Mohammed Zakarya Baba-Ahmed
email: zaki.babaahmed@gmail.com

Badr Benmammar
email: badr.benmammar@gmail.com

Keywords: Cognitive Radio, Autonomic Networking, Spectral Handover, Self-Management

Cognitive Radio (CR) operates in different fields as varied, one of these is cognitive radio networks. In this paper, we propose a new approach used CR, which aims to manage potential failures of computer systems and applications through the introduction of two aspects of autonomous networks to make systems capable of managing themselves with minimum human intervention.

I. INTRODUCTION

The idea of cognitive radio was officially presented by Joseph Mitola III in a seminar at KTH, the Royal Institute of Technology in 1998, later published in an article by Mitola and Gerald Q. Maguire in 1999 [START_REF] Benmammar | Radio Resource Allocation and Dynamic Spectrum Access[END_REF].

The term cognitive radio was frequently used to refer to a system able to recognize its environment and to take advantage of this information. Sometimes, it is considered more restrictive as a system with high frequency agility to explore opportunities that may exist in the frequency spectrum [START_REF] Ngom | La radio cognitive[END_REF]. CR provides a good spectrum management occupy or operate in the unoccupied bands of radio spectrum, and of course, thus improving spectrum management. And that this is due to the Software Defined Radio (SDR).

The SDR is a radio communication system which can adapt to any frequency band and receive any modulation using the same material [START_REF] Benmammar | Radio Resource Allocation and Dynamic Spectrum Access[END_REF].

Autonomous networks are mainly represented by the selfmanagement, which aims to become less dependent computer systems users. The latter is characterized by four distinct points: Self-optimization, self-configuration, self-protection and self-healing. Tlemcen, Algeria ftbendimerad@gmail.com An autonomous system will never settle for the present situation. It will be constantly monitoring the achievement of predefined system or performance levels to ensure that all systems are running at an optimal level, it must be able to install and configure the software automatically. It must also identify, detect, and protect valuable corporate resources from many threats, as he will be able to find and repair any problems to ensure that the systems function properly [3].

In this paper we begin by establishing the link between CR and autonomous network by integrating the specifications of self-management in cognitive systems, then we present an approach to manage potential interference problems may appear between a Primary User (PU) with a license on the spectrum and a Secondary User (SU) will allocated channels in the spectrum.

II. AUTONOMY IN THE CONTEXT OF CR

Autonomy in CR networks is mainly focus on spectrum management which allows the improvement of the throughput without degrade the communications of others. Several studies were presented on the different characteristics of self-management:

A. Self-optimization of cognitive engine

This work has been already achieved by [START_REF] Colson | Mécanismes d'adaptation autonome pour la radio cognitive[END_REF] which allow complete and autonomous adaptation of cognitive engine to:

• Respect the regulatory framework that controls access to the spectrum ;

• Meet the needs of the user in terms of quality of service ;

• Let's make an optimized management of available resources [START_REF] Colson | Mécanismes d'adaptation autonome pour la radio cognitive[END_REF]. In Figure 2, the elements of CR communicate with SDR support through the cognitive engine. This latter is responsible of the optimization and control of the SDR module based on some input part parameters such as information from sensory perception to reasoning with the system or the learning of the radio environment, user context and network status.

Self-protection and self-healing in the context of cognitive radio

Here, the knowledge base keeps the system states and actions available. The reasoning engine uses the knowledge base to select the best action. The Learning Engine performs the manipulation of knowledge based on the observed information (information on channel availability, the error rate in the channel, etc.) [START_REF] Benmammar | Radio Resource Allocation and Dynamic Spectrum Access[END_REF].

B. Layer-2 Self-Configuration Algorithms

In a network with compatible devices, self-configuration Layer 2 in the CR involves determining a common set of channels to facilitate communication between participating nodes. It is a unique challenge because maybe the CR network nodes ignore:

• Their neighbors ;

• The channels on which they can communicate with a neighbor.

The authors in [START_REF] Krishnamurthy | Time-e cient distributed layer-2 auto-configuration for cognitive radio networks[END_REF] have proposed a CR network algorithm with a distributed time-efficient for selfconfiguration of Layer 2. During the layer-2 auto-configuration process, the Time Division Multiple Access (TDMA) scheme is used for communication among nodes. Time is divided into O(D) rounds. A round is defined as the time taken for every node to communicate with each one of its neighbors using a (local) broadcast mechanism. Each round consists of equal-sized intervals referred to as frames. The number of frames in a round may vary as shown in Figure . 3. A round in phase 1 consists of M frames while a round in phase 2 consists of only one frame. Each frame is further divided into N timeslots, each of equal length. Node i transmit during the ith timeslot in each frame (see Figure . 3) and all other nodes are in receiving mode during the ith timeslot. We next present more details of the algorithm [START_REF] Krishnamurthy | Time-e cient distributed layer-2 auto-configuration for cognitive radio networks[END_REF].

C. Self-Awareness in cycle of cognition

Self-awareness demonstrates the development of concepts that constitute a major problem in the design of cognitive self-managed networks according to the vision of the Future Internet networks being developed in the Self-NET project [START_REF] Mihailovic | An Approach for Designing Cognitive Self-Managed Future Internet[END_REF].

Figure 4 unifies and makes some of the key issues for the development of cognitive systems in self-awareness and currently performs the implementation of these with real test beds that will reveal the practical implementation and coordination problems in the deployment of cognitive cycles for use cases and application of the theoretical framework presented in Figure 4. III. PROPOSES SOLUTION As you've probably noticed, we have not discussed two characteristics of autonomous systems, self-protection and self-healing. Our approach in this paper will focus on these two points. But for now we'll see what kind of transmission will be dedicated this autonomy.

Quality of Service (QoS) is part of the complexity of data transmission, as Bandwidth, Delay, Loss and Jitter. What better than to ensure the transmission of the video conference who has a very high priority to insure in CR. Table I illustrates the varied application data types and their respective network sensitivities regarding bandwidth, packet loss, delay. The characteristics of the data types are denoted as very high -5, high -4, medium -3, low -2, and very low -1 [START_REF] Wiggins | Data-Centric Prioritization in a Cognitive Radio Network: a Qualityof-Service based Design and Integration[END_REF].

A. Proposes architecture

B. Principle

The principle of this approach is that a SU will detect and monitor the QoS thresholds periodically to find a free part of a spectrum licensed of a PU. It is assumed in this case that the PU has 8 channels and thus to ensure the QoS of the video conference with a sensitivity rated 4 (i.e. high); the SU will allocate 4 of these 8 channels with the agreement of PU. Now the PU has three possible scenarios:

1/ If the PU uses one, two or three of its channels, the SU is in Normal Mode. So the SU directs, plans and decides the best action to take after the SU uses/share the spectrum with the PU.

2/ If the PU uses 4 channels available, in this case we are in Warning Mode, using its 8 channels can cause interference between the SU and the PU during transmission, which will seriously disturb the SU as it is supposed guaranteed QoS for video conferencing, then a negotiation phase will be handled between the PU and SU is the tamper, after negotiation we would have two possible cases:

• The first case will occur if the PU cooperates with the SU and agrees to assign at least one transmission channel, so the SU will return to the first scenario (i.e. in Normal mode). • The second case occurs if the PU refuses any type of negotiation, the SU will entered a new phase that is mentioned in the third scenario.

3/ If the PU decides to use more than four channels, in this case, not enough channels to allocated by the SU, it is in the Failure Mode, the interference will be guaranteed and no negotiation phase will be given, so a new phase is required, this is called the Self-healing. In this phase we made a changing spectrum or in other words we need a Spectral handover.

Auto-protection is generated by the negotiation phase that seeks a dynamic spectrum access; there are various existing approaches for dynamic spectrum allocation:

• Medium Access Control (MAC);

• Local Negotiation;

• Game Theory;

• Multi Agent Systems (MAS); • MARKOV's chains.

The latter approach will be more conducive to the negotiation of our work because normally the most known approaches based on MARKOV's chains; the authors model the interactions between users and calculate the probabilities of blocking and non-completion as the main parameters evaluation [START_REF] Mir | Utilization of Cooperative Multiagent Systems for Spectrum Sharing in Cognitive Radio Networks[END_REF].

The Self-healing is caused by the phase of the changing spectrum (Spectral Handover) to conduct a new detection of another spectrum with the same requirements in terms of quality of service for video conferencing.

Learning is done by the events that occurred during the three modes and the results obtained by the management of the two phases, which leads to enrich the knowledge base by Automatic Learning.

IV. CONCLUSION

In this paper we presented several parts of the selfmanagement link cognitive radio with the autonomous networks, but our work was mainly focused on two parts of autonomy, of course we speak about self-protection and self -healing of a cognitive radio node on which we studying the case of maturity or non-cooperation between primary and secondary user with the use of negotiation in case of threat of interference between the latter two, and in case of changing spectrum caused by out of cooperation.

In our future work, we will seek to minimize the failure rate negotiation between PUs and SUs whom coexist in the same spectrum and we will study the impact of selfmanagement in cognitive radio networks.

Figure 1 .

 1 Figure 1. The four basic elements

Figure 2 .

 2 Figure 2. Self-optimization of cognitive engine

Figure 3 .

 3 Figure 3.Operation cycle of a diameter-aware node.

Figure 4 .

 4 Figure 4. Self-NET logical architecture of the cognitive cycle for selfmanagement [6]

Figure 5 .

 5 Figure 5. Self-protection and self-healing of a cognitive radio node to management failures

TABLE I .

 I QOS NETWORKS DATA TYPE AND SENSITIVITIES

			characteristic
	Transmission Type	Bandwidth	Delay	Loss	Jitter
	Voice	1	4	3	4
	E-Commerce	2	4	4	2
	Transactions	2	4	4	2
	E-mail	2	2	4	2
	Telnet	2	3	4	2
	Casual browsing	2	3	3	2
	Serious browsing	3	4	4	2
	File transfers	4	2	3	2
	Video conferencing	4	4	3	4
	Multicasting	4	4	4	4


Hung V Souganidis
email: souganidis@math.uchicago.edu

Tran

Adina Ciomaga

Panagiotis E Souganidis ♯,♮

Hung V Tran

STOCHASTIC HOMOGENIZATION OF INTERFACES MOVING WITH CHANGING SIGN VELOCITY

Keywords: stochastic homogenization, Hamilton-Jacobi equations, viscosity solutions, noncoercive Hamiltonian, random media, level sets, front propagation. AMS Classification: 35B27 70H20 37A50 49L25 78A48

HAL is a

Introduction

We study the averaged behavior of interfaces moving in stationary ergodic environments, with oscillatory normal velocity which changes sign. The problem can be formulated using the level-set method, as the homogenization of a Hamilton-Jacobi equation with a positively homogeneous noncoercive Hamiltonian. For a general overview and further details we refer to Osher and Sethian [START_REF] Osher | Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations[END_REF] (see also Souganidis [START_REF] Souganidis | Front propagation: theory and applications[END_REF], Barles and Souganidis [START_REF] Barles | A new approach to front propagation problems: theory and applications[END_REF] and the references therein). The interface is defined as the zero level set Γ ε (t, ω) of the solutions u ε of the initial value problem

u ε t + a x ε , ω |Du ε | = 0 in R n × (0, ∞), u ε = u 0 on R n × {0}, (1.1)
where u 0 ∈ UC (R n ), the space of uniformly continuous functions on R n . The random environment is modeled by a probability space (Ω, F, P) endowed with an ergodic group of measure preserving transformations (τ z ) z∈R n .

