Loïc Chaumont

Salem Lamine

On R d -valued multi-self-similar Markov processes

Keywords: Multi-self-similarity, Markov additive process, Lévy process, time change AMS MSC 2010: 60J45

An R d -valued Markov process X

 in the aim of extending the Lamperti transformation of positive self-similar Markov processes to R d + -valued processes. This paper aims at giving a complete description of all R d -valued multi-self-similar Markov processes. We show that their state space is always a union of open orthants with 0 as the only absorbing state and that there is no finite entrance law at 0 for these processes. We give conditions for these processes to satisfy the Feller property. Then we show that a Lamperti-type representation is also valid for R d -valued multi-self-similar Markov processes. In particular, we obtain a one-to-one relationship between this set of processes and the set of Markov additive processes with values in {-1, 1} d × R d . We then apply this representation to study the almost sure asymptotic behavior of multi-self-similar Markov processes.

Introduction

A Markov process {(X t ) t≥0 , P x } with state space E ⊂ R d satisfies the multi-scaling property of index α = (α 1 , . . . , α d ) ∈ [0, ∞) d if for all x = (x 1 , . . . , x d ) ∈ E and c = (c 1 , . . . , c d )

∈ (0, ∞) d , {(X c α t ) t≥0 , P c•x } = {(c • X t ) t≥0 , P x } , (1.1)
where c α := d i=1 c α i i and • denotes the Hadamard product, that is c•x := (c 1 x 1 , . . . , c d x d ). These processes, called multi-self-similar Markov processes, were introduced by Jacobsen and Yor in [START_REF] Jacobsen | Multi-self-similar Markov processes on R n + and their Lamperti representations[END_REF] in order to extend the famous Lamperti representation to (0, ∞) d -valued Markov processes. They proved that any (0, ∞) d -valued multi-self-similar Markov process {(X (1) t , . . . , X (d) t ) t≥0 , P x } can be represented as

X (i) t = exp ξ (i) τt , (1.2)
where (ξ (1) , . . . , ξ (d) ) is a d-dimensional Lévy process issued from (log x 1 , . . . , log x d ) and τ t = inf{s : s 0 exp(α 1 ξ

(1)

u + • • • + α d ξ (d)
u ) du > t}. They also proved that conversely, for any Lévy process (ξ (1) , . . . , ξ (d) ), the transformation (1.2) defines a (0, ∞) d -valued multiself-similar Markov process.

On the other hand, in the recent work [START_REF] Alili | Inversion, duality and Doob h-transforms for self-similar Markov processes[END_REF], the authors showed that there is a way to extend the Lamperti representation to all standard self-similar Markov processes with values in R d \ {0}. This extension defines a bijection between the set of these processes and this of S d-1 ×R-valued Markov additive processes, where S d-1 is the sphere in dimension d. Actually this representation does not provide a complete description of R d -valued self-similar Markov processes since it does not give any information on the existence of an entrance law at 0 or a recurrent extension after the first passage time at 0. Regarding these questions, only the real case has been investigated up to now, see [START_REF] Rivero | Entrance laws for positive self-similar Markov processes[END_REF], [START_REF] Dereich | Real self-similar processes starting from the origin[END_REF], [START_REF] Pantí | Recurrent extensions of real-valued selfsimilar Markov processes[END_REF] and the references therein.

We show in this paper that unlike for self-similar Markov processes, a complete description of R d -valued multi-self-similar Markov processes can be given through a Lamperti-type representation. The multi-scaling property generates quite specific properties of the process. In particular any element x ∈ R d such that x 1 x 2 . . . x d = 0 is absorbing and the state space can always be reduced to a union of open orthants with 0 as the only absorbing state. It implies that there is no continuous multi-self-similar Markov process whose state space covers the whole set R d . Moreover there is no finite multi-self-similar entrance law for these processes. We will then prove that, provided the process has infinite lifetime, the Feller property is satisfied on the whole state space, see Theorem 1. These features will be proved in Section 2 and will be used in Section 3 to show that (1.2) can be extended for R d -valued processes, so that

X (i) t = J (i) τt exp ξ (i) τt , (1.3)
where (J (i) , ξ (i) ) 1≤i≤d is a Markov additive process with values in {-1, 1} d × R d , and τ t = inf{s :

s 0 exp(α 1 ξ (1)
u + • • • + α d ξ (d) u ) du > t}, see Theorem 2.
Markov additive processes can be considered as generalizations of d-dimensional Lévy processes. Roughly speaking, (ξ (i) ) 1≤i≤d behaves like a new Lévy process in between each pair of jumps of the continuous time Markov chain (J (i) ) 1≤i≤d . Together with the representation (1.3) they provide an easy means to construct many concrete examples of multi-self-similar Markov processes. This quite simple structure will also be exploited for the description of their path properties. We will show in Subsection 3.3, see Theorem 3, that actually these processes cannot reach the set {x ∈ R d \ {0} : x 1 x 2 . . . x d = 0} in a continuous way. Moreover, contrary to self-similar Markov processes, the finiteness of their lifetime depends on their scaling index. Then we will describe the behavior of multiself-similar Markov processes on the left neighborhood of their lifetime. In particular, we will study the existence of a limit and give conditions for this limit to be 0, when it exists.

2 General properties of mssMp's , where sign(0) = 0, and for all s ∈ {-1, 0, 1} d , we define the set,

Q s = {x ∈ R d : sign(x) = s}.
(2.1)

If s ∈ {-1, 1} d , then Q s is called an open orthant.
In all the remainder of this work, unless explicitly stated, we will assume that d ≥ 2. Let us emphasize that most of our results would not apply for d = 1. However in the latter case multi-self-similar Markov processes coincide with self-similar Markov processes which have already been extensively studied in the literature, see [START_REF] Lamperti | Semi-stable Markov processes[END_REF], [START_REF] Chaumont | The Lamperti representation of realvalued self-similar Markov processes[END_REF], [START_REF] Dereich | Real self-similar processes starting from the origin[END_REF], [START_REF] Rivero | Entrance laws for positive self-similar Markov processes[END_REF] or [START_REF] Pantí | Recurrent extensions of real-valued selfsimilar Markov processes[END_REF], for instance.

In this subsection, we give a proper definition of multi-self-similar Markov processes and describe the general form of their state space. Let E be a subset of R d which is locally compact with a countable base and let E be its Borel σ-field. Let (Ω, F, (F t ) t≥0 , (X t ) t≥0 , (P x ) x∈E ) be a Markov process with values in (E, E), where (Ω, F) is some measurable space, (P x ) x∈E are probability measures on (Ω, F) such that P x (X 0 = x) = 1, for all x ∈ E and (F t ) t≥0 is some filtration to which X = (X t ) t≥0 is adapted and completed with respect to the measures (P x ) x∈E . Such a process will be denoted by {X, P x } and will simply be referred to as an E-valued Markov process.

We will now consider E-valued Markov processes which satisfy the multi-scaling property (1.1). We stress the fact that this property supposes the condition:

Q sign(x) ⊂ E, for all x ∈ E.
Define T = inf{t : X t = X 0 }. We say that x ∈ E is a holding state if P x (T > 0) = 1 and that it is an absorbing state if P x (T = ∞) = 1.

Proposition 1. Let {X, P x } be an E-valued Markov process. Assume moreover that {X, P x } is right continuous and satisfies (1.1). Then,

1. each x ∈ E such that x 1 x 2 . . . x d = 0 is an absorbing state.

If

x ∈ E is a holding sate (resp. an absorbing state), then all elements of the set Q sign(x) are holding sates (resp. absorbing sates).

Proof. Let x ∈ E such that x 1 x 2 . . . x d = 0. With no loss of generality we can assume that

x 1 = 0. Let c ∈ (0, ∞) d such that c 2 = c 3 = • • • = c d = 1.
Then the multi-scaling property (1.1) entails that the following identity in law holds for all c 1 > 0 and t ≥ 0,

{(X (1) c α 1 t , X (2)
c α 1 t , . . . , X (d) c α 1 t ), P (0,x 2 ,...,x d ) } = {(X (1)
t , X

t , . . . , X

t ), P (0,x 2 ,...,x d ) } .

(2.2)

But letting c 1 go to 0 and using the fact that {X, P x } is right continuous at 0, we obtain that for all t ≥ 0,

P (0,x 2 ,...,x d ) ((X (1)
t , X

t , . . . , X

t ) = (0, x 2 , . . . , x d )) = 1 .

Therefore the state x must be an absorbing state.

The second assertion follows directly from the multi-scaling property (1.1).

Let {X, P x } be a Markov process satisfying the conditions of Proposition 1. Since all

x ∈ E such that x 1 x 2 . . . x d = 0 are absorbing, we can send the process to 0 whenever it reaches the set {x ∈ E : x 1 x 2 . . . x d = 0}. Moreover, from part 2 of Proposition 1, if

x ∈ E is an absorbing state such that x i = 0, for all i = 1, . . . , d, then all states of the orthant Q sign(x) are absorbing. Then we will remove absorbing orthants as well as the set {x ∈ E : x 1 x 2 . . . x d = 0} \ {0} from the state space so that with no loss of generality we can claim that any right continuous Markov process satisfying (1.1) has a state space of the form:

E ∪ {0} where E = ∪ s∈S Q s , (2.3)
and where S is some subset of {-1, 1} d . Moreover, 0 is the only absorbing state.

From now on E will always be a set of the form given in (2.3). In order to define multi-self-similar Markov processes, we need further usual assumptions. In particular, we will consider the set E 0 := E ∪ {0} as the Alexandroff one-point compactification of E. This means that the open sets of E 0 are those of E and all sets of the form 0 ∪ K c , where K is a compact subset of E. The later sets form a neighborhood system for 0 and this particular state is called the point at infinity. Then E 0 endowed with this topology is a compact space. In particular, for a sequence x (n) of E 0 , lim n x (n) = 0 if and only if

lim n min |x (n) i |, |x (n) i | -1 , i = 1, . . . , d = 0 . (2.4)
We denote by E 0 , the Borel σ-fields of E 0 . Let us set ζ := inf{t : X t = 0}. The random time ζ is called the lifetime of {X, P x }, and the latter process is said to be absorbed at 0.

In all the remainder this paper, we will be dealing with Hunt processes which we recall the definition from Section I.9 of [START_REF] Blumenthal | Markov processes and potential theory[END_REF] and Section A.2 of [START_REF] Fukushima | Dirichlet forms and symmetric Markov processes[END_REF]. An E-valued Markov process {X, P x } absorbed at 0 is a Hunt process if:

(i) it is a strong Markov process, (ii) its paths are right continuous on [0, ∞) and have left limits on (0, ∞), (iii) it has quasi-left continuous paths on (0, ∞).

Such a process will be called an E-valued Hunt process absorbed at 0. Definition 1. A multi-self-similar Markov process (mssMp) with index α ∈ [0, ∞) d is an E-valued Hunt process absorbed at 0, which satisfies the multi-scaling property (1.1). The state space of mssMp's is always of the form given in (2.3) and 0 is the only absorbing state. In the sequel, such a process will simply be referred to as an E-valued mssMp with index α ∈ [0, ∞) d . Some examples of mssMp's with state space E = (0, ∞) d are given in [START_REF] Jacobsen | Multi-self-similar Markov processes on R n + and their Lamperti representations[END_REF]. For more general sets E of the form (2. For each x ∈ E, let P x be the probability measure which assigns to the process X the law of (x 1 J

(1)

t/x ᾱ , . . . , x d J (d) t/x ᾱ ) , t ≥ 0. (2.5)
Then we readily check that {X, P x } is an E-valued mssMp with index α.

2) A slightly more sophisticated example is given by the law of

x 1 (1 + ᾱtx -α ) 1/ ᾱJ (1) ln(1+ ᾱtx -α ) 1/ ᾱ , . . . , x d (1 + ᾱtx -α ) 1/ ᾱJ (d) ln(1+ ᾱtx -α ) 1/ ᾱ , t ≥ 0, (2.6)
where we assume that ᾱ > 0. Again the process {X, P x } is an E-valued mssMp with index α.

