Mikhail Menshikov
email: khail.menshikov@durham.ac.uk

Dimitri Petritis
email: dimitri.petritis@univ-rennes1.fr

Explosion, implosion, and moments of passage times for continuous-time Markov chains : a semimartingale approach 1

Keywords: 1991 Mathematics Subject Classification : 60J10, 60K20 Continuous-time Markov chain, recurrence criteria, explosion criteria, moments of passage times, implosion. 1

We establish general theorems quantifying the notion of recurrencethrough an estimation of the moments of passage times -for irreducible continuous-time Markov chains on countably infinite state spaces. Sharp conditions of occurrence of the phenomenon of explosion are also obtained. A new phenomenon of implosion is introduced and sharp conditions for its occurrence are proven. The general results are illustrated by treating models having a difficult behaviour even in discrete time.

1.

Résumé

Nous établissons des théorèmes généraux qui quantifient la notion de récurrence -à travers l'estimation des moments de temps de passagepour des chaînes de Markov à temps continu sur des espaces d'états dénombrablement infinis. Des conditions fines garantissant l'apparition du phénomène d'explosion sont obtenues. Un nouveau phénomène d'implosion est introduit et des conditions fines pour son apparition sont aussi démontrées. Les résultats généraux sont illustrés sur des modèles ayant un comportement non-trivial même lorsqu'ils sont considérés en temps discret.

Introduction, notation, and main results

Introduction

In this paper we establish general theorems quantifying the notion of recurrence -by studying which moments of the passage time exist -for irreducible continuous-time Markov chains ξ = (ξ t ) t ∈[0,∞[ on a countable space X in critical regimes.

Models of discrete-time Markov chains with non-trivial behaviour include reflected random walks in wedges of dimension d = 2 [START_REF] Aspandiiarov | Passage-time moments for nonnegative stochastic processes and an application to reflected random walks in a quadrant[END_REF]5,[START_REF] Menshikov | Markov chains in a wedge with excitable boundaries[END_REF], Lamperti processes [START_REF] Lamperti | Criteria for the recurrence or transience of stochastic process[END_REF][START_REF] Lamperti | Criteria for stochastic processes. II. Passage-time moments[END_REF], etc. These chains exhibit strange polynomial behaviour. In the null recurrent case some (but not all) moments of the random time needed to reach a finite set are obtained by transforming the discrete-time Markov chain into a discrete-time semimartingale via its mapping through a Lyapunov function [5].

There exist in the literature powerful theorems [START_REF] Aspandiiarov | Passage-time moments for nonnegative stochastic processes and an application to reflected random walks in a quadrant[END_REF], applicable to discretetime critical Markov chains, allowing to determine which moments of the passage time exist. Beyond their theoretical interest, such results can be used to estimate the decay of the stationary measure [START_REF] Menshikov | Exact power estimates for countable Markov chains. Markov Process[END_REF], and even the speed of convergence towards the stationary measure. The first aim of this paper is to show that theorems concerning moments of passage times can be usefully and instrumentally extended to the continuous time situation.

Continuous-time Markov chains have an additional feature compared to discrete-time ones, namely, on each visited state they spend a random holding time (exponentially distributed) defined as the difference between successive jump times. We consider the space inhomogeneous situation where the parameters γ x ∈ R + of the exponential holding times (the inverse of their expectation) are unbounded, i.e. sup x∈X γ x = +∞. In such situations, the phenomenon of explosion can occur for transient chains. Chung [START_REF] Lai | Markov chains with stationary transition probabilities[END_REF] has established that the condition ∞ n=1 1/γ ξn < +∞, where ξn is the position of the chain immediately after the n-th jump has occurred, is equivalent to explosion. However this condition is very difficult to check since is global i.e. requires the knowledge of the entire trajectory of the embedded Markov chain. Later, sufficient conditions for explosion -whose validity can be verified by local estimates -have been introduced. Sufficiently sharp conditions of explosion and non-explosion, applicable only to Markov chains on countably infinite subsets of non-negative reals, are given in [7,8]. In [18], a sufficient condition of explosion is established for Markov chains on general countable sets; similar sufficient conditions of explosion are established in [START_REF] Wagner | Explosion phenomena in stochastic coagulation-fragmentation models[END_REF] for Markov chains on locally compact separable metric spaces.

The second aim of this paper is to show that the phenomenon of explosion can also be sharply studied by the use of Lyapunov functions and to establish locally verifiable conditions for explosion/non explosion for Markov chains on arbitrary graphs. This method is applied to models that even without explosion are difficult to study. More fundamentally, the development of the semimartingale method has been largely inspired with these specific critical models in mind (such as the cascade of k-critically transient Lamperti models or of reflected random walks on quarter planes) that seem refractory to known methods.

Finally, we demonstrate a new phenomenon, we termed implosion (see definition 1.1 below), reminiscent of the Döblin's condition for general Markov chains [START_REF] Doblin | Éléments d'une théorie générale des chaînes simples constantes de Markoff[END_REF], occurring in the case sup x∈X γ x = ∞. We show that this phenomenon can also be explored with the help of Lyapunov functions.

Notation

Throughout this paper, X denotes the state space of our Markov chains; it denotes an abstract denumerably infinite set, equipped with its full σ-algebra X = P (X). It is worth stressing here that, generally, this space is not naturally partially ordered. The graph whose edges are the ones induced by the stochastic matrix, when equipped with the natural graph metric on X need not be isometrically embedable into Z d for some d . Since the definition of a continuous-time Markov chain on a denumerable set is standard (see [START_REF] Lai | Markov chains with stationary transition probabilities[END_REF], for instance), we introduce below its usual equivalent description in terms of holding times and embedded Markov chain merely for the purpose of establishing our notation.

Denote by Γ = (Γ x y ) x,y∈X the generator of the continuous Markov chain, namely the matrix satisfying: Γ x y ≥ 0 if y = x and Γ xx = -γ x , where γ x = y∈X\{x} Γ x y . We assume that for all x ∈ X, we have γ x < ∞.

We construct a stochastic Markovian matrix P = (P x y ) x,y∈X out of Γ by defining

P x y = Γ x y γ x if γ x = 0 0 if γ x = 0,
for y = x, and

P xx = 0 if γ x = 0 1 if γ x = 0.
The kernel P defines a discrete-time (X, P )-Markov chain ξ = ( ξn ) n∈N termed the Markov chain embedded at the moments of jumps. To avoid irrelevant complications, we always assume that this Markov chain is irreducible. Define a sequence σ = (σ n ) n≥1 of random holding times distributed, conditionally on ξ, according to an exponential law. More precisely, consider

P(σ n ∈ d s| ξ) = γ ξn-1 exp(-sγ ξn-1 )1 R + (s)d s, n ≥ 1, so that E(σ n | ξ) = 1/γ ξn-1 .
The sequence J = (J n ) n∈N of random jump times is defined accordingly by J 0 = 0 and for n ≥ 1 by J n = n k=1 σ k . The life time is denoted ζ = lim n→∞ J n and we say that the (not yet defined continuous-time) Markov chain explodes on {ζ < ∞}, while it does not explode (or is regular, or conservative) on {ζ = ∞}.

Remark . The parameter γ x must be interpreted as the proper frequency of the internal clock of the Markov chain multiplicatively modulating the local speed of the chain. We always assume that for all x ∈ X, 0 < γ x < ∞. The case 0 < γ := inf x∈X γ x ≤ sup x∈X =: γ < ∞ is elementary because the chain can be stochastically controlled by two Markov chains whose internal clocks tick respectively at constant pace γ and γ. Therefore the sole interesting cases are -sup x γ x = ∞: the internal clock ticks unboundedly fast (leading to an unbounded local speed of the chain), -inf x γ x = 0: the internal clock ticks arbitrarily slowly (leading to a local speed that can be arbitrarily close to 0).

To have a unified description of both explosive and non-explosive processes, we can extend the state space into X = X ∪ {∂} by adjoining a special absorbing state ∂. The continuous-time Markov chain is then the càdlàg process ξ = (ξ t ) t ∈[0,∞[ defined by

ξ 0 = ξ0 and ξ t = n∈N ξn 1 [J n ,J n+1 [ (t ) for 0 < t < ζ ∂ for t ≥ ζ.
Note that although X is merely a set (i.e. no internal composition rule is defined on it), the above "sum" is well-defined since for every fixed t only one term survives. We refer the reader to standard texts (for instance [START_REF] Lai | Markov chains with stationary transition probabilities[END_REF][START_REF] Norris | Markov chains, volume 2 of Cambridge Series in Statistical and Probabilistic Mathematics[END_REF]) for the proof of the equivalence between ξ and ( ξ, J ). The natural right continuous filtration (F t ) t ∈[0,+∞[ is defined as usual through F t = σ(ξ s : s ≤ t ); similarly F t -= σ(ξ s : s < t ), and Fn = σ( ξk , k ≤ n) for n ∈ N. For an arbitrary (F t )-stopping time τ, we denote as usual its past σ-algebra

F τ = {A ∈ F ∞ : A ∩ {τ ≤ t } ∈ F t } and its strict past σ-algebra F τ-= σ{A ∩ {t < τ}; t ≥ 0, A ∈ F t } ∨ F 0 . Since it is imme- diate to show that τ is F τ--measurable,
we conclude that the only information contained in F J n+1 but not in F J n+1 -is contained within the random variable ξn+1 , i.e. the position where the chain jumps at the moment J n+1 .

If A ∈ X , we denote by τ A = inf{t ≥ 0 : ξ t ∈ A} the (F t )-stopping time of reaching A.

A dual notion to explosion is that of implosion:

Definition 1.1. Let (ξ t ) t ∈[0,
∞[ be a continuous-time Markov chain on X and let A ⊂ X be a proper subset of X. We say that the Markov chain implodes towards

A if ∃K > 0 : ∀x ∈ A c , E x (τ A ) ≤ K .
Remark . It will be shown in proposition 2.11 that in the case the set A is finite and the chain is irreducible, implosion towards A means implosion towards any state. In this situation, we speak about implosion of the chain.

It is worth noticing that some other definitions of implosion can be introduced; all convey the same idea of reaching a finite set from an arbitrary initial point within a random time that can be uniformly (in the initial point) bounded in some appropriate stochastic sense. We stick at the form introduced in the previous definition because it is easier to establish necessary and sufficient conditions for its occurrence and is easier to illustrate on specific problems (see §3).

We use the notational conventions of [START_REF] Revuz | Markov chains[END_REF] to denote measurable functions, namely mX = { f : X → R| f is X -measurable} with all possible decorations: bX to denote bounded measurable functions, mX + to denote non-negative measurable functions, etc. For f ∈ mX + and α > 0, we denote by S α ( f ) the

sublevel set of f of height α defined by S α ( f ) := {x ∈ X : f (x) ≤ α}. We recall that a function f ∈ mX + is unbounded if sup x∈X f (x) = +∞ while tends to infinity ( f → ∞) when for every n ∈ N the sublevel set S 1/n ( f ) is finite.
Measurable functions f defined on X can be extended to functions f , defined on X, by f (x) = f (x) for all x ∈ X and f (∂) := 0 (with obvious extension of the σ-algebra).

We denote by Dom(Γ) = { f ∈ mX : y∈X\{x} Γ x y | f (y)| < +∞, ∀x ∈ X} the domain of the generator and by Dom + (Γ) the set of non-negative functions in the domain. The action of the generator Γ on f ∈ Dom(Γ) reads then: Γ f (x) := y∈X Γ x y f (y).

Main results

We recall once more that in the whole paper we make the following Global assumption 1.2. The chain embedded at the moments of jumps is irreducible and 0 < γ x < ∞ for all x ∈ X.

We are now in position to state our main results concerning the use of Lyapunov function to obtain, through semimartingale theorems, precise and locally verifiable conditions on the parameters of the chain allowing us to establish existence or non-existence of moments of passage times, explosion or implosion phenomena. The proofs of these results are given in section 2; the section 3 treats some critical models (especially 3.1 and 3.3) that are difficult to study even in discrete time, illustrating thus both the power of our methods and giving specific examples on how to use them.

Existence or non-existence of moments of passage times

Theorem 1.3. Let f ∈ Dom + (Γ) be such that f → ∞.

1. If there exist constants a > 0, c > 0 and p > 0 such that f p ∈ Dom + (Γ) and

Γ f p (x) ≤ -c f p-2 (x), ∀x ∈ S a ( f ), then E x (τ q S a ( f ) )
< +∞ for all q < p/2 and all x ∈ X. 2. Let g ∈ mX + . If there exist (a) a constant b > 0 such that f ≤ bg , (b) constants a > 0 and c 1 > 0 such that Γg (x) ≥ -c 1 for x ∈ S a (g ), (c) constants c 2 > 0 and r > 1 such that g r ∈ Dom(Γ) and Γg r (x) ≤ c 2 g r -1 (x) for x ∈ S a (g ), and

(d) a constant p > 0 such that f p ∈ Dom(Γ) and Γ f p ≥ 0 for x ∈ S ab ( f ), then E x (τ q S a ( f ) ) = +∞ for all q > p and all x ∈ S a ( f ).

In many cases, the function g , whose existence is assumed in statement 2, of the above theorem can be chosen as g = f (with obviously b = 1). In such situations we have to check Γ f r ≤ c 2 f r -1 for some r > 1 and find a p > 0 such that Γ f p ≥ 0 on the appropriate sets. However, in the case of the problem studied in §3.3, for instance, the full-fledged version of the previous theorem is needed.

Note that the conditions f ∈ Dom + (Γ) and f p ∈ Dom + (Γ) for some p > 0 holding simultaneously imply that f q ∈ Dom + (Γ) for all q in the interval with end points 1 and p. When τ A is integrable, the chain is positive recurrent. In the null recurrent situation however, τ A is almost surely finite but not integrable; nevertheless, some fractional moments E(τ q A ) with q < 1 can exist. Similarly, in the positive recurrent case, some higher moments E(τ q A ) with q > 1 may fail to exist.

When p = 2, the first statement in the above theorem 1.3 simplifies to the following: if Γ f (x) ≤ -, for some > 0 and for x outside a finite set F , then the passage time E x (τ q F ) < ∞ for all x ∈ X and all q < 1. As a matter of fact, in this situation, we have a stronger result, expressed in the form of the following Theorem 1.4. The following are equivalent:

1. The chain is positive recurrent.

2. There exist a function f ∈ Dom + (Γ) tending to infinity, a constant > 0 and a finite set F such that Γ f (x) ≤for all x ∈ F .

