Paul Doukhan

Lionel Truquet

Weakly dependent random fields with infinite interactions

Keywords: 60G60: Random fields keywords: 60B12: Limit theorems for vector-valued random variables, 60F25: L p -limit theorems, 60K35: Interacting random processes, statistical mechanics type models, percolation theory, 62M40: Random fields, image analysis, 60B99: Weak dependence, 60K99: Bernoulli shifts

We introduce new models of stationary random fields, solutions of

the input random field ξ is stationary, e.g. ξ is independent and identically distributed (iid). Such models extend most of those used in statistics. The (nontrivial) existence of such models is based on a contraction principle and Lipschitz conditions are needed; those assumptions imply Doukhan and Louhichi (1999)'s [6] weak dependence conditions. In contrast to the concurrent ones, our models are not set in terms of conditional distributions. Various examples of such random fields are considered. We also use a very weak notion of causality of independent interest: it allows to relax the boundedness assumption of inputs for several new heteroscedastic models, solutions of a nonlinear equation.

Introduction

Description of random fields is a difficult task, a very deep reference is [START_REF] Georgii | Gibbs measures and phase transition[END_REF] book [START_REF] Georgii | Gibbs measures and phase transition[END_REF]; a synthetic presentation is given by Föllmer (1985) [START_REF] Föllmer | Random Fields and Diffusion Processes[END_REF]. The usual way to describe interactions makes use of conditional distributions with respect to large sets of indices. This presentation is natural for discrete valued random fields as in Comets et alii (2002), [START_REF] Comets | Processes with Long Memory: Regenerative Construction and Perfect Simulation[END_REF]. The existence of conditional densities is a more restrictive assumption for continuous state spaces. The existence of random fields is often based on conditional specifications, see Föllmer (1985) ([9], pages 109-119) and [START_REF] Dobrushin | Prescribing a system of random variables by conditional distributions[END_REF] [START_REF] Dobrushin | Prescribing a system of random variables by conditional distributions[END_REF], through Feller continuity assumptions. The uniqueness of Gibbs measures is often based on projective conditional arguments; it follows with a mixing type argument. Such conditions rely on the regularity of conditional distributions; applications to resampling exclude such hypotheses. Various applications to image, geography, agronomy, physic, astronomy, electromagnetism may for instance be considered, see [START_REF] Georgii | Gibbs measures and phase transition[END_REF] or [START_REF] Loubaton | Champs stationnaires au sens large sur Z 2 : Propriétés structurelles et modèles paramétriques[END_REF]. [START_REF] Comets | Processes with Long Memory: Regenerative Construction and Perfect Simulation[END_REF] We omit here any assumption relative to the conditional distributions. Our idea is to define random fields through more algebraic and analytic arguments. We present here the new models of stationary random fields subject to the relation:

X t = F (X t-j ) j∈Z d \{0} ; ξ t (1)
where ξ = (ξ t ) t∈Z d is an independent identically distributed (iid) random field. The independence of inputs ξ may also be relaxed to a stationarity assumption.

For the models with infinite interactions [START_REF] Comets | Processes with Long Memory: Regenerative Construction and Perfect Simulation[END_REF], the existence and uniqueness rely on the contraction principle. Lipschitz type conditions are thus needed, they are closely related to weak dependence, see [START_REF] Doukhan | A new weak dependence condition and applications to moment inequalities[END_REF]. Analogue weak dependence conditions are already proved in [START_REF] Shashkin | A weak dependence property of a spin system[END_REF] for spin systems, [START_REF] Shashkin | A weak dependence property of a spin system[END_REF]. A causal version of such models, random processes solutions of an equation X t = F (X t-1 , X t-2 , . . . ; ξ t ) (t ∈ Z) is considered in [START_REF] Doukhan | Weakly dependent chains with infinite memory Preprint available in[END_REF], [START_REF] Doukhan | Weakly dependent chains with infinite memory Preprint available in[END_REF]; in this paper the results are proved in a completely different way fitting to coupling arguments. Our results state existence and uniqueness of a solution of (1) as a Bernoulli shift X t = H((ξ t-s ) s∈Z d ) as well as the weak dependence properties of this solution.

Our models are not necessarily Markov, neither linear or homoskedastic. Moreover the inputs do not need additional distributional assumptions (like for Gibbs random fields). They extend on ARMA random fields which are special linear random fields (see [START_REF] Loubaton | Champs stationnaires au sens large sur Z 2 : Propriétés structurelles et modèles paramétriques[END_REF] or [START_REF] Guyon | Random fields on a network: modeling, statistics, and applications[END_REF]). A forthcoming paper will be aimed at developing statistical issues of those models. Identification and estimation of random fields with integer values is considered in [START_REF] Doukhan | Integer valued random fields with infinite interactions[END_REF].

The paper is organized as follows. We first recall weak dependence from [START_REF] Doukhan | A new weak dependence condition and applications to moment inequalities[END_REF] in § 2. General results are then stated for stationary (non necessarily independent) inputs. Those results imply heavy restrictions on the innovations in some cases: a convenient notion of causality is thus used. A last subsection addresses the problem of simulating such models. A following section details examples of such models. They are natural extensions of the standard times series models. We shall especially consider LARCH(∞) and doubly stochastic linear random fields for which this causality allows to relax the boundedness assumptions.

Proofs are rejected at in a last section of the paper.

Main results

In order to state our dependence results, we first introduce the concepts of weak dependence. Our main results will be stated in the following subsection. After this, causality will be proved to imply other powerful results. A last subsection is aimed at describing a way to simulate those very general random fields.

Weak dependence

We recall here the weak dependence conditions introduced in Doukhan & Louhichi (1999). They may replace heavy mixing assumptions.

Definition 1 Set (s 1 , . . . , s d ) = max{|s 1 |, . . . , |s d |} for s 1 , . . . , s d ∈ Z. The E = R k -valued random field (X t ) t∈Z d is weakly dependent if for a sequence (ǫ(r)) r∈N with limit 0 |Cov (f (X s 1 , . . . , X su ) , g (X t 1 , . . . , X tv )| ≤ ψ(u, v, Lip f, Lip g)ǫ(r), where indices s 1 , . . . , s u , t 1 , . . . , t v ∈ Z d are such that s k -t l ≥ r for 1 ≤ k ≤ u and 1 ≤ l ≤ v. Moreover, the real valued functions f, g defined on R k u and R k v , satisfy f ∞ , g ∞ ≤ 1 and Lip f, Lip g < ∞ where a norm • is given on R k and, Lip f = sup (x 1 ,...,xu) =(y 1 ,...,yu) |f (x 1 , . . . , x u ) -f (y 1 , . . . , y u )| x 1 -y 1 + • • • + x u -y u .
If ψ(u, v, a, b) = au + vb, this is denoted as η-dependence and the sequence ǫ(r) will be written η(r).

If ψ(u, v, a, b) = abuv, this is denoted as κ-dependence and the sequence ǫ(r) will be written κ(r).

If ψ(u, v, a, b) = au + vb + abuv, this is denoted as λ-dependence and the sequence ǫ(r) will be written λ(r).

Random fields with infinite interactions

Let ξ = (ξ t ) t∈Z d be a stationary random field with values in E ′ (usually E ′ = R k ′ for some k ′ ≥ 1 but in some cases E ′ is a denumerable tensor product of such sets). We shall consider stationary E = R k valued random fields driven by the implicit equation [START_REF] Comets | Processes with Long Memory: Regenerative Construction and Perfect Simulation[END_REF]. For a topological space S, B(S) denote the Borel σ-algebra on S.

We denote I = Z d \ {0}. In the sequel, F :

E (I) × E ′ , B(E I ) ⊗ B(E ′ ) → E, B(E)
denotes a measurable function defined for each sequence with a finite number of nonvanishing arguments (1 ). In this paper • will be arbitrary norms on E (or E ′ when needed). We will always use the suppremum norm on Z d and this norm will be also denoted by • . We prove that simple assumptions entail existence of a unique solution as a Bernoulli shift

X t = H (ξ t-j ) j∈Z d
Let µ denote ξ's distribution; this is a probability measure on the measurable space

E ′ Z d , B E ′ Z d .
For some m ≥ 1, we denote • m the usual norm of L m and the space of

µ-measurable H : E ′ Z d , B E ′ Z d → (E, B(E)) with finite moments is denoted L m (µ) = {H E H(ξ) m < ∞}.
We shall use the assumptions:

(H1) F (0; ξ 0 ) m < ∞.
(H2) There exist constants a j ≥ 0, for j ∈ α > 0 with, for each ∀z,

z ′ ∈ E (Z d \{0}) , F (z; ξ 0 ) -F (z ′ ; ξ 0 ) ≤ j∈Z d \{0} a j z j -z ′ j , a.s. ( 2
)
j∈Z d \{0} a j = e -α < 1.
We now extend the function F to the trajectories of a stationary random field:

Lemma 1 Assume (H1) and (H2). Let X and X ′ be two E-valued stationnary random fields in L m , then:

1) lim p→∞ F (X j 1 0< j ≤p ) j =0 ; ξ 0 exists in L m and a.s., we denote it F (X j ) j∈Z d \{0} ; ξ 0 .

2)

F (X j ) j∈Z d \{0} ; ξ 0 -F (X ′ j ) j∈Z d \{0} ; ξ 0 m ≤ j∈Z d \{0} a j X j -X ′ j m .
Theorem 1 Assume that ξ is stationary and (H1) and (H2) hold. Then there exists a unique stationary solution of equation ( 1). This solution writes

X t = H (ξ t-j ) j∈Z d for some H ∈ L m (µ).
Lemma 8 below, will also provide us with an approximation of this solution with finitely many interactions.

Weak dependence of the solution (iid inputs)

In the general case we shall restrict to independent inputs to derive η-weak dependence of the previous solution.

Theorem 2 Assume that ξ is iid and (H1) and (H2) hold. Then the stationary solution of equation ( 1) obtained in theorem 1 is η-weakly dependent and there exists a constant

C > 0 with η(r) ≤ C • inf p∈N * e -α r 2p + i >p a i . (3)
If

a i = 0 for i > p then η(r) ≤ C • e -α r 2p .
Explicit (sub-geometric) rates are now derived from more specific decay rates of the coefficients:

Lemma 2 (Geometric decays) If a i ≤ Ce -β i there exists a constant C ′ > 0 with η(r) ≤ C ′ r d-1 2 e - √ αβr/2 . Lemma 3 (Riemanian decays) If a i ≤ C i -β for some β > d then there exists C ′ > 0 such that η(r) ≤ C ′ r ln r d-β
.

Thus a large range of decay rates may be considered for such models of random fields.

Weak dependence of the solution (dependent inputs)

If ξ is either η or λ-dependent it may be proved in specific examples that weak dependence is hereditary. Here follows a general result. The following assumption will be necessary:

(H2') There exist a subset Ξ ⊂ E ′ with P (ξ 0 ∈ Ξ) = 1, nonnegative constants with

j∈Z d \{0}
a j = e -α < 1 and a constant b > 0 such that

F (x; u) -F (x ′ ; u ′ ) ≤ j∈Z d \{0} a j x j -x ′ j + b u -u ′ , for all x, x ′ ∈ E (Z d \{0}) and u, u ′ ∈ Ξ.
We quote that assumption (H2') is more restrictive than (H2) Proposition 1 Assume (H1) and (H2').

1) If the random field ξ is η-weakly dependent, with weak dependence coefficients η ξ (r), then X is η-weakly dependent with

η(r) ≤ C inf p∈N * j >p a j + inf n∈N * a n + p n η ξ ((r -2pn) ∨ 0)
where C > 0 is a constant.

2) If the random field ξ is λ-weakly dependent, with dependence coefficients denoted λ ξ (r), then X is λ-weakly dependent with

λ(r) ≤ C inf p∈N *    j >p a j + inf n∈N * a n + p 2n λ ξ ((r -2pn) ∨ 0)    for some constant C > 0.
Remark. For models with finite interactions, i.e. F (x; u) = f (x j 1 , . . . , x j k ; u) for x = (x j ) j =0 , this simply writes

η(r) ≤ c inf n∈N * {a n + k n η ξ ((r -2ρn) ∨ 0)} , λ(r) ≤ c inf n∈N * a n + k 2n λ ξ ((r -2ρn) ∨ 0) ,
here ρ = max{ j 1 , . . . , j k }. If η ξ (r) or λ ξ (r) have geometric or Riemannian decay the same holds for the output random field. More precisely set a = e -α and k = e κ under η-dependence and k 2 = e κ under λ-dependence, then decay rates of the outputs (X t ) write

Geometric decays: e -αβ α+2ρβ+κ r , for dependence decays of the inputs with order e -βr , Riemannian decays: r -αb α+κ , for dependence decays of the inputs with order r -b .

Causality

For d = 1, the recurrence equation X t = ξ t (a + bX t-1 ) is given with F (x; u) = u(a + bx 1 ). There exist a stationary solution with ξ t and X t-1 independent. Here (H2) implies that innovations are bounded, which seems unrealistic. In this example, instead of H ((ξ t ) t∈Z )) ∈ L m (µ), this is enough to exhibit solutions H ((ξ t ) t≥0 ) ∈ L m (µ) (which is independent of (ξ s ) s<0 ). This allows to replace suprema by integrals in (H2) in order to derive a contraction principle. Causality of random fields has been considered in [START_REF] Helson | Prediction theory and Fourier series in several variables[END_REF] [START_REF] Helson | Prediction theory and Fourier series in several variables[END_REF]; we adapt this idea in order to relax the previous assumption.

Definition 2 (causality)

If A ⊂ Z d \{0}, we denote c(A) the convex cone of R d generated by A, c(A) = k i=1 r i j i (j 1 , . . . , j k ) ∈ A k , (r 1 , . . . , r k ) ∈ R k + , k ≥ 1 . 1) The set A is a causal subset of Z d if c(A) ∩ -c(A) = {0}.
2) If F is measurable with respect to the σ-algebra F A ⊗ B(E ′ ) for some causal set A, then the equation

X t = F ((X t-j ) j∈I ; ξ t ) is A-causal.
For a causal set A ⊂ Z d , we denote by A the subset c(A) ∩ Z d .