3) We called our third example "the jumping spider". Let {X, P x } be a process which starts at time t = 0, at some point x ∈ E and runs along the axis (0, x) as a reflected Brownian motion. Then at some time before hitting 0, the process jumps out to some other state of y ∈ E and runs in the same way along the axis (0, y), and so on. More specifically let (R t ) t≥0 be a reflected Brownian motion independent of the process (J

t , . . . , J

t ) t≥0 defined above and such that R 0 = 1, a.s. Let α be such that ᾱ = 2 and define

X (i) t = x i J (i) t/x α 0 R -2 s ds R t/x α , 0 ≤ t < x α ζ, 0 , t ≥ x α ζ,
where ζ = inf{s : R s = 0}. Then we can check that {X, P x } is a mssMp with index α, which is absorbed at 0, at time ζ.

Note that in these three examples, the decomposition (2.3) of the space E is determined by the state space S of the continuous time Markov chain (J

t , . . . , J

t ) t≥0 . An extension of the above constructions of mssMp's will be given in Section 3 through a Lamperti type transformation, see Theorem 2.

Remark 1. It is straightforward from (1.1) that mssMp's of index α are also self-similar Markov processes with index α 1 + • • • + α d . However multi-self-similarity imparts to Markov processes much richer properties which could not be derived from the study of self-similar Markov processes given in [START_REF] Alili | Inversion, duality and Doob h-transforms for self-similar Markov processes[END_REF]. See for instance Theorem 1 and Proposition 3 below.

On the Feller property of mssMp's.

Proposition 1 means in particular that given any mssMp {X, P x }, there does not exist any Markov process starting from 0 and with the same semigroup as {X, P x }. We will see in the next proposition that actually there is no finite multi-self-similar entrance law.

Let us now fix some definitions. In what follows, {X, P x } will always be an E-valued mssMp with index α ∈ [0, ∞) d . We will denote by (P t ) the transition function of {X, P x }. An entrance law for {X, P x } is a family of non zero measures {η t , t > 0} on E satisfying the identity η s P t = η t+s , that is for all nonnegative Borel function f defined on E and all s, t > 0,

E E x (f (X t ), t < ζ) η s (dx) = E f (x) η t+s (dx) .
We say that {η t , t > 0} is a multi-self-similar entrance law if moreover there is a multi-

index γ ∈ [0, ∞) d such that for all c ∈ (0, ∞) d , η s = c -γ η sc -α H c , (2.7)
where H c denotes the dilation operator

H c f (x) = f (c • x).
This definition is the natural extension of self-similar entrance laws introduced in the framework of positive self-similar Markov processes in [START_REF] Rivero | Entrance laws for positive self-similar Markov processes[END_REF], see (EL-i) and (EL-ii). In this paper the existence of self-similar entrance laws has been fully studied. Then in part 4.2 of [START_REF] Rivero | Entrance laws for positive self-similar Markov processes[END_REF] the author extended his study to the case of mssMp's with values in E = (0, ∞) d whose radial part tends to infinity. In this particular case, the definition of multi-self-similarity of the entrance law corresponds to (2.7) for γ = 0. An expression for the corresponding multi-self-similar entrance law can be found in [START_REF] Rivero | Entrance laws for positive self-similar Markov processes[END_REF]. In the next proposition we complete this result by showing that there cannot exist any finite such entrance law in general.

Proposition 2. MssMp's do not admit finite multi-self-similar entrance laws.

Proof. Let {η t , t > 0} be a multi-self-similar entrance law for {X, P x } and let f be any positive Borel function defined on E. Let t > 0 and a > 0, then from (2.7) applied to c = (1, a, 1, . . . , 1), we obtain

E f (x 1 , . . . , x d )a γ 2 η t (dx) = E f (x 1 , ax 2 , . . . , x d )η t/a α 2 (dx) . (2.8)
Let π i : x → x i be the projection on the i-th coordinate and set

U i = π i (E). Denote by η t = η t • π -1 1 the image of η t by π 1 on U 1 .
It follows from the above identity that

a γ 2 η t = η t/a α 2 .
(2.9)

If α 2 = 0 and γ 2 > 0, then η t is necessarily the infinite measure. Denote by

η t = η t • π -1 2 the image of η t by π 2 on U 2 . If α 2 = γ 2 = 0, then from (2.8), η t satisfies U 2 g(x 2 )η t (dx) = U 2 g(ax 2 )η t (dx) ,
for all positive Borel function g defined on U 2 . Recalling that either U 2 or -U 2 is the multiplicative group R\{0} or (0, ∞), the latter identity shows that if η t is finite on all compact sets, then η t corresponds to the Haar measure on U 2 , that is η t (dx) = (cst) • |x| -1 dx, which has infinite mass. Now suppose that α 2 > 0. Then applying (2.7) again with c = (a, 1, . . . , 1), we obtain

E f (x 1 , . . . , x d )a γ 1 η t (dx) = E f (ax 1 , x 2 , . . . , x d )η t/a α 1 (dx) ,
that is for all positive Borel function g defined on U 1 ,

U 1 g(x 1 )a γ 1 η t (dx) = U 1 g(ax 1 )η t/a α 1 (dx) . (2.10)
Then replacing a by a α 1 /α 2 in (2.9) gives

a γ 2 α 1 /α 2 η t = η t/a α 1 , (2.11)
so that from (2.10) and (2.11)

with κ = γ 1 -γ 2 α 1 /α 2 , U 1 g(ax 1 )η t (dx) = U 1 g(x 1 )a κ η t (dx) .
But again, either U 1 or -U 1 is the multiplicative group R \ {0} or (0, ∞), hence the latter identity shows that if η t is finite on all compact sets, then η (dx) = (cst) • |x| -(κ+1) dx, which has infinite mass.

Knowing the form of the state space of mssMp's and their entrance boundaries, we can now investigate their Feller property. Actually there is no universally agreed definition of the Feller property. This varies depending on which space the transition function

(P t ) of {X, P x } should be defined. Let C b (E 0 ) (resp. C b (E))
be the space of continuous and bounded functions on E 0 (resp. E). In our case, the most natural definition should require the following two conditions:

(a) For all f ∈ C b (E 0 ) and t ≥ 0, P t f ∈ C b (E 0 ). (b) For all f ∈ C b (E 0 ), lim t→0 P t f = f , uniformly.
If the transition function of {X, P x } satisfies (a) and (b), we say that {X, P x } is a Feller process on E 0 . As will be seen later on, this property is actually very strong and rarely satisfied by mssMp's. We will actually focus our attention on mssMp's with an infinite lifetime. A mssMp {X, P x } is said to have an infinite lifetime if

P x (ζ = ∞) = 1, for all x ∈ E.
When the lifetime is infinite, the restriction of the transition function (P t ) of {X, P x } to the space E is still Markovian. In this case, we say that {X, P x } has the Feller property on E if it satisfies:

(a ) For all f ∈ C b (E) and t ≥ 0, P t f ∈ C b (E). (b ) For all f ∈ C b (E), lim t→0 P t f = f , uniformly on compact subsets of E.
Examples of mssMp's whose lifetime is infinite can be found in [START_REF] Jacobsen | Multi-self-similar Markov processes on R n + and their Lamperti representations[END_REF].

Theorem 1. Let {X, P x } be a mssMp with an infinite lifetime. Then, {X, P x } has the Feller property on E.

Proof. Let us define the scaling operator S c by

S c (X) = (c • X t/c α ) t≥0 , where c ∈ (0, ∞) d . Then let x ∈ E and x (n) = (x (n) 1 , . . . , x (n)
d ) be any sequence of E which converges towards x ∈ E. Recall the form (2.3) of E and set

Q (i) = Q sign(x i ) so that x i ∈ Q (i) . Since the Q (i)
's are open sets, there is n 0 such that for all n ≥ n 0 and all i = 1, . . . , d, x

(n) i ∈ Q (i) and x (n) i
has the same sign as x i . Now for n ≥ n 0 , define c

(n) i = x (n) i /x i and note that c (n) = (c (n) 1 , . . . , c (n) d ) ∈ (0, ∞) d . Let f ∈ C b (E), then from (1.1), for all t ≥ 0, E x (n) (f (X t )) = E x (f (S c (n) (X) t )) .
(2.12)

Since c (n) tends to 1 and {X, P x } is almost surely continuous at time t, see [START_REF] Blumenthal | Markov processes and potential theory[END_REF], it follows from (2.12) and dominated convergence that

lim n→∞ E x (n) (f (X t )) = E x (f (X t )) .
This proves (a ).

Let us now prove (b ). First observe that for all x ∈ E,

{X, P x } = {S c(x) (X), P sign(x) } , (2.13)
where

c i (x) = |x i |. Let K be some compact subset of E. For s ∈ {-1, 1} d , define the compact subsets K s = Q s ∩ K. From (2.13
) and the right continuity of {X, P x } at 0, we have for all y ∈ K s ,

lim t→0 |f (S c(y) (X) t ) -f (y)| = 0 , P s -almost surely,
so that from the uniform continuity of f on K s ,

lim t→0 sup y∈Ks |f (S c(y) (X) t ) -f (y)| = 0 , P s -almost surely.
Then the result follows from the inequality for all x ∈ K,

|P t f (x) -f (x)| ≤ max s∈{-1,1} d E s ( sup y∈Ks |f (S c(y) (X) t ) -f (y)|) ,
the boundeness of f and dominated convergence.

Remark 2. Let us emphasize that Theorem 1 highlights a great difference between selfsimilarity and multi-self-similarity. Indeed, it is no true that all d-dimensional self-similar Markov processes with infinite lifetime satisfy the Feller property on E. As can been seen in the previous proof, self-similarity on all axis allows us to consider the limit of P t f (x n ) for any sequence (x n ) converging to x in E, whereas this convergence would only hold for sequences of the type x n = c n x, where c n > 0 → 1 for self-similar Markov processes.

We will say that a mssMp {X, P x } is symmetric if it satisfies the two following conditions:

(i) E = s • E for all s ∈ {-1, 1} d such that (s • E) ∩ E = ∅. (ii) P x (X t ∈ A) = P s•x (X t ∈ s • A)
, for all t ≥ 0, A ∈ E and s satisfying (i).

Note that if E satisfies condition (i), then conditions (1.1) and (ii) are equivalent to

{(X |c| α t ) t≥0 , P c•x } = {(c • X t ) t≥0 , P x } , (2.14)
where

|c| α := |c 1 | α 1 . . . |c d | α d , for all x ∈ E and c ∈ (R \ {0}) d such that c • x ∈ E.
Note also that if E consists in a single orthant, that is E = Q s , for some s ∈ {-1, 1} d , then {X, P x } is symmetric according to this definition. Moreover, we easily construct symmetric mssMp's with E as the union of at least two orthants from the examples given in Subsection 2.1.

We will see in the next proposition that when {X, P x } is symmetric, the lifetime is either a.s. finite or a.s. infinite, independently of the starting state. Moreover, either the process hits 0 continuously, a.s. or by a jump, a.s. In the next proposition, we will use the notation, lim t↑ζ X t = X ζ- when this limit exists. Note that when ζ is finite, the existence of X ζ-is guaranteed by the fact that {X, P x } is a Hunt process. Moreover, the fact that X ζ-= 0 is to be understood in the topology of E 0 . It means that,

lim t↑ζ min(|X (i) t |, |X (i) t | -1 , i = 1, . . . , d) = 0, a.s., see (2.4).
Proposition 3. Assume that {X, P x } is a symmetric mssMp. Then,

(i) either P x (ζ = ∞) = 1, for all x ∈ E, or P x (ζ < ∞) = 1, for all x ∈ E. (ii) Assume that P x (ζ < ∞) = 1, for all x ∈ E. Then either P x (X ζ-= 0) = 1, for all x ∈ E or P x (X ζ-= 0) = 1, for all x ∈ E.
Proof. Let us note that from our assumptions, for all x, y ∈ E, there is c

∈ (R \ {0}) d such that y = c • x and {(X |c| α t ) t≥0 , P y } = {(c • X t ) t≥0 , P x }, which yields the identity in law, {|c| α ζ, P x } = {ζ, P y }, (2.15)
since ζ is the first passage time at 0 by X, i.e. ζ = inf{t : X t = 0}. This observation allows us to extend the case of positive self-similar Markov processes which is treated in [START_REF] Lamperti | Semi-stable Markov processes[END_REF] and from which our proof is inspired.