Obviously, positive recurrence implies a fortiori that E x (τ F ) < ∞. If only establishing occurrence of recurrence or transience is sought, the first generalisation of Foster's criteria to the continuous-time case was given in the unpublished technical report [15]. Notice however that the method in that paper is subjected to the same important restriction as in the original paper of Foster [6], namely the semi-martingale condition must be verified everywhere but in one point.

If γ x is bounded away from 0 and ∞, then since the Markov chain can be stochastically controlled by two Markov chains with constant γ x reading respectively γ x = γ and γ x = γ for all x, the previous result is the straightforward generalisation of the theorems 1 and 2 of [START_REF] Aspandiiarov | Passage-time moments for nonnegative stochastic processes and an application to reflected random walks in a quadrant[END_REF] established in the case of discrete time; as a matter of fact, in the case of constant γ x , the complete behaviour of the continuous time process is encoded solely into the jump chain and since results in [START_REF] Aspandiiarov | Passage-time moments for nonnegative stochastic processes and an application to reflected random walks in a quadrant[END_REF] were optimal, the present theorem introduces no improvement. Only the interesting cases of sup x∈X γ x = ∞ or inf x∈X γ x = 0 are studied in the sequel; the models studied in §3, illustrate how the theorem can be used in critical cases to obtain locally verifiable conditions of existence/non-existence of moments of reaching times. The process X t = f (ξ t ), the image of the Markov chain through the Lyapunov function f , can be shown to be a semimartingale; therefore, the semimartingale approach will prove instrumental as was the case in discrete time chains.

Explosion

The next results concern explosion obtained again using Lyapunov functions. It is worth noting that although explosion can only occur in the transient case, the next result is strongly reminiscent of the Foster's criterion [6] for positive recurrence! Theorem 1.5. The following are equivalent:

1. There exist f ∈ Dom + (Γ) strictly positive and > 0 such that Γ f (x) ≤for all x ∈ X.

The explosion time ζ satisfies

E x ζ < +∞ for all x ∈ X.
Remark . Comparison of statements 1 of theorem 1.3 and 1 of theorem 1.5 demands some comments. The conditions of theorem 1.3 imply that S a ( f ) is a finite set and necessarily not empty. For p = 2 and F = f p , the condition reads ΓF (x) ≤outside some finite set and this implies recurrence. In theorem 1.5 the condtion Γ f ≤must hold everywhere and this implies transience.

The one-side implication [Γ f (x) ≤ -, ∀x] ⇒ [P x (ζ < +∞) = 1, ∀x] is already established, for f ≥ 0, in the second part of theorem 4.3.6 of [18]. Here, modulo the (seemingly) slightly more stringent requirement f > 0, we strengthen the result from almost sure finiteness to integrability and prove equivalence instead of mere implication. Proposition 1.6. Let f ∈ Dom + (Γ) be a strictly positive bounded function and denote b = sup x∈X f (x); assume there exists an increasing -not necessarily strictly -function g : R + \{0} → R + \{0} such that its inverse has an integrable singularity at 0, i.e. b 0

1 g (y) d y < ∞. If we have Γ f (x) ≤ -g ( f (x)) for all x ∈ X, then E x ζ < ∞ for all x.
The previous proposition, although stating the conditions on Γ f quite differently than in theorem 1.5, will be shown to follow from the former. This proposition is interesting only when inf x∈X g • f (x) = 0 because then the condition required in 1.6 is weaker than the uniform requirement Γ f (x) ≤for all x of theorem 1.5.

If for some x ∈ X, explosion (i.e. P x (ζ < +∞) > 0) occurs, irreducibility of the chain implies that the process remains explosive for all starting points x ∈ X. However, since the phenomenon of explosion can only occur in the transient case, examples (see §3.4) can be constructed with non-trivial tail boundary so that for some initial x ∈ X, we have both 0 < P x (ζ < +∞) < 1. Additionally, the previous theorems established conditions that guarantee E x ζ < ∞ (implying explosion). However examples are constructed where P x (ζ = ∞) = 0 (explosion does not occur) while E x ζ = ∞. It is therefore important to have results on conditional explosion.

Theorem 1.7. Let A be a proper (finite or infinite) subset of X and f

∈ Dom + (Γ) such that -there exists x 0 ∈ A with f (x 0 ) < inf x∈A f (x), -Γ f (x) ≤ -on A c Then E x (ζ|τ A = ∞) < ∞.
The previous results (theorems 1.5 and 1.7) -through unconditional or conditional integrability of the explosion time ζ -give conditions establishing explosion. For the theorem 1.5 these conditions are even necessary and sufficient. It is nevertheless extremely difficult in general to prove that a function satisfying the conditions of the theorems does not exist. We need therefore a more manageable criterion guaranteeing non-explosion. Such a result is provided by the following

Theorem 1.8. Let f ∈ Dom + (Γ). If -f → ∞,
-there exists an increasing (not necessarily strictly) function g : R + → R + whose inverse is locally integrable but has non integrable tail (i.e. G(z) :=

z 0 d y g (y) < +∞ for all z ∈ R + but lim z→∞ G(z) = ∞), and -Γ f (x) ≤ g ( f (x)) for all x ∈ X, then P x (ζ = +∞) = 1 for all x ∈ X.
The idea of the proof of the theorem 1.8 relies on the intuitive idea that if f (x) ≤ g ( f (x)) for all x, then E( f (X t )) cannot grow very fast with time and since f → ∞ the process itself cannot grow fast either. The same idea has been used in [4] to prove non-explosion for Markov chains on metric separable spaces. Our result relies on the powerful ideas developed in the proof of theorems 1 and 2 of [8] and of theorem 4.1 of [4] but improves the original results in several respects. In first place, our result is valid on arbitrary denumerably infinite state spaces X (not necessarily subsets of R); in particular, it can cope with models on higher dimensional lattices (like random walks in Z d or reflected random walks in quadrants). Additionally, even for processes on denumerably infinite subsets of R, our result covers critical regimes such as those exhibited by the Lamperti model (see §3.1), a "crash test" model, recalcitrant to the methods of [8].

Implosion

Finally, we state results about implosion. Theorem 1.9. Suppose the embedded chain is recurrent.

The following are equivalent:

-There exist a function f

∈ Dom + (Γ) such that sup x∈X f (x) = b < ∞, an a ∈]0, b[, such that S a ( f ) is finite, and an > 0 such that x ∈ S a ( f ) ⇒ Γ f (x) ≤ -.
-For every finite A ∈ X , there exists a constant C :

= C A > 0 such that x ∈ A ⇒ E x τ A ≤ C ,
(hence there is implosion towards A and subsequently the chain is implosive).

2. Let f ∈ Dom + (Γ) be such that f → ∞ and assume there exist constants a > 0, c > 0, > 0, and r

> 1 such that f r ∈ Dom + (Γ). If further -Γ f (x) ≥ -, for all x ∈ S a ( f ), and -Γ f r (x) ≤ c f r -1 (x)
, for all x ∈ S a ( f ), then the chain does not implode towards S a ( f ).

We conclude this section by the following Proposition 1.10. Suppose the embedded chain is recurrent. Let f ∈ Dom + (Γ)

be strictly positive and such that sup x∈X f (x) = b < ∞; assume further that for any a such that 0 < a < b, the sublevel set S a ( f

) is finite. Let g : [0, b] → R + be an increasing function such that B := b 0 d y g (y) < ∞. If Γ f (x) ≤ -g ( f (x)) for all x ∈ S a ( f ) then E x τ S a ( f ) ≤ B for all x ∈ S a ( f ), i.e. the chain implodes towards S a ( f ).
In some applications, it is quite difficult to guess immediately the form of the function f satisfying the uniform condition Γ f (x) ≤required for the first statement of the previous theorem 1.9 to apply. It is sometimes more convenient to check merely that Γ f (x) ≤ -g ( f (x)) for some function g vanishing at 0 in some controlled way. The proposition 1.10 -although does not improve the already optimal statement 1 of the theorem 1.9 -provides us with a convenient alternative condition to be checked.

Proof of the main theorems

We have already introduced the notion of Dom(Γ). A related notion is that of locally p-integrable functions, defined as

p (Γ) = { f ∈ mX : y∈X Γ x y | f (y) - f (x)| p <
+∞, ∀x ∈ X}, for some p > 0. Obviously 1 (Γ) = Dom(Γ). In accordance to our notational convention on decorations, p + (Γ) will denote positive p-integrable functions. For f ∈ 1 (Γ), we define the local f -drift of the embedded Markov chain as the random variable

∆ f n+1 := ∆ f ( ξn+1 ) := f ( ξn+1 ) -f ( ξn+1-) = f ( ξn+1 ) -f ( ξn ), the local mean f -drift by m f (x) := E(∆ f n+1 | ξn = x) = y∈X P x y ( f (y) -f (x)) = E x ∆ f 1 , and for ρ ≥ 1 and f ∈ ρ (Γ) the ρ-moment of the local f -drift by v (ρ) f (x) := E(|∆ f n+1 | ρ | ξn = x) = y∈X
Since (ξ t ) t is a pure jump process, the process (X t ) t transformed by f ∈ Dom(Γ), i.e. X t = f (ξ t ), is also a pure jump process reading, for t < ζ,

X t = f (ξ t ) = ∞ k=0 f ( ξk )1 [J k ,J k+1 [ (t ) = X 0 + ∞ k=0 ∆ f k+1 1 ]0,t ] (J k ).
If there is no explosion, the process (X t ) is a (F t )-semimartingale admitting the decomposition X t = X 0 + M t + A t , where M t is a martingale vanishing at 0 and the predictable compensator reads [9]

A t = ]0,t ] Γ f (ξ s-)d s = [0,t [ Γ f (ξ s )d s.
Notice that, although not explicitly marked, (X t ), (M t ), and (A t ) depend on f . Therefore, for any admissible f we have

d X t = d M t + d A t = d M t + Γ f (ξ t -)d t .

Proof of the theorems 1.3 and 1.4 on moments of passage times

We start by the theorem 1.4 that -although stronger -is much simpler to prove. Proof of the theorem 1.4.

[2 ⇒ 1] : Let X t = f (ξ t ); then the condition 2 reads E(d X t ∧τ F |G t -) ≤ -1 {τ F ≥t } d t .
Hence, in accordance with the previous lemma 2.1, we get

E x (τ F ) ≤ f (x)
for every x ∈ F . Now, let x ∈ F . Then

E x (τ F ) = E x (τ F | ξ1 ∈ F )P x ( ξ1 ∈ F ) + E x (τ F | ξ1 ∈ F )P x ( ξ1 ∈ F ) ≤ sup x∈F 1 γ x + y∈X P x y f (y) < ∞,
the finiteness of the last expression being guaranteed by the conditions f ∈ Dom + (Γ) and the finiteness of the set F . Positive recurrence follows then from irreducibility of the chain.

[1 ⇒ 2] : Let F = {z} for some fixed z ∈ X; positiive recurrence of the chain implies that E x (τ F ) < ∞ for all x ∈ X. Fix some > 0 and define

f (x) = E x (τ F ) if x ∈ F, 0 otherwise. Then, m f (x) ≤ y =x P x y E y (τ F )-E x (τ F ) = E x (τ F -σ 1 )-E x (τ F ) = -E x (σ 1 ) = -γ x , for all x ∈ F . It follows that Γ f (x) = γ x m f (x) ≤ -outside F .
Writing X t = f (ξ t ), we check immediately that (X t ) satisfies the requirements of the lemma 2.1 above. The proof of the theorem 1.3 is quite technical and will be split into several steps formulated as independent lemmata and propositions on semimartingales that may have an interest per se. As a matter of fact, we use these intermediate results to prove various results of very different nature. Lemma 2.2. Let f ∈ Dom + (Γ) tending to infinity, p ≥ 2, and a > 0. Use the abbreviation A := S a ( f ). Denote X t = f (ξ t ) and assume further that f p ∈ Dom + (Γ).

If there exists c > 0 such that

Γ f p (x) ≤ -c f p-2 (x), ∀x ∈ A,
then the process defined by Z t = (X 2 τ A ∧t + c p/2 τ A ∧ t ) p/2 is a non-negative supermartingale.

Proof. Introducing the predictable decomposition 1 = 1 {τ

A <t } + 1 {τ A ≥t } , we get E(d Z t |F t -) = E(d (X 2 t + c 1 p/2 t ) p/2 |F t -)1 {τ A ≥t } . Now, (X t
) is a pure jump process, hence by applying Itô formula, reading for any g ∈ C 2 and (S t ) a semimartingale, d g (S t ) = g (S t -)d S c t +∆g (S t ), where (S c t ) denotes the continuous component of (S t ), we get

d (X 2 t + c p/2 t ) p/2 = c(X 2 t -+ c p/2 t ) p/2-1 d t + (X 2 t + c p/2 t ) p/2 -(X 2 t -+ c p/2 t ) p/2 .
Writing the semimartingale decomposition for the process (X p t ), we remark that the hypothesis of the lemma implies that

E(d X p t |F t -) = Γ f p (ξ t -)d t ≤ -c X p-2 t -d t on the event {τ A ≥ t }.
Applying conditional Minkowski inequality and the supermartingale hypothesis, we get, on the set {τ A ≥ t },

E((X 2 t + c p/2 t ) p/2 |F t -) ≤ [E((X p t |F t -) 2/p + c p/2 t ] p/2 = [(X p t -+ E(d X p t |F t -) 2/p + c p/2 t ] p/2 ≤ ((X 2 t -(1 - c X 2 t - d t ) 2/p + c p/2 t ) p/2 ≤ (X 2 t -- c p/2 d t + c p/2 t ) p/2 .
Hence, on the event {τ A ≥ t }, we have the estimate

E(d Z t |F t -) ≤ c(X 2 t -+ c p/2 t ) p/2-1 d t +(X 2 t -+ c p/2 t - c p/2 d t ) p/2 -(X 2 t -+ c p/2 t ) p/2 .
Simple expansion of the remaining differential forms (containing now only

F t -- measurable random variables) yields E(d Z t |F t -) ≤ 0.
Corollary 2.3. Let f ∈ Dom + (Γ) tending to infinity, p ≥ 2, and a > 0. Use the abbreviation A := S a ( f ). Denote X t = f (ξ t ) and assume further that f p ∈ Dom + (Γ).

