Philippe Carmona

Yueyun Hu

The Spread of a Catalytic Branching Random Walk

Keywords:

We consider a catalytic branching random walk on that branches at the origin only. In the supercritical regime we establish a law of large number for the maximal position M n : For some constant α, M n n → α almost surely on the set of infinite number of visits of the origin. Then we determine all possible limiting laws for M n -αn as n goes to infinity.

Introduction

A catalytic branching random walk (CBRW) on branching at the origin only is the following particle system: When a particle location x is not the origin, the particle evolves as an irreducible random walk (S n ) n∈ on starting from x.

When a particle reaches the origin, say at time t, then a time t + 1 it dies and gives birth to new particles positioned according to a point process 0 . Each particle (at the origin at time t) produces new particles independently of every particle living in the system up to time t. These new particles evolve as independent copies of (S n ) n∈ starting from their birth positions.

The system starts with an initial ancestor particle located at the origin. Denote by the law of the whole system ( also governs the law of the underlying random walk S), and by x if the initial particle is located at x (then = 0 ).

Let X u , |u| = n denote the positions of the particles alive at time n (here |u| = n means that the generation of the particle u in the Ulam-Harris tree is n). We assume that

0 = X u , |u| = 1 d = S (i) 1 , 1 ≤ i ≤ N
where N is an integer random variable describing the offspring of a branching particle, with finite mean m = [N ], and (S (i) n , n ≥ 0) i≥1 are independent copies of (S n , n ≥ 0), and independent of N .

Let τ be the first return time to the origin τ := inf n ≥ 1 : S n = 0 with inf = +∞ .

The escape probability is q esc := (τ = +∞) ∈ [0, 1) (q esc < 1 because S is irreducible). Assume that we are in the supercritical regime, that is m(1q esc ) > 1 .

(1.1)

An explanation of assumption (1.1) is given in Section 7, Lemma 7.3.

Since the function defined on (0, ∞) by r → ρ (r) = m e -rτ is of class C ∞ , strictly decreasing, lim r→0 ρ (r) = m (τ < +∞) = m(1q esc ) > 1 and lim r→+∞ ρ (r) = 0, there exists a unique r > 0, a Malthusian parameter such that m e -rτ = 1 .

(1.2)

Let ψ be the logarithmic moment generating function of S 1 : ψ(t) := log e tS 1 ∈ (-∞, +∞], t ∈ .

Let ζ := sup{t > 0 : ψ(t) < ∞}. We assume furthermore that ζ > 0 and there exists some t 0 ∈ (0, ζ) such that ψ(t 0 ) = r. (1.3) Observe that by convexity ψ ′ (t 0 ) > 0.

Let M n := sup |u|=n X u be the maximal position at time n of all living particles (with convention sup := -∞). Since the system only branches at the origin 0, we define the set of infinite number of visits of the catalyst by := ω : lim sup n→∞ u : |u| = n, X u = 0 = .

Remark that (dω)-almost surely on c , for all large n ≥ n 0 (ω), either the system dies out or the system behaves as a finite union of some random walks on , starting respectively from X u (ω) with |u| = n 0 . In particular, the almost sure behavior of M n is trivial on c . It is then natural to consider M n on the set . Our first result on M n is Theorem 1.1 (Law of large numbers). Assume (1.1) and (1.3). On the set , we have the convergence

lim n→+∞ M n n = α := ψ(t 0 )
t 0 a.s.

In Theorem 1.1, the underlying random walk S can be periodic. In order to refine this convergence to a fluctuation result by centering M n , we shall need to assume the aperiodicity of S. However, we cannot expect a convergence in distribution for M n -αn since M n is integer-valued whereas αn in general is not.

For x ∈ , let ⌊x⌋ be the integer part of x and {x} := x -⌊x⌋ ∈ [0, 1) be the fractional part of x. Theorem 1.2. Assume (1.1) and (1.3). Assume furthermore that (N 2 ) < ∞ and that S is aperiodic. Then there exists a constant c * > 0 and a random variable Λ ∞ such that for any fixed y ∈ , M n -αn > y =

1e -c * e -t 0 y (e t 0 {αn+ y} +o(1))Λ ∞ , (1.4) where o(1) denotes some deterministic term which goes to 0 as n → ∞. The random variable Λ ∞ is non negative and satisfies that

Λ ∞ > 0 = a.s. (1.5)
Consequently for any subsequence n j → ∞ such that {αn j } → s ∈ [0, 1) for some s ∈ [0, 1), we have that lim j→∞ M n j -⌊αn j ⌋ = y = e -c * e -t 0 ( y-s) Λ ∞e -c * e -t 0 ( y-1-s) Λ ∞ (∀ y ∈ .) (1.6) Let us make some remarks on Theorem 1.2:

Remark 1.

1. The random variable Λ ∞ is the limit of the positive fundamental martingale of Section 4. The value of constant c * is given in (6.14) at the beginning of Section 6.

2. The hypothesis (N 2 ) < ∞ might be weakened to (N log(N + 1)) < ∞, just as the classical L log L-condition (see e.g. Biggins [START_REF] Biggins | Martingale convergence in the branching random walk[END_REF]) in the branching random walk.

3. We do need the aperiodicity of the underlying random walk S in the proof of Theorem 1.2. However, for the particular case of the nearest neighborhood random walk (the period equals 2), we can still get a modified version of Theorem 1.2, see Remark 5 of subsection 6.1.

Theorems 1.1 and 1.2 are new, even though a lot of attention has been given to CBRW in continuous time. In papers [3-5, 10, 28-31] very precise asymptotics are established for the moments of η t (x) the number of particles located at x at time t, in every regime (sub/super/critical). Elaborate limit theorems were obtained for the critical case by Vatutin, Topchii and Yarovaya in [START_REF] Topchii | Individuals at the origin in the critical catalytic branching random walk, Discrete random walks[END_REF][START_REF]Two-dimensional limit theorem for a critical catalytic branching random walk[END_REF][START_REF] Vatutin | A limit theorem for critical catalytic branching random walks[END_REF][START_REF] Vatutin | Catalytic branching random walks and queueing systems with a random number of independently operating servers[END_REF].

Concerning on the maximal/minimal position of a branching random walk (BRW) on , some important progress were made in recent years, in particular a convergence in law result was proved in Aïdékon [START_REF] Aïdékon | Convergence in law of the minimum of a branching random walk[END_REF] when the BRW is not lattice-valued. It is expected that such convergence dos not hold in general for lattice-valued BRW, for instance see Bramson [START_REF] Bramson | Minimal displacement of branching random walk[END_REF] where he used a centering with the integer part of some (random) sequence. In the recent studies of BRW, the spine decomposition technique plays a very important role. It turns out that a similar spine decomposition exists for CBRW (and more generally for branching Markov chains), and we especially acknowledge the paper [START_REF] Döring | Catalytic branching processes via spine techniques and renewal theory[END_REF] that introduced us the techniques of multiple spines, see Section 3.

We end this introduction by comparing our results to their analogue for (non catalytic) branching random walks (see e.g. [START_REF] Aïdékon | Convergence in law of the minimum of a branching random walk[END_REF][START_REF] Aïdékon | Weak convergence for the minimal position in a branching random walk: a simple proof[END_REF][START_REF] Hu | Minimal position and critical martingale convergence in branching random walks, and directed polymers on disordered trees[END_REF][START_REF] Révész | Random walks of infinitely many particles[END_REF]). We shall restrict ourselves to simple random walk on , that is S 1 = ±1 = ) the rate function for large deviations of the simple random walk and ψ(t) = log cosh(t). For CBRW, we can do explicit computations : Since for x = 0, x e -rτ = e -t 0 |x| the Malthusian parameter satisfies r + t 0 = log(m). Combined with log cosh(t 0 ) = r this implies e t 0 = 2m -1 and α = 2 log(m) log(2m-1) -1. Numerically, for m = 1.83 we find b = 0.9 and α = 0.24. The second order results emphasize the difference between BRW and CBRW : for BRW, M (br w) n bn is of order O(log n), whereas for CBRW, M n -αn is of order O [START_REF] Aïdékon | Convergence in law of the minimum of a branching random walk[END_REF], see Remark 5.