The velocity a = a(y, ω) : R n × Ω → R is a stationary random process with respect to (Ω, F, P) (precise definitions are given later) and continuous in y for each ω. The main challenge is to analyze the averaged behavior of the u ε 's when a(•, ω) changes sign. In this case, the Hamiltonian corresponding to (1.1), given by H(p, y, ω) = a(y, ω)|p|,

is neither coercive nor convex and, hence, the stochastic homogenization of (1.1) cannot be handled by the theory developed so far in the stationary ergodic environments. When a is strictly positive and, for all ω ∈ Ω, inf R n a(•, ω) = a 0 > 0, the Hamiltonian is convex and coercive. The typical results yield that, as ε → 0, the u ε 's converge, locally uniformly in R n × [0, ∞) and a.s. in Ω, to a deterministic u which is the solution of the effective initial value problem u t + H(Du) = 0 in R n × (0, ∞),

u = u 0 on R n × {0}. (1.3)
The main step is, of course, to determine the effective Hamiltonian H = H(p), which is often called the ergodic constant.

The periodic homogenization of coercive Hamilton-Jacobi equations was first studied by Lions, Papanicolaou and Varadhan [START_REF] Lions | Homogenization of Hamilton-Jacobi equations[END_REF] and later by Evans [START_REF] Evans | The perturbed test function method for viscosity solutions of nonlinear PDE[END_REF][START_REF] Evans | Periodic homogenisation of certain fully nonlinear partial differential equations[END_REF]. Ishii established in [START_REF] Ishii | Almost periodic homogenization of Hamilton-Jacobi equations[END_REF] the homogenization of Hamilton-Jacobi equations in almost periodic settings. The stochastic homogenization for Hamilton-Jacobi equations with convex and coercive Hamiltonians was established independently by Souganidis [START_REF] Souganidis | Stochastic homogenization of Hamilton-Jacobi equations and some applications[END_REF] and Rezakhanlou and Tarver [START_REF] Rezakhanlou | Homogenization for stochastic Hamilton-Jacobi equations[END_REF]. Results for viscous Hamilton-Jacobi equations were obtained by Lions and Souganidis [START_REF] Lions | Homogenization of "viscous" Hamilton-Jacobi equations in stationary ergodic media[END_REF] and Kosygina, Rezakhanlou and Varadhan [START_REF] Kosygina | Stochastic homogenization of Hamilton-Jacobi-Bellman equations[END_REF], while problems with space-time oscillations were considered by Kosygina and Varadhan [START_REF] Kosygina | Homogenization of Hamilton-Jacobi-Bellman equations with respect to time-space shifts in a stationary ergodic medium[END_REF] and Schwab [START_REF] Schwab | Stochastic homogenization of Hamilton-Jacobi equations in stationary ergodic spatio-temporal media[END_REF]. In [START_REF] Lions | Stochastic homogenization of Hamilton-Jacobi and "viscous"-Hamilton-Jacobi equations with convex nonlinearities-revisited[END_REF] Lions and Souganidis gave a simpler proof for homogenization in probability using weak convergence techniques. Their program was completed by Armstrong and Souganidis in [START_REF] Armstrong | Stochastic homogenization of Hamilton-Jacobi and degenerate Bellman equations in unbounded environments[END_REF][START_REF] Armstrong | Stochastic homogenization of level-set convex Hamilton-Jacobi equations[END_REF], who also introduced the metric approach. The viscous case was later refined by Armstrong and Tran in [START_REF] Armstrong | Stochastic homogenization of viscous Hamilton-Jacobi equations and applications[END_REF].

When the velocity in (1.1) changes sign, the Hamiltonian (1.2) is neither convex nor coercive. In this case, we consider the connected components (U i (ω)) i∈I of {x ∈ R n : a(x, ω) = 0}. In each U i (ω), the Hamiltonian is either convex or concave. Therefore, we study the evolution of the u ε 's in each rescaled connected component U i,ε (ω) := {εx : x ∈ U i (ω)} and obtain, for each i ∈ I, an effective equation u i,t + H i (Du i ) = 0 in R n × (0, ∞),

u i = u 0 on R n × {0}, (1.4)
with a deterministic effective Hamiltonian H i = H i (p). It is, of course, necessary to assume that the sets U i (ω) are themselves stationary ergodic (definitions are given later). The result is that, as ε → 0, the u ε 's converge locally uniformly in U i,ε (ω) × [0, ∞) to the solution u i of (1.4) and locally in L ∞ (R n × (0, ∞))-weak ⋆ to u := θ 0 u 0 + i∈I

θ i u i , (1.5)
where, for all i ∈ I, θ i := E[1 U i (ω) ] and θ 0 := E[1 {a(•,ω)=0} ]. This result was already established in the periodic setting by Cardaliaguet, Lions and Souganidis [START_REF] Cardaliaguet | A discussion about the homogenization of moving interfaces[END_REF]. We recall that, in the periodic case, each effective Hamiltonian H i (p) is found by solving the associated cell problem in the connected component U i , that is, for each p ∈ R n , there is a unique constant H i (p) such that there exists a periodic solution w, called corrector, of

a(y)|p + Dw| = H i (p) in U i .
(1.6)

The main difficulty in random environments is the lack of correctors. When the media is stationary ergodic, one can establish existence of subcorrectors for convex and coercive Hamiltonians [START_REF] Lions | Homogenization of "viscous" Hamilton-Jacobi equations in stationary ergodic media[END_REF], that is, subsolutions of (1.6) with strictly sublinear decay at infinity. To overcome the lack of correctors, a new approach was initiated in [START_REF] Armstrong | Stochastic homogenization of Hamilton-Jacobi and degenerate Bellman equations in unbounded environments[END_REF][START_REF] Armstrong | Stochastic homogenization of level-set convex Hamilton-Jacobi equations[END_REF] to study the stochastic homogenization of Hamilton-Jacobi equations by introducing the metric problem.

We follow here this approach and consider, for µ > 0 and each unbounded connected component U i (ω) of {x ∈ R n : a(x, ω) > 0} (respectively for µ < 0 and each unbounded connected component

U i (ω) of {x ∈ R n : a(x, ω) < 0}) and z ∈ U i (ω), a(y, ω)|Dm µ | = µ in U i (ω) \ {z}, m µ (•, z, ω) = 0 at {z}. (1.7)
Problem (1.7) has a stationary and subadditive maximal subsolution (respectively superadditive minimal supersolution) m µ in U i (ω), which can be thought of as the minimal travel time from z to y, with passage velocity a(•, ω), while staying within the connected domain U i (ω). The difficulty is that the m µ 's are defined only in U i (ω) and not in R n . The lack of coercivity of (1.2) creates an additional problem. Since a(•, ω) = 0 on each ∂U i (ω), the m µ 's are not Lipschitz continuous up to the boundary of U i (ω). In fact, as we show, the m µ 's blow up on ∂U i (ω) with a logarithmic rate. In general, Lipschitz estimates for solutions of Hamilton-Jacobi equations are given in terms of the geodesic distance within the domain (see for example Lions [START_REF] Lions | Generalized solutions of Hamilton-Jacobi equations[END_REF]). Here, using the local structure of the equation, we give sharp Lipschitz bounds of the m µ 's, independent of the geodesic distance. As a consequence, homogenization holds for a wide class of domains, such as classical percolation structures from probability (see for example Garet and Marchand [START_REF] Garet | Asymptotic shape for the chemical distance and first-passage percolation on the infinite bernoulli cluster[END_REF], Antal and Pisztora [START_REF] Antal | On the chemical distance for supercritical Bernoulli percolation[END_REF]) and Poisson point environments.

The tool to study the behavior of the solution to the metric problem is the subadditive ergodic theorem, which, however, cannot be applied directly to the m µ 's, since they are not defined everywhere. To overcome this drawback, we first extend m µ on each ray starting from the origin, apply the subadditive ergodic theorem to the extension and then relate back the averaged limit to m µ . This difficulty is amplified by the blow up of the m µ 's on ∂U i (ω). To overcome the latter, we restrict the m µ 's to the sets

U δ i (ω) = {x ∈ U i (ω) : |a(x, ω)| > δ}.
We show that the averaging takes place in U δ i (ω) and the limit is independent of δ. Moreover, in view of the Lipschitz estimates, we obtain an averaging of m µ , in the liminf, up to the boundary. More precisely, we show that, for each connected component of {x ∈ R n : a(x, ω) > 0} lim inf

t→∞ ty∈U i (ω) m µ (ty, 0, ω) t = lim t→∞ ty∈U δ i (ω) m µ (ty, 0, ω) t = m µ (y).
When we consider connected components corresponding to {x ∈ R n : a(x, ω) < 0} the liminf becomes limsup. The effective Hamiltonian is then characterized by the m µ 's as the smallest (respectively the largest) constant for which the metric problem (1.7) has a subsolution with strictly sublinear decay at infinity. We provide an inf-sup representation formula for each deterministic effective Hamiltonian H i (p) and establish the homogenization result in each connected component. The weak convergence (1.5) then follows.

Few results are available for non-coercive Hamiltonians and they all rely on some reduction property that compensates for the lack of coercivity; see, for example, Alvarez and Bardi [START_REF] Alvarez | Ergodicity, stabilization, and singular perturbations for Bellman-Isaacs equations[END_REF], Barles [START_REF] Barles | Some homogenization results for non-coercive Hamilton-Jacobi equations[END_REF] and Imbert and Monneau [START_REF] Imbert | Homogenization of first-order equations with (u/ǫ)-periodic Hamiltonians. I. Local equations[END_REF]. A different approach, based on nonresonance conditions, was initiated by Arisawa and Lions [START_REF] Arisawa | On ergodic stochastic control[END_REF] and extended to periodic noncoercive-nonconvex Hamiltonians by Cardaliaguet in [START_REF] Cardaliaguet | Ergodicity of Hamilton-Jacobi equations with a noncoercive nonconvex Hamiltonian in R 2 /Z 2[END_REF]. Of special interest is the study of noncoercive Hamilton-Jacobi equations associated to moving interfaces. The homogenization of (1.1) in the periodic setting was established by Cardaliaguet, Lions and Souganidis [START_REF] Cardaliaguet | A discussion about the homogenization of moving interfaces[END_REF]. Prior to their work, Craciun and Bhattacharya [START_REF] Craciun | Homogenization of a Hamilton-Jacobi equation associated with the geometric motion of an interface[END_REF] discussed formally and gave numerical examples for periodic homogenization of such problems.