If there exists c > 0 such that

Γ f p (x) ≤ -c f p-2 (x), ∀x ∈ A,
then there exists c > 0 such that 2q for all q ≤ p/2 and all x ∈ X. Proof. Without loss of generality, we can assume that x ∈ A c since otherwise the corollary holds trivially. Denoting by Y t = X 2 t ∧τ A + c p/2 t ∧τ A , we observe that

E x (τ q A ) ≤ c f (x)
Z t = Y p/2 t
is a non-negative supermartingale by virtue of the lemma 2.2. Since the function R + w → w 2q/p ∈ R + is increasing and concave for q ≤ p/2, it follows that Y q t is also a supermartingale. Hence,

c p/2 E x [(t ∧ τ A ) q ] ≤ E x (Y q t ) ≤ E x (Y q 0 ) = f (x) 2q .
We conclude by the dominated convergence theorem on identifying c = p 2c .

Proposition 2.4. Let f ∈ Dom + (Γ) tending to infinity, 0 < p ≤ 2, and a > 0.

Use the abbreviation A := S a ( f ). Denote X t = f (ξ t ) and assume further that f p ∈ Dom + (Γ). If there exists c > 0 such that

Γ f p (x) ≤ -c f p-2 (x), ∀x ∈ A, then the process, defined by Z t = X p τ A ∧t + c q (τ A ∧ t ) q , satisfies E x (Z t ) ≤ c f (x) p , for all q ∈]0, p/2]. 13 Proof. Since d (X p t + c q t q ) = d X p t + ct q-1 d t , we have E(d Z t |F t -) ≤ E(d Z t |F t -)1 {τ A ≥t } ≤ cd t 1 {τ A ≥t } (-X p-2 t -+ t q-1 ). Now, q p ≤ 1 2 ≤ 1-q 2-p . Choosing v ∈] q p , 1-q 2-p [, we write E(d Z t |F t -) ≤ cd t 1 {τ A ≥t } (-X p-2 t -+ t q-1 )1 {X t -∈]a,t v ]} +cd t 1 {τ A ≥t } (-X p-2 t -+ t q-1 )1 {X t -∈]t v ,+∞]} .
For X t -≤ t v , the first term of the right hand side of the previous inequality is non-positive; as for the second, the condition X t -> t v implies that -X p-2 t -+ t q-1 ≤ t q-1 . Combining the latter with Markov inequality guarantees that, on the set {X t -> t v }, we have

E x (d Z t ) ≤ cd t 1 {τ A ≥t } f (x) p t v p t q-1 . Integrating this dif- ferential inequality yields E x (Z t ) ≤ c f p (x) ∞ a 1/v d t
t v p+1-q ; the condition v > q/p insures the finiteness of the last integral proving thus the lemma with c = c

∞ a 1/v d t t v p+1-q .
Corollary 2.5. Under the same conditions as in proposition 2.4, there exists c >

0 such that E x (τ q A ) ≤ c f (x) p , ∀q ∈]0, p/2]. Proof. Since X t is non-negative, q c Z t ≥ (t ∧ τ A ) q . By the previous proposition, E x [(t ∧ τ A ) q ] ≤ q c E x (Z t ) ≤ c q c f (x) p .
We conclude by the dominated convergence theorem on identifying c = c q c . Remark . All the propositions, lemmata, and corollaries shown so far allow to prove statement 1 of the theorem 1.3. The subsequent propositions are needed for statement 2 of this theorem. Notice also that the following proposition 2.6 is very important and tricky. It provides us with a generalisation of the theorem 3.1 of Lamperti [START_REF] Lamperti | Criteria for stochastic processes. II. Passage-time moments[END_REF] and serves twice in this paper: one first time to establish conditions for some moments of passage time to be infinite (statement 2 of theorem 1.3) and once more in a very different context, namely for finding conditions for the chain not to implode (statement 2 of theorem 1.9). Proposition 2.6. Let (Ω, G , (G t ) t , P) be a filtered probability space and (Y t ) be a (G t )-adapted process taking values in an unbounded subset of R + . Let a > 0 and T a = inf{t ≥ 0 : Y t ≤ a}. Suppose that there exist constants c 1 > 0 and c 2 > 0 such that

1. E(d Y t |G t -) ≥ -c 1 d t on the event {T a > t }, and 2. there exists r > 1 such that E(d Y r t |G t -) ≤ c 2 Y r -1 t -d t on the event {T a > t }.
Then, for all α ∈]0, 1[, there exist > 0 and δ > 0 such that ∀t > 0 :

P(T a > t + Y t ∧T a |G t ) ≥ 1 -α, on the event {T a > t ; Y t > a(1 + δ)}.
Remark . The meaning of the previous proposition 2.6 is the following. If the process (Y t ) has average increments bounded from below by a constant -c 1 , it is intuitively appealing to suppose that the average time of reaching 0 is of the same order of magnitude as Y 0 . However this intuition proves false if the increments are wildly unbounded since then 0 can be reached in one step. The condition 2, by imposing some control on r -moments of the increments with r > 1, prevents this from occurring. It is in fact established that if T a > t , the remaining time T at to reach A a := [0, a] exceeds Y t with probability 1 -α; more precisely, for every α we can chose such that

P(T a -t > Y t |G t ) ≥ 1 -α.
Proof of proposition 2.6. Let σ = (T at )1 {T a ≥a} ; then for all s > 0 we have {σ < s} = {T a ≥ t } ∩ {T a < t + s} ∈ G t +s-. To prove the proposition, it is enough to establish P(σ > Y t |G t ) ≥ 1 -α on the set {σ > 0; Y t > a(1 + δ)}. On this latter set: P(σ > Y t |G t ) = P(Y t +( Y t )∧σ > a|G t ), because once the process Y t +( Y t )∧σ enters in A a , it remains there for ever, due to the stopping by σ. On defining

U := Y ( Y t )∧σ+t one has E(U |G t ) = E(U 1 {U ≤a} |G t ) + E(U 1 U >a} |G t ) ≤ a + (E(U r |G )) 1/r (P(U > a|G t )) 1-1/r ; therefore P(U > a|G t )) ≥ (E(U |G t ) -a) + (E(U r |G )) 1/r r /(r -1)
.

To minorise the numerator, we observe that

E(U |G t ) = E(Y t + Y t -Y t |G t ) + Y t = t + Y t t E(d Y s |G t ) + Y t ≥ -c 1 Y t + Y t .
To majorise the denominator

E(U r |G ) = E(Y r t +( Y t )∧σ |G t ),
we must be able to majorise E(Y r t +s∧σ |G t ) for arbitrary s > 0. Let t > 0 be arbitrary and S be a G toptional random variable, S > 0. For c 3 = c 2 /r and any s ∈]0, S], define

F S (s) = E[(Y t +s∧σ + c 3 S -c 3 s ∧ σ) r |G t ].
We shall show that F S (s) ≤ F S (s-) for all s ∈]0, S]. It is enough to show this inequality on {σ > s} since otherwise F S (s) = F S (s-) and there is nothing to prove. To show that F S is decreasing in ]0, S], it is enough to show that E(d Ξ s |G t +s-) ≤ 0 for all s ∈]0, S], where Ξ s = (Y t +s∧σ + c 3 Sc 3 s ∧ σ) r . Now, on {σ > s}, by use of Itô's formula, we get

d Ξ s = -r c 3 (Y t +s-+ c 3 S -c 3 s) r -1 d s + (Y t +s + c 3 S -c 3 s) r -(Y t +s-+ c 3 S -c 3 s) r .
Moreover, using Minkowski inequality, we get

E[(Y t +s + c 3 S -c 3 s) r |G t +s-] ≤ E(Y r t +s |G t +s-) 1/r + c 3 S -c 3 s r ,
and by use of the hypothesis

E(Y r t |G t -) ≤ Y r t -+c 2 Y r -1 t -1 {ρ>t } d t = Y r t -1 + c 2 Y t - 1 {T a >t } d t ≤ Y t -+ c 3 1 {T a >t } d t r .
Therefore, E(d Ξ s |G t +s-) ≤ 0 for all s ∈]0, S]. Subsequently, for all S > 0;

Y r t +S = F S (S) ≤ lim s→0+ F S (s) = (Y t + c 3 S) r .
Since S is an arbitrary G t -optional random variable, on choosing S = Y t , we get finally

(E(Y r t + Y t |G t )) 1/r ≤ Y t + c 3 Y t .
Substituting yields, that for any α ∈]0, 1[, parameters > 0 and δ > 0 can be chosen so that the following inequality holds:

P(U > a|G t ) ≥ (1 -c 1 ) -a Y t ) + 1 -c 3 r r -1 ≥ (1 -c 1 - 1 1 + δ ) r r -1 + (1 + c 3 ) -r r -1 ≥ 1 -α.
Lemma 2.7. Let (Ω, G , (G t ), P) be a filtered probability space, (Y t ) and (Z t ) two R + -valued, (G t )-adapted processes on it. For a > 0, we denote S a = inf{t ≥ 0 : Y t ≤ a} and T a = inf{t ≥ 0 : Z t ≤ a}. Suppose that there exist positive constants a, b, p, K 1 such that 1. Y t ≤ bZ t almost surely, for all t , and

2. E(Z p t ∧T a ) ≤ K 1 . Then, there exists K 2 > 0 such that E(Y p t ∧S ab ) ≤ K 2 .
Proof. For arbitrary s > 0, the condition Z s < a implies X s ≤ bZ s < ab almost surely. Hence, {S ab ≥ t } ⊆ {T a ≥ t }. Then,

E(Y p t ∧S ab ) ≤ E Y p t ∧S ab (1 {S ab ≥t } + 1 {S ab >t } ) ≤ E(Y p t 1 {S ab ≥t } ) + (ab) p ≤ b p K 1 + (ab) p := K 2 .
Proposition 2.8. Let (Ω, G , (G t ), P) be a filtered probability space, (Y t ) and (Z t ) two R + -valued, (G t )-adapted processes on it. For a > 0, we denote S a = inf{t ≥ 0 : Y t ≤ a} and T a = inf{t ≥ 0 : Z t ≤ a}. Suppose that 1. there exist positive constants a, c 1 , c 2 , r such that -

E(d Z 2 t |G t -) ≥ -c 1 d t on the event {T a ≥ t }, -E(d Z r t |G t -) ≤ c 2 Z r -1 t -d t on the event {T a ≥ t }, -Z 0 = z > a.
2. Y 0 = y and there exists a constant b > 0 such that -ab < y < bz and -Y t ≤ bZ t almost surely for all t .

If for some p > 0, the process (Y

p t ∧S ab
) is a submartingale, then E(T q a ) = ∞ for all q > p.

Proof. We can without loss of generality examine solely the case P(S ab < ∞) = 1, since otherwise E(T q a ) = ∞ holds trivially for all q > 0. Assume further that for some q > p, it happens E(T q a ) < ∞. Hypothesis 1 allows applying proposition 2.6; for α = 1/2, we can thus determine positive constants and δ such that

P(T a > t + Z t ∧T a |G t ) ≥ 1/2 on the event {Z t ∧T a > a(1 + δ)}. Hence E(T q a ) ≥ E(T q a 1 {Z t ∧T a >a(1+δ)} ) ≥ 1 2 E (t + Z t ∧T a ) q 1 {Z t ∧T a >a(1+δ)} ≥ q 2 E(Z q t ∧T a ) - q 2 a q (1 + δ) q .
Now, finiteness of the q moment of T a implies the existence of a constant K 1 > 0 such that E(Z q t ∧T a ) ≤ K 1 . From the previous lemma 2.7 follows that there exists some K 2 > 0 such that E(Y q t ∧S ab ) ≤ K 2 . Since the time S ab is assumed almost surely finite, lim t →∞ E(Y

q t ∧S ab ) = E(Y q S ab
) ≤ (ab) q . On the other hand, (Y

p t ∧S ab ) is a submartingale, so is thus a fortiori (Y q t ∧S ab ). Hence, E(Y q t ∧S ab
) ≥ E(Y q 0 ) = y q , leading to a contradiction if we chose y > ab.

Corollary 2.9. Let (Ω, G , (G t ), P) be a filtered probability space, (X t ) a R + -valued, (G t )-adapted processes on it. For a > 0, we denote T a = inf{t ≥ 0 : X t ≤ a}. Suppose that there exist positive constants a, c 1 , c 2 , p, r such that -

X 0 = x > a, -E(d X t |G t -) ≥ -c 1 d t on the event {T a ≥ t }, -E(d X r t |G t -) ≤ c 2 X r -1 t -d t on the event {T a ≥ t }, -(X p t ∧T a ) is a submartingale. Then E(T q a ) = ∞ for all q > p.
After all this preparatory work, the proof of the theorem 1.3 is now immediate.

Proof of the theorem 1.3. Write X t = f (ξ t ) and use the abbreviation A := S a ( f ).

Notice moreover that τ A = T a .

1. Since f → ∞ the set A is finite. The integrability of the passage time follows from corollaries 2.3 and 2.5.

2. On identifying Z t and Y t in proposition 2.8 with g (ξ t ) and f (ξ t ) respectively, we see that the conditions of the theorem imply the hypotheses of the proposition. The non-existence of moments immediately follows.

Proof of theorems on explosion and implosion

As stated in the introduction, the result concerning integrability of explosion time is reminiscent to Foster's criterion for positive recurrence! The reason lies in the lemma 2.1 and the fact that theorem 1.5, establishing explosion, is equivalent to the proposition 2.10 below.

Since we wish to treat explosion, we work with a Markov chain evolving on the augmented state space X = X ∪ {∂} and with augmented functions f as explained in the introduction. Note that if f > 0 then f X = f > 0 while f (∂) = 0. We extend also naturally X t = f (ξ t ) (and stricto sensu also the generator Γ). We use the "hat" notation • to denote quantities referring to the extended chain. Proposition 2.10. Let f : X → [0, ∞[ belong to Dom + ( Γ) and verify f (x) > 0 for all x ∈ X (it vanishes only at ∂). We denote f the restriction of f on X. The following are equivalent:

1. There exists > 0 such that Γ f (x) ≤for all x ∈ X.

Êx

( τ∂ ) < ∞ for all x ∈ X. Proof. Let Y t = f ( ξt ), for t ≥ 0. Then (Y t ) is a [0, ∞[-valued, ( Ft )-adapted pro- cess. Note also that τ∂ = T 0 := inf{t ≥ 0 : Y t = 0} [1 ⇒ 2:] The condition Γ f (x) ≤ -for all x ∈ X is equivalent to the condition E(d Y t ∧T 0 |G t -) ≤ -1 T 0 ≥t d t .
The conclusion follows from direct application of the lemma 2.1.