The organization of the rest of this paper is as follows: We first give in Section 2 the heuristics explaining the differences between CBRW and ordinary BRW (branching random walk). Then we proceed (in Section 3) to recall many to one/few lemmas, we exhibit a fundamental martingale (in Section 4) and prove Theorems 1.1 and 1.2 in sections 5 and 6 respectively, with the help of sharp asymptotics derived from renewal theory. Finally, Section 7 is devoted to an extension to the case of multiple catalysts. There the supercritical assumption (1.1) appears in a very natural way.

Finally, let us denote by C, C ′ or C ′′ some unimportant positive constants whose values can be changed from one paragraph to an another.

Heuristics

Assume for sake of simplicity that we have a simple random walk. The existence of the fundamental martingale Λ n = e -r n |u|=n φ(X u ), See Section 4, such that Λ ∞ > 0 = , shows that on the set of non extinction , we have roughly e r n particles at time n.

If we apply the usual heuristic for branching random walk (see e.g. [START_REF] Révész | Random walks of infinitely many particles[END_REF] Section II.1), then we say that we have approximately e r n independent random walks positioned at time n, and therefore the expected population above level an > 0 is roughly:

  ⌊e r n ⌋ i=1 1 (S (i) n ≥an)   = ⌊e⌋ r n ⌋ S n ≥ an = e -n(ψ * (a)-r)(1+o(1))
where ψ * (a) = sup t≥0 (ta -ψ(t)) is the large deviation rate function (for simple random walk, e ψ(t) = e tS 1 = ch(t)). This expected population is of order 1 when ψ * (a) = r and therefore we would expect to have M n n → γ on , where ψ * (γ) = r. However, for CBRW, this is not the right speed, since the positions of the independent particles cannot be assumed to be distributed as random walks.

Instead, the ⌊e r n ⌋ independent particles may be assumed to be distributed as a fixed probability distribution, say ν. If η n (x) = |u|=n 1 (X u =x ) is the number of particles at location x at time n, we may assume that for a constant C > 0, e -r n η n (x) → Cν(x) and thus, ν inherits from η n the relation :

ν(x) = e -r y c( y)p( y, x)(m 1 ( y=0) + 1 ( y =0) )
with p(x, y) the random walk kernel. For simple random walk, this implies that for |x| ≥ 2 we have 1 2 (ν(x + 1) + ν(x -1)) = e r ν(x) and thus ν(x) = C e -t 0 |x| for |x| ≥ 2, with ψ(t 0 ) = log cosh(t 0 ) = r.

Therefore the expected population with distance to the origin at least an is roughly

   |x|≥an η n (x)    = e r n |x|≥an e -r n η n (x) ∼ e r n C |x|≥an e -t 0 |x| ∼ C ′ e r n e -t 0 an
This expectation is of order 1 when a = r This heuristically gives the law of large numbers in Theorem 1.1.

Many to one/few formulas for multiple catalysts branching random walks (MCBRW)

For a detailed exposition of many to one/few formulas and the spine construction we suggest the papers of Biggins and Kyprianou [START_REF] Biggins | Measure change in multitype branching[END_REF], Hardy and Harris [START_REF] Hardy | A spine approach to branching diffusions with applications to p -convergence of martingales[END_REF], Harris and Roberts [START_REF] Harris | The many-to-few lemma and multiple spines[END_REF] and the references therein. For an application to the computations of moments asymptotics in the continuous setting, we refer to Döring and Roberts [START_REF] Döring | Catalytic branching processes via spine techniques and renewal theory[END_REF]. We state the many to one/two formulas for a CBRW with multiple catalysts and will specify the formulas in the case with a single catalyst.

Multiple catalysts branching random walks (MCBRW)

The set of catalysts is a some subset of . When a particle reaches a catalyst x ∈ it dies and gives birth to new particles according to the point process

x d = (S (i) 1 , 1 ≤ i ≤ N x )
where (S (i) n , n ∈ ) i≥1 are independent copies of an irreducible random walk (S n , n ∈ ) starting form x, independent of the random variable N x which is assumed to be integrable. Each particle in produces new particles independently from the other particles living in the system. Outside of a particle performs a random walk distributed as S. The CBRW (branching only at 0) corresponds to = {0}.

The many to one formula for MCBRW

Some of the most interesting results about first and second moments of particle occupation numbers that we obtained come from the existence of a "natural" martingale. An easy way to transfer martingales from the random walk to the branching processes is to use a slightly extended many to one formula that enables conditioning. Let

m 1 (x) := N x < ∞, x ∈ . (3.1)
On the space of trees with a spine (a distinguished line of descent) one can define a probability via martingale change of probability, that satisfies

   Z |u|=n f (X u )    =   Z f (X ξ n ) 0≤k≤n-1 m 1 (X ξ k )   , (3.2)
for all n ≥ 1, f : → + a nonnegative function and Z a positive n measurable random variable, and where ( n , n ≥ 0) denotes the natural filtration generated by the MCBRW (it does not contain information about the spine).

On the right-hand-side of (3.2) (ξ k ) is the spine, and it happens that the distribution of (X ξ n ) n∈ under is the distribution of the random walk (S n ) n∈ . Specializing this formula to CBRW for which

m 1 (x) = m 1 (x=0) + 1 (x =0) yields    |u|=n f (X u )    = f (S n )m L n-1 , (3.3)
where

L n-1 = n-1 k=0 1 (S k =0)
is the local time at level 0.

The many to two formula for MCBRW

Recall (3.1). Let us assume that

m 2 (x) := N 2 x < ∞, x ∈ . (3.4)
Then for any n ≥ 1 and f : × → + , we have

   |u|=|v|=n f (X u , X v )    =    f (S 1 n , S 2 n ) 0≤k<T d e ∧n m 2 (S 1 k ) T d e ∧n≤k<n m 1 (S 1 k )m 1 (S 2 k )    , (3.5
) where under , S 1 and S 2 are coupled random walks that start from 0 and stay coupled (in particular at the same location) until the decoupling time T d e and after T d e , they behave as independent random walks.

More precisely, we have a three component Markov process (S 1 n , S 2 n , I n , n ≥ 0) where I n ∈ {0, 1} is the indicator that is one iff the random walks are decoupled: when the two random walks are coupled at time n, and at site x, the they stay coupled at time n+1 with probability (x) . That means that the transition probability are the following:

m 1 (x) m 2
• S 1 n+1 = y, S 2 n+1 = y, I n+1 = 0 | S 1 n = S 2 n = x, I n = 0 = m 1 (x) m 2 (x) p(x, y) • S 1 n+1 = y, S 2 n+1 = z, I n+1 = 1 | S 1 n = S 2 n = x, I n = 0 = (1- m 1 (x)
m 2 (x) )p(x, y)p(x, z)

• S 1 n+1 = y, S 2 n+1 = z, I n+1 = 1 | S 1 n = x 1 , S 2 n = x 2 , I n = 1 = p(x 1 , y)p(x 2 , z).
The random walks are initially coupled and at the origin. The decoupling time T d e = inf n ≥ 1 : I n = 1 satisfies for any k ≥ 0,

T d e ≥ k + 1 | σ{S 1 j , S 2 j , I j , j ≤ k} = 0≤l≤k-1 m 1 (S 1 l ) m 2 (S 2 l ) 1 (I k =0) , (3.6)
where we keep the usual convention ≡ 1. This formula is proved in [START_REF] Hardy | A spine approach to branching diffusions with applications to p -convergence of martingales[END_REF][START_REF] Harris | The many-to-few lemma and multiple spines[END_REF] by defining a new probability on the space of trees with two spines.

An alternative proof, that makes more natural the coupling of (S 1 , S 2 ) is to condition on the generation of the common ancestor w = u ∧ v of the two nodes, then use the branching to get independence, and plug in the many to one formula in each factor. We omit the details.

