Basile De Loynes
email: basile.deloynes@ensai.fr

Local Limit Theorem for Markov Additive Processes on Z d with Null Recurrent Internal Markov Chain

Keywords: Inhomogeneous random walks, Markov Additive processes, Semi-Markov processes, Markov Random Walks, Random Walks with Internal Degrees of Freedom, Local Limit Theorem, Recurrence, Transience, Fourier analysis

In the classical framework, a random walk on a group is a Markov chain with independent and identically distributed increments. In some sense, random walks are time and space homogeneous. This paper is devoted to a class of inhomogeneous random walks on Z d termed Markov Additive processes (also known as Markov Random Walks, Random Walks with Internal Degree of Freedom or semi-Markov Processes). In this model, the increments of the walk are still independent but their distributions are dictated by a Markov chain termed the internal Markov chain. Whereas this model is largely studied in the literature, most of the results involve internal Markov chains whose operator is quasi-compact. This paper extends two results for more general internal operators: a Local Limit Theorem and a sufficient criterion for their transience. These results are thereafter applied to a new family of models of drifted random walks on the lattice Z d .

Introduction

In the classical framework, a random walk on a group G is a discrete time stochastic process (Z n ) n≥0 defined as the product of independent and identically distributed (i.i.d.) random variables (ξ n ) n≥1 . Random walks on groups are Markov chains that are adapted to the group structure in the sense that the underlying Markov operator is invariant under the group action of G on itself. Thus, this homogeneity naturally gives rise to deep connections between stochastic properties of the random walk and algebraic properties of the group. Starting with the seminal paper of Pòlya, [START_REF] Pólya | Über eine Aufgabe der Wahrscheinlichkeitsrechnung betreffend die Irrfahrt im Straßennetz[END_REF], many of these connections have been studied (see [64] and references therein) and this research area has remained in constant progress over the last two decades (without claiming to be exhaustive, see [START_REF] Salaün | Marche aléatoire sur un groupe libre: théorèmes limite conditionnellement à la sortie[END_REF][START_REF] Vadim | Boundary and entropy of space homogeneous Markov chains[END_REF][START_REF] Pittet | On random walks on wreath products[END_REF][START_REF] Brofferio | Green kernel estimates and the full Martin boundary for random walks on lamplighter groups and Diestel-Leader graphs[END_REF]4,42,[START_REF] Blachère | Asymptotic entropy and Green speed for random walks on countable groups[END_REF][START_REF] Guivarc | Recurrence and ergodicity of random walks on linear groups and on homogeneous spaces[END_REF][START_REF] Mathieu | Differentiating the entropy of random walks on hyperbolic groups[END_REF][START_REF] Gouëzel | Sharp lower bounds for the asymptotic entropy of symmetric random walks[END_REF][START_REF] Bartholdi | Poisson-Furstenberg boundary and growth of groups[END_REF]).

In this paper, we aim at investigating inhomogeneous random walks. It turns out that there are at least two ways to introduce inhomogeneity. First, we can consider spatial inhomogeneity by weakening the group structure, replacing it, for instance, by a directed graph as in [START_REF] Campanino | Random walks on randomly oriented lattices[END_REF]32,[START_REF] Guillotin | A functional limit theorem for a 2D-random walk with dependent marginals[END_REF][START_REF] Pène | Transient random walk in Z 2 with stationary orientations[END_REF][START_REF] Basile De | Marche aléatoire sur un di-graphe et frontière de Martin[END_REF][START_REF] Brémont | Markov chains in a stratified environment[END_REF][START_REF] Brémont | Planar random walk in a stratified quasi-periodic environment[END_REF][START_REF] Bosi | Recurrence and windings of two revolving random walks[END_REF]. Secondly, we can study temporal inhomogeneous random walks by introducing a notion of memory as in the model of reinforced [START_REF] Pemantle | Vertex-reinforced random walk[END_REF][START_REF] Volkov | Vertex-reinforced random walk on arbitrary graphs[END_REF], excited [START_REF] Raimond | Random walks with occasionally modified transition probabilities[END_REF]5], self-interacting [22,[START_REF] Peres | On recurrence and transience of selfinteracting random walks[END_REF], persistent random walks [17,[START_REF] Cénac | Persistent random walks, variable length Markov chains and piecewise deterministic Markov processes[END_REF][START_REF] Cénac | Persistent random walks. I. Recurrence versus transience[END_REF][START_REF] Cénac | Persistent random walks. II. Functional scaling limits[END_REF]18] or also Markov additive processes (abbreviated by MAP) that shall be at the core of this paper. All these models belong to the larger class of stochastic processes with long range dependency.

Markov additive processes are also known as Random Walk With Internal Degree of Freedom -see [46] -semi-Markov processes or Markov random walks -see for instance [45,[START_REF] Guivarc | Application d'un théorème limite local à la transience et à la récurrence de marches de Markov[END_REF][START_REF] Erhan | Markov additive processes. I, II. Z. Wahrscheinlichkeitstheorie und Verw[END_REF][START_REF] Ney | Markov additive processes. I. Eigenvalue properties and limit theorems[END_REF][START_REF] Ney | Markov additive processes II. Large deviations[END_REF]2,61,[START_REF] Kazami | Random walks on periodic graphs[END_REF]. Roughly speaking, Markov additive processes are discrete time Z d -valued (or R d ) processes whose increments are still independent but no longer stationary. The distribution of an increment is then driven by a Markov chain termed the internal Markov chain. Most of results in the context of standard random walks are generalized to Markov additive processes when the Markov operator of the internal chain is assumed to be quasi-compact on a suitable Banach space: among them, a renewal theorem [45, 2, 30, 31], local limit theorem [46,[START_REF] Guivarc | Application d'un théorème limite local à la transience et à la récurrence de marches de Markov[END_REF][START_REF] Hervé | Théorème local pour chaînes de Markov de probabilité de transition quasi-compacte. Applications aux chaînes V -géométriquement ergodiques et aux modèles itératifs[END_REF][START_REF] Hervé | The Nagaev-Guivarc'h method via the Keller-Liverani theorem[END_REF]26,[START_REF] Hervé | A local limit theorem for densities of the additive component of a finite Markov additive process[END_REF], central limit theorem [START_REF] Hervé | The Nagaev-Guivarc'h method via the Keller-Liverani theorem[END_REF]26], results on the recurrence set [START_REF] Rogers | Recurrence of additive functionals of Markov chains[END_REF][START_REF] Alsmeyer | Recurrence theorems for Markov random walks[END_REF][START_REF] Hervé | On the recurrence set of planar Markov random walks[END_REF], large deviations [START_REF] Ney | Markov additive processes. I. Eigenvalue properties and limit theorems[END_REF][START_REF] Ney | Markov additive processes II. Large deviations[END_REF], asymptotic expansion of the Green function [61,[START_REF] Kazami | Random walks on periodic graphs[END_REF], one-dimensional Berry-Essen theorem [START_REF] Hervé | The Nagaev-Guivarc'h method via the Keller-Liverani theorem[END_REF]26,[START_REF] Hervé | A uniform Berry-Esseen theorem on M -estimators for geometrically ergodic Markov chains[END_REF] with applications to M -estimation, first passage time [START_REF] Fuh | Asymptotic expansions in multidimensional Markov renewal theory and first passage times for Markov random walks[END_REF].

However, assuming the internal operator to be quasi-compact is rather strong (see [START_REF] Lin | On quasi-compact Markov operators[END_REF]). Actually, beyond the technical difficulties inherent to the infinite dimension, there is no real difference in nature with the finite dimension under this assumption. On the other hand, relax this assumption and the study of Markov Additive processes can be very challenging. Besides, it is worth noting that many (interesting) Markov Additive processes do not admit a quasicompact internal operator as it will be illustrated by the examples considered in this paper.

In the context of Markov additive processes, classical Fourier analysis can be extended by introducing the Fourier transform operator which is a perturbation of the internal Markov operator in an appropriate Banach space. As in the classical context, the Fourier transform operator characterizes the distribution of the additive part of Markov Additive processes. By a continuous perturbation argument (see for instance [43]), when the internal Markov operator is quasi-compact, the Fourier transform operator remains quasi-compact for all sufficiently small perturbations. It allows, under suitable moment conditions on the distribution of increments, to derive a Taylor expansion at the second order of the perturbed dominating eigenvalue, say λ(t), t ∈ [-π, π) d , whose coefficients are given, roughly speaking, by the mean and the variance operators. Finally, under an assumption on the spectrum of the Fourier transform operator for large perturbations, it can be concluded that all the needed stochastic information is actually contained in the nature of the singularity at zero of (1 -λ(t)) -1 (note that λ(0) = 1). For instance, an integral test criterion, similar to the Chung-Fuchs criterion (see [START_REF] Chung | On the distribution of values of sums of random variables[END_REF] or [60]), involving a singularity of this kind is given in [18].

In this paper, the quasi-compacity condition is dropped and the internal Markov chain is only assumed to be irreducible recurrent. The condition on the spectrum of Fourier transform operator for large perturbations remains similar but the nature of the singularity at the origin is analyzed considering the Taylor expansion of the quenched characteristic exponent (in a wide sense) defined in Section 1.2. The terms of order 1 and 2 are termed the quenched drift and the quenched dispersion respectively. These quantities are characteristics of the increments of the process and naturally appears in the Local Limit in Theorem 2.1 and the transience condition in Corollary 2.4.

The paper is organized as follows. In Section 1 are gathered the main notions involved in the statement of Theorems 2.1 and its Corollary 2.4. Section 2 is devoted to the statement of these two results. In Section 3 is dedicated to the proof of the Theorem 2.1. In Section 4, Theorems 2.1 and Corollary 2.4 are applied on various families of Markov Additive processes that extends the models considered in [48,[START_REF] Campanino | Random walks on randomly oriented lattices[END_REF][START_REF] Brémont | On planar random walks in environments invariant by horizontal translations[END_REF]. Those models are simple random walks on directed graphs built upon Z 2 . Various phenomena are observed whether the directions are fixed periodically or not. As it shall be clear for the model periodically directed, it is possible to factorize a Markov Additive process without changing the distribution of the additive part (see Section 4). It will be concluded that the simple random walk on the periodically directed graph is a Markov Additive process with a finite internal Markov chain. As such, the internal Markov operator is a matrix and is quasi-compact. For more general directions (random directions for instance), such a reduction is no longer possible.

Markov additive processes 1.Definitions

Let (Ω, F, P) be a probability space and X a countable set. The set X is naturally endowed with the σ-algebra consisting of all subsets of X. Definition 1.1 (Markov additive process). Let d ≥ 1 be an integer. A Markov additive process (MAP for short) is a Markov chain ((X n , Z n )) n≥0 taking values in X × Z d defined on (Ω, F, P) satisfying for all n ≥ 0 and all bounded functions f :

X × Z d → R E[f (X n+1 , Z n+1 -Z n )|((X k , Z k )) 0≤k≤n ] = E[f (X n+1 , Z n+1 -Z n )|X n ]. (1)
From equality (1), it follows immediately that (X n ) n≥0 is a Markov chain on X. The latter is termed the internal Markov chain. The corresponding Markov kernel shall be denoted by P , namely P f (x) := E[f (X 1 )|X 0 = x] for any bounded function f : X → R (in symbols, f ∈ ∞ (X)). Generally speaking, there exists a σ-finite measure m dominating the family of probabilities (P (x, •)) x∈X , i.e. m(y) = 0 implies P (x, y) = 0 for all x ∈ X. If the internal Markov chain is irreducible and recurrent, the invariant measure (unique up to a positive constant) is a natural choice for m.