[2 ⇒ 1:] Define f (x) = E x ( τ∂ ). Obviously f (∂) = 0 while 0 < f (x) < ∞ for all x ∈ X.
Conditioning on the first move of the embedded Markov chain ξ we get:

m f (x) = E( f ( ξ1 ) -f (x)| ξ0 = x) = E( f (ξ J 1 ) -f (ξ 0 )|ξ 0 = x) = y∈X\{x} P x y E y ( τ∂ ) -E x ( τ∂ ) = (E x ( τ∂ ) -E x (σ 1 )) -E x ( τ∂ ) = -E x (σ 1 ) = -γ x .
Hence Γ f (x) ≤for all x ∈ X.

Proof of the theorem 1.5. Evident by proposition 2.10, since ζ = τ∂ .

Proof of the proposition 1.6.

Let G(z) = z 0 d y g (y) . Then G is differentiable, with G (z) = 1 g (z) > 0 hence an increasing function of z ∈ [0, b]. Since g is increas- ing, G is decreasing and hence G is concave satisfying lim z→0 G(z) = 0 and lim z→∞ G(z) < ∞. Additionally, boundedness of G imply that G • f ∈ 1 + (Γ)
. Due to differentiability and concavity of G, we have:

ΓG • f (x) = γ x E G( f ( ξn ) + ∆ f n+1 ) -G( f ( ξn ))| ξn = x ≤ γ x 1 g ( f (x)) m f (x) = Γ f (x) g ( f (x)) ≤ -c;
we conclude by theorem 1.5 because G • f is strictly positive and bounded.

Proof of the theorem 1.7. The conditions of the theorem imply transience of the chain. In fact, let

Y t = f (ξ t ∧τ A ). The condition Γ f (x) ≤ -on A c implies that (Y t ) is a non-negative supermartingale, converging almost surely towards Y ∞ . For every x ∈ A c , Fatou's lemma implies E x (Y ∞ ) ≤ lim t →∞ E x (Y t ) ≤ f (x).
Suppose now that the chain is not transient, hence

P x (τ A < ∞) = 1. In that case lim t →∞ Y t = Y τ A almost surely.
We get then by monotone convergence theorem inf y∈A f (y) ≤ E x ( f (ξ τ A )) ≤ f (x), in contradiction with the hypothesis that there exists x ∈ A c such that f (x) < inf y∈A f (y). Hence, P x (τ A = ∞) > 0. Let now T := τ A ∧ζ; on defining Z t = f (ξ t ∧T ), we obtain then that E(d Z t |F t -) ≤ -1 {T >t } d t . We conclude by proposition 2.10 and the fact that the event {τ A = ∞} is not negligible.

Proof of the theorem 1.8. Let G(z) = z 0 1 g (y) d y; this function is differentiable with G (z) = 1 g (z) > 0, hence increasing. Since g is increasing, G is decreasing, hence G is concave. Non integrability of the infinite tail means that G is un- boundedly increasing towards ∞ as z → ∞, hence G • f → ∞. Concavity and differentiability of G imply that G( f (x) + ∆) -G( f (x)) ≤ ∆G ( f (x)); integrability of f guarantees then 0 ≤ E(G( f ( ξn ) + ∆ f n+1 )| ξn = x) ≤ G( f (x)) + m f (x) g ( f (x)) < ∞ so that F := G • f ∈ 1 + (Γ) as well. Now ΓG • f (x) = γ x E G( f ( ξn ) + ∆ f n+1 ) -G( f ( ξn ))| ξn = x = γ x E f (x)+∆ f n+1 f (x) 1 g (y) d y| ξn = x ≤ γ x 1 g ( f (x)) m f (x) ≤ c.
Let X t = F (ξ t ) be the process obtained form the Markov chain after transformation by F . Using the semimartingale decomposition of X t = X 0 +M t + ]0,t ] ΓF (ξ s-)d s, it becomes obvious then that E x X t ≤ F (x) + ct , showing that for every finite t , P x (X t = ∞) = 0. But since F → ∞ the process itself ξ t cannot explode.

Proposition 2.11. Suppose the chain is recurrent and there exists a finite set A ∈ X and a constant C > 0 such that, for all x ∈ X, the uniform bound E x τ A ≤ C holds. Then the chain implodes towards any state z ∈ X.

Proof. First remark that obviously σ 0 ≤ τ A , where σ 0 is the holding time at the initial state, i.e. 1

γ x = E x (σ 0 ) ≤ E x τ A ≤ C . Hence γ := inf x∈X γ x > 0.
Let a be an arbitrary point in A and z an arbitrary state in X. Irreducibility means that there exists a path of finite length, say k, satisfying a ≡ x 0 , . . . , x k ≡ z and

P x 0 ,x 1 • • • P x k-1 ,x k = δ a > 0. Then, for K > 1
2γ sup a∈A k a , we estimate

P a (τ z ≤ K ) ≥ P a ( k a -1 i =0 σ i ≤ K | ξ1 = x 1 , . . . , ξk = x k = z)P a ( ξ1 = x 1 , . . . , ξk = x k = z). Now, Markov's inequality yields P a (τ z ≤ K ) ≥ (1 -k a K γ )δ a ≤ 1 2 inf a δ a := δ > 0.
Using exactly the same arguments, we show that P a (τ A c ≤ K ) ≥ δ > 0 for some parameters K < ∞ and δ > 0.

Denote K = max(K , K ) < ∞, δ = min(δ , δ ) > 0, and r = sup a∈A E a (τ z ). The hypothesis of the proposition implies

E x τ z = E x (τ z |τ z ≤ τ A )P a (τ z ≤ τ A ) + E x (τ z |τ z > τ A )P a (τ z > τ A ) ≤ C + E x (τ z |τ z > τ A )P a (τ z > τ A ). Now E x (τ z |τ z > τ A ) ≤ E x τ A P x (τ z >τ z ) +E x (E(τ z -τ A |F τ A )|τ z < τ A ). We get finally E x τ z ≤ 2C + r .
On the other hand,

E a (τ z ) = E x (τ z |τ z ≤ K )P a (τ z ≤ K ) + E x (τ z |τ z > K )P a (τ z > K ) = K + E a (E X K (τ z )). Now, on the set {τ z > K }, X K can be any a ∈ A or any x ∈ A c \ {z}. Hence, E a (τ z ) ≤ K + δ max x ∈A c \{z} E x τ z P a (X K = x ), a ∈A E a τ z P a (X K = a ) .
Using the previously obtained majorisations, we get E a τ z ≤ K + δ(2C + r ) and taking the max a of the l.h.s. we obtain finally r (1 -δ) ≤ K + 2C δ proving thus that sup x E x τ z ≤ 2C + K +2C 1-δ that guarantees implosion towards any z ∈ X.

Proof of the theorem 1.9.

1. We first show implosion. Since f is bounded, the condition

Γ f (x) = γ x m f (x) ≤ -guarantees that γ = inf x γ x > 0.
[⇒:] The condition Γ f (x) ≤for x ∈ S a ( f ) guarantees -by virtue of the proposition 2.10 -that

E x τ S a ( f ) ≤ f (x) ≤ sup z ∈S a ( f ) f (z) .
Abbreviate B := S a ( f ) and remark that B is finite by hypothesis. Now, due to recurrence and irreducibility, if there exists a constant C such that x ∈ B ⇒ E x τ B < C , then, for every z ∈ X, there exists a constant C such that x = z ⇒ E x τ {z} < C , by virtue of the previous proposition 2.11.

[⇐:] Suppose now that for a finite A ∈ X , there exists a constant C such that

E x τ A ≤ C . Consequenlty 1 γ x ≤ E x τ A ≤ C , leading necessarily to the lower bound γ > 0. Define f (x) = 0 if x ∈ ȃA E x τ A if x ∈ ȃA.
Then it is immediate to show that A = S 1 ( f ) and that for x ∈ A, we have Γ f (x) ≤ -1.

2.

To show non-implosion, use lemma 2.1 to guarantee that if at some time t the process ξ t is at some point x 0 sufficiently large, then the time needed for the process X t = f (ξ t ) to reach S a ( f ) exceeds f (x 0 ) with some substantially large probability. More precisely, there exists α ∈]0, 1[ such that

P x 0 (τ S a ( f ) -t > f (x 0 )) ≥ 1 -α. Therefore E x 0 (τ S a ( f ) ) ≥ (1 -α) f (x 0
) and since f → ∞ then this expectation cannot be bounded uniformly in x 0 .

Proof of the proposition 1.10.

Let G(z) = z 0 d y g (y) . Since G (z) = 1 g (z) > 0 the func- tion G is increasing with G(0) = 0 and G(b) = B . Since g is increasing G = 1 g is decreasing, hence the function G is concave. Then concavity leads to the majorisation m G• f (x) ≤ G ( f (x))m f (x) = m f (x) g ( f (x)) .
The condition imposed on the statement of the proposition implies that ΓG • f (x) ≤ -1. We conclude from lemma 2.1.

We need first some technical conditions. Let f ∈ 2+δ 0 for some δ 0 > 0. For every g : R + → R + a C 3 function, we get

m g • f (x) = E(g ( f ( ξn+1 ))-g ( f ( ξn ))| ξn = x) = g ( f (x))m f (x)+ 1 2 g ( f (x))v f (x)+R g (x),
where R g (x) is the conditional expectation of the remainder occurring in the Taylor expansion2 .

Definition 3.1. Let F = (F n ) n∈N be an exhaustive nested sequence of sets F n ↑ X, f ∈ 2+δ + (Γ) and g ∈ C 3 (R + ; R + ).
We say that the chain (or its jumps) satisfies the remainder condition (or shortly condition R) for F, f , and g , if

lim n→∞ sup x∈F c n R g (x)/[g ( f (x))m f (x) + 1 2 g ( f (x))v f (x)] = 0. The quantity D g ( f , x) := g ( f (x))m f (x) + 1 2 g ( f (x))v f (x)
in the expression above is the effective average drift at the scale defined by the function g . If the function f is non-trivial, there exists a natural exhaustive nested sequence determined by the sub-level sets of f . When we omit to specify the nested sequence, we shall always consider the natural one.

Introduce the notation ln (0) s = s and recursively, for all integers k ≥ 1, ln (k) s = ln(ln (k-1) s) and denote by L k (s) = k i =0 ln (i ) s, for k ≥ 0, and L k (s) := 1 for k < 0. Equivalently, we define exp (k) as the inverse function of ln (k) . In most occasions we shall use (Lyapunov) functions g (s) := ln η (l ) s with some integer l ≥ 0 and some real η = 0, defined for s sufficiently large, s ≥ s 0 := exp (l ) (2) say. It is cumbersome but straightforward to show then that for f ∈ 2+δ 0 (Γ) with some δ 0 > 0, the condition R is satisfied. It will be shown in this section that condition R and Lyapunov functions of the aforementioned form play a crucial role in the study of models where the effective average drift D g ( f , x) tends to 0 in some controlled way with n when x ∈ F c n , where (F n ) is an exhaustive nested sequence; models of this type lie in the deeply critical regime between recurrence and transience.

Critical models on denumerably infinite and unbounded subsets of R +

Consider a discrete time irreducible Markov chain ( ξn ) on a denumerably infinite and unbounded subset X of R + . Since now X inherits the composition properties stemming from the field structure of R, we can define directly m(x) := m id (x) and v(x) := v id (x), where id is the identity function on X. The model is termed critical when the drift m tends to 0, in some precise manner, when x → ∞.

In [START_REF] Lamperti | Criteria for the recurrence or transience of stochastic process[END_REF] Markov chains on X = {0, 1, 2, . . .} were considered; it has been established that the chain is recurrent if m

(x) ≤ v(x)/2x while is transient if m(x) > θv(x)/2x for some θ > 1. In particular, if m(x) = O ( 1
x ) and v(x) = O (1) the model is in the critical regime.

The case with arbitrary degree of criticality

m(x) = k i =0 α i L i (x) + o 1 L k (x) and v(x) = k i =0 xβ i L i (x) + o x L k (x)
, with α i , β i constants, has been settled in [START_REF] Men Shikov | Markov processes with asymptotically zero drift[END_REF] by using Lyapunov functions. Under the additional conditions lim sup n→∞ ξn = ∞ and lim inf

x∈X v(x) > 0
and some technical moment conditions -guaranteeing the condition R for this model -that can straightforwardly be shown to hold if id ∈ 2+δ 0 (Γ), for some δ 0 > 2, it has been shown in [START_REF] Men Shikov | Markov processes with asymptotically zero drift[END_REF] (theorem 4) that -if 2α 0 < β 0 the chain is recurrent while if 2α 0 > β 0 the chain is transient; -if 2α 0 = β 0 and 2α i -β i -β 0 = 0 for all i : 0 ≤ i < k and there exists i : 0 ≤ i < k such that β i > 0 then -if 2α k -β k -β 0 ≤ 0 the chain is recurrent, -if 2α k -β k -β 0 > 0 the chain is transient. We assume that the moment conditions id ∈ 2+δ 0 (Γ) -guaranteeing the condition R for this model -are satisfied through out this section.

Let ( ξn ) be a Markov chain on X = {0, 1, . . .} satisfying for some k ≥ 0

m(x) = k-1 i =0 β 0 2L i (x) + α k L k (x) + o 1 L k (x) and v(x) = β 0 + o(1),
with 2a k > β 0 . The aforementioned result guarantees the transience of the chain; we term such a chain k-critically transient. When 2α k < β 0 the previous result guarantees the recurrence of the chain; we term such a chain k-critically recurrent.

In spite of its seemingly idiosyncratic character, this model proves universal as Lyapunov functions used in the study of many general models in critical regimes map those models to some k-critical models on denumerably infinite unbounded subsets of R + .

