Stochastic Homogenisation of Free-Discontinuity Problems

In this paper we study the stochastic homogenisation of free-discontinuity functionals. Assuming stationarity for the random volume and surface integrands, we prove the existence of a homogenised random free-discontinuity functional, which is deterministic in the ergodic case. Moreover, by establishing a connection between the deterministic convergence of the functionals at any fixed realisation and the pointwise Subadditive Ergodic Theorem by Akcoglou and Krengel, we characterise the limit volume and surface integrands in terms of asymptotic cell formulas.


Introduction
In this article we prove a stochastic homogenisation result for sequences of free-discontinuity functionals of the form where f and g are random integrands, ω is the random parameter, and ε > 0 is a small scale parameter. The functionals Eε are defined in the space SBV (A, R m ) of special R m -valued functions of bounded variation on the open set A ⊂ R n . This space was introduced by De Giorgi and Ambrosio in [22] to deal with deterministic problemse.g. in fracture mechanics, image segmentation, or in the study of liquid crystals -where the variable u can have discontinuities on a hypersurface which is not known a priori, hence the name free-discontinuity functionals [21]. In (1.1), Su denotes the discontinuity set of u, u + and u − are the "traces" of u on both sides of Su, νu denotes the (generalised) normal to Su, and ∇u denotes the approximate differential of u.
Our main result is that, in the macroscopic limit ε → 0, the functionals Eε homogenise to a stochastic free-discontinuity functional of the same form, under the assumption that f and g are stationary with respect to ω, and that each of the realisations f (ω, ·, ·) and g(ω, ·, ·, ·) satisfies the hypotheses considered in the deterministic case studied in [16] (see Section 3 for details). Moreover, we show that under the additional assumption of ergodicity of f and g the homogenised limit of Eε is deterministic. Therefore, our qualitative homogenisation result extends to the SBV -setting the classical qualitative results by Papanicolaou and Varadhan [30,31], Kozlov [27], and Dal Maso and Modica [17,18], which were formulated in the more regular Sobolev setting.

1.1.
A brief literature review. The study of variational limits of random free-discontinuity functionals is very much at its infancy. To date, the only available results are limited to the special case of discrete energies of spin systems [2,14], where the authors consider purely surface integrals, and u is defined on a discrete lattice and takes values in {±1}.
In the case of volume functionals in Sobolev spaces, classical qualitative results are provided by the work by Papanicolaou and Varadhan [30,31] and Kozlov [27] in the linear case, and by Dal Maso and Modica [17,18] in the nonlinear setting. The need to develop efficient methods to determine the homogenised coefficients and to estimate the error in the homogenisation approximation, has recently motivated an intense effort to build a quantitative theory of stochastic homogenisation in the regular Sobolev case.
The first results in this direction are due to Gloria and Otto in the discrete setting [25,26]. In the continuous setting, quantitative estimates for the convergence results are given by Armstrong and Smart [8], who also study the regularity of the minimisers, and by Armstrong, Kuusi, and Mourrat [5,6]. We also mention [7], where Armstrong and Mourrat give Lipschitz regularity for the solutions of elliptic equations with random coefficients, by directly studying certain functionals that are minimised by the solutions.
The mathematical theory of deterministic homogenisation of free-discontinuity problems is well established. When f and g are periodic in the spatial variable, the limit behaviour of Eε can be determined by classical homogenisation theory. In this case, under mild assumptions on f and g, the deterministic functionals Eε behave macroscopically like a homogeneous free-discontinuity functional. If, in addition, the integrands f and g satisfy some standard growth and coercivity conditions, the limit behaviour of Eε is given by the simple superposition of the limit behaviours of its volume and surface parts (see [13]). This is, however, not always the case if f and g satisfy "degenerate" coercivity conditions. Indeed, while in [10,15,24] the two terms in Eε do not interact, in [9,20,11,32,33] they do interact and produce rather complex limit effects. The study of the deterministic homogenisation of free-discontinuity functionals without any periodicity condition, and under general assumptions ensuring that the volume and surface terms do "not mix" in the limit, has been recently carried out in [16].