A fundamental Martingale

Martingale arguments have been used for a long time in the study of branching processes. For example, for the Galton Watson process with mean progeny m and population Z n at time n, the sequence W n = Z n m n is a positive martingale converging to positive finite random variable W . The Kesten-Stigum theorem implies that if N log(N + 1) < +∞, we have the identity a.s., {W > 0} equals the survival set. A classical proof can be found in the reference book of Athreya and Ney [START_REF] Athreya | Branching processes[END_REF], Section I.10. A more elaborate proof, involving sizebiased branching processes, may be found in Lyons-Pemantle-Peres [START_REF] Lyons | Conceptual proofs of L log L criteria for mean behavior of branching processes[END_REF].

Similarly, the law of large numbers for the maximal position M n of branching random walks system may be proved by analyzing a whole one parameter family of martingales (see Shi [START_REF] Shi | Branching random walks, Saint-Flour summer's course[END_REF] for a detailed exposition on the equivalent form of Kesten-Stigum's theorem for BRW). Recently, the maximal position of a branching brownian motion with inhomogeneous spatial branching has also been studied with the help a a family of martingale indexed this time by a function space (see Berestycki, Brunet, Harris and Harris [START_REF] Berestycki | The almost-sure population growth rate in branching Brownian motion with a quadratic breeding potential[END_REF] or Harris and Harris [START_REF] Harris | Branching Brownian motion with an inhomogeneous breeding potential[END_REF]).

We want to stress out the fact that for catalytic branching random walk, since we branch at the origin only, we only have one natural martingale, which we call the fundamental martingale.

Let T = inf n ≥ 0 : S n = 0 be the first hitting time of 0, recall that τ = inf n ≥ 1 : S n = 0 and let

φ(x) := x e -r T (x ∈ ) , (4.1)
where r is given in (1.2). Finally let p(x, y) = x S 1 = y and P f (x) = y p(x, y) f ( y) be the kernel and semigroup of the random walk S.

Proposition 4.1. Under (1.1) and (1.3).

The function φ satisfies

Pφ(x) = e r φ(x) 1 m 1 (x=0) + 1 (x =0) .

The process

∆ n := e -r n φ(S n )m L n-1 is a martingale, where L n-1 = 0≤k≤n-1 1 (S k =0)
is the local time at level 0.

The process

Λ n := e -r n |u|=n φ(X u )
is a martingale called the fundamental martingale.

4. If N 2 < +∞, then the process Λ n is bounded in L 2 , and therefore is a uniformly integrable martingale.

Proof. (1) If x = 0, then T ≥ 1, therefore, by conditioning on the first step:

φ(x) = y p(x, y)e -r
y e -r T = e -r Pφ(x) .

On the other hand, τ ≥ 1 so conditioning by the first step again,

φ(0) = 1 = m e -rτ = m y p(0, y)e -r y e -r T = me -r Pφ(0) .
(2) Denote by S n := σ{S 1 , ..., S n } for n ≥ 1. We have,

∆ n+1 | S n = e -r(n+1) m L n φ(S n+1 ) | S n = e -r(n+1) m L n Pφ(S n ) = e -r(n+1) m L n e r φ(S n )( 1 m 1 (S n =0) + 1 (S n =0) ) = ∆ n .
(3) Recall that ( n ) n≥0 denotes the natural filtration of the CBRW. By the many to one formula, if Z is n-1 measurable positive, then

Λ n Z = e -r n    |u|=n φ(X u )Z    = e -r n Zφ(S n )m L n-1 = Z∆ n = Z∆ n-1 (the martingale property of ∆ n ) = Λ n-1 Z . ( 4
)
The proof is given in Section 7 in the case of multiple catalysts and uses heavily the many to two formula.

Let us introduce η n (x) the number of particles located at x at time n:

η n (x) := |u|=n 1 (X u =x) .
Corollary 4.2. Under (1.1) and (1.3).

1. We have sup x,n e -r n φ(x)η n (x) < +∞ a.s.

2. If N has finite variance then there exists a constant 0 < C < ∞ such that

η n (x)η m ( y) ≤ C φ(x)φ( y) e r(n+m) (n, m ∈ , x, y ∈ d ). Proof. (1) Let us write Λ n = e -r n x φ(x)η n (x).
Since it is a positive martingale it converges almost surely to a finite integrable positive random variable

Λ ∞ . Therefore Λ * ∞ := sup Λ n < +∞ a.s.and sup x,n e -r n φ(x)η n (x) ≤ Λ * ∞ .
(2) Assume for example that n ≤ m and let

C = sup n Λ 2 n < +∞. We have, since Λ n is a martingale, e -r(n+m) φ(x)φ( y) η n (x)η m ( y) ≤ Λ n Λ m = Λ n Λ m | n = Λ 2 n ≤ C.
For the proof of the following result instead of using large deviations for L n , we use renewal theory, in the spirit of [START_REF] Carmona | A large deviation theory via the renewal theorem[END_REF][START_REF] Döring | An application of renewal theorems to exponential moments of local times[END_REF]. Let d be the period of the return times to 0:

d := gcd {n ≥ 1 : (τ = n) > 0} . (4.2)
Proposition 4.3. Assume (1.1) and (1.3). For every x ∈ there exists a constant c x ∈ (0, ∞) and a unique l x ∈ {0, 1,

• • •, d -1} such that lim n→+∞ e -r(d n+l x ) η nd+l x (x) = c x .
Moreover, for any l ≡ l x (mod d), η nd+l (x) = 0 for all n ≥ 0. In particular, for x = 0, l x = 0 and c 0 = d m . Proof. By the many to one formula (3.2),

v n (x) := η n (x) =    |u|=n 1 (X u =x )    = 1 (S n =x) e A 0 (ξ n ) = 1 (S n =x ) m L n-1 .
We decompose this expectation with respect to the value of τ = inf n ≥ 1 : S n = 0 :

v n (x) = m 1 (S n =x) 1 (τ≥n) + 1≤k≤n-1 1 (S n =x) m L n-1 1 (τ=k) .
By the Markov property, if

u k := (τ = k), then v n (x) = m τ ≥ n, S n = x + 1≤k≤n-1 mu k v n-k (x) = m τ ≥ n, S n = x + mv.(x) * u(n) ,
Recall that the Malthusian parameter r is defined by

1 = m e -rτ = m k≥1 e -r k u k .
Hence if we let ṽn (x) = e -r n v n (x) and ũk = me -r k u k then,

ṽn (x) = me -r n τ ≥ n, S n = x + ṽ• (x) * ũ(n) .
By the periodicity, we have u n = 0 if n is not a multiple of d and for x ∈ d there is a unique

l x ∈ {0, 1, . . . , d -1} such that ν n (x) = 0 if n ≡ l x (mod d).
Therefore the sequence t n = ṽnd+l (x) satisfies the following renewal equation

t n = y n + t * s n
with s n = ũnd and y n = e -r(nd+l x ) τ ≥ d n + l x , S d n+l x = x . Since the sequence s is aperiodic, the discrete renewal theorem (see Feller [START_REF] Feller | An Introduction to Probability Theory and Its Applications[END_REF], section XIII.10, Theorem 1) implies that

t n → ∞ n=1 y n ∞ n=1 ns n =: c x . Remark that ∞ n=1 ns n = ∞ n=1 ne -r nd mu nd = 1 d . We have c x = d ∞ n=1 e -r(nd+l x ) τ ≥ d n + l x , S d n+l x = x > 0.
This is exactly the desired result.

Finally for x = 0,

ℓ x = 0 and c 0 = d ∞ n=1 e -r nd τ ≥ d n, S d n = 0 = d ∞ n=1 e -r nd (τ = d n) = d (e -rτ ) = d
m by the choice of r. This completes the proof of Proposition 4.3.

Remark 2. The family (c x ) x∈ satisfies a system of linear equations, dual to the one (see Proposition 4.1) satisfied by the function φ: Recalling that p(x, y) =

x (S 1 = y) is the kernel of the random walk, we have the recurrence relation

η n+1 (x) = y η n ( y) p( y, x)(m 1 ( y=0) + 1 ( y =0) ).
Assuming for simplicity d = 1 and multiplying by e -r(n+1) and letting n → +∞, we obtain the following functional equation for the function x → c x :

c x = e -r y c y p( y, x)(m 1 ( y=0) + 1 ( y =0) ) , x ∈ .
We end this section by the following lemma which yields the part (1.5) in Theorem 1.2. Lemma 4.4. Assume (1.1) and (1.3). Assume furthermore that N has finite variance. Then we have

Λ ∞ > 0 = a.s.
Remark that in this Lemma we do not need the aperiodicity of the underlying random walk S.