Let F = {ζ < ∞}. Then from (2.15), P x (F ) does not depend on x ∈ E. Let us set P x (F ) = p. From the Markov property, one has for all t > 0,

P x (t < ζ < ∞) = E x (1I {t<ζ} P x (∃s ∈ (t, ∞), X s = 0 | F t )) = E x (1I {t<ζ} P Xt (ζ < ∞)) = pP x (t < ζ) ,
which leads to

p = P x (ζ ≤ t) + P x (t < ζ < ∞) = P x (ζ ≤ t) + pP x (t < ζ) , so that (1 -p)P x (ζ ≤ t) = 0.
We conclude that either p = 1, or P x (ζ ≤ t) = 0 for all t ≥ 0 and all x ∈ E, that is P x (ζ = +∞) = 1 for all x ∈ E.

Let us now prove (ii). Set G = {X ζ-= 0}. Again, from our assumptions, for all x, y ∈ E, there is c

∈ (R \ {0}) d such that y = c • x and {c • X ζ-, P x } = {X ζ-, P y },
so that P x (G) does not depend on x ∈ E. Set q = P x (G) and let K be any compact subset of E. Set T = inf{t : X t ∈ K c }, where K c denotes the complementary set of K in E. Since G ⊂ {T < ζ} and G • θ T = G, it follows from the strong Markov property that for all x ∈ E,

q = P x (G) = P x (G, T < ζ) = E x 1I {T <ζ} P x (G • θ T | F T ) = P x 1I {T <ζ} P X T (G) = qP x (T < ζ) .
If q = 0, then P x (T < ζ) = 1, for all x ∈ E. Since this is true for all compact subsets of E, it follows that {X, P x } doest not reach 0 by a jump and hence q = 1.

The Lamperti-type representation established in Subsection 3.2 will allow us to give many other examples of mssMp's with infinite lifetime, see Theorem 3.

Note that in order to have the Feller property on E 0 , the process {X, P x } should also satisfy

lim x→0 E x (f (X t )) = f (0) , (2.16)
for all t ≥ 0 and f ∈ C b (E 0 ), where again this limit is to be understood in the topology of E 0 , see (2.4). If x tends to 0 (in E 0 ) in such a way that |x i | > a for some a > 0 and all i = 1, . . . , d, then (2.16) holds. Indeed, from the multiscaling property,

E sgn(x) (f (|x| • X t/|x| α )).
In this case, t/|x| α tends to 0 as x tends to 0 in E 0 , so that from the right continuity of X at 0, lim x→0 X t/|x| α = sgn(x), P sgn(x) -a.s. and hence lim x→0 |x| • X t/|x| α = 0, P sgn(x) -a.s. Then (2.16) follows from the fact that f ∈ C b (E 0 ) and dominated convergence. However, it seems that (2.16) may fail when lim inf |x i | = 0 for some coordinates x i of x since in this case, we can have lim inf x→0 |x| α = 0 and lim sup x→+∞ |x| α = +∞.

Multiplicative agglomeration property of mssMp's.

We will prove in this subsection that symmetric mssMp's enjoy the multiplicative agglomeration property, namely the process obtained by multiplying some of its coordinates is still a mssMp (with lower dimension). This property has been highlighted for (0, ∞) dvalued mssMp's in [START_REF] Jacobsen | Multi-self-similar Markov processes on R n + and their Lamperti representations[END_REF] as a direct consequence of the Lamperti representation, see Corollary 5 therein. Although Lamperti representation will be generalized to all mssMp's later on in this paper, we prefer to study multiplicative agglomeration property of mssMp's in a more direct way by using Dynkin's criterion.

For 1 ≤ d ≤ d and a partition I = {I 1 , . . . , I d } of {1, 2, . . . , d}, we define

Π I (x) = Π i∈I 1 x i , . . . , Π i∈I d x i , x ∈ E .
Let us also define E (I) = Π I (E). Then clearly E (I) is a subspace of R d which has the form described in (2.3). We denote by E (I) the corresponding Borel σ-field and by E (I) 0

the Alexandroff compactification of E (I) which is defined as for E, see Subsection 2.1.

Given any E-valued mssMp absorbed at 0, {X, P x }, we define the E (I) -valued process absorbed at 0, X (I) by

X (I) = Π I (X) ,
and we set

F (I) = σ(X (I) t ∈ B, t ≥ 0, B ∈ E (I)
) and for all t ≥ 0, F

(I) t = F t ∩ F (I) .
Proposition 4. Let {X, P x } be a symmetric mssMp with index α ∈ [0, ∞) d such that for all i = 1, . . . , d and for all j, k ∈ I i , α j = α k . We set α i = α j , for j ∈ I i . Then the process X (I) defined on the space (Ω, F (I) , (F

(I) t ) t≥0
) is an E (I) -valued mssMp absorbed at 0 with index α = (α 1 , . . . , α d ). Moreover, the family of probability measures P (I) y , y ∈ E (I) 0 associated with X (I) is given by

P (I) y (Γ) = P x (Γ), Γ ∈ F (I) , (2.17)
for any x ∈ Π -1 I (y).

Proof. From Dynkin's criterion, see Theorem 10.23, p.325 in [START_REF] Dynkin | Markov Processes[END_REF], in order to prove that X (I) is a Markov process defined on (Ω, F (I) , (F (I) t ) t≥0 ), with respect to the family of probability measures (P (I) y ) given in (2.17), it suffices to prove that for all x, x ∈ E such that Π I (x) = Π I (x ), and for all t ≥ 0 and B ∈ E (I) ,

P x (X t ∈ Π -1 I (B)) = P x (X t ∈ Π -1 I (B)).
(2.18)

Let t ≥ 0, B ∈ E (I) and x, x ∈ E be such that Π I (x) = Π I (x ) and let us set

a i = x i x i .
Since all coordinates of Π I (a), with a = (a 1 , . . . , a d ), are equal to 1 and from our assumption on α, we have |a| -α = 1. Hence, from (2.14),

P x (X t ∈ Π -1 I (B)) = P a•x (X t ∈ Π -1 I (B)) = P x (a • X |a| -α t ∈ Π -1 I (B)) = P x (X t ∈ Π -1 I (B)),
which is Dynkin's criterion (2.18).

It remains to check that the process {X (I) , P

x } satisfies the multi-self-similarity property of index α defined in the statement. This follows directly from the definition of {X (I) , P (I)

x } and the multi-self-similarity property of {X, P x }. Indeed, recall that d is the dimension of E (I) and let c ∈ (0, ∞) d , y ∈ E (I) and x ∈ E, c ∈ (0, ∞) d such that for all j ∈ I i , c j = (c i ) card(I i ) -1 and x ∈ Π -1 I (y). Then, {c • X (I) , P (I)

y } = {Π I (c • X), P x } = {Π I (X c α • ), P c•x } = {X (I) (c ) α • , P (I) c •y },
which achieves the proof of the proposition.

Note that we recover Jacobsen and Yor's result from Proposition 4 since the process is always symmetric when E = (0, ∞) d . Let us also mention that symmetry is not a necessary condition for the process {X (I) , P (I)

x } to be a mssMp. Examples of non symmetric mssMp's which satisfy the multiplicative agglomeration property can be obtained from the Lamperti type representation presented in the next sections. We also emphasize the importance for the coordinates of the index α to be constant on each element of the partition I. The above proof shows that it necessary for the process X (I) to be Markovian. This fact is easier to see from the Lamperti type representation, see Theorem 2.

3 Time changes in mssMp's.

Markov Additive Processes.

We will now consider Markov processes with values in a state space of the form S × R d , where S is some topological set such that S × R d is locally compact with a countable base. As usual we define the Alexandroff compactification of S × R d by adding a point at infinity which we denote by δ. Definition 2. A Markov additive process (MAP) {(J, ξ), P y,z } is an S × R d -valued Hunt process absorbed at some extra state δ, such that for any y ∈ S, z ∈ R d , s, t ≥ 0, and for any positive measurable function f , defined on S × R d ,

E y,z (f (J t+s , ξ t+s -ξ t ), t + s < ζ * | G t ) = E Jt,0 (f (J s , ξ s ), s < ζ * )1I {t<ζ * } , (3.1)
where ζ * = inf{t : (J t , ξ t ) = δ} is the lifetime of {(J, ξ), P y,z } and (G t ) t≥0 is some filtration to which (J, ξ) is adapted and completed by the measures P y,z , (y, z)

∈ S × R d .
In what follows, we will always consider the case where S is a subset of {-1, 1} d . Then the space S × R d is always locally compact with a countable base. Moreover, while the structure of general MAP's can turn out to be quite complicated (see [START_REF] Ezhov | Markov processes with homogeneous second component: I[END_REF] and [START_REF] Çinlar | Markov additive processes. I, II. Z. Wahrscheinlichkeitstheorie und Verw[END_REF] where these processes were first introduced) the case where S is a finite set is rather intuitive and can be plainly described. Since in this case, the process (J t ) t≥0 is nothing but a possibly absorbed continuous time Markov chain, it is readily seen from (3.1) that in between two successive jump times of J, the process ξ behaves like a Lévy process. Let us state this result more formally.

It is straightforward from (3.1) that the law of {J, P y,z } does not depend on z. Moreover, since S is finite, {J, P y,z } is an S-valued continuous time Markov chain with lifetime ζ * , which may be sent to some extra state δ for t ≥ ζ * . Let us set n := 2 d = card(S), then the law of the MAP {(J, ξ), P y,z } is characterized by the intensity matrix Q = (q ij ) i,j∈S of J, the laws of n possibly killed R d -valued Lévy processes ξ(1) , . . . , ξ(n) , and the R d -valued random variables ∆ ij , such that ∆ ii = 0 and where, for i = j, ∆ ij represents the size of the jump of ξ when J jumps from i to j. More specifically, for u ∈ C d , define for i, j ∈ S and k = 1, . . . , n when these expectations exist, E(e u, ξ(k) 1 ) = e ψ k (u) and G i,j (u) = E(exp( u, ∆ i,j )) .

Then a trivial extension of Proposition 2.2 in Section XI.2 of [START_REF] Asmussen | Applied probability and queues[END_REF] shows that the law of {(J, ξ), P y,z } is given by

E i,0 (e u,ξt , J t = j) = (e A(u)t ) i,j , i, j ∈ S , u ∈ C d , (3.2)
where A(u) is the matrix,

A(u) = diag(ψ 1 (u), . . . , ψ n (u)) + (q ij G i,j (u)) i,j∈S .
The matrix-valued mapping u → A(u) will be called the characteristic exponent of the MAP {(J, ξ), P y,z }. We also refer to Sections A.1 and A.2 of [START_REF] Dereich | Real self-similar processes starting from the origin[END_REF] for more details. Throughout the remainder of this paper, the coordinates of a MAP with values in S × R d will be denoted by (J, ξ) = (J (i) , ξ (i) ) 1≤i≤d .

Let {(J, ξ), P y,z } be an S × R d -valued MAP with infinite lifetime, that is P y,z (ζ * = ∞) = 1, for all y, z ∈ S ×R d . Then it is readily seen that for each k = 1, . . . , d, the process (J, ξ (k) ) is itself an S × R-valued MAP with infinite lifetime. Let P (k) y,z , y, z ∈ S × R be the corresponding family of probability measures, i.e. {(J, ξ (k) ), P

(k) y,z } is an S × R-valued MAP with infinite lifetime. Denote by A k the corresponding characteristic exponent, that is from (3.2), A k (u) = A(u • e k ) , u ∈ C , (3.3)
where e k is the k-th unit vector of R d . Fix k = 1, . . . , d, assume that the Markov chain (J t ) is irreducible and that there exists u ∈ R \ {0} such that A k (u), i = 1, . . . , d is well defined (i.e. all entries of A k (u) exist and are finite). Then from Perron-Frobenius theory, the matrix A k (u) has a real simple eigenvalue χ k (u) which is larger than the real part of all its other eigenvalues. Let I = [u, 0] if u < 0 and I = [0, u] if u > 0. Then the function u → χ k (u) is convex on I. Let us denote by χ k (0) the left (respectively, the right) derivative at 0 of χ k if u < 0 (respectively, if u > 0). The following result can be found in Section XI.2 of [START_REF] Asmussen | Applied probability and queues[END_REF].