The conditional distribution of Z n+1 -Z n given (X n , X n+1 ) = (x, y) will be denoted by µ x,y and the Fourier transform of µ x,y by µ x,y . Then, for any t ∈ R d , the Fourier transform operator P t acting on the bounded function f : X → C is defined as follows:

P t f (x) := E[f (X 1 )e i t,Z 1 -Z 0 |X 0 = x] = y∈X P (x, y)f (y) µ x,y (t).
From Markov property, it follows for all n ≥ 1

P n t f (x) = E[f (X n )e i t,Zn-Z 0 |X 0 = x].
Moreover,

P n t 1(x) = z∈Z d e i t,z P(Z n -Z 0 = z|X 0 = x). (2)
Consequently, the function

R d t → P n t 1(x) ∈ C is the Fourier transform of the conditional distribution of Z n -Z 0 given X 0 = x.
By periodicity, it is sufficient to consider the operator P t for t ∈ T d = [-π, π) d . In addition, T d * will stand for T d \ {0}.

Conditional characteristic exponent, conditional drift and conditional dispersion

Proposition 1.2. For any MAP, the following identity holds for all n ≥ 0:

P(Z n -Z 0 = z|X 0 = x) = 1 (2π) d T d e -i t,z P n t 1(x) dt (3)
and

P n t 1(x) = E[Π n (t)|X 0 = x] where Π n (t) := n-1 k=0 E e i t,Z k+1 -Z k X k , X k+1 . (4)
The proof of Proposition 1.2 relies on a result from [25,Equation (8)] whose statement is recalled below.

Lemma 1.3. For all m ≥ n ≥ p ≥ 0, E e i t,Zn-Zp σ(X j , p ≤ j ≤ n) E e i t,Zm-Zn σ(X j , n ≤ j ≤ m) = E e i t,Zm-Zp σ(X j , p ≤ j ≤ m) . (5) Proof of Proposition 1.2. By inverse Fourier transform, (2π) d P(Z n -Z 0 = z|X 0 = x) = T d P n t 1(x)e -i t,z dt.
Then, by Lemma 1.3, setting G n = σ(X , 0 ≤ ≤ n), n ≥ 1,

P n t 1(x) = E E n-1 k=0 e i t,Z k+1 -Z k G n X 0 = x = E n-1 k=0 E e i t,Z k+1 -Z k X k , X k+1 X 0 = x
As a matter of fact, Π n is a almost-surely continuous C-valued function of T d satisfying Π n (0) = 1. Hence, in a neighborhood of the origin, the logarithm of Π n is well defined. We may call log Π n the quenched characteristic exponent (with a slight abuse of terminology since log Π n is not defined on R d in general). Its second order Taylor expansion at t = 0 then exhibits two important quantities:

U n := n-1 =0 E[Z +1 -Z |X , X +1 ] and V n := 1 n n-1 =0 Cov(Z +1 -Z |X , X +1 ).
The latter are called respectively the quenched drift and the quenched dispersion of the additive component.

Remark 1.4. The normalization factor for the quenched dispersion is chosen as the approximate growth rate of the sum. In particular, its eigenvalues remain bounded and away from zero almost-surely as illustrated in Section 4, Equation (16).

Spectral condition

Definition 1.5 (Spectral condition). A MAP is said to satisfy the spectral condition if, for any compact K ⊂ T d with 0 / ∈ K there exist constants C > 0 and γ ∈ (0, 1) such that,

∀n ≥ 1, P n t 1 ∞ (X) ≤ Cγ n .
Proposition 1.6. Assume that the family (µ x,y ) x,y∈X of probability measures is uniformly aperiodic i.e, for all t ∈ T d * , sup x,y∈X | µ x,y (t)| < 1, then the MAP fulfills the spectral condition.

Proof. For all x ∈ X and all f ∈ ∞ (X),

|P t f (x)| = y∈X P (x, y) µ x,y (t)f (y) ≤ f ∞ (X) sup x,y∈X | µ x,y (t)|.
Therefore,

P n t 1 ∞ (X) ≤ sup x,y∈X | µ x,y (t)| n .
In the literature, it is usually assumed a stronger condition than the spectral condition of Definition 1.5. It involves the spectral radius of each P t acting on some Banach subspace of ∞ (X). Such assumptions imply the spectral condition of Definition 1.5 as shown in Proposition 1.7 just below.

Proposition 1.7. Let (B, • B ) be a Banach subspace of ∞ (X) such that (C1)
1 ∈ B and the canonical injection B → ∞ (X) is continuous;

(C2) for each t ∈ T d , the operator P t acts continuously on B;

(C3) the map T d t → P t ∈ L(B) is continuous for the subordinated operator norm induced by • B .
Suppose that the spectral radius of P t defined, for all t ∈ T d , by

r B (t) = inf{ P n t 1/n B , n ≥ 1},
satisfies r B (t) < 1 as soon as t ∈ T d \ {0}. Then, the resulting MAP satisfies the spectral condition of Definition 1.5.

Proof. Since T d t → P t ∈ L(B) is continuous, the spectral radius t → r B (t) is upper semicontinuous as the infimum of continuous functions. Consequently, if K ⊂ T d with 0 / ∈ K then there exists

t K ∈ K such that 1 > r B (t K ) = sup t∈K r B (t).
Choose γ ∈ (r B (t K ), 1) and denote by Γ ⊂ C the circle of the complex plan centered at 0 of radius γ. As a matter of fact, for any (λ, t) ∈ Γ × K the operator λ -P t is invertible. By [24, Theorem 10, p. 560], it follows

P n t B = γ n 2π Γ (λ -P t ) -1 B dλ. Now, the map Γ × K (λ, t) → λ -P t ∈ L(B) is continuous. Besides, by [24, Lemma 1, p. 584], the map A → A -1 is a homeomorphism of the open subset of invertible operators in L(B). Consequently, the map Γ × K (λ, t) → (λ -P t ) -1 ∈ L(B) is continuous on the compact Γ × K hence sup (λ,t)∈Γ×K (λ -P t ) -1 B < ∞.
The spectral condition of Definition 1.5 follows immediately from the continuity of the canonical injection of B in ∞ (X).

The uniform aperiodicity condition introduced in Proposition 1.6 is far from being necessary. Below, we introduce the usual notion of aperiodic MAP (we refer to [61] for instance).

Definition 1.8 (Aperiodic Markov Additive process).

A MAP is said to be aperiodic if there exists no proper subgroup H of the additive group Z d such that, for every positive integer n and every x, y ∈ X with m(x)P n (x, y) > 0, there exists a = a n (x, y) ∈ Z d satisfying

P[Z n -Z 0 ∈ a + H|X 0 = x, X n = y] = 1.
Proposition 1.9. Let (X, Z) be an aperiodic MAP for which the internal Markov chain X is irreducible and recurrent. Let (B, • B ) be a Banach subspace of ∞ (X) satisfying the assumptions (C1), (C2) and (C3) of Proposition 1.7. Then, for any t ∈ T d \ {0}, the operator P t ∈ L(B) has no eigenvalues of modulus one.

Proof. Suppose on the contrary that there exists t 0 ∈ T d \ {0}, f ∈ B \ {0} and θ ∈ R such that

P t 0 f (x) = y∈X P (x, y) µ x,y (t 0 )f (y) = e iθ f (x), x ∈ X. (6)
By Jensen's inequality and the fact that

| µ x,y (t 0 )| ≤ 1, it follows that |f | ≤ P |f |. Consequently, f ∞ (X) -|f
| is a non-negative superharmonic function. Since P is irreducible and recurrent, the function |f | is constant (see [64, Theorem 1.16, p.5] for instance). Hence, for all x ∈ X such that m(x) > 0,

1 = y∈X P (x, y) µ x,y (t 0 ) ≤ y∈X P (x, y)| µ x,y (t 0 )| ≤ 1, x ∈ X.
This means that, for all x, y ∈ X with m(x)P (x, y) > 0, | µ x,y (t 0 )| = 1. Since |f | is constant and f = 0, Equation ( 6) can be rewritten as follows

y∈X P (x, y) µ x,y (t 0 ) f (y) f (x) = e iθ , x ∈ X.
This convex combination is again extremal so that, for all x, y ∈ X, µ x,y (t 0 ) f (y) f (x) = e iθ . More generally,

∀n ≥ 1, ∀(x 0 , . . . , x n ), n-1 k=0 µ x k ,x k+1 (t 0 ) = e inθ f (x 0 ) f (x n ) . (7)
Now, fix n ≥ 1 and x, y ∈ X and choose once for all a n (x, y) ∈ R d such that e inθ f (x) f (y) = e i t 0 ,an(x,y) . In fact, a n (x, y) ∈ Z d since left handside of Equation ( 7) is nothing but the Fourier transform of a measure supported by Z d . Thereafter, denote by S n (x, y) the support of the distribution of Z n -Z 0 -a n (x, y) conditionally on the event {X 0 = x} ∩ {X n = y} and set H the group generated by the family of sets S n (x, y), n ≥ 1 and x, y ∈ X. By construction of H, one has that

∀n ≥ 1, ∀x, y ∈ X, ∃a n (x, y) ∈ Z d : P[Z n -Z 0 ∈ a n (x, y) + H|X 0 = x, X n = y] = 1. By aperiodicity, H = Z d .
Finally, take the expectation on both sides of Equation ( 7) so that

e i t 0 ,an(x,y) = E e i t 0 ,Zn-Z 0 |X 0 = x, X n = y .
Then, the extremality in the convex combination yields that for all w ∈ S n (x, y), t 0 , w = 0 modulo 2π. By linearity, this identity extends to the whole group H = Z d . Taking w = e i , i = 1, . . . , d, the vectors of the standard basis of Z d , it follows that t 0 = 0 leading to a contradiction.

Let us point out that an arithmetic condition such as the aperiodicity fails to give information on spectral values that are not eigenvalues. Nonetheless, if in addition, it is known that the peripheral spectrum consists of eigenvalues then the spectral condition of Definition 1.5 is clearly fulfilled. Such an assumption on the peripheral spectrum which is rather technical can nonetheless be difficult to verify in practice. That is why, it can be often preferable to check directly the spectral condition of Definition 1.5.

Main results

Let us first summarize the assumptions involved in the statement of the Local Limit Theorem and the transience sufficient criterion.

Assumptions 1. The internal Markov chain P is irreducible and recurrent.

Assumptions 2. The family of probability measures (µ x,y ) x,y∈X admits a uniform third order moment: sup

x,y∈X z∈Z d z 3 µ x,y (z) < ∞ Assumptions 3. The quenched dispersion is uniformly elliptic: there exists a (deterministic) constant α > 0 such that

∀t ∈ R d , ∀n ≥ 1, t, V n t ≥ α t 2 , a.s..
Assumptions 4. The MAP satisfies the spectral condition.

A Local Limit Theorem

Theorem 2.1 (Local Limit Theorem). Under Assumptions 1, 2, 3 and 4, for all x ∈ X, ∀ε ∈ (0, 1 2 ), sup

z∈Z d (2πn) d/2 P(Z n -Z 0 = z|X 0 = x) -E 1 det(V n ) exp - 1 2n U n -z, V -1 n (U n -z) X 0 = x = O(n -( 1 2 -ε) ).

A sufficient criterion for transience: some potential theory

This section is devoted to a sufficient criterion for the transience of the additive part of Markov additive processes. The recurrence or transience in this context is defined in [18] as follows.

