{'fraction_non_alphanumeric': 0.0624784008439586, 'fraction_numerical': 0.012386547590898344, 'mean_word_length': 3.7035674565831123, 'pattern_counts': {'":': 0, '<': 8, '<?xml version=': 0, '>': 20, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 27, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Stochastic unfolding and homogenization of spring network models

The aim of our work is to provide a simple homogenization and discrete-to-continuum procedure for energy driven problems involving stochastic rapidly-oscillating coefficients. Our intention is to extend the periodic unfolding method to the stochastic setting. Specifically, we recast the notion of stochastic two-scale convergence in the mean by introducing an appropriate stochastic unfolding operator. This operator admits similar properties as the periodic unfolding operator, leading to an uncomplicated method for stochastic homogenization. Secondly, we analyze the discrete-to-continuum (resp. stochastic homogenization) limit for a rate-independent system describing a network of linear elasto-plastic springs with random coefficients.


Introduction
The motivation for this paper is twofold: First, we introduce the method of stochastic unfolding as an analogon to the periodic unfolding method, which in recent years has been successfully applied in the analysis and modeling of multiscale problems with periodic microstructure. Our intention is to provide an easily accessible method for stochastic homogenization and discreteto-continuum analysis that enjoys many parallels to periodic homogenization via unfolding. Secondly, we analyze the macroscopic behavior (based on stochastic unfolding) of a rate-independent system describing a network of elasto-plastic springs with random coefficients. Our result derives (via stochastic homogenization) a continuum evolutionary rate-independent (linear) plasticity system starting from a discrete model. Discrete spring networks depict solid media as collection of material points that interact via one-dimensional elements with certain constitutive laws. They are widely used in material science and the mechanical engineering community. On the one hand, they are used to model materials with an intrinsic discreteness (on a scale larger than the atomistic scale), such as granular media, truss-like structures, composites. On the other hand, spring network models are used as a numerical approximation scheme for continuum models. We refer to [36,24,16] and the references therein. In this introduction we first give a brief overview of the stochastic unfolding method, that we develop in this paper, and then discuss the discrete-to-continuum limits of random spring networks.
In order to give a brief overview of the stochastic unfolding, which we develop in this paper, let us consider for a moment a prototypical example of convex homogenization: Let O ⊂ R d be open and bounded, p ∈ (1, ∞), ε > 0, and consider the minimization problem min u∈W 1,p (subject to suitable boundary conditions).
Above V ε (x, F ) denotes a family of energy densities that are convex in F , and which we think to rapidly oscillate in x on a scale ε. The objective of homogenization is to derive a simpler minimization problem, say min u∈W 1,p (O) O V 0 (∇u(x)) dx (subject to suitable boundary conditions), with an effective (and simpler) energy density V 0 that captures the behavior of (1) for small ε. This is done by an asymptotic analysis for ε → 0, and a classical way to approach this type of problems is based on two-scale convergence methods. The notion of (periodic) twoscale convergence was introduced and developed by Nguetseng [35] and Allaire [3]. A sequence u ε ∈ L p (O) is said to two-scale converge to u(x, y)ϕ(x, y)dydx as ε → 0, for all ϕ ∈ L q (O, C per (2)). Here, 2 := [0, 1) d is the unit box and C per (2) is the space of continuous, 2-periodic functions. The two-scale limit of a sequence refines its weak limit by capturing oscillations on a prescribed scale ε. It is therefore especially useful in homogenization problems involving linear (or monotone) operators and convex potentials with periodic coefficients. With regard to problem (1), two-scale convergence methods apply, e.g. if V ε (x, F ) = V (x, x ε , F ) with V being periodic in its second component and sufficiently regular (e.g. continuous) in its first component.
In [12] the method of periodic unfolding was introduced based on the dilation technique [5]. The idea of unfolding, which is closely related to two-scale convergence, is to introduce an operator (the unfolding operator), which embeds sequences of oscillating functions to a larger two-scale space, to the effect that two-scale convergence can be characterized by the usual weak convergence in the two-scale space. In some cases, this method facilitates a more straightforward, and operator theory flavored analysis of periodic homogenization problems. In recent years periodic unfolding has been applied to a large variety of multiscale problems, e.g. see [11,15,32,33,29,38,10,37]. For a systematic investigation of two-scale calculus associated with the use of the periodic unfolding method we refer to [13,39,32].
Motivated by periodic two-scale convergence, in [8] a related notion of stochastic two-scale convergence in the mean was introduced. It is taylor-made for the study of stochastic homogenization problems. In particular, it applies to a stochastic version of problem (1): Let Ω be a probability space with a corresponding measure-preserving dynamical system {T x } x∈R d (see Section 2.2). In the context of stochastic homogenization, we might view Ω as a configuration space, and · (the associated expected value) as an ensemble average w.r.t. configurations. Then we might consider a stochastic version of (1), namely min u∈L p (Ω)⊗W 1,p where the potential V (ω, F ) is parametrized by ω ∈ Ω, and thus minimizers of (2) are random fields, i.e. they depend on ω ∈ Ω. The random potential V ε (x, ·) = V (T x ε ω, ·) in (2) is rapidly oscillating and its statistics is homogeneous in space (i.e. for any finite number of points x 1 , ..., x m ∈ R d and all F ∈ R d , the joint distribution of {V (T xi+z ·, F )} i=1,...,m is independent of the shift z ∈ R d ). In contrast to periodic two-scale convergence, stochastic two-scale convergence requires test-functions defined not only on the physical space O ⊂ R d , but also on the probability space Ω (see Remark 3.1, where we recall the definition of stochastic two-scale convergence in the mean of [8,4] in a discrete version).
Stochastic unfolding. In this paper, we introduce a stochastic unfolding method, that (analogously to the periodic case) allows to characterize stochastic two-scale convergence in the mean by mere weak convergence in an extended space. Having discrete-to-continuum problems in mind, we concentrate in this paper on a discrete setting: E.g. in (2) O is replaced by the discrete set O ε := O ∩εZ d , and instead of the gradient we consider difference quotients. As we shall demonstrate, the stochastic unfolding method features many analogies to the periodic case; as a consequence it allows to lift systematically and easily homogenization results and multiscale models for periodic media to the level of random media. In the following, in particular for readers familiar with periodic unfolding, we briefly summarize the main properties of stochastic unfolding and its analogies to the periodic case: • We introduce an operator T ε : L p (Ω × εZ d ) → L p (Ω × R d ) which is a linear isometry and we call it stochastic unfolding operator (see Section 3.1).
• Two-scale convergence in the mean for u ε ∈ L p (Ω × εZ d ) reduces to weak convergence of the unfolding T ε u ε in L p (Ω × R d ) (see Remark 3.1).
• We define weak (strong) stochastic two-scale convergence as weak (strong) convergence of the unfolded sequence T ε u ε (see Definition 3.1).
• (Compactness). Bounded sequences converge (up to a subsequence) in the weak stochastic two-scale sense (see Lemma 3.1).