Proposition 5. Assume that J is irreducible. Let k = 1, . . . , d and assume that there exists u ∈ R \ {0} is such that A k (u) is well defined. Then the asymptotic behavior of ξ (k) does not depend on the initial state of {(J, ξ (k) ), P

y,z } and is given by

lim t→∞ ξ (k) t t = χ k (0), P (k)
y,z -a.s. for all y, z ∈ S × R.

In that case, for all y, z ∈ S × R, lim t→∞ ξ

(k) t = ∞, P (k)
y,z -a.s. or lim t→∞ ξ

(k) t = -∞, P (k) y,z -a.s. or lim sup t→∞ ξ (k) t = -lim inf t→∞ ξ (k) t = ∞, P (k)
y,z -a.s., according as χ k (0) > 0, χ k (0) < 0 or χ k (0) = 0, respectively.

Note that more generally, if M : R d → R d is a linear mapping, where d is any integer, then the process {(J, M (ξ)), P y,z } is an S × R d -valued MAP. This property will be used in the next subsections with

M (x) = α, x , for some α ∈ [0, ∞) d .
Examples of MAP's can easily be obtained by coupling any continuous time Markov chain on S together with any d-dimensional Lévy process and by killing the couple at some independent exponential time. More specifically, the transition probabilities of the process {(J, ξ), P y,z } have the following particular form:

P y,z (J t ∈ dy 1 , ξ t ∈ dz 1 ) = e -λt P J y (J t ∈ dy 1 )P ξ z (ξ t ∈ dz 1 ) , P y,z ((J t , ξ t ) = δ) = 1 -e -λt , (3.4)
for all t ≥ 0, (y, z), (y 1 , z 1 ) ∈ S × R d , where λ > 0 is some constant, {ξ , P ξ z } is any non killed d-dimensional Lévy process and {J , P J y } is any continuous time Markov chain on S with infinite lifetime. Then it is easy to check that this process {(J, ξ), P y,z } is an S × R d -valued MAP which is absorbed at δ, in the sense of Definition 2. The law of such a MAP is characterized by the fact that ψ k = ψ l for all k, l = 1, . . . , n and G i,j (u) = 1 for all i, j ∈ S in (3.2). Assume that the above process has infinite lifetime, that is λ = 0. Then the condition of Proposition 5 is satisfied if and only if there exists u ∈ R such that ψ 1 (u) exists and is finite. In this example one may check that χ (0) > 0, < 0 or = 0 according as ψ 1 (0) > 0, < 0 or = 0. Of course this result is intuitively clear since J and ξ are independent.

The Lamperti representation for mssMp's.

Recall that S and E are any sets such that

S ⊂ {-1, 1} d and E = ∪ s∈S Q s ,
where Q s is defined in (2.1). Then let us define the one-to-one transformation ϕ : S×R d → E and its inverse as follows:

ϕ(y, z) = (y i e z i ) 1≤i≤d , (y, z) ∈ S × R d , ϕ -1 (x) = (sgn(x i ), log(|x i |)) 1≤i≤d , x ∈ E .
In the remainder of this work, for α ∈ [0, ∞) d , we will denote,

ξ = α, ξ ,
where ξ is the second coordinate of the S × R d -valued MAP {(J, ξ), P y,z }.

The next theorem extends Theorem 1 of [START_REF] Jacobsen | Multi-self-similar Markov processes on R n + and their Lamperti representations[END_REF]. It provides a one to one relationship between the set of R d -valued mssMp's and this of MAP's with values in {-1,

1} d × R d .
Theorem 2. Let α ∈ [0, ∞) d and {(J, ξ), P y,z } be a MAP in S × R d , with lifetime ζ * and absorbing state δ. Define the process X by

X t = ϕ(J τt , ξ τt ) , if t < ζ * 0 e ξs ds , 0 , if t ≥ ζ * 0 exp( ξs ) ds ,
where τ t is the time change τ t = inf{s :

s 0 e ξu du > t}, for t < ζ * 0 e ξs ds. Define the probability measures P x := P ϕ -1 (x) , for x ∈ E and P 0 := P δ . Then the process {X, P x } is an E-valued mssMp, with index α and lifetime ζ * 0 e ξs ds. Conversely, let {X, P x } be an E-valued mssMp, with index α ∈ [0, ∞) d and denote by ζ its lifetime. Define the process (J, ξ) by

(J t , ξ t ) =    ϕ -1 (X At ) , if t < ζ 0 ds |X (1) s | α 1 ...|X (d) s | α d , δ , if t ≥ ζ 0 ds |X (1) s | α 1 ...|X (d) s | α d ,
where δ is some extra state, and A t is the time change A t = inf{s :

s 0 du |X (1) u | α 1 ...|X (d) u | α d > t}, for t < ζ 0 ds |X (1) s | α 1 ...|X (d)
s | α d . Define the probability measures, P y,z := P ϕ(y,z) , for (y, z) ∈ S × R d and P δ := P 0 . Then the process {(J, ξ), P y,z } is a MAP in S × R d , with lifetime

ζ 0 ds |X (1) s | α 1 ...|X (d) s | α d .
Proof. Let us denote by (G t ) t≥0 the filtration associated to the process (J, ξ) and completed with respect to the measures (P y,z ) y,z∈S×R d , see the beginning of Subsection 2.1. Then define the process

Y t = ϕ(J, ξ) t , if t < ζ * , 0 , if t ≥
)) t≥0 , P y,ln c+z } = {(c i J (i) τt exp(ξ (i) τt )) t≥0 , P y,z } .

On the other hand, it is straightforward from the definitions that

X (i) c α t = c i J (i) τ (c) t exp(ξ (c,i) τ (c) t
) .

Then by taking x = ϕ(y, z) so that c • x = ϕ(y, ln c + z) and P x = P y,z , P c•x = P y,ln c+z , we derive from (3.7) and (3.8) that

{(X c α t ) t≥0 , P c•x } = {(c • X t ) t≥0 , P x } .
Conversely, let {X, P x } be an E-valued mssMp, with index α ∈ [0, ∞) d and lifetime ζ. Then by arguing exactly as in the direct part, we prove that the process {(J, ξ), P y,z } defined in the statement is an S × R d -valued Hunt process, with lifetime

ζ * := ζ 0 ds |X (1) s | α 1 ...|X (d)
s | α d . Now we have to check that the Hunt process {(J, ξ), P y,z } is a MAP. Let (F t ) t≥0 be the filtration associated to X and completed with respect to the measures

(P x ) x∈E . Define A t as in the statement if t < ζ 0 ds |X (1) s | α 1 ...|X (d) s | α d , set A t = ∞, if t ≥ ζ 0 ds |X (1) s | α 1 ...|X (d)
s | α d and note that for each t, A t is a stopping time of (F t ) t≥0 . Let us denote by θ t the usual shift operator at time t and note that for all s, t ≥ 0,

A t+s = A t + θ At (A s ) .
Then let us prove that {(J, ξ), P y,z } is a MAP in the filtration G t := F At . First observe that (J, ξ) is clearly adapted to this filtration. Then from the strong Markov property of {X, P x } applied at the stopping time A t , we derive from the definition of {(J, ξ), P y,z } that for any positive, Borel function f and x ∈ E, From this theorem, it is now easy to construct many examples of non trivial mssMp's. We can use for instance the MAP which is defined in (3.4) by coupling any continuous time Markov chain with an independent Lévy process.

E ϕ -1 (x) (f (J t+s , ξ t+s -ξ t ), t + s < ζ * | G t ) = E x f ϕ -1 (X At+θ A t (As) ) -(0, ln |X At |) , A t + θ At (A s ) < ζ | F At = E X A t f ϕ -1 (X As ) -(0, ln z) , A s < ζ z=|X A t | 1I {At<ζ} = E sign(X A t ) f ϕ -1 (X As ) , A s < ζ 1I {At<ζ} = E Jt,0 (f (J s , ξ s ), s < ζ * )1I {t<ζ * } ,

Asymptotic behavior of mssMp's.

In this subsection we derive from Theorem 2 the behavior of a mssMp {X, P x } as t tends to ζ. From this theorem and the construction of MAP's given in Subsection 3.1, if the lifetime ζ * of the underlying MAP (J, ξ) under P y,z is finite with positive probability, then so is ζ and for x = ϕ(y, z), the process X under P x , jumps to 0 on the set ζ < ∞, that is X ζ-= 0, with positive probability. This situation has no interest for the problem we are studying and we will skip it. Moreover, we will always assume that J is irreducible so that if E is composed of at least two orthants, then {X, P x } does not pass through some of them a finite number of times. The reducible case can always be boiled down to the irreducible one from classical arguments. Therefore, we will assume that {X, P x } is an Evalued mssMp absorbed at 0 whose underlying MAP {(J, ξ), P y,z } in the transformation given in Theorem 2 satisfies:

(a) For all y, z ∈ S × R d , ζ * = ∞, P y,z -a.s. (b) J is irreducible.
Under assumption (a) and from Theorem 2, the lifetime of {X, P x } is given by ζ = ∞ 0 e ξs ds. In order to determine conditions for this lifetime to be finite or infinite, we need reasonable assumptions on the asymptotic behavior of ξ. These conditions will be ensured by Proposition 5, so we will also assume:

(c) For each k = 1, . . . , d, there is u ∈ R \ {0} such that A k (u) is well defined.
Let α ∈ [0, ∞) d and note that the law of the process (J, ξ) under P y,z only depends on y and α, z . As already observed at the end of Subsection 3.1, this process is actually an S × R-valued MAP under the family of probability measures defined by P y,t := P y,z , where z is any vector such that t = α, z . The characteristic exponent of {(J, ξ), P y,t } is then given by Ā Recall from Proposition XI.2.10 of [START_REF] Asmussen | Applied probability and queues[END_REF] that lim t→∞ ξt = -∞, P y,z -a.s. for all y, z ∈ S × R d or lim t→∞ ξt = +∞, P y,z -a.s. for all y, z ∈ S × R d or lim inf t→∞ ξt = -∞ and lim sup t→∞ ξt = +∞, P y,z -a.s. for all y, z ∈ S × R d , according as κ α < 0, > 0 or = 0.

(u) = A(u • α), u ∈ C. Since A k (u) = A(u • e k ),
Then it follows from Theorem 2 that ζ < ∞, P x -a.s. for all x ∈ E or ζ = ∞, P x -a.s. for all x ∈ E according as κ α < 0 or κ α > 0. The case where κ α = 0 requires a bit more care and is proved in the following lemma which we have not found explicitly stated in the literature.

Lemma 1. Assume that conditions (a), (b) and (c) are satisfied.

(i) If κ α < 0, then ∞ 0 e ξs ds < ∞, P y,z -a.s. for all y, z ∈ S × R d , (ii) If κ α ≥ 0, then
∞ 0 e ξs ds = ∞, P y,z -a.s. for all y, z ∈ S × R d . Proof. The cases where κ α < 0 and κ α > 0 follow directly from (3.9). Let us assume now that κ α = 0. Then as recalled above, lim inf t→∞ ξt = -∞ and lim sup t→∞ ξt = +∞, P y,z -a.s. for all y, z ∈ S × R d . Fix a > 0 and let τ

(n) a = inf{t ≥ σ (n-1) a : ξt > a} and σ (n) a = inf{t ≥ τ (n) a : ξt < a}, n ≥ 0, with σ (0) a = 0. Then τ (n) a
is a sequence of stopping times in the filtration (G t ) t≥0 of {(J, ξ), P y,z } such that lim n→∞ τ Recall that J is irreducible and let π be its invariant measure on S. We derive from the definition of MAP's that the sequence σ

(n) a -τ (n)
a , n ≥ 0 is stationary under P π,0 and since P π,0 (σ

(1) a -τ (1)
a > 0) = 1, we obtain that P π,0 ( n=1 (σ

(n) a -τ (n) a ) = ∞) = 1.
On the other hand, from the Markov property, for all y ∈ S and n ≥ 1,

P y,0 ∞ k=1 (σ (k) a -τ (k) a ) = ∞ = E y,0 P J τ (n) a ,0 ∞ k=1 (σ (k) a -τ (k) a ) = ∞ .
Since, the finite valued Markov chain (J τ (n) a ) n≥1 converges in law to π, by letting n go to ∞ in this equality, we obtain that for all y ∈ S, P y,0

∞ k=1 (σ (k) a -τ (k) a ) = ∞ = P π,0 ( ∞ n=1 (σ (n) a -τ (n)
a ) = ∞) = 1, so that from inequality (3.10),

∞ 0 e ξs ds = ∞, P y,0 -a.s. for all y ∈ S. But the definition of MAP's clearly implies that this holds P y,z -a.s. for all y, z ∈ S × R d .