Stationary random integrands.
Before giving the precise statement of our results, we need to recall some definitions. The random environment is modelled by a probability space (Ω, T , P ) endowed with a group τ = (τz) z∈Z n of T -measurable P -preserving transformations on Ω. That is, the action of τ on Ω satisfies P (τ (E)) = P (E) for every E ∈ T .
We say that f : Ω × R n × R m×n → [0, +∞) and g : Ω × R n × (R m \ {0}) × S n−1 → [0, +∞) are stationary random volume and surface integrands if they satisfy the assumptions introduced in the deterministic work [16] (see Section 3 for the complete list of assumptions) for every realisation, and the following stationarity condition with respect to τ : When, in addition, τ is ergodic, namely when any τ -invariant set E ∈ T has probability zero or one, we say that f and g are ergodic.
1.3. The main result: Method of proof and comparison with previous works. Under the assumption that f and g are stationary random integrands, we prove the convergence of Eε to a random homogenised functional E hom (Theorem 3.13), and we provide representation formulas for the limit volume and surface integrands (Theorem 3.12). The combination of these two results shows, in particular, that the limit functional E hom is a free-discontinuity functional of the same form as Eε. If, in addition, f and g are ergodic, we show that E hom is deterministic.
Our method of proof consists of two main steps: a purely deterministic step and a stochastic one, in the spirit of the strategy introduced in [18] for integral functionals of volume type defined on Sobolev spaces.
In the deterministic step we fix ω ∈ Ω and we study the asymptotic behaviour of Eε(ω). Our recent result [16,Theorem 3.8] ensures that Eε(ω) converges (in the sense of Γ-convergence) to a free-discontinuity functional of the form , see (f) in Section 2) attaining a piecewise constant boundary datum near ∂Q ν r (rx) (see (1.5)), and Q ν r (rx) is obtained by rotating Qr(rx) in such a way that one face is perpendicular to ν.
In the stochastic step we prove that the limits (1.2) and (1.3) exist almost surely and are independent of x. To this end, it is crucial to show that we can apply the pointwise Subadditive Ergodic Theorem by Akcoglou and Krengel [1]. Since our convergence result [16] ensures that there is no interaction between the volume and surface terms in the limit, we can treat them separately.
More precisely, for the volume term, proceeding as in [18] (see also [29]), one can show that the map defines a subadditive stochastic process for every fixed ξ ∈ R m×n (see Definition 3.10). Then the almost sure existence of the limit of (1.2) and its independence of x directly follow by the n-dimensional pointwise Subadditive Ergodic Theorem, which also ensures that the limit is deterministic if f is ergodic. For the surface term, however, even applying this general programme presents several difficulties. One of the obstacles is due to a nontrivial "mismatch" of dimensions: On the one hand the minimisation problem appearing in (1.3) is defined on the n-dimensional set Q ν r (rx); on the other hand the integration is performed on the (n − 1)-dimensional set Su ∩ Q ν r (rx) and the integral rescales in r like a surface measure. In other words, the surface term is an (n − 1)-dimensional measure which is naturally defined on ndimensional sets. Understanding how to match these different dimensions is a key preliminary step to define a suitable subadditive stochastic process for the application of the Subadditive Ergodic Theorem in dimension n − 1.
To this end we first set x = 0. We want to consider the infimum in (1.5) as a function of (ω, I), where I belongs to the class In−1 of (n − 1)-dimensional intervals (see (3.9)). To do so, we define a systematic way to "complete" the missing dimension and to rotate the resulting n-dimensional interval. For this we proceed as in [2], where the authors had to face a similar problem in the study of pure surface energies of spin systems.
Once this preliminary problem is overcome, we prove in Proposition 5.2 that the infimum in (1.5) with x = 0 and ν with rational coordinates is related to an (n − 1)-dimensional subadditive stochastic process µ ζ,ν on Ω × In−1 with respect to a suitable group (τ ν z ′ ) z ′ ∈Z n−1 of P -preserving transformations (see Proposition 5.2). A key difficulty in the proof is to establish the measurability in ω of the infimum (1.5). Note that this is clearly not an issue in the case of volume integrals considered in [17,18]: The infimum in (1.4) is computed on a separable space, so it can be done over a countable set of functions, and hence the measurability of the process follows directly from the measurability of f . This is not an issue for the surface energies considered in [2] either: Since the problem is studied in a discrete lattice, the minimisation is reduced to a countable collection of functions. The infimum in (1.5), instead, cannot be reduced to a countable set, hence the proof of measurability is not straightforward (see Proposition A.1 in the Appendix).
The next step is to apply the (n − 1)-dimensional Subadditive Ergodic Theorem to the subadditive stochastic process µ ζ,ν , for fixed ζ and ν. This ensures that the limit g ζ,ν (ω) := lim t→+∞ µ ζ,ν (ω)(tI) t n−1 L n−1 (I) , (1.6) exists for P -a.e. ω ∈ Ω and does not depend on I. The fact that the limit in (1.6) exists in a set of full measure, common to every ζ and ν, requires some attention (see Proposition 5.1), and follows from the continuity properties in ζ and ν of some auxiliary functions (see (5.9) and (5.10) in Lemma 5.4). As a final step, we need to show that the limit in (1.3) is independent of x, namely that the choice x = 0 is not restrictive. We remark that the analogous result for (1.2) follows directly by Γ-convergence and by the Subadditive Ergodic Theorem (see also [18]). The surface case, however, is more subtle, since the minimisation problem in (1.5) depends on x also through the boundary datum u rx,ζ,ν . To prove the x-independence of g hom we proceed in three steps. First, we exploit the stationarity of g to show that (1.6) is τ -invariant. Then, we prove the result when x is integer, by combinining the Subadditive Ergodic Theorem and the Birkhoff Ergodic Theorem, in the spirit of [2, Proof of Theorem 5.5] (see also [14,Proposition 2.10]). Finally, we conclude the proof with a careful approximation argument.