• (Compactness for gradients). If u ε ∈ L p (Ω × εZ d ) is bounded and its (discrete) gradient is bounded, then (up to extracting a subsequence) u ε weakly two-scale converges to U 0 ∈ L p inv (Ω) ⊗ W 1,p (R d ). Moreover, its gradient weakly two-scale converges and the limit has a specific structure: ∇U 0 + χ where χ ∈ L p pot (Ω) ⊗ L p (R d ). Here, L p pot (Ω) := {Dϕ : ϕ ∈ L p (Ω)} where the closure is taken in L p (Ω), and L p inv (Ω) is the space of shiftinvariant functions (see Section 2.2). In the ergodic case, the latter reduces to the space of constant functions, and thus the two-scale limit U 0 is deterministic, i.e. it does not depend on ω ∈ Ω. The general structure of this compactness statement for gradients is quite similar to its analogon in the periodic case, which we briefly recall: Up to a subsequence, a bounded sequence u ε ∈ W 1,p (R d ) converges weakly to some u 0 ∈ W 1,p (R d ), and the gradient ∇u ε weakly two-scale converges to ∇u 0 ( pot,per (2) with 2 denoting the reference cell of periodicity, say 2 := [0, 1) d . In the periodic case, thanks to Poincaré's inequality on 2, any periodic, conservative field v ∈ L p pot,per (2), can be represented as v = ∇ y ϕ for some potential ϕ ∈ W 1,p per (2). Thus, the weak two-scale limit of ∇u ε takes the form ∇u 0 (x) + ∇ y ϕ(x, y) with ϕ ∈ L p (R d , W 1,p per (2)). In the stochastic case, typically it is not possible to represent v ∈ L p pot (Ω) with help of a potential defined on Ω. This is one of the main differences between stochastic and periodic homogenization.
• (Recovery sequences). For U 0 and χ as above, we construct a sequence u ε ∈ L p (Ω × εZ d ) which converges as in the previous item and additionally in the strong two-scale sense.
• The following transformation formula holds Using this formula and the previous properties, the Γ-convergence analysis of the discrete version of (2) becomes straightforward, and relies (as only noteworthy additional ingredient) on the weak lower-semicontinuity of convex integral functionals, see Proposition 3.3 and Theorem 4.1.
We would like to remark that some of the statements above (in particular those that involve only weak two-scale convergence) have already been established in the continuum setting in [8].
Yet, the arguments that we present have a different twist, since they are based on the unfolding operator and apply to the discrete setting.
In contrast to periodic (deterministic) two-scale convergence, in the stochastic case different meaningful notions for two-scale convergence exist, since one may ask for convergence in the L p (Ω)-sense (Ω being the probability space) or in a quenched sense, i.e. for a.e. ω ∈ Ω. The former corresponds to stochastic two-scale convergence in the mean and the notion of stochastic unfolding that we introduce in this paper. The latter corresponds to a finer notion of stochastic two-scale convergence, as considered in [41,19] .
Discrete-to-continuum limits of random spring networks. In the second part of this paper, we study the macroscopic, rate-independent behavior of periodic networks formed of elasto-plastic springs with random material properties. In the following, we briefly summarize our result in the simplest (non-trivial) two-dimensional setting. In Section 4 we shall treat a general, multi-dimensional case. To explain the model, we first consider a single spring that in a natural state has endpoints x 0 , x 1 ∈ R d , and thus is aligned with b := x 1 − x 0 . We describe a deformation of the spring with help of a displacement function u that maps an endpoint x i to its new position x i + u(x i ). As the measure of relative elongation (resp. compression) of the spring, we consider the Cauchy strain |b+|b|∂ b u|−|b| If the displacement is (infinitesimally) small, we arrive (by rescaling and passing to the limit |∂ b u| → 0) at the linearized strain b |b| · ∂ b u. As it is usual in linear elasto-plasticity (see e.g. [17, Section 3]), we assume that the strain admits an additive decomposition b |b| · ∂ b u = e + z, where e and z are its elastic and plastic parts respectively. The force (its intensity) exerted by the spring is linear in the elastic strain: σ = ae, a > 0 being the spring constant. We define a free energy (describing materials with linear kinematic hardening) where h > 0 denotes a hardening parameter. The rate-independent evolution of the elasto-plastic spring under a loading l : Above, σ y ≥ 0 is the yield stress of the spring, ∂I [−σy,σy] denotes the convex subdifferential of I [−σy,σy] , which is the indicator function of the set [−σ y , σ y ]. Note that the first two equations are force balance equations (inertial terms are disregarded), reasonable in regimes of small displacements, and the second expression is a flow rule for the variable z. We consider a network of springs where the nodes x ∈ εZ 2 represent the reference configuration of particles connected by springs. The displacement of the network is described with help of a map u : εZ 2 → R 2 and the plastic strains of the springs are accounted by an internal variable z : εZ 2 → R 3 (e.g. z 1 (x) is the plastic strain of the spring [x, x + εe 1 ]). We assume that the particles outside of a set O ε = O ∩ εZ d are fixed, i.e. u = 0 in εZ d \ O ε ; furthermore we suppose that z is supported in O + ε := x ∈ εZ 2 : (x, x + εb) ∩ O = ∅ for some b ∈ {e 1 , e 2 , e 1 + e 2 } . A small external force εl ε : [0, T ] × O ε → R 2 acts on the system. According to the evolution law (3) for single springs, the evolution of the network is determined by which is a superposition of (3). We tacitly identify b ∈ {e 1 , e 2 , e 1 + e 2 } with indices i = 1, 2, 3.
The above equations may be equivalently recast in the global energetic formulation for rateindependent systems (see Appendix A.1) with the help of energy and dissipation functionals, Above, ∇ ε s stands for the discrete symmetrized gradient ∇ ε , h(x, e 1 + e 2 )). We assume that the coefficients are random fields oscillating on a scale ε. In particular, the deterministic coefficients A(x), H(x) and σ y (x, b) in the above functionals are replaced by realizations of rescaled stationary random fields A(T x ε ω), H(T x ε ω) and σ b y (T x ε ω). As a consequence of this, the solutions of the corresponding evolutionary equation at each time instance are not deterministic functions but rather random fields on Ω×εZ 2 . Under suitable assumptions (cf. Section 4.3), there exists a unique solution (u ε , 3 )) to the above described microscopic evolutionary rate-independent system. Applying the method of stochastic unfolding, we are able to capture the averaged (w.r.t. the probability measure) behavior of the solution (u ε , z ε ) in the limit ε → 0. Particularly, we show that, upon assuming suitable strong convergence for the initial data and forces, there exists which solves an effective rate-independent system on a continuum physical space (see Section 4.3) and for every t ∈ [0, T ] where c2 → denotes "cross" two-scale convergence and it is explained in Section 4.3. In the continuum case, similar results have been obtained, for deterministic periodic materials in [32,18] (via periodic unfolding), and for random materials recently in [20] (using quenched stochastic two-scale convergence) and [22,23]. We discuss the literature on problems involving discrete-to-continuum transition in more detail in Section 4. If we consider the constraint z ε = 0 and time-independent force l ε (t) = l ε , the above problem boils down to the homogenization of the functional u → E ε (0, u, 0), which corresponds to the discrete-to-continuum limit of the static equilibrium of a spring network with (only) elastic interactions. We remark that the methods in this paper apply as well to systems with different constitutive laws, e.g. one might consider an energy functional with an additional term depending on the gradient of the internal variable z ε , as it is the case in gradient plasticity (see Section 4.4). In the ergodic case, we even obtain a deterministic elasto-plastic limiting model. Another interesting extension of our method (which we do not discuss in this paper) is the discrete-to-continuum analysis of random spring networks featuring damage or fracture. The convergence result that we establish can be seen as a justification of continuum models for microstructural spring networks that feature uncertainty in the constitutive relations on the microscopic scale. In this context, the method could also be applied to prove the consistency of computational schemes based on the lattice method as discussed in the mechanical engineering community (e.g. see [16]).
Structure of the paper. In Section 2, we introduce a convenient setting for problems involving homogenization and the passage from discrete to continuum systems. Section 3 is devoted to the introduction of the stochastic unfolding operator and its most important properties. In Section 4, we apply the stochastic unfolding method to an example of a multi-dimensional network of elastic/elasto-plastic springs with random coefficients.