Definition 2.2 (Recurrence versus Transience). A MAP ((X n , Z n ) n≥0 is said to be recurrent if for any (x, z) ∈ X × Z d , there exists r > 0 such that

P lim inf n→∞ Z n < r X 0 = x, Z 0 = z = 1.
It is said to be transient if for any

(x, z) ∈ X × Z d P lim n→∞ Z n = ∞ X 0 = x, Z 0 = z = 1.
In [18], it is proved that a MAP is either recurrent or transient under Assumption 1. In particular, the recurrence or transience of such a MAP does not depend on the initial state (x, z) ∈ X × Z d .

For any positive function f on X × Z d , recall that the potential of the charge f is given by

Gf (x, z) := E   n≥0 f (X n , Z n ) X 0 = x, Z 0 = z   .
By analogy with the classical context of random walks, it is natural to look for a criterion for the recurrence or transience of a MAP that involves the mean sojourn time of the set X × {z}:

G1 X×{z} (x, z) = E   n≥0 1 X×{z} (X n , Z n ) X 0 = x, Z 0 = z   . (8)
Naturally, if the quantity in ( 8) is finite, the additive component of the MAP hits z ∈ Z d only finitely many times almost surely by applying the Markov inequality.

Definition 2.3 (Irreducibility).

A MAP is said to be irreducible if for any x ∈ X, for any z, z ∈ Z d , there exists n ≥ 0 such that

P[Z n = z |X 0 = x, Z 0 = z] > 0.
Therefore, if a MAP is irreducible and if the quantity in Equation ( 8) is finite for all x ∈ X and for some (equivalently any) z ∈ Z d then the MAP is transient by the Markov property. Consequently, we obtain the following criterion as a Corollary of Theorem 2.1.

Corollary 2.4 (Sufficient criterion for transience). Let d ≥ 2. Suppose that the MAP is irreducible. Then, under Assumptions 1, 2, 3 and 4, if for any

x ∈ X n≥1 1 n d/2 E 1 det(V n ) exp - 1 2n U n , V -1 n U n X 0 = x < ∞ (9)
then, the MAP is transient.

Remark Proof. Fix ≥ 0. Under Assumption 2, there exists a (deterministic) constant κ > 0 such that

E e i t,Z +1 -Z X , X +1 -1 ≤ κ t ∞ . (10)
Then, for all δ ∈ (0, 1/2κ) ∩ (0, 1) and all t ∈ δT d , the C-valued function

π : t -→ log E e i t,Z +1 -Z X , X +1
is three times continuously differentiable. Observe that π (0) = 0. Also, the partial derivative with respect to the p th coordinate, p ∈ {1, . . . , d}, is given by

∂ p π (t) = i E e i t,Z +1 -Z (Z (p) +1 -Z (p) ) X , X +1 E e i t,Z +1 -Z X , X +1
.

Finally, the partial derivative with respect to the p th and q th coordinates, p, q ∈ {1, . . . , d}, writes

∂ 2 p,q π (t) = E e i t,Z +1 -Z X , X +1 -2 -E e i t,Z +1 -Z (Z (p) +1 -Z (p) )(Z (q) +1 -Z (q) ) X , X +1 E e i t,Z +1 -Z X , X +1 + E e i t,Z +1 -Z (Z (p) +1 -Z (p) ) X , X +1 E e i t,Z +1 -Z (Z (q)
+1 -Z (q) ) X , X +1 .

Hence, the Taylor expansion at t = 0 yields for all t ∈ δT d

π (t) = i t, E[Z +1 -Z |X , X +1 ] - 1 2 t, Cov(Z +1 -Z |X , X +1 )t + R (t)
where R stands for the Lagrange's reminder in the Taylor expansion. Then, for any t ∈ δT d , almost surely,

|R (t)| ≤ t 3 ∞ sup t∈δT d
sup |∂ 3 p,q,r π (t)|, p, q, r ∈ {1, . . . , d} .

In fact, Equation [START_REF] Brémont | Planar random walk in a stratified quasi-periodic environment[END_REF] implies that ∀δ ∈ (0, 1/2k) ∩ (0, 1), ∀t ∈ δT d , E e i t,Z +1 -Z X , X +1 ≥ 1 2 .

Thus, computing explicitly the third order derivatives of π , it follows, for any p, q, r ∈ {1, . . . , d} and any t ∈ δT d , that

|∂ 3 p,q,r π (t)| ≤ 2 4 E Z +1 -Z 3 1 X , X +1 + 2E Z +1 -Z 1 X , X +1 3 + 5E Z +1 -Z 2 1 X , X +1 E Z +1 -Z 1 X , X +1 ≤ 2 7 E Z +1 -Z 3 1 X , X +1 .
The existence of the deterministic constant K follows immediately from Assumption 2. Now,

Π n (t) = n-1 =0 π (t) = exp i t, U n - 1 2 t, nV n t + Θ n (t) , where Θ n (t) = n-1 =0 R (t) and |Θ n (t)| ≤ nK t 3 ∞ .
Proof of Theorem 2.1. Let δ ∈ (0, 1) be such that Proposition 3.1 holds. Then, by Proposition 1.2, (2π

) d P(Z n -Z 0 = z|X 0 = x) = T d P n t 1(x)e -i t,z dt = T d \δT d P n t 1(x)e -i t,z dt + δT d P n t 1(x)e -i t,z dt. ( 11
)
The change of variables u = t/ √ n in the second term yields

δT d P n t 1(x)e -i t,z dt = E δT d e -i t,z Π n (t)dt X 0 = x = n -d/2 E δ √ nT d e -i √ n t,z Π n (t/ √ n)dt X 0 = x . (12)
Let a ∈ (0, δ √ n). The quantity in Equation 12 can be decomposed as follows:

n -d/2 E R d exp i 1 √ n t, U n -z - 1 2 t, V n t dt X 0 = x + n -d/2 E δ √ nT d \aT d e -i √ n t,z Π n (t/ √ n)dt X 0 = x + n -d/2 E aT d e -i 1 √ n t,z Π n (t/ √ n) -exp i 1 √ n t, U n -z - 1 2 t, V n t dt X 0 = x -n -d/2 E R d \aT d exp i 1 √ n t, U n -z - 1 2 t, V n t dt X 0 = x
Finally, remarking that t/ √ n ∞ ≤ δ as soon as t ∈ aT d so that Proposition 3.1 holds, Equations ( 11) and ( 12) imply

(2π √ n) d P(Z n -Z 0 = z|X 0 = x) -E R d exp i 1 √ n t, U n -z - 1 2 t, V n t dt X 0 = x = A 1 (z, n, a) + A 2 (z, n, a) + A 3 (z, n, a, δ) + A 4 (z, n, δ)
where

A 1 (z, n, a) := E aT d exp i 1 √ n t, U n -z - 1 2 t, V n t e Θn(t/ √ n) -1 dt X 0 = x , A 2 (z, n, a) := -E R d \aT d exp i 1 √ n t, U n -z - 1 2 t, V n t dt X 0 = x , A 3 (z, n, a, δ) := E δ √ nT d \aT d e -i √ n t,z Π n (t/ √ n)dt X 0 = x , A 4 (z, n, δ) := n d/2
T d \δT d P n t 1(x)e -i t,z dt.

Lemma 3.2. Under Assumptions 2 and 3, it holds that

∀n ≥ 1, ∀a ∈ 0, δ √ n , sup z∈Z d |A 1 (z, n, a)| = O a 3 K √ n exp a 3 K √ n
where K is defined in Proposition 3.1.

Proof. Let t ∈ aT d . By Proposition 3.1 under Assumptions 2 and 3,

e i 1 √ n t,Un-z -1 2 t,Vnt e Θn(t/ √ n) -1 ≤ e -1 2 t,Vnt Θ n (t/ √ n) exp Θ n (t/ √ n) ≤ e -1 2 t,Vnt a 3 nK n 3/2 exp a 3 nK n 3/2 ≤ e -α 2 t 2 2 a 3 K √ n exp a 3 K √ n a.s..
The result follows by integrating with respect to t ∈ aT d .

Lemma 3.3. Under Assumption 3, ∀n ≥ 1, ∀a > 0, sup

z∈Z d |A 2 (z, n, a)| ≤ 2 d e -αda 2 (αda) d .
Proof. Assumption 3 implies sup

z∈Z d |A 2 (z, n, a)| ≤ R d \aT d exp(- 1 2 α t 2 2 ) dt ≤ t 2 ≥a exp(- 1 2 α t 2 2 ) dt ≤ 2 d e -αda 2 (αa) d .
Lemma 3.4. Under Assumptions 2 and 3, for any δ > 0 sufficiently small and any a ∈ (0, δ √ n)

sup z∈Z d |A 3 (z, n, a, δ)| ≤ K 3 exp -K 4 a 2 .
Proof. As a matter of fact,

E δ √ nT d \aT d e -i √ n t,z Π n (t/ √ n)dt X 0 = x ≤ E δ √ nT d \aT d Π n (t/ √ n) dt X 0 = x . ( 13
)
Under Assumptions 2 and 3, by Proposition 3.1, the integrand in Equation ( 13) satisfies

Π n (t/ √ n) ≤ exp - 1 2 t, V n t + |Θ n (t/ √ n)| . Moreover, since t ∈ δ √ nT d \ aT d |Θ n (t/ √ n)| ≤ nK t/ √ n 2 ∞ t/ √ n ∞ = K t 2 ∞ t/ √ n ∞ ≤ Kδπ t 2 ∞ ≤ Kδπd t 2 2 , and -1 2 t, V n t ≤ -1 2 α t 2 2 .
Thus, choosing any δ > 0 so small that Kδπd ≤ α/4 (and that Proposition 3.1 holds true), the integrand in Equation ( 13) is bounded above by exp(-β t 2

2 ) for some β > 0. Consequently, there exists K 3 , K 4 > 0 such that |A 3 (z, n, a, δ)| ≤ K 3 e -K 4 a 2 . Lemma 3.5. Under Assumptions 4, for any δ ∈ (0, 1), there exist γ ∈ (0, 1) and a constant C > 0 such that sup

z∈Z d |A 4 (z, n, δ)| ≤ (2π) d Cγ n .
Proof. Fix δ ∈ (0, 1). Then, under Assumption 4, there exists γ ∈ (0, 1) and a constant C > 0 such that, for all t ∈ T d \ δT d , P n t 1 ∞ ≤ Cγ n . The result follows immediately.

The four lemmas above imply that sup

z∈Z d n d/2 (2π) d P(Z n -Z 0 = z|X 0 = x) -E R d e i 1 √ n t,Un-z -1 2 t,Vnt dt X 0 = x = O a 3 K √ n e a 3 K √ n + 2 d (αa) d e -αda 2 /2 + K 3 e -K 4 a 2 + Cγ n = O(n -( 1 2 -ε) ),
by setting a = δn ε/3 for any ε ∈ (0, 1 2 ). Finally, noting that V n is invertible by Assumption 3, the proof of Theorem 2.1 ends with the help of the Fourier transform

R d exp i 1 √ n t, U n -z - 1 2 t, V n t dt = (2π) d/2 det(V n ) exp - 1 2n U n -z, V -1 n (U n -z) .