Proof. We first prove that c ⊂ Λ ∞ = 0 a.s. In fact, on c , either the system dies out then Λ n = 0 for all large n, or for all large n ≥ n 0 (ω) : η n (0) = 0 . Then, if η n = x η n (x) is the total population, η n = η n 0 for all n ≥ n 0 since the system only branches at 0. Since Λ n = e -r n φ(X u ) ≤ e -r n η n = e -r n η n 0 , we still get Λ ∞ = 0.

Let s = Λ ∞ = 0 and ŝ := ( c ). If we can prove s = ŝ, then the Lemma follows. We shall condition on the number of children of the initial ancestor N . For k ≥ j ≥ 0, let Υ k, j be the event that amongst k particles of the first generation there are exactly j particles which will return to 0. Then

s = Λ ∞ = 0 = ∞ k=0 (N = k) k j=0 Υ k, j ∩ {Λ ∞ = 0} N = k = ∞ k=0 (N = k) k j=0 k j q k-j esc (1 -q esc ) j s j = ∞ k=0 (N = k)(q esc + s(1 -q esc ) k = f (q esc + s(1 -q esc )), with f (x) =
x N the generating function of the reproduction law. Exactly in the same way, we show that ŝ satisfies the same equation as s.

It remains to check the equation x = f (q esc + x(1q esc )) has a unique solution in [0, 1) (ŝ ≤ s and s < 1 thanks to Proposition 4.1). To this end, we consider the function g(x)

:= f (q esc + x(1-q esc ))-x. The function g is strictly convex on [0, 1], g(0) = f (q esc ) > 0, g(1) = 0 and g ′ (1) = m(1 -q esc ) -1 > 0.
Thus g has a unique zero on [0, 1), proving the Lemma.

5 The law of large numbers : proof of Theorem 1.1.

Proof of the upper bound.

Let θ > 0, x > 0. By the many to one formula,

M n > x n =    |u|=n 1 (X u >x n) = 0    ≤    |u|=n 1 (X u >x n)    = 1 (S n >nx ) m L n-1 ≤ e θ (S n -x n) m L n-1 = e -θ nx h n , with h n = e θ S n m L n-1 .
As in Proposition 4.3, we are going to use the renewal theory to study the asymptotics of v n . Let us condition on τ = inf n ≥ 1 : S n = 0 :

h n = e θ S n m L n-1 1 (τ≥n) + 1≤k≤n-1 e θ S n m L n-1 1 (τ=k) = e θ S n 1 (τ≥n) + 1≤k≤n-1 m (τ = k)h n-k = z n + mh * u(n) ,
with z n := e θ S n 1 (τ≥n) and u n := (τ = n). Assume now that θ > t 0 so that ψ(θ ) > ψ(t 0 ) = r. We let hn := e -nψ(θ ) h n , zn := e -nψ(θ ) z n , ũn := me -nψ(θ ) u n .

On the one hand, by definition of the Malthusian parameter we have 1 = m e -rτ = m n e -r n u n so that k ũk < 1.

On the other hand,

zn = e θ S n -nψ(θ ) 1 (τ≥n) = θ (τ ≥ n)
with θ defined by the martingale change of probability

d θ d = e θ S n -nψ(θ ) (on n ) .
Since under θ , (S n ) n≥0 is a random walk with mean θ S 1 = ψ ′ (θ ) ≥ ψ ′ (t 0 ) > 0, we have zn → z∞ := θ (τ = +∞) .

If we make the aperiodicity assumption d = 1, then by the discrete renewal theorem, we have hn → ỹ∞ 1k ũk .

In the general case, we can prove exactly as in the proof of Proposition 4.3 that for every l ∈ {0, . . . , d -1} there exists a finite constant K l such that

lim n→+∞ hnd+l → K l .
Therefore in any case, the sequence hn is bounded, and if x >

ψ(θ ) θ M n > x n ≤ e -n(θ x-ψ(θ )) hn satisfies n M n > x n < +∞.
t 0 = α a.s.

Proof of the lower bound, under the hypothesis

(N 2 ) < ∞.
The strategy of proof is as follows: Let 0 < s < 1, a > 0 and consider the event n,a,s (with c ′ a positive constant): "the particles survive forever, there are at least 1 2 c ′ e rsn particle alive at time sn, and one of these particle stays strictly positive until time n and reaches a position larger that (1s)an at time n".

We shall prove that for a suitable constant c ′ , we can choose a, s such that on the set of infinite number of visits to 0, for large n we are in n,a,s . This implies that almost surely on , lim inf

M n n ≥ a(1 -s).
Optimizing over the set of admissible couples (a, s) will yield the desired lower bound : lim inf

η d n (0) ≥ c ′ 2 e r d n ,
for all large n.

Proof. We shall write the proof for the aperiodic case d = 1. The generalization to a period d ≥ 2 is straightforward by considering d n instead of n throughout the proof of this Lemma. Let η n = x η n (x) be the total population at time n. Since 0 ≤ φ(x) ≤ 1 we have Λ n = e -r n x φ(x)η n (x) ≤ e -r n η n . Furthermore, a particle living at time n has to have an ancestor at location 0 at some time k ≤ n, and if N i is the number of children of this ancestor, then

η n ≤ 1≤i≤Γ n N i with Γ n = 0≤k≤n η k (0)
where the (N i ) i≥1 are independent random variables distributed as N and independent of Γ n . Since [N ] < +∞, by Borel Cantelli's Lemma, there exists i 0 = i 0 (ω) such that

N i ≤ i 2 for i ≥ i 0 .
Hence, almost surely for n large enough,

η n ≤ 1≤i≤i 0 N i + Γ 2 n ≤ 1≤i≤i 0 N i + n 2 ( sup 0≤k≤n η k (0)) 2
By Lemma 4.4, almost surely on the survival set , we have Λ ∞ > 0 and thus, for n large enough η n ≥ 1 2 Λ ∞ e r n and therefore for n large enough, on ,

sup 0≤k≤n η k (0) > e r n/4
Considering the stopping time (for the branching system endowed with the natural filtration) T := inf{n : η n (0) > e r n/4 }.

We have established that on , T < ∞ a.s. It follows from the branching property and (5.1) that

η n+T (0) ≤ c ′ e r n , ≤ η n (0) ≤ c ′ e r n e r n/4
≤ (1c ′ ) e r n/4 , whose sum on n converges. By Borel-Cantelli's lemma, on , a.s. for all large n,

η n (0) ≥ c ′ e r(n-T ) ≥ c ′ 2 e r n .
This proves the Lemma.

Proof of the lower bound of M

n . Let 0 < s < 1. Define k = k(n) := d⌊ sn d ⌋.
By the preceding Lemma, on the survival set , at time k, there are at least ⌊ c ′ 2 e r k ⌋ particles at 0, which move independently. Letting these particles move as the random walk S staying positive up to time nk, then M n is bigger than ⌊ c ′ 2 e r k ⌋ i.i.d. copies of S n-k with S 1 > 0, ..., S n-k > 0. By a large deviations estimate (Theorem 5.2.1 of Dembo and Zeitouni [START_REF] Dembo | Large deviations techniques and applications[END_REF], see the forthcoming Remark 3), for any fixed a ∈ (0, ∞),

S n-k > a(1 -s)n, S 1 > 0, ..., S n-k > 0 = e -(1-s)nψ * (a)+o(n) ,
where we denote as before,

ψ * (a) = sup θ >0 (aθ -ψ(θ )) .
It follows that

M n ≤ (1 -s)an, η k (0) ≥ c ′ 2 e r k ≤ 1 -(S n-k > a(1 -s)n, S 1 > 0, ..., S n-k > 0 ⌊ c ′ 2 e r k ⌋ = exp(-e rsn-ψ * (a)(1-s)n+o(n) ).
Choose (a, s) ∈ (0, +∞) × (0, 1) such that

rs > ψ * (a)(1 -s),
we apply Borel-Cantelli's lemma and get that a.s. for all large n, either M n > (1s)an or η k (0) < c ′ 2 e r k . Hence on the set , by Lemma 5.1, a.s.,

lim inf n→∞ M n n ≥ γ := sup{(1 -s)a : (a, s) ∈ (0, ∞) × (0, 1), rs > ψ * (a)(1 -s)}. (5.2) Recalling r = ψ(t 0 ), then γ = sup a>0 aψ(t 0 ) ψ * (a) + ψ(t 0 )
.