General framework
In this section, we introduce the setting for functions on a discrete/continuum physical space suited for problems involving discrete-to-continuum transitions. In addition, we present the standard setting for stochastic homogenization problems.
2.1 Functions and differential calculus on εZ d and R d Throughout the paper we consider p, q ∈ (1, ∞) exponents of integrability that are dual, i.e. 1 p + 1 q = 1. We denote by {e i } i=1,...,d the standard basis of R d . For ε > 0, we denote the Banach space of p-summable functions by L p (εZ d ) : our purposes it is convenient to view L p (εZ d ) as the L p -space of p-integrable functions on the In particular, we use the notation and we call ∇ ε discrete gradient and ∇ ε, * negative discrete divergence. In the case ε = 1, we simply write ∇ and ∇ * instead of ∇ ε and ∇ ε, * . For u ∈ L p (εZ d ), g ∈ L q (εZ d ) d , we have the discrete integration by parts formula Definition 2.1 (Weak and strong convergence). Consider a sequence u ε ∈ L p (εZ d ) and U ∈ L p (R d ). We say that: • u ε strongly converges to U (denoted by It is convenient to consider piecewise-constant and piecewise-affine interpolations of functions in L p (εZ d ).
The proof of the following lemma is an uncomplicated exercise and therefore we omit it.
The following claims are equivalent.
2.2 Description of random media and a differential calculus for random variables As it is standard in stochastic homogenization, we describe random configurations (e.g. coefficients of a PDE or energy densities that describe properties of some medium with quenched disorder) using a probability space (Ω, F, P ) together with a measure preserving dynamical system T x : Ω → Ω (x ∈ Z d ) such that: We write · for the expectation and L p (Ω) for the usual Banach space of p-integrable random variables. Throughout the paper we assume that (Ω, F, P, T ) satisfies the properties above, and that (Ω, F, P ) is a separable measure space; the latter implies the separability of L p (Ω).
The dynamical system T is called ergodic (we also say · is ergodic), if for any A ∈ F the following implication holds: Remark 2.2. A multiparameter version of Birkhoff 's ergodic theorem (see [1,Theorem 2.4]) states that if · is ergodic and ϕ ∈ L p (Ω) then for P-a.e. ω ∈ Ω.
Example 2.1. Let (Ω 0 , F 0 , P 0 ) denote a separable probability space. Define (Ω, F, P ) as the Z dfold product of (Ω 0 , F 0 , P 0 ), i.e. Ω : Note that a configuration ω ∈ Ω can be seen as a function ω : Then it follows that (Ω, F, P, T ) satisfies the assumptions above and defines an ergodic dynamical system. With regard to the example in the introduction, (2), we might consider random potentials of the form where a 0 : Ω 0 → (λ, 1) is a random variable, with λ > 0 denoting a positive constant of ellipticity. We remark that the coefficients appearing in the corresponding energy a 0 (T x ε ω(0)) x∈εZ d are independent and identically distributed random variables.
Differential calculus for random variables. For ϕ : Ω → R and ψ = (ψ 1 , ..., ψ d ) : Ω → R d measurable, we introduce the horizontal derivative D and negative horizontal divergence D * : are linear and bounded operators. Furthermore, for any ϕ ∈ L p (Ω) and ψ ∈ L q (Ω) d the integration by parts formula holds. Hence, D (defined on L p (Ω)) and D * (defined on L q (Ω) d ) are adjoint operators.
We denote the set of shift-invariant functions in L p (Ω) by and note that L p inv (Ω) R if and only if · is ergodic. We denote by P inv : L p (Ω) → L p inv (Ω) the conditional expectation w.r.t. the σ-algebra generated by the family of shift invariant sets A ∈ F : T x A = A for every x ∈ Z d . It is a contractive projection and in the ergodic case we simply have P inv f = f . The adjoint of P inv is denoted by P * inv : L q (Ω) → L q (Ω) and a simple computation shows that P * inv ϕ ∈ L q inv (Ω) for any ϕ ∈ L q (Ω). It is easily verified that L p inv (Ω) = kerD and by standard arguments (see [9, Section 2.6]) we have the orthogonality relations The above relations hold in the sense of L p − L q duality (we identify L q (Ω) with L p (Ω)). Namely, D : L p (Ω) → L p (Ω) d and D * : L q (Ω) d → L q (Ω) and the orthogonal of a set A ⊂ L q (Ω) is given by Random fields. In this paper measurable functions defined on Ω × εZ d or on Ω × R d are called random fields. We mainly consider the space of p-integrable random fields L p (Ω × εZ d ), and frequently use the following notation: . In particular, since L p (Ω) is separable (thanks to our assumption on the underlying measure space), we have . In this respect, we tacitly identify linear and bounded operators on X (or Y ) by their obvious extension to X ⊗ Y . The applications involve problems with homogeneous Dirichlet boundary conditions, and therefore the following space is convenient: for O ⊂ R d we set 3 Stochastic unfolding

Definition and properties
For u : Ω × εZ d → R we define the unfolding of u via The above expression defines an isometric isomorphism T ε : We call both operators T ε and T ε stochastic unfolding operator. Note that T ε (defined on L p ) and T −1 Definition 3.1 (Two-scale convergence). We say that a sequence u ε ∈ L p (Ω × εZ d ) strongly (weakly) stochastically two-scale converges in the mean to and we use the notation u ε For vector valued functions, the convergence is defined componentwise.
Remark 3.1. Note that the adaptation of the two-scale convergence in the mean from [8,4] to the discrete setting reads: for all ϕ ∈ L q (Ω) and all η ∈ C ∞ c (R d ). This notion is equivalent to our notion of weak stochastic two-scale convergence.
The following lemma is obtained easily by exploiting the fact that the unfolding is a linear isometry and thanks to the usual properties of weak convergence, and therefore we omit its proof.
As in the periodic setting, it is possible to define a suitable "inverse" of the unfolding operator -the stochastic folding operator : Lemma 3.2. F ε is linear and it satisfies: The proof of this lemma is omitted since it mostly relies on the definition of the folding operator.