1.4.
Outline of the paper. The paper is organised as follows. In Section 2 we introduce some notation used throughout the paper. In the first part of Section 3 we state the assumptions on f and g and we introduce the stochastic setting of the problem; the second part is devoted to the statement of the main results of the paper. The behaviour of the volume term is studied in the short Section 4, while Sections 5 and 6, as well as the Appendix, deal with the surface term.

Notation
We introduce now some notation that will be used throughout the paper. For the convenience of the reader we follow the ordering used in [16].
(a) m and n are fixed positive integers, with n ≥ 2, R is the set of real numbers, and R m 0 := R m \ {0}, while Q is the set of rational numbers and Q m 0 := Q m \ {0}. The canonical basis of R n is denoted by e1, . . . , en. For a, b ∈ R n , a · b denotes the Euclidean scalar product between a and b, and | · | denotes the absolute value in R or the Euclidean norm in R n , R m , or R m×n , depending on the context. (g) For A ∈ A and p > 1 we define (h) For A ∈ A and p > 1 we define it is known that GSBV p (A, R m ) is a vector space and that for every u ∈ GSBV p (A, R m ) and for [19, page 172]). (i) For every L n -measurable set A ⊂ R n let L 0 (A, R m ) be the space of all (L n -equivalence classes of) L n -measurable functions u : A → R m , endowed with the topology of convergence in measure on bounded subsets of A; we observe that this topology is metrisable and separable. (j) For x ∈ R n and ρ > 0 we define We omit the subscript ρ when ρ = 1. (k) For every ν ∈ S n−1 let Rν be an orthogonal n×n matrix such that Rν en = ν; we assume that the restrictions of the function ν → Rν to the sets S n−1 ± defined in (b) are continuous and that R−νQ(0) = Rν Q(0) for every ν ∈ S n−1 ; moreover, we assume that Rν ∈ O(n) ∩ Q n×n for every ν ∈ Q n ∩ S n−1 . A map ν → Rν satisfying these properties is provided in [16, Example A.1 and Remark A.2]. (l) For x ∈ R n , ρ > 0, and ν ∈ S n−1 we set (o) For x ∈ R n and ν ∈ S n−1 , we set Π ν 0 := {y ∈ R n : y · ν = 0} and Π ν x := {y ∈ R n : (y − x) · ν = 0}. (p) For a given topological space X, B(X) denotes the Borel σ-algebra on X. In particular, for every integer k ≥ 1, B k is the Borel σ-algebra on R k , while B n S stands for the Borel σ-algebra on S n−1 . (q) For every t ∈ R the integer part of t is denoted by ⌊t⌋; i.e., ⌊t⌋ is the largest integer less than or equal to t.