Let us study the derivative of a → aψ(t 0 ) ψ * (a)+ψ(t 0 ) . Recall that ψ * (a) = aθ (a)ψ(θ (a)) with a = ψ ′ (θ (a)), and (ψ * ) ′ (a) = θ (a). Since the derivative of a → aψ(t 0 ) I (a)+ψ(t 0 ) has the same sign as ψ * (a) + ψ(t 0 )a(ψ * ) ′ (a) = ψ(t 0 ) -ψ(θ (a)), it is negative if a > ψ ′ (t 0 ) (i.e. θ (a) > t 0 ), positive if a < ψ ′ (t 0 ) and vanishes at ψ ′ (t 0 ). Therefore

γ = ψ(t 0 ) t 0 = α,
which in view of (5.2) yields the lower bound of Theorem 1.1 under the hypothesis that (N 2 ) < ∞.

Remark 3. Mogulskii's theorem (Theorem 5.2.1 of Dembo and Zeitouni [START_REF] Dembo | Large deviations techniques and applications[END_REF]) implies that S j > a j, S 1 > 0, ..., S j > 0 = e -jK(a)+o( j) , with

K(a) = inf 1 0 ψ * ( ḟ (t)) d t, f ∈ , = φabsolutely continuous, f (0) = 0, f (1) = a, f (s) > 0∀s ∈ (0, 1) .
Let us check that K(a) = ψ * (a). In fact, since the function f (t) = at is in , we have K(a) ≤ 1 0 ψ * (a) d t = ψ * (a). On the other hand, the function ψ * is convex, therefore, by Jensen's inequality, if φ ∈ ,

1 0 ψ * ( ḟ (t)) d t ≥ ψ * 1 0 ḟ (t)d t = ψ * ( f (1) -f (0)) = ψ * (a) .
We can thus conclude that K(a) = ψ * (a).

Proof of the lower bound, without the hypothesis

(N 2 ) < ∞.
The proof relies on a coupling for the general N with mean m: Let N (L) := min(N , L) with a sufficiently large integer L such that m L := (N (L) ) satisfies m L (1q esc ) > 1 (this is possible since m L → m). Consider a new CBRW (X (L) u , |u| ≥ 0) with N (L) as the number of offsprings and the same random walk (S n ) as the displacements, i.e. on each step of branching at 0 we keep at most L-children and their displacements in the original CBRW. The associated maximum at generation n is denoted by M (L) n . Then by construction

M n ≥ M (L) n , a.s.
By the lower bound for M (L) n established in Section 5.2, if we denote by

L := ω : lim sup n→∞ {u : |u| = n, X (L) u = 0} = , then a.s. on L , lim inf n→∞ M (L) n n ≥ α L , with α L = ψ(t 0 (L))
t 0 (L) , and where t 0 (L) is defined in the same way as t 0 in (1.3) and (1.2) by replacing m by m L . We remark that by continuity such solution t 0 (L) exists for all sufficiently large L, say L ≥ L 0 . Moreover α L → α as L → ∞, and L ⊂ L+1 for any L ≥ 1. Then on the set := ∪ L≥1 L , a.s. Let us check (5.3) in the same way as in the proof of Lemma 4.4. Plainly ⊂ . To prove the reverse inclusion, we remark at first that by Lemma 4.4, L equals a.s. the non-zero set of the corresponding limit of the fundamental martingale (which is bounded in L 2 ), hence L = for all large L. Consequently = . Let t := ( c ) and t := ( c ). Then t ≤ t < 1. As in the proof of Lemma 4.4, by conditioning on the number of offsprings N , we obtain that

t = ∞ k=0 (N = k) k j=0 C j k q k-j esc (1 -q esc ) j ( t) j = f (q esc + t(1 -q esc )),
with f (x) = (x N ). The constant t satisfies the same equation as t and we have already proved in the proof of Lemma 4.4 the uniqueness of solutions in [0, 1). Hence t = t and (5.3) follows. This completes the proof of the lower bound in Theorem 1.1.

6 Refining the Convergence : proof of Theorem 1.2.

The key of the proof of Theorem 1.2 is the following double limit of Proposition 6.1. Then we shall prove its uniform version (uniformly on the starting point of the system) in Proposition 6.2, from which Theorem 1.2 follows easily (see Section 6.3). (1-e -t 0 ) (H 1 ) and (H 1 ) is given in equation (6.9). We also mention that we can not replace M n > αn + z by M n ≥ αn + z in the above Proposition, since M n is integer-valued.

The proof of Proposition 6.1 is divided into the upper and lower bounds, proved respectively in Section 6.1 and Section 6.2.

Upper bound in Proposition 6.1

Recall that α := For |u| = n with X u > αn+z > 0 (as n is large), there exists some k < n such that X u k = 0 and X u j > 0 for all k < j ≤ n. Therefore

M n > αn + z = 0≤k≤n-1 B k (6.1)
with

B k := |v|=k A v (k, n) , and
A v (k, n) := ∃|u| = n : v = u k , X v = 0, X u j > 0, ∀k < j ≤ n, X u n-k > αn + z .
Denote as before by η n (x) the number of particles at x at time n. Then, conditioning on k , B k is an union of η k (0) i.i.d. events, and each event holds with probability

p(k, n) := ∃|u| = n -k, X u 1 > 0, ..., X u n-k > 0, X u > αn + z .
It is easy to compute p(k, n): by conditioning on the number of offspring N = l, p(k, n) is the probability that among these l particles in the first generation there exists at least one particle which remains positive up to generation nk and lives in (αn + z, ∞) at (nk)-th generation. It follows that

p(k, n) = ∞ l=0 (N = l) 1 -(1 -q(k, n)) l = 1 -f (1 -q(k, n)), ( 6.2)
where f (x) := (x N ) is the generating function of N and q(k, n) is defined as follows:

q(k, n) := S 1 > 0, ..., S n-k > 0, S n-k > αn + z .
Let ǫ > 0 be small. By Proposition 4.3 (with d = 1), lim n→∞ e -r n η n (0

) = c 0 = 1 m . It follows that for any n > k ≥ k 0 ≡ k 0 (ǫ), B k ≤ η k (0)p(k, n) ≤ (c 0 + ǫ)e r k p(k, n).
Hence for any n > k 0 ,

M n > αn + z ≤ n-1 k=0 (B k ) ≤ (c 0 + ǫ) n-1 k=k 0 e r k p(k, n) + C k 0 k 0 -1 k=1 p(k, n), (6.3)
where C k 0 := max 1≤k≤k 0 (η k (0)). Recalling f ′ (1) = m and (6.2), we deduce from the convexity of f that for all k < n,

f ′ (1 -q(k, n)) q(k, n) ≤ p(k, n) ≤ m q(k, n). (6.4)
It is easy to see that the sum

k 0 -1 k=1 in (6.3) is negligible as n → ∞. In fact, for any 1 ≤ k ≤ k 0 , q(k, n) ≤ (S n-k-1 > αn + z). But (S 1 ) = ψ ′ (0) < α = ψ(t 0 ) t 0
by the (strict) convexity of ψ. Then p(k, n) ≤ mq(k, n) → 0 as n → ∞ (exponentially fast by the large deviation principle).

To estimate the probability q(k, n) for k 0 ≤ k < n, we introduce a new probability d d σ{S 0 ,...,S n } = e t 0 S n -nψ(t 0 ) .