Two-scale limits of gradients
In this section, we treat two-scale limits of gradients. First we present some compactness results and later we show that weak two-scale limits can be recovered in the strong two-scale sense by convenient linear constructions.
See Section 3.4 for the proof. The above statement can be adapted to sequences supported in a domain: Since the unfolding operator is naturally defined for functions on R d , we tacitly identify functions in L p (O) and W 1,p 0 (O) with their trivial extension by 0 to R d . As a corollary of Proposition 3.1 we obtain: . Then additionally to the convergence statements in Proposition 3.1, the two-scale limits (from Proposition 3.1) satisfy: The proof of the above corollary is left to the reader. We remark that in Proposition 3.1 (i) and (ii) the two-scale limit U is shift invariant and therefore in the ergodic setting it is deterministic, i.e. U = P inv U = U . Corollary 3.2. Let γ ∈ [0, 1) and · be ergodic. Let u ε satisfy the assumptions in Proposition 3.1. Then the claims in Proposition 3.1 (i) and (ii) hold and we have: For the proof of this corollary, see Section 3.4.
In the following, we show that weak two-scale accumulation points can be recovered in the strong two-scale sense.
Moreover, the operator norm of G γ ε can be bounded independently of 0 < ε ≤ 1.
is linear and bounded and it holds Moreover, its operator norm is bounded uniformly in 0 < ε ≤ 1.
(iv) Let γ ∈ (0, 1). There exists a mapping which is linear and bounded and it holds Moreover, its operator norm is bounded uniformly in 0 < ε ≤ 1.
For the proof of the above two results, see Section 3.4. We remark that in the case γ ≥ 1, the recovery sequence for U ∈ L p (Ω × R d ) is simply given by F ε U . Moreover, in the case of prescribed boundary data for the recovery sequence, we might consider a cut-off procedure as in (iv) above.
Remark 3.2. Note that the construction of the recovery sequence in the whole-space case (i) and (ii) (and if γ ∈ (0, 1) for a domain (iv)) is linear in the sense that the mapping (U, χ) → u ε is linear. In contrast, the construction for domains (iii) is non-linear, since it relies on a cut-off procedure applied to the whole-space construction. We remark that the cut-off procedure can be avoided in certain cases: For p = 2, we can construct the recovery sequence, similarly as in the proof of Proposition 3.2 (i), by defining u ε as the unique solution of ∇ ε, * ∇ ε u ε = ∇ ε, * (∇ ε F ε U + F ε χ) in the interior of O ∩ εZ d and with prescribed homogeneous Dirichlet boundary data. For p = 2 the same strategy applies as long as the above discrete elliptic equation satisfies maximal L p -regularity. The latter depends on the regularity of the domain O.

Unfolding and (lower semi-)continuity of convex energies
Let V : Ω × R k → R be jointly measurable (i.e. w.r.t. F ⊗ B(R k )) and for P -a.e. ω ∈ Ω, V (ω, ·) convex. Moreover, we assume that there exists C > 0 such that for P -a.e. ω ∈ Ω and all F ∈ R k . Our Γ-convergence results for convex energies exploit the following result.
The proof of this result is in Section 3.4. We have applications in mind, where such integral functionals are treated and the role of u ε is played by a discrete (symmetrized) gradient (see Section 4).