Examples. A singleton is causal, as well as {i, j} if and only if

-j / ∈ i • R + . The half plane {(i, j) ∈ Z 2 /i > 0} {(0, j); j > 0} ⊂ Z 2 is also causal.
One consequence of this notion is the elementary lemma:

Lemma 4 If A is a causal subset of Z d , then ∀(j, j ′ ) ∈ A × A we have j + j ′ = 0. For a linear basis b = (b 1 , . . . , b d ) of R d , (x 1 , . . . , x d ) → x 1 b 1 + • • • + x d b d , defines an isomorphism f : R d → R d .
We denote by ≤ b the total order relation on R d defined by:

u ≤ b v ⇔ f -1 (u) ≤ lex f -1 (v) with ≤ lex the lexicographic order on R d . Proposition 2 (characterization of causal sets) If B is a convex cone of R d such that B ∩ (-B) = {0} there exists a basis b of R d such that B ⊂ {j ∈ R d /0 ≤ b j}. Moreover if b is a basis of R d , {j ∈ Z d /0 < b j} is a causal set of Z d witch will be called maximal causal subset.
Remarks.

• The maximal causal subsets of Z are {1, 2, 3, . . .} and {-1, -2, . . .}. An example of maximal causal subset of Z 2 is {(i, j) ∈ Z 2 /i > 0 or (i = 0, j > 0)}.
• Helson and Lowdenslager (1959) [START_REF] Helson | Prediction theory and Fourier series in several variables[END_REF] define symmetric half planes as subsets S ⊂ Z 2 such that S is stable by addition and S ∪ (-S) = Z 2 , S ∩ (-S) = {0}. A nice review of this causality condition is given in Loubaton (1989) [START_REF] Loubaton | Champs stationnaires au sens large sur Z 2 : Propriétés structurelles et modèles paramétriques[END_REF], applications are essentially given in terms of linear random fields.

Note that S \ {0} is a maximal causal subset of Z 2 . This notion plays a prominent part in prediction theory of 2-D stationary process (see [START_REF] Loubaton | Champs stationnaires au sens large sur Z 2 : Propriétés structurelles et modèles paramétriques[END_REF]).

If D ⊂ Z d , we denote by π s (respectively π ′ s ) the coordinate applications in E Z d (resp. in (E ′ ) Z d ), F D = σ(π s ; s ∈ D) and F ′ D = σ(π ′ s ; s ∈ D).
Hence we denote by L m D (µ) the subspace of L m (µ) of functions µ-measurable with respect to F ′ D . The following result takes this definition into account to relax the assumptions in theorem 1, Theorem 3 Let X t = F (X t-j ) j∈Z d \{0} ; ξ t be a A-causal equation with iid inputs ξ. Besides the assumption (H1) we assume the following condition: (H3) there exist nonnegative constants with j∈A a j = e -α < 1 and

F (x; ξ 0 ) -F (x ′ ; ξ 0 ) m ≤ j∈A a j x j -x ′ j , ∀x, x ′ ∈ E (Z d \{0}) .
Then there exists a unique strictly stationary solution X of this equation in

L m if for each t ∈ Z d , X t is measurable wrt σ ξ t-j /j ∈ A .
This solution writes

X t = H (ξ t-j ) j∈Z d where H ∈ L m e A
. and it is η-weakly dependent; moreover relation (3) still holds for a constant C > 0. Now the function F is extended as follows:

Lemma 5 Suppose (H1) and (H3). If ξ 0 is independent of σ (X j , X ′ j )/j ∈ A for two random fields X and X ′ in L m then, 1) lim p→∞ F (X j 1 0< j ≤p ) j =0 ; ξ 0 exists in L m and it is denoted F (X j ) j =0 ; ξ 0 . 2) F (X j ) j =0 ; ξ 0 -F (X ′ j ) j =0 ; ξ 0 m ≤ j∈A a j X j -X ′ j m .

Simulation of the model

Simulations of those models are deduced from the proof of the existence theorems based on the Picard fixed point theorem. Consider the shift operators θ j :

E ′ Z d → E ′ Z d defined as (x k ) k∈Z d → (x k+j ) k∈Z d . For H ∈ L m (µ) we note Φ p (H) = F (H • θ j )1 j ≤p j ; π 0 It is shown in theorem 1's proof that the application Φ : L m (µ) → L m (µ) given by Φ(H) = F ((H • θ j ) j =0 ; π 0 ).
is well defined and has a fixed point in L m (µ). The proof of theorem 3 shows that it is also the case for a A-causal equation if we replace

L m (µ) by L m e A (µ). For n, p ∈ N * , t ∈ Z d we denote X n t = Φ (n) (0) (ξ t-j ) j∈Z d and X n p,t = Φ (n) p (0) (ξ t-j ) j∈Z d .
Lemma 6 We assume that conditions in theorem 1 or in theorem 3 hold for some m ≥ 1.

Let n ∈ N then:

1. For every t ∈ Z d , X t -X n t m ≤ a n X 0 m , hence lim n→∞ X n t = X t a.s. 2. if p ∈ N we have, X t -X n p,t m ≤ X 0 m a n + 1 1-a j >p a j . Thus if p = p n is chosen such that n≥1 j >pn a j m < ∞ then lim n→∞ X n pn,t = X t , a.s. (4)
Remarks.

• If the random field has finitely many interactions, then 1. provides a simulation scheme.

• For each finite p the operator Φ p can be calculated thus relation ( 4) provides an explicit simulation scheme even for infinitely many interactions.

• A.s. convergence rates may also be evaluated in the previous lemma. They write o a.s. (a n n ǫ ) in the first point for each ǫ > 1/m and o a.s. (n -ǫ ) for 0 < ǫ < α -1/m if j >pn a j ≤ Cn -α for some C > 0, α > 1/m in the point 2.

• If T ⊂ Z d is a finite set the random field X may be analogously simulated over T and (X t ) t∈T is estimated by X n pn,t t∈T .

Simulation scheme for finitely many interactions

Let F (x; u) = f (x j 1 , . . . , x j k ; u). The sequence of random fields X n is defined from:

X 1 t = f (0; ξ t ), t ∈ Z d , X n+1 t = f X n t-j 1 , . . . , X n t-j k ; ξ t , for n ≥ 0
We now simulate samples (X 10 t ) 1≤t 1 ,t 2 ≤15 of LARCH models with d = 2, k = k ′ = 1 and p = 10: 2) Figure 2 deals with the causal case with a j = 0.05 j 2 1 + j 2 2 if 0 ≤ j 1 , j 2 ≤ 10 and a j = 0 otherwise.

X t = ξ t 1 + 0< j ≤p a j X t-j
In this case, ξ 0 is N (0, 1)-distributed.

Examples

Theorems 2 and 3 are now applied to examples of random fields with infinite interactions. Causality will allow to weaken moment conditions. In fact, theorem 2 proves a contraction principle in L m for each value of m while theorem 3 only works with one fixed value of m.

Finite interactions random fields

If ξ t = (ζ t , γ t ) with ζ t ∈ R p and γ t a p × q matrix, and functions

f (•) ∈ R p and g(•) ∈ R q X t = f (X t-ℓ 1 , . . . , X t-ℓ k ) + γ t g(X t-ℓ 1 , . . . , X t-ℓ k ) + ζ t (5)
with ℓ 1 , . . . , ℓ k = 0. E.g. non linear auto-regression corresponds to γ t ≡ 0 and ARCH type models are obtained with ζ t = 0 (classically p = q = 1, f is linear and g 2 (x 1 , . . . , x k ) is an affine function of x 2 1 , . . . , x 2 k ). Theorems 1, 2, 3 imply the following lemma,

Corollary 1 Suppose ζ 0 m < ∞ and f (x 1 , . . . , x k ) -f (y 1 , . . . , y k ) ≤ k i=1 b i x i -y i , g(x 1 , . . . , x k ) -g(y 1 , . . . , y k ) ≤ k j=1 c j x j -y j . 1. If ξ is iid and k i=1 b i + γ 0 ∞ c i = e -α < 1, then η(r) ≤ C e -α 2k r
for the model [START_REF] Doukhan | Integer valued random fields with infinite interactions[END_REF].

If the equation ( 5) is causal and

k i=1 b i + γ 0 m c i = e -α < 1 the same relation holds.
2. If now ξ is η or λ-weakly dependent, g bounded and

k i=1 b i + γ 0 ∞ c i = e -α < 1,
then X is η or λ-weakly dependent. Decay rates are given according to proposition 1.

The remark following proposition 1 states precise decay rates under standard decay rates (i.e. Riemannian or geometric decays) of the weak dependence coefficients of the input process. The volatility coefficients γ t need to be bounded in the general case and they only have finite moments under causality. Note that functions f and g may only depend on a strict subset of the indices 1, . . . , k.

Linear fields

Let X be a solution of the equation

X t = j∈A α j t X t-j + ζ t , (6)
innovations ζ t are vectors of E = R k and coefficients α j t are k × k matrices, • is a norm of algebra on this set of matrices and X will be an E valued random field. Let A ⊂ Z d \ {0}, we assume that the iid random field ξ = (α j t ) i∈A , ζ t t∈Z d takes now its values in (M k×k ) A × E; here M k×k denotes the set of k × k matrices.

Proposition 3 If b = j∈A α j 0 ∞ < 1,
then theorem 2 applies with a j = α j 0 ∞ . For a causal equation if b = j∈A α j 0 m < 1 theorem 3 applies with a j = α j 0 m . In both cases the solution of equation ( 6) writes a.s. and in L m ,

X t = ζ t + j∈A α j t ξ t-j + ∞ i=2 j 1 ,...,j i ∈A α j 1 t α j 2 t-j 1 • • • α j i t-j 1 -•••-j i-1 ζ t-(j 1 +•••+j i ) .
This means that the random coefficients are bounded in the general case and they need only to have have finite moments under causality.

Examples. If the sequence (α j t ) t is deterministic then those models extend on linear auto-regressive models. If only a finite number of coefficients α j t do not vanish we obtain auto-regressive models with random coefficients, see [START_REF] Tjostheim | Some doubly stochastic time series models[END_REF].

LARCH(∞) random fields

Stationary innovations ξ t are now k × k ′ matrices and • will denote a norm k × k ′ or k ′ × k matrices while X t ∈ R k . For bounded innovations we first recall Theorem 4 (Doukhan, Teyssière, Winant (2006)) Let α j be a k ′ × k matrix for j ∈ Z d \ {0}, note A(x) = j ≥x α j and suppose that λ = A(1) ξ 0 ∞ < 1, then

X t = ξ t   a + ∞ k=1 j 1 ,...,j k =0 α j 1 ξ t-j 1 • • • α j k ξ t-j 1 -•••-j k a   (7)
is a solution of the equation

X t = ξ t   a + j =0 α j X t-j   , t ∈ Z d (8) if moreover ξ is iid, then η(r) ≤ E ξ 0   E ξ 0 k<r/2 λ k-1 A r k + λ [r/2] 1 -λ   a .
If we use theorem 2 we also obtain that eqn. ( 8) admits a unique Bernoulli shift L m solution. Note that this solution is bounded. Notice that for Riemannian decay the previous A(u) ≤ Cu -c relation yields η(r) = O(r -c ) while theorem 3 only provides us this bound up to a log-loss; geometric decays yield the same result for both cases. Bounded innovations ξ t look unnatural hence we investigate below the causal case. Let A a causal subset of Z d and

X t = ξ t a + s∈A a s X t-s (9)
Proposition 4 If b ξ 0 m < 1 with b = s∈A a s , theorem 3 applies with a j = ξ 0 m α j to the solution (7) of eqn. ( 9) (we set α j = 0 for j / ∈ A).

Non linear ARCH(∞) random fields

Models with

X t = ξ t a + j =0 g j (X t-j )
clearly extend on LARCH(∞) models; bounded functions g j provide robust models.

Corollary 2 If g j (x) -g j (y) ≤ α j x -y and ξ 0 ∞ j =0 α j < 1, theorem 2 holds with a i = ξ 0 ∞ α i (innovations are bounded here). Assume now that g i ≡ 0 for i / ∈ A, causal set then theorem 3 holds with a i = ξ 0 m α i (and now the innovations do not need anymore to be bounded).

This causality argument improves on [START_REF] Doukhan | Vector valued ARCH(∞) processes[END_REF] by only assuming finite moments for innovations instead of boundedness.

Mean field type model

Consider innovations in R k ′ and k × k matrices α i ,

X t = f ξ t , s =t α s-t X s ( 10
)
Corollary 3 Assume that f : R k ′ × R k → R k satisfies sup u∈R k ′ f (u, x) -f (u, y) ≤ b x -y , ∀x, y ∈ R k , b i =0 α i < 1.
then equation ( 10) admits a unique solution in L m written as a Bernoulli shift and this solution is η-weakly dependent with a i = b α i 1 .

The same results hold if now a i = 0 for i / ∈ A with A is causal in Z d and,

f ξ 0 , x) -f ξ 0 , y m ≤ b x -y , ∀x, y ∈ R k , b i =0 α i < 1.
LARCH(∞) model is still special cases of this one.

Proofs

We begin with the proof of some lemmas which relate the assumptions to contraction conditions in the space of Bernoulli shifts. Then we give separated proofs for existence and weak dependence properties. Those proofs always follow two steps since we first consider models with a finite range. For shortness we write here I = Z d \ {0}.

Proof of lemma 1

For p ∈ N * , we set Y p = F (X j 1 0< t ≤p ) j , ξ 0 and Y ′ p = F (X ′ j 1 0< t ≤p ) j , ξ 0 .