Moments of passage times

Proposition 3.2. Let (ξ t ) be a continuous-time Markov chain on X and A be the finite set A := [0, x 0 ] ∩ X for some sufficiently large x 0 . Suppose that for some integer k ≥ 0, its embedded chain is k-critically recurrent, i.e.

m(x) = k-1 i =0 β 0 2L i (x) + α k L k (x) + o 1 L k (x) and v(x) = β 0 + o(1), with 2a k < β 0 . Denote C = α k /β 0 (hence C < 1/2)
and assume there exists a

constant κ > 0 such that 3 γ x = O ( L 2 k (x) ln κ (k) (x) ) for large x. Define p 0 := p 0 (C , κ) = (1 -2C )/κ. 1. Assume that v (ρ) (x) = O (1) with ρ = max(2, 1 -2C ) + δ 0 . If q < p 0 then E x τ q A < ∞. 2. Assume that v (ρ) (x) = O (1) with ρ = max(2, p 0 ) + δ 0 . If q ≥ p 0 then E x τ q A = ∞.
Remark . We remark that when κ ↓ 0 (for fixed k and C ) then p 0 ↑ ∞ implying that more and more moments exist.

Proof of the proposition 3.2. For the function f

(x) = ln η (k) x we determine m f (x) = 1 2 β 0 η(2C + η -1 + o(1)) ln η (k) (x) L 2 k (x)
.

1. For a p > 0, we remark that

[0 < pη < 1 -2C and η ≥ κ 2 ] ⇒ Γ f p (x) ≤ -c f p-2 (x),
for some constant 4 c > 0. Hence p < 2(1 -2C )/κ and statement 1 of the theorem 1.3 implies for any q ∈]0, (1 -2C )/κ[, the q-th moment of the passage time exists. Optimising over the accessible values of q we get that E x (τ q ) < ∞ for all q < p 0 .

2. We distinguish two cases:

[1 -2C < κ:] We verify that, choosing η ∈]1 -2C , κ] and p > 1-2C κ , the three conditions conditions of statement 2 of the theorem 1.3 (with f = g ), namely Γ f ≥ -c 1 , Γ f r ≤ c 2 f r -1 , for some r > 1, and Γ f p ≥ 0, outside some finite set A.

3. i.e. there exists a constant c 3 > 0 such that 1 c 3 ≤

γ x ln κ (k) (x) L 2 k (x) ≤ c 3 for x ≥ x 0 . 4. The constant c can be chosen c ≥ 1 2 β 0 η(1 -2C )c 3 .
[κ ≤ 1 -2C :] In this situation, choosing the parameters η ∈]0, κ] and p > 1-2C κ implies simultaneous verification of the three conditions of statement 2 of the theorem 1.3.

In both situations, we conclude that for all q > 1-2C κ , the corresponding moment does not exist. Optimising over the accessible values of q, we get that E x (τ q ) = ∞ for all q > p 0 .

Explosion and implosion

Proposition 3.3. Let (ξ t ) be a continuous time Markov chain. Suppose that for some integer k ≥ 0, its embedded chain is k-critically transient.

1. If there exist a constant d 1 > 0, an integer l > k, and a real κ > 0 (arbitrarily small) such that

γ x ≥ d 1 L k (x)L l (x)(ln (l ) x) κ , x ≥ x 0 , then P y (ζ < ∞) = 1 for all y ∈ X.
2. If there exist a constant d 2 > 0 and an integer l > k such that

γ x ≤ d 2 L k (x)L l (x), x ≥ x 0 ,
then the continuous-time chain is conservative.

Proof.

1. For a k-critically transient chain, chose a function f behaving at large x as f (x) = 1 ln η (l ) (x) . Denote C = α k /β 0 (hence C > 1/2 for the chain to be transient). We estimate then

m f (x) = -2β 0 η(2C -1) 1 L k (x)L l (x) ln η (l ) x + o( 1 L k (x)L l (x) ln η (l ) x
).

We conclude by theorem 1.5.

2. Choosing as Lyapunov function the identity function f (x) = x and estimate

m ln (l +1) • f (x) = 2β 0 (2C -1) ln (l +1) x L k (x)L l +1 (x) = 2β 0 (2C -1) 1 L k (x)L l (x) .
We conclude by theorem 1.8 by choosing g (s) = L l (s).

Proposition 3.4. Let (ξ t ) be a continuous time Markov chain. Suppose that for some integer k ≥ 0, its embedded chain is k-critically recurrent. Denote C = α k /β 0 (hence C < 1/2 for the chain to be recurrent). Let A be the finite set A := [0, x 0 ] ∩ X for some sufficiently large x 0 .

1. If there exist a constant d 1 >, an integer l > k, and an arbitrarily small real κ > 0 such that

γ x ≥ d 1 L k (x)L l (x)(ln (l ) x) κ , x ≥ x 0 ,
then there exists a constant B such that E y τ A ≤ B , uniformly in y ∈ A c , i.e. the chain implodes.

2. If there exist a constant d 2 > 0 and an integer l > k such that

γ x ≤ d 2 L k (x)L l (x), x ≥ x 0 ,
then the continuous time chain does not implode.

Proof.

1. Use the function f defined for sufficiently large x by the formula

f (x) = 1 -1 ln η (l ) x
, for some l > k and η > 0. We estimate then

m f (x) = 2β 0 η(2C -1) 1 L k (x)L l (x) ln η (l ) x + o( 1 L k (x)L l (x) ln η (l ) x
).

We conclude by statement 1 of theorem 1.9.

2. Using the function f defined for sufficiently large x by the formula f (x) = ln η (l +1) x, for some l ≥ k. We estimate then

m f (x) = 2β 0 η(2C -1) ln η (l +1) x L k (x)L l +1 (x) + o( ln η (l +1) x L k (x)L l +1 (x)
).

If γ x ≤ d 2 L k (x)L l (x) for large x, then, using the above estimate for the case η = 1 and the case η = r for some small r > 1, we observe that the conditions Γ f ≥and Γ f r ≤ f r -1 are simultaneously verified. We conclude by the statement 2 of the theorem 1.9.

Simple random walk on Z d for d = 2 and d ≥ 3

Here the state space X = Z d and the embedded chain is a simple random walk on X. Since in dimension 2 the simple random walk is null recurrent while in dimension d ≥ 3 is transient, a different treatment is imposed.

Dimension d ≥ 3

For the Lyapunov function f defined by Z d x → f (x) := x , we can show that there exist constants α 0 > 0 and β 0 > 0 such that lim x →∞ x m f (x) = α 0 and lim x →∞ v f (x) = β 0 such that C = α 0 /β 0 > 1/2. Therefore the one dimensional process X t = f (ξ t ) has 0-critically transient Lamperti behaviour.

We get therefore that (ξ t ) t ∈R + is a (quite unsurprisingly) transient process and that if there exist a constant a > 0 and -a constant d 1 > 0, an integer l > 0, and a real κ > 0 (arbitrarily small) such that

γ x ≥ d 1 x L l ( x )(ln (l ) x ) κ , x ≥ a, then P y (ζ < ∞) = 1
for all y ∈ X; -a constant d 2 > 0 and an integer l > 0 such that

γ x ≤ d 2 x L l ( x ), x ≥ a,
then the continuous time chain is conservative.

Dimension 2

Using again the Lyapunov function f (x) = x , we show that the one dimensional process X t = f (ξ t ) is of the 1-critically recurrent Lamperti type. Hence, again using the results obtained in §3.1, we get that if there exist a constant a > 0 and -a constant d 1 > 0, an integer l > 1, and an arbitrarily small real κ > 0 such that

γ x ≥ d 1 L 1 ( x )L l ( x )(ln (l ) x ) κ , x ≥ a,
then there exists a constant C such that E y τ A ≤ C , uniformly in y for y : y ≥ a, i.e. the chain implodes; -a constant d 2 > 0 and an integer l > 1 such that

γ x ≤ d 2 L 1 ( x )L l ( x ), x ≥ a,
then the continuous time chain does not implode.

Random walk on Z 2

+ with reflecting boundaries

The model in discrete time

Here X = Z 2 + . We denote by

• X = {x ∈ Z 2 + : x 1 > 0, x 2 >
0} the interior of the wedge and by ∂ 1 X = {x ∈ Z 2 + : x 2 = 0} (and similarly for ∂ 2 X) its boundaries.

Since X is a subset of a vector space, we can define directly the increment vector D := ξn+1 -ξn and the average conditional drift m(x) := m id (x) = E(D| ξn =

x) ∈ R 2 . We assume that for all x ∈ • X, m(x) = 0 so that we are in a critical regime. For x ∈ ∂ X; with = 1, 2, the drift m (x) does not vanish but is a constant vector m that forms angles φ with respect to the normal to ∂

X. For x ∈ • X, the con- ditional covariance matrix C (x) := (C (x) i j ), with C (x) i j (= E[D i D j | ξn = x], is the constant 2 × 2 matrix C , reading C := Cov(D,D) = s 2 1 λ λ s 2 2 .
There exists an isomorphism Φ on R 2 such that Cov(ΦD,ΦD) = ΦC Φ t = I ; it is elementary to show that

Φ = s 2 d -λ s 2 d 0 1 s 2 ,
where d = detC , is a solution to the aforementioned isomorphism equation. This isomorphism maps the quadrant R 2 + into a squeezed wedge Φ(R 2 + ) having an angle ψ at its summit reading ψ = arctan(-

d 2 /λ). Obviously ψ = π/2 if λ = 0, while ψ ∈]0, π/2[ if λ < 0 and ψ ∈]π/2, π[ if λ > 0.
We denote Y = Φ(X) the squeezed image of the lattice. The isomorphism Φ transforms the average drifts at the boundaries into n = Φm forming new angles, ψ , with the normal to the boundaries of Φ(R 2 + ). The discrete time model has been exhaustively treated in [START_REF] Aspandiiarov | Passage-time moments for nonnegative stochastic processes and an application to reflected random walks in a quadrant[END_REF] and its extension to the case of excitable boundaries carrying internal states in [START_REF] Menshikov | Markov chains in a wedge with excitable boundaries[END_REF]. Here we recall the main results of [START_REF] Aspandiiarov | Passage-time moments for nonnegative stochastic processes and an application to reflected random walks in a quadrant[END_REF] under some simplifying assumptions that allow us to present them here without redefining completely the model or considering all the technicalities. The assumptions we need are that the jumps -are bounded from below, i.e. there exists a constant K > 0 such that D 1 ≥ -K and D 2 ≥ -K , -satisfy a sufficient integrability condition, for instance E( D 2+δ 0 2+δ 0 ) < ∞ for some δ 0 > 0, -are such that their covariance matrix is non degenerate. Under these assumptions we can state the following simplified version of the results in [START_REF] Aspandiiarov | Passage-time moments for nonnegative stochastic processes and an application to reflected random walks in a quadrant[END_REF].

Denote by χ = (ψ 1 + ψ 2 )/ψ and A = {x ∈ X : x ≤ a}.

1. If χ ≥ 0 the chain is recurrent.

2. If χ < 0 the chain is transient.

3. If 0 < χ < 2 + δ 0 , then for every p < χ/2 and every

x ∈ A, E x τ p A < ∞. 4. If 0 < χ < 2 + δ 0 , then for every p > χ/2 and every x ∈ A, E x τ p A = ∞. Let f : R 2 + → R + be a C 2 function. Define f (y) = f (Φ -1 y).
Although the Hessian operator does not in general transform as a tensor, the linearity of Φ allows however to write Hess f (x) = Φ t Hess f (Φx)Φ. For every x ∈ • X we establish then 5 the identity:

E(〈 D, Hess f •Φ (x)D 〉| ξn = x) = Lap f •Φ (x).
We denote by h β,β 1 (x) = x β cos(β arctan( x 2

x 1 ) -β 1 ). Then this function is harmonic, i.e. Lap h β,β 1 = 0. We are interested in harmonic functions that are positive on Φ(R 2 + ); positivity and geometry impose then conditions on β and β 1 . In fact, sign(β)β 1 is the angle of ∇h β,β 1 (x) at x ∈ ∂ 1 X, with the normal to ∂ 1 X. Similarly, if β 2 = βψβ 1 , then sign(β)β 2 is the angle of the gradient with the normal to ∂ 2 X. Now, it becomes evident that β i , i = 1, 2, must lie in the interval ] -π/2, π/2[ and subsequently β = β 1 +β 2 ψ . Notice also that the datum of two admissible angles β 1 and β 2 uniquely determines the harmonic function whose gradient at the boundaries forms angles as above. Hence, 〈 ∇h β,β 1 (y), n 〉 = y β β sin(ψ -β ), for y ∈ ∂ Y and = 1, 2. Let now g : R + → R + be a C 3 function and h = h β,β 1 an harmonic function that remains positive in Φ(R 2 + ). On denoting y = Φx, abbreviating Ξ := g (h(Φ ξn+1 ))g (h(Φ ξn )), and using the fact that h is harmonic, we get ∇h(y) 2 .

The model in continuous-time

Proposition 3.5. Let 0 < χ = (ψ 1 + ψ 2 )/ψ (hence the chain is recurrent) and A := A a = {x ∈ X : x ≤ a} for some a > 0, and γ x = O ( x 2-κ ); denote p 0 = χ/κ. Suppose further that id ∈ ρ (Γ) for some ρ > 2.

5. Since we have used the symbol ∆ to denote the jumps of the process, we introduce the symbol Lap to denote the Laplacian.

1. If q < p 0 , then E x (τ q A ) < ∞. 2. If q > p 0 , then E x (τ q A ) = ∞.

Proof. Consider the Lyapunov function f (x) = h β,β 1 (x) η .

1. If 0 < pη < 1 then m f p (x) < 0. The condition Γ f p ≤ -c f p-2 reads then

γ x ≥ C h pη-2η β (x) h pη-2 β ∇h β 2 = C x 2β-2βη
x 2β-2 = C x 2-2βη from which follows that 2βη > κ. This inequality, combined with 0 < pη < 1, yields that for all q < β κ < χ κ , E x (τ q A ) < ∞. Hence, on optimising on the accessible values of q we obtain the value of p 0 . Using again Lyapunov functions of the form f = h η β we can show that the drift of the chain in the transient case can be controlled by two 0-critically transient Lamperti processes in the variable x that are uniformly comparable. We can thus show, using methods developed in §3.1 the following Proposition 3.6. Let χ < 0 (hence the chain is transient).

1. If there exist a constant d 1 > 0 and an arbitary integer l > 0 such that γ x ≥ d 1 x L l ( x ) ln κ (l ) x , for some arbitrarily small κ > 0, then the chain explodes.