Application: random walks with local drift

This section is dedicated to the illustration of Theorems 2.1 and its Corollary 2.4. The examples presented below are largely inspired by those studied in [START_REF] Campanino | Random walks on randomly oriented lattices[END_REF], which in turn were previously introduced in [48]. The discrete time processes considered in [START_REF] Campanino | Random walks on randomly oriented lattices[END_REF] are basically simple random walks on directed graphs built upon Z 2 . More precisely, whereas the random walker can move freely toward North or South at each step, he can only move toward East or West depending on a prescribed environment. As shown in [START_REF] Campanino | Random walks on randomly oriented lattices[END_REF], different environments lead to different behaviors for the simple random walk explaining that a significant part of the literature considers these models (see for instance [32,[START_REF] Pène | Transient random walk in Z 2 with stationary orientations[END_REF][START_REF] Castell | A local limit theorem for random walks in random scenery and on randomly oriented lattices[END_REF][START_REF] Basile De | Marche aléatoire sur un di-graphe et frontière de Martin[END_REF][START_REF] Hervé | On the recurrence set of planar Markov random walks[END_REF][START_REF] Campanino | Type transition of simple random walks on randomly directed regular lattices[END_REF][START_REF] Brémont | On planar random walks in environments invariant by horizontal translations[END_REF][START_REF] Brémont | Markov chains in a stratified environment[END_REF][START_REF] Brémont | Planar random walk in a stratified quasi-periodic environment[END_REF] or even more recently [START_REF] Bosi | Recurrence and windings of two revolving random walks[END_REF]).

In the model considered below, the random walker can move simultaneously vertically and horizontally in one of the quarter-plane North West, North East, South East or South West. Whereas the choice between North or South remains unrestricted, the choice between East and West is dictated by a prescribed environment. Thus, the case of a periodic environment considered in Section 4.2 has to be compared to the model L of [START_REF] Campanino | Random walks on randomly oriented lattices[END_REF]. The two-directed half planes of Section 4.3 is expected to be similar to the model H of [START_REF] Campanino | Random walks on randomly oriented lattices[END_REF]. Finally, the randomly directed case of Section 4.4 is reminiscent of the model O.

The model of random walks with local drift

In this section the internal Markov chain X of the MAP (X, Z) shall be the simple random walk on X = Z. More precisely, let (ξ k ) k≥1 be a sequence of i.i.d. random variables where ξ 1 is uniformly distributed on {-1, 1}. Let X 0 be a Z-valued random variable and, for n ≥ 1,

X n := X 0 + ξ 1 + • • • + ξ n .
Now, let us define the additive part Z of the MAP (X, Z). To this end, introduce a sequence ((T h,k , T v,k )) k≥0 of independent copies of some N 2 -valued random vector (T h , T v ) and a sequence ε = (ε x ) x∈Z of Z-valued random variables. Then, the additive part Z taking values in Z 2 is defined as follows:

∀n ≥ 0, Z n -Z 0 := n-1 k=0 ε X k T h,k , n-1 k=0 ξ k+1 T v,k . (14)
Assumptions 5.

1. The internal Markov chain X, the sequences ((T h,k , T v,k )) k≥0 and ε are independent,

E[T

h ] = m h ∈ (0, ∞) and E[T v ] = m v ∈ (0, ∞), 3. V[T h ] = σ 2 v > 0 and V[T v ] = σ 2 h > 0,
4. T h and T v are independent.

Proposition 4.1. Under Assumptions 5, conditionally on the environment ε, the quenched drift and quenched dispersion are respectively given by

U n (ε) = m h n-1 k=0 ε X k m v (X n -X 0 ) and V n (ε) = 1 n σ 2 h n-1 k=0 ε 2 X k 0 0 nσ 2 v .
Proof. Conditionally on the environment ε, the quenched drift and quench dispersion are respectively given by

U n (ε) = n-1 =0 E[Z +1 -Z |X , X +1 , ε] and V n (ε) = 1 n n-1 =0 Cov(Z +1 -Z |X , X +1 , ε).
By independence, for all ≥ 0,

E[Z +1 -Z |X , X +1 , ε] = ε X E[T h ] ξ +1 E[T v ] and Cov(Z +1 -Z |X , X +1 , ε) = 1 n V(T h )ε 2 X 0 0 V(T v )ξ 2 +1 .
The result follows immediately.

Proposition 4.2. Under Assumptions 5, conditionally on the environment ε, the Fourier transform operator is given by

∀x ∈ Z, ∀t ∈ T 2 , P t f (x) = y∈Z P (x, y) µ x,y ε (t)f (y),
where P is the Markov operator associated with the simple random walk X and

µ x,y ε (t) = ϕ T h (ε x t 1 )ϕ Tv ((y -x)t 2 ), x, y ∈ Z s.t. P (x, y) > 0, t = (t 1 , t 2 ) ∈ T 2 ,
where ϕ T h and ϕ T h are the characteristic functions of T h and T v respectively.

Proof. Let t = (t 1 , t 2 ) ∈ T 2 . Then,

P t f (x) = E[e it 1 ε X 0 T h,0 e it 2 (X 1 -X 0 )T v,0 f (X 1 )|X 0 = x, ε] = y∈Z P (x, y)E[e it 1 ε X 0 T h,0 e it 2 (X 1 -X 0 )T v,0 |X 0 = x, X 1 = y, ε]f (y) = y∈Z P (x, y)ϕ T h (ε x t 1 )ϕ T h ((y -x)t 2 )f (y).
In order to apply Theorem 2.1 and Corollary 2.4, it is necessary to make further assumptions.

Assumptions 6.

(UB) There exists B > 0 such that P[sup x∈Z |ε x | ≥ B] = 0;

(ND) For all x ∈ Z, P[ε x = 0] = 0;

(M) E[T 3 h ], E[T 3 v ] are finite;
(AP) the distribution of T h and T v are aperiodic.

Proposition 4.3. Under Assumptions 5 and 6, conditionally on the environment ε, Assumptions 1, 2, 3 and 4 are fulfilled.

Proof. It is well known that the simple random walk X is irreducible recurrent (Assumption 1). The uniform moment condition involved in Assumption 2 is equivalent to

sup x∈Z |ε x |E[T 3 h ] + E[T 3 v ] < ∞.
The latter is straightforward by (UB) and (M) of Assumption 6.

The uniform ellipticity condition of Assumption 3 follows immediately of (ND) of Assumption 6.

Finally, since T h and T v are aperiodic, for all t ∈ [-π, π) \ {0}, |ϕ T h (t)| < 1 and |ϕ Tv (t)| < 1. Consequently, conditionally on the environment ε, the family of probability measure (µ x,y ε ) x,y∈Z is uniformly aperiodic. By Proposition 1.6, Assumption 4 follows.

As a matter of fact, the quantity

1 2n (U n (ε) -z), V -1 n (U n -z) , n ≥ 1, z = (z 1 , z 2 ) ∈ Z 2 , appearing in Theorem 2.1 is given by 1 2n (U n (ε) -z), V -1 n (U n -z) = 1 2    m h n-1 k=0 ε X k -z 1 2 σ 2 h n-1 k=0 ε 2 X k + (m v (X n -X 0 ) -z 2 ) 2 nσ 2 v    . (15)
Also, it is worth noting that, under Assumptions (UB) and (ND) of Assumption 6,

∀n ≥ 1, det(V n ) σ 2 h σ 2 v = 1 n n-1 k=0 ε 2 X k ∈ [1, B 2 ]. (16)
Consequently, this factor will be omitted in the sequel when applying Corollary 2.4. The end of this section is devoted to the application of Theorem 2.1 or Corollary 2.4 for three different environments ε. The first two are deterministic whereas the last one is random.

Periodic environment

In this example, the environment ε is supposed periodic. Namely, it is assumed that, for all x ∈ Z, ε x = (-1) x . Proposition 4.4. Let z ∈ Z 2 . Then, under Assumptions 5 and 6, for all x ∈ Z,

P[Z n -Z 0 = z|X 0 = x] ∼ n→∞ 1 2πσ h σ v n 1 + m 2 v σ 2 v -1/2 . Proof. First observe that ∀n ≥ 1, n-1 k=0 ε X k = (-1) X 0 1 -(-1) n 2 and det(V n ) = σ 2 h σ 2 v .
Then, setting z = (z 1 , z 2 ),

σ h σ v E 1 det(V n ) exp - 1 2n U n -z, V -1 n (U n -z) X 0 = x = exp - 1 2σ 2 h n m h (-1) x 1 -(-1) n 2 -z 1 2 E exp - (m v (X n -X 0 ) -z 2 ) 2 2nσ 2 v X 0 = x .
As a matter of fact, the first exponential factor converges toward 1 as n → ∞.

For the second factor, it suffices to remark that n

-1/2 [m v (X n -X 0 ) -z 2 ] converges in distribution to a centered standard Gaussian random variable. Since t → e -t 2 /2σ 2 v is continuous, it follows lim n→∞ E exp - (m v (X n -X 0 ) -z 2 ) 2 2nσ 2 v X 0 = x = R e -y 2 2 1+ m 2 v σ 2 v dy √ 2π = 1 + m 2 v σ 2 v -1/2
. By Theorem 2.1, the result of the proposition follows immediately.

The estimate of Proposition 4.4 implies that the series in Corollary 2.4 if infinite. However, since the internal Markov chain is null recurrent, the recurrence of the MAP can not be deduced directly. Actually, it turns out that, taking advantage of the periodicity of the environment, the distribution of the additive component can be described with a simpler internal Markov chain. Proposition 4.5. Suppose the MAP (X, Z) is irreducible. Then, under Assumptions 5 and 6, the MAP Z is recurrent.

Proof. The key idea consists in remarking that M = ((ε Xn , ξ n+1 )) n≥0 is a Markov chain taking values in the finite space {-1, 1} 2 . The transition probabilities are given, for all x, x , y, y ∈ {-1, 1}, as follows:

Q((x, y), (x , y )) := Q 1 (x, x )Q 2 (y, y ) with Q 1 := 0 1 1 0 and Q 2 := 1 2 1 2 1 2 1 2 . Now, let us set for all n ≥ 1, Z n -Z 0 = n-1 k=0 M (1) k T h,k , n-1 k=0 M (2) k T v,k ,
where M (1) and M (2) are respectively the first and second component of M . From Equation ( 14), the following equality in distribution holds

L((Z n -Z 0 ) n≥0 |X 0 = x) = L(( Z n -Z 0 ) n≥0 |M (1) 0 = ε x ).
From this equality in distribution, the MAP (M, Z) inherits irreducibility from the MAP (M, Z). Moreover, applying [18], the MAP (M, Z) is recurrent noting that M is positive recurrent. Consequently, (X, Z) is recurrent.

Two directed half planes

The reduction made in the proof of Proposition 4.5 is naturally not always possible as illustrated by the example of this section. This obstruction was the main motivation of the extension to the null recurrent case.

The two directed half planes presented below corresponds to the environment ε defined, for all x ∈ Z, by ε x = 21 Z + (x) -1 where Z + stands for the set of non negative integers.

Proposition 4.6. Assume that the MAP (X, Z) is irreducible. Then, under Assumptions 5 and 6, (X, Z) is transient.

Proof. By irreducibility, Equation [START_REF] Erhan | Markov additive processes. I, II. Z. Wahrscheinlichkeitstheorie und Verw[END_REF] with z 1 = z 2 = 0 and the fact that, for all n ≥ 1, det(V n ) = σ 2 h σ 2 v , the transience of (X, Z) follows Corollary 2.4 by proving the series

n≥1 1 n E   exp    - m 2 h 2nσ 2 h n-1 k=0 ε X k 2    X 0 = x   . ( 17
)
is finite for all x ∈ Z. Thereafter, a simple computation yields, for all n ≥ 1, that However, the singularity at y = 1 coming from the density function of the arc-sine distribution is integrable and so is the singularity at y = 1 2 since y → log y is locally integrable in the positive neighborhood of 0.

Randomly directed random walks

The last family of examples involve a random environment. More precisely, it will be assumed that ε is a sequence of i.i.d. random variables. In the sequel, the common marginal distribution will be denoted by π.