1. If q ∈ N * from assumption (H2),

Y p -Y p+q ≤ p< j ≤p+q a j X j , a.s.
Since the serie i∈I a j X j m is convergent the serie i∈I a j X j converges a.s. Hence, we deduce that a.s (Y p ) p∈N * is a Cauchy sequence in E and then converges. We denote by Y = F (X j ) j =0 ; ξ 0 this limit. Moreover, for p ∈ N * , we have:

Y p -F (0; ξ 0 ) m ≤ 0< j ≤p a j X j m
This proves that Y p ∈ L m . Hence the convergence in L m is a simply consequence of the Fatou lemma since:

Y -Y p m ≤ lim inf q→∞ Y q -Y p m ≤ lim inf q→∞ p< j ≤q a j X 0 m = j >p a j X 0 m
2. If p ∈ N * , we have using (H2):

Y p -Y ′ p m ≤ j =0 a j X j -X ′ j m
Hence the result follows with p → ∞.

Proof of the existence theorem 1

Assuming (H1) and (H2) we set a = j =0 a j . We adopt the notations of paragraph 2.4. For H ∈ L m (µ) we note

Φ p (H) = F (H • θ j )1 j ≤p j ; π 0
A direct consequence of lemma 1 is that lim p→∞ Φ p (H) exists in L m (µ). Denote this limit by F ((H • θ j ) j =0 ; π 0 ), the application Φ : L m (µ) → L m (µ) is defined as

Φ(H) = F ((H • θ j ) j =0 ; π 0 ) .
Let show that Φ is a contraction of L m (µ).If H, H ′ ∈ L m (µ), then applying the lemma 1 to the random fields X and X ′ defined as X j = H • θ j (ξ) and X ′ j = H ′ • θ j (ξ), we obtain:

Φ(H)(ξ) -Φ(H ′ )(ξ) m ≤ j =0 a j H • θ j (ξ) -H ′ • θ j (ξ) m ≤ j =0 a j H(ξ) -H ′ (ξ) m
Picard fixed point theorem applies since the space L m (µ) is complete. There exists a unique

H ∈ L m (µ) with Φ(H) = H thus H(ξ) = F (H • θ j (ξ)) j∈Z d ; ξ 0 , a.s. Set X t = H ((ξ t-i ) i∈Z d )
then with stationarity of ξ and since Z d is denumerable we get

X t = F (X t-j ) j∈Z d \{0} ; ξ t , ∀t ∈ Z d a.s.
Let Y be a stationary solution of this equation, we denote

u t = X t -Y t 1 for each t ∈ Z d . We obtain u t ≤ j =0 a j u t-j
As sup t u t ≤ X 0 1 + Y 0 1 < ∞ we note that the previous relation implies sup t u t ≤ a sup t u t . Hence u t = 0 for each t. Thus X t = Y t a.s for each t.

Proof of theorem 2

For an independent copy (ξ ′ t ) t∈Z d of ξ = (ξ t ) t∈Z d and s ∈ R + , we set ξ (s) = (ξ

(s) t ) t∈Z d with ξ (s) t = ξ t if t < s and ξ (s) s = ξ ′
s else. For a Bernoulli shift defined by H a straightforward extension of a result in [START_REF] Doukhan | A new weak dependence condition and applications to moment inequalities[END_REF] to random fields implies

η(r) ≤ 2δ r/2 , where δ r = H (ξ) -H ξ (r) 1 (11)

Weak dependence under finite interactions

We first assume that F depends finitely many variables

X t = F (X t-j 1 , . . . , X t-j k ; ξ 0 ) Lipschitz coefficients of F in condition (H2) write a 1 , . . . , a k and we set a = k i=1 a i < 1. Let H be the element of L m (µ) with X t = H ((ξ t-i ) i∈Z d ) and δ r = E H(ξ) -H(ξ (r) ) = E X 0 -X (r) 0 with X (r) t = H((ξ (r) t-i ) i∈ Z d ).
Lemma 7 Assume that (H1) and (H2) hold, then δ r ≤ 2 X 0 1 a r ρ hence δ r → r→∞ 0.

Proof of lemma 7. Set ρ = max{ j 1 , . . . , j k } and r > 0. Since ξ and ξ (r) admit the same distribution, we have for each t:

X t = F (X t-j 1 , . . . , X t-j k ; ξ t ) X (r) t = F (X (r) t-j 1 , . . . , X (r) t-j k ; ξ (r) t )
If t < r then ξ (r) t = ξ t and using (H2), we have :

X t -X (r) t 1 ≤ k l=1 a l X t-j l -X (r) t-j l 1 (12) Set now i = -[-r ρ ] if r ≥ ρ, then if u ≤ i -1 and l 1 , . . . , l u ∈ {1, . . . , k}: j l 1 + j l 2 + • • • + j lu < r.
We use inequality [START_REF] Helson | Prediction theory and Fourier series in several variables[END_REF] to derive recursively the bounds

X 0 -X (r) 0 1 ≤ k l 1 =1 a l 1 X -j l 1 -X (r) -j l 1 1 ≤ • • • ≤ k l 1 =1 a l 1 k l 2 =1 a l 2 • • • k l i =1 a l i X -(j l 1 +j l 2 +•••+j l i ) -X (r) -(j l 1 +j l 2 +•••+j l i ) 1 ≤ 2 X 0 1 a i From i ≥ r/ρ we get X 0 -X (r) 0 1 ≤ 2 X 0 1 a r ρ thus δ r ≤ 2 X 0 1 a r ρ . If now r < ρ, X 0 -X (r) 0 1 ≤ k l 1 =1 a l 1 X -j l 1 -X (r) -j l 1 1
Thus δ r ≤ 2 X 0 1 a i ≤ 2 X 0 1 a r ρ . The result follows with a < 1.

We now set a useful result. (X t ) t∈Z d and (X p,t ) t∈Z d will denote for p ≥ 0 the previous unique solution of the equations (1) and

Z t = F (Z t-j 1 {0< j ≤p} ) j∈Z d \{0} ; ξ t .
Lemma 8 Assume that the conditions in theorem 1 hold. Then X p,t → s→∞ X t in L m , for each t ∈ Z d .

Proof.

X p,0 -X 0 m ≤ 0< j ≤p a j X p,-j -X -j + j >p a j X -j m ≤ a X p,0 -X 0 m + X 0 m j >p a j thus X p,0 -X 0 m ≤ (1 -a) -1
X 0 m j >p a j which entails the first result. We also quote that sup p X p,0 m < ∞.

Weak dependence

Lemma 9 Assume that the conditions in theorem 1 hold. Then the random field (X t ) t∈Z d is η-weakly dependent.

Proof. Recall that sup p X p,0 m < ∞; if m ≥ 1, weak dependence follows from

E X (r) 0 -X 0 ≤ E X (r) 0 -X (r) p,0 + E X (r) p,0 -X p,0 + E X p,0 -X 0 = 2E X p,0 -X 0 + E X (r) p,0 -X p,0 For r ≥ p, from lemma 7 we derive E X (r) p,0 -X p,0 ≤ 2 X p,0 1 j ≤p a j r p Hence δ r = E X (r) 0 -X 0 ≤ 2 • X 0 1 1 -a j >p a j + 2 X p,0 1 j ≤p a j r p ≤ 2 • X 0 1 1 -a j >p a j + 2 X p,0 1 a r p
Using the fact that sup p X p,0 1 < ∞ there exists a constant C > 0 with

δ r ≤ C • inf p j >p a j + a r p
For any ε > 0 we choose p 0 > 0 such that j >p 0 a j ≤ ε/(2C) and r 0 such that r ≥ r 0 implies a r p ≤ ε/(2C). Thus δ r ≤ ε if r ≥ r 0 and δ r → r→∞ 0. Using [START_REF] Georgii | Gibbs measures and phase transition[END_REF] we prove that (X t ) t is η-weakly dependent and η(r) ≤ δ r/2 .

Decay rates

Using the representation of the solution as a Bernoulli shift and the inequality [START_REF] Georgii | Gibbs measures and phase transition[END_REF] this will be enough to bound the expression of δ r . Set b p = #{i ∈ Z d / i ≤ p} and

s p = #{i ∈ Z d / i = p} for i = max{|i 1 |, . . . , |i d |} we obtain b p = (2p + 1) d and s p = b p -b p-1 ≤ Kp d-1 for a constant K > 0.
Proof of lemma 2.

j >p a j = q>p j =q e -βq ≤ Kp d-1 q>p e -βq ≤ K q≥p+1 q d-1 e -βq .
The function x → x d-1 e -βx decreases on (p, +∞) for large enough p, thus

q≥p+1 q d-1 e -βq ≤ +∞ p x d-1 e -βx dx = p d-1 e -βp β + (d -1) +∞ p x d-2 e -βx dx = O p d-1 e -βp .
We thus find a constant C 1 such that

δ r ≤ C 1 inf p p d-1 e -βp + e -α r p = C 1 inf p e -βp+(d-1) ln p + e -α r p .
Assume βp -(d -1) ln p ∼ αr/p (r → ∞) which implies βp 2 ∼ αr, then p ∼ αr/β. Thus, there is a constant C 2 such that:

δ r ≤ C 2 r d-1 2 e - √ αβr .
Proof of lemma 3. As before, j >p

a j ≤ K q>p q d-β-1 ≤ K +∞ p x d-β-1 dx = K p d-β β -d . Hence δ r ≤ c • inf

Proof of proposition 1

Models with finite interactions

We assume first that there exist ≥ 1 and j 1 , . . . , j k ∈ I such F (x; u) only depends on x j 1 , . . . , x j k for each x = (x j ) j =0 ∈ E I . Hence writing a i instead of a j i for 1 ≤ i ≤ k, we have inequality

F (x; u) -F (y; u ′ ) ≤ k i=1 a i x j i -y j i + b u -u ′ , a = k i=1 a i < 1 Now h : E k × E ′ → Eis such that F (x; u) = h(x j 1 , . . . , x j k , u
). We will denote ρ = max{ j 1 , . . . , j k }.

Lemma 10 1) If the random field ξ is η-weakly dependent (the weak dependence coefficients are denoted η ξ (r)) then X is η-weakly dependent with

η(r) ≤ C inf n∈N * a n + k n η ξ ((r -2ρn) ∨ 0) ,
where C is a positive constant.

2) If the random field ξ is λ-weakly dependent (the weak dependence coefficients are denoted λ ξ (r)) then X is λ-weakly dependent with

λ(r) ≤ C inf n∈N * a n + k 2n λ ξ ((r -2ρn) ∨ 0) ,
for some constant C > 0.

Proof of lemma 10. We will use the lemma 6 and the following useful lemma 11.

Lemma 11 1. For every x and y ∈ C (Z d ) we have

Φ(0)(x) -Φ(0)(y) ≤ b x 0 -y 0 and if n ≥ 2, Φ (n) (0)(x) -Φ (n) (0)(y) ≤ n-1 l=1 k i 1 ,...,i l =1 a i 1 • • • a i l b x j i 1 +•••+j i l -y j i 1 +•••+j i l + b x 0 -y 0 2. Fix x ∈ C (Z d ) .
Then Φ(0)(x) only depends on x 0 and Φ(0) defines a b-Lipschitz function on C. We set K 1 = b and p 1 = 1.

For n ≥ 2 we set

A n = n-1 l=1 {j i 1 + • • • + j i l / 1 ≤ i 1 , . . . , i l ≤ k} ∪ {0}, p n = card A n and K n = b 1-a n 1-a . Then Φ (n) (0)(x) only depends on x j for j ∈ A n Moreover Φ (n) (0) defines a Lipschitz function on C pn and Lip Φ (n) (0) ≤ K n .
Proof of lemma 11.

• The first point is easy to check. For n ≥ 2 we use induction. For n = 2

Φ (2) (0)(x) -Φ (2) (0)(y) = h (Φ(0) • θ 1 (x), . . . , Φ(0) • θ k (x), x 0 ) -h (Φ(0) • θ 1 (y), . . . , Φ(0) • θ k (y), y 0 ) ≤ k i=1 a i F (0, x i ) -F (0, y i ) + b x 0 -y 0 ≤ k i=1 a i b x i -y i + b x 0 -y 0
Assuming that the inequality holds for an integer n ≥ 2, we estimate φ n,x,y = Φ (n+1) (0)(x) -Φ (n+1) (0)(y) :

φ n,x,y = h Φ (n) (0)(θ j 1 x), . . . , Φ (n) (0)(θ j k x), x 0 -h Φ (n) (0)(θ j 1 y), . . . , Φ (n) (0)(θ j k y), y 0 ≤ k i=1 a i Φ (n) (0)(θ j i x) -Φ (n) (0)(θ j i y) + b x 0 -y 0 ≤ k i=1 a i n-1 l=1 1≤i 1 ,...,i l ≤k a i 1 • • • a i l b x j i +j i 1 +...+j i l -y j i +j i 1 +...+j i l + b x j i -y j i + b x 0 -y 0 = n l=1 1≤i 1 ,...,i l ≤k a i 1 • • • a i l b x j i 1 +...+j i l -y j i 1 +...+j i l + b x 0 -y 0
Hence inequality holds for n + 1.

• The case n = 1 is easy to check. For the first point we use induction. For n = 2 the result is a consequence of:

Φ (2) (x)(0) = h (h(0, x j 1 ), . . . , h(0, x j k ), x 0 )
Suppose now the result true for an integer n ≥ 2. Then the identity

Φ (n+1) (0)(x) = h Φ (n) (0)(θ j 1 x), . . . , Φ n (0)(θ j k x), x 0
shows that Φ (n+1) (0)(x) only depends of coordinates (x j i +j ) 1≤i≤k,j∈An and x 0 that is to say coordinates (x j ) j∈A n+1 .