Setting of the problem and statements of the main results
This section consists of two parts: in Section 3.1 we introduce the stochastic free-discontinuity functionals and recall the Ergodic Subadditive Theorem; in Section 3.2 we state the main results of the paper.
Given f ∈ F and g ∈ G, we consider the integral functionals F, G : is reversed when the orientation of νu is reversed, the functional G is well defined thanks to (g7).
Moreover, for G as in (3.2), and w ∈ L 0 (R n , R m ) with w|A ∈ SBVpc(A, R m ), we set In (3.3) and (3.4), by "u = w near ∂A" we mean that there exists a neighbourhood U of ∂A such that As a consequence we may readily deduce the following.
is not restrictive follows from assumption (g6) by using w as a competitor in the minimisation problem We are now ready to introduce the probabilistic setting of our problem. In what follows (Ω, T , P ) denotes a fixed probability space.
Let f be a random volume integrand. For ω ∈ Ω the integral functional F (ω) : , with g(·, ·, ·) replaced by g(ω, ·, ·, ·). Finally, for every ε > 0 we consider the free-discontinuity functional Eε(ω) : In the study of stochastic homogenisation an important role is played by the notions introduced by the following definitions. Definition 3.6 (P -preserving transformation). A P -preserving transformation on (Ω, T , P ) is a map T : Ω → Ω satisfying the following properties: If, in addition, every set E ∈ T which satisfies T (E) = E (called T -invariant set) has probability 0 or 1, then T is called ergodic.
Definition 3.7 (Group of P -preserving transformations). Let k be a positive integer. A group of Ppreserving transformations on (Ω, T , P ) is a family (τz) z∈Z k of mappings τz : Ω → Ω satisfying the following properties: , for every E ∈ T and every z ∈ Z k ; (d) (group property) τ0 = idΩ (the identity map on Ω) and If, in addition, every set E ∈ T which satisfies τz(E) = E for every z ∈ Z k has probability 0 or 1, then (τz) z∈Z k is called ergodic.
Remark 3.8. In the case k = 1 a group of P -preserving transformations has the form (T z ) z∈Z , where T := τ1 is a P -preserving transformation.
We are now in a position to define the notion of stationary random integrand.
Definition 3.9 (Stationary random integrand). A random volume integrand f is stationary with respect to a group (τz) z∈Z n of P -preserving transformations on (Ω, T , P ) if for every ω ∈ Ω, x ∈ R n , z ∈ Z n , and ξ ∈ R m×n . Similarly, a random surface integrand g is stationary with respect to (τz) z∈Z n if We now recall the notion of subadditive stochastic processes as well as the Subadditive Ergodic Theorem by Akcoglu and Krengel [1, Theorem 2.7].
We now state a variant of the pointwise ergodic Theorem [1, Theorem 2.7 and Remark p. 59] which is suitable for our purposes (see, e.g., [18, Proposition 1]).