Proofs
Proof of Proposition 3.1. Before proving the claim, we present a couple of auxiliary lemmas.
The following commutator identity for u ε : Ω × εZ d → R, obtained by direct computation, is practical: for any ϕ ∈ L q (Ω), η ∈ C ∞ c (R d ) and i ∈ {1, ..., d}. We consider the sequence v ε = F ε (ϕη) ∈ L q (Ω × εZ d ) and by Lemma 3.2 (iii), we have v ε 2 → ϕη in L q (Ω × R d ). Therefore, using Lemma 3.1 (iv), we obtain Indeed, the first term on the right-hand side of (8) vanishes in the strong two-scale limit since η is compactly supported and smooth. The second term strongly two-scale converges to D * i ϕη. This and Lemma 3.1 (iv) imply which, together with (7), yields (6).
Then there exists U ∈ L p inv (Ω) ⊗ W 1,p (R d ) such that, up to a subsequence, where we used that P * inv ϕ is a shift invariant function. Consequently, by a density argument it follows that P inv • T ε = P inv .
Step 2. Convergence of P inv u ε . Using boundedness of P inv and the fact that ∇ ε and P inv commute, we obtain lim sup Applying Lemma 3.1 (ii) and Lemma 3.
for a (not relabeled) subsequence. Note that, additionally, we have V ∈ L p inv (Ω) ⊗ L p (R d ) d . Let ϕ ∈ L q (Ω) and η ∈ C ∞ c (R d ) and denote v ε = F ε (ϕη). Since v ε 2 → ηϕ (Lemma 3.2 (iii)), for i = 1, ..., d, we have On the other hand, it holds Using this, we conclude that V ∈ L p inv (Ω) ⊗ W 1,p (R d ) and ∇V = V . Step 3. We show that u ε 2 V in L p (Ω × R d ) (up to another subsequence). Using Lemma 3.1 (ii) and Lemma 3.3, we conclude that there exist another subsequence (not relabeled) and Since P inv is a linear and bounded operator, it follows that P inv (T ε u ε ) P inv U in L p (Ω × R d ), and P inv U = U by shift invariance of U . Furthermore, by Step 1 and Step 2 we have that P inv T ε u ε = T ε P inv u ε V and therefore U = V . This completes the proof.
Proof of Proposition 3.1. (i) By Lemma 3.4 we deduce that there exists U ∈ L p inv (Ω)⊗W 1,p (R d ) such that u ε 2 U and by boundedness of ∇ ε u ε (Lemma 3.1 (ii)) there exists V ∈ L p (Ω × R d ) d such that ∇ ε u ε 2 V (up to a subsequence). In order to prove the claim, it suffices to show that for any ϕ ∈ L q (Ω) d with D * ϕ = 0 and η ∈ C ∞ c (R d ). Indeed, this implies that χ := V − ∇U ∈ L p pot (Ω) ⊗ L p (R d ) and thus the claim of the proposition. For the argument consider v ε = F ε (ηϕ), the folding acting componentwise.
On the other hand, the commutator identity (5) and the definition of F ε yield Since D * ϕ = 0, the contribution from the second term on the right-hand side above vanishes. After a discrete integration by parts, the right-hand side reduces to which is concluded by using that u ε 2 U and that η is smooth and compactly supported. Since U is shift invariant, the second term on the right-hand side vanishes. After an integration by parts, we are able to infer (9) and conclude the proof of part (i).
(ii) By Lemma 3.1 (ii), there exists U ∈ L p (Ω × R d ) such that u ε 2 U (up to a subsequence).
Since γ ∈ (0, 1), u ε satisfies the assumptions in Lemma 3.3 and therefore U ∈ L p inv (Ω) ⊗ L p (R d ). With help of (i), we obtain that for the sequence v ε : However, using that u ε 2 U , we conclude that V = 0 and the claim is proved.
Following the same strategy as in Lemma 3.3 it can be obtained that V = DU .
(up to a subsequence). Also, using part (iii), for the sequence v ε := ε γ−1 u ε , there exists The fact that u ε 2 U implies that V = 0 and the proof is complete.
Proof of Corollary 3.2.
Proof. (i) The claim follows directly from Lemmas 3.4 and 2.1.
(ii) Exploiting linearity and boundedness of P inv and Step 1 in the proof of Lemma 3.4, we obtain that The above, Lemma 2.1 and the fact that χ = 0 allow us to conclude the proof.
As a result of this, u ε u weakly in W 1,p (R d ). Rellich's embedding theorem implies that u ε → U strongly in L p (Q) and using Lemma 2.1 we conclude that u ε → U strongly in L p (Q).
(iv) We have by Jensen's inequality and by boundedness of ϕ: The right-hand side of the above inequality equals R d |T ε u ε (ω, x) − U (x)| p dx and therefore it vanishes as ε → 0.
Proof of Proposition 3.2. Before presenting the proof, we provide an auxiliary lemma providing a nonlinear approximation for χ in the case γ = 0. Lemma 3.5 (Nonlinear approximation). For χ ∈ L p pot (Ω) ⊗ L p (R d ) and δ > 0, there exists a sequence g δ,ε ∈ L p (Ω × εZ d ) such that We define g ε := εF ε V and remark that g ε L p (Ω×εZ d ) ≤ ε V L p (Ω×R d ) , which follows from the boundedness of F ε . This proves the first part.
The first and the last term on the right-hand side above vanish as ε → 0 and therefore the claim follows (since we can choose δ arbitrarily small). Indeed, for the first term it is sufficient to note that DV is smooth and has compact support w.r.t. its x-variable. Also, the last term vanishes thanks to boundedness of π ε and the boundedness of difference quotients by gradients; specifically (for i = 1, ..., d) Proof of Proposition 3.2. In the following proof we appeal to maximal L p -regularity for the equation in form of which is uniform in ε. For p = 2 this is a standard a priori estimate.
The estimate above follows by the same argument from the Calderón-Zygmund estimate for the discrete laplacian on εZ d , for the latter see e.g. [14,7]. (i) Let 2γ < α < 2. For a given χ ∈ L p pot (Ω) ⊗ L p (R d ) we define G γ ε χ := u ε as the unique solution to the following equation in L p (Ω × εZ d ) (for P -a.e. ω ∈ Ω) The discrete maximal L p -regularity theory implies that As a result of this, we have u ε L p (Ω×εZ d ) ≤ ε α 2 −γ C χ L p (Ω×R d ) d and therefore u ε 2 → 0 as ε → 0. We consider the sequence g δ,ε from Lemma 3.5 corresponding to χ. Note that w δ,ε := u ε −ε −γ g δ,ε is the unique solution in L p (Ω × εZ d ) to (for P -a.e. ω ∈ Ω) We employ again the discrete maximal L p -regularity theory to obtain Multiplication of the above inequality by ε γ yields As a result of this and with help of the isometry property of T ε , we obtain Letting first ε → 0 and then δ → 0 the right-hand side of the above inequality vanishes using Lemma 3.5. This completes the proof of (i).
(ii) We consider a sequence U δ = Using the triangle inequality, it follows that First, we treat the first term on the right-hand side. For i = 1, ..., d, by the isometry property of T ε and contraction property of F ε , In the last inequality we use the fact that difference quotients are bounded by gradients. Second, we compute (i = 1, ..., d) )) .
(12) The second part of the right-hand side of the above equality vanishes (for P -a.e. ω ∈ Ω) by shift invariance of U δ . Further, we have For any δ > 0 the last expression converges to 0 as ε → 0 since U δ is smooth in its x-variable. Finally, in (10) we first let ε → 0 and then δ → 0 to conclude the proof.
(iii) Let 0 < α < 2γ. For a given U ∈ L p inv (Ω) ⊗ L p (R d ) we define F γ ε U := u ε as the unique solution to the following equation in L p (Ω × εZ d ) (for P -a.e. ω ∈ Ω) The maximal L p -regularity theory and boundedness of F ε imply that As a result of this and the isometry property of T ε , we obtain that ε γ ∇ ε u ε Note that w δ,ε := u ε − F ε U δ is the unique solution in L p (Ω × εZ d ) to (for P -a.e. ω ∈ Ω) The maximal L p -regularity theory implies that We multiply the above inequality by ε α 2 and use boundedness of F ε , to obtain Using the above inequality and the isometry property of T ε , we obtain Letting ε → 0 the last two terms on the right-hand side of the above inequality vanish. Indeed, the middle term is bounded by Cε α 2 ∇U δ L p (Ω×R d ) d (cf. part (ii) (11)) and the last term vanishes using Lemma 3.2 (iii). Finally, letting δ → 0 we conclude that u ε 2 → U .
We have The first term on the right-hand side above is bounded by ε γ−1 C U − U δ L p (Ω×R d ) (using boundedness of all of the appearing operators). We compute, as in (12) As a result of this, we obtain The first term on the right-hand side above is bounded by ε γ C ∇U δ L p (Ω×R d ) d and therefore it vanishes in the limit ε → 0. The second term vanishes as well in the limit ε → 0. Collecting the above claims and letting first ε → 0 and then δ → 0 in (13), we conclude the proof.