For the second point, we use inequality in 1. We have:

φ n,x,y ≤ n-1 l=1 1≤i 1 ,...,i l ≤k a i 1 • • • a i l b x j i 1 +...+j i l -y j i 1 +...+j i l + b x 0 -y 0 ≤ b n-1 l=1 a l + 1 j∈An x j -y j = b • 1 -a n 1 -a j∈An x j -y j
End of the proof of lemma 10. We recall the notation

X n t = Φ (n) (0) (ξ t-j ) j for n ∈ N * and t ∈ Z d . Set f 1 = f (X s 1 , . . . , X su ), g 1 = g(X t 1 , . . . , X tv ) and f ′ 1 = f X n s 1 , . . . , X n su , g ′ 1 = g X n t 1 , . . . , X n tv For each t ∈ Z d , if n ∈ N \ {0, 1} then A t,n = {t} -A n . If s i -t l ≥ r for 1 ≤ i ≤ u and 1 ≤ l ≤ v then d(A s i ,n , A t l ,n ) ≥ (r -2ρn) ∨ 0 = d r,n . Thus |Cov(f 1 , g 1 )| ≤ |Cov(f 1 -f ′ 1 , g 1 )| + |Cov(f ′ 1 , g 1 -g ′ 1 )| + |Cov(f ′ 1 , g ′ 1 )| ≤ 4E|f 1 -f ′ 1 | + 4E|g 1 -g ′ 1 | + ψ(up n , vp n , K n Lip (f ), K n Lip (g))ε ξ (d r,n ) ≤ 4Lip (f ) u i=1 X s i -X n s i 1 + 4Lip (g) v i=1 X t i -X n t i 1 + ψ(up n , vp n , K n Lip (f ), K n Lip (g))ε ξ (d r,n ) ≤ (4Lip (f )u + 4Lip (g)v)a n X 0 1 + ψ(up n , vp n , K n Lip (f ), K n Lip (g))ε ξ (d r,n )
Note that this result is still true for n = 1.

1) Under η-weak dependence, ψ(u, v, a, b) = au + bv,

|Cov(f 1 , g 1 )| ≤ (uLip f + vLip g)(4a n X 0 1 + K n p n η ξ (d r,n ) Thus |Cov(f 1 , g 1 )| ≤ (uLip f + vLip g)η(r) where η(r) ≤ inf n∈N * {4a n X 0 1 + K n p n η ξ (d r,n )} 2) With λ-weak dependence ψ(u, v, a, b) = au + bv + abuv, |Cov(f 1 , g 1 )| ≤ (uLip f + vLip g + uvLip f Lip g)(4a n X 0 1 + K n p n λ ξ (d r,n ) Now |Cov(f 1 , g 1 )| ≤ (uLip f + vLip g + uvLip f Lip g)λ(r) with λ(r) ≤ inf n∈N * {4a n X 0 1 + K n p 2 n λ ξ (d r,n )} As (K n ) n is bounded and p n ≤ n-1 l=1 k l = k-k n 1-k for n ≥ 2
, we obtain the proposed bounds.

We now prove that lim r→∞ λ(r) = 0. We suppose that the sequence (λ ξ (r)) r nonincreasing without loss of generality. We use the bound

λ(r) ≤ C inf N +2ρn=r,n∈N * a n + k 2n λ ξ (N ) If N ∈ N, we choose n N = [log(λ ξ (N ))/(log a -2 log k)]. Note that lim N →∞ n N = ∞ and lim N →∞ a n N +k 2n N λ ξ (N ) = 0. For r ≥ r N = N +2ρn N , we have N +r -r N +2ρn N = r, hence: λ(r) ≤ a n N + k 2n N λ ξ (N + r -r N ) ≤ a n N + k 2n N λ ξ (N ) → N →∞ 0
Hence lim r→∞ λ(r) = 0. Analogously, lim r→∞ η(r) = 0.

General case

Recall that we have denoted (X p,t ) t∈Z d for s > 0 the unique solution of the equation Z t = F Z t-j 1 0< j ≤p} j =0 ; ξ t . Denote f 1 = f (X s 1 , . . . , X su ), g 1 = g(X t 1 , . . . , X tv ), f ′ 1 = f (X p,s 1 , . . . , X p,su ) and g ′ 1 = f (X p,t 1 , . . . , X p,tv ), then

|Cov(f 1 , g 1 )| ≤ |Cov(f 1 -f ′ 1 , g 1 )| + |Cov(f ′ 1 , g 1 -g ′ 1 )| + |Cov(f ′ 1 , g ′ 1 )| ≤ 4 Lip f u i=1 X s i -X p,s i 1 + 4 Lip g v i=1 X t i -X p,t i 1 + |Cov(f ′ 1 , g ′ 1 )| ≤ 4 X 0 -X s,0 1 (uLip f + vLip g) + |Cov(f ′ 1 , g ′ 1 )|
Recall that from the proof of lemma 8 we have:

X p,0 -X 0 1 ≤ X 0 1 1 -a j >p a j
Moreover, the field X p,t is k-dependent with k = (2p) d .

• Suppose first that the random field ξ is η-weakly dependent. From proposition 10,

|Cov(f ′ 1 , g ′ 1 )| ≤ (uLip f + vLip g)C inf n∈N * a n + p dn η ξ ((
d-1 , then g(C) is a convex cone of R d-1 such that g(C) ∩ -g(C) = {0}. Hence there exists a basis c = (c 2 , . . . , c d ) such that g(C) ⊂ {x ∈ R d-1 /0 ≤ c x}. For i = 2, . . . , d we set b i = g -1 (c i ). Then b = (b 1 , . . . , b d ) is a basis of R d and if x = x 1 b 1 + . . . + x d b d ∈ B,
we have by the preceding two points x 1 ≥ 0. Suppose that x 1 = 0, then x ∈ C and g

(x) ≥ c 0 ⇒ (x 2 , . . . , x d ) ≥ lex 0 in R d-1 . Hence (x 1 , . . . , x d ) ≥ lex 0 in R d , in other word x ≥ b 0. 4.6.2 Proof of lemma 5 Proof of lemma 5 Denote for p ∈ N * , Y p = F (X j 1 0< j ≤p ) j ; ξ 0 . 1) We first prove that for p ∈ N * , Y p ∈ L m .
Recall that here F is measurable wrt F e A ⊗B(E ′ ). Let x ∈ E I , then using the independence between ξ 0 and σ X j , j ∈ A and the condition (H3), we have:

E Y p -F (0; ξ 0 ) m X j = x j , j ∈ A = E F ((x j 1 0< j ≤p ) j ; ξ 0 ) -F (0; ξ 0 ) m ≤   0< j ≤p a j x j   m
Hence by integration:

Y p -F (0; ξ 0 ) m ≤ 0< j ≤p a j X j m ≤ X 0 m
As F (0; ξ 0 ) ∈ L m , we obtain the result.

It is enough to prove that (Y p ) p∈N * is a Cauchy sequence in L m . Using the same method as in 1), we obtain if q > 0:

Y p+q -Y p m ≤ X 0 m j >p a j
This inegality imply the result.

2) Using the same method as in 1), we have for p ∈ N * :

Y p -Y ′ p m ≤ 0< j ≤p a j Y j -Y ′ j m
Hence the result follows with p → ∞. (µ), we denote by Y the random field defined as

Y j = H • θ j (ξ) for j ∈ Z d . If j ∈ A, then H • θ j is measurable wrt σ(π ′ j+j ′ /j ′ ∈ A) ⊂ F ′ e A . Hence if p ∈ N * , Φ p (H) ∈ L m e A
(µ) and by the lemma 4, ξ 0 is independant of σ(Y j /j ∈ A). By application of the lemma 5, Φ p (H) converges to an element of L m e A (µ) witch is F (H • θ j ) j =0 ; π 0 . Lets show that the application Φ :

L m e A (µ) → L m e A (µ) defined as Φ(H) = F (H • θ j ) j =0 ; π 0 is contraction in L m e A (µ). If H, H ′ ∈ L m e A
(µ) then the two random fields Y and Y ′ defined as Y j = H • θ j (ξ) and Y ′ j = H ′ • θ j (ξ) for j ∈ Z d verify the assumptions of lemma 5. Indeed σ(Y j , Y ′ j /j ∈ A) ⊂ σ(ξ j+j ′ /j ∈ A, j ′ ∈ A) and using the lemma 4 we deduce the independence between ξ 0 and σ(Y j , Y ′ j /j ∈ A). Hence, we have:

Φ(H)(ξ) -Φ(H ′ )(ξ) m ≤ j∈A a j H • θ j (ξ) -H ′ • θ j (ξ) m = j∈A a j H(ξ) -H ′ (ξ) m
witch shows the result. The construction of X t comes from theorem 2. The variable H(ξ) being measurable wrt σ ξ j ; j ∈ A measurability of X t is simply deduced. Then unicity is a consequence of the application of the fixed point theorem.

Weak dependence

Weak dependence of the solution is as in § 4.3.1 where (H2) replaces (H3'). The case of finite range corresponds to k-Markov systems on a finite causal set. To prove lemma 7, we use (H3') and independence of rvs X t-j 1 , . . . , X t-j k , X

t-j 1 , . . . , X

t-j k and ξ t to derive [START_REF] Helson | Prediction theory and Fourier series in several variables[END_REF]. In the general case we note (X p,t ) t the solution of Z t = F (Z t-j 1 {j∈Ap} ) j ; ξ t with A p = {t ∈ A t ≤ p} and we conclude as in lemma 9.

Proof of lemma 6

1) The proposed bound is a consequence of the fixed point theorem. Thus we deduce that for each ε > 0:

n≥1 P X t -Φ (n) (0)(ξ t-j , j ∈ Z d ) ≥ ε ≤ 1 ε n≥1 X t -Φ (n) (0)(ξ t-j , j ∈ Z d ) 1 < ∞
Hence by the Borel-Cantelli lemma, we deduce lim n→∞ Φ (n) (0) ξ t-j , j ∈ Z d = X t a.s.

2) We use induction. For n = 1

X t -Φ p (0)(ξ t-j , j ∈ Z d ) 1 = F ((X t-j ) j , ξ t ) -F (0, ξ t ) 1 ≤ a X 0 1 ≤ a X 0 1 + X 0 1 j >p a j
Suppose the result true for an integer n ≥ 1, then

X t -X n+1 p,t 1 = F ((X t-k ) k , ξ t ) -F Φ (n+1) p (0)(ξ t-j-k , j ∈ Z d )1 k ≤p k , ξ t 1 ≤ k ≤p a k X t-k -Φ (n) p (0)(ξ t-j-k , j ∈ Z d ) 1 + X 0 1 k >p a k ≤ a a n X 0 1 + 1 -a n 1 -a X 0 1 k >p a k + X 0 1 k >p a k = a n+1 X 0 1 + 1 -a n+1 1 -a X 0 1 k >p a k
4.9 Proofs for the section 3 4.9.1 Proof of corollary 1

Here F (x; u) = f (x ℓ 1 , . . . , x ℓ k ) + h(u)g(x ℓ 1 , . . . , x ℓ k ) + u. Condition (H1) is easy to check and e.g. in the first case,

F (z; ξ 0 ) -F (z ′ ; ξ 0 ) m ≤ f (z ℓ 1 , . . . , z ℓ k ) -f z ′ ℓ 1 , . . . , z ′ ℓ k + γ t ∞ g (z ℓ 1 , . . . , z ℓ k ) -g z ′ ℓ 1 , . . . , z ′ ℓ k ≤ k i=1 b i z ℓ i -z ′ ℓ i + γ 0 ∞ k i=1 c i z ℓ i -z ′ ℓ i .
For dependent inputs, we remark that (z, u) → F (z; u) is a Lipschitz function in order to apply proposition 1.

Proof of proposition 3

Normal convergence in L m will justify all the forthcoming manipulations of series. We only consider the more complicated causal case. In order to prove that X t ∈ L m we will prove the normal convergence of the series.

Set S = ξ t + +∞ i=1 j 1 ,...,j i ∈A

α j 1 t • • • α j i t-j 1 -•••-j i-1 ξ t-(j 1 +•••+j i ) m
, we notice from causality that indices t and (t -(j 1 + • • • + j ℓ )) are distinct if 1 ≤ ℓ ≤ i hence the independence of inputs implies

α j 1 t • • • α j i t-j 1 -•••-j i-1 ξ t-(j 1 +•••+j i ) m ≤ α j 1 t • • • α j i t-j 1 -•••-j i-1 ξ t-(j 1 +•••+j i ) m = α j 1 t m • • • α j i t-j 1 -•••-j i-1 m ξ t-(j 1 +•••+j i ) m S ≤ ζ t m + j∈A j 1 ,...,j i ∈A α j 1 t m • • • α j i t-j 1 -•••-j i-1 m ζ t-(j 1 +•••+j i ) m = ζ 0 m 1 + j i nA j 1 ,...,j i ∈A α j 1 0 m • • • α j i 0 m = ζ 0 m 1 + +∞ i=1 j∈A a j 0 m i = ζ 0 m 1 + +∞ i=1 b i = ζ 0 m 1 + b 1 -b < ∞.
In order to prove that X t is solution of the equation, we expand it:

X t = ζ t + j 1 ∈A α j 1 t ζ t-j 1 + ∞ i=2 j 1 ,...,j i ∈A α j 1 t • • • α j i t-j 1 -•••-j i-1 ζ t-(j 1 +•••+j i ) = ζ t + j 1 ∈A α j 1 t ζ t-j 1 + j 1 ∈A α j 1 t ∞ i=2 j 2 ,...,j i ∈A α j 2 t-j 1 • • • α j i t-j 1 -•••-j i-1 ζ t-j 1 -(j 2 +•••+j i ) = ζ t + j 1 ∈A α j 1 t X t-j 1
Here F (x; (u, v)) = j∈A u j x j + v and we use notations in (H3). As ξ is iid, the variables (Z(ξ), Z ′ (ξ)) are (α j 0 ) j∈A are independent and

F (z; ζ 0 ) -F z ′ ; ζ 0 m ≤ j∈A α j 0 m z j -z ′ j
Since b = j∈A a j 0 m < 1, (H3) holds. In the first non-causal case the above inequalities are only changed by using the bound

α j 1 t • • • α j i t-j 1 -•••-j i-1 ξ t-(j 1 +•••+j i ) m ≤ α j 1 t ∞ • • • α j i t-j 1 -•••-j i-1 ∞ ξ t-(j 1 +•••+j i ) m .