Proposition 4.7. Additionally to Assumptions 5 and 6, suppose that ε = (ε x ) x∈Z is a sequence of i.i.d. random variables such that E[ε 0 ] = 0. Then, for π ⊗Z -a.e. sequences ε, if the MAP (X, Z) in the environment ε is irreducible, then it is transient.

Proof. Applying Markov inequality to the probability measure π ⊗Z , it is only needed to prove that

n≥1 1 n E   exp      - m 2 h n-1 k=0 ε X k 2 2σ 2 h n-1 k=0 ε 2 X k      X 0 = x    . (19)
Without loss of generality, one may suppose that x = 0 since π ⊗Z is shift invariant. Remarking that n ≤ n-1 k=0 ε 2 X k ≤ nB 2 by conditions (UB) and (ND) of Assumption 6, then, for any α ∈ ( 1 2 , 3 4 ), where the o(n -3/4 ) is uniform in x ∈ R. Noting that C is bounded (for instance [14, Lemma 4]), the probability above behaves like O(n α-3/4 ) which ends the proof of the proposition.

E   exp      - m 2 h n-1 k=0 ε X k 2 2σ 2 h n-1 k=0 ε 2 X k         ≤ P n-1 k=0 ε X k ≤ n α + exp e - m 2 h σ 2 h B 2 n 2α-1 . ( 20

n- 1 k=0ε 1 k=0 1 Z(+ E exp -n m 2 h 2σ 2 h(

 1112 X k = 2N n (Z + ) -n where N n (Z + ) = n-+ (X k ).Then, verifying that the function[0, ∞) t → exp -2N n (Z + ) -n) 2 X 0 = x ≤ k n d W (L(N n (Z + )/n), L(Γ)) 2Γ -1) 2 X 0 = x (18)where Γ is distributed as an arc-sine law supported by [0, 1] and d W stands for the Kantorovich-Rubinstein metric on probability measures (see[62, Chapter 6, p.94]). It is shown in [28, Theorem 1.2] that d W (L(N n (Z + )/n), L(Γ)) = O(1/n) so that the series in Equation (17) is finite if and only if, applying Fubini's Theorem,

- 1 k=0ε

 1 ) Now, set D n = x ∈ R : n 3/4 x ∈ [-n α , n α ] ∩ Z . By [14,Theorem 1], with the very same function C, the following estimate holdsP nX k ≤ n α = x∈Dn C(x) n 3/4 + o(n -3/4 )

  2.5. Under the assumptions of Corollary 2.4, the quantities in Equations (8) and (9) are simultaneously finite or infinite. When the internal Markov chain is positive recurrent, the MAP is recurrent as soon as the quantity in Equation (8) is infinite as shown in Proposition [18, Proposition 2.2] giving rise to a complete characterization of the transient or recurrent type. Such a characterization remains an open question in the case of a null recurrent internal Markov chain since the possibility of a transient MAP for which the quantity in Equation (8) remains infinite can not be ruled out.

Remark 2.6. Intuitively, Assumption 3 means that the MAP remains genuinely d-dimensional and is not attracted by a sub-manifold. In addition, it is worth noting that, under Assumption 3, det(V n ) ≥ α d for all n ≥ 1 and, under Assumption 2, sup n≥1 V n 2 < ∞ so that the factor det(V n ) does not play any role in the nature of the series (9).

3 Proofs

Proposition 3.1. Under Assumption 2, for any sufficiently small δ ∈ (0, 1) and any t ∈ δT d , where δT d is the magnification of the hypercube T d by δ, there exist a (deterministic

) constant K and Θ n (t) with |Θ n (t)| ≤ nK t 3 ∞ such that

Π n (t) = exp i t, U n -1 2 t, nV n t + Θ n (t) .

Acknowledgment: The author thanks the two anonymous referees for their very detailed and thorough reviews that significantly improve the readability and the content of the paper.


P Batocchi

F Mauvy

S Fourcade

M Parco

Electrical and electrochemical properties of architectured electrodes based on perovskite and A 2 MO 4 -type oxides for Protonic Ceramic Fuel Cell

Two mixed ionic-electronic conducting oxides (MIEC) have been investigated as potential cathode materials for protonic ceramic fuel cell (PCFC): the perovskite Ba0.5Sr0.5Co0.8Fe0.2O3-δ (BSCF) and the Ruddlesden Popper Pr2NiO4+δ (PrN). Their electrical properties have been studied over a large range of water vapour partial pressure.

All compounds exhibit high electronic conductivities (σ ≥ 40 S.cm -1 at 600 °C) whatever the pH2O of the surrounding atmosphere. Electrochemical characterizations have been performed as a function of pH2O, under zero dc conditions and under dc polarization using symmetrical cells based on BaCe0.9Y0.1O3-δ (BCY10) as electrolyte. For this purpose, two electrode architectures have been elaborated: a single phase electrode and a composite cathode/BCY10 architectured electrode. All electrodes showed pH2O-dependence with promising polarization resistance values lower than 0.8 Ω cm 2 at 600 °C under air whatever the gas humidification rate. The use of architectured electrodes led to a significant decrease of the polarization resistance with values as low as 0.23 and 0.19 Ω cm 2 for PrN and BSCF respectively, at 600 °C and pH2O = 0.20 bar. Concerning the oxygen reduction reaction (ORR) mechanisms, rate determining steps involving protons have been identified. They have been respectively assigned to the proton interface transfer and to the water formation and/or desorption for single phase and architectured electrodes. This change has been attributed to an extent of the electrochemically active area and to an enhancement of the protonic transport properties in the architectured electrodes. However electrodes performances seem to be governed by the dissociative adsorption of oxygen species and/or the charge transfer. Concerning performances under dc current, cathodic polarization is reduced when architectured electrodes are used. An enhancement of the electrodes performances has been also evidenced with water content increasing which corroborates the protonic conduction process into the cathode materials.

Introduction

Solid oxide fuel cells (SOFC) are considered as promising alternative power sources due to their high efficiency and low pollution emission. However their commercialization is limited by high operating temperatures (800-1000 °C) which cause accelerated thermal ageing of the materials and high cost of the components.

Consequently, great efforts have been devoted to reduce their operating temperature in the 400-600 °C range.

Such restriction mainly implies the development of alternative electrolyte and cathode materials able to operate in this intermediate temperature range without any drastic effect on cell performances. Since more than two decades, protonic ceramic fuel cells (PCFC) have been considered as an interesting alternative due to the lower activation energy of the protonic conduction (0.4-0.6 eV) [START_REF] Kreuer | Proton-Conducting Oxides[END_REF] compared to the classical oxygen ion conductors.

Iwahara et al. [START_REF] Iwahara | Proton conduction in sintered oxides and its application to steam electrolysis for hydrogen production[END_REF][START_REF] Uchida | Relation between proton and hole conduction in SrCeO 3 -based solid electrolytes under water-containing atmospheres at high temperatures[END_REF][START_REF] Iwahara | Proton Conduction in Sintered Oxides Based on BaCeO 3[END_REF] first reported protonic conduction in oxygen deficient perovskite oxides such as doped SrCeO3 and BaCeO3 by water incorporation into the structure according to the relation : Since then, acceptor-doped perovskite oxides formulated AM1-xM'xO3-δ such as barium cerates are considered as the best protonic conductors but suffer from poor chemical stability under acidic (CO2) and/or wet atmospheres [START_REF] Matsumoto | Relation Between Electrical Conductivity and Chemical Stability of BaCeO3-Based Proton Conductors with Different Trivalent Dopants[END_REF] . In order to enhance stability of these compounds, solid solutions between BaCeO3 and BaZrO3 have been widely investigated [START_REF] Ryu | Chemical stability and proton conductivity of doped BaCeO3-BaZrO3 solid solutions[END_REF][START_REF] Katahira | Protonic conduction in Zr-substituted BaCeO3[END_REF][START_REF] Guo | Zirconium doping effect on the performance of proton-conducting BaZryCe0.8-yY0.2O3-δ (0.0 ≤ y ≤ 0.8) for fuel cell applications[END_REF] . However, high conductivity and low reactivity are antagonist properties and so optimisation of compositions is crucial in order to find the best compromise for a PCFC electrolyte application.

Unlike SOFCs, water vapour is formed at the cathode side of a PCFC according to the relation [START_REF] Peng | Cathode processes and materials for solid oxide fuel cells with proton conductors as electrolytes[END_REF] : This specific feature prevents fuel dilution which can lead to a reduction of the cell efficiency in SOFCs, but makes the oxygen reduction reaction (ORR) and the water exhaust management more complex in the PCFC technology.

In addition, the lack of proper cathode materials in PCFCs often leads to the use of conventional mixed ionicelectronic (O 2-/e -) conducting oxides (so-called MIEC). Consequently, the triple phase boundary area (TPB) is confined to the electrolyte/electrode interface, leading to large cathodic polarization resistance. To extend the electrochemically active area, a suitable cathode material should possess a mixed H + /e -conductivity with a good electrocatalytic activity toward ORR. However up to now, protonic conductors with an electronic conductivity large enough have not been reported [START_REF] Fabbri | Electrode materials: a challenge for the exploitation of protonic solid oxide fuel cells[END_REF] .

In order to overcome this problem, several approaches have been considered. The most widespread one is to elaborate composite electrodes by mixing electrolyte and cathode materials. In addition to the enlargement of the active sites for the ORR, this approach has the advantages to improve the electrode adherence to the electrolyte and to reduce thermal expansion coefficient (TEC) mismatching between electrolyte and cathode layer. For this purpose, various perovskite-type oxides have been investigated such as Ba0.5Sr0.5Co0.8Fe0.2O3-δ (BSCF) [START_REF] Lin | Intermediate-to-low temperature protonic ceramic membrane fuel cells with Ba0.5Sr0.5Co0.8Fe0.2O3-δ idem for BaZ0.1Ce0.7Y0.2O3-δ composite cathode[END_REF] , La0.6Sr0.4Fe0.8Co0.2O3-δ (LSFC) [START_REF] Fabbri | Composite Cathodes for Proton Conducting Electrolytes[END_REF][START_REF] Yang | A mixed proton, oxygen ion, and electron conducting cathode for SOFCs based on oxide proton conductors[END_REF] or Sm0.5Sr0.5CoO3-δ (SSC ) [START_REF] Yang | A Novel Composite Cathode for Low-Temperature SOFCs Based on Oxide Proton Conductors[END_REF][START_REF] Wu | Sm0.5Sr0.5CoO3 -δ-BaCe0.8Sm0.2O3-δ composite cathodes for proton-conducting solid oxide fuel cells[END_REF][START_REF] He | Cathode reaction models and performance analysis of Sm0.5Sr0.5CoO3-δ-BaCe0.8Sm0.2O3-δ composite cathode for solid oxide fuel cells with proton conducting electrolyte[END_REF][START_REF] Yang | High-performance anode-supported Solid Oxide Fuel Cells based on Ba(Zr0.1Ce0.7Y0.2)O3-δ (BZCY) fabricated by a modified co-pressing process[END_REF][START_REF] Wu | Nano-sized Sm0.5Sr0.5CoO3-δ as the cathode for solid oxide fuel cells with proton-conducting electrolytes of BaCe0.8Sm0.2O2.9[END_REF] . Development of efficient composite electrodes requires optimization in terms of composition and elaboration process. Indeed, percolation of both proton conducting and MIEC phases is essential to not affect the electrocatalytic activity of the electrode. Wu et al. [START_REF] Wu | Sm0.5Sr0.5CoO3 -δ-BaCe0.8Sm0.2O3-δ composite cathodes for proton-conducting solid oxide fuel cells[END_REF] studied the influence of the SSC content on the P a g e | 3 polarization resistance using composite cathodes based on BaCe0.8Sm0.2O3-δ (BCS) electrolyte, elaborated by screen-printing. Values decreased with MIEC insertion until a minimum of 0.67 Ω cm 2 at 600 °C at a 60 wt.% content. According to the literature data, it can be considered that a weight ratio between 60 and 70% of cathode material should be enough to create conduction pathways for oxygen ions, protons and electrons and so to reach good electrode performances. Obviously, this ratio depends on the density and particle size of the materials, and on the elaboration process.