Under , S 1 has the mean ψ ′ (t 0 ) > 0. Therefore for 1 ≤ k ≤ n and for all z ≥ 0,

q(k, n) = S 1 > 0, ..., S n-k > 0, S n-k > αn + z = e -t 0 S n-k +(n-k)ψ(t 0 ) 1 (S j >0,∀ j≤n-k,S n-k >αn+z) = e -r k e(t 0 ) ≥ S n-k -αn, S j > 0, ∀ j ≤ n -k, S n-k > αn + z ,
where e(t 0 ) denotes an independent exponential random variable with parameter t 0 and we also used the fact that α = ψ(t 0 ) t 0 and r = ψ(t 0 ). Plainly in the event of the above probability term, e(t 0 ) must be bigger than z. Thanks to the loss of memory property of e(t 0 ), we get that for 1 ≤ k ≤ n and for all z ≥ 0, e r k q(k, n) = e -t 0 z S j > 0, ∀ j ≤ nk, αn + z < S n-k ≤ αn + z + e(t 0 ) . (6.5) Summing (6.5) over 0 ≤ k ≤ n -1 and letting i = nk, we obtain that

n-1 k=0 e r k q(k, n) = e -t 0 z n i=1 S j > 0, ∀ j ≤ i, αn + z < S i ≤ αn + z + e(t 0 ) = e -t 0 z U(αn + z, αn + z + e(t 0 )] -s n , (6.6)
where for any x < y,

U( y) := ∞ k=1 S j > 0, ∀1 ≤ j ≤ k, S k ≤ y , U(x, y] := U( y) -U(x), ( 6.7)
and

s n := ∞ k=n S j > 0, ∀ j ≤ k, αn + z < S k ≤ αn + z + e(t 0 ) . (6.8)
Under , S j is a random walk with positive mean. Define by T 0 := 0, T j := inf{i > T j-1 : S i > S T j-1 } and H j := S T j for j ≥ 1. Then 0 < T 1 < ... < T j < ... and 0 < H 1 < • • • < H j < • • • are the strict ladder epochs and ladder heights of the random walk S (under ). The duality lemma says that for any y > 0,

U( y) = ∞ l=1 H l ≤ y . Since S 2 1 < +∞, (H 1 )
< ∞ and we have the Wald identity (see [START_REF]An introduction to probability theory and its applications[END_REF] Feller Volume II, Chapter XVIII, Theorem 1)

(H 1 ) = (S 1 ) (T 1 ) . (6.9)
We are going to apply the renewal theorem (see [START_REF]An introduction to probability theory and its applications[END_REF] Feller, pp. 360) to U and prove that there exists some constant c H > 0 such that lim x→∞ e -t 0 {x} U(x, x + e(t 0 )] = c H . (6.10)

To check (6.10), we remark that the span of H 1 equals 1 (because S is aperiodic). By the renewal theorem, for any j ≥ 1, U(x, x + j] → j (H 1 ) as x → ∞. Moreover there exists some constant C > 0 such that for all y > x ≥ 0, U(x, y] ≤ C(1 + yx). Let x > 0. Observe that almost surely,

U(x, x + e(t 0 )] = U(⌊x⌋, x + e(t 0 )] = ∞ j=1 1 ( j<{x}+e(t 0 )< j+1) U(⌊x⌋, ⌊x⌋ + j].
Taking expectation gives that

U(x, x + e(t 0 )] = ∞ j=1 e -t 0 ( j-{x}) (1 -e -t 0 )U(⌊x⌋, ⌊x⌋ + j],
which proves (6.10) after an application of the dominated convergence theorem, with

c H := ∞ j=1 e -t 0 j (1 -e -t 0 ) j (H 1 ) = e -2t 0 (1 -e -t 0 ) (H 1 ) . ( 6.11)
Now we prove that s n → 0, where s n is defined in (6.8). Remark that E(S 1 ) = ψ ′ (t 0 ) > α := ψ(t 0 ) t 0 by convexity. Pick up some small positive constant δ < (ψ ′ (t 0 ) -α)/2. There exists some sufficiently small constant b ∈ (0, t 0 ) such that e -bS 1 ≤ e -b(ψ ′ (t 0 )-δ) . Then by Chebychev's inequality, for any t > 0 and k ≥ n, (S k ≤ z + αn + t) ≤ e bz+bt e bαn e -bS k ≤ e bz+bt e -δbk . It follows that

s n ≤ ∞ k=n
e bz e -δbk (e be(t 0 ) ) = t 0

(1e -δb )(t 0b) e bz e -δbn .

In particular s n → 0. This together with (6.10), (6.6) yields that for any z ≥ 0, lim n→∞ e -t 0 {αn+z} n-1 k=0 e r k q(k, n) = e -t 0 z c H . (6.12)

Now by using the lower bound of (6.4) and (6.5), for any k < n and z ≥ 0, k+1) e -t 0 z ≤ e -r k . Then for any small δ > 0, there exists some

p(k, n) ≥ f ′ (1 -e -r k )q(k, n) because q(k, n) ≤ e -r(
k 0 (δ) such that f ′ (1 -e -r k ) ≥ m(1 -δ) for all k ≥ k 0 (recalling f ′ (1) = m). It follows that for any k 0 ≤ k < n and z ≥ 0, p(k, n) ≥ (1 -δ)mq(k, n). On the other hand, p(k, n) ≤ mq(k, n) for any k < n,
and lim n→∞ k 0 k=0 q(k, n) = 0. This in view of (6.12) implies that for any z ≥ 0, lim n→∞ e -t 0 {αn+z} n-1 k=0 e r k p(k, n) = me -t 0 z c H . (6.13)

Applying the above limit to (6.3) gives that for any z ≥ 0, lim sup n→∞ e -t 0 {αn+z} (M n > αn + z) ≤ (c 0 + ǫ) m c H e -t 0 z , which implies the upper bound in Proposition 6.1 by letting ǫ → 0 and

c * := c 0 mc H = c 0 m e -2t 0 (1 -e -t 0 ) (H 1 ) = e -2t 0 (1 -e -t 0 ) (H 1 ) , (6.14)
since c 0 = 1/m (the period d = 1) as stated in Proposition 4.1.

Remark 4. Let us mention an uniform estimate: for some constant C > 0,

(M n > αn + z) ≤ C e -t 0 z , ∀z ∈ , n ≥ 1. (6.15)
In fact, there exists some constant C ′ > 0 such that (η k (0)) ≤ C ′ e r k for any k ≥ 1, hence by the first inequality in (6.3),

(M n > αn+z) ≤ C ′ n-1 k=0 e r k p(k, n) ≤ C ′ m n-1
k=0 e r k q(k, n). Using (6.6) and the fact that ∃C ′′ > 0: U(x, y] ≤ C ′′ (1 + yx) for all x < y, we immediately get (6.15).

Remark 5. If the underlying random walk S is of period d ≥ 2, then in (6.1),

B k = if k is not multiple of d (namely if d | k).
Then instead of n-1 k=0 e r k p(k, n), we have to deal with n-1 k=0,d|k e r k p(k, n), which in turn leads to the study of n k=0,d|k e r k q(k, n). An equality similar to (6.6) holds with U replaced by

U (d,ℓ) ( y) := ∞ k=0 S j > 0, ∀1 ≤ j ≤ kd + ℓ, S kd+ℓ ≤ y ,
where ℓ ∈ {0, ..., d -1} comes from the rest of division of n by d [ℓ being fixed and we let n → ∞ with n-1 ≡ ℓ(mod d)]. Technically we are not able to prove any renewal theorem for U (d,ℓ) ( y) for a general random walk S.

In the particular case when S is a nearest neighbor random walk on , we can use parity to handle U (d,ℓ) ( y). Considering for instance ℓ = 0 (d = 2). Thanks to parity, we have that for any k ≥ 1 and y > 0,

S j > 0, ∀1 ≤ j ≤ 2k, S 2k ≤ y = S 2 j > 0, ∀1 ≤ j ≤ k, S 2k ≤ y ,
which implies that U (2,0) ( y) is the renewal function for the random walk (S 2n ) n≥0 (under ). Then we can apply the standard renewal theorem to U (2,0) ( y). The term U (2,1) ( y) can be dealt with in the same way. Then we get a result similar to Proposition 6.1 and the forthcoming Proposition 6.2, and finally a modified version of Theorem 1.2 for the nearest neighbor random walk. The details are omitted.