Note that the trichotomy of Lemma 1 holds for any MAP with values in F × R, where F is any finite set.

Recall from Subsection 2.2 the notation lim t↑ζ X t = X ζ-, when this limit exists and where ζ is supposed to be finite or infinite. We remind that this limit is to be understood in the topology of the compact space E 0 and that, as already observed before Proposition 3, when ζ < ∞, the existence of X ζ-is guaranteed by the fact that {X, P x } is a Hunt process. Recall also the definition of χ k (0) from Subsection 3.1. The first assertion is an immediate consequence of the expression ζ = ∞ 0 e ξs ds in Theorem 2 and Lemma 1. The second one follows from the expression X t = ϕ(J τt , ξ τt ). Indeed, if χ k (0) < 0 or χ k (0) > 0, for some k = 1, . . . , d, then from Proposition XI.2.10 of [START_REF] Asmussen | Applied probability and queues[END_REF], as t tends to ζ, the coordinate X (k) t = J (k) τt e ξ (k) τ t tends to 0 if χ k (0) < 0 and to ∞ if χ k (0) > 0, P y,z -a.s. for all (y, z) ∈ S × R d . Therefore X t tends to 0 as t tends to ζ, in the topology of the compact set E 0 , P x -a.s. for all x ∈ E. If χ k (0) = 0, for all k = 1, . . . , d and κ α < 0 (resp. κ α > 0), then from Proposition XI.2.10 of [START_REF] Asmussen | Applied probability and queues[END_REF], the process |(X Let us now prove (iii). Assume that x = ϕ(y, z) is such that X ζ-exists P x -a.s. then for any k = 1, . . . , d, from (3.11) and Proposition XI.2.10 of [START_REF] Asmussen | Applied probability and queues[END_REF], the process ξ (k) τt either tends to -∞ or to +∞ or oscillates P x -a.s. as t tends to ζ. Therefore the limit lim t↑ζ X (k) t = lim t↑ζ J (k) τt e ξ (k) τ t cannot belong to R \ {0}.

Note that parts (i) and (ii) of Theorem 3 extend Proposition 3 to the case where E is any state space, but with additional assumptions. It seems that it is not possible to conclude in the case where χ k (0) = 0, for all k = 1, . . . , d and κ α = 0. We are only able to construct examples such that for all x ∈ E, X t has no limit P x -a.s., when t tends ∞. Note also that part (iii) of Theorem 3 completes part 1. of Proposition 1 where it was

2. 1

 1 The Markovian framework. Let us first set some notation. For a = (a 1 , . . . , a d ) ∈ [0, ∞) d and b = (b 1 , . . . , b d ) ∈ R d , we set a b = a b 1 1 . . . a b d d . For x ∈ R d , we set sign(x) := (sign(x 1 ), . . . , sign(x d ))

  3), let us mention the following simple examples. 1) Let (J (1) t , . . . , J (d) t ) t≥0 be any continuous time Markov chain with values in some subset S of {-1, 1} d and starting at (1, 1, . . . , 1) ∈ S. Let α ∈ [0, ∞) d and set ᾱ = α 1 + • • • + α d .

  where we have set |x| = (|x 1 | . . . , |x d |), ln |x| = (ln |x 1 | . . . , ln |x d |) and where the third equality follows from the multi-self-similarity property of {X, P x }. We have obtained (3.1) and this ends the proof of the theorem.

  P y,z -a.s. for all y, z ∈ S × R d . Moreover,

Theorem 3 .

 3 Assume that conditions (a), (b) and (c) are satisfied.(i) If κ α < 0, then P x (ζ < ∞) = 1, for all x ∈ E and if κ α ≥ 0, then P x (ζ = ∞) = 1,for all x ∈ E.(ii) If one of the following conditions is satisfied:(a) χ k (0) < 0 or χ k (0) > 0, for some k = 1, . . . , d,(b)χ k (0) = 0, for all k = 1, . . . , d and κ α < 0, (c) χ k (0) = 0, for all k = 1, . . . , d and κ α > 0,then P x (X ζ-= 0) = 1, for all x ∈ E. (iii) For all x ∈ E, P x (X ζ-exists and X ζ-∈ R d \ {0}) = 0.Proof. First is it useful to observe that under our assumptions, since ζ * = ∞, P y,z -a.s., for all y, z ∈ S × R d , lim t↑ζ τ t = ∞, P y,z -a.s., for all y, z ∈ S × R d .(3.11)

( 1 )

 1 t ) α 1 . . . (X (d) t ) α d | = e ξτ ttends to 0 (resp. to ∞) as t tends to ∞. Hence X t tends to 0 in the topology of E 0 .

  ζ * , ) as usual. Recall from Subsection 2.1 the definition of E 0 . Since ϕ is a continuous one-to-one transformation from S × R d to E, we readily check that the process (Ω, F, (G t ) t≥0 , (Y t ) t≥0 , (P x ) x∈E 0 ), where P x is defined as in the statement, is an E-valued Hunt process absorbed at 0. Now let τ t be as in the statement if t < ζ * 0 e ξs ds, and set τ t = ∞ and Y τt = 0, if | α d , (τ t ) t≥0 is the right continuous inverse of the continuous, additive functional t → | α d ds of {Y, P x }, which is strictly increasing on (0, ζ * ). It follows from Theorem A.2.12, p.406 in [7] that the time changed process(Ω, F, (G τt ) t≥0 , (X t ) t≥0 , (P x ) x∈E 0 ),is an E-valued Hunt process absorbed at 0. Moreover ζ := Now let us show that {X, P x } fulfills the multi-scaling property. Let c ∈ (0, ∞) d , then for t < c -α ζ * 0 e ξs ds, τ c α t = inf{s : = (ln c 1 , . . . , ln c d ). Let us set τ

	t ≥	ζ * 0 e ξs ds. Since e ξs = |Y s | α 1 |Y (1) s | α 2 . . . |Y (2)	(d)	t∧ζ * 0	|Y s | α 1 |Y (1)	(2) s | α 2 . . . |Y	(d)
								ζ * 0 e ξs ds is the lifetime of
	{X, P x }.
								s	e α,ξ (c) v	dv > t} ,	(3.5)
								0
	where ξ t (c)	= (ξ t (c,1)	, . . . , ξ	(c,d) t	) and ξ t (c,i)	= -ln c i + ξ t . It follows from Definition 2 of (i)
	MAP's that
							{(J, ξ (c) ), P y,ln c+z } = {(J, ξ), P y,z },	(3.6)
	where ln c (c) t	:= τ c α t , then we derive from (3.5) and (3.6)
	that
			{(c i J
	and set Y t = (Y	(1) t , . . . , Y	(d)

t s s

  under assumption (c), there is u ∈ R \ {0} such that Ā(u) is well defined. Note that the Perron-Frobenius eigenvalue of Ā(u) depends on α. Let us denote it by κ α . Applying Proposition 5, we obtain , P y,z -a.s., for all y, z ∈ S × R d .

		ξt
	lim t→∞	t	= κ α (3.9)

LAREMA UMR CNRS 6093, Université d'Angers,

2, Bd Lavoisier Angers Cedex 01, 49045, France. Email: loic.chaumont@univ-angers.fr 2 University of Monastir, Faculty of sciences, Monastir and LAREMA UMR CNRS 6093, Université d'Angers, 2, Bd Lavoisier Angers Cedex 01, 49045, France. Email: salem.lamine@hotmail.fr

proved that the set {x ∈ R d \ {0} : x 1 x 2 . . . x d = 0} is absorbing. Actually under our assumptions this set is a.s. never attained.

Remark 3. It is important to note that {X, P x } has finite or infinite lifetime depending on the valued of α. Changing α may change a finite lifetime to an infinite one. This makes another difference with self-similar Markov processes, where the index can be changed simply by raising the process to some power. Then the lifetime remains unchanged. However, recall that {X, P x } is a self-similar Markov process with index

Therefore the finiteness of the lifetime of {X, P x } does not depend on the sum α 1 +• • •+α d .


R Garnier
email: robin.garnier@insa-lyon.frhelene.casterbrunet@insa-lyon.frsylvie.barraud@insa-lyon.fr

H Castebrunet

A Fardel

K Flanagan
email: kelsey.flanagan@enpc.frmarie-christine.gromaire@enpc.fr

M-C Gromaire

D Ramier
email: david.ramier@cerema.fr

F Rodriguez
email: fabrice.rodriguez@ifsttar.fr

S Barraud

Monitoring and sampling methodology of source control systems for environmental assessment in Lyon, Nantes and Paris

Keywords: micropollutant, monitoring, source control, stormwater control measures

Stormwater control measures (SCM) are nowadays widespread and municipalities encourage their implementation in order to reduce runoff volumes and loads of micropollutants (MP), such as PAHs and metals. However, very few studies have highlighted their removal efficiency regarding less known MP such as pesticides, PBDEs, alkyphenols, etc. Several research projects in France were undertaken in order to better understand the capacity of SCMs to handle these emerging micropollutants and prescribe tools and guidelines to improve their design and construction. Three of these projects, MATRIOCHKAS (Nantes), MICROMEGAS (Lyon) and ROULÉPUR (Paris), decided to work together in order to cover the largest possible range of systems and contexts of use. The current paper presents the monitoring and sampling procedures for several different source control experimental sites across France.

Reducing the impact of runoff water (flows and pollution) on receiving water bodies is currently one of the greatest challenges in the urban drainage research field. Since the promulgation of the Water Framework Directive (WDF 2000), many studies have been conducted, in Europe and across the world, to highlight and quantify the contamination due to micropollutants (MP) such as heavy metals and hydrocarbons (e.g. [START_REF] Ellis | Micropollutants in urban stormwater: occurrence, concentrations and atmospheric contribution for a wide range of contaminants on three French catchments[END_REF].. More recently, other studies have detected and quantified emerging chemicals, such as alkylphenols, pesticides, phthalates, Bisphenol A, PCBs and PBDEs, in runoff water (e.g., [START_REF] Birch | Monitoring of priority pollutants in dynamic stormwater discharges from urban areas[END_REF][START_REF] Ellis | Micropollutants in urban stormwater: occurrence, concentrations and atmospheric contribution for a wide range of contaminants on three French catchments[END_REF].

More and more, French municipalities encourage the implementation of stormwater control measures (SCM), and more particularly source control systems based on onsite infiltration, known to reduce water flows and heavy metal and hydrocarbon contamination (e.g. [START_REF] Silva | SWITCH in Belo Horizonte, Brazil: infiltration and detention systems for more sustainable stormwater control in Belo Horizonte[END_REF][START_REF] Hatt | Hydrologic and pollutant removal performance of storwater biofiltration systems at the field scale[END_REF]). However, insufficient knowledge of in-situ systems and their efficiency regarding MP contributes to the general lack of confidence in their true ecological potential. . Moreover, as the global environmental performance of these SCMs is not well-known, combined with a poor maintenance consideration, urban planners continue to recommend end of pipe centralized systems.