Proof of proposition 4

Here (H1) holds and with the notation in (H3):

F (z, ξ 0 ) -F (z ′ , ξ 0 ) m ≤ j∈Z d \{0}
α j ξ 0 m z j -z ′ j .

The proposed solution is in L m from normal convergence of series

X t m ≤ ξ t m a + ∞ k=1 j 1 ,...,j k ∈A α j 1 ξ t-j 1 m • • • α j k ξ t-j 1 -•••-j k m a = ξ 0 m a 1 + ∞ k=1 j 1 ,...,j k ∈A α j 1 ξ t-j 1 m • • • α j k ξ 0 m = ξ 0 m a 1 + ∞ k=1 b i ξ 0 i m = ξ 0 m a 1 + b ξ 0 m 1 -b ξ 0 m < ∞.
Substitutions prove that this process is a solution of the equation.

X t = ξ t a + ∞ k=1 j 1 ,...,j k ∈A α j 1 ξ t-j 1 • • • α j k ξ t-j 1 -•••-j k a = ξ t a + j 1 ∈A
α j 1 ξ t-j 1 a + +∞ k=2 α j 2 ξ t-j 1 -j 2 . . . α j k ξ t-j 1 -j 2 -•••-j k = ξ t a + j 1 ∈A α j 1 X t-j 1 .

Figure 1 :Figure 2 : 2 1 + j 2 2

 1222 Figure 1: Non causal LARCH field

pe

  -αr/p + p d-β β -d . Choosing (β -d) ln p ∼ αr/p thus (β -d)p ln p ∼ αr we derive ln p ∼ ln r, thus p ∼ αr (β-d) ln r . There exists some constant C 3 with δ r ≤ C 3 r ln r d-β .

4. 6

 6 Results on causality4.6.1 Proof of proposition 2We will use here the Euclidean norm on R d . We proceed by induction on d. For d = 1, if there exists r 1 , r 2 ∈ B such that r 1 > 0 and r 2 < 0 then B ∩ (-B) = {0}.Then we can choose b1 = 1 if B ⊂ R + or b 1 = -1 if B ⊂ R -. Suppose the result true for d -1. We first define b 1 . 1) If B • is empty, since B is convex and contain 0 there exists b 1 ∈ R d \ {0} such that B ⊂ H = {x ∈ R d /x.b 1 = 0}(• denotes the scalar product in R d ). 2) Now if B • is not empty, like B ∩ (-B) = {0} it is clear that 0 / ∈ B • . Moreover B • is still convexand by application of the Hahn-Banach theorem ([2], theorem 3.3, page 108), there exists b 1 ∈ R d \ {0} such that B • ⊂ {x ∈ R d /x.b 1 ≥ 0}. Like for a convex B • = B, then the same inclusion holds for B. We set here H = {x ∈ R d /x.b 1 = 0}. We consider now the convex cone C = B ∩ H. If g denote an isomorphism between H and R

4. 7

 7 Proof of theorem 3 4.7.1 Existence If H ∈ L m e A

  + vLip g + uvLip f Lip g) inf n∈N * a n + p 2dn λ ξ ((r -2pn) ∨ 0) up a suitable positive constant C. Hence we bound |Cov(f 1 , g 1 )| by, (uLip f + vLip g + uvLip f Lip g) a n + p 2dn λ ξ ((r -2pn) ∨ 0)up to another positive constant C. Then we obtain the proposed bound.

	for a suitable positive constant C.
	Hence we bound |Cov(f 1 , g 1 )| by, 			
	(uLip f + vLip g)C		j >p	a j + inf n∈N * a n + p dn η ξ ((r -2ρn) ∨ 0)	
	for another positive constant denoted C. Then we obtain the proposed bound.
	• Suppose that the random field ξ is λ-weakly dependent. From proposition 10, |Cov(f ′ 1 , g ′ 1 )| is bounded by
	(uLip f ×	j >p	a j + inf n∈N
					r -2pn) ∨ 0)

*

If V denotes a vector space and B an arbitrary set then V (B) ⊂ V B denotes the set of v = (v b ) b∈B such that there is some finite subset B1 ⊂ B with v b = 0 for each b / ∈ B1.


Paul Doukhan
email: doukhan@ensae.fr

Gilles Teyssière

Pablo Winant
email: pablo.winant@ens-lyon.org

A LARCH(∞) Vector Valued Process

. We finally consider a long range dependent version of this model; in this case we provide an existence and uniqueness result.

Introduction

The purpose of this chapter is to propose a unified framework for the study of ARCH(∞) processes that are commonly used in the financial econometrics literature. We extend the study, based on Volterra expansions, of univariate ARCH(∞) processes by Giraitis et al. [START_REF] Giraitis | Stationary ARCH models: dependence structure and central limit theorems[END_REF] and Giraitis and Surgailis [START_REF] Giraitis | ARCH-type bilinear models with double long memory[END_REF] to the multi-dimensional case.

Let {ξ t } t∈Z be a sequence of real valued random matrices independent and identically distributed of size d × m, {a j } j∈N * be a sequence of real matrices m × d, and a be a real vector of dimension m. The vector ARCH(∞) process is defined as the solution to the recurrence equation:

X t = ξ t   a + ∞ j=1 a j X t-j   . (1)
The following section 2 displays a chaotic expansion solution to this equation; we also consider a random fields extension of this model. Some approximations of this solutions are listed in the next section 3, where we consider approximations by m-dependent sequences, coupling results and approximations by Markov sequences. Section 4 details the weak dependence properties of the model and section 5 provides an existence and uniqueness condition for the solution of the previous equation; in that case, long range dependence may occur. The end of this section is dedicated to review examples of this vector valued model.

The vector ARCH(∞) model nests a large variety of models, the two first extensions being obvious:

1. The univariate linear ARCH(∞) (LARCH) model, where the X t and a j are scalar, 2. The bilinear model, with

X t = ζ t   α + ∞ j=1 α j X t-j   + β + ∞ j=1 β j X t-j ,
where all variables are scalar, and ζ t are iid centered innovations. We set

ξ t = (ζ t , 1) , a = α β , a j = α j β j .
In that case, the expansion (3) is the same as the one used by Giraitis and Surgailis [START_REF] Giraitis | ARCH-type bilinear models with double long memory[END_REF]. 3. With a suitable re-parameterization, this vector ARCH(∞) nests the standard GARCH-type processes used in the financial econometrics literature for modeling the non-linear structure of the conditional second moments. The GARCH(p, q) model is defined as

r t = σ t ε t σ 2 t = p j=1 β j σ 2 t-j + γ 0 + q j=1 γ j r 2 t-j γ 0 > 0, γ j 0, β i 0,
where the ε are centered and iid. This model is nested in the class of bilinear models with the following re-parameterization

α 0 = γ 0 1 -β i , α i z i = γ i z i 1 -β i z i ,
see Giraitis et al. [START_REF] Giraitis | Recent advances in ARCH modelling[END_REF]. The covariance function of the sequence {r 2 t } has an exponential decay, which is implied by the exponential decay of the sequence of weights α j ; see Giraitis et al. [START_REF] Giraitis | Stationary ARCH models: dependence structure and central limit theorems[END_REF]. 4. The ARCH(∞) model, where the sequence of weights β j might have either a exponential decay or a hyperbolic decay.

r t = σ t ε t , σ 2 t = β 0 + ∞ j=1 β j r 2 t-j ,
with the following parameterization

X t = r 2 t , ξ t = ε 2 t -λ 1 κ , 1 , a = κβ 0 λ 1 β 0 , a j = κβ j λ 1 β j ,
where the ε are centered and iid, λ 1 = E(ε 2 0 ), and κ 2 = Var(ε 2 0 ). Note that the first coordinate of ξ 0 is thus a centered random variable. Conditions for stationarity of the unidimensional ARCH(∞) model have been derived using Volterra expansions by Giraitis et al. [START_REF] Giraitis | Stationary ARCH models: dependence structure and central limit theorems[END_REF] and Giraitis and Surgailis [START_REF] Giraitis | ARCH-type bilinear models with double long memory[END_REF]. The present paper is a multidimensional generalization of these previous works. 5. We can consider models with several innovations and variables like:

Z t = ζ 1,t   α + ∞ j=1 α 1 j Z t-j   + µ 1,t   β + ∞ j=1 β 1 j Y t-j   + γ + ∞ j=1 γ 1 j Z t-j Y t = ζ 2,t   α + ∞ j=1 α 2 j Y t-j   + µ 2,t   β + ∞ j=1 β 2 j Z t-j   + γ + ∞ j=1 γ 2 j Y t-j
This model is straightforwardly described through equation ( 1) with d = 2 and m = 3. Here 3 and the process

ξ t = ζ 1,t µ 1,t 1 ζ 2,t µ 2,t 1 is a 2×3 iid sequence, a j =   α 1 j α 2 j β 1 j β 2 j γ 1 j γ 2 j   is a 3 × 2 matrix and a =   α β γ   is a vector in IR
X t = Z t Y t
is a vector of dimension 2. Dimensions m = 3 and d = 2 are only set here for simplicity. Replacing m = 3 by m = 6 would allow to consider different coefficients α, β and γ for both lines in this system of two coupled equations. This generalizes the class of multivariate ARCH(∞) processes, defined in the p-dimensional case as:

R t = Σ 1 2 t ε t ,
where R t is a p-dimensional vector, Σ t is a p × p positive definite matrix, and ε t is a p-dimensional vector. Those models are formally investigated by Farid Boussama in [START_REF] Boussama | Ergodicité, mélange, et estimation dans les modèles GARCH[END_REF]; published references include [START_REF] Boussama | Normalité asymptotique de l'estimateur du pseudo maximum de vraisemblance d'un modèle GARCH[END_REF] and [START_REF] Engle | Multivariate simultaneous generalized ARCH[END_REF]. This model is of interest in financial econometrics as the volatility of asset prices of linked markets, e.g., major currencies in the Foreign Exchange (FX) market, are correlated, and in some cases display a common strong dependence structure; see [START_REF] Teyssière | Modelling Exchange rates volatility with multivariate longmemory ARCH processes[END_REF]. This common dependence structure can be modeled with the assumption that the innovations ε 1 , . . . , ε p are correlated. An (empirically) interesting case for the bivariate model (X t , Y t ) is obtained with the assumption that the (ζ 1,t , ζ 2,t ) are cross-correlated.

Existence and Uniqueness in L p

In the sequel, we set A(x) = j x a j , A = A(1), where • denotes the matrix norm.

Theorem 2.1 Let p > 0, we denote

ϕ = j 1 a j p∧1 (E ξ 0 p ) 1 p∧1 . (2)
If ϕ < 1, then a stationary solution in L p to equation (1) is given by:

X t = ξ t   a + ∞ k=1 j1,...,j k 1 a j1 ξ t-j1 • • • a j k ξ t-j1-•••-j k a   . (3)
Proof. The norm used for the matrices is any multiplicative norm. We have to show that expression (3) is well defined under the conditions stated above, converges absolutely in L p , and that it satisfies equation (1).

Step 1. We first show that expression (3) is well defined (after the second line we omit to precise the norms). For p 1, we have j1,...,j k 1

a j1 ξ t-j1 • • • a j k ξ t-j1-•••-j k m×m j1,...,j k 1 a j1 m×d • • • a j k m×d ξ t-j1 d×m • • • ξ t-j1-•••-j k d×m
The series thus converges in norm L p because ∞ k=1 j1,...,j k 1

(E a j1 ξ t-j1 • • • a j k ξ t-j1-•••-j k p ) 1/p ∞ k=1 j1,...,j k 1 a j1 • • • a j k (E ξ t-j1 p ) 1/p • • • (E ξ t-j1-•••-j k p ) 1/p ∞ k=1 j1,...,j k 1 a j1 • • • a j k (E ξ 0 p ) k p ∞ k=1 ϕ k
The series ∞ k=1 ϕ k is finite since ϕ < 1, hence the series (3) converges in L p . For p < 1, the convergence is defined through the metric d p (U, V ) = E U -V p between vector valued L p random variables U, V and we start from

  j1,...,j k 1 a j1 ξ t-j1 • • • a j k ξ t-j1-•••-j k   p j1,...,j k 1 a j1 ξ t-j1 • • • a j k ξ t-j1-•••-j k p ,
and we use the same arguments as for p = 1.

Step 2. We now show that equation ( 3) is solution to equation ( 1):

X t = ξ t   1 + ∞ k=1 j1,...j k 1 a j1 ξ t-j1 • • • a j k ξ t-j1-•••-j k   a = ξ t   a + j1 1 a j1 ξ t-j1 + + ∞ k=2 j1 1 a j1 ξ t-j1 j2,...,j k 1 a j2 ξ t-j1-j2 • • • a j k ξ t-j1-j2-•••-j k a   = ξ t   a + j1 1 a j1 ξ t-j1 a+ + ∞ k=2 j2,...,j k 1 a j2 ξ (t-j1)-j2 • • • a j k ξ (t-j1)-j2-•••-j k a   = ξ t   a + j 1 a j X t-j   .
Remark 2.1 The uniqueness of this solution is not demonstrated without additional condition; see Theorem 2.2 and section 5 below.

Theorem 2.2 Assume that p 1 then from (2), ϕ = j a j ξ 0 p . Assume ϕ < 1. If a stationary solution (Y t ) t∈Z Z to equation (1) exists (a.s.), if Y t is independent of the sigma-algebra generated by {ξ s ; s > t}, for each t ∈ Z Z, then this solution is also in L p and it is (a.s.) equal to the previous solution (X t ) t∈Z Z defined by equation (3).

Proof.