Statement of the main results.
In this section we state the main result of the paper, Theorem 3.13, which provides a Γ-convergence and integral representation result for the random functionals (Eε(ω))ε>0 introduced in (3.7), under the assumption that the volume and surface integrands f and g are stationary.
The volume and surface integrands of the Γ-limit are given in terms of separate asymptotic cell formulas, showing that there is no interaction between volume and surface densities by stochastic Γ-convergence. The next theorem proves the existence of the limits in the asymptotic cell formulas that will be used in the statement of the main result. The proof will be given in Sections 4-6.
Thanks to Theorem 3.13 we can also characterise the asymptotic behaviour of some minimisation problems involving Eε(ω). An example is shown in the corollary below. Corollary 3.14 (Convergence of minimisation problems). Let f and g be stationary random volume and surface integrands with respect to a group (τz) z∈Z n of P -preserving transformations on (Ω, T , P ), let Ω ′ ∈ T (with P (Ω ′ ) = 1), f hom , and g hom be as in Theorem 3.12. Let ω ∈ Ω ′ , A ∈ A , h ∈ L p (A, R m ), and let (uε)ε>0 ⊂ GSBV p (A, R m ) ∩ L p (A, R m ) be a sequence such that Proof. The proof follows from Theorem 3.13, arguing as in the proof of [16, Corollary 6.1].

Proof of the cell-formula for the volume integrand
In this section we prove (3.10).
We can now give the proof of Proposition 4.1.
Proof of Proposition 4.1.
The existence of f hom and its independence of x follow from Proposition 4.2 and [18, Theorem 1] (see also [29,Corollary 3.3]). The fact that f hom is a random volume integrand can be shown arguing as in [16, Lemma A.5 and Lemma A.6], and this concludes the proof.

Proof of the cell-formula for the surface integrand: a special case
This section is devoted to the proof of (3.11) in the the special case x = 0. Namely, we prove the following result.
Theorem 5.1. Let g be a stationary random surface integrand with respect to a group (τz) z∈Z n of Ppreserving transformations on (Ω, T , P ). Then there exist Ω ∈ T , with P ( Ω) = 1, and a random surface integrand g hom : Ω × R m 0 × S n−1 → R such that for every ω ∈ Ω, ζ ∈ R m 0 , and ν ∈ S n−1 .
The proof of Theorem 5.1 will need several preliminary results. A key ingredient will be the application of the Ergodic Theorem 3.11 with k = n − 1. This is a nontrivial task, since it requires to define an (n − 1)dimensional subadditive process starting from the n-dimensional set function A → m pc G(ω) (u 0,ζ,ν , A). To this end, we are now going to illustrate a systematic way to transform (n − 1)-dimensional intervals (see (3.9)) into n-dimensional intervals oriented along a prescribed direction ν ∈ S n−1 .
Let A ′ ∈ In−1; we define the (rotated) n-dimensional interval Tν (A ′ ) as The next proposition is the analogue of Proposition 4.2 for the surface energy, and will be crucial in the proof Theorem 5.1.
To conclude the proof of Proposition 5.1 we need two preliminary lemmas. Then g, g ∈ G.
Proof. It is enough to adapt the proof of [16,Lemma A.7].
We will also need the following result.
Proof. The proof of (g2) can be obtained by adapting the proof of [16,Lemma A.7].
An analogous argument, now using the cube Q and hence the continuity of g (x, ζ, ·) in S n−1 + . The proof of the continuity in S n−1 − , as well as that of the continuity of g are similar.
We are now ready to prove Theorem 5.1.
Therefore the T -measurability of the function ω → g(ω, ζ, ν) in Ω for every ζ ∈ R m 0 and ν ∈ S n−1 implies that the restriction of g to Ω × R m 0 × S n−1 ± is measurable with respect to the σ-algebra induced in Ω×R m 0 × S n−1 ± by T ⊗B m ⊗B n S . This implies the (T ⊗B m ⊗B n S )-measurability of g hom on Ω×R m 0 ×S n−1 , thus showing that g hom satisfies property (c) of Definition 3.5.
Note now that for every ω ∈ Ω the function (x, ζ, ν) → g hom (ω, ζ, ν) defined in (5.17) belongs to the class G. Indeed, for ω ∈ Ω this follows from Lemma 5.3 while for ω ∈ Ω \ Ω this follows from the definition of g hom . Thus, g hom satisfies property (d) of Definition 3.5, and this concludes the proof. 6. Proof of the formula for the surface integrand: the general case In this section we extend Theorem 5.1 to the case of arbitrary x ∈ R n , thus concluding the proof of (3.11). More precisely, we prove the following result.
We now state some classical results from Probability Theory, which will be crucial for the proof of Theorem 6.1. For every ψ ∈ L 1 (Ω, T , P ) and for every σ-algebra T ′ ⊂ T , we will denote by E[ψ|T ′ ] the conditional expectation of ψ with respect to T ′ . This is the unique random variable in L 1 (Ω, T ′ , P ) with the property that We start by stating Birkhoff's Ergodic Theorem (for a proof, see, e.g., [28, Theorem 2.1.5]). Theorem 6.4 (Birkhoff's Ergodic Theorem). Let (Ω, T , P ) be a probability space, let T : Ω → Ω be a P -preserving transformation, and let IP (T ) be the σ-algebra of T -invariants sets. Then for every ψ ∈ L 1 (Ω, T , P ) we have for P -a.e. ω ∈ Ω.
We also recall the Conditional Dominated Convergence Theorem, whose proof can be found in [12,Theorem 2.7]. Theorem 6.5 (Conditional Dominated Convergence). Let T ′ ⊂ T be a σ-algebra and let (ϕ k ) be a sequence of random variables in (Ω, T , P ) converging pointwise P -a.e. in Ω to a random variable ϕ. Suppose that there exists ψ ∈ L 1 (Ω, T , P ) such that |ϕ k | ≤ ψ P -a.e. in Ω for every k.
We are now ready to prove the main result of this section.
We divide the proof into several steps. We use the notation for the integer part introduced in (q), Section 2.
Moreover, if (τz) z∈Z n is ergodic, then by Corollary 6.3 the function g hom does not depend on ω and (6.2) can be obtained by integrating (5.1) on Ω, and using the Dominated Convergence Theorem thanks to (5.4).