In this context, the French government, through the French Agency of Biodiversity, decided to finance several projects aiming to highlight and quantify a large range of micropollutants in stormwater runoff treated using source control systems. Three of these projects decided to work jointly to assess and quantify the MP removal efficiency of many different types of source control systems: MATRIOCHKAS in the city of Nantes, MICROMEGAS in Lyon and ROULÉPUR in Paris (see Figure 1).

Across the seventeen sites (see Table 1), covering a wide range of existing source control systems and many different contexts of use, the three projects face similar issues intrinsic to in-situ data acquisition for both i) flow monitoring and ii) water sampling. Source control systems receive small flowrates, due to the small size of their catchment; consequently, usual monitoring technologies can't be implemented without huge uncertainties in the acquired data. New monitoring devices and sampling procedures have to be specifically designed and implemented on each site to address the high variability in fluxes as well as local constraints (see "situational constraints" column in Table 1).

Depending on the systems, various instrumentations have been implemented to monitor runoff flows. The devices acquire data using tipping bucket flowmeters, electromagnetic flowmeters, water level/speed laws, weirs… (see "flow monitoring" column in Table 1). However, each one of these devices is developed and implemented according to many specific constraints such as space restriction, expected flow ranges and conditions required for water collection.

Most of the micropollutants studied by the projects (more than 50 in total for the three projects) are found at trace concentrations (a few ng/L). All micropollutants analyses must thus be run in specific labs meeting several detection and quantification needs. Analytical requirements, such as water sample volume and quality, and the determination of biases and uncertainties associated with the instrumentation introduce further constraints regarding sampling procedures and devices.

The paper will present the specificities of the SCMs studied and will suggest recommendation and illustration to adapt the monitoring to small system, small catchment and associated local characteristics (e.g. What can be assimilated to an "Inlet" or an "Oulet" for a source control system?). Such monitoring is dedicated to runoff flow and micropollutants loads under high variability of events (How a harvestable rain event is defined in terms of flow and/or sampling?).


Corentin Pigot

Marc Bocquet

Fabien Gilibert

Marina Reyboz

Olga Cueto

V Della Marca

Paola Zuliani

Comprehensive Phase-Change Memory Compact Model for Circuit Simulation

Keywords: Compact modeling, Integrated circuit modeling, Phase-change memory, PCM, PCRAM

published or not. The documents may come

I. INTRODUCTION

HASE-CHANGE memory (PCM) is a non-volatile resistive memory device relying on the phase transition of a small volume in a chalcogenide layer (typically GST). It can be switched from an amorphous high resistive phase to a crystalline low resistive phase by applying a short voltage pulse [1], [START_REF] Raoux | Phase Change Materials[END_REF]. This technology is overcoming all of its potential roadblocks and now exhibits rapid switching, extended endurance, high temperature data retention, low power consumption, and good thermal stability under solder reflow conditions [START_REF] Burr | Recent Progress in Phase-Change Memory Technology[END_REF]- [START_REF] Xiong | Towards ultimate scaling limits of phasechange memory[END_REF].

Device compact models are mandatory to simulate the circuit performance during the back-end design flow step. Consequently, they need to be accurate while ensuring a fast convergence time and a minimal memory usage. Macromodels [START_REF] Junsangsri | A New Comprehensive Model of a Phase Change Memory (PCM) Cell[END_REF]- [START_REF] El-Hassan | Improved SPICE model for Phase Change Memory cell[END_REF] and piecewise linear models including a decision module or negative differential resistance [START_REF] Cai | A Compact Spice Model with Verilog-A for Phase Change Memory[END_REF]- [START_REF] Xu | Multi-domain compact modeling for GeSbTe-based memory and selector devices and simulation for largescale 3-D cross-point memory arrays[END_REF] present discontinuities, which are likely to generate convergence issues. Compact models based on rate equations [START_REF] Covi | Compact model for phase change memory cells[END_REF]- [START_REF] Jung | Compact Verilog-A model of phase-change RAM transient behaviors for multi-level applications[END_REF], can be fast and robust. However, despite the enhanced accuracy of some of them, a good correlation with experimental data for any random pulse applied has never been published. Such validations are especially important in a multilevel context, where intermediate resistance level states are exploited. This paper presents a new continuous compact model of phase-change memory, based on comprehensive rate equations. The proposed model is extensively validated by experimental results, in a wide range Manuscript received; revised. The review of this paper was arranged by C. Pigot and F. Gilibert are with STMicroelectronics, 850 rue Jean Monnet,38926 Crolles, France (e-mail: corentin.pigot@st.com; fabien.gilibert@st.com).

P. Zuliani is with STMicroelectronics, Via C. Olivetti 2, 20041 Agrate Brianza, Italy (e-mail: paola.zuliani@st.com).

of time and temperatures. Using a single model card, the simulation of any random shaped pulse can be achieved, for the very first time. The modeling approach relies on simplified temperature computation, compensated by detailed considerations of the nanophysics inside a PCM cell during programming. The model is efficiently implemented in a Verilog-A code without any decision module to ensure convergence and short simulation time.

Section I describes the model equations and the measurements performed to validate it. Section II presents the correlation between simulations and silicon measurements through a brief extraction flow. Section III is focused on model card parameters with emphasis on their influences and physical meanings. A summary of obtained performances is given in Section IV.

II. MODEL PRESENTATION

A. Model architecture

A block diagram of the modeling process is shown in Fig. 1. First, the temperature is computed based on inputs and PCM state from the previous simulation step. The temperature computation module feeds two other related modules: one processing the portion of the active melted volume and the other one determining the crystalline fraction of the remaining nonmelted active part. The crystalline fraction and the temperature are then re-input to the DC module, which computes the cell resistance for the current simulation step.

M. Reyboz and O. Cueto are with CEA Leti, Minatec Campus, 38054 Grenoble, France (e-mail: marina.reyboz@cea.fr; olga.cueto@cea.fr).

M. Bocquet, V. Della Marca, and J.M. Portal are with IM2NP, Aix-Marseille Université, 5 rue Enrico Fermi bât Fermi, 13453 Marseille, France (e-mails: marc.bocquet@univ-amu.fr; vincenzo.della-marca@univ-amu.fr; jeanmichel.portal@univ-amu.fr). Self-heating temperature inside the PCM cell TSH, follows the first order differential equation (1) [START_REF] Covi | Compact model for phase change memory cells[END_REF], [START_REF] Marcolini | Modeling the dynamic self-heating of PCM[END_REF]:

Comprehensive Phase-Change Memory Compact Model For Circuit Simulation

!" # $ !" %& '( %! + ) *, = !" # -.
/ 012

(1)

Where U and RPCM are respectively the voltage across and resistance of the cell, Rth and Cth are effective thermal resistance and capacitance respectively, and t is the time. The ambient temperature Tamb is then added to the self-heating temperature TSH to obtain the final temperature T used in the next calculations. This temperature has no spatial dimension and we consider that it is calculated on a single point in the system, which is the hottest spot of the device. Thermal calculations performed using Cueto et al.'s electro thermal solver [START_REF] Cueto | Coupling the Phase-Field Method with an electrothermal solver to simulate phase change mechanisms in PCRAM cells[END_REF] shown in Fig. 2 indicate that this hottest spot inside the PCM layer is located at the interface between the heater and the amorphous dome.

The effective thermal resistance depends on the phase of the material [START_REF] Sonoda | A compact model of phase-change memory based on rate equations of crystallization and amorphization[END_REF], and has been implemented in the model via the equation (2):

!" = !"3 4 56 3 + 6 7 8 + !"9 4 6 9

(2)

Rthc is the crystalline thermal resistance and Rtha the amorphous thermal resistance. Fc, Fm and Fa are respectively the crystalline, melted, and amorphous fraction of the material.

2) Melting Module

Despite its simplified calculation, the temperature is not considered uniform inside the active volume. Hence, some portion of the material can be melted whereas other parts stay solid during a pulse. This partial melting is implemented by the introduction of the state variable Fm, computed as the solution of the first order differential equation (3):

: 7 ;6 7 ;< + 6 7 = >? + exp @ ) 7 A ) B 7 CD EF GGGGGGGGGGGGGG(3)
Where Hm is the melting time constant, ) 7 is the melting temperature of the GST, and Im is a parameter reflecting the spatial non-uniformity of the temperature inside the cell. The form of the right hand side of (3) and the fitting parameter Im ensure a smooth and continuous transition from solid to melted.

3) Crystalline/Amorphous Module

The crystalline fraction Fc is calculated by modifying the right hand side of the classical Johnson-Mehl-Avrami-Kolmogorov (JMAK) equation [START_REF] Avrami | Granulation, phase change, and microstructure kinetics of phase change. III[END_REF] to take the Fm variable into account:

: 3 ;6 3 ;< + 6 3 = ? A 6 7 G (4)
Where Hc is the characteristic time of crystallization and "1" stands for the whole active area. The amorphous fraction is then calculated by: 6 9 = ? A 6 7 G A 6 3 GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG(5)

When the time becomes infinite, the variable Fc tends to the solid fraction, which is 1-Fm, then Fa tends to 0. Indeed, the amorphous phase is seen as a metastable phase, crystallizing even at room temperature. However, this dynamics do not impact the simulation because the crystallization time is long enough.

4) Resistance Module

The resistance of the whole cell is calculated as a series of resistances for each phase, weighted by using their respective fraction: JKL = 6 3 4 3 + 6 7 4 3 + 6 9 4 9 + "M9!MN (6

)
Where Rheater is the resistance of the heater, which is constant for a given technology. Rc is the resistance of the crystalline phase, and Ra is the resistance of the amorphous phase. We assume that the resistance of the melted phase is Rc because in this state, the current is mainly controlled by the resistance of the poly-crystalline GST surrounding the active area.

The current IPF in the amorphous phase is modeled by Poole-Frenkel conduction [START_REF] Shih | Understanding amorphous states of phase-change memory using frenkel-poole model[END_REF]- [START_REF] Pigot | Phase-change memory: A continuous multilevel compact model of subthreshold conduction and threshold switching[END_REF] as given by equation [START_REF] Cobley | Parameterized SPICE model for a phase-change RAM device[END_REF]:

O JP = Q RJP G 4 6 4 exp SA T 0U EV 0U WP R& XGGGG ( 7
)
where AkPFYGZPF and [PF are fitting parameters with physical meanings described previously [START_REF] Pigot | Phase-change memory: A continuous multilevel compact model of subthreshold conduction and threshold switching[END_REF], and k is the Boltzmann constant. Ra is calculated using the Ohm's law, assuming that the voltage drop inside the PCM is mainly located inside the amorphous dome.

The calculation of the field F, as given by equation ( 8), is impacted by the thickness of the amorphous cap ua. ua is defined in (8) as a fraction of the maximum size ua,max of the dome, the latter being treated as a fitting parameter.

6 = - \ ] GGGGGGGwithGGGGGG^9 = 6 9 ^9Y79_ (8)
The ZPF parameter of ( 7) follows Varshni's empirical law [START_REF] Varshni | Temperature dependence of the energy gap in semiconductors[END_REF], [START_REF] Gallo | Evidence for thermally assisted threshold switching behavior in nanoscale phase-change memory cells[END_REF], such that,

`JP = a 9b A 9 c] & . d c] f& (9)
Where Ea0 is the activation energy at 0K, considered as a fitting parameter, and ava and bva are thermal parameters, set to the values found in Le Gallo et al. [START_REF] Gallo | Evidence for thermally assisted threshold switching behavior in nanoscale phase-change memory cells[END_REF]: ava=600µeV.K -1 , bva=800K. The temperature dependence of the resistance during the semi-conducting crystalline phase Rc, follows the expression [START_REF] Becker | Properties and Uses of Thermistors -Thermally Sensitive Resistors[END_REF]:

Thermostatic Top Electrode

Amorphous dome heater c-GST

! = !" e #$ %& ' ( )* %+, -#- ( )* . ----------------------------------( 10
)
Where Eac is an activation energy and / 0 = !12 when 3 = 3 456 ; both are model card parameters.

B. Model innovations

Equation ( 4) gives the time dependency of Fc, the parameter 7c contains all other dependencies. In this study, the crystallization speed is considered dependent on the temperature and the amorphous fraction.