Proof of Corollary 3.3
Proof. (i) and (ii) follow directly from Lemma 3.2 (iii) and Proposition 3.2.
(iii) For δ > 0 we consider a cut-off function Let u ε,δ = F ε U δ + η δ G 0 ε χ, where G 0 ε denotes the operator given in Proposition 3.2. We have Above on the right-hand side, the first term can be bounded by U δ − U L p (Ω×R d ) (by boundedness of F ε ), the second term is bounded by T ε G 0 ε χ L p (Ω×R d \O −δ ) (using the properties of η δ ) and the third term is bounded by C ∇U δ − ∇U L p (Ω×R d ) d (similarly as in (11)). The last term is treated as follows. We take advantage of the following product rule: for f, g : The first term on the right-hand side of (15) is bounded by T ε ∇ ε G 0 ε χ L p (Ω×R d \O −δ ) d and for small enough ε, the second term is bounded by . Collecting all the above bounds for the inequality (14), using the isometry property of T ε and with help of part (i), we obtain that and C(d) is the diameter of 2. Hence, there exists a diagonal sequence u ε := u ε,δ(ε) which satisfies the claim of the corollary.
is the cut-off function from part (iii) with δ(ε) = ε γ 2 . For notational convenience, we write u ε instead of u ε (U, χ). We have The second and last terms on the right-hand side above vanish as ε → 0 using the claim of part (ii). The first term is bounded by T ε (F γ ε U + G γ ε χ) L p (Ω×R d \O −δ(ε) ) (cf. part (iii)) and therefore it vanishes as ε → 0 using the fact that T ε (F γ ε U + G γ ε χ) converges strongly (and therefore it is uniformly integrable). For small enough ε, the third term is bounded (up to a constant) by (15) in part (iii)). The last expression vanishes in the limit ε → 0 using the properties of F γ ε U + G ε ε χ. The proof is complete.

Proof of Proposition 3.3
Proof. For notational convenience, let E ε (u ε ) : for a suitable small boundary layer set L ε,k ⊂ {x : dist(x, O +ε \ A k ) < C 1 ε}, where C 1 is a fixed constant depending only on the dimension d (see Figure 2). The growth conditions of V yield Figure 2: Small boundary layer.
Letting ε → 0, we obtain The last inequality is obtained using the fact that the functional U → A k V (ω, U ) is weakly lower-semicontinuous (since it is convex and continuous w.r.t. the strong L p -topology). Finally, letting k → ∞ we obtain the claimed result.
(ii) Similarly as in part (i), we find suitable boundary layer sets L + ε and L − ε (with |L ± ε | → 0 as ε → 0), such that where we use the growth conditions of the integrand V . The terms in the middle on the righthand side vanish as ε → 0, as well as the last term (using strong convergence of T ε u ε ). As a result of this and using strong continuity of U → O V (ω, U ) , we obtain that lim sup ε→0 E ε (u ε ) ≤

Stochastic homogenization of spring networks
In this section, we illustrate the capabilities of the stochastic unfolding operator in homogenization of energy driven problems that invoke convex functionals. We treat a multidimensional analogon of the problem presented in the introduction -a network of springs which exhibit either elastic or elasto-plastic response. The material coefficients are assumed to be rapidly oscillating random fields and we derive effective models in the sense of a discrete-to-continuum transition.
In Section 4.1, we briefly present the setting of periodic lattice graphs and the corresponding differential calculus. If the springs display only elastic behavior and the forces acting on the system do not depend on time, the static equilibrium of the spring network is determined by a convex minimization problem. Accordingly, in Section 4.2 we present homogenization results for convex functionals. On the other hand, in the case of elasto-plastic springs, the evolution of the system is embedded in the framework of evolutionary rate-independent systems (ERIS). A short description of that framework can be found in Appendix A.1, and for a detailed study we refer to [27,30]. In the limit, as the characteristic size of the springs vanishes, we obtain a homogenized model, which is also described by an ERIS on a continuum physical space (Section 4.3 and 4.4).
We remark that homogenization results concerning minimization problems in the discrete-tocontinuum setting are already well-established. Earlier works (e.g. [2]) treat more general problems than we do in this paper. Namely, the considered potentials might be nonconvex and if the media is ergodic, previous works feature quenched homogenization results. This means that for almost every configuration, the energy functional Γ-converges to a homogenized energy functional (cf. Remark 4.3). In our results for convex minimization problems, we obtain weaker, averaged homogenization results (see Section 4.2). Despite our results being weaker in a general situation, we would like to point out that our strategy relies only on the simple idea of the unfolding operator, namely the compactness properties of the unfolding (which are closely related to compactness statements in usual L p -spaces) and lower semicontinuity of convex functionals play a central role in our analysis. On the other hand, previous works are based on more involved techniques, such as the subadditive ergodic theorem [1] ( [2], [34]), or the notion of quenched stochastic two-scale convergence ( [21]).

Functions on lattice graphs
be the edge generating set and we assume that E 0 includes {e i } i=1,...,d . We consider the rescaled periodic lattice graph (εZ d , εE), where the set of edges is where the discrete gradient ∇ ε u is defined as in Section 2.1. We define the discrete symmetrized gradient as ∇ ε s u : Furthermore, we introduce a suitable symmetrization operator for random fields as follows. For a matrix F ∈ R d×d , we denote by F s ∈ R k the vector with entries (F s ) i = bi |bi| ·F bi |bi| (i = 1, ..., k). Analogously, for F : Ω → R d×d measurable, we set F s : Ω → R k , Throughout the paper, we assume that (Z d , E) satisfies a discrete version of Korn's inequality: There exists C(d, p) > 0 such that for all compactly supported u : Remark 4.1. An example of a lattice satisfying the above Korn's inequality is Z d , E with The assumption (Korn) implies a continuum version of Korn's inequality. Namely, let O ⊂ R d be open and bounded, there exists C(p) > 0 such that This inequality is obtained applying (Korn) to π ε u δ , where u δ is a smooth approximation of u, and passing to the limits ε → 0 and δ → 0 (cf. Lemma 4.1 in Section 4.5). Note that (Korn) implies another, stochastic version of Korn's inequality -see Lemma 4.2 in Section 4.5.

Static problem
As a preparation for the rate-independent evolutionary problem, we first discuss a homogenization procedure for a static convex minimization problem, and discuss different notions of convergence in the homogenization result. We consider a set of particles with reference positions at εZ d . It is assumed that the edges εE represent springs with elastic response (cf. Introduction with internal variable z = 0 and loading l(t) = l). The equilibrium state of the system is determined by a minimization problem which (in a slightly more general setting) reads: We assume the following: (A1) V : Ω × R k → R is jointly measurable (i.e. w.r.t. the product σ-algebra F ⊗ B(R k )).
(A3) There exists C > 0 such that for P -a.e. ω ∈ Ω and all F ∈ R k .
In the case that the loading l ε converges in a sufficiently strong sense (see Remark 4.2), in order to describe the asymptotic behavior of minimizers in (17), it is sufficient to consider the energy functional As shown below for ε → 0, we derive the effective two-scale functional Moreover, if we assume that · is ergodic, the effective energy reduces to a single-scale functional where the homogenized energy density V hom : R d×d → R is defined by the corrector problem Theorem 4.1 (Two-scale homogenization). Assume (A0) − (A3).
(ii) (Lower bound) Assume that the convergence (19) holds for the whole sequence u ε . Then → l where l ∈ L p (Ω × O) d , the above theorem implies (by a standard argument from Γ-convergence) that minimizers u ε in (17) satisfy (up to a subsequence) ) d is a minimizer of the two-scale functional In the ergodic case, the limit is deterministic: Theorem 4.2 (Ergodic case). Assume (A0) − (A3) and that · is ergodic.