Appendix. Measurability issues
The main result of this section if the following proposition, which gives the measurability of the function ω → m pc G(ω) (w, A). This property was crucial in the proof of Proposition 5.2.
Proposition A.1. Let (Ω, T , P ) be the completion of the probability space (Ω, T , P ), let g be a stationary random surface integrand, and let A ∈ A . Let G(ω) be as in (3.2), with g replaced by g(ω, ·, ·, ·). Let w ∈ L 0 (R n , R m ) be such that w|A ∈ SBVpc(A, R m ) ∩ L ∞ (A, R m ), and for every ω ∈ Ω let m pc G(ω) (w, A) be as in (3.4), with G replaced by G(ω). Then the function ω → m pc G(ω) (w, A) is T -measurable. The main difficulty in the proof of Proposition A.1 is that, although ω → G(ω)(u, A) is clearly Tmeasurable, m pc G(ω) (w, A) is defined as an infimum on an uncountable set. This difficulty is usually solved by means of the Projection Theorem, which requires the completeness of the probability space. It also requires joint measurability in (ω, u) and some topological properties of the space on which the infimum is taken, like separability and metrisability. In our case (see (3.4)) the infimum is taken on the space of all functions u ∈ L 0 (R n , R m ) such that u|A ∈ SBVpc(A, R m ) and u = w near ∂A, and it is not easy to find a topology on this space with the above mentioned properties and such that (ω, u) → G(ω)(u, A) is jointly measurable. Therefore we have to attack the measurability problem in an indirect way, extending (an approximation of) G(ω)(u, A) to a suitable subset of the space of bounded Radon measures, which turns out to be compact and metrisable in the weak * topology.
We start by introducing some notation that will be used later. For every every A ∈ A we denote by