Considering [START_REF] Xiong | Towards ultimate scaling limits of phasechange memory[END_REF], the crystallization speed 89 ! :8; can be extracted from ( 4) as a function of Fa and 7c:

89 ! 8; = 9 4 < ! >3? 9 4 @ ----------------------------------------------(11)
To isolate the dependencies, we introduce vg as the normalized amorphous-fraction-dependent growth speed and τset the temperature-dependent crystallization time:

< ! >3? 9 4 @ = < ABC >3@ D 9 4 E F >9 4 @ -------- (12)

1) Temperature dependent crystallization time

Ciocchini et al. [START_REF] Ciocchini | Evidence for non-arrhenius kinetics of crystallization in phase change memory devices[END_REF] show that 7set follows a non-Arrhenius behavior, because of an incompatibility of the activation energies between high and low temperatures. This is why in our approach, high-and low-temperature crystallization kinetics are separated, as given in equation ( 13):

< ABC = < G* H < I* = '< "G* J K LMN ON H < "I* J K LPN ON . ( 13
)
Thereby, equation ( 13) enables the model to provide a good retention at room temperature and a short crystallization time during high-voltage pulses. Plotted in Fig. 3 is < ABC as a function of 1/kT, with the temperature scale reported on the upper axis. This plot validates that the crystallization time is short at high temperature, whereas it is many orders of magnitude higher at room temperature.

2) Amorphous-fraction-dependent growth speed

The normalized growth speed vg is inserted in the model to reflect the non-uniformity of the crystallization as a function of the size of the amorphous dome. As shown in Fig. 2, the simplified temperature calculated by the model is considered to be the one at the interface between the heater and the GST. However, the temperature determining the crystallization process, considering re-growth from the surrounding crystalline GST ( [START_REF] Cueto | Coupling the Phase-Field Method with an electrothermal solver to simulate phase change mechanisms in PCRAM cells[END_REF], [START_REF] Bastard | Crystallization study of 'melt quenched' amorphous GeTe by transmission electron microscopy for phase change memory applications[END_REF]), is more likely the one at the external interface of the active volume. To take into account this effect while keeping a simple temperature calculation, we chose to let the growth speed vary with the amorphous fraction. Indeed, assuming a constant temperature gradient between the hot spot and the top electrode kept at room temperature, the temperature at the external interface moves with the position of this interface for a given input power. The smaller the amorphous dome, the closer the Amorphous/Crystalline interface is from the heater, and the warmer it is. The normalized growth speed vg is defined by the expression shown in ( 14):

E F >9 4 @ = Q D 9 4 D J (#6R % ----- ( 14
)
Where b is a fitting parameter. According to this expression, vg at a given temperature is higher for low amorphous fractions, as shown in Fig. 4 where vg versus Fa is plotted.

C. Electrical characterization method

The Transmission Electron Microscopy (TEM) crosssection, along with the equivalent scheme of the test structure is shown in Fig. 5. All characterizations are performed using this test structure. A wall-type PCM structure [START_REF] Annunziata | Phase Change Memory technology for embedded non volatile memory applications for 90nm and beyond[END_REF] is connected in series with a MOSFET selector, where the gate is used to limit the current flowing through the cell. ua is the radius of the active volume, where phase transitions are located. This material switches from amorphous to crystalline and back to amorphous under the application of pulses. The whole device is integrated in a CMOS technology and the node between the PCM cell and the MOSFET selector is not accessible from outside. Therefore, the following results present the simulation of both devices in series, the MOSFET being previously extracted on a separately.

The characterization method aims to test every possible operating condition of the device in order to validate the model extensively. First, the current versus voltage is acquired for several intermediate states by setting the PCM into the desired state and applying a slow ramp voltage on the top electrode [START_REF] Pigot | Phase-change memory: A continuous multilevel compact model of subthreshold conduction and threshold switching[END_REF]. Then, the phase transitions are studied through three different characteristics, shown in Fig. 6; the RESET-SET-RESET is a staircase-up measurement with long squared pulses, the SET Low is a staircase-up with increasing pulse width in the low current regime and the Rampdown SET is high current pulses with increasing fall times [START_REF] Sousa | Operation fundamentals in 12Mb Phase Change Memory based on innovative Ge-rich GST materials featuring high reliability performance[END_REF], [START_REF] Puzzilli | On the RESET-SET transition in phase change memories[END_REF].

Each measured point reported in these R-time or R-current characteristics is always composed of the same set of pulses, pictured in Fig. 7. First, a RESET operation is performed to fully control the initial state. Then a SET pulse, with a variable width, fall time and current amplitude, is applied to the cell and the programming current is measured. The programming current is averaged in the second half of the pulse width to avoid being disturbed by overshoots, when the current settles. As sketched in Fig. 7, the programming Word Line (WL) voltage is tuned in order to set the current value whereas the Bit Line (BL) voltage controls time width and fall time (FT) parameters. Finally the resistance value of the cell, which is a single point of measurement on the R-time or R-current characteristics, is extracted with 0.1V on the BL and 1.2V on the WL through a DC measurement.

The measurement conditions used for each plotted characteristics are summarized in Table I. To get rid of the drift effect [START_REF] Pirovano | Low-field amorphous state resistance and threshold voltage drift in chalcogenide materials[END_REF], the delay between the different pulses is constant. The impact of the ambient temperature is studied and error bars are obtained by repeating three times the exact same set of measurements.

III. MODEL VALIDATION THROUGH MODEL CARD EXTRACTION

The entire voltage range measured, including temperature and pulse time dependencies, is fitted using a single model card, which is shown in Table II. Given the simplifications made to ensure the fast convergence of the compact model, the model parameters do not correspond precisely to the physical parameters to which they are related. Yet, all model parameters have been kept to reasonable values during the extraction and they remain coherent with their associated physical parameters. They can be classified into three main categories: the first one is composed of the parameters related to the conductivity of the amorphous and crystalline phases, the second one includes the calculation of the temperature and the melting fraction inside the phase-change area, and the last category refers to timedependent crystallization.

As the resistance is conditional upon the low field conduction, the I-V characteristics must be modeled first. The parameters AkPF, uamax and Ea0 act respectively on the level, slope and temperature dependence of the amorphous subthreshold conduction. The threshold switching is modeled using a thermal runaway inside a Poole-Frenkel mechanism.

Hence, tuning Rtha helps fitting the threshold switching. The SET resistance is modeled using Rc0 and Eac. The PCM current versus the Bit Line voltage for several intermediate states is shown in Fig. 8. No snapback is shown because the MOSFET voltage drop is not de-embedded. However, the MOSFET model has been previously validated on a dedicated test structure, and the snapback capability of the model has already been demonstrated in our previous paper [START_REF] Pigot | Phase-change memory: A continuous multilevel compact model of subthreshold conduction and threshold switching[END_REF]. The model can simulate correctly the subthreshold conduction and the threshold switching for several intermediate states, with only one varying model parameter from one state to another, namely ua. It validates the model for multilevel simulations.

The parameters belonging to the second category act on the RESET-SET-RESET characteristics, presented in Fig. 9. In this figure, the resistance is plotted as a function of the programming current for three different temperatures. The transitions RESET to SET and SET to RESET are well fitted considering the self-heating effect. We demonstrate that our model is able to reproduce the weak impact of external temperature as shown by experimental characterizations. All parameters dealing with both SET and RESET resistances have been previously extracted, using curves shown in Fig. 8. On the SET to RESET transition visible on the right hand side of these curves, the melting parameters Tm and sm are extracted. The time constant m has not been measured, because it is shorter than the shortest measured pulse width, i.e. 200ns. It is set to 1ns, as previously described in the literature [START_REF] Gallo | Evidence for thermally assisted threshold switching behavior in nanoscale phase-change memory cells[END_REF]. The RESET to SET transition is mostly determined by the parameters responsible for the crystallization at low temperature, namely 0LT and EALT. The parameter Rthc acts on the whole curve, since it is accountable for the temperature computation.

Once all static parameters are extracted, the dynamic behavior of the crystallization is studied on the Rampdown SET plot, presented in Fig. 10. The read resistance as function of the fall time is plotted for different programming currents on the left hand side, and for different temperatures on the right hand side. The transition for several temperatures and current levels is captured by the model using the parameters b, 0HT and EAHT.

The dynamics of the crystallization must also be checked on the SET Low characteristics, where the SET operation is realized at relatively low current but depends on the pulse width. The SET Low results are shown in Fig. 11, exhibiting the phase transition for pulse widths ranging from 200ns to 800ns for three different ambient temperatures. The ambient temperature variation has an important impact on the high resistive conduction, in both the model and the measurements, and it seems that crystallization happens earlier at high ambient temperature in both the measurements and in the model. However, error bars in the transition regime means that the transition happens abruptly and stochastically in the transition regime. The modeling of this transition is smooth, because the model is built on state variables that always vary continuously. This stochasticity, which has been reported by Le Gallo et al. [START_REF] Gallo | Inherent stochasticity in phase-change memory devices[END_REF], will be integrated in future development using corners. The modeling of the SET Low characteristics is done using the same crystallization parameters as the Rampdown SET, i.e b and HT. Hence, a tradeoff has to be found between those two fits. The thermal activation EAHT can be optimized, as well as the ratio Rtha/Rthc, so that all features are well fitted.

IV. PARAMETER PHYSICAL MEANING

After the model is validated, the variation of the model card parameters is shown in order to illustrate their impact and physical meanings. Let us consider the RESET-SET-RESET characteristics, as pictured in Fig. 12. The level of the crystalline resistance is modified by Rc0 and the high resistance level is dependent on AkPF, the Poole-Frenkel pre-factor. Rthc plays on every aspect of this curve, as it rules the internal temperature. The SET to RESET transition proceeds of the amorphisation of the material, which involves the melting of it first. As the fall time is short in this experiment, the crystal does not have time to grow and the material stays in the disordered phase. The level of resistance achieved only depends on the amount of material that has been melted during the pulse. A change in the melting temperature Tm shifts the transition whereas a modification of the sm parameter modifies the slope of the transition. Finally, the parameter 0LT dominates the RESET-SET transition because the operation happens in the model at a quite low temperature. b accentuates the growth speed for low amorphous fraction, thus accelerates the crystallization process as the crystallization fraction increases. This auto-positive feedback sets the abruptness of the transition that can be seen in Fig. 13(b).

Both Rampdown SET and SET Low operations are modeled by the same mechanism, which is the growth from the surrounding crystalline GST. In order to satisfy both crystallization dynamics with the same set of parameters, two levers are used. First, the temperature activation of the crystallization time can be tuned because both crystallizations do not happen at the same internal temperature. Second, we can use the concurrency between both thermal resistances, Rtha and Rthc. Inspired by finite element simulations done by Cueto et al. [START_REF] Cueto | Coupling the Phase-Field Method with an electrothermal solver to simulate phase change mechanisms in PCRAM cells[END_REF], the lever consists of enhancing the crystallization speed at low current by artificially increasing the internal temperature immediately after the threshold switching using a strong Rtha. On the contrary, a lower Rthc will slow the crystallization at high current, as the temperature achieved in the cell will be lower.

V. PERFORMANCE ASSESSMENT

The performance of the model, measured in simulation time, is now discussed. To check this aspect, addressable matrixes of PCM, including the MOSFET selector, have been simulated using the Mentor Graphics' simulator Eldo on a single 2.60GHz Intel® Xeon® CPU E5-2697 v3. Simulating 10 bits takes half a second and the multiple thread processing of a 10 Kbits matrix on 4 CPU can take just about 12 minutes, for a full RESET/SET/READ sequence. This simple example proves that this model is completely suitable for circuit simulation, because it is accurate and allows fast simulation, even for multiple cells.

VI. CONCLUSION

A new comprehensive compact model is proposed. It is compared to extensive measurements covering a wide range of times, currents, and operating temperatures. Our model fits well to all the electrical characterization results using a single model card, including for the first time a compact modeling of two different crystallization dynamics, named Rampdown SET and SET Low. This is possible using several modeling innovations that have never been used before, such as the explicit use of a melting fraction, a non-Arrhenius form of the crystallization time and a growth speed enhanced for low amorphous fractions. The physical effects described in this model help to model the different crystallization dynamics independently on the temperature and programming current variations. A switchless Verilog-A implementation of the physical-based equations ensures the fast convergence of the model, even out of the fitting range. Tests run on multiple addressable matrixes exhibit reasonable simulation times, proving that the model is suitable for circuit simulation, thanks to the efficient implementation which enables short convergence time.