Considering that most of MIEC oxides exhibit oxygen non-stoichiometry, an other approach could be to consider a possible mixed H + /O 2-/e -conduction in wet conditions. For this purpose, recent works [START_REF] Dailly | Perovskite and A2MO4-type oxides as new cathode materials for protonic solid oxide fuel cells[END_REF][START_REF] Grimaud | Hydration Properties and Rate Determining Steps of the Oxygen Reduction Reaction of Perovskite-Related Oxides as H + -SOFC Cathodes[END_REF] have investigated electrochemical properties of several perovskite-related compounds under moist atmosphere. The highly oxygen deficient perovskite BSCF showed attractive water insertion abilities with respect to a large amount of vacant sites [20]. On the other hand, over-stoichiometric layered compounds such as Ruddlesden-Popper series Ln2NiO4+δ and double perovskites LnBaCo2O5+δ, have drawn the attention because of their interesting oxygen transport properties with respect to their anisotropic oxygen diffusion and cation ordered structures [START_REF] Bassat | Anisotropic ionic transport properties in La2NiO4+δ single crystals[END_REF][START_REF] Boehm | Oxygen diffusion and transport properties in non-stoichiometric Ln2-xNiO4+δ oxides[END_REF][START_REF] Tarancon | Layered perovskites as promising cathodes for intermediate temperature solid oxide fuel cells[END_REF][START_REF] Taskin | Achieving fast oxygen diffusion in perovskites by cation ordering[END_REF][START_REF] Boehm | Oxygen transport properties of La2Ni1-xCuxO4+δ mixed conducting oxides[END_REF] . These oxides exhibit high values of surface exchange and oxygen diffusion coefficients with low activation energies which make them suitable for a PCFC cathode application. Particularly, the Pr2NiO4+δ (PrN) composition has been found to be a possible mixed H + /O 2-/e -conducting oxide with a low polarization resistance when cooperates with a proton conductor (BCY10) [START_REF] Grimaud | Hydration Properties and Rate Determining Steps of the Oxygen Reduction Reaction of Perovskite-Related Oxides as H + -SOFC Cathodes[END_REF] .

In this work, a combination of these two approaches is proposed in order to reach good electrode performances.

Referring to previous results [START_REF] Dailly | Perovskite and A2MO4-type oxides as new cathode materials for protonic solid oxide fuel cells[END_REF], [START_REF] Grimaud | Hydration Properties and Rate Determining Steps of the Oxygen Reduction Reaction of Perovskite-Related Oxides as H + -SOFC Cathodes[END_REF], [START_REF] Dailly | Electrochemical properties of perovskite and A2MO4-type oxides used as cathodes in protonic ceramic half cells[END_REF] , two promising MIEC oxides have been selected as cathode materials: the perovskite Ba0.5Sr0.5Co0.8Fe0.2O3-δ (BSCF) and the Ruddlesden-Popper Pr2NiO4+δ (PrN) with the K2NiF4-type structure. As a first step, their electrical properties have been studied over a large range of water partial pressure (pH2O). In a second step, electrochemical studies under zero dc conditions have been performed in wet air using symmetrical cells with BaCe0.9Y0.1O3-δ (BCY10) as electrolyte material. For this purpose two electrode architectures have been investigated: a single phase electrode based on pure MIEC oxide, and an architectured one based on composite BCY10-cathode materials. Concerning the architectured electrodes, the goal is not only to improve the mechanical adherence near the electrode/electrolyte interface and to reduce thermal expansion coefficient (TEC) mismatching but also to optimize the electrochemical properties by the help of the microstructure tuning.

For each cell, electrochemical properties against the pH2O have been analysed in terms of polarization resistances and rate-determining steps involved in the ORR mechanisms. Finally, electrochemical measurements under dc polarization have been carried out to study architectured electrodes performances.

Experimental

Structural characterization

Commercial powders provided by the company Marion Technologies were used as electrolyte and cathode materials. The powders were characterized by X-ray diffraction (XRD) at room temperature on a PANalytical X'pert MPD Bragg-Brentano θ-θ geometry diffractometer, over an angular range of 2θ = 8-120°, using Cu-Kα radiation. All materials were found to be single phases and were refined in orthorhombic (BCY10, PrN) or cubic (BSCF) space groups with the FullProf program to obtain the lattice parameters. Results from this work are listed in Table 1 and are in agreement with the literature data [START_REF] Takeuchi | The crystal structures and phase transitions in Y-doped BaCeO3: their dependence on Y concentration and hydrogen doping[END_REF][START_REF] Allançon | Influence of oxygen on structural transitions in Pr2NiO4+δ[END_REF][START_REF] Mcintosh | Oxygen stoichiometry and chemical expansion of Ba0. 5Sr0. 5Co0. 8Fe0. 2O3-δ measured by in situ neutron diffraction[END_REF] . P a g e | 4

Chemical reactivity

From a general point of view, the chemical reactivity between electrolyte and cathode materials is considered as an important issue for the fuel cell performances [START_REF] Fabbri | Electrode materials: a challenge for the exploitation of protonic solid oxide fuel cells[END_REF] . Thus, it is well known that a high reactivity can promote the formation of insulating phases at the electrolyte/electrode interface leading to a lowered ionic transfer and consequently to a drastic increase of the polarization resistance. In addition, a poor cathode layer adhesion can have a detrimental effect on the interface area, contributing to increase the polarization resistance and inducing a delamination process.

According to these considerations, the chemical reactivity between BCY10 electrolyte and selected cathode materials has been investigated under sintering conditions. Powders of electrolyte and cathode materials were thoroughly mixed in a (1:1) weight ratio and then pressed into pellets in order to exacerbate the reactivity. The pellets were then fired in air, at 1200 °C and 1100 °C for 1 hour for the PrN-BCY10 and BSCF-BCY10 mixtures, respectively. XRD patterns of powder mixtures before and after heat treatment have been reported in Fig. 1. Results show that BSCF does not obviously react with BCY10 after heat treatment while reaction between electrolyte material and PrN led to the formation of a small amount of secondary phases like Ce1-xPrxO2-δ and NiO. In addition, the presence of a phase-type Pr2-xBaxNiO4+δ suggests possible barium diffusion in the nickelate structure. Ce1-xPrxO2-δ fluorite-type oxides are considered to be mixed ionic-electronic conductors and have been previously investigated as potential active layers in SOFCs [START_REF] Chiba | High temperature properties of Ce1-xPrxO2-δ as an active layer material for SOFC cathodes[END_REF] . Thus, it can be supposed in this work that the

P a g e | 5
formation of such a phase is not detrimental to the ORR and in addition, could promote the electrode adherence to the electrolyte. Similarly a limited diffusion of barium into the nickelate structure (x ≤ 0.3) should not significantly affect electrochemical properties of the electrode [START_REF] Chung | Synthesis, structural and electrical characterizations of Pr2-xBaxNiO4+δ[END_REF] .

Preparation of dense ceramics

In order to perform electrical and electrochemical measurements described below, ceramics with compacities higher than 94% are required to avoid the presence of open porosity [START_REF] Kingery | Introduction to Ceramics[END_REF] . For this purpose, fine powders with a small particle size (Dv(0.5) ∼ 0.40 μm for all compositions) as well as optimized shaping and thermal treatments remain basic conditions.

For the cathode materials, powders were ground using an agate mortar and isostatically pressed into pellets (20 mm diameter, 1-2 mm thick) at 3000 bars for 5 min. The pellets were then sintered in air at 1350 °C for 4 hours and 1200 °C for 12 hours respectively for PrN and BSCF. After sintering, they were manually polished with a 4000 grade SiC paper in order to get mirror surfaces. The densities of the as-prepared ceramics were determined by the geometric method and were higher than 95% of the theoretical values.

In the case of electrolyte ceramics, BCY10 powder was in the same way isostatically pressed at 3000 bars for 5 min into 20 mm diameter pellets and then sintered in air at 1400 °C for 10 hours. The surface of the pellets was finally roughened with a 600 grade SiC paper to get uniform surface conditions. All the sintered BCY10 ceramics have a diameter of ∼ 15.5 mm, a thickness of ∼ 1 mm and a densification rate upper than 95%.

Elaboration of symmetrical cells

In order to study different architectured electrodes, two symmetrical cells have been elaborated as described in Fig. 2. The first one, named single phase electrode, consists of a two-layer deposition of the cathode material, while the so-called architectured electrode is made of a composite BCY10-cathode interface layer associated with a single phase top layer. In the second configuration, the composite interface layer is expected to increase the protonic conductivity in the cathode and to enhance both ionic transfer and mechanical compatibility (adhesion, TEC) at the electrolyte/electrode interface as demonstrated in the litterature [START_REF] Yang | A mixed proton, oxygen ion, and electron conducting cathode for SOFCs based on oxide proton conductors[END_REF], [START_REF] Wu | Sm0.5Sr0.5CoO3 -δ-BaCe0.8Sm0.2O3-δ composite cathodes for proton-conducting solid oxide fuel cells[END_REF], [START_REF] Zhao | Reaction model for cathodes cooperated with oxygen-ion conductors for solid oxide fuel cells using protonconducting electrolytes[END_REF] . Finally, a single phase top layer is added to ensure a good current-collection. In order to get homogeneous composite electrodes, electrolyte and cathode powders have been thoroughly mixed in a (3:7) weight ratio into a mortar and then ball-milled at 500 rpm for 1 hour.

Screen-printing method was selected for the electrodes deposition. Inks were first elaborated by mixing cathode materials powders into a solvent with dispersant (terpineol) and binder (ethylcellulose) commercial agents. The ink formulation has been optimized both in terms of particle size distribution and composition [START_REF] Ferchaud | High performance praseodymium nickelate oxide cathode for low temperature solid oxide fuel cell[END_REF] . Before deposition, a three-cylinder grinding step was done to avoid powder agglomeration and to ensure homogeneous inks. Electrodes were then screen-printed on both sides of dense BCY10 pellets and sintered in air for 1 hour at 1100 °C for BSCF single phase and architectured BSCF-BCY10 electrodes, and at 1170 and 1200 °C for PrN single phase and PrN-BCY10 architectured electrodes respectively. The sintering conditions have been optimized to get porous cathodes without excessive reactivity between materials, while keeping a good adhesion to the electrolyte pellet. An initial ramp of 1 °C min -1 up to 400 °C is used to eliminate organic binders, followed by a heating rate of 3 °C min -1 up to the sintering temperature.

Fig. 2. Schematic representation of the two symmetrical cell configurations tested in this study.