(i) (Compactness and lower bound) Let
There exists U ∈ W 1,p 0 (O) d such that, up to a subsequence, Remark 4.3. In the ergodic case, E ε Γ-converges to the deterministic functional E hom . In fact, it is known that for P -a.e. ω ∈ Ω the functional Γ-converges to E hom . This quenched convergence result can be found e.g. in [2], where even nonconvex integrands are treated. Based on stochastic unfolding, we obtain the weaker "averaged" result of Theorem 4.2 as a corollary of Theorem 4.1 and Corollary 3.2. While our argument is relatively easy, the analysis of the stronger quenched convergence result is based on the subadditive ergodic theorem [1] and it is more involved. We remark that minimizers ω → u ε (ω) of the functionals E ε (ω, ·) present a random field (under the above assumptions), which minimizes the averaged energy E ε (and vice versa).
If we, additionally, assume that V ≥ 0 and for P -a.e. ω ∈ Ω it is uniformly convex with modulus (·) p , i.e. there exists C > 0 such that for all F, G ∈ R k and t ∈ [0, 1] then we obtain strong two-scale convergence for minimizers: where U ∈ W 1,p 0 (O) d is the unique minimizer of Remark 4.4 (Periodic homogenization). As mentioned earlier, one knows that E ε (ω, ·) Γ → E hom in the quenched sense, whereas we obtain convergence for minimizers in an averaged sense (as above in (20)). Yet if we consider the setting for periodic homogenization, using the above convergence in the mean, we recover a standard (pointwise) periodic homogenization result. In particular, for N ∈ N we set Ω = N 2 ∩ Z d /N Z d the discrete N -torus with a corresponding (rescaled) counting measure. The dynamical system (T x ) is defined as T x ω = ω+x mod N . The above example of the probability space with the dynamical system (T x ) satisfies the assumptions given in Section 2.2 and it is ergodic. We remark that in this case Ω is a finite set and therefore (20) implies that u ε (ω) → U in L p (R d ) d for all ω ∈ Ω.

Rate-independent evolutionary problem
Let us first describe the system we have in mind. A system of particles connected with springs is represented using (εZ d , εE). Namely, εZ d ∩O serves as the reference configuration of particles. The edges εE represent springs with elasto-plastic response (cf. Introduction). Upon an external loading l, the system evolves according to an ERIS (see Appendix A.1). Let O, O +ε ⊂ R d be open (see below for specific assumptions). The following model is a random and discrete counterpart of the model considered in [32], where the periodic, continuum case is treated.
placement u ε and the internal variable z ε are merged into a joint variable y ε = (u ε , z ε ). We equip Y ε with the scalar product We consider the ERIS (see Appendix) associated with (E ε , Ψ ε ) and we denote the set of stable states at time t corresponding to these functionals by S ε (t). We assume the following: (B1) A ∈ L ∞ (Ω, R 2k×2k sym ) and it satisfies: there exists C > 0 such that A(ω)F · F ≥ C|F | 2 for P -a.e. ω ∈ Ω and every F ∈ R 2k .
The passage to the limit model (as ε → 0) is conducted in the setting of evolutionary Γconvergence [31] and it involves a discrete-to-continuum transition. The homogenized model as well is described by an ERIS: • The state space is given by (Ω) ⊗ L 2 (O)) d and we denote the state variable by y = (U, Z, χ).
• The energy functional is • The limit dissipation functional is given by S(t) denotes the set of stable states w.r.t. (E 0 , Ψ 0 ) at time t. For notational convenience we introduce the following abbreviation: for y ε ∈ Y ε and y ∈ Y y ε c2 y ⇔ u ε 2 U, ∇ ε u ε 2 ∇U + χ and z ε 2 Z (in the corresponding L 2 -spaces).
Also, we use c2 → if the above quantities stronly two-scale converge. The "c" in this shorthand refers to "cross" convergence as in the periodic case. The proof of the following homogenization theorem follows closely the strategy developed in [32]. In that paper, the periodic unfolding method is applied to a similar (continuum) problem with periodic coefficients. Let y ε ∈ C Lip ([0, T ], Y ε ) be the unique energetic solution associated with (E ε , Ψ ε ) and y ε (0) = y 0 ε . Then y 0 ∈ S(0) and for every t ∈ [0, T ] : where y ∈ C Lip ([0, T ], Y ) is the unique energetic solution associated with (E 0 , Ψ 0 ) and y(0) = y 0 .
Remark 4.7. We remark that the above result holds true in the case that · is not ergodic (with minor changes in the proof ) with a modified state space for the continuum model, specifically

Gradient plasticity
The limit rate-independent system from the previous section cannot be equivalently recast as a rate-independent system with deterministic properties as in the case of convex minimization (Theorem 4.2). The reason for this is that the limiting internal variable Z is in general not deterministic. The microscopic problem might be regularized by adding a gradient term of the internal variable z ε and in that way homogenization yields a deterministic limit problem. This strategy was demonstrated in [18], where periodic homogenization of gradient plasticity in the continuum setting is discussed. In the following, we show that the same applies in our stochastic, discrete-to-continuum setting.
Let γ ∈ (0, 1). The new microscopic system involves the same dissipation potential Ψ ε as before, as well as the same state space Y ε , yet now equipped with the scalar product We consider a modified energy functional E γ ε : where G : Ω → R k×d . We assume the following: (B3) G ∈ L ∞ (Ω, Lin(R k×d , R k×d )) and it satisfies: there exists C > 0 such that G(ω)F 1 : F 1 ≥ C|F 1 | 2 and G(ω)F 1 : F 2 = F 1 : G(ω)F 2 for P -a.e. ω ∈ Ω and all F 1 , F 2 ∈ R k×d .
The set of stable states at time t corresponding to (E γ ε , Ψ ε ) is denoted by S γ ε (t).
In the limit ε → 0, we obtain a deterministic rate-independent system described as follows.
• The state space is Q = H 1 (O) d × L 2 (O) k and the state variable is denoted by q = (U, Z).
where V hom is given by the corrector problem: For F 1 , F 2 ∈ R k , In fact, it can be shown that V hom is quadratic: There exists A hom ∈ R 2k×2k The set of stable states at time t corresponding to the functionals (E hom , Ψ hom ) is denoted by S hom (t).