VII. REFERENCES [1]

S. R. Ovshinsky and H. Fritzsche, "Amorphous semiconductors for switching, memory, and imaging applications," IEEE Trans. Electron

Corentin Pigot ,Fig

 Pigot Fig. 1. Modeling flow block diagram

Fig. 2 .

 2 Fig. 2. Finite element simulations of the temperature inside the cell. A temperature gradient is induced between the heater and the top electrode. The simplified temperature calculated in the model refers to the hottest spot.

Fig. 3 .Fig. 4 .

 34 Fig. 3. Crystallization time as a function of temperature. < ABC is the sum of < I* and < G* , so that the process is quick at high temperature and slow at low temperature Fig. 4. Growth speed versus the amorphous fraction. This function is implemented to enhance the growth speed for low amorphous fraction.

Fig. 5 .

 5 Fig. 5. TEM cross section of the test structure on the left hand side and equivalent schematics on the right hand side.

Fig. 6 .Fig. 7 .

 67 Fig. 6. Example of typical characteristics of PCM. Each point on the characteristics is composed of a RESET Pulse, a SET pulse and a DC reading: (a) RESET-SET-RESET is long squared pulses with increasing pulse current; (b) Rampdown SET is high current pulses with increasing fall times; (c) Set Low is a staircase-up with increasing pulse width in the low current regime.

Fig. 8 .Fig. 9 .

 89 Fig. 8. Current of the PCM versus Bit Line voltage for several intermediate states, allowing the extraction of the conduction parameters. Dots are measurements, lines are simulations. The only parameter varying is the amorphous thickness ua.

Fig. 10 .

 10 Fig. 10. Rampdown SET characteristics: Transition from RESET to SET using the fall time increase for three different programming current on the left, under three operating temperatures on the right. Dots are measurements, lines are simulations. Error bars are obtained by repeating three times the exact same set of measurements. The model reacts appropriately for both dependencies.

Fig. 11 .

 11 Fig. 11. SET Low characteristics: Transition from RESET to SET using low current density for four pulse widths under three operating temperatures. Dots are measurements, lines are simulations. Error bars are obtained by repeating three times the exact same set of measurements. The dynamics is hard to fit with the same model card as the Rampdown SET, as tradeoff has to be found to fit both characteristics.

Fig. 13

 13 Fig. 13 highlights the details of the Rampdown SET operation through the variation of two effective parameters 0HT and b. 0HT impacts the average crystallization time, as illustrated in Fig. 13(a). As displayed in the inset of Fig. 13(b),

Fig. 12 .

 12 Fig. 12. Impact of the static parameters on the RESET-SET-RESET characteristics. (a) Rc0 rules the conduction in the SET regime; (b) AkPF impacts the RESET resistance; (c) Rthc increases the internal temperature, which is mostly seeable in the melting phase; (d) The average melting temperature Tm impacts the SET to RESET transition current; (e) sm changes SET to RESET transition spread; (f) 0LT deals with RESET to SET transition.

Fig. 13 .

 13 Fig. 13. Impact of the variation of the parameters 0HT and b on the growth speed. (a) 0HT acts on the mean crystallization time; (b) b increases the growth speed for low amorphous fraction (inset), so enhances the positive feedback on the Rampdown SET crystallization speed.

  348KThese measurement conditions cover a wide range of the device operating and allow the full extraction of the model card parameters.

		TABLE I
		MEASUREMENT CONDITIONS
	Variable	RESET-SET-RESET	Rampdown SET	SET Low
	VWL	0 -2V	2V	0 -1.2V
	VBL	2V	2V	2V
	Current	0 -300µA	300µA	0 -200µA
	Width	10µs	10µs	200n -800ns
	Fall time	10ns	10n -600ns	10ns
	Temperature	298 -348K	298 -348K	298 -


Diego Pérez-Galacho

Laurent Bramerie

Charles Baudot

Mohamed Chaibi

Sonia Messaoudène

Nathalie Vulliet

Laurent Vivien

Christophe Peucheret

Delphine Marris-Morini

O-Band QPSK modulation based on a silicon dual-drive Mach-Zehnder

Keywords: Silicon modulators, silicon photonics, integrated optics, advanced modulation formats, coherent optical communications

Advanced modulation formats will be required in next generation optical interconnects in order to keep up with future bandwidth demand. In this work QPSK modulation in the O-band is experimentally demonstrated using a silicon-based dual-drive Mach-Zehnder modulator.

INTRODUCTION

Significant research effort has been dedicated over the past years to the development of silicon based photonic technologies. Nowadays, it is believed that silicon photonics is the best suited for next generation short range optical fiber communications [START_REF] Jalali | Silicon photonics[END_REF], [START_REF] Paniccia | A perfect marriage: optics and silicon[END_REF]. Developing high performance silicon modulators based on the free-carrier plasma dispersion (FCPD) effect [START_REF] Soref | Electrooptical effects in silicon[END_REF] has been an important milestone. These modulators are commonly based on carrier depletion in PN junctions and outstanding performances have already been demonstrated [START_REF] Perez-Galacho | Low voltage 25Gbps silicon Mach-Zehnder modulator in the O-band[END_REF]- [START_REF] Li | High-speed BPSK modulation in silicon[END_REF]. However, in order to keep up with the growing bandwidth demand in datacom applications, it is necessary to move from binary on-off keying (OOK) modulation formats to more advanced formats such as 4-level pulse amplitude modulation (PAM-4) or quadrature phase-shift keying (QPSK) [START_REF] Shacham | Photonic networks-on-chip for future generations of chip multiprocessors[END_REF]. Advanced modulation formats using silicon modulators have already been demonstrated [START_REF] Wang | Silicon high-speed binary phase-shift keying modulator with a single-drive push-pull highspeed traveling wave electrode[END_REF]- [START_REF] Li | High-speed BPSK modulation in silicon[END_REF]. However, most of these demonstrations have been carried out in the C-band of optical communication systems (1530 nm -1565 nm) where the FCPD is stronger. Nevertheless, current standards for short range optical fiber communications favour the use of the O-band (1260 nm -1360 nm) in order to exploit the zero dispersion region of standard single-mode fibers [START_REF]Physical Layer Specifications and Management Parameters for 40 Gb/s and 100 Gb/s Operation over Fiber Optic Cables[END_REF]. In this work we present for the first time a QPSK silicon-based dual-drive Mach-Zehnder modulator (DDMZM) operating in the O-band. QPSK generation is normally based on nested MZM structures, therefore requiring four driving signals. However, the QPSK generation based on DDMZM requires only two driving signals, exhibiting a better power efficiency. As first demonstration, 10 GBd (20 Gbps) is shown.

MODULATOR DESIGN AND PRINCIPLE OF OPERATION

The cross-section of the designed phase shifter is depicted in Fig. 1(a). It comprises a PN junction implemented in a rib waveguide. It was designed following the simplified modeling approach proposed in [START_REF] Perez-Galacho | Simplified modeling and optimization of silicon modulators based on free-carrier plasma dispersion effect[END_REF]. The targeted doping concentrations of the junction were 𝑃 = 5 ⋅ 10 17 𝑐𝑚 -3 and 𝑁 = 1.4 ⋅ 10 18 𝑐𝑚 -3 . The designed phase shifter was integrated in both arms of a Mach-Zehnder interferometer in order to form a DDMZM, schematically represented in Fig. 1(b) in which the Δ𝑛 𝑖 boxes represent the phase shifting elements and the Φ box symbolizes a fixed phase shift intended to set the operation point of the DDMZM.

In order to generate QPSK modulation using this architecture it is necessary to bias the modulator at the quadrature point (Φ = 𝜋/2) and to apply the binary in-phase and quadrature-phase modulating signals to each phase shifter so that a phase shift of 𝜋 can be achieved in each arm [START_REF] Ho | Generation of arbitrary quadrature signals using one dual-drive modulator[END_REF]. In the designed DDMZM a 1×2 MMI coupler was used as combiner whereas the input splitter was implemented by means of a 2×2 MMI, naturally setting the modulator to the quadrature point. Heaters were introduced in each arm of the DDMZM in order to compensate for fabrication errors and fine tune the bias of the modulator to the quadrature point. GSGSG coplanar traveling wave electrodes were used to apply the two modulating signals to the phase shifters. The active length of the DDMZM was 2 mm. The modulator was fabricated using the 300 mm technological platform from STMicroelectronics [START_REF] Baudot | DAPHNE silicon photonics technological platform for research and development on WDM applications[END_REF].

CHARACTERIZATION SETUP

In order to assess QPSK modulation, the setup of Fig. 2 was used. TE-polarized light from an O-band external laser was injected in/out of the fabricated chip using grating couplers. RF and DC electrical probes were used to electrically drive the phase shifters and heaters, respectively. Two 2 15 -1 long de-correlated pseudo random binary sequence (PRBSs) at 10 Gbps were used as modulating signals. The two PRBSs modulating signal were amplified and then a DC bias was added to ensure that the PN junctions were operated in depletion. The driving signals applied to the phase-shifters had peak-to-peak voltages of 7Vpp with a DC bias of 4V. 50Ω loads were added at the end of the GSGSG coplanar electrodes. The signal at the output of the modulator was then input to an integrated coherent receiver through a variable optical attenuator in order to control the received power. For the purpose of this first demonstration, the same laser was used as local oscillator in the receiver, therefore implementing an homodyne detection scheme. The I/Q output signals of the receiver were fed to a 40 Gsps digital sampling oscilloscope. 10 5 symbols were acquired and then processed offline. For digital processing, an adaptive finite impulse response (FIR) filter with decision-directed least mean square (DD-LMS) tracking was used.

CHARACTERIZATION SETUP

The DDMZM was first characterized in DC. The measured fiber-to-fiber attenuation was 17 dB, including the loss of the grating couplers, on-chip passive losses and modulator insertion loss (IL). Using a straight waveguide on the chip as normalization, the IL of the modulator was estimated to be 3.8 dB. The Vπ of each phase shifter was measured to be between 6 V and 7 V. Taking into account the 2 mm active length, this leads to a modulation efficiency between 1.2 Vcm and 1.4 Vcm.

High-speed measurements were then performed on the modulator. Using the setup previously described, the bit error rate (BER) was measured as a function of the signal average received power, as shown in Fig. 3(a). Each point in the BER curve of Fig. 3(a) is the result of the averaging of five BER measurements, each carried out on 10 5 transmitted symbols. Using a BER value of 10 -3 as the limit for error free transmission based on forward error correction, the required received power is of -20dBm. The received constellation diagram after offline digital signal processing is shown in Fig. 3(b) for a received power of -22 dBm. A clear QPSK constellation diagram can be seen, and the measured EVM for this constellation is 33%.

CONCLUSIONS

Short range optical communication systems in the O-Band are facing important demands of bandwidth nowadays. As a first step towards more efficient modulation schemes, QPSK modulation in the O-band using a silicon modulator has been proposed and experimentally demonstrated at 10GBd (20GBps) for the first time using a DDMZM. The benefit of this modulator is that, relying on a single Mach-Zehnder interferometer, it only requires two driving signals, whereas standard QPSK modulation based on nested MZMs require four driving signals. This reduction on power consumption will be an important advantage in framework of short range communications.

Figure 1 .

 1 Figure 1. Cross section of the phase shifter (a) and schematic representation of the dual drive Mach-Zehnder modulator (b).

Figure 2 .

 2 Figure 2. Experimental setup employed for QPSK signal characterization.

Figure 3

 3 Figure 3 BER as a function of the received power (a) and measured constellation diagram for a received power of -22dBm (b).

ACKNOWLEDGEMENTS

This work was supported by the European projects Plat4m (FP7-2012-318178), Cosmicc (H2020-ICT-27-2015-688516). We acknowledge the support of French Industry Ministry/DGE through the Nano2017 program.