Step 1. We first prove that Y 0 p < ∞. From equation ( 1) and from {Y t } t∈Z Z 's stationarity, we derive

Y 0 p ξ 0 p   a + ∞ j=1 a j Y 0 p   < ∞,
hence, the first point in the theorem follows from:

Y 0 p ξ 0 p a 1 -ϕ < ∞.
Step 2. As in [START_REF] Giraitis | Stationary ARCH models: dependence structure and central limit theorems[END_REF] we write recursively

Y t = ξ t a + j 1 a j Y t-j = X m t +S m t , with X m t = ξ t   a + m k=1 j1,••• ,j k 1 a j1 ξ t-j1 • • • a j k ξ t-j1•••-j k a   , S m t = ξ t   j1,••• ,jm+1 1 a j1 ξ t-j1 • • • a jm ξ t-j1•••-jm a jm+1 Y t-j1•••-jm   .
We have

S m t p ξ p j1,••• ,jm+1 1 a j1 • • • a jm+1 ξ m p Y 0 p = Y 0 p ϕ m+1 .
We recall the additive decomposition of the chaotic expansion X t in equation ( 3) as a finite expansion plus a negligible remainder that can be controlled

X t = X m t + R m t where R m t = ξ t   k>m j1,••• ,j k 1 a j1 ξ t-j1 • • • a j k ξ t-j1•••-j k a   , satisfies R m t p a ξ 0 p k>m ϕ k a ξ 0 p ϕ m 1 -ϕ → 0.
Then, the difference between those two solutions is controlled as a function of

m with X t -Y t = R m t -S m t , hence X t -Y t p R m t p + S m t p ϕ m 1 -ϕ a ξ 0 p + Y 0 p ϕ m 2 ϕ m 1 -ϕ a ξ 0 p thus, Y t = X t a.s.
We also consider the following extension of equation ( 1) to random fields {X t } t∈Z Z D : Lemma 2.1 Assume that a j are m × d-matrices now defined for each j ∈ Z Z D \ {0}. Fix an arbitrary norm • on Z Z D . We extend the previous function A to A(x) = j x a j , A = A(1) and we suppose with p = ∞ that ϕ = A ξ 0 ∞ < 1. Then the random field

X t = ξ t   a + ∞ k=1 j1 =0 • • • j k =0 a j1 ξ t-j1 • • • a j k ξ t-j1-•••-j k a   (4)
is a solution to the recursive equation:

X t = ξ t   a + j =0 a j X t-j   , t ∈ Z Z D . (5)
Moreover, each stationary solution to this equation is also bounded and equals X t , a.s.

The proof is the same as before, we first prove that any solution is bounded and we expand it as the sum of the first terms in this chaotic expansion, up to a small remainder (wrt to sup norm); the only important modification follows from the fact that now j 1 + • • • + j ℓ may really vanish for nonzero j i 's which entails that the bound with expectation has to be replaced by upper bounds.

Remark 2.2 In the previous lemma, the independence of the ξ's does not play a role. We may have stated it for arbitrary random fields {ξ t } such that ξ t ∞ M for each t ∈ Z Z D ; such models with dependent inputs are interesting but assumptions on the innovations are indeed very strong. This means that such models are heteroscedastic but with bounded innovations: according to [START_REF] Mandelbrot | The Misbehavior of Markets: A Fractal View of Risk, Ruin, and Reward[END_REF], this restriction excludes extreme phenomena like crashes and bubbles. Mandelbrot school has shown from the seminal paper [START_REF] Mandelbrot | The variation of certain speculative prices[END_REF] that asset prices returns do not have a Gaussian distribution as the number of extreme deviations, the so-called "Noah effects", of asset returns is far greater than what is allowed by the Normal distribution, even with ARCH-type effects. It is the reason why this extension is not pursued in the present paper.

Approximations

This section is aimed to approximate a sequence {X t } given by (3), solution to eqn. (1) by a sequence { Xt }. We shall prove that we can control the approximation error E X t -Xt within reasonable small bounds.

Approximation by Independence

The purpose is to approximate X t by a random variable independent of X 0 . We set

Xt = ξ t   a + ∞ k=1 j1+•••+j k <t a j1 ξ t-j1 • • • a j k ξ t-j1-•••-j k a   .
Proposition 3.1 Define ϕ from (2). A bound for the error is given by:

E X t -Xt E ξ 0 E ξ 0 t-1 k=1 kϕ k-1 A t k + ϕ t 1 -ϕ a .
Furthermore, we have as particular results that if b, C > 0 and q ∈ [0, 1), then for a suitable choice of constants K, K ′ :

E X t -Xt K (log(t)) b∨1 t b
, for Riemannian decays A(x) Cx -b , K ′ (q ∨ ϕ) √ t , for geometric decays A(x) Cq x . Remark 3.1 Note that in the first case this decay is essentially the same Riemannian one while it is sub-geometric (like t → e -c √ t ) when the decay of the coefficients is geometric. Remark 3.2 In the paper Riemannian or Geometric decays always refer to the previous relations.

Idea of the Proof. A careful study of the terms in X t 's expansion which do not appear in Xt entails the following bound with the triangular inequality. For this, quote that if j 1 + • • • + j k t for some k 1 then, at least, one of the indices j 1 , . . . , or j k is larger than t/k. The additional term corresponds to those terms with indices k > t in the expansion [START_REF] Boussama | Normalité asymptotique de l'estimateur du pseudo maximum de vraisemblance d'un modèle GARCH[END_REF].

The following extension to the case of the random fields determined in lemma 2.1 is immediate by setting

Xt = ξ t   a + ∞ k=1 j 1 , . . . , j k = 0 j 1 + • • • + j k < t a j1 ξ t-j1 • • • a j k ξ t-j1-•••-j k a    .
Proposition 3.2 The random field (X t ) t∈Z Z D defined in lemma 2.1 satisfies:

E X t -Xt E ξ 0   ξ 0 ∞ 1 k< t kϕ k-1 A t k + ϕ t 1 -ϕ   a .

Coupling

First note that the variable Xt which approximates X t does not follow the same distribution. For dealing with this issue, it is sufficient to construct a sequence of iid random variables ξ ′ i which follow the same distribution as the one of the ξ i , each term of the sequence being independent of all the ξ i . We then set

ξ * t = ξ t if t > 0 ξ ′ t if t 0
, and

X * t = ξ t   a + ∞ k=1 j1,...,j k a j1 ξ * t-j1 • • • a j k ξ * t-j1-•••-j k a   .
Coefficients τ t for the τ -dependence introduced by Dedecker and Prieur [START_REF] Dedecker | Coupling for τ -dependent sequences and applications[END_REF] are easily computed. In this case, we find the upper bounds from above, up to a factor 2:

τ t = E X t -X * t 2E ξ 0 E ξ 0 t-1 k=1 kϕ k-1 A t k + ϕ t 1 -ϕ a ;
see also Rüschendorf [START_REF] Rüschendorf | The Wasserstein distance and approximation theorems[END_REF], Prieur [START_REF] Prieur | Recent results on weak dependence[END_REF]. These coefficients τ k are defined as τ k = τ (σ(X i , i 0), X k ) where for each random variable X and each σalgebra M one sets

τ (M, X) = E sup Lip f 1 f (x)IP X|M (dx) -f (x)IP X (dx)
where IP X and IP X|M denotes the distribution and the conditional distribution of X on the σ-algebra M and Lip f = sup x =y |f (x) -f (y)|/ x -y .

Markovian Approximation

We consider equation ( 1) truncated at the order N :

Y t = ξ t (a + N j=1 a j Y t-j
). The solution considered above can be rewritten as

X N t = ξ t   a + ∞ k=1 N j1,...,j k 1 a j1 ξ t-j1 • • • a j k ξ t-j1-•••-j k a   .
We can easily find an upper bound of the error:

E X t -X N t ∞ k=1 A(N ) k .
As in proposition 3.1, in the Riemannian case, this bound of the error writes as C ∞ k=1 N -bk C/(N b -1) with b > 1, while in the geometric case, this writes as Cq N /(1 -q N ) Cq N /(1 -q), 0 < q < 1.

Weak Dependence

Consider integers u, v 1. Let i 1 < • • • < i u , j 1 < • • • < j v be
integers with j 1 -i u r, we set U and V for the two random vectors U = (X i1 , X i2 , . . . , X iu ) and V = (X j1 , X j2 , . . . , X jv ). We fix a norm • on R d . For a function

h : R d w → R we set Lip(h) = sup x1,y1,...,xw,yw∈R d |h(x 1 , . . . , x w ) -h(y 1 , . . . , y w )| w i=1 x i -y i .
Theorem 4.1 Assume that the coefficient defined by (2) satisfies ϕ < 1. The solution (3) to the equation ( 1) is θ-weakly dependent, see [START_REF] Dedecker | A new covariance inequality and applications[END_REF]. This means that:

|Cov(f (U ), g(V ))| 2v f ∞ Lip(g)θ r , for any integers u, v 1, i 1 < • • • < i u , j 1 < • • • < j v such that j 1 -i u r; with θ r = E ξ 0 E ξ 0 r-1 k=1 kϕ k-1 A r k + ϕ r 1 -ϕ a .
Proof. For calculating a weak dependence bound, we approximate the vector V by the vector V = ( Xj1 , Xj2 , . . . , Xjv ), where we set

Xt = ξ t   a + ∞ k=1 j1+•••+j k <s a j1 ξ t-j1 • • • a j k ξ t-j1-•••-j k a   .
Note that for each index j ∈ Z Z, Xj is independent of (X j-s ) s r . Note that for 1 k v, E X j k -Xj k θ r defined in theorem 4.1. Then

|Cov(f (U ), g(V ))| E f (U )(g(V ) -g( V ) -E(f (U ))E(g(V ) -g( V )) 2 f ∞ E g(V ) -g( V ) 2 f ∞ Lip(g) v k=1 E X j k -Xj k 2v f ∞ Lip(g)θ r .
Remark 4.1 We obtain explicit expressions for this bound in proposition 3.1 for the Riemannian and geometric decay rates.

Remark 4.2 In the case of random fields the η-weak dependence condition in [START_REF] Doukhan | A new weak dependence condition and applications to moment inequalities[END_REF] or [START_REF] Doukhan | Rates of convergence in the weak invariance principle for the empirical repartition process of weakly dependent sequences[END_REF] holds in a similar way with

η r = 2E ξ 0   ξ 0 ∞ k<r/2 kϕ k-1 A r k + ϕ [r/2] 1 -ϕ   a ,
which means that the previous bound now writes as

|Cov(f (U ), g(V ))| u g ∞ Lip(f ) + v f ∞ Lip(g) η r .
The argument is the same except for the fact that now Û and V are independent vectors with truncations at a level s = [r/2] but V and U are not necessarily independent (recall that independence of U and V follows from s r in the proof for the causal case). This point makes the previous bound a bit more complicated than the one in theorem 4.1 and it explains the appearance of the factor 2 in the expression of η r .

Remark 4.3 Those weak dependence conditions imply various limit theorems both for partial sums processes and for the empirical process (see [START_REF] Doukhan | A new weak dependence condition and applications to moment inequalities[END_REF], [START_REF] Dedecker | A new covariance inequality and applications[END_REF] and [START_REF] Doukhan | Rates of convergence in the weak invariance principle for the empirical repartition process of weakly dependent sequences[END_REF]).

L 2 Properties

For the univariate case, the uniqueness of a stationary solution to equation (1) has been demonstrated by Giraitis et al. [START_REF] Giraitis | Stationary ARCH models: dependence structure and central limit theorems[END_REF]. We first present an existence and uniqueness condition for the model in L 2 . The situation is then no longer necessarily weakly dependent.

Theorem 5.1 Assume that the iid sequence {ξ t } satisfies E(ξ k ) = 0.

Assume that the matrix S = ∞ k=1 a ′ k E(ξ ′ k ξ k )a k has a spectral radius which satisfies ρ(S) < 1.
Then there exists a stationary solution in L 2 to equation (1) given by (3). Moreover the solution in L 2 to equation (1) is unique.

Remark 5.1 • The assumption ρ(S) < 1 implies ξ t ∈ L 2 for t ∈ Z Z.

• In [START_REF] Giraitis | ARCH-type bilinear models with double long memory[END_REF], the example 2 of the bilinear model displays the double long memory property when the corresponding series α j and β j are not summable but

∞ j=1 α 2 j Eζ 2 0 + β 2 j < 1.
As a particular case, the squares of the LARCH(∞) process, example 1, display long-range dependence as well. Those authors prove that the corresponding partial sums process converges to the fractional Brownian Motion, appropriately normalized (with normalization ≫ √ n). • Models GARCH(p, q), in example 3, are always weakly dependent, in the sense of [START_REF] Doukhan | A new weak dependence condition and applications to moment inequalities[END_REF]. • Note that [START_REF] Giraitis | Stationary ARCH models: dependence structure and central limit theorems[END_REF] and [START_REF] Giraitis | ARCH-type bilinear models with double long memory[END_REF] prove that the stationary ARCH(∞) model, described as example 4 in section 1, is not long range dependent in the previous sense; more precisely the sequence of partial sums processes, normalized with √ n, converges to the Brownian Motion.

Proof.

Step 1: existence. Define T = E(ξ ′ k ξ k ). Considering the chaotic solution (3) and setting

C t (k 2 , . . . , k ℓ ) = ξ t a k2 ξ t-k2 • • • a k ℓ ξ t-k2•••-k ℓ a we write E(X ′ t X t ) = a ′ Eξ ′ t ξ t a + B = a ′ T
E(ξ 0 a) ′ (ξ 0 a) ∞ ℓ=1 ρ(S) ℓ (recursively) a ′ aρ(T ) ∞ ℓ=1 ρ(S) ℓ , hence, E(X ′ t X t ) a ′ T a + a ′ a ρ(T ) 1 -ρ(S) < ∞ (6)
In the previous relations we both use the fact that the ξ t are centered and iid and the relation v ′ Av v ′ vρ(A) which holds if A denotes a non-negative d× d matrix and v ∈ IR d . This conclude the proof of the existence of a solution in L 2 .

