ∈ W 1,2 0 (O) is an eigenfunction corresponding to λ, i.e. − ω v = λv. (For λ = ν j 0 ,r 0 argument is similar.) Then v is of the form (61), where − v ω = λv ω in Y 1, ω 0 , whenever ω 0 ∈ [r 1 , r 2 ]. But λ is only an eigenvalue of the operator − acting in W 1,2 0 (Y 1, ω 0 ) if ω 0 = r 0 , hence v(ω) = v( ω, y) = ϕ j 0 ,r 0 (y), ω 0 = r 0 , y ∈ Y 1,r 0 , 0 o t h e r w i s e .
Next, we focus on the spectrum of A.
which clearly blows up as ω 0 → r 0 . We show that the corresponding v is not an element of L 2 (O), leading to a contradiction. Indeed, using the identity

Funding
KC is grateful for the support of the Engineering and Physical Sciences Research Council [grant number EP/L018802/2] 'Mathematical foundations of metamaterials: homogenisation, dissipation and operator theory'. IV has been supported by the Croatian Science Foundation -Hrvatska Zaklada za Znanost [grant number 9477 (MAMPITCoStruFl)].