2. If there exist a constant d 2 > 0 and an arbitary integer l > 0 such that γ x ≤ d 2 x L l ( x ), then the chain does not explode.

With the help of similar arguments we can show the following Proposition 3.7. Let 0 < χ (hence the chain is recurrent).

1. If there exist a constant d 1 > 0 and an arbitrary integer l > 0 such that γ x ≥ d 1 x L l ( x ) ln κ (l ) x , for some arbitrarily small κ > 0, then the chain implodes.

2. If there exist a constant d 2 > 0 and an arbitrary integer l > 0 such that γ x ≤ d 2 x L l ( x ), then the chain does not implode.

a subsequent publication). Suppose that for every n ∈ N there exist an integer l n ≥ 0, a real δ n > 0, and a K n > 0 such that for x = (x 1 , x 2 ) ∈ N × N for x 1 large enough, γ x satisfies γ (x 1 ,x 2 ) = K x 2 O (L l x 2 (x 1 ) ln

δ x 2
(l x 2 ) x 1 ). By applying theorem 1.7, we establish that Z x 2 := E (x 1 ,x 2 ) (ζ|τ o = ∞) < ∞ for all x 1 > 0 and all x 2 ∈ N, hence P (x 1 ,x 2 ) (ζ = ∞|τ o = ∞) = 0. Irreducibility implies then that P o (ζ = ∞) = 0. However,

E o (ζ) ≥ x 2 ∈N π x 2 E (1,x 2 ) (ζ|τ o = ∞)P (1,x 2 ) (τ o = ∞) = 2p -1 p x 2 ∈N π x 2 Z x 2 .
Since the sequences (l n ) n , (δ n ) n , and (K n ) n are totally arbitrary, while the positive sequence (π n ) n must solely satisfy the probability constraint n∈N π n = 1, all possible behaviour for E o ζ can occur. In particular, we can choose, for all n ∈ N, l n = 0 and δ n = 1; this choice gives γ (x 1 ,n) = K n O ( 1

x 2 1
) for every n and for large x 1 , leading to the estimate Z n ≥ C K n , for all n.

Choosing now, for instance, π n = O (1/n 2 ) and K n = O (n) for large n, we get

P o (ζ = ∞) = 0 and E o ζ = ∞.
This remark leads naturally to the question whether for transient exploding chains with non-trivial tail boundary, there exists some critical q > 0 such that E(ζ p ) < ∞ for p < q while E(ζ p ) = ∞ for p > q. Such models include continuous time random walks on the homogeneous tree and more generally on non-amenable groups. These questions are currently under investigation and are postponed to a subsequent publication.

E 2 E

 2 (Ξ| ξn = x) = g (h(y))E(〈 ∇h(y), ΦD 〉| ξn = x)+ g (h(y)) 2 E(〈 ∇h(y), ΦD 〉 2 | ξn = x) + [g (h(y))] 2 (〈 ΦD, Hess h (y)ΦD 〉| ξn = x) + R 3 = g (h(y))〈 ∇h(y), n(y) 〉 + g (h(y)) 2 ∇h(y) 2 + R 3 (y),where R 3 is the remainder of the Taylor expansion. The value of the conditional increment depends on the position of x. If x ∈ ∂ X the dominant term of the right hand side is g (h(y))〈 ∇h(y), n 〉, while in the interior of the space, that term strictly vanishes because there n(y) = 0; hence the dominant term becomes the term g (h(y)) 2

2 .

 2 We proceed similarly; we need however to use the full-fledged version of the statement 2 of theorem 1.3, with both the function f and g = h η χ,ψ 1 . Then f (x) ≤ C g (x) and [ηβ < κ and ηp > 1] ⇒ [Γg ≥and Γg r ≤ cg r -1 for r > 1, and Γ f p ≥ 0]. Simultaneous verification of these inequalities yields p 0 = χ/κ.

P x y | f (y)f (x)| ρ = E x |∆ f 1 | ρ .We always write shortly v f (x) := v[START_REF] Lai | Markov chains with stationary transition probabilities[END_REF] f (x). The action of the generator Γ on f reads Γ f (x) := y∈X Γ x y f (y) = γ x m f (x).

This section intends to treat some problems that even with without the explosion phenomenon have critical behaviour illustrating thus how our methods can be applied and showing their power.

The most convenient form of the remainder is the Roche-Schlömlich one.

Collection of one-dimensional complexes

We introduce some simple models to illustrate two phenomena: -it is possible to have 0 < P x (ζ < ∞) < 1, -it is possible to have P x (ζ = ∞) = 0 and E x ζ = ∞. The simplest situation corresponds to a continuous-time Markov chain whose embedded chain is a simple transient random walk on X = Z with non trivial tail boundary. For instance, choose some p ∈]1/2, 1[ and transition matrix

Then, for every x = 0, 0 < P x (τ 0 = ∞) < 1. Suppose now that γ x = c for x < 0 while there exists a sufficiently large integer l ≥ 0 and an arbitrarily small δ > 0 such that γ x ≥ cL l (x) ln δ (l ) x for x ≥ x 0 . Then, using theorem 1.7, we establish that

This result combined with irreducibility of the chain leads to the conclusion:

It is worth noting that bending the axis Z at 0 allows considering the state space as the gluing of two one-dimensional complexes X 2 = {0} ∪ N × {-, +}; every point x ∈ Z \ {0} is now represented as x = (|x|, sgn(x)). This construction can be generalised by gluing a denumerably infinite family of one-dimensional complexes through a particular point o and introducing the state space X ∞ = {o}∪N×N; every point x ∈ X ∞ \{o} can be represented as x = (x 1 , x 2 ) with x 1 , x 2 ∈ N.

Let (ξ t ) t ∈[0,∞[ be a continuous time Markov chain evolving on the state space X ∞ . Its embedded (at the moments of jumps) chain ( ξn ) n∈N has transition matrix given by

where 1/2 < p < 1 and π = (π n ) n∈N is a probability vector on N, satisfying π n > 0 for all n. The chain is obviously irreducible and transient. The space X ∞ must be thought as a "mock-tree" since, for transient Markov chains, it has a sufficiently rich boundary structure without any of the complications of the homogeneous tree (the study on full-fledged trees is postponed in


Jean-Yves Enjalbert
email: jean-yves.enjalbert@lipn.univ-paris13.fr

Hoang Ngoc Minh

Combinatorial study of colored Hurwitz polyzêtas

A combinatorial study discloses two surjective morphisms between generalized shuffle algebras and algebras generated by the colored Hurwitz polyzêtas. The combinatorial aspects of the products and co-products involved in these algebras will be examined.

Introduction

Classically, the Riemann zêta function is ζ(s) = n>0 n -s , the Hurwitz zêta function is ζ(s; t) = n>0 (n -t) -s and the colored zêta function is ζ s q = n>0 q s n -s , where q is a root of unit. The three previous functions are defined over Z >0 but can be generalized over any composition (sequence of positive integers) s = (s 1 , . . . , s r ), like, respectively, the Riemann polyzêta function ζ(s) = n1>...>nr>0 n -s1 1 . . . n -sr r , the Hurwitz polyzêta function ζ(s; t) = n1>...>nr>0 (n 1 -t 1 ) -s1 . . . (n r -t r ) -sr and the colored polyzêta function ζ s q i = n1>...>nr>0 q i1n1 . . . q ir nr n -s1 1 . . . n -sr r , with q a root of unit and i = (i 1 , . . . , i r ) a composition. These sums converge when s 1 > 1.

To study simultaneously these families of polyzêtas, the colored Hurwitz polyzêtas, for a composition s = (s 1 , . . . , s r ) and a tuple of complex numbers ξ = (ξ 1 , . . . , ξ r ) and a tuple of parameters in ] -∞; 1[, t = (t 1 , . . . , t r ), are defined by [START_REF] Enjalbert | Propriétés combinatoires et prolongement analytique effectif de polyzêtas de Hurwitz et de leurs homologues[END_REF] Di(F ξ,t ; s) = n1>...>nr >0 ξ n1 1 . . . ξ nr r (n 1 -t 1 ) s1 . . . (n r -t r ) sr .

(

) 1
Note that, for l = 1 . . . , r, the numbers ξ l are not necessary roots of unity q i l . We are working, in this note, with the condition

(E) ∀i, | i k=1 ξ k | ≤ 1 and t i ∈] -∞; 1[.
Hence, Di(F ξ,t ; s) converges if s 1 > 1. We note E the set of C-tuples verifying (E). These polyzêtas are obtained as special values of iterated integrals 1 over singular differential 1-forms introduced in [START_REF] Ngoc | Fonctions de Dirichlet d'ordre n et de paramètre t[END_REF]. As iterated integrals, they are encoded by words or by non commutative formal power series [START_REF] Ngoc | Fonctions de Dirichlet d'ordre n et de paramètre t[END_REF] and are used to construct bases for asymptotic expanding [START_REF] Ngoc | Algebraic Combinatoric Aspects of Asymptotic Analysis of Nonlinear Dynamical System with Singular Input[END_REF] or symbolic integrating fuchian differential equations [START_REF] Hoang | Symbolic integration of meromorphic differential systems via Dirichlet functions[END_REF] exactly or approximatively [START_REF] Hoang | Input/Output behaviour of nonlinear analytic systems : rational approximations, nilpotent structural approximations[END_REF]. The meromorphic continuation of the colored Hurwitz polyzêtas 2 is already studied in [START_REF] Enjalbert | Analytic and combinatoric aspects of Hurwitz polyzêtas[END_REF][START_REF] Enjalbert | Propriétés combinatoires et prolongement analytique effectif de polyzêtas de Hurwitz et de leurs homologues[END_REF]. In our studies, we constructed an integral representation 3 of colored Hurwitz polyzêtas and a distribution treating simultanously two singularities and our methods permit to make the meromorphic continuation commutatively over the variables s 1 , . . . , s r [START_REF] Enjalbert | Analytic and combinatoric aspects of Hurwitz polyzêtas[END_REF][START_REF] Enjalbert | Propriétés combinatoires et prolongement analytique effectif de polyzêtas de Hurwitz et de leurs homologues[END_REF]. Moreover, [START_REF] Enjalbert | Propriétés combinatoires et prolongement analytique effectif de polyzêtas de Hurwitz et de leurs homologues[END_REF] gives another way to obtain the meromorphic continuation thanks to translation equations [START_REF] Gari | la dimorphie et l'arithmétique des multizêtas : un premier bilan[END_REF]. Our methods give the structure of multi-poles [START_REF] Enjalbert | Analytic and combinatoric aspects of Hurwitz polyzêtas[END_REF] (Theorem 4.2) and two ways to calculate algorithmically the multi-residus 4 .

In this note, in continuation with our previous works [START_REF] Ngoc | Fonctions de Dirichlet d'ordre n et de paramètre t[END_REF][START_REF] Hoang | Symbolic integration of meromorphic differential systems via Dirichlet functions[END_REF][START_REF] Hoang | De l'algèbre des ζ de Riemann multivariées à l'algèbre des ζ de Hurwitz multivariées[END_REF][START_REF] Ngoc | Finite polyzêtas, Poly-Bernoulli numbers, identities of polyzêtas and noncommutative rational power series[END_REF][START_REF] Enjalbert | Analytic and combinatoric aspects of Hurwitz polyzêtas[END_REF][START_REF] Enjalbert | Propriétés combinatoires et prolongement analytique effectif de polyzêtas de Hurwitz et de leurs homologues[END_REF], we are focusing on Hofp algebra, for a class of products as minusstuffle ( ), mulstuffle ( q ), . . . , and in particular for the new product duffle ( q ), obtained as "tensorial product" of q and the well known stuffle ( ), of symbolic representations of these polyzêtas (see Definition 2.1 and Proposition 2.1 bellow).

Combinatorial objects 2.1 Some products and their algebraic structures

Let X be an encoding alphabet and the free monoid over X is denoted by X * . The length of any word w ∈ X * is denoted by |w| and the unit of X * is denoted by 1 X * . For any unitary commutative algebra A, a formal power series S over X with coefficients in A can be written as the infinite sum w∈X * S|w w. The set of polynomials (resp. formal power series) over X with coefficients in A is denoted by A X (resp. A X ). The set of degree 1 monomials is AX = {ax/a ∈ A, x ∈ X}. Definition 2.1 We note P the set of products ⋆ over A X verifying the conditions :

1 They are presented as generalized Nielsen polylogarithms in [START_REF] Ngoc | Fonctions de Dirichlet d'ordre n et de paramètre t[END_REF] (Definition 2.3) and as generalized Lerch functions in [START_REF] Hoang | De l'algèbre des ζ de Riemann multivariées à l'algèbre des ζ de Hurwitz multivariées[END_REF] (Definition 3). 2 See also references and a discussion about meromorphic continuation of Riemann polyzêtas in [START_REF] Enjalbert | Analytic and combinatoric aspects of Hurwitz polyzêtas[END_REF]. 3 This integral representation is obtained by applying successively the polylogarithmic transform [START_REF] Ngoc | Fonctions de Dirichlet d'ordre n et de paramètre t[END_REF]. It is an application of non commutative convolution as shown in [START_REF] Ngoc | Summations of Polylogarithms via Evaluation Transform[END_REF] (Section 2.4). Other integral representations can be also deduced easily by change of variables, for example t = zr and then r = e -u [START_REF] Enjalbert | Analytic and combinatoric aspects of Hurwitz polyzêtas[END_REF]. 4 Other meromorphic continuations can also be obtained by Mellin transform as already done in [START_REF] Matsumoto | Analytic properties of multiple zeta-functions in several variables[END_REF] or by classical estimation on the imaginary part [START_REF] Essouabri | Multiple zeta functions associated with linear recurrence sequences and the vectorial sum formula[END_REF] but these later work reccursively, depth by depth, and the commutativity of this process over the variables s 1 , . . . , sr must be proved. Unfortunately, the structure of multi-poles as well as multi-residus are missing in both works [START_REF] Essouabri | Multiple zeta functions associated with linear recurrence sequences and the vectorial sum formula[END_REF][START_REF] Matsumoto | Analytic properties of multiple zeta-functions in several variables[END_REF]. In [START_REF] Komori | An integral representation of multiple Hurwitz-Lerch zeta functions and generalized multiple Bernoulli numbers[END_REF], to make the meromorphic continuation (giving the expression of non positive integers multi-residus via a generalization of Bernoulli numbers -but not of all multi-residus) of the specialization at roots of unity of colored Hurwitz polyzêtas Di(F ξ,t ; s), the author bases on the integral representation, on the contours, of the multiple Hurwitz-Lerch which corresponds mutatis mutandis to the integral representation of generalized Lerch functions introduced earlier in [5] (Corollary 3.3).