Lower bound in Proposition 6.1

Let ǫ > 0 be small. Let λ ≡ λ(ǫ) be a large constant whose value will be determined later on. Recall (6.1). Consider

E n := n-1 k=0 B ′ k , with B ′ k := B k ∩{η k (0) ≤ λe r k } := B k ∩ F k .
Then by Cauchy-Schwarz' inequality,

M n > αn + z ≥ E n ≥ 0≤k<n (B ′ k ) 2 0≤k 1 ,k 2 <n (B ′ k 1 ∩ B ′ k 2
) . (6.16)

Conditioning on k , B k is an union of η k (0) i.i.d. events,

B k k = 1 -(1 -p(k, n)) η k (0) .
Let 0 ≤ k < n. By (6.4) and (6.5), p(k, n) ≤ me -r(k+1) e -t 0 z . On F k , η k (0) ≤ λ e r k hence p(k, n)η k (0) ≤ e -r-t 0 z mλ. Therefore for all z ≥ z 0 (λ, ǫ) and for all k < n,

1 -(1 -p(k, n)) η k (0) ≥ (1 -ǫ)p(k, n)η k (0) hence B ′ k k ≥ (1 -ǫ)p(k, n)η k (0)1 F k .
In particular,

n-1 k=0 (B ′ k ) ≥ (1 -ǫ) n-1 k=0 p(k, n) η k (0)1 F k .
Since η k (0)e -r k is bounded in L 2 , we deduce from Proposition 4.3 that we can choose (and then fix) some large λ and some k 0 ≡ k 0 (ǫ) such that

η k (0)1 F k ≥ c 0 (1 -ǫ)e r k for all k ≥ k 0 . It follows that for all n > k 0 , n-1 k=0 (B ′ k ) ≥ c 0 (1 -ǫ) 2 n-1 k=k 0 e r k p(k, n).
Consequently, for all z ≥ z 0 there exists some n 0 (z, ǫ) such that for all n ≥ n 0 ,

n-1 k=0 (B ′ k ) ≥ c 0 m (1 -ǫ) 3 n-1
k=k 0 e r k q(k, n) ≥ c * (1 -ǫ) 4 e -t 0 z e t 0 {αn+z} , (6.17) by applying (6.12) [recalling c * = c 0 mc H , c 0 = 1/m and that for any fixed k q(k, n) → 0 as n → ∞]. The probability (B k ) has already been estimated in the proof of upper bound of Proposition 6.1, see (6.3) and (6.13): for all z ≥ 0 and n ≥ n 0 (z, ǫ),

n-1 k=1 (B ′ k ) ≤ n-1 k=1 (B k ) ≤ c * (1 + ǫ)e -t 0 z e t 0 {αn+z} . (6.18)
Now we estimate the denominator in (6.16). Let k 1 < k 2 . On B k 1 ∩ B k 2 , there are at least two different v = v ′ at generation k 1 such that A v (k 1 , n) holds and for v ′ , there exists some descendant u (denoted by u > v ′ ) at generation k 2 such that A u (k 2 , n) holds. Then,

B k 1 ∩ B k 2 ⊂ v =v ′ ,|v|=|v ′ |=k 1 A v (k 1 , n) ∩ {∃|u| = k 2 , u > v ′ : A u (k 2 , n) holds} .
Since different particles branch independently, we get that

B k 1 ∩ B k 2 k 1 ≤ v =v ′ ,|v|=|v ′ |=k 1 p(k 1 , n) |u|=k 2 ,u>v ′ p(k 2 , n) k 1 .
Taking the expectations, we obtain that for k 1 < k 2 ,

B k 1 ∩ B k 2 ≤ p(k 1 , n)p(k 2 , n) (η k 1 (0)η k 2 (0)) ≤ C p(k 1 , n)p(k 2 , n)e r(k 1 +k 2 )
, by Corollary 4.2. Therefore for all z ≥ z 0 and n > n 0 (z, ǫ),

0≤k 1 ,k 2 <n B ′ k 1 ∩ B ′ k 2 ≤ n-1 k=0 (B ′ k ) + C n-1 k=0 e r k p(k, n) 2 ≤ c * (1 + ǫ)e -t 0 z e t 0 {αn+z} + C ′ e -2t 0 z ,
for some numerical constant C ′ > 0. In view of (6.16), we have that for all z ≥ z 0 and n > n 0 (z, ǫ),

M n > αn + z ≥ c 2 * (1 -ǫ) 8 e -t 0 z e t 0 {αn+z} c * (1 + ǫ) + C ′ e -t 0 z .
It follows that lim inf z→∞ lim inf n→∞ e t 0 z-t 0 {αn+z} M n > αn + z ≥ c * (1 -ǫ) 8 1 + ǫ .

Letting ǫ → 0, we obtain the lower bound in Proposition 6.1. The proof of Proposition 6.1 is complete.

Recall that φ(x) is defined in (4.1) and φ(x) > 0 thanks to the aperiodicity. Let us establish an uniform version of Proposition 6.1: Proposition 6.2. Under the assumptions in Theorem 1.2. Uniformly on x ∈ , lim sup n→∞ e t 0 z e -t 0 {αn+z} φ(x)

x M n > αn + z -c * → 0 , as z → ∞.
Proof. Assume x = 0 and let S * = max 0≤i≤τ S i , where τ is the first return time to 0. Then

x M n > αn + z ≤ x (S * > αn + z) + n k=1 x (τ = k) M n-k > αn + z .
Let ǫ > 0 be small (in particular ǫ < c * ). Let ℓ be some integer whose value will be fixed later on. By Proposition 6.1, there exists some y 0 (ǫ) > 0 such that for all y ≥ y 0 (ǫ), there exists some j 0 ( y, ǫ) such that for all j ≥ j 0 ( y, ǫ), e t 0 y e -t 0 {α j+ y} M j > α j + yc * < ǫ. (6.19) Observe that for any k

< n, M n-k > αn + z = M n-k > α(n -k) + z +
αk . We shall apply (6.19) to y = αk + z and j = nk. Then for all z ≥ y 0 (ǫ), there exists some j 1 (z, ℓ) such that for all 1 ≤ k ≤ ℓ and n ≥ j 1 (z, ℓ),

e t 0 (z+αk) e -t 0 {αn+z} M n-k > αn + z -c * < ǫ. (6.20)
We stress that y 0 (ǫ) does not depend on ℓ. Then for all n > j 1 (z, ℓ),

ℓ k=1 x (τ = k) M n-k > αn + z ≤ (c * + ǫ)e -t 0 z e t 0 {αn+z} ℓ k=1 x (τ = k)e -αt 0 k ≤ (c * + ǫ)e -t 0 z e t 0 {αn+z} φ(x),
since αt 0 = ψ(t 0 ) = r and φ(x) = x e -rτ . For k > ℓ, we apply (6.15) and get that For the lower bound, we have from (6.20) that for any z ≥ y 0 (ǫ) and all n > j 1 (z, ℓ),

ℓ k=1 x (τ = k) M n-k > αn + z ≥ (c * -ǫ)e -t 0 z e t 0 {αn+z} ℓ k=1 x (τ = k)e -αt 0 k
= (c * -ǫ)e -t 0 z e t 0 {αn+z} x e -rτ 1 (τ≤ℓ) .

Hence for any z ≥ y 0 (ǫ) and any x ∈ , lim inf n→∞ e t 0 z e -t 0 {αn+z} x M n > αn + z ≥ (c * -ǫ) x e -rτ 1 (τ≤ℓ) .

Letting ℓ → ∞ the above lim inf inequality and in (6.21) gives that for any z ≥ y 0 (ǫ) and uniformly for all x ∈ , lim sup n→∞ e t 0 z e -t 0 {αn+z}

x M n > αn + zc * φ(x) ≤ ǫφ(x), (6.22) proving Proposition 6.2 since ǫ can be arbitrarily small.