Step 2: L 2 uniqueness. Let now X 1 t and X 2 t be two solutions to equation (1) in L 2 . Define Xt = X 1 t -X 2 t , then Xt is solution to

Xt = ξ t Ãt , Ãt = ∞ k=1 a k Xt-k . (7)
Now we use [START_REF] Doukhan | Rates of convergence in the weak invariance principle for the empirical repartition process of weakly dependent sequences[END_REF] and the fact that Xt is centered and thus E Xs Xt = 0 for s = t to derive

E ( Xt g) ′ ( Xt g) = ∞ k=1 g ′ E X′ t-k a ′ t-k T a t-k Xt-k g = ∞ k=1 g ′ E X′ t a ′ t-k T a t-k Xt g = g ′ E X′ t S Xt g = E ( Xt g) ′ S( Xt g) ρ(S)E ( Xt g) ′ ( Xt g)
From equation ( 6), this expression is finite and thus the assumption ρ(S) < 1 concludes the proof.

Remark 5.2 The proof does not extend to the case of random fields because in this case the previous arguments of independence cannot be used. In that case we cannot address the question of uniqueness.

The previous L 2 existence and uniqueness assumptions do not imply that j 1 a j < ∞, thus this situation is perhaps not a weakly dependent one. Giraitis and Surgailis [START_REF] Giraitis | ARCH-type bilinear models with double long memory[END_REF], prove results both for the partial sums processes of X t and X 2 t -EX 2 t . In our vector case the second problem is difficult and will be addressed in a forthcoming work. However X t is now the increment of a (vector valued-)martingale and thus we partially extend Theorem 6.2 in [START_REF] Giraitis | ARCH-type bilinear models with double long memory[END_REF], providing a version of Donsker's theorem for partial sums processes of {X t }. Proposition 5.1 Assume that the assumptions in theorem 5.1 hold. Then S n (t)/ VarS n (t) converges to ΣW (t), if S n (t) = 1 i nt X i for 0 t 1 and where W (t) is a IR d valued Brownian motion and Σ is a symmetric non negative matrix such that Σ 2 is the covariance matrix of X 0 . The convergence holds for finite dimensional distributions.

Remark 5.3 • The convergence only holds for any k-tuples (t 1 , . . . , t k ) ∈ [0, 1] k and since the section is related to L 2 properties we cannot use the tightness arguments in [START_REF] Giraitis | ARCH-type bilinear models with double long memory[END_REF] to obtain the Donsker theorem; indeed tightness is obtained through moment inequalities of order p > 2. L p existence conditions are obtained in [START_REF] Giraitis | ARCH-type bilinear models with double long memory[END_REF] for the bilinear case if p = 4; the method is based on the diagram formula and does not extend simply to this vector valued case. A bound for the moments of order p > 2 of the partial sum process S n (t) can be obtained using Rosenthal inequality, Theorem 2.11 in [START_REF] Hall | Martingale Limit Theory and Its Applications[END_REF], if E X t p < ∞. This inequality would imply the functional convergence in the Skohorod space D[0, 1] if p > 2.

• If Eξ 0 = 0 (as for the case of the bilinear model in [START_REF] Giraitis | ARCH-type bilinear models with double long memory[END_REF]), we may write

X t = ∆M t + Eξ 0 a + ∞ j=1 a j X t-j where ∆M t = (ξ t -Eξ t )   a + ∞ j=1 a j X t-j  
is a martingale increment. This martingale also obeys a central limit theorem. then, n -1/2 S n (t) → ΣW (t),

where W (t) is a vector Brownian motion, where Σ′ Σ = Σ. If Eξ 0 = 0 this is a way to prove proposition 5.1, which is a multi-dimensional extension of the proof in [START_REF] Giraitis | ARCH-type bilinear models with double long memory[END_REF].

For the case of the bilinear model, Giraitis and Surgailis also prove the (functional) convergence of the previous sequence of process to a Fractional Brownian Motion in [START_REF] Giraitis | ARCH-type bilinear models with double long memory[END_REF]. For this, Riemannian decays of the coefficients are assumed. The covariance function of the process is also completely determined to prove such results; this is a quite difficult point to extend to our vector valued frame. • A final comment concerns the analogue for powers of X t which, if suitably normalized, are proved to converge to some higher order Rosenblatt process in [START_REF] Giraitis | ARCH-type bilinear models with double long memory[END_REF] for the bilinear case. We have a structural difficulty to extend it; the only case which may reasonably be addressed is the real valued one (d = 1), but it also presents very heavy combinatorial difficulties. Computations for the covariances of the processes (X k t ) t∈Z Z will be addressed in a forthcoming work in order to extend those results.

  EC ′ t-k1 (k 2 , . . . , k ℓ )a ′ k1 T a k1 C t-k1 (k 2 , . . . , k ℓ ) EC ′ t-k1 (k 2 , . . . , k ℓ )a ′ k1 Eξ ′ t-k1 ξ t-k1 a k1 C t-k1 (k 2 , . . . , k ℓ ) , . . . , k ℓ )C t (k 2 , . . . , k ℓ )

	B =
	ℓ,k1,...,k ℓ 1
	=
	ℓ,k1,...,k ℓ
	=	EC ′ t-k1 (k 2 , . . . , k ℓ ) Ea ′ k1 ξ ′ t-k1 ξ t-k1 a k1 C t-k1 (k 2 , . . . , k ℓ )
	ℓ,k1,...,k ℓ
	=	EC ′ t (k 2 , . . . , k ℓ ) Ea ′ k1 ξ ′ t-k1 ξ t-k1 a k1 C t (k 2 , . . . , k ℓ )
	ℓ,k1,...,k ℓ
	=	EC ′ t (k 2 , . . . , k ℓ )SC t (k 2 , . . . , k ℓ )
	ℓ,k2,...,k ℓ
	ρ(S)	EC ′ t (k 2
	ℓ,k2,...,k ℓ
		a + B where

Acknowledgements. The authors are grateful to the referees for their valuable comments.


Philippe Laurenc ¸ot

Christoph Walker
email: walker@ifam.uni-hannover.de

FINITE TIME SINGULARITY IN A MEMS MODEL REVISITED

Keywords: Mathematics Subject Classification, 35M30 -35R35 -35B44 -35Q74 MEMS -free boundary problem -finite time singularity, Partially

A free boundary problem modeling a microelectromechanical system (MEMS) consisting of a fixed ground plate and a deformable top plate is considered, the plates being held at different electrostatic potentials. It couples a second order semilinear parabolic equation for the deformation of the top plate to a Laplace equation for the electrostatic potential in the device. The validity of the model is expected to break down in finite time when the applied voltage exceeds a certain value, a finite time singularity occurring then. This result, already known for non-positive initial configurations of the top plate, is here proved for arbitrary ones and thus now includes, in particular, snap-through instabilities.

Introduction

An important feature of microelectromechanical (MEMS) devices is the so-called pull-in instability which occurs in principle when the potential difference applied across the device exceeds a certain threshold value [START_REF] Jazar | Nonlinear mathematical modeling of microbeam MEMS[END_REF][START_REF] Pelesko | Modeling MEMS and NEMS[END_REF]. For an idealized electrostatic MEMS actuator consisting of an elastic plate coated with a thin dielectric film held at potential one (after normalization) and suspended above a rigid conducting ground plate held at potential zero, this phenomenon corresponds to the touchdown of the elastic plate on the ground plate. It is of utmost importance to figure out whether and when it does take place. Indeed, though pull-in might be a sought-for behavior (in switching applications for instance) or not (for micro-mirrors, in particular), its possible occurrence anyway has a strong influence on the operating conditions of the MEMS device. From a mathematical viewpoint, touchdown takes place when the vertical deflection of the elastic plate goes beyond a certain value at some time T * . More precisely, we assume that the ground and elastic plates have the same shape D, which is a smooth bounded domain of R n , n = 1, 2. The ground plate is assumed to be located at height z = -1 (so that it corresponds to the surface D × {-1} in R n+1 ) while the elastic plate at time t ≥ 0 is the surface

{(x, z) ∈ D × R ; z = u(t, x)} ⊂ R n+1
, where u(t, x) denotes the deflection of the elastic plate in the vertical direction at position x ∈ D and time t. The touchdown phenomenon then takes place at time

T * > 0 if lim t→T * min x∈ D {u(t, x)} = -1 . (1.1)
It is actually rather well-understood for the so-called vanishing aspect ratio model

∂ t u -∆u = - λ (1 + u) 2 , t > 0 , x ∈ D , u = 0 , t > 0 , x ∈ ∂D , u(0) = u 0 , x ∈ D , (1.2)
where the parameter λ is proportional to the square of the potential difference applied across the device before rescaling, and the initial condition satisfies u 0 > -1 in D. A threshold value λ s of the parameter λ is found for which there is no stationary solution to (1.2) when λ exceeds λ s while there is at least one stable stationary solution for λ ∈ (0, λ s ). Similarly, for the evolution equation (1.2), there is a threshold value λ e (u 0 ) > 0 depending on the initial condition u 0 with the following properties: if λ ∈ (0, λ e (u 0 )), then there is a unique classical solution to (1.2) which exists for all times and is well-separated from -1 on each finite time interval. On the contrary, if λ > λ e (u 0 ), then the unique classical solution to (1.2) exists only on a finite time interval [0, T * ) and touchdown occurs at time T * as described in (1.1). Several results are also available for variants of (1.2) where ∂ t u is replaced by ∂ 2 t u and/or ∆u is replaced by ∆ 2 u with either clamped (u = ∂ ν u = 0 on D) or pinned (u = ∆u = 0 on D) boundary conditions. The results, though, are less complete and several gray areas persist. We refer to [3-5, 8, 10, 11, 13, 15] and the references therein for a more complete description of the available results.

Far less is known for a more complex and more precise model for MEMS devices, which describes not only the dynamics of the deflection in the vertical direction u, but also that of the electrostatic potential ψ u between the two plates. When D := (-1, 1), it reads

∂ t u -∂ 2 x u = -λ g(u) , x ∈ D , t > 0 , (1.3a)
with clamped boundary conditions

u(t, ±1) = 0 , t > 0 , (1.3b)
and initial condition

u(0, x) = u 0 (x) , x ∈ D , (1.3c)
the electrostatic force g(u) being given by

g(u(t))(x) := ε 2 |∂ x ψ u (t, x, u(t, x))| 2 + |∂ z ψ u (t, x, u(t, x))| 2 , t > 0 , x ∈ D . (1.3d)
The dimensionless electrostatic potential

ψ u = ψ u (t, x, z) is defined in the region Ω(u(t)) := {(x, z) ∈ D × (-1, ∞) : -1 < z < u(t, x)}
between the ground plate and the elastic plate and satisfies a rescaled Laplace's equation

ε 2 ∂ 2 x ψ u + ∂ 2 z ψ u = 0 , (x, z) ∈ Ω(u(t)) , t > 0 , (1.4a)
the parameter ε > 0 being the so-called aspect ratio of the device, that is, the ratio between the vertical and horizontal directions. Laplace's equation is supplemented with non-homogeneous boundary

conditions ψ u (t, x, z) = 1 + z 1 + u(t, x) , (x, z) ∈ ∂Ω(u(t)) , t > 0 . (1.4b)
In particular, ψ u (t, x, u(t, x)) = 1 and ψ u (t, x, -1) = 0 for x ∈ D as required. The equation (1.2) for n = 1 is formally derived from (1.3), (1.4) by setting ε = 0 since then the solution to (1.4) is given explicitly by (1.4b) for all x ∈ Ω(u(t)) and t > 0. In contrast to (1.2), the initial-boundary value problem (1.3), (1.4) features a rather intricate coupling between the two unknowns u and ψ u . Indeed, the source term in (1.3a) governing the evolution of u is proportional to the square of the trace of the gradient of ψ u on the elastic plate. The electrostatic potential, in turn, solves an elliptic equation which involves a non-smooth domain varying with u. Thus, the source term in (1.3a) depends in a nonlocal and nonlinear way on u.

Still, it is shown in [2, Theorem 1] that (1.3), (1.4) is locally well-posed in S 2 q (D) := v ∈ W 2 q (D) ; v(±1) = 0 and -1 < v(x) for x ∈ D , when q > 2.
It was further reported in [2, Theorem 2] that, under the additional assumption that u 0 is non-positive, the corresponding solution to (1.3), (1.4) cannot exist globally if λ exceeds some value λ e (u 0 , ε) depending on u 0 and ε: indeed, for λ > λ e (u 0 , ε) there is T m ∈ (0, ∞) depending on λ, u 0 , and ε such that

lim t→Tm min x∈[-1,1] {u(t, x)} = -1 or lim t→Tm u(t) W 2 q = ∞ . (1.5)
According to the discussion above, only the first singularity in (1.5) corresponds to the pull-in instability (recall (1.1)), and this is the sole expected to occur. Though we have been unable to rule out the blowup of the Sobolev norm up to now, we point out that the non-positivity of the initial condition is important in the proof of (1.5) as it entails the upper bound u(t) ≤ 0 as long as it exists, thanks to the non-positivity of the right-hand side of (1.3a) and the comparison principle. The purpose of this note is to prove that the non-positivity assumption on u 0 can be relaxed and improved to the weaker assumption that u 0 is simply bounded from above. It is worth mentioning that relaxing the sign condition on u 0 is not just a mere mathematical improvement, but is physically relevant, for instance, in the study of the so-called snap-through instability, where the shape of the plate is initially an arch (such as u 0 (x) = h(1 + cos(πx)) for x ∈ D) or a bell [START_REF] Das | Pull-in and snap-through instabilities in transient deformations of microelectromechanical systems[END_REF][START_REF] Krylov | The pull-in behavior of electrostatically actuated bistable microstructures[END_REF][START_REF] Ouakad | The dynamic behavior of MEMS arch resonators actuated electrically[END_REF]. Also, nonpositivity plays a crucial role in the proof of the occurrence of a finite time singularity in a related MEMS model [START_REF] Ch | On qualitative properties of solutions to microelectromechanical systems with general permittivity[END_REF]. More precisely, we prove the following result.