After sintering, the electrodes microstructure was observed by field emission scanning electron microscopy (SEM) using a JEOL JSM 6330F equipped with an energy dispersive spectrometer (EDS) detector. Micrographs of PrNbased symmetrical cells are given as an example in Fig. 3. From cross-section views, a good adherence of the electrodes has been evidenced with an average thickness of about 30 μm. Foremost, homogeneous microstructures were found in any cases with a fine distribution of grain size and pore volume. However coarser particles have been observed for single phase layers, with a mean size around 1 μm, against 0.5 μm for the composite ones. The porosity has been calculated from image analysis using Image J software, and is close to 30 and 20% for the single phase and the composite layers respectively. Such porosity levels can be considered high enough for gas diffusion and water vapour exhaust. Otherwise, one can note the presence of an interface layer for the PrN single phase symmetrical cells, which was identified as a cerium-rich phase Ce1-xPrxO2-δ-type by EDS analysis. This result confirms the reactivity of the nickelate with the BCY10 electrolyte previously highlighted with XRD measurements. However the impact of this interface layer on the electrochemical performances has not been identified [START_REF] Grimaud | Hydration Properties and Rate Determining Steps of the Oxygen Reduction Reaction of Perovskite-Related Oxides as H + -SOFC Cathodes[END_REF], [START_REF] Dailly | Electrochemical properties of perovskite and A2MO4-type oxides used as cathodes in protonic ceramic half cells[END_REF] in so far as this ceria based compound seems to be a good ionic conductor and to enhance the electrode adherence. It can be noticed that this phase does not appear in the case of the architectured electrode.

P a g e | 7

Electrochemical measurements

The polarization resistance and overpotential versus current density curves of the as-prepared electrodes were determined by electrochemical impedance spectroscopy (EIS) and voltamperometry measurements.

Measurements were performed either in two-electrode configuration (under zero dc conditions) or in threeelectrode configuration (under dc polarization) as depicted in Fig. 4. For the three-electrode configuration (Fig. 4b), a specific cell-design was adopted: working and counter electrodes were screen-printed on both sides of the electrolyte pellet in a symmetrical way, covering the half area of the ceramic. A platinum electrode was then painted on the same side of the working electrode, at a distance larger than five times the electrolyte thickness in order to reduce the current lines interactions [START_REF] Adler | Reference Electrode Placement in Thin Solid Electrolytes[END_REF][START_REF] Cimenti | Distortions in Electrochemical Impedance Spectroscopy Measurements Using 3-Electrode Methods in SOFC. I-Effect of Cell Geometry[END_REF][START_REF] Adler | Reference electrode placement and seals in electrochemical oxygen generators[END_REF] . EIS measurements were achieved with an Autolab PGSTAT 30 equipped with a frequency response analyzer (FRA), in the range of 0.01 Hz to 1 MHz applying 50 mV signal amplitude. Gas humidification rate was adjusted in the range 0.03 ≤ pH2O ≤ 0.30 bars, by bubbling air into a water tank heated at a given temperature.

Measurements were carried out with respect to an equilibrium time of about 2 hours and during heating and cooling thermal cycles. Impedance diagrams were fitted using Zview™ software (Scribner Associates).

Results and discussion

Electrical measurements

The total electrical conductivity of the cathode materials was determined by the direct current four-probe method on dense pellets [38][39] . Measurements were performed under air, in the temperature range 100-700 °C and over a large range of water partial pressure (0.10-0.30 bars). Electrical conductivity has been reported in Fig. 5 as a function of the inverse temperature, for cooling and heating cycles. Both phases exhibit semi-conducting type behaviour: electronic conductivity is increasing until 500-550 °C, and then slightly decreases due to the loss of oxygen in structures and consequently to the diminution of the charge carriers density as evidenced in previous studies [START_REF] Dailly | Perovskite and A2MO4-type oxides as new cathode materials for protonic solid oxide fuel cells[END_REF], [START_REF] Boehm | Oxygen diffusion and transport properties in non-stoichiometric Ln2-xNiO4+δ oxides[END_REF], [START_REF] Bassat | The Semiconductor-to-Metal Transition in Question in La2-xNiO4+δ (δ > 0 or δ < 0)[END_REF] . In addition, an hysteresis was observed for BSCF curves until 600 °C resulting from an oxygen stoichiometry variation during cooling and heating cycles, previously reported by thermogravimetric analyses [START_REF] Dailly | Perovskite and A2MO4-type oxides as new cathode materials for protonic solid oxide fuel cells[END_REF][START_REF] Grimaud | Hydration Properties and Rate Determining Steps of the Oxygen Reduction Reaction of Perovskite-Related Oxides as H + -SOFC Cathodes[END_REF] . From data analysis, conductivity of PrN was found to be approximately three times higher than the BSCF one, with values reaching respectively 167 and 54 S cm -1 at 600 °C under dry atmosphere. Furthermore, water steam introduction did not significantly affect electrical properties as shown by the conductivities measured under high humidification rates (0.30 bar H2O), resulting in 153 and 47 S cm -1 at 600 °C for PrN and BSCF compounds, respectively.

These results suggest that expected hydroxyle species formation inside the structures due to water insertion [19- 20] does not modify the concentration of holes and so electrical properties of these compounds remain unchanged. Moreover, these materials exhibit an electrical conductivity higher than 40 S cm -1 at 600 °C whatever the pH2O, which makes them suitable for application in a fuel cell.

Electrochemical measurements under zero dc conditions

EIS measurements were performed under zero dc conditions i.e. at thermodynamical equilibrium, as a function of temperature and water partial pressure. From a general point of view, impedance diagrams show several contributions which can be assigned to specific phenomena. A typical Nyquist plot recorded at 600 °C under wet air (0.20 bar H2O) for a BCY10/PrN symmetrical cell is reported in Fig. 6a. Referring to previous studies [START_REF] Dailly | Perovskite and A2MO4-type oxides as new cathode materials for protonic solid oxide fuel cells[END_REF], [START_REF] Grimaud | Hydration Properties and Rate Determining Steps of the Oxygen Reduction Reaction of Perovskite-Related Oxides as H + -SOFC Cathodes[END_REF], [START_REF] Mauvy | Electrode properties of Ln2NiO4+δ (Ln = La, Nd, Pr): AC Impedance and DC Polarization Studies[END_REF] , experimental data were fitted using equivalent circuits constituted of a series resistance (Rs) and resistanceconstant phase elements (R-CPE) in parallel associated in series. In the high frequency (HF) range, the electrolyte response can be modelled in most diagrams, by a series resistance whose value is in agreement with the protonic conductivity of the BCY10 in wet atmosphere (∼ 10 -2 S cm -1 at 600 °C) [START_REF] Ma | Ionic conduction and nonstoichiometry in BaxCe0.90Y0.10O3-δ[END_REF][START_REF] Coors | Proton Conductivity Measurements in Yttrium Barium Cerate by Impedance Spectroscopy[END_REF][START_REF] Oishi | Defect structure analysis of B-site doped perovskite-type proton conducting oxide BaCeO3: Part 2: The electrical conductivity and diffusion coefficient of BaCe0.9Y0.1O3-δ[END_REF] . The two depressed semi-circles observed in the middle (MF) and low (LF) frequency ranges corresponds to the electrode response and have been fitted using two R-CPE elements. Polarization resistance (Rp) is calculated using the following formula:

where RMF and RLF are the polarization resistances of MF and LF contributions respectively, and A the electrode surface area.

From fitted data, equivalent capacitances (Ceq) and relaxation frequencies (frelax) of each contribution have been determined using relations (4) and ( 5) [START_REF] Dailly | Perovskite and A2MO4-type oxides as new cathode materials for protonic solid oxide fuel cells[END_REF] .

where p is a parameter used to compensate for non-homogeneity in the system (with a value between 0.9 and 1).

Calculated values have been systematically plotted as a function of the temperature, in the Schouler-type representation, which can be considered as a reference for interpreting an experimental impedance diagram [START_REF] Schouler | In situ study of the sintering process of yttria stabilized zirconia by impedance spectroscopy[END_REF] .

Results obtained for the PrN single phase electrode are reported in Fig. 6b and6c.

With respect to equivalent capacitance and relaxation frequency values, two electrochemical processes can be distinguished. Constant capacitances and a linear temperature dependence of the associated frequencies indicate also that involved phenomena seem to remain the same all over the studied temperature range. The middle-frequency contribution (10 3 -10 5 Hz) is characterized by low capacitance values, approximately 10 -5 F.

According to previous results, the related electrochemical process could be interpreted as the electrolyte/electrode interface ionic transfer [START_REF] Dailly | Perovskite and A2MO4-type oxides as new cathode materials for protonic solid oxide fuel cells[END_REF], [START_REF] Mauvy | Electrode properties of Ln2NiO4+δ (Ln = La, Nd, Pr): AC Impedance and DC Polarization Studies[END_REF] . The low frequency contribution (0.01-10 Hz) which is defined by higher capacitance values (∼ 10 -1 F), could be assigned to the charge transfer and to the electrode reaction process.

While quite similar results have been found for the BCY10/BSCF symmetrical cell, differences were highlighted in the case of the architectured ones. More precisely, higher capacitance values (∼ 10 -2 F) and a lower frequency range (10 1 -10 3 Hz) were obtained for the MF contribution. Such a difference could be explained by an improvement of the electrolyte/cathode ionic transfer due to the composite interface layer. Otherwise, no differences were found regarding the LF contribution, suggesting that the involved process is similar.

Thermal variation of the MF and LF resistances has been investigated. Results obtained for the PrN single phase electrode are shown in Fig. 6d. On the whole, RMF and RLF decrease with increasing temperature due to the thermal activation of the related electrochemical reactions. Concerning single phase electrodes, the simulated activation energies of RMF are around 0.6 eV which are closed to the values reported for proton conductivity in BCY10 under wet air [START_REF] Ma | Ionic conduction and nonstoichiometry in BaxCe0.90Y0.10O3-δ[END_REF], [START_REF] Grimaud | Transport properties and in-situ Raman spectroscopy study of BaCe0.9Y0.1O3-δ as a function of water partial pressures[END_REF] . This implies that the associated electrochemical process might involve protons, which is consistent with the interface ionic transfer assumption. In any cases, the low-frequency contribution exhibits the highest resistance and activation energy values, suggesting that electrode reaction might be the rate-limiting step.

Comparison of the electrochemical performances of single phase and architectured electrodes at fixed pH2O (0.20 bar)

Arrhenius plots of the polarization resistances of PrN and BSCF single phase electrodes are reported in Fig. 7a.

Despite higher activation energy, PrN electrode exhibits the lowest polarization resistance for temperatures above 500 °C. Corresponding values reach respectively 0.32 and 0.60 Ω cm 2 for PrN and BSCF electrodes at 600 °C, which are significantly lower than those reported in previous works [START_REF] Dailly | Electrochemical properties of perovskite and A2MO4-type oxides used as cathodes in protonic ceramic half cells[END_REF], [START_REF] Quarez | Optimization of the Lanthanum Tungstate/Pr2NiO4 Half Cell for Application in Proton Conducting Solid Oxide Fuel Cells[END_REF], [START_REF] Lin | Silver-modified Ba0.5Sr0.5Co0.8Fe0.2O3-δ as cathodes for a proton conducting solid-oxide fuel cell[END_REF] .