The homogenization of the G-equation (used as model for fronts propagating with normal velocity and advection) was recently established by Cardaliaguet, Nolen and Souganidis citeCNS:11 in spatio-temporal periodic environments and by Cardaliaguet and Souganidis in [START_REF] Cardaliaguet | Homogenization and enhancement of the g-equation in random environments[END_REF] in random media. Homogenization of general non-convex Hamiltonians in random environments has remained, until now, completely open. A first extension to level-set convex Hamiltonians was proven by Armstrong and Souganidis in [START_REF] Armstrong | Stochastic homogenization of level-set convex Hamilton-Jacobi equations[END_REF] and, more recently, Armstrong, Tran and Yu established stochastic homogenization for double-well type Hamiltonians in [START_REF] Armstrong | Stochastic homogenization of a nonconvex Hamilton-Jacobi equation[END_REF].

The organization of the paper. In Section 2 we introduce the assumptions on the random media and the Hamiltonian and state the main results. In Section 3 we discuss some examples. We study the metric problem and give optimal Lipschitz regularity results of the maximal subsolution in Section 4. We prove in Section 5 the homogenization of the metric problem and obtain a deterministic effective Hamiltonian in Section 6. The main homogenization result is shown in Section 7.

Notations. We work in the n-dimensional Euclidean space R n and we denote by Q n and Z n respectively the sets of points with rational and integer coordinates. For a unit vector y ∈ R n , we write R + y := {ty : t ≥ 0}. We denote by B r (x 0 ) the ball of radius r centered at x 0 and by B R the ball of radius R centered at the origin. If U ⊂ R n , we denote by 1 U the characteristic function of U . For all ε > 0 we consider the rescaled sets

U ε = {εx : x ∈ U }. Let C 1 (U ) and C ∞ c (R n
) be respectively the sets of continuously differentiable functions on U and of infinitely differentiable functions on R n with compact support. We denote by L the class of Lipschitz functions on R n .

Assumptions and Main Results

We describe here the general setting, introduce the assumptions and present the main theorems. For the reader's convenience, we recall some preliminary results to be used later in the paper.

The general random setting. We consider a probability space (Ω, F, P) endowed with an ergodic group of measure preserving transformations (τ z ) z∈R n , that is, a family of maps τ z : Ω → Ω satisfying, for all z, z ′ ∈ R n and all U ∈ F,

τ z+z ′ = τ z • τ z ′ and P[τ z U ] = P[U ] and if τ z (U ) = U for every z ∈ R n , then either P[U ] = 1 or P[U ] = 0.
The assumptions. Let a : R n × Ω → R be measurable with respect to the σ-algebra generated by B × F, where B denotes the Borel σ-algebra of R n and assume that (A1) a is stationary with respect to (τ z ) z∈R n , that is, for every y, z ∈ R n and each ω ∈ Ω, a(y, τ z ω) = a(y + z, ω), (A2) the family a(•, ω) ω∈Ω is equi-bounded and equi-Lipschitz continuous, with Lipschitz constant L > 0, that is, for every y, z ∈ R n and ω ∈ Ω, |a(y, ω)a(z, ω)| ≤ L|y -z|, (A3) for each ω ∈ Ω, a(•, ω) changes sign.

Let U i (ω) i∈I + be the connected components of {a(•, ω) > 0} and U i (ω) i∈I -be the connected components of {a(•, ω) < 0} such that, for all i ∈ I ± ,

∂U i (ω) ⊂ U 0 (ω) := {x ∈ R n : a(x, ω) = 0}.
Denote by I := I + ∪ U -and I 0 := I ∪ {0}.

We introduce next the assumptions on the random structure of U i (ω) i∈I 0 . At first look, the conditions appear uncheckable since there is no topology in Ω, but later in the paper (Section 3) we give several examples of random media that satisfy these assumptions. We assume that: (S1) there exists a family {U i } i∈I 0 of measurable subsets of Ω such that P[U 0 ] = θ 0 ≥ 0 and, for all i ∈ I, P[U i ] = θ i > 0 and, for each ω ∈ Ω, U i (ω) is stationary, that is,

U i (ω) = x ∈ R n : τ x ω ∈ U i , (
S2) for each ω ∈ Ω and all i ∈ I ± , U i (ω) is δ-connected, that is, there exists δ 0 > 0 such that, for every δ ∈ (0, δ 0 ), U δ i (ω) is connected and (S3) if U i (ω) with i ∈ I is unbounded, then, for every δ ∈ (0, δ 0 ) and every unit vector y ∈ R n , there exists an (at most) countable family of random intervals (s j (ω), t j (ω))

∞ j=1 , which depend on y and δ, so that

I δ y (ω) := {t ≥ 0 : ty ∈ U δ i (ω)} = ∞ j=1 (s j (ω), t j (ω))
and, uniformly in y and δ,

lim j→∞ s j (ω) t j (ω) = 1 a.s. in ω. (2.8)
In view of Borel-Cantelli Lemma, checking (2.8) is equivalent to showing that, for every ε > 0,

∞ j=1 P ω ∈ Ω : t j (ω) -s j (ω) t j (ω) > ε < ∞.
The main result. We now present the homogenization result, which is established in several steps. First we show that, for each connected component U i (ω) with i ∈ I, there exists a deterministic effective Hamiltonian and then we prove that, as ε → 0, the u ε 's converge locally uniformly in each U i,ε (ω) to a deterministic ūi . The L ∞ -weak ⋆ convergence of the u ε 's to an averaged profile ū follows.

For each µ > 0, ω ∈ Ω and U (ω) a connected component of {x ∈ R n : a(x, ω) = 0}, let S µ (U (ω)) be the class of functions w such that, for each η > 0 sufficiently small, there exists δ = δ(η) > 0, which is independent of ω, satisfying lim η→0 δ(η) = 0 and, for all δ ∈ (0, δ) and all y 1 , y 2 ∈ U δ (ω),

|w(y 1 ) -w(y 2 )| ≤ µ δ + η |y 1 -y 2 |.
Define L µ (U (ω)) := L∩S µ (U (ω)). We also consider the classes S ± of Lipschitz continuous functions which are strictly sublinear from below and from above respectively, that is

S + := w ∈ L : lim inf |y|→∞ w(y) |y| ≥ 0 and S -:= w ∈ L : lim sup |y|→∞ w(y) |y| ≤ 0 .
Theorem 2.1 (The effective Hamiltonians). Assume (A1), (A2), (A3), (S1), (S2) and (S3). For each i ∈ I there exists a deterministic, 1-positively homogeneous, continuous effective Hamiltonian

H i : R n → R such that (i) if U i (ω) is bounded, then H i ≡ 0, (ii) if U i (ω) is unbounded and i ∈ I + , then H i is convex, nonnegative and, for all p ∈ R n , H i (p) := inf (w,λ)∈S + ×(0,∞) w+p•y∈L λ (U (ω))
ess sup

y∈U i (ω) a(y, ω)|p + Dw| , (2.9)
(iii) if U i (ω) is unbounded and i ∈ I -, then H i is concave, nonpositive and, for all p ∈ R n , H i (p) := sup (w,λ)∈S -×(0,∞) w+p•y∈L λ (U (ω))
ess inf

y∈U i (ω)
a(y, ω)|p + Dw| .

(2.10)

For each i ∈ I we solve the averaged equation

u i,t + H i (Du i ) = 0 in R n × (0, ∞), u i = u 0 on R n × {0} (2.11)
and establish the following homogenization result.

Theorem 2.2 (Homogenization). Assume (A1), (A2), (A3), (S1), (S2) and (S3). There exists δ 0 > 0 and an event of full probability Ω ⊆ Ω such that, for each ω ∈ Ω, δ ∈ (0, δ 0 ) and T > 0, the unique solution

u ε = u ε (•, •, ω) of (1.1) converges locally uniformly in U δ i,ε (ω) × [0, T ] to the unique solution u i of (2.11), that is, for (x, t) ∈ R n × (0, T ), lim inf ε→0 y→x,s→t y∈U i,ε (ω) u ε (y, t, ω) = lim sup ε→0 y→x,s→t y∈U δ i,ε (ω) u ε (y, t, ω) = ūi (x, t) if i ∈ I + , (2.12) lim sup ε→0 y→x,s→t y∈U i,ε (ω) u ε (y, t, ω) = lim inf ε→0 y→x,s→t y∈U δ i,ε (ω) u ε (y, t, ω) = ūi (x, t) if i ∈ I -. (2.13) (2.14)
Moreover, for each ω ∈ Ω and R > 0, as ε → 0,

u ε * ⇀ u := θ 0 u 0 + i∈I θ i u i in L ∞ (B R × (0, T )). ( 2

.15)

Some preliminary results. An important tool in our analysis is the subadditive ergodic theorem for continuous subadditive processes. For the reader's convenience we give below the precise statement and we refer to Akcoglu and Krengel [START_REF] Akcoglu | Ergodic theorems for superadditive processes[END_REF] for its proof. Let (σ t ) t≥0 and I be respectively a semi-group of measure-preserving transformations on Ω and the class of subsets of [0, ∞) which are finite unions of intervals of the form [a, b). A continuous subadditive process on (Ω, F, P) is a map

Q : I → L 1 (Ω, P) which is (i) stationary invariant, that is, Q(I)(σ t ω) = Q(t + I)(ω)
for all t > 0, I ∈ I, a.s. in ω, (ii) uniformly integrable, that is, there exists C > 0 such that, E|Q(I)| ≤ C|I| for all I ∈ I, and (iii) subadditive with respect to unions of disjoint intervals, that is, if

I 1 , I 2 , ..., I k ∈ I are disjoint, then Q( k i=1 I i ) ≤ k i=1 Q(I i ).
Theorem 2.3 (The subadditive ergodic theorem). Let Q : I → L 1 (Ω, P) be a continuous subadditive process on (Ω, F, P) with respect to the semi-group of measure-preserving transformations (σ t ) t≥0 . Then there exists a random variable q such that lim

t→∞ 1 t Q [0, t) (ω) = q(ω) a.s. in ω.
If, in addition, (σ t ) t≥0 is ergodic, then q(ω) ≡ q a.s., for some constant q.