Theorem 1.1. Let ε > 0 and q > 2 and consider u 0 ∈ S 2 q (D). There is λ * (u 0 , ε) > 0 depending on ε and u 0 such that, if λ > λ * , then the maximal existence time T m of the corresponding solution (u, ψ u ) to (1.3), (1.4) is finite and

lim t→Tm min x∈[-1,1] {u(t, x)} = -1 or lim t→Tm u(t) W 2 q = ∞ .
A well-known technique to prove the occurrence of a finite time singularity in evolution equations is the so-called eigenfunction method [START_REF] Kaplan | On the growth of solutions of quasi-linear parabolic equations[END_REF]Theorem 8]. Owing to the nonlocality of the right-hand side of (1.3a), a direct application of this technique to (1.3), (1.4) seems to fail. Nevertheless, we developed in [START_REF] Escher | A parabolic free boundary problem modeling electrostatic MEMS[END_REF] a nonlinear version of this technique which allowed us therein to prove Theorem 1.1 under the additional assumption that u 0 is non-positive, the latter being used in particular to control the supplementary nonlinear terms. Not surprisingly, the proof of Theorem 1.1 herein relies on the same technique and borrows some steps of the proof of [2, Theorem 2]. Besides some technical variations, the main new step in the proof is to use the non-positivity of the right hand side of (1.3a) and the decay properties of the linear heat equation to control in a suitable way the contribution of the positive part of u, see Lemma 2.1 below.

Finite time singularity

Let ε > 0, λ > 0, q > 2, and consider u 0 ∈ S 2 q (D). Denoting the corresponding solution to (1.3), (1.4) by (u, ψ u ) and its maximal existence time by T m , we recall from [2, Proposition 5] that

u(t) ∈ S 2 q (D) , t ∈ [0, T m ) , (2.1)
and ψ u (t) ∈ H 2 (Ω(u(t)) for all t ∈ [0, T m ). Consequently,

γ m (t, x) := ∂ z ψ u (t, x, u(t, x)) , (t, x) ∈ [0, T m ) × D ,
is well-defined and, since

∂ x ψ u (t, x, u(t, x)) = -∂ x u(t, x)γ m (t, x) , (t, x) ∈ [0, T m ) × D ,
by (1.4b), the right-hand side of (1.3a) also reads

-λg(u) = -λ(1 + ε 2 |∂ x u| 2 )γ 2 m .
Introducing the principal eigenvalue µ 1 := π 2 /4 of the Laplace operator -∂ 2

x in L 2 (D) with homogeneous Dirichlet boundary conditions and the corresponding eigenfunction ζ 1 (x) := π cos(πx/2)/4, x ∈ D, we define

E α (t) := D ζ 1 (x) u(t, x) + α 2 u(t, x) 2 dx , t ∈ [0, T m ) ,
where α ∈ [0, 1] is to be determined later on. Observe that (2.1) and the properties of

ζ 1 guarantee that E α (t) > -1 , t ∈ [0, T m ) , (2.2)
for all α ∈ [0, 1]. We shall show that for any sufficiently large value of λ we can choose α ∈ [0, 1] such that (2.2) can only hold true when T m < ∞. This then implies Theorem 1.1.

The starting point of the analysis is the following upper bound on E α .

Lemma 2.1. Let α ∈ [0, 1] with α ≤ 2/(1 + (max u 0 ) + ).
There is a positive constant C 0 depending only on max u 0 such that

E α (t) ≤ C 0 e -µ 1 t , t ∈ [0, T m ) . (2.3)
Proof. Let v and w be the solutions to

∂ t v -∂ 2 x v = 0 , (t, x) ∈ (0, T m ) × D , v(t, ±1) = 0 , t ∈ (0, T m ) , (2.4) ∂ t w -∂ 2 x w = -λg(u) , (t, x) ∈ (0, T m ) × D , w(t, ±1) = 0 , t ∈ (0, T m ) , (2.5)
with initial conditions v(0, x) = max{u 0 (x), 0} , w(0, x) = u 0 -max{u 0 (x), 0} , x ∈ D .

(2.6)

Since w(0, x) ≤ 0 ≤ v(0, x) ≤ (max u 0 ) + for x ∈ D, we infer from (2.4), (2.5), (2.6), and the comparison principle that

w(t, x) ≤ 0 ≤ v(t, x) ≤ (max u 0 ) + , (t, x) ∈ (0, T m ) × D , and
v(t) 2 ≤ e -µ 1 t v(0) 2 ≤ √ 2(max u 0 ) + e -µ 1 t , t ∈ (0, T m ) . Also, w(t, x) = u(t, x) -v(t, x) ≥ -1 -(max u 0 ) + , (t, x) ∈ (0, T m ) × D , so that 1 + α 2 w(t, x) ≥ 1 - α 2 1 + (max u 0 ) + ≥ 0 , (t, x) ∈ (0, T m ) × D ,
thanks to the constraint on α. Therefore, for t ∈ (0, T m ),

E α (t) = D ζ 1 v(t) + α 2 v(t) 2 + αv(t)w(t) + w(t) 1 + α 2 w(t) dx ≤ D ζ 1 v(t) + α 2 v(t) 2 dx ≤ π 4 √ 2 v(t) 2 + v(t) 2 2 2 ≤ π (max u 0 ) + + (max u 0 ) 2 + e -µ 1 t
, and the proof is complete.

We next derive a differential inequality for E α . Though the proof is quite similar to that performed in [START_REF] Escher | A parabolic free boundary problem modeling electrostatic MEMS[END_REF], we nevertheless recall it here not only for the sake of completeness but also to shed some light on the importance of choosing α > 0.

Lemma 2.2. Let p ≥ 1 and δ > 0. If

α := λε 2 λε 2 + 4δ 2 ∈ (0, 1) , (2.7)
then d dt E α ≤ F p,δ (E α ) , t ∈ [0, T m ) , (2.8)
where

F p,δ (y) := µ 1 + 4δλ p(λε 2 + 4δ 2 ) µ 1 ε 2 p + p 4δ + pµ 1 ε 2 p + 1 y - 1 1 + y , y > -1 .
(2.9)

Proof. It readily follows from (1.3a) that, for t ∈ [0, T m ), d dt E α = D ζ 1 (1 + αu)(∂ 2 x u -λg(u)) dx = -µ 1 E α -α D ζ 1 |∂ x u| 2 dx -λ D ζ 1 (1 + αu)g(u) dx . Since ζ 1 ≥ 0 and 1 + αu ≥ 1 -α in D, we conclude that d dt E α + µ 1 E α + α D ζ 1 |∂ x u| 2 dx ≤ -λ(1 -α) D ζ 1 g(u) dx , t ∈ [0, T m ) . (2.10)
The next step is to estimate the last term on the right hand side of (2.10) in terms of E α . To this end, we recall the following results established in [2, Lemma 9 & Lemma 10] which do not rely on the sign of u 0 . For t ∈ [0, T m ) and p ∈ [1, ∞), we have

4p (p + 1) 2 D ζ 1 (x) 1 + u(t, x) dx ≤ p Ω(u(t)) ζ 1 (x) ψ u (t, x, z) p-1 |∂ z ψ u(t) (t, x, z)| 2 d(x, z) (2.11) and D ζ 1 (x) 1 + ε 2 |∂ x u(t, x)| 2 γ m (t, x) dx = p Ω(u(t)) ζ 1 (x) ψ u (t, x, z) p-1 ε 2 |∂ x ψ u (t, x, z)| 2 + |∂ z ψ u (t, x, z)| 2 d(x, z) + µ 1 ε 2 p + 1 Ω(u(t)) ζ 1 (x)ψ u (t, x, z) p+1 d(x, z) - µ 1 ε 2 (p + 1)(p + 2) - µ 1 ε 2 p + 1 D ζ 1 (x) u(t, x) dx .
(2.12)

Since ψ u ≥ 0 by the comparison principle, it follows from (2.11), (2.12), and the non-negativity of ζ 1 that, for t ∈ [0, T m ),

D ζ 1 (1 + ε 2 |∂ x u| 2 )γ m dx ≥ p Ω(u(t)) ζ 1 ψ p-1 u |∂ z ψ u | 2 d(x, z) - µ 1 ε 2 (p + 1)(p + 2) - µ 1 ε 2 p + 1 D ζ 1 u dx ≥ 4p (p + 1) 2 D ζ 1 1 + u dx - µ 1 ε 2 p 2 - µ 1 ε 2 p + 1 E α ≥ 4p (p + 1) 2 D ζ 1 1 + u + αu 2 /2 dx - µ 1 ε 2 p 2 - µ 1 ε 2 p + 1 E α ≥ 1 p D ζ 1 1 + u + αu 2 /2 dx - µ 1 ε 2 p 2 - µ 1 ε 2 p + 1 E α .
Using Jensen's inequality we end up with

D ζ 1 (1 + ε 2 |∂ x u| 2 )γ m dx ≥ 1 p 1 1 + E α - µ 1 ε 2 p 2 - µ 1 ε 2 p + 1 E α . (2.13)
We now deduce from Young's inequality that, for any δ > 0,

D ζ 1 (1 + ε 2 |∂ x u| 2 )γ m dx ≤ δ D ζ 1 (1 + ε 2 |∂ x u| 2 )γ 2 m dx + 1 4δ D ζ 1 (1 + ε 2 |∂ x u| 2 ) dx .
Therefore, by (2.13),

D ζ 1 g(u) dx = D ζ 1 (1 + ε 2 |∂ x u| 2 )γ 2 m dx ≥ 1 δ D ζ 1 (1 + ε 2 |∂ x u| 2 )γ m dx - 1 4δ 2 D ζ 1 (1 + ε 2 |∂ x u| 2 ) dx ≥ 1 δp 1 1 + E α - µ 1 ε 2 δp 2 - µ 1 ε 2 δ(p + 1) E α - 1 4δ 2 - ε 2 4δ 2 D ζ 1 |∂ x u| 2 dx .
Combining the above inequality with (2.10) gives

d dt E α + µ 1 E α + α D ζ 1 |∂ x u| 2 dx ≤ - λ(1 -α) δp 1 1 + E α - µ 1 ε 2 p - p 4δ - pµ 1 ε 2 p + 1 E α + λ(1 -α)ε 2 4δ 2 D ζ 1 |∂ x u| 2 dx .
Owing to the choice (2.7) of α, we see that

α = λ(1 -α)ε 2 4δ 2 ,
so that the terms involving |∂ x u| 2 cancel in the previous inequality, and we end up with

d dt E α ≤ -µ 1 E α + 4δλ p(λε 2 + 4δ 2 ) µ 1 ε 2 p + p 4δ + pµ 1 ε 2 p + 1 E α - 1 1 + E α , hence (2.8) as E α > -1 by (2.2).
Proof of Theorem 1.1. Let p ≥ 1 and δ > 0. We investigate the properties of the function F p,δ defined in (2.9) and aim at finding values of the yet undetermined parameters p ≥ 1 and δ > 0 which are suitable for our purpose. To begin with, we set α = λε 2 λε 2 + 4δ 2 as in (2.7) and require that λε

2 λε 2 + 4δ 2 ≤ 2 1 + (max u 0 ) + , that is, δ ≥ χ ε √ λ 2 , χ ε := ε ((max u 0 ) + -1) + 2 . (2.14)
Then, according to Lemma 2.1, the positive part of E α (t) decays rapidly as time increases, so that only the behavior of F p,δ in a neighborhood of the interval (-1, 0] is expected to matter in the following. We first notice that F p,δ is increasing on (-1, ∞) with F p,δ (0) = µ 1 + 4δλ p(λε 2 + 4δ 2 )

µ 1 ε 2 p + p 4δ -1 .
We now choose p = 1 + 2µ 1 ε 2 and δ = χ √ λ/2 with χ := max{1, χ ε }, the parameter χ ε being defined in (2.14). Then

F p,δ (0) = µ 1 + 2χ √ λ (1 + 2µ 1 ε 2 )(χ 2 + ε 2 ) µ 1 ε 2 1 + 2µ 1 ε 2 + 1 + 2µ 1 ε 2 2χ √ λ -1 ≤ µ 1 + 2χ √ λ (1 + 2µ 1 ε 2 )(χ 2 + ε 2 ) 1 + 2µ 1 ε 2 2χ √ λ - 1 2 ≤ µ 1 + χ (1 + 2µ 1 ε 2 )(χ 2 + ε 2 ) 1 + 2µ 1 ε 2 χ - √ λ ≤ χ (1 + 2µ 1 ε 2 )(χ 2 + ε 2 ) 1 + 2µ 1 ε 2 χ 1 + µ 1 (χ 2 + ε 2 ) - √ λ . Consequently, if √ λ > 1 + 2µ 1 ε 2 χ 1 + µ 1 (χ 2 + ε 2 ) , (2.15)
then F p,δ (0) < 0 and there is y p,δ > 0 such that F p,δ (y p,δ ) < 0 .

(2.16)

Now, assume for contradiction that T m = ∞. According to Lemma 2.1, there is t p,δ > 0 such that E α (t) ≤ y p,δ for t ≥ t p,δ and the monotonicity of F p,δ further entails that F p,δ (E α (t)) ≤ F p,δ (y p,δ ) for t ≥ t p,δ . We then infer from Lemma 2.2 that d dt E α (t) ≤ F p,δ (y p,δ ) < 0 , t ≥ t p,δ , hence, after integration, E α (t) ≤ E α (t p,δ ) + F p,δ (y p,δ )(t -t p,δ ) for t ≥ t p,δ . Thus, there is T > t p,δ such that E α (T ) < -1, which contradicts (2.2). Therefore, T m is finite and the proof of Theorem 1.1 is complete.

Acknowledgments

Part of this work was done while PhL enjoyed the hospitality of the Institut für Angewandte Mathematik, Leibniz Universität Hannover.