(i) the map

⋆ : A X × A X → A X is bilinear, (ii) for any w ∈ X * , 1 X * ⋆ w = w ⋆ 1 X * = w, (iii) for any a, b ∈ X and u, v ∈ X * , au ⋆ bv = a(u ⋆ bv) + b(au ⋆ v) + [a, b](u ⋆ v),
where [., .] : AX × AX → AX is a function verifying :

(S1) ∀a ∈ AX, [a, 0] = 0 , (S2) ∀(a, b) ∈ (AX) 2 , [a, b] = [b, a], (S3) ∀(a, b, c) ∈ (AX) 3 , [[a, b], c] = [a, [b, c]].
Example 1 (see [START_REF] Reutenauer | Free Lie Algebras[END_REF]) Product of interated integrals. The shuffle is a bilinear product such that :

∀w ∈ X * w ⊔⊔ 1 X * = 1 X * ⊔⊔ w = w and ∀(a, b) ∈ X 2 , ∀(u, v) ∈ X * 2 , au ⊔⊔ vb = a(u ⊔⊔ bv) + b(au ⊔⊔ v).
For example, for any letter x 0 , x and x ′ in X, [START_REF] Hoffman | Quasi-shuffle products[END_REF]) Product of quasi-symmetric functions. Let X be an alphabet indexed by N. The stuffle is a bilinear product such that :

x 0 x ′ ⊔⊔ x 2 0 x = x 0 x ′ x 2 0 x + 2x 2 0 x ′ x 0 x + 3x 3 0 x ′ x + 3x 3 0 xx ′ + x 2 0 xx 0 x ′ . Example 2 (see
∀w ∈ X * , w 1 X * = 1 X * w = w and ∀(x i , x j ) ∈ X 2 , ∀(u, v) ∈ X * 2 , x i u x j v = x i (u x j v) + x j (x i u v) + x i+j (u v).
In particular, with the alphabet Y = {y 1 , y 2 , y 3 , . . .}, (y 3 y 1 ) y 2 = y 3 y 1 y 2 + y 3 y 2 y 1 + y 3 y 3 + y 2 y 3 y 1 + y 5 y 1 .

Example 3 ([3]

) Product of large multiple harmonic sums.

Let X be an alphabet indexed by N.

The minus-stuffle is a bilinear product such that :

∀w ∈ X * , w 1 X * = 1 X * w = w and ∀(x i , x j ) ∈ X 2 , ∀(u, v) ∈ X * 2 , x i u x j v = x i (u x j v) + x j (x i u v) -x i+j (u v). Example 4 ([6]) Product of colored sums.
Let X be an alphabet indexed by a monoid (I, ×).

The mulstuffle is a bilinear product such that :

∀w ∈ X * w q 1 X * = 1 X * q w = w and ∀(x i , x j ) ∈ X 2 , ∀(u, v) ∈ X * 2 , x i u q x j v = x i (u q x j v) + x j (x i u q v) + x i×j (u q v).
For example, with X indexed by

Q * , x 2 3 x -1 q x 1 2 = x 2 3 x -1 x 1 2 + x 2 3 x 1 2 x -1 + x 2 3 x -1 2 + x 1 2 x 2 3 x -1 + x 1 3 x -1 .
Remark 2.1 Thanks to the one-to-one correspondence (i 1 , . . . , i r ) → x i1 . . . x ir between tuples of I and word over X, the calculus of x 2 3 x -1 q x 1 2 can be written as

2 3 , -1 q 1 2 = 2 3 , -1, 1 2 + 2 3 , 1 2 , -1 + 2 3 , -1 2 + 1 2 , 2 3 , -1 + 1 3 , -1 .
1 A * = (1 Y * , 1 E * ). If Y is indexed by N and E by a monoid (I, ×), the duffle is a bilinear product such that ∀w ∈ A * , w q 1 A * = 1 A * q w = w, ∀(y i , y j ) ∈ Y 2 , ∀(e l , e k ) ∈ E 2 , ∀(u, v) ∈ A * 2
, (y i , e l ).u q (y j , e k ).v = (y i , e l ).

(u q (y j , e k ).v) + (y j , e k ). ((y i , e l ).u q v) + (y i+j , e l×k ).(u q v).

Proposition 2.1 The shuffle, the stuffle, the minus-stuffle and the mulstuffle are elements of P, with respectively,

[x i , x j ] = 0, [x i , x j ] = x i+j , [x i , x j ] = -x i+j , [x i , x j ] =
x i×j for any letters x i and x j of X.

The duffle is in P, with [(y i , e l ), (y j , e k )] = (y i+j , e l×k ) for all y i , y j in Y , e l , e k in E.

Proposition 2.2 Let ⋆ ∈ P, then (A X , ⋆) is a commutative algebra.

Proof. We just have to show the commutativity and the associativity of ⋆.

To obtain w 1 ⋆ w 2 = w 2 ⋆ w 1 for all w 1 , w 2 in X * , we use an induction on

|w 1 | + |w 2 |. It is true when |w 1 | + |w 2 | ≤ 1 thanks to (i) since w 1 or w 2 is 1 X * .
The equality (iii), the condition (S2) and the commutative of + give the induction. In the same way, an induction on

|w 1 | + |w 2 | + |w 3 | gives w 1 ⋆ (w 2 ⋆ w 3 ) = (w 1 ⋆ w 2 ) ⋆ w 3 thanks to (iii)
and (S3). ✷ If we associate to each letter of X an integer number called weight, the weight of a word is the sum of the weight of its letters. In this case X is graduated. In [START_REF] Hoffman | Quasi-shuffle products[END_REF], Hoffman works over X = X ∪ {0} with [., .] : X × X → X and call quasiproduct any product in P with the additional condition : Example 6 1. The shuffle is a quasi-product.

2. Let X be an alphabet indexed by N and define the weight of x i , i ∈ N, by i . Then the stuffle is a quasi-product.

Theorem 2.1 ( [START_REF] Hoffman | Quasi-shuffle products[END_REF]) If X is graduated and has a quasi-product ⋆, then (A X , ⋆) is a commutative graduated A-algebra..

We can define (i) a comultiplication

∆ : A X → A X ⊗ A X , (ii) a counit ǫ : A X → A, by : ∀w ∈ X * , ∆w = uv=w u ⊗ v and ǫ(w) = 1 if w = 1 X * 0 otherwise. The coproduct ∆ is coassociative so (A X , ∆, ǫ) is a coalgebra. Lemma 2.1 For any w ∈ X * and x ∈ X, (x ⊗ 1 X * )∆w + 1 X * ⊗ xw = ∆xw. Proof. ∀w ∈ X * , ∀x ∈ X, ∆xw = uv=xw u ⊗ v = u ′ v=w xu ′ ⊗ v + 1 X * ⊗ xw so ∆xw = x ⊗ 1 X * u ′ v=w u ′ ⊗ v + 1 X * ⊗ xw = (x ⊗ 1 X * )∆w + 1 X * ⊗ xw. ✷ Proposition 2.3 If ⋆ ∈ P, then (A X , ⋆, ∆, ǫ) is a bialgebra. Remember that ⋆ acts over A X ⊗ A X by (u ⊗ v) ⋆ (u ′ ⊗ v ′ ) = (u ⋆ u ′ ) ⊗ (v ⋆ v ′ ).
Proof. ǫ is obviously a ⋆-homomorphism. It still has to be show ∆(w 1 ) ⋆ ∆(w 2 ) = ∆(w 1 ⋆ w 2 ) over X * . This equality is true if w 1 or w 2 is equal to 1 X * . Assume now that ∆(u)⋆∆(v) = ∆(u⋆v) for any word u and v such that |u|+|v| ≤ n, n ∈ N, and let w 1 and w 2 be in X * with |w 1 | + |w 2 | = n + 1. We note w 1 = au and w 2 = bv, with a and b two letters of X, u and v two words of X * . Thus, by definition,

∆w 1 = u1u2=u au 1 ⊗ u 2 + 1 X * ⊗ au and ∆w 2 = v1v2=v bv 1 ⊗ v 2 + 1 X * ⊗ bv. ∆(w 1 ) ⋆ ∆(w 2 ) = u1u2=u,v1v2=v (au 1 ⋆ bv 1 ) ⊗ (u 2 ⋆ v 2 ) + u1u2=u (au 1 ) ⊗ (u 2 ⋆ bv) + v1v2=v (bv 1 ) ⊗ (au ⋆ v 2 ) + 1 X * ⊗ (au ⋆ bv) = u1u2=u,v1v2=v (a(u 1 ⋆ bv 1 ) ⊗ (u 2 ⋆ v 2 ) + b(au 1 ⋆ v 1 ) ⊗ (u 2 ⋆ v 2 ) +([a, b](u 1 ⋆ v 1 )) ⊗ (u 2 ⋆ v 2 )) + u1u2=u (au 1 ) ⊗ (u 2 ⋆ bv) + v1v2=v (bv 1 ) ⊗ (au ⋆ v 2 ) + 1 X * ⊗ a(u ⋆ bv) +1 X * ⊗ b(au ⋆ v) + 1 X * ⊗ [a, b](u ⋆ v) = u1u2=u,v1v2=v a(u 1 ⋆ bv 1 ) ⊗ (u 2 ⋆ v 2 ) + u1u2=u (au 1 ) ⊗ (u 2 ⋆ bv) + u1u2=u,v1v2=v b(au 1 ⋆ v 1 ) ⊗ (u 2 ⋆ v 2 ) + v1v2=v (bv 1 ) ⊗ (au ⋆ v 2 ) +[a, b] ⊗ 1 X * u 1 u 2 =u v1v2=v (u 1 ⊗ u 2 ) ⋆ (v 1 ⊗ v 2 ) +(1 X * ⊗ a(u ⋆ bv) + 1 X * ⊗ b(au ⋆ v) + 1 X * ⊗ [a, b](u ⋆ v)) = (a ⊗ 1 X * )(∆(u) ⋆ ∆(w 2 )) + 1 X * ⊗ a(u ⋆ bv) + (b ⊗ 1 X * )(∆(w 1 ) ⋆ ∆(v)) +1 X * ⊗ b(au ⋆ v) + ([a, b] ⊗ 1 X * )(∆(u) ⋆ ∆(v)) + 1 X * ⊗ [a, b](u ⋆ v).
Using the induction hypothesis then the lemma 2.1 (since [a, b] ∈ AX) gives

∆(w 1 ) ⋆ ∆(w 2 ) = ∆(a(u ⋆ w 2 )) + ∆(b(w 1 ⋆ v)) + ∆([a, b](u ⋆ v)) = ∆(a(u ⋆ w 2 ) + b(w 1 ⋆ v) + [a, b](u ⋆ v)) = ∆(w 1 ⋆ w 2 ). ✷ Remark 2.2
In particular, ∆ is a ⊔⊔ -homomorphism, a -homomorphism and a qhomomorphism.

Let C n be the set of positive integer sequences (i 1 , . . . , i k ) such that i 1 + . . .

+ i k = n.
Theorem 2.2 Define a ⋆ by, for all x 1 , . . . , x n in X,

a ⋆ (x 1 . . . x n ) = (i1,...,i k )∈Cn (-1) k x 1 . . . x i1 ⋆ x i1+1 . . . x i1+i2 ⋆ . . . ⋆ x i1+...+i k-1 +1 . . . x n then, if ⋆ ∈ P, (A X , ⋆, ∆, ǫ, a ⋆ ) is a Hopf algebra.
Proof. With the applications :

µ : A → A X λ → λ 1 X * and m : A X ⊗ A X → A X u ⊗ v → u ⋆ v , the antipode must verify m • (a ⋆ ⊗ Id) • ∆ = µ • ǫ, or, in equivalent terms uv=w a ⋆ (u) ⋆ v = w|1 X * 1 X * . i.e. a ⋆ (1 X * ) = 1 X * n ∀x ∈ X, a ⋆ (x) = -x and, if w = x 1 . . . x n with n ≥ 2, x 1 , . . . , x n ∈ X, a ⋆ (w) = - n-1 k=1 a ⋆ (x 1 . . . x k ) ⋆ x k+1 . . . x n .
An induction over the length n shows that a ⋆ defined in theorem verifies these equalities, and, in the same way,

a ⋆ verifies m • (Id ⊗ a ⋆ ) • ∆ = µ • ǫ. ✷ Corollary 2.
1 If ⋆ is ⊔⊔ or or q or q , then this construction gives an Hopf algebra. Moreover, for ⊔⊔ or , we obtain a graduated Hopf algebra.

Iterated integral

Let us associate to each letter x i in X a 1-differential form ω i , defined in some connected open subset U of C. For all paths z 0 z in U, the Chen iterated integral associated to w = x i1 • • • x i k along z 0 z, noted is defined recursively as follows

α z z0 (w) = z0 z ω i1 (z 1 )α z1 z0 (x i2 • • • x i k ) and α z z0 (1 X * ) = 1, (2)
verifying the rule of integration by parts [START_REF] Chen | Iterated path integrals[END_REF] :

α z z0 (u ⊔⊔ v) = α z z0 (u)α z z0 (v). (3)
We extended this definition over A X (resp. A X ) by

α z z0 (S) = w∈X * S|w α z z0 (w). ( 4
)

Shuffle relations 2.3.1 First encoding for colored Hurwitz polyzêtas

Let ξ = (ξ n ) be a sequence of complex numbers and T a family of parameters. Put X ′ an alphabet indexed over N * × C N × T and X = {x 0 } ∪ X ′ . To each x in X we associate the differential form :

       ω 0 (z) = dz z si x = x 0 ω i,ξ,t (z) = i k=1 ξ k 1 - i k=1 ξ k z × dz z t if x = x i,ξ,t with i 1.
(

) 5
For any T -tuple t = (t 1 , . . . , t r ) we associate the T -tuple t = (t 1 , . . . , t r ) given by

           t 1 = t 1 -t 2 , t 2 = t 2 -t 3 , . . . t r = t r-1 -t r in this way            t 1 = t 1 + t 2 + . . . + t n , t 2 = t 2 + . . . + t n , . . . t r = t r (6)
We choose the sequence ξ and the family t such that the condition (E) is satisfied.