Solutions of the Hamilton-Jacobi equations are to be understood in the viscosity sense (see, for example, the "User's Guide" of Crandall, Ishii and Lions [START_REF] Crandall | User's guide to viscosity solutions of second order partial differential equations[END_REF] and the books by Barles [11] and Bardi and Capuzzo-Dolcetta [START_REF] Bardi | Optimal control and viscosity solutions of Hamilton-Jacobi-Bellman equations[END_REF]). The following well known fact will be used several times in the paper.

Lemma 2.4. Let p ∈ R n , µ ∈ R, U ⊂ R n open and a : U → R such that a > 0 in U and a = 0 on ∂U . Then w ∈ L is a viscosity solution of a(y)|p + Dw| ≤ µ in U (2.16)
if and only if w satisfies (2.16) almost everywhere in U .

Examples of stationary random environments satisfying (S1), (S2) and (S3)

We present several examples of random media, some occurring in percolation theory, that satisfy our assumptions.

Example 1 (Site percolation). We briefly review a site percolation structure in R n . The space is divided into cubes of size 1 with vertices on Z n and each cube is painted, independently, white or black with probability p or 1p respectively, where 0 ≤ p ≤ 1. Cubes are always assumed to be closed, unless otherwise stated.

For each ω ∈ Ω, let F (ω) be the union of all black cubes. The interior of F (ω) and R n \ F (ω) consist of mutually disjoint open connected components that we denote respectively by U i (ω) i∈I - and U i (ω) i∈I + . It is known (see Grimmett [START_REF] Grimmett | Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences[END_REF]) that there exists a critical threshold p c > 0 such that if p > p c , then a.s. there only exists one unbounded connected component U ∞ (ω) of R n \F (ω). Figure 1 illustrates a site percolation model where U ∞ (ω) is painted in blue. Let Π n be the set of unit cubes whose vertices are points of Z n . The probability space (Ω, F, P) is Ω := {0, 1} Π n , F = {U : U ⊂ Ω} and P = (pδ 1 + (1p)δ 0 ) ⊗Π n . The translation operators (τ z ) z∈Z n , given by τ z : Ω → Ω with (τ z ω)(π) := ω(z + π), for all π ∈ Π n , are ergodic. Theorem 2.2 holds for any velocity a(•, ω) such that a(•, ω) > 0 in U i (ω) with i ∈ I + , a(•, ω) < 0 in U i (ω) with i ∈ I -and a(•, ω) = 0 on ∂U i (ω) with i ∈ I ± , which is bounded, Lipschitz continuous and stationary with respect to (τ z ) z∈Z n . A particular choice of a(•, ω) is the distance function to the boundaries of U i (ω), that is,

a(x, ω) = min d(x, ∂U i (ω)), 1/2 if x ∈ U i (ω), with i ∈ I + , -min d(x, ∂U i (ω)), 1/2 if x ∈ U i (ω), with i ∈ I -. Figure 1. Bernoulli percolation.
Clearly a satisfies (A1), (A2), (A3) and (S1), (S2) hold for all U i (ω) with i ∈ I 0 , while U ∞ (ω) satisfies (S3). Indeed, for any y ∈ R n with |y| = 1 and ε > 0, we have

P ω ∈ Ω : t j (ω) -s j (ω) t j (ω) > ε = P ω ∈ Ω : t 1 (ω) -s 1 (ω) t j (ω) > ε ≤ P {ω ∈ Ω : |t 1 (ω) -s 1 (ω)| > jε} ≤ p jε
and, thus,

∞ j=1 P ω ∈ Ω : t j (ω) -s j (ω) t j (ω) > ε ≤ ∞ j=1 p εj = 1 1 -p ε .
In view of Theorem 2.1 all effective Hamiltonians are null, except for H ∞ corresponding to U ∞ (ω). Theorem 2.2 then yields, the L ∞ -weak ⋆ convergence, as ε → 0, of the u ε 's to u = (1θ)u 0 + θu ∞ with θ = E[1 U∞(ω) ], where u ∞ is the solution of (2.11) corresponding to H ∞ .

In the particular case of a periodic checkerboard configuration and of a periodic velocity a, all H i ≡ 0 and, hence, u ε converges uniformly to the initial data u 0 . We recover thus the result of [START_REF] Cardaliaguet | A discussion about the homogenization of moving interfaces[END_REF], also known as trapping.

Example 2 (Site percolation on regular latices). Theorem 2.2 holds for general percolation structures given by regular lattices, such as those displayed in Figure 2, where as before, each cell of the lattice is painted white or black with probability p and 1p respectively. Example 3 (Site percolation with isolated obstacles). Consider a site percolation Π n as in Example 1, where now each cube π ∈ Π n is either empty with probability p or compactly contains a ball of fixed size with probability 1-p. Let a be a Lipschitz continuous, bounded and translation invariant velocity, which is negative inside each ball and non-negative outside. Then there exists one infinite connected component U ∞ (ω) with effective Hamiltonian H ∞ > 0 and, in view of Theorem 2.2, as ε → 0, the u ε 's converge weakly to u = (1θ)u 0 + θu ∞ , where θ = E[1 U∞(ω) ] and u ∞ is the solution of (2.11) associated to U ∞ (ω). In particular, in the periodic setting, this construction yields a periodic configuration of small holes and we recover the result of [START_REF] Cardaliaguet | A discussion about the homogenization of moving interfaces[END_REF].

Example 4 (Poisson cloud). Theorem 2.2 applies to Poisson environments. Consider a random set

F (ω) = ∞ i=1 B r i (x i , ω)
, where the points (x i ) ∞ i=1 are given by a Poisson point process and the radii (r i ) ∞ i=1 are independent and identically distributed on (0, ∞) (see Figure 3). Let (U i (ω)) i∈I + and (U i (ω)) i∈I -be respectively the connected components of R n \ F (ω) and the interior of F (ω) and a be a Lipschitz continuous, bounded and translation invariant velocity which is negative inside each ball and positive outside. One can easily check, as in Example 1, that assumptions (S1), (S2) and (S3) are satisfied.

The Metric Problem

Fix ω ∈ Ω and assume that U (ω) is an unbounded connected component of {x ∈ R n : a(x, ω) > 0} with ∂U (ω) ⊆ {x ∈ R n : a(x, ω) = 0}. For each fixed constant µ ≥ 0 and each z ∈ U (ω) we consider the eikonal-type problem a(y, ω)

|Dm µ | = µ in U (ω) \ {z}, m µ (•, z, ω) = 0 at {z}. (4.17)
We show below that there exists a maximal, positive, subadditive, locally Lipschitz continuous subsolution, which blows-up on the boundary of U (ω). Similar arguments remain true for the unbounded connected components of {x ∈ R n : a(x, ω) < 0}, in which case (4.17) is well posed for each fixed constant µ ≤ 0 and has instead a minimal, negative, superadditive supersolution.

Maximal subsolutions. We first define the maximal subsolution of (4.17) and establish some basic properties.

For each µ ≥ 0 and z ∈ U (ω) let m µ (•, z, ω) : U (ω) → [0, ∞) be given by

m µ (y, z, ω) := sup w(y, ω) : w(•, ω) ∈ L µ (U (ω)), w(z, ω) = 0, a(y, ω)|Dw| ≤ µ in U (ω) . (4.18)
Recall that, in view of Lemma 2.4, the differential inequality in (4.18) may be interpreted either in the viscosity sense or in the almost everywhere sense. Note that m µ (•, z, ω) is well defined in U (ω) for all µ ≥ 0. Indeed, the set of admissible w in (4.18) is nonempty, since w ≡ 0 satisfies all the requirements, a fact which also implies that m µ (•, z, ω) ≥ 0. Moreover, in view of the Lipschitz bounds we establish below, m µ (y, z, ω) is finite for all y ∈ U (ω). Also, observe that for µ = 0, m µ (•, z, ω) ≡ 0 in U (ω). Henceforth we only deal with µ > 0.

The next lemma summarizes two of the most important properties of m µ , namely that is maximal and locally Lipschitz continuous in U (ω).

Lemma 4.1. Assume (S2). For each z ∈ U (ω), m µ (•, z, ω) ∈ S µ (U (ω)) and it is a maximal subsolution of (4.17

), that is, if w ∈ L µ (U (ω)) is a subsolution of a(y, ω)|Dw| ≤ µ in U (ω), then w(•) -w(z) ≤ m µ (•, z, ω).
Proof. The result is a direct consequence of the definition of L µ (U (ω)) and (S2). The fact that m µ (•, z, ω) is a subsolution of (4.17) is immediate, since it is the supremum of a collection of subsolutions (see [START_REF] Crandall | User's guide to viscosity solutions of second order partial differential equations[END_REF].) Since a(•, ω) is degenerate on ∂U (ω), the Hamiltonian is not coercive and thus the solutions are not Lipschitz up to the boundary. We show next that the m µ 's blow up on the boundary of U (ω) with a logarithmic rate. Proof. Fix z ∈ U (ω) and δ ∈ (0, min(δ 0 , a(z, ω)/2)) with δ 0 > 0 given by (S2). Recall that in view of (S2), U δ (ω) = {x ∈ U (ω) : a(x, ω) > δ} is connected and note that z ∈ U δ (ω). Define the barrier

b δ (y, ω) = -µ L log a(y, ω) + c in U δ (ω), -µ L log δ + c in R n \ U δ (ω),
where c = (µ/L) log a(z, ω). Since a(•, ω) is differentiable a.e. in U δ (ω),

Db δ (•, ω) L ∞ (U δ (ω)) ≤ µ L Da(•, ω) L ∞ inf U δ (ω) a(•, ω) = µ δ . Thus b δ (•, ω) ∈ L µ (U (ω)) with a global Lipschitz constant equal to µ/δ, b δ (z, ω) = 0 and satisfies a(y, ω)|Db δ | = 0 in U (ω) \ U δ (ω). Therefore a(y, ω)|Db δ | ≤ µ in U δ (ω)
and, hence, b δ is an admissible subsolution in (4.18). In view of the maximality of m µ , for all y ∈ U (ω),

m µ (y, z, ω) ≥ sup δ b δ (y, ω) = - µ L log a(y, ω) + c.
The claimed bound is now obvious from the positivity of m µ (•, •, ω) and the definition of c.

is also a subsolution of (4.17) in U (ω), w(z, ω) = 0 and, in view of Lemma 4.1, w ∈ S µ (U (ω)). But w(•, ω) cannot be extended to a function in L due to its behavior at ∂U (ω). To overcome this, we use the following cut-off argument. For each M > 0, let w M (y, ω) := min{M, w(y, ω)} for y ∈ U (ω),

M for y ∈ R n \ U (ω).
In light of Proposition 4.2, for M large enough, there exists δ M ∈ (0, δ 0 ) such that w M = M in R n \ U δ M (ω). It follows from Barron and Jensen [START_REF] Barron | Semicontinuous viscosity solutions for Hamilton-Jacobi equations with convex Hamiltonians[END_REF] and Barles [10] that w M (•, ω) is a subsolution of (4.17) in U (ω), since it is the minimum of two subsolutions associated to a convex Hamiltonian in U (ω). Furthermore w M ∈ L µ (U (ω)) and, hence, in view of the maximality of m µ (•, z, ω), for all y ∈ U (ω),

w M (y, ω) ≤ m µ (y, z, ω).
Letting M → ∞ yields the claim.