Proofs
Before presenting the proofs, we consider a couple of auxiliary lemmas.
If we have strong two-scale convergence for ∇ ε u ε , strong two-scale convergence for ∇ ε s u ε follows.
Proof. For any i ∈ {1, ..., k} we compute For any fixed y ∈ Z d , the function (ω, x) → T ε ∇ ε u ε (T −y ω, x − εy)B i (y) weakly converges to (ω, x) → F (T −y ω, x)B i (y). If we assume strong two-scale convergence for the gradient, the previous quantities converge in the strong sense. Using this and the fact that B i (y) = 0 for all but finitely many y ∈ Z d , the claim follows.

Lemma 4.2 (Stochastic Korn's inequality).
Recall that it is assumed that (Z d , E) satisfies (Korn) (see Section 4.1). There exists C > 0 such that Proof. We show the inequality in the case χ = Dϕ for ϕ ∈ L p (Ω) d . For general χ ∈ L p pot (Ω) d , it is obtained by an approximation argument. We denote by ϕ : Ω × Z d → R d the stationary extension of ϕ, i.e. ϕ(ω, x) = ϕ(T x ω).
Let R > 0 and K > 0 such that K > sup {|b i | : b i ∈ E 0 }. Let η R be a cut-off function given by η R : Z d → R with η R = 1 in B R+K ∩ Z d and η R = 0 otherwise (B R ⊂ R d is a ball of radius R with center in 0). Using the properties of η R and the discrete Korn's inequality, we obtain By invariance of P the left-hand side of the above inequality equals |Dϕ| p for any R. Moreover, the first term on the right-hand side equals C |D s ϕ| p . Therefore, it is sufficient to show that the second term vanishes in the limit R → ∞. To obtain that, we estimate (for P -a.e. ω ∈ Ω) In the first inequality above, we used the fact that ∇ s : operator. An integration of (21) over Ω yields This concludes the proof.
The above result and the direct method of calculus of variations imply the following.
Therefore, the claim follows from Proposition 3.1 (i) and Corollary 3.1.
(ii) The claim directly follows from Lemma 4.1 and Proposition 3.3 (i).

Proof of Theorem 4.2
Proof. (i) By Theorem 4.1 there exist U ∈ W 1,p 0 (O) d and χ ∈ (L p pot (Ω)⊗L p (O)) d and a two-scale convergent subsequence such that The corresponding convergence for u ε follows from Corollary 3.2.
(ii) It is sufficient to show that for U ∈ W 1,p 0 (O), there exists χ ∈ (L p pot (Ω) ⊗ L p (O)) d such that Indeed, if this holds, Theorem 4.1 (iii) implies that there exists u ε ∈ L p (Ω) and the corresponding convergence for u ε and ∇ ε u ε follows by Corollary 3.2 and Lemma 2.1.
To show the above claim, we note that ∇U ∈ L p (O) d×d and consider a sequence of piecewise . Note that χ n ∈ (L p pot (Ω) ⊗ L p (O)) d and with the help of the growth assumptions (A3) and the stochastic Korn's inequality (Lemma 4.2), we obtain As a result of this, there exist a subsequence n and χ ∈ (L p pot (Ω) ⊗ L p (O)) d such that Above, in the first inequality we use weak lower-semicontinuity of the functional G ∈ L p (Ω × O) d×d → O V (ω, G(ω, x))dx and the facts that F n → ∇U , χ n χ and that (·) s is a linear and bounded operator. In order to justify the last equality, we remark that V hom is convex (and therefore continuous) and satisfies the growth assumptions −C − C|F | p ≤ V hom (F ) ≤ C|F | p + C which implies that the functional G ∈ L p (O) d×d → O V hom (G(x))dx is strongly continuous. On the other hand, it is easy to see that O V hom (∇U (x))dx ≤ O V (ω, ∇ s U (x) + χ s (ω, x))dx . This concludes the proof.

Proof of Proposition 4.1
Proof. The uniqueness of the minimizer in (17) follows by uniform convexity assumption on the integrand V . Theorem 4.2 implies that (up to a subsequence) u ε U is a minimizer of I hom . As in the proof of Theorem 4.2, we find χ ∈ (L p (Ω) ⊗ L p (O)) d with I hom (U ) = E 0 (U, χ) + O f · U dx. By Theorem 4.1, there exists a strong two-scale recovery The second term on the right-hand side vanishes in the limit ε → 0 by the properties of v ε . The first term is treated as follows.
where we use the discrete Poincaré-Korn inequality and the convexity assumption on V . It follows that This yields the claim for a subsequence. Convergence for the whole sequence follows by a contradiction argument and using the uniqueness of the minimizer U . . There exists C > 0 such that Ay, y Y * ,Y ≥ C y 2 Y for all y ∈ Y .
Proof. First, we have Ay, Using the continuum Korn's inequality (16) (Section 4.1) and the stochastic Korn's inequality Lemma 4.2, we conclude the proof.
Proof of Theorem 4.3. First, we prove as an auxiliary result, the existence of joint recovery sequences, which implies the stability of two-scale limits of solutions.
Lemma 4.5. Let t ∈ [0, T ] and y ε ∈ S ε (t) such that y ε c2 y ∈ Y . For any y ∈ Y there exists y ε ∈ Y ε such that y ε c2 y and This implies y ∈ S(t).
Proof. Corollary 3.3 (i) implies that there exists a sequence v ε ∈ L 2 (Ω) We define y ε componentwise: u ε = u ε + v ε and z ε = z ε + g ε . By weak two-scale convergence of y ε , we have that y ε c2 y, and furthermore y ε − y ε c2 → y − y. The energy functional is quadratic and thus it satisfies We rewrite the first term on the right-hand side as This expression is a scalar product of strongly and weakly convergent sequences (see Lemma 4.1), and therefore it converges to (as ε → 0) Similarly, the second term on the right-hand side of (22) converges to − O l(t) · ( U − U ) . Furthermore, by Jensen's inequality we obtain ρ(ω, Z(ω, y) − Z(ω, y))dydm ε (x) = Ψ 0 ( y − y).
This shows that the first two terms on the right-hand side of (26) vanishes in the limit ε → 0 and therefore the claim about strong convergence follows. To show that the convergence holds for the whole sequence, for a fixed t ∈ [0, T ], we consider e ε (t) := v ε (t) − V (t) H . For any subsequence ε of ε, we can find a further subsequence ε such that e ε (t) → 0 by the uniqueness of the solution y. From this follows that the whole sequence converges in the sense given in the statement of the theorem.
Proof of Theorem 4.4. The proof of this theorem follows the same strategy and it is very similar to the proof of Theorem 4.3. Therefore, we only sketch the proof, emphasizing the differences to the prior setting.
As a result of this, we conclude that χ 1 (t) is the corrector corresponding to q(t), i.e.