Proposition 2.4 For any s = (s 1 , . . . , s r ) with

s 1 > 1 if ξ = (ξ 1 , . . . , ξ r ) ∈ E r and t = (t 1 , . . . , t r ) ∈ T r , then Di(F ξ,t ; s) = α 1 0 (x s1-1 0 x 1,ξ,t1 . . . x sr -1 0 x r,ξ,tr ).
Proof. Since ω i,ξ,t (z) =

(F ξ,t ; s)} ξ∈C ′ ,t∈T ′ is the A modulus generated by {Di(F ξ,t ; s)} ξ∈C ′ ,t∈T ′ .
Proof. We have express the product Di(F ξ,t ; s) Di(F ξ ′ ,t ′ ; s ′ ), with s = (s 1 , . . . , s r ),

s ′ = (s ′ 1 , . . . , s ′ r ′ ), ξ, ξ ′ ∈ C ′ and t = (t 1 , . . . , t r ), t ′ = (t ′ 1 , . . . , t ′ r ) ∈ T ′ ,
as linear combination of colored Hurwitz polyzêtas. This is an iterated integral associated

to x s1-1 0 x 1,ξ,t1 . . . x sr -1 0 x r,ξ,tr ⊔⊔ x s ′ 1 -1 0 x 1,ξ ′ ,t ′ 1 . . . x s ′ r ′ -1 0 x r ′ ,ξ ′ ,t ′ r ′
which is a sum of terms of the form x

s ′′ 1 -1 0 x 1,ξ (1) ,t (1) . . . x s ′′ i -1 0 x ji,ξ (i) ,t (i) . . . x s ′′ r -1 0
x j r ′′ ,ξ (r ′′ ) ,t (r ′′ ) , with

s ′′ i ∈ N, ξ (i)
is ξ or ξ ′ and t (i) is t ji or t ′ ji for all i; and r ′′ = r + r ′′ . Note that

α z 0 (x i,ξ,ti x s-1 0 x j,ξ ′ ,tj ) = z 0 m>0 i k=1 ξ m k z m-ti-1 1 dz 1 z1 0 dz 2 z 2 ... zs+1 0 n>0 i k=1 ξ ′n k z n-t ′ j -1 s+1 dz s+1 = m,n>0 (ξ 1 . . . ξ i ) m ξ ′ 1 . . . ξ ′ j n (m + n -t i -t ′ j )(n -t ′ j ) s z n+m , α 1 0 x s ′′ 1 -1 0 x 1,ξ (1) ,t (1)
. . . x

s ′′ i -1 0 x ji,ξ (i) ,t (i) . . . x s ′′ r -1 0 x j r ′′ ,ξ (r ′′ ) ,t (r ′′ ) = m1,...,m r ′′ >0 r ′′ i=1 (ξ (i) 1 . . . ξ (i) ji ) mi (m i + . . . + m r ′′ -t (i) -. . . -t (r ′′ ) ) s ′′ i = n1>...>n r ′′ >0 r ′′ i=1 ξ ′′ ni i (n i -t ′′ i ) s ′′ i with n i = m i + . . . + m r ′′ , t ′′ i = t (i) + . . . + t (r ′′ ) for all i, so t ′′ ∈ T ; ξ ′′ 1 = ξ (1) 1 and ξ ′′ i = ξ (i) 1 ...ξ (i) j i ξ (i-1) 1 ...ξ (i-1) j i-1
for i > 1 so ξ ′′ ∈ C : we can express each term of the shuffle product as Di(F ξ ′′ ,t ′′ ; s ′′ ). ✷ Note that the shuffle product over two words of X * X ′ acts separately over (C ′ , .), (T ′ , +) and the convergent compositions. We can describe the situation with the shuffle algebra 5 : Theorem 2.4 Let H be the Q-algebra generated by the colored Hurwitz polyzêtas. The map ζ : (Q (x * 0 x i,ξ,t ) * , ⊔⊔ ) ։ (H, .), x s1 0 x 1,ξ,t1 . . . x sr 0 x r,ξ,tr → Di(F ξ,t ; s + 1) is a surjective algebra morphism.

Example 7 Since Di(F ξ,t ; 3) = α 1 0 (x 2 0 x 1,ξ,t ) and Di(F ξ ′ ,t ′ ; 2) = α 1 0 (x 0 x 1,ξ ′ ,t ′ ) then Di(F xi,t ; 3) Di(F xi ′ ,t ′ ; 2) = α 1 0 (x 0 x 1,ξ ′ ,t ′ ⊔⊔ x 2 0 x 1,ξ,t ). Example 1 with x = x 1,ξ,t and x ′ = x 1,ξ ′ ,t ′ gives the expression of x 0 x 1,ξ ′ ,t ′ ⊔⊔ x 2 0 x 1,ξ,t . But the first term obtained is

α 1 0 (x 0 x 1,ξ ′ ,t ′ x 2 0 x 1,ξ,t ) = 1 0 dz 1 z 1 z1 0 m>0 ξ ′ m z m-t ′ -1 2 dz 2 z2 0 dz 3 z 3 z3 0 dz 4 z 4 z4 0 n>0 ξ n z n-t-1 5 dz 5 = n,m>0 ξ ′ m ξ n (m + n -t ′ -t) 2 (n -t) 3 = n1>n2>0 (ξ ′ ) n1 (ξ/ξ ′ ) n2 (n 1 -t ′ -t) 2 (n 2 -t) 3
= Di(F (ξ,ξ/ξ ′ );(t+t ′ ,t) ; (2, 3)). 5 Working in Q x * 0 x i,ξ,t * implies working in the graduated Hopf algebra (Q X * , ⊔⊔ , ∆, ǫ, a ⊔⊔ ).

We can make similar calculus for the other terms and find :

Di(F ξ,t ; 3) Di(F ξ ′ ,t ′ ; 2) = Di(F (ξ ′ ,ξ/ξ ′ );(t+t ′ ,

Second encoding for colored Hurwitz polyzêtas

For the Hurwitz polyzêtas, we can obtain an encoding indexed by a finite alphabet. Let the alphabet X = {x 0 ; x 1 } and associate to x 0 the form ω 0 (z) = z -1 dz and at x 1 the form ω 1 (z) = (1 -z) -1 dz.

For each x ∈ X and λ ∈ C, we note (λx) * = k≥0 (λx) k . Then, (see [START_REF] Ngoc | Fonctions de Dirichlet d'ordre n et de paramètre t[END_REF], [START_REF] Hoang | Symbolic integration of meromorphic differential systems via Dirichlet functions[END_REF]),

α 1 0 x s1-1 0 (t 1 x 0 ) * s1 x 1 . . . x sr -1 0 (t r x 0 ) * sr x 1 = ζ(s; t).
Theorem 2.5 Let H ′ be the Q-algebra generated by the Hurwitz polyzêtas and X the Q-algebra generated by (t 1 x 0 ) * s1 x 1 . . . (t r x 0 ) * sr x r . Then, ζ : (X , ⊔⊔ ) ։ (H ′ , .) is a surjective morphism of algebras.

Note that we can apply the idea of encoding of "simple" colored Hurwitz zetas functions (with depth one : r = 1). Let ξ = (ξ n ) be a sequence of complex numbers in the unit ball B(0; 1) and T a family of parameters. Let X = {x 0 , x 1 , . . .} be a alphabet indexed by N. Associate to x 0 the differential form ω 0 (z) = z -1 dz and to

x i , i ≥ 1, the differential form ω i (z) = ξ i (1 -ξ i z) -1 dz. Proposition 2.5 With this notation, α 1 0 ((tx 0 ) * x 0 ) s-1 (tx 0 ) * x i = n>0 ξ n i (n -t) s . Proof. Since ξ i dz 0 1 -ξ i z 0 = ξ i n≥0
(ξ i z 0 ) n dz 0 , we can write

α z 0 (tx 0 ) k x i = t k z 0 dz k z k z k 0 . . . z1 0 ξ i n≥0 (ξ i z 0 ) n dz 0 = n>0 t k ξ n i z n n k+1 ,
for z ∈ B(0; 1) and for k ∈ N. Thanks to the absolute convergence,

α z 0 ((tx 0 ) * x i ) = n>0 ξ n i z n n k≥0 t n k = n>0 ξ n i z n n -t .
In the same way, if z ∈ B(0; 1) :

∀k ∈ N, α z 0 (tx 0 ) k x 0 (tx 0 ) * x i = n>0 t k ξ n i n -t z n n k+1 , so α z 0 ((tx 0 ) * x 0 (tx 0 ) * x i ) = n>0 ξ n i z n (n -t) 2
and α z 0 ((tx 0 ) * x 0 ) s-1 (tx 0 ) * x i = n>0 ξ n i z n (n -t) s . ✷ Remark 2.3 Note that, with the same notation,

α z 0 x 1 ((t 2 x 0 ) * x 0 ) s-1 (t 2 x 0 ) * x 2 = n,m>0 ξ n 2 ξ m 1 z n+m (n -t 2 ) s (m + n) = n1>n2>0 ξ n2 2 ξ n1-n2 1 z n1 n 1 (n 2 -t 2 ) s .
In other words, this encoding appears to be widespread only as couples of the type ξ = (1, 1, . . . , 1, ξ r ) : with ξ 1 = 1 and ω 1 = (1 -z) -1 dz, We can export the duffle over the tuples s = (s 1 , . . . , s r ) ∈ Z r >0 and ξ ∈ C r with :

(s, ξ) q ((), 1) = ((), 1) q (s, ξ) = (s, ξ) and (s 1 , s; ξ 1 , ξ) q (r 1 , r; ρ 1 , ρ) = (s 1 ; ξ 1 ). ((s; , ξ) q (r 1 , r; ρ 1 , ρ)) + (r 1 ; ρ 1 ). ((s 1 , s; ρ 1 , ξ) q (r; ρ)) +(s 1 + r 1 ; ξ 1 ρ 1 ). ((s; ξ) q (r; ρ))

Proposition 2.6 Let s = (s 1 , . . . , s l ) and r = (r 1 , . . . , r k ) be two compositions, ξ ∈ C l , ρ ∈ C k . Then ∀n ∈ N, M n s,ξ (λ) M n r,ρ (λ) = M n (s,ξ) q (r,ρ) (λ).

Proof. Put the compositions s ′ = (s 2 , . . . , s l ), r ′ = (r 2 , . . . , r k ), the tuples of complex numbers ξ ′ = (ξ 2 , . . . , ξ l ) and ρ ′ = (ρ 2 , . . . , ρ k ), then

M n s,ξ (λ) M n r,ρ (λ) = n>n1,n>n ′ 1 ξ n1 1 λ s1 n1 M n1 s ′ ,ξ ′ (λ) ρ n ′ 1 1 λ r1 n ′ 1 M n ′ 1 r ′ ,ρ ′ (λ) = n>n1 ξ n1 1 λ s1 n1 M n1 s ′ ,ξ ′ (λ) M n1 r,ρ (λ) + n>n ′ 1 ρ n ′ 1 1 λ r1 n ′ 1 M n ′ 1 s,ξ (λ) M n ′ 1 r ′ ,ρ ′ (λ)

Example 5 (

 5 [START_REF] Enjalbert | Propriétés combinatoires et prolongement analytique effectif de polyzêtas de Hurwitz et de leurs homologues[END_REF]) Product of colored Hurwitz polyzêtas. Let Y and E be two alphabets and consider the alphabet A = Y × E with the concatenation defined recursively by (y, e).(w Y , w E ) = (yw Y , ew E ) for any letters y ∈ Y , e ∈ E, and any word w Y ∈ Y * , w E ∈ E * . The unit of the monoide A * is given by

(

  S4) Either [a, b] = 0 for all a, b in X; or the weight of [a, b] is the sum of the weight of a and the weight of b for all a, b in X.

1 0x 1 1 0

 111 tr (n -t r ) sr . Hence, α 1 0 (x s1-,ξ,t1 . . . x sr -+ . . . + m r -t j -. . . -t r ) sj , and then, by change of variables, n1>...>nr >0 ξ n1 1 . . . ξ nr r (n 1 -t 1 ) s1 . . . (n r -t r ) sr . ✷ Theorem 2.3 Let T be the group of parameters generated by T ; + , C be a subgroup of (C * , .) and A a sub-ring of C. Put C ′ = C N ∩ E and T ′ the set of finite tuple with elements in T . Then the A algebra generated by {Di

α 1 0 x s1- 1 0(t 1 2 . 4

 1124 x 0 ) * s1 x 1 . . . x sr-1-1 0 (t r-1 x 0 ) * sr-1 x r-1 x sr -1 0 (t r x 0 ) * sr x r = n1>...>nr ξ nr r (n 1 -t 1 ) s1 . . . (n r -t r ) sr . Duffle relations Let λ = (λ n ) be a set of parameters, s = (s 1 , . . . , s r ) a composition, ξ ∈ C r . Then ∀n ∈ Z >0 , M n s,ξ (λ) =

A recurrence ended the demonstration. ✷ Theorem 2.6 Let s = (s 1 , . . . , s l ) and r = (r 1 , . . . , r k ) be two compositions, ξ a ltuple and ρ a k-tuple of E, t = (t, . . . , t) a l-tuple and t ′ = (t, . . . , t) a k-tuple, both formed by the same parameter t diagonally. Then

Remark 2.4 Extend the duffle product to triplets (s, t, ξ) ∈ ∪ r∈N * N r × {t} r × C r by (s 1 , s; t, t; ξ 1 , ξ) q (r 1 , r; t, t ′ ; ρ 1 , ρ) = (s 1 ; t; ξ 1 ). ((s; t; ξ) q (r 1 , r; t, t ′ ; ρ 1 , ρ)) + (r 1 ; t; ρ 1 ). ((s 1 , s; t, t; ρ 1 , ξ) q (r; t ′ ; ρ)) + (s 1 + r 1 ; t; ξ 1 ρ 1 ). ((s; t; ξ) q (r; t ′ ; ρ)) , and define the function F over I = ∪ r∈N * N r × {t} r × C r by F (s, t, ξ) = Di(F ξ,t ; s). Then, by Theorem 2.6, the function F : (I, q ) → (C, .) is morphism of algebras.