Remark 4.5. In order to construct admissible test functions w which are globally Lipschitz, we will frequently use the cut-off argument above.

Well-posedness. It follows from Perron's method for viscosity solutions ( [START_REF] Ishii | Perron's method for Hamilton-Jacobi equations[END_REF]) that (4.17) is wellposed for each µ > 0 and m µ is actually a solution of (5.24). Since the proof requires some special care due to the fact that the equation is restricted to U (ω) and solutions are not globally Lipschitz, we present next some of the details. Proof. In view of Lemma 4.1, it is only necessary to show that m µ is a supersolution of (4.20) in U (ω) \ {z}. We argue by contradiction. Assume m µ (•, z, ω) is not a supersolution in U (ω) \ {z}. Then there exists y 0 ∈ U (ω) \ {z} and φ(•, ω) ∈ C 1 (U (ω)) such that m µ (y 0 , z, ω) = φ(y 0 , ω), m µ (•, z, ω) > φ(•, ω) in U (ω) \ {z} and a(y 0 , ω)|Dφ(y 0 , ω)| < µ.

(4.21)

For any ε > 0, let ψ ε (y, ω) := φ(y, ω) + ε -|yy 0 | 2 and consider, for all y ∈ U (ω),

w ε (y, ω) := sup m µ (y, z, ω), ψ ε (y, ω) .
Note that w ε (y, ω) = ψ ε (y, ω) only if y ∈ B r (y 0 ), with r = √ ε. For M large enough, let

w ε M (y, ω) := min{M, w ε (y, ω)} for y ∈ U (ω), M for y ∈ R n \ U (ω).
We show below that, for ε > 0 sufficiently small, w ε M (•, ω) ∈ L µ (U (ω)) and, hence, is a subsolution of (4.20) in U (ω). Since it satisfies w ε M (y 0 , ω) = φ(y 0 , ω) + ε > m µ (y 0 , z, ω), the claim contradicts the maximality of m µ (•, z, ω) at y 0 .

It suffices to check that, for ε > 0 sufficiently small, w ε (•, ω) ∈ S µ (U (ω)). Since m µ ∈ S µ (U (ω)), for any η > 0, let δ(η) > 0 such that, for all δ ∈ (0, δ) and all y 1 , y 2 ∈ U δ (ω),

|m µ (y 1 , z, ω) -m µ (y 2 , z, ω)| ≤ µ δ + η |y 1 -y 2 |.
If y 1 , y 2 ∈ U δ (ω) \ B r (y 0 ), then the estimate above holds for w ε (•, ω), since w ε (•, ω) = m µ (•, z, ω), while if y 1 , y 2 ∈ B r (y 0 ), the smoothness of ψ ε (•, ω) gives

|ψ ε (y 1 , ω) -ψ ε (y 2 , ω)| ≤ Dψ ε L ∞ (Br(y 0 )) |y 1 -y 2 |.
Note also that (4.21) and the continuity of a(•, ω), taking if necessary a smaller ε, yield that ψ ε is a smooth subsolution of (4.17) in B r (y 0 ), hence,

|ψ ε (y 1 , ω) -ψ ε (y 2 , ω)| ≤ µ inf Br(y 0 ) a(•, ω) |y 1 -y 2 |.
For δ > 0 sufficiently small we have B r (y 0 )∩U δ (ω) = ∅. The continuity of a(•, ω) yields that, for any ν > 0, there exists some small r > 0 such that inf Br(y 0 ) a(•, ω) ≥ δ -ν. Choosing ν = δ 2 η/(2µ+2δη) so that ν → 0 as η → 0, we further get

|ψ ε (y 1 , ω) -ψ ε (y 2 , ω)| ≤ µ δ -ν |y 1 -y 2 | < µ δ + η |y 1 -y 2 |.
Lastly if y 1 ∈ B r (y 0 ) ∩ U δ (ω) and y 2 ∈ U δ (ω) \ B r (y 0 ) are such that w ε (y 1 , ω) = ψ ε (y 1 , ω) and w ε (y 2 , ω) = m µ (y 2 , z, ω), consider y 3 ∈ B r (y 0 ) on the segment determined by y 1 and y 2 , that is

y 3 = αy 2 + (1 -α)y 2 for some α ∈ (0, 1), such that w ε (y 3 , ω) = ψ ε (y 3 , ω) = m µ (y 3 , z, ω).
Then, in view of the previous cases, we get

|w ε (y 1 , ω) -w ε (y 2 , ω)| ≤ |w ε (y 1 , ω) -w ε (y 3 , ω)| + |w ε (y 3 , ω) -w ε (y 2 , ω)| ≤ µ δ + η |y 1 -y 3 | + µ δ + η |y 2 -y 3 | = µ δ + η |y 1 -y 2 |.
We continue with several properties of m µ with respect to µ that are used later in the paper. Since the proof is similar to the one in [START_REF] Armstrong | Stochastic homogenization of level-set convex Hamilton-Jacobi equations[END_REF] we omit it here. Proposition 4.7 (Continuity and monotonicity in µ). Assume (A2) and (S2). For µ 1 < µ 2 , let m µ 1 and m µ 2 be the maximal subsolutions of (4.17). Then, for any z ∈ U (ω),

m µ 1 (•, z, ω) ≤ m µ 2 (•, z, ω) in U (ω). Moreover µ → m µ (y, z, ω) is continuous on (0, ∞), locally uniformly in (y, z) ∈ U (ω) × U (ω), in the sense that, if µ j → µ as j → ∞, then for every compact K ⊂ U (ω), m µ j (•, •, ω) → m µ (•, •, ω) uniformly in K × K.

The Homogenization of the Metric Problem

We use the subadditive ergodic theorem to establish an averaging result for the solution of the metric problem in each unbounded connected component of {x ∈ R n : a(x, ω) = 0}. The argument is not standard, since m µ is not defined everywhere in R n . To overcome this difficulty, we first extend m µ along rays starting from the origin, apply the subadditive ergodic theorem to the extension and then relate back the limit to the original function. This difficulty is amplified by the blow up of the solution on the boundary of the connected components. To overcome the latter, we construct an averaged metric δ away from the boundary, then prove that the limit is independent of δ.

To fix the ideas, we assume U (ω) is an open unbounded connected component of {x ∈ R n : a(x, ω) > 0} with ∂U (ω) ⊆ {x ∈ R n : a(x, ω) = 0}. We show the homogenized limit exists, it is subadditive, 1-positively homogeneous and globally Lipschitz. When U (ω) is an unbounded connected component of {x ∈ R n : a(x, ω) < 0}, we construct similarly a superadditive averaged metric.

Theorem 5.1 (Averaging of m µ ). Assume (A1), (A2), (A3), (S1), (S2) and (S3). There exist δ 0 > 0 sufficiently small, an event of full probability Ω ⊆ Ω and, for each µ > 0, m µ : R n → R, such that, for every ω ∈ Ω, y, z ∈ R n and every δ ∈ (0, δ 0 ),

lim t→∞ sup |x|≤R lim sup y-z→x ty,tz∈U δ (ω) 1 t m µ (ty, tz, ω) -m µ (x) = 0. (5.22)
Since the proof of Theorem 5.1 is long, we divide it in several lemmata, which we state and prove first. We begin by establishing the averaging result at the origin z = 0 for each fixed direction y. Lemma 5.2. There exists δ 0 > 0 such that, for each µ > 0, δ ∈ (0, δ 0 ) and y ∈ R n , there exists a set of full probability Ω µ,δ y ⊆ Ω and a Lipschitz extension

m δ µ (•, •ω) : R + y × R + y → R of m µ (•, •, ω) such that, for every ω ∈ Ω µ,δ y , m δ µ (y, ω) := lim t→∞ 1 t m δ µ (ty, 0, ω).
Proof. We first define an extension of m µ in each direction y ∈ R n and then we introduce a new random process which satisfies the assumptions of the subadditive ergodic theorem. Fix y ∈ R n with |y| = 1. For each t ≥ 0 let t * y and t * y be the exit point from U δ (ω) and the entrance point in U δ (ω) before and after ty respectively, where t * := sup 0 ≤ s ≤ t : sy ∈ U δ (ω) and t * := inf s ≥ t : sy ∈ U δ (ω) .

It is immediate that t * and t * are measurable with respect to F, 0 ≤ t * ≤ t ≤ t * and there exists α ∈ (0, 1) such that t = (1α) t * + αt * . Note that if ty ∈ U δ (ω), then ty = t * y = t * y.

We define the extension

m δ µ (•, •, ω) : R + y × R + y → R of m µ (•, •, ω) as the bilinear interpolation m δ µ (ty, sy, ω) := (1 -α) (1 -β) m µ (t * y, s * y, ω) + β m µ (t * y, s * y, ω) + α (1 -β) m µ (t * y, s * y, ω) + β m µ (t * y, s * y, ω) , (5.23)
where

s := (1 -β)s * + βs * with β ∈ (0, 1). In particular, if 0 ∈ U δ (ω), m δ µ (•, 0, ω) : R + y → R is given by m δ µ (ty, 0, ω) = (1 -α) m µ (t * y, 0, ω) + α m µ (t * y, 0, ω). (5.24)
In view of (S3),

m δ µ (•, •, ω) is well defined. Moreover, if U δ y (ω) = U δ (ω) ∩ R + y, then m δ µ (•, •, ω) = m µ (•, •, ω) on U δ y (ω) × U δ y (ω). Given m δ µ (•,
•, ω), we define the random process Q : I → L 1 (Ω, P) by Q([s, t))(ω) := m δ µ (ty, sy, ω), which is a continuous subadditive process on (Ω, F, P) endowed with τ ty t≥0 .