Influence of the water partial pressure on the electrochemical properties

In order to study the influence of the water partial pressure on the electrochemical properties, the variation of the polarization resistance has been investigated over a large range of pH2O (0.03-0.30 bars). Fig. 8 shows the results obtained under air at 600 °C. All cathodes exhibit pH2O-dependence and whatever the gas humidification rate, promising Rp values < 0.8 Ω cm 2 have been obtained. Because of water formation at the cathode side, the oxygen reduction reaction is assumed to be more complex in PCFCs than in SOFCs. Using platinum electrode as PCFC cathode, Uchida et al. [START_REF] Uchida | Polarization at Pt electrodes of a fuel cell with a high temperature-type proton conductive solid electrolyte[END_REF] proposed at first the elementary steps involved in the ORR, by adding the formation and evolution of water to the dissociative adsorption and diffusion of oxygen species. Since then, many works have been dedicated to the description of the ORR mechanisms in cathodes based on MIEC O 2-/e -oxides [START_REF] Peng | Cathode processes and materials for solid oxide fuel cells with proton conductors as electrolytes[END_REF], [START_REF] He | Cathode reaction models and performance analysis of Sm0.5Sr0.5CoO3-δ-BaCe0.8Sm0.2O3-δ composite cathode for solid oxide fuel cells with proton conducting electrolyte[END_REF], [START_REF] Zhao | Reaction model for cathodes cooperated with oxygen-ion conductors for solid oxide fuel cells using protonconducting electrolytes[END_REF], [START_REF] Lin | Silver-modified Ba0.5Sr0.5Co0.8Fe0.2O3-δ as cathodes for a proton conducting solid-oxide fuel cell[END_REF] . Various models have been suggested

including additional reactions related to the transfer and reaction of protons and implying a variable number of elementary steps. The polarization resistance of each of these steps depends on the oxygen and/or water vapour partial pressures, and can be expressed as follow:

where Ri is the polarization resistance related to the step i, and mi and ni the corresponding reaction orders with respect to pO2 and pH2O respectively.

With respect to the scheme proposed by Grimaud et al. [START_REF] Grimaud | Hydration Properties and Rate Determining Steps of the Oxygen Reduction Reaction of Perovskite-Related Oxides as H + -SOFC Cathodes[END_REF] , the oxygen reduction reaction at a PCFC cathode can be decomposed into six elementary steps which are listed in Table 2 with their respective reaction orders.

These elementary steps include the adsorption of oxygen gas on the electrode surface (step 1) and the oxygen molecular dissociation (step 2), followed by the oxygen diffusion with charge transfer reaction (step 3). Then, the proton transfer from the electrolyte to the cathode is described by the step 4. Finally the water formation and desorption are defined by steps 5 and 6 respectively.

P a g e | 13

To go further into the interpretation of the electrochemical processes previously evidenced, the variation of the polarization resistance versus water partial pressure has been investigated at reduced temperature (450 °C) in order to better identify the different contributions of the impedance diagrams. As EIS measurements were achieved under air, only the pH2O influence has been studied, considering that the oxygen partial pressure did not vary significantly (pO2 ∼ 0.21 bar).

Impedance diagrams recorded for the architectured PrN electrode are reported in Fig. 9. In order to simplify the comparison the ohmic drop, related to the electrolyte contribution, has been subtracted. As seen in Fig. 8, it can be noticed a notable reduction of the polarization resistance when water partial pressure is increasing with values starting from 6.81 Ω cm 2 at 0.03 bar to 2.88 Ω cm 2 at 0.30 bar of steam water. indicate that there are no significant changes in the cathode materials composition and in the ionic transport properties. It can be added that a good mechanical strength of the electrodes and interfaces microstructures is obtained, even at high water content. As described before, the MF contribution of the architectured electrode shows higher capacitance and lower frequency values (Fig. 10b) probably due to a modification in the electrochemical process. Concerning MF polarization resistance, one can observe a pH2O -1/2 dependence for the PrN single phase electrode (Fig. 10c), suggesting that the associated rate determining step can be assigned to the proton transfer at the electrolyte/electrode interface (step 4). This result is consistent with the activation energy of RMF (0.6 eV), closed to the BCY10 conductivity. Focusing on the PrN architectured electrode, a pH2O -1 dependence is nearly observed, indicating that the rate determining step is more related to the water formation (step 5) or desorption (step 6). In addition, linear variations of the MF frequencies reported in Fig. 10b are in agreement with the calculated reaction orders. For BSCF-based electrodes, pH2O dependences were evidenced for RMF but related changes could not be correlated to specific reaction orders.

For both electrodes, the LF contribution seems to be associated to the same process with respect to capacitances and relaxation frequencies and does not depend on pH2O. So, the associated elementary reaction could be assigned to the oxygen dissociative adsorption or to the charge transfer (step 1-3). In addition, the electrodes performances seem to be governed by the low frequency contribution which exhibits in any cases the highest resistance values. Similar results have been obtained for the BSCF-based electrodes.

On the whole, performances of single phase and architectured electrodes are significantly promoted by the gas hydration rate in the temperature range studied. Concerning the MF contribution, the rate determining step seems to be the proton interface transfer in the case of single phase electrodes while for architectured electrodes, it can rather be assigned to the water formation/desorption. These results suggest firstly a possible protonic conduction in the MIEC phases and secondly an enhancement of the protonic transport properties into the electrodes when composite interface layers are used. On the other hand, with respect to the LF contribution features, the main part of the polarization resistance can be attributed to elementary reactions involving oxygen species. It can be concluded that even if a protonic conduction appears in the BSCF and PrN phases, it should be limited, considering that these oxides are predominantly mixed O 2-/e -conductors.

Electrochemical measurements under dc polarization

Cathodic overpotential versus current density curves of the architectured electrodes were determined by EIS measurements under dc polarization using the three-electrode configuration described before (Fig. 4b).

Impedance diagrams were recorded over a defined dc potential range (0 ≤ EWR ≤ 1.5 V) applied between working and reference electrodes. The resulting cathodic overpotentials (ηcath) were calculated as follow:

where Rs (Ω cm 2 ) is the ohmic drop across the electrolyte measured by EIS and idc (A cm -2 ), the resulting current density. Fig. 11 shows the polarization curves of architectured electrodes under wet air at 600 °C. Referring to the work of Grimaud et al. [START_REF] Grimaud | Hydration Properties and Rate Determining Steps of the Oxygen Reduction Reaction of Perovskite-Related Oxides as H + -SOFC Cathodes[END_REF] , comparison has been made with results obtained on cells based on PrN and BSCF single phase cathode materials associated with BCY10 electrolyte. From polarization curves analysis, it can be seen a large gain on cathodic overpotential when using optimized electrodes. On the one hand, as seen for EIS measurements under zero dc conditions, PrN single phase led to the best performances in the quoted study [START_REF] Grimaud | Hydration Properties and Rate Determining Steps of the Oxygen Reduction Reaction of Perovskite-Related Oxides as H + -SOFC Cathodes[END_REF] . On the other hand, when architectured electrodes are used, the BSCF one exhibits the lowest overpotential. In this case positive overpotential values can be obtained for current densities lower than 0.1 A cm -2 . Such behaviour is due to the close values of ohmic losses (Rs × idc) and applied potential (EWR). It can be deduced that good electrochemical performances have been reached. From polarization curves, apparent current exchange densities i0 have been estimated around 22 mA cm -2 for PrN and upper than 100 mA cm -2 for BSCF architectured electrodes at 600 °C under wet air. These values are significantly larger than those reported by Dailly at al. [START_REF] Dailly | Electrochemical properties of perovskite and A2MO4-type oxides used as cathodes in protonic ceramic half cells[END_REF] for single phase PrN and BSCF cathodes in the same conditions, 8.1 and 6.5 mA cm -2 respectively.

In addition, the influence of the water vapour partial pressure has been investigated. For both electrodes, cathodic overpotential clearly decreases with increasing water content, as illustrated in the case of the PrN architectured electrode (Fig. 12). For a given current density of 0.07 A cm -2 , measured ηcath values reach -0.029, -0.017 and -0.009 V for 0.03, 0.15 and 0.25 bar of steam water, respectively. It can be concluded that high water partial pressures promote electrodes performances as previously evidenced, suggesting a protonic conduction process in the electrode materials.

Conclusion

On the basis of previous studies, two promising MIEC oxides have been selected as cathode materials for PCFC application: the perovskite Ba0.5Sr0.5Co0.8Fe0.2O3-δ (BSCF) and the Ruddlesden Popper Pr2NiO4+δ (PrN). Chemical reactivity in sintering conditions with the reference BaCe0.9Y0.1O3-δ (BCY10) electrolyte material revealed that BSCF was stable while small amount of secondary phases has been evidenced for PrN. Electrical conductivity of the selected materials was investigated as a function of water vapour partial pressure. Steam introduction did not affect the electrical conductivity and compositions kept values higher than 40 S cm -1 at 600 °C.

Electrochemical characterizations were performed on symmetrical cells BCY10/cathode elaborated by screenprinting. Two electrode architectures have been elaborated: a so-called single phase electrode, consisting of pure cathode material, and an architectured one, including a composite interface layer. Electrochemical measurements were carried out over a large range of pH2O under zero dc conditions and under dc polarization.

PrN single phase electrode exhibited the lowest polarization resistance with 0.32 Ω cm 2 at 600 °C under wet air (0.20 bar H2O), against 0.60 Ω cm 2 for BSCF. A large decrease has been obtained using architectured electrodes with values as low as 0.23 and 0.19 Ω cm 2 , for PrN and BSCF, respectively tested under the same conditions.

The characterization of electrochemical properties as a function of water vapour partial pressure revealed that all cathodes showed pH2O-dependence. Even if electrodes performances seem to be governed by the dissociative adsorption of oxygen species and/or the charge transfer, a proton conduction mechanism can be supposed for both MIEC phases. In addition, the use of composite architectured electrodes led to an improvement of the protonic transport properties into the cathodes and consequently to an extent of the electrochemically active area. Concerning performances under dc current, the same conclusion can be done as so far as cathodic polarization is reduced when architectured electrodes are used and when increasing water content.

Fig. 1 .

 1 Fig. 1. XRD patterns of PrN-BCY10 (a) and BSCF-BCY10 (b) as-prepared mixtures, and PrN-BCY10 (c) and BSCF-BCY10 (d) after heat treatment.

Fig. 3 .

 3 Fig. 3. Typical SEM micrographs of the elaborated symmetrical cells after sintering: BCY10/PrN single phase (a), BCY10/PrN architectured (b) as an example.

Fig. 4 .

 4 Fig. 4. Schematic representation of the two-electrode (a) and three-electrode (b) configurations used for EIS measurements.

Fig. 5 .

 5 Fig. 5. Thermal variation of the electrical conductivity under dry (full symbols) and humidified-0.30 bar H2O (open symbols) air.

Fig. 6 .

 6 Fig. 6. Typical impedance diagram obtained at 600 °C (a), thermal variation of capacitances (b) and relaxation frequencies (c) of the identified middle and low frequency phenomena and Arrhenius plots of the total, middle and low frequency resistances (d) under wet air (0.20 bar H2O) for BCY10/PrN symmetrical cell.

P a g e | 11 Fig. 7 .

 117 Fig. 7. Arrhenius plots of the polarization resistances for single phase (a) and architectured (b) PrN and BSCF electrodes under wet air (0.20 bar H2O).

Fig. 8 .

 8 Fig. 8. Polarization resistances variations as a function of water partial pressure at 600 °C.

Fig. 9 .

 9 Fig. 9. Impedance diagrams recorded at 450 °C in the water partial pressure range 0.03-0.30 bar, for the architectured PrN electrode.

P a g e | 14 P a g e | 15 Fig. 10 .

 141510 Fig. 10. Variations as a function of pH2O at 450 °C, of MF and LF capacitances (a), relaxation frequencies (b) and resistances (c) for single phase (■) and architectured (□) PrN electrodes.

P a g e | 16 Fig. 11 .

 1611 Fig. 11. Cathodic overpotential vs. idc under wet air (0.03 bar H2O) at 600 °C for PrN and BSCF-based electrodes.

Fig. 12 .

 12 Fig. 12. Cathodic overpotential vs. idc as a function of pH2O at 600 °C for PrN architectured electrode.

Acknowledgments

The research leading to these results has received funding from the European Community's Seventh Framework Programme (FP7/2011-2014) for the Fuel Cells and Hydrogen Joint Technology Initiative under grant agreement METPROCELL n°277916.