Lipschitz continuity of m µ yield the Lipschitz continuity of m δ µ as follows:

m δ µ (y) = lim j→∞ 1 r j (ω) m µ (r j (ω)y, 0, ω) ≤ lim j→∞ 1 r j (ω) m µ (l j (ω)y ′ , 0, ω) + m µ (r j (ω)y, l j (ω)y ′ , ω) ≤ lim j→∞ 1 r j (ω) m µ (l j (ω)y ′ , 0, ω) + µ δ + η |r j (ω)||y -y ′ | + |r j (ω) -l j (ω)||y ′ | ≤ m δ µ (y ′ ) + µ δ + η |y -y ′ |.
The subadditivity of m δ µ follows directly from the subadditivity of m µ (•, •, ω) in U (ω). For l j (ω)y j≥0 , r j (ω)(y + z) j≥0 ⊂ U δ y (ω) as before, we have

m µ (y + z) = lim j→∞ 1 r j (ω) m µ (r j (ω)(y + z), 0, ω) ≤ lim j→∞ 1 r j (ω) m µ (r j (ω)(y + z), l j (ω)y, ω) + m µ (l j (ω)y, 0, ω) = m µ (y) + m µ (z).
That the average holds for an event of full probability for all y ∈ R n is an immediate consequence of the the density of Q n , the Lipschitz continuity of m µ and assumption (S3).

Having established Lemma 5.2 and Lemma 5.4, the proof of Theorem 5.1 is a consequence of Egoroff's theorem, the subadditive ergodic theorem and the Lipschitz estimates of m µ . Although the argument has already appeared in several references [START_REF] Lions | Stochastic homogenization of Hamilton-Jacobi and "viscous"-Hamilton-Jacobi equations with convex nonlinearities-revisited[END_REF][START_REF] Armstrong | Stochastic homogenization of Hamilton-Jacobi and degenerate Bellman equations in unbounded environments[END_REF], for the benefit of the reader we present some of the details, since averaging takes place only locally in U (ω) and not in R n .

Proof of Theorem 5.1. To simplify the arguments we drop the dependence on µ of the probability events and assume, in view of Remark 5.3, that 0 ∈ U (ω) a.s.. For each R > 0, consider the process

M([s, t))(ω) := sup |x|≤R m δ,x µ (tx, sx, ω) -(t -s)m δ µ (x) ,
where m δ,x µ (•, •, ω) is the bilinear extension in direction x of m µ (•, •, ω), given by (5.23). Arguing as in the proof of Lemma 5.2, it is easy to see that M is a continuous subadditive process. The subadditive ergodic theorem then yields, a.s. in ω ∈ Ω δ , lim

t→∞ sup |x|≤R 1 t m δ,x µ (tx, 0, ω) -m δ µ (x) = lim t→∞ sup |x|≤R tx∈U δ (ω) 1 t m µ (tx, 0, ω) -m δ µ (x) = 0.
It follows from Egoroff theorem that, for any ε > 0, there exists t ε > 0 and an event

W δ,ε ⊂ Ω δ such that P[Ω δ \ W δ,ε ] < ε n /4 and, for all t ≥ t ε , ess sup ω∈W δ,ε sup |x|≤R tx∈U δ (ω) 1 t m δ µ (tx, 0, ω) -m δ µ (x) ≤ ε. (5.27)
Applying the ergodic theorem to 1 W δ,ε we find an event of full probability Ω δ,ε ⊆ Ω δ , such that, for all ω ∈ Ω δ,ε , lim

r→∞ 1 |B r | Br 1 W δ,ε (τ z ω)dz = P[W δ,ε ] ≥ 1 - ε n 4 .
Consider the event Ωδ = ε∈(0,1)∩Q Ω δ,ε such that P[ Ωδ ] = 1. Then, for each ε > 0 and ω ∈ Ω δ , there exists r ε > 0 such that, for all r > r ε ,

z ∈ B r : τ z ω ∈ W δ,ε > 1 - ε n 2 |B r |, (5.28)
which implies that no ball of radius rε is contained in {z ∈ B r :

τ z ω ∈ W δ,ε }.
In view of (S3), for any z ∈ B r \ {0}, there exist a sequence t j (ω)z j≥0 ⊂ U δ (ω) such that, as j → ∞, t j (ω) → ∞. We deduce from (5.28) that, for each j sufficiently large, there exists z j ∈ B r , such that τ t j (ω)z j ω ∈ W δ,ε and |zz j | < ε. Then (5.27) yields

M([0, t j (ω)))(τ t j (ω)z ω) = sup |x|≤R t j (ω)x∈U δ (τ t j (ω)z ω) m µ (t j (ω)x, 0, τ t j (ω)z ω) -t j (ω)m δ µ (x) ≤ sup |x|≤R t j (ω)(x+z j )∈U δ (ω) m µ (t j (ω)(x + z j ), t j (ω)z j , ω) -t j (ω)m δ µ (x) + 2t j (ω) µ δ + η |z -z j | = sup |x|≤R t j (ω)x∈U δ (τ t j (ω)z j ω) m µ (t j (ω)x, 0, τ t j (ω)z j ω) -t j (ω)m δ µ (x) + 2t j (ω) µ δ + η ε ≤ t j (ω)ε + 2t j (ω) µ δ + η ε.
Letting ε → 0 we find

lim j→∞ 1 t j (ω) M([0, t j (ω)))(τ t j (ω)z ω) = lim t→∞ sup |y-z|≤R ty,tz∈U δ (ω) 1 t m µ (ty, tz, ω) -m δ µ (y -z) = 0.
Similarly we obtain an event of full probability independent of µ.

Finally, we check that m δ µ is independent of δ. Let 0 < δ 2 < δ 1 < δ 0 such that U δ i (ω) ⊂ U (ω) for i = 1, 2, which, by (S2), remain connected. In view of (S3), for each fixed y ∈ R n there exists a sequence (t j (ω)) j≥0 such that (t j (ω)y) j≥0 ⊂ U δ 2 (ω) ⊂ U δ 1 (ω) and as j → ∞, t j (ω) → ∞. Consider the event of full probability Ω = n≥1 Ω 1 n . Then, for every ω ∈ Ω ,

m δ 1 µ (y) = lim j→∞ 1 t j (ω) m µ (t j (ω)y, 0, ω) = m δ 2 µ (y).
We show in the following lemma that the liminf of the averaged m µ 's holds all the way up to the boundary. However, the result does not hold for the limsup, in view of the loss of Lipschitz estimates of m ′ µ s near ∂U (ω). In the case when U (ω) is a connected component of {x ∈ R n : a(x, ω) < 0} the arguments are reverted: the metric m µ is negative and the limsup of the averaged m µ holds up to the boundary. Lemma 5.5. Assume that (A1), (A2), (A3), (S1), (S2) and (S3) hold and let U be given by (S1). For µ > 0, let m µ (•, •, ω) be the solution of the metric problem (4.17) and m µ be the averaged metric corresponding to U . For y, z ∈ R n \ {0}, ω ∈ Ω and δ > 0 sufficiently small, lim inf In view of Lemma 2.4 the differential inequality in (6.29) may be interpreted either in the viscosity sense or in the almost everywhere sense. Note that w(•, ω) is defined in R n , but it is only required to be subsolution in the domain U (ω).

It follows from the stationarity of a and of the level sets U (ω) and the ergodicity assumption that the effective Hamiltonian is deterministic. More precisely, the following holds. Proposition 6.1. Assume (A1) and (S1). There exists a set of full probability Ω ⊆ Ω such that, for every ω ∈ Ω and p ∈ R n ,

H(p) = H(p, ω) = inf (w,λ)∈S + ×(0,∞) w+p•y∈L λ (U (ω))
ess sup

y∈U (ω)
a(y, ω)|p + Dw| .

In view of the inf-sup representation formula, we establish some immediate properties of H, which we state without a proof. Proposition 6.2 (Convexity, Homogeneity). Assume (A1), (S1) hold and let U ∈ F be given by (S1). The effective Hamiltonian H corresponding to U is convex, 1-positively homogeneous and

min p∈R n H(p) = H(0) = 0.
The next result is essential to our analysis and establishes that the effective Hamiltonian corresponding to bounded connected components is null. The connection between the effective Hamiltonian and the metric problem. We establish here the connection between the effective Hamiltonian H(p) and the solution m µ (y, z, ω)p • (yz) of the corresponding metric problem with µ = H(p). It turns out that the effective Hamiltonian is the smallest constant µ for which m µ (y, z, ω)p • (yz) has strictly sublinear decay at infinity, from below. We illustrate in Figure 4 the profile of the maximal solution m µ , which, although blows up at ∂U i (ω), has a sublinear decay from below. Lemma 4.3 is essential in establishing this result. Proposition 6.4. Assume that (A1) and (S1) hold and let U be given by (S1). For µ > 0 and z ∈ U (ω), let m µ (•, z, ω) and H be respectively the maximal subsolution of the metric problem (4.17 Furthermore, for each y ∈ R n , m µ (y) = sup{y • q : q ∈ R n such that H(q) ≤ µ} (6.32)

and it is the solution of

H(Dm µ ) = µ in R n \ {0}
, with m µ (0) = 0. (6.33)

Note that Corollary 6.5 can be seen as a homogenization result for the metric problem. Indeed, by rescaling m ε µ (x, z, ω) = εm µ (x/ε, z/ε, ω) we observe that m ε µ satisfies, for

z ε = z/ε ∈ U ε (ω), a x ε , ω |Dm ε µ | = µ in U ε (ω) \ {z ε }, m ε µ (•, z ε , ω) = 0 at {z ε }
and, as ε → 0, m ε µ converges to m µ , with m µ a solution of problem (6.33). We prove in the next section, how this implies the homogenization of the original problem (1.1).

Finally, we give a characterization of the subgradient of the averaged metric m µ corresponding to each ergodic constant µ = H(p). This result is used in the main proof of homogenization. Corollary 6.6. Assume that (A1), (A2), (A3), (S1), (S2) and (S3) hold. Let U be given by (S1), H the corresponding effective Hamiltonian and m µ be the averaged metric. Then, for each p ∈ R n with H(p) > 0, there exists y ∈ R n with |y| = 1 such that p ∈ ∂m H(p) (y).

The Proof of Homogenization

It is well known (see for example [START_REF] Armstrong | Stochastic homogenization of Hamilton-Jacobi and degenerate Bellman equations in unbounded environments[END_REF]) that an intermediate step in the proof of homogenization of (1.1) is the homogenization of the time-independent problem

w ε + a x ε , ω |p + Dw ε | = 0 in R n × Ω. ( 7

.34)

The next proposition summarizes the properties of the solution to (7.34).

Proposition 7.1. Assume (A2) and (A3). For each ε > 0 and p ∈ R n , (7.34) has a unique solution w ε = w ε (•, ω; p) such that, for all ω ∈ Ω,

||w ε (•, ω)|| ∞ ≤ a ∞ |p|. Moreover, for each compact set K ⊂ U i,ε (ω), sup K |Dw ε (•, ω; p)| ≤ a ∞ inf K |a(•, ω)| + 1 |p| and, for all p, q ∈ R n w ε (•, ω; p) -w ε (•, ω; q) L ∞ ≤ a ∞ |p -q|.
We next establish a homogenization result for (7.34), from where Theorem 2.2 follows. For the proof of convergence we deal with the entire partition U i (ω) i∈I 0 of R n . We recall that, for each bounded connected component, the effective Hamiltonian H i ≡ 0, while for unbounded components

H i > 0 if i ∈ I + and H i < 0 if i ∈ I -.
Proposition 7.2. Assume (A1), (A2), (A3), (S1), (S2) and (S3). For each i ∈ I, let U i be given by (S1), θ i = P(U i ) and H i the effective Hamiltonian corresponding to U i . There exists δ 0 > 0 and an event of full probability Ω ⊆ Ω such that, for each ω ∈ Ω and p ∈ R n , the unique solution w ε = w ε (•, ω; p) of (7.34) satisfies, for all δ ∈ (0, δ 0 ), R > 0 and i ∈ I,

lim ε→0 sup x∈U δ i,ε (ω)∩B R |w ε (x, ω; p) + H i (p)| = 0 (7.35)
and, as ε → 0,

w ε (•, ω; p) * ⇀ w := i∈I θ i H i (p) in L ∞ (B R ). (7.36)
We follow the approach developed in [START_REF] Armstrong | Stochastic homogenization of level-set convex Hamilton-Jacobi equations[END_REF] and present a direct argument to show that Theorem 5.1 and Corollary 6.5 imply homogenization of (7.34). We use the perturbed test function method introduced in [START_REF] Evans | The perturbed test function method for viscosity solutions of nonlinear PDE[END_REF], where we face the two usual difficulties, namely the blow up of the m ε µ 's near the boundaries of U i,ε (ω) and their restriction to U δ i,ε (ω).

Proof of Proposition 7.2. We divide the proof into three different steps. The first is about the sub-solution property, the second about the super-solution and the last about the weak convergence result. For the first we follow the analogous proof of [START_REF] Armstrong | Stochastic homogenization of level-set convex Hamilton-Jacobi equations[END_REF]. For the second step it is usually necessary to consider another class of metric problems corresponding to H(-p, x, ω). Here, however, this is not necessary, in view of the symmetry of the Hamiltonian.

To fix ideas, we consider, as usual, one connected component U (ω) of {x ∈ R n : a(x, ω) > 0} and we drop the subscript i in the following. Note that, in this case, w ε ≤ 0.

Step 1. We first show that, for all ω ∈ Ω, p ∈ R n , δ > 0 sufficiently small and R > 0, lim ε→0 sup x∈Uε(ω)∩B R w ε (x, ω) ≤ -H(p).

(7.37)

The proof uses a comparison argument similar to [START_REF] Armstrong | Stochastic homogenization of Hamilton-Jacobi and degenerate Bellman equations in unbounded environments[END_REF][START_REF] Armstrong | Stochastic homogenization of level-set convex Hamilton-Jacobi equations[END_REF], but special care is needed to handle the restriction of w ε to U ε (ω). Proposition 6.4 is essential in establishing (7.37). Since w ε ≤ 0, the result is immediate if H(p) = 0, hence we only consider the case H(p) > 0. We argue by contradiction. If (7.37) is false, then there exists η > 0 and δ > 0 such that, for every ε > 0 sufficiently small there exists z ε ∈ U δ ε (ω) ∩ B R so that w ε (z ε , ω; p) > -H(p) + η. We may assume further, that as ε → 0, z ε → z 0 . For c > 0 a positive constant, to be conveniently chosen, define

v ε (x, ω; p) = w ε (x, ω; p) -w ε (z ε , ω; p) -cη 1 + |x -z ε | 2 1/2 + cη. It follows that, for c = 1/(4 a ∞ ), v ε (•, ω; p) satisfies, in the viscosity sense, a x ε , ω |p + Dv ε | ≤ H(p) - η 2 in D η,ε (ω), (7.38)
where D η,ε (ω) := {x ∈ U ε (ω) : v ε (x, ω; p) ≥ -η/4} . Furthermore, there exists r > 0 depending on η, a ∞ and independent of ε such that D η,ε (ω) ⊆ B r (z ε ).

Let µ = H(p) > 0. Corollary 6.6 implies that there exists a vector |e| = 1 so that p ∈ ∂m µ (e) and 0 = m µ (e)p • e ≤ m µ (y)p • y for all y ∈ R n . (7.39)

In view of (S3), there exists a sequence ε j → 0, as j → ∞, so that ẑ := z 0re ∈ U δ ε j (ω) for all ε j > 0. Since m µ (•, •, ω) is the maximal solution of the metric problem (4.20) for p = 0, µ (z ε j , ẑ, ω; p)v ε j (z ε j , ω; p)

≤ ε j m µ z ε j ε j , ẑ ε j , ω -p • (z ε j -ẑ).
On the other hand m µ (•, ẑ, ω) ≡ ∞ on ∂(U ε j (ω)) and therefore the infimum cannot be achieved on ∂(U ε j (ω)). Thus, since v ε j (•, ω; p) = -η/4 on ∂D η,ε j (ω) \ ∂U ε j (ω), inf ∂Dη,ε j (ω) m ε j µ (x, ẑ, ω; p)v ε j (x, ω; p) = inf ∂Dη,ε j (ω)

ε j m µ x ε j , ẑ ε j , ω -p • (x -ẑ) + η 4 .
The above and the fact that D η,ε (ω) ⊆ B r (z ε ) imply inf Br(zε j )∩Uε j (ω)

ε j m µ x ε j , ẑ ε j , ω -ε j m µ z ε j ε j , ẑ ε j , ω -p • (x -z ε j ) ≤ - η 4 .
Taking inferior limit as j → ∞ and recalling that z ε j → z 0 , we obtain, in light of Theorem 5. We arrived to a contradiction with (7.39).

Step 2. Arguing similarly, we establish the converse inequality In view of the symmetry of m µ (•, •, ω), we now compare w ε (•/ε, •/ε, ω) with the downward cone -m µ (•/ε, •/ε, ω). Note, however, that symmetry plays a crucial role in this case and that, in general, one could not use -m µ (•/ε, •/ε, ω), but rather consider a different set of metric problems (see for example the case of level set convex Hamilton-Jacobi equations [START_REF] Armstrong | Stochastic homogenization of level-set convex Hamilton-Jacobi equations[END_REF]).

Step 3. Finally, the weak convergence result (7.36) is an immediate consequence of the two previous steps. We present a sketch of the proof here for the sake of completeness. For details, we send the reader to [START_REF] Cardaliaguet | A discussion about the homogenization of moving interfaces[END_REF]. Recall that I = I + ∪ I -∪ {0} with H 0 ≡ 0. Fix η > 0, let J be a finite subset of I so that B R \ j∈J U j,ε (ω) ≤ η 2 .

Let δ > 0 sufficiently small so that

B R \ j∈J U δ j,ε (ω) ≤ η.
In light of (7.35) and the ergodic theorem, we have that j∈J (U δ j,ε (ω)) dx ≤ η φ ∞ . Letting η → 0, we deduce the desired result. This concludes the proof of homogenization for the time-independent problem (7.34).

It is immediate to show that Theorem 7.2 implies Theorem 2.2. We only remind the ideas below.

Proof of Theorem 2.2. It is enough to establish the uniform convergence of time-dependent solutions in each connected component U i (ω). We use the classical perturbed test function method to show that, when i ∈ I + φ(x, t, ω) := lim sup ε→0 y→x,s→t y∈U δ i,ε (ω)

u ε (y, t, ω)
is a subsolution of the initial value problem (2.11) and, hence, by the comparison principle deduce that φ(x, t) ≤ ūi (x, t). Similarly, one can prove the reverse inequality, where, in view of Lemma 5.5, the above liminf is taken on U i (ω), unlike limsup above which is restricted to U δ i (ω). The weak convergence for time-dependent solutions follows from (7.36). This completes the proof of homogenization for (1.1).

Figure 2 .

 2 Figure 2. Percolation on Archimedean Lattices (all polygons are regular and each vertex is surrounded by the same sequence of polygons).

Figure 3 .

 3 Figure 3. Homogenization holds for Poisson point process.

Proposition 4 . 2 (

 42 Blow-up on the boundary). Assume (A2) and (S2). For all y, z ∈ U (ω),m µ (y, z, ω) ≥ µ L |log a(y, ω)log a(z, ω)|and, for each fixed z ∈ U (ω), lim dist(y,∂U(ω))→0, y∈U (ω) m µ (y, z, ω) = +∞.

Proposition 4 . 6 (

 46 Well-posedness). Assume (A2) and (S2). For each µ > 0 and z ∈ U (ω), the maximal subsolution m µ (•, z, ω) is a solution of        a(y, ω)|Dm µ | = µ in U (ω) \ {z}, a(y, ω)|Dm µ | ≤ µ in U (ω), m µ (•, z, ω) = 0 at {z}.

  t→∞ ty,tz∈U (ω) m µ (ty, tz, ω) t = lim t→∞ ty,tz∈U δ (ω) m µ (ty, tz, ω) t = m µ (yz).

Proposition 6 . 3 .

 63 Assume (A1), (S1) hold and let U ∈ F be given by (S1). If H is the effective Hamiltonian corresponding to U and U (ω) is bounded, then H(p) = 0.Proof. Since U (ω) is bounded, there exists R = R(ω) > 0 such that U (ω) ⊂ B R . Let φ(•, ω) ∈ C ∞ c (R n ) suchthat φ(y, ω) = -p • y in U (ω) and φ(•, ω) ≡ 0 in R n \ B R+1 . Then φ(•, ω) ∈ S + , for all µ > 0, φ(•, ω) + p • y ∈ L µ (U (ω))) and, in view of Proposition 6.1, 0 ≤ H(p) ≤ ess sup y∈U (ω) a(y, ω)|p + Dφ| = 0.

  ) and the effective Hamiltonian corresponding to U . Then, for each p ∈ R n and ω ∈ Ω, µ ≥ H(p) if and only if lim inf |y|→∞ y∈U (ω) m µ (y, z, ω)p • (yz) |y| ≥ 0. (6.30)

µ

  (x, ẑ, ω; p) = ε j m µ ẑ, ω)| = H(p) in U ε j (ω) \ {ẑ}.(7.40) Recalling that D η,ε j (ω) ⊂ ε j U (ω) \ {ẑ}, by classical comparison arguments between viscosity suband super-solution, we have infDη,ε j (ω) m ε j µ (x, ẑ, ω; p)v ε j (x, ω; p) = inf ∂Dη,ε j (ω) m ε j µ (x, ẑ, ω; p)v ε j (x, ω; p) .Note that z ε j ∈ D η,ε j (ω) and thus inf Dη,ε j (ω) m ε j µ (x, ẑ, ω; p)v ε j (x, ω; p) ≤ m ε j

  1 and Lemma 5.5, that inf x∈Br(z 0 )m µ (xẑ)m µ (z 0ẑ)p • (xz 0 ) ≤m µ (xz 0 + re)m µ (re)p • (xz 0 ) = r inf y∈B 1 (0) m µ (e)m µ (y)p • y ≤ -η 4 .

  R w ε (x, ω; p) ≥ -H(p).

1 j∈J

 1 (U δ j,ε (ω)) w ε = j∈J 1 U δ j,ε (ω) w ε * ⇀ j∈J θ j H j (p) in L ∞ ( B R ). (7.41) Thus, for any test function φ ∈ L ∞ (B R ), it is easy to check that B R φw dx -B R φw ε 1

♮ Partially supported by the NSF grants DMS -0901802 and DMS -1266383

We next remark that the Lipschitz continuity of a(•, ω) and the fact that a(•, ω) = 0 on ∂U (ω) imply, for all y ∈ U (ω) and d(•, ω) := dist(•, ∂U (ω)), 0 ≤ a(y, ω) ≤ Da(•, ω) ∞ d(y, ω) ≤ Ld(y, ω).

It follows that, for each fixed z ∈ U (ω), We analyze further the behavior of the m µ 's near ∂U (ω) and show that they are controlled from below by a cone. This fact is used in Proposition 6.4 to investigate the properties of the effective Hamiltonian and is an indispensable estimate to establish the proof of homogenization. Lemma 4.3 (Lipschitz growth near the boundary). Assume (A2) and (S2). For any η > 0, there exists δ 0 = δ 0 (η) > 0 such that, for all δ ∈ (0, δ 0 ) and all y 1 , z ∈ U δ (ω), y 2 ∈ U (ω) \ U δ (ω),

Proof. Fix η > 0, let δ(η) > 0 and δ 0 > 0 be given by S µ (U (ω)) and (S2) and set δ 0 (η) := min(δ 0 , δ(η)). In light of Lemma 4.1, m µ (•, z, ω) is Lipschitz continuous on U δ (ω) with Lipschitz constant at most µ/δ + η. We extend m µ (•, z, ω) to the whole space R n by

We use the maximality of m µ and its behavior near the boundary to show next that m µ is a pseudo-metric and thus a subadditive quantity in U (ω).

Proof. Let z ∈ U (ω) and rewrite the formula for the maximal subsolution as

For each w(•, ω) ∈ L µ (U (ω)), consider its reflection w(y, ω) := -w(y, ω) and observe that m µ (y, z, ω) = sup { w(z, ω)w(y, ω) : w(•, ω) ∈ L µ (U (ω)) and a(y, ω)|D w| ≤ µ in U (ω)} = m µ (z, y, ω).

Fix now x, z ∈ U (ω) and recall that m µ (•, z, ω) is the maximal subsolution of (4.17) in U (ω) such that m µ (z, z, ω) = 0. Since the supremum of viscosity subsolutions is still a subsolution, note that

Without any loss of generality, it is enough to check the claim for the linear interpolation (5.24), since the bilinear interpolation (5.23) is linear in each of its arguments. We assume hence that 0 ∈ U δ (ω), otherwise we consider 0 * y the closest point to the origin in direction y and work directly with the bilinear interpolation (5.23). The m δ µ 's preserve the translation invariance of m µ , since m δ µ (ty, 0,

note that we used the translation invariance of the level sets of a(•, ω) to say that sy ∈ U δ (ω) if and only if 0 ∈ U δ (τ sy ω) and

The m δ µ 's also preserve the Lipschitz constants of m µ 's on U δ y (ω). Indeed, for all t > 0,

Finally the m δ µ 's remain subadditive. Indeed, let t > s > 0 and assume again, without loss of generality, that sy ∈ U δ (ω). Then

= m δ µ (ty, sy, ω) + m δ µ (sy, 0, ω). All the above properties remain true for the bilinear extension m δ µ = m δ µ (ty, sy, ω). Then, by the subadditive ergodic theorem, there exists an event Ω µ,δ y of full probability such that, for all ω ∈ Ω µ,δ y , there exists

Remark 5.3. Note that we may assume, without any loss of generality, that 0 ∈ U (ω) a.s., which yields that 0 ∈ U δ (ω) for δ > 0 sufficiently small. Otherwise, for any unit vector y ∈ R n and δ > 0, we shift the origin in direction y ∈ R n to the closest point which lies inside the domain U δ (ω), namely we replace 0 by 0 * y, where

In this case, instead of arguing for the linear interpolation (5.24), we work directly with the bilinear interpolation (5.23).

We show next that m δ µ is deterministic and establish the a.s. convergence.

Lemma 5.4. Let η > 0 and δ 0 = δ 0 (η) > 0 be given by Lemma 4.1. For each µ > 0 and δ ∈ (0, δ 0 ), there exists a set of full probability Ω µ,δ ∈ F and m δ µ : R n → R such that, for every ω ∈ Ω µ,δ and y ∈ R n ,

1 t m µ (ty, 0 * y, ω).

(5.25)

For fixed µ > 0 and δ ∈ (0, δ 0 ), m δ µ is subadditive, 1-positively homogeneous and Lipschitz continuous with Lipschitz constant at most µ/δ + η.

Proof. In light of Remark 5.3, we may assume that 0 ∈ U (ω) a.s.. It remains to show that m δ µ (y, •) is deterministic. In view of the ergodicity this would follow once we show that m δ µ is translation invariant, that is, it satisfies for all z ∈ R n , m δ µ (y, ω) = m δ µ (y, τ z ω).

(5.26)

We establish (5.26) first for z ∈ U δ (ω) and then deduce the general case.

Let z ∈ U δ (ω). In view of (S3), for each δ ∈ (0, δ 0 ) and y ∈ R n \ {0}, there exist l j (ω)y j≥0 ⊂ U δ y (ω) and r j (τ z ω)y + z j≥0 ⊂ U δ y (ω) such that, as j → ∞, l j (ω) → ∞, r j (τ z ω) → ∞ and l j (ω)/r j (τ z ω) → 1. Then, by the subadditivity and Lipschitz continuity of m µ ,

(m µ (r j (τ z ω)y + z, l j (ω)y, ω) + m µ (l j (ω)y, 0, ω) + m µ (0, z, ω))

A similar argument gives the reverse inequality and, hence, for all z ∈ U δ (ω), (5.26) holds.

Let

It is clear, from the construction, that the average metric m δ µ is 1-positively homogeneous. We show next that, in view of Lemma 5.2, m δ µ is Lipschitz continuous. Indeed, let y, y ′ ∈ R n \{0} and ω ∈ Ω µ,δ y ∩ Ω µ,δ y ′ . In view of (S3), there exist two sequences l j (ω)y ′ j≥0 , r j (ω)y j≥0 ⊂ U δ (ω) such that, as j → ∞, l j (ω) → ∞, r j (ω) → ∞, and l j (ω)/r j (ω) → 1. Then the subadditivity and Proof. In view of Remark 5.3 and the Lipschitz continuity of m µ , we may assume for simplicity that z = 0. Let η > 0, δ 0 (η) be given by Lemma 4.1 and fix δ ∈ (0, δ 0 ). We only need to check the result for points ty ∈ U (ω) \ U δ (ω).

In view of (S3), there exist (l j (ω)y) j≥0 ⊂ U (ω) \ U δ (ω) and (r j (ω)y) j≥0 ⊂ U δ (ω) such that, as j → ∞, l j (ω) → ∞, r j (ω) → ∞ and l j (ω)/r j (ω) → 1. In light of Lemma 4.3 we have m µ (l j (ω)y, 0, ω) ≥ m µ (r j (ω)y, 0, ω) -µ δ + η |l j (ω)r j (ω)||y|.

Theorem 5.1 yields further that lim inf

The conclusion follows.

The next properties follow immediately from Lemma 5.4 and Proposition 4.7.

Proposition 5.6 (Properties of the averaging metric). Assume that (A1), (A2), (A3), (S1), (S2) and (S3) hold and let U ∈ F given by (S1). For every µ > 0, let m µ : R n → R be the averaged metric corresponding to U . Then (i) y → m µ (y) is Lipschitz continuous with Lipschitz constant depending only on µ.

(ii) y → m µ (y) is subadditive, positively homogeneous and, hence, convex.

(iii) µ → m µ (y) is increasing and locally uniformly continuous in y ∈ R n .

The Effective Hamiltonian

Since the connected components (U i (ω)) i∈I 0 are generated by a family (U i ) i∈I 0 of the probability space Ω, it is possible to show that, for each U i (ω) with i ∈ I, there exists a deterministic, effective Hamiltonian H i .

As usual, we fix one connected component U (ω) of {x ∈ R n : a(x, ω) > 0} and characterize the corresponding effective Hamiltonian H(p). Similar arguments remain true for the connected components of {x ∈ R n : a(x, ω) < 0}.

The characterization of the effective Hamiltonian. We first define the effective Hamiltonian as the smallest constant for which the metric problem has a subsolution with stricly sublinear decay at infinity. We then give an inf-sup representation formula for the effective Hamiltonian, which by the stationarity assumption holds a.s. in ω. Unlike previous results [START_REF] Armstrong | Stochastic homogenization of level-set convex Hamilton-Jacobi equations[END_REF][START_REF] Lions | Homogenization of "viscous" Hamilton-Jacobi equations in stationary ergodic media[END_REF] where subsolutions are in R n , now they are restricted to satisfy the equation in the connected component U (ω). Moreover, in view of the well posedness of the metric problem, they must satisfy the constraint imposed by the class L µ (U (ω)).

The effective Hamiltonian H(p, ω) corresponding to U (ω) is given by

such that a(y, ω)|p + Dw| ≤ µ in U (ω) . (6.29) From the definition of S + we deduce the inequality in (6.30) holds for µ > H(p) and, in view of the continuity of m µ with respect to µ, it holds for any µ ≥ H(p).

Conversely assume the inequality in (6.30) holds for some µ > 0. For η > 0 and δ ∈ (0, δ 0 (η)) let w δ µ (•, ω) ∈ S µ (U (ω)) so that w We next use this result to give a dual formulation for H(p) and the averaged metric m µ corresponding to U ∈ F. Since the proof is similar to the one in [START_REF] Armstrong | Stochastic homogenization of level-set convex Hamilton-Jacobi equations[END_REF], we omit it. Corollary 6.5 (The effective Hamiltonian and the averaged metric). Assume that (A1), (A2), (A3), (S1), (S2) and (S3) hold and let U ∈ F be given by (S1). For each µ > 0, let m µ (•) and H be respectively the averaged metric and the effective Hamiltonian corresponding to U . Then H(p) = inf{µ > 0 : m µ (y) ≥ p • y for all y ∈ R n }. (6.31)