Proof of Theorem 1.2

The part (1.5) of Theorem 1.2 was already proved in Lemma 4.4. We now prove (1.4). Let ǫ, δ > 0 be small. For any k ≥ 1, there exists some integer

ℓ k = ℓ k (ǫ) such that max |u|=k |X u | ≤ ℓ k ≥ 1 -ǫ.
Recalling the martingale Λ n defined in Proposition 4.1. Since a.s. Λ n → Λ ∞ , there exists some

k 1 = k 1 (ǫ, δ) such that for any k ≥ k 1 , (1 -δ)Λ ∞ ≤ Λ k ≤ (1 + δ)Λ ∞ ≥ 1 -ǫ.
By (6.22), there exists some z 0 (δ) such that for all z ≥ z 0 (δ) and for all x ∈ , there exists some n 0 (z, x, δ) such that for all j ≥ n 0 (z, x, δ), e t 0 z e -t 0 {α j+z} x M j > α j + zc * φ(x) ≤ δφ(x). (6.23)

Elementarily there exists some s 0 (δ) > 0 such that 1s ≥ e -(1+δ)s for all 0 ≤ s < s 0 (δ). Let k 2 = k 2 (δ, y) be some integer satisfying (c * + δ)e -t 0 (αk 2 + y-1) < s 0 (δ). Define k :

= k 1 + k 2 + ⌊ z 0 (δ) α ⌋ + 1. Let n 1 := max x∈ ,|x|≤ℓ k n 0 (z, x, δ) + k. Considering n ≥ n 1 .
Conditioning on k and on the set {max |u|=k |X u | ≤ ℓ k }, the particles in the k-th generation move independently, hence for any n > n 1 ,

M n > αn + y k = 1 - x∈ ,|x|≤Lk x M n-k ≤ αn + y η k (x) . ( 6.24)
Applying (6.23) to j = nk, z = αk + y yields that for any |x| ≤ ℓ k (and x ∈ ), (c * -δ)φ(x)e -t 0 (αk+ y)+t 0 {αn+ y} ≤ x M n-k > αn+ y ≤ (c * +δ)φ(x)e -t 0 (αk+ y)+t 0 {αn+ y} .

Since 1 s ≥ e -(1+δ)s for all 0 ≤ s < s 0 (δ), we deduce from (6.24) that on the set

{max |u|=k |X u | ≤ ℓ k }, M n > αn + y k ≤ 1 -exp - x∈ ,|x|≤ℓ k (c * + δ)(1 + δ)φ(x)η k (x)e -t 0 (αk+ y) e t 0 {αn+ y} = 1 -exp -(c * + δ)(1 + δ)Λ k e -t 0 y e t 0 {αn+ y} . (6.25)
Then by taking the expectation, we get

M n > αn + y ≤ 1 -exp -(c * + δ)(1 + δ)Λ k e -t 0 y e t 0 {αn+ y} + max |u|=k |X u | > ℓ k ≤ 1 -exp -(c * + δ)(1 + δ) 2 Λ ∞ e -t 0 y e t 0 {αn+ y} + 2ǫ,
where the factor 2 in 2ǫ comes from Λ k which is replaced by (1 + δ)Λ ∞ . Since ǫ and δ can be arbitrarily small, we get the upper bound in (1.4). The lower bound in (1.4) can be proved in the same way. Finally, let y ∈ . Observe that for any n j ≥ 1, (M n j -⌊αn j ⌋ ≥ y + 1) = (M n j -⌊αn j ⌋ > y + {αn j }) = (M n j -αn j > y). We apply (1.4) to y and y -1, (1.6) follows immediately. This completes the proof of Theorem 1.2. .

Extension to multiple catalysts branching random walk (MCBRW)

Recall Section 3.1 for the definition of MCBRW. Let us assume that the set of catalysts is a finite subset of . By forgetting/erasing the time spent between the catalysts, we obtain an underlying Galton-Watson process which is multitype with the moment matrix M x y := mean number of particles born at x that reach site y

= m 1 (x) x τ = τ y , τ < ∞ (x, y ∈ ) ,
where m 1 (x) = N x is the mean offspring at site x, τ y := inf n ≥ 1 : S n = y is the first return time at y, and τ = τ = inf y∈ τ y is the first return time to .

We assume to be the supercritical regime, that is ρ > 1, where ρ is the maximal eigenvalue of matrix M , which by assumption is irreducible. We let ρ (r) be the maximum eigenvalue of the matrix M (r) x y := m 1 (x) x e -rτ 1 (τ=τ y ,τ<∞) (x, y ∈ ).

The function r → ρ (r) is continuous, strictly decreasing, C ∞ on (0, +∞), ρ (0) = ρ > 1 and lim r→+∞ ρ (r) = 0 since M (r) x y ≤ m 1 (x)e -r . Therefore there exists a unique r > 0, a Malthusian parameter, such that ρ (r) = 1. We shall fix this value of r in the sequel.

Let v = v (r) be a right eigenvector of M (r) associated to ρ (r) = 1: For any x ∈ , v(x) > 0 and Proof. The proof is similar to that of Proposition 4.1 by using the Markov property of the random walk. The details are omitted.

We are now ready to introduce the fundamental martingale. (2) For random walk, the process

∆ n := e -r n φ(S n ) x∈ m 1 (x) L x n-1
is a martingale where L x n-1 = 0≤k≤n-1 1 (S k =x ) is the local time at level x at time n -1.

(3) If N has finite variance, then the process Λ n is bounded in L 2 and therefore a uniformly integrable martingale.

Proof. Based on the many-to-one formula, the parts (1) and ( 2) can be proved in the same way as in Proposition 4.1. Let us only give the details of the proof of (3). To compute the second moment, we use the many to two formula (3.5) of Section 3

Λ 2 n = e -2r n    |u|=|v|=n φ(X u )φ(X v )    = e -2r n   φ(S 1 n )φ(S 2 n ) 0≤k<T ∧n m 2 (S 1 k ) T ∧n≤k<n m 1 (S 1 k )m 1 (S 2 k )   .
Recall (3.6). We have that Observe that since 0 ≤ φ ≤ 1 we have 0 ≤ φ(x) 2 ≤ φ(x), and combine it with

x ∆ p = φ(x)m 1 (x) ≤ C and m 1 (x) m 2 (x) ≤ 1 to obtain the upper bound

Λ 2 n ≤ 1 + C 2 1≤k≤n-1 e -2r(k-1) ≤ C ′ < ∞,
which completes the proof of this Lemma.

We are now able to give an explanation of the supercritical regime assumption of the introduction. Lemma 7.3. When there is only one catalyst at the origin, the supercritical regime is m(1q esc ) > 1.

Proof. Indeed, M is a one dimensional matrix and ρ = M 00 = m (τ < +∞) = m(1q esc ) .

We end this section by stating the law of large numbers. Intuitively, if c is the rightmost catalyst, the maximal position at time n comes from particles born at location c. Proof. First observe that the heuristics do not change at all since by applying the optional stopping theorem to the martingale e t 0 S n -nr to the time T , we obtain that for x > c e t 0 x = e t 0 c x e -r T , and thus φ(x) = v(c) x e -r T c = v(c)e t 0 (x-c) , and we approximate the expected number of particles above level an in the same way, and hence obtain the same guess for the asymptotics.

Furthermore, the proofs are mutatis mutandis the same as the one given in section 5. The only difference would come from the use of renewal theorems: we get a system of renewal equations, e.g., for ( a (e θ S n b∈ m 1 (b) L b n-1 )) a,b∈ as n → ∞, which can be dealt with an application of a matrix version of renewal theorems (see [START_REF] Crump | On systems of renewal equations[END_REF][START_REF] De Saporta | Renewal theorem for a system of renewal equations[END_REF]). We feel free to omit the details.

1 2 .

 2 For supercritical BRW (m > 1), almost surely on the set of non extinction lim n→+∞ M (br w) n n = b, where b is the unique solution of ψ * (b) = log m, with ψ * (b) := sup t (bt -ψ(t)

Lemma 5 . 1 .

 51 d n η d n (0) = c 0 , sup n e -2r d n η d n (0) 2 < +∞ .Therefore Paley-Zygmund's inequality entails thatη d n (0) ≥ c ′ e r d n ≥ c ′ , (5.1)for some constant c ′ > 0. The following lemma aims at describing the a.s. behavior of η n (0): Under (1.1) and (1.3). Almost surely on ,

  This will yield the lower bound in Theorem 1.1 once we have checked the equality:

Proposition 6 . 1 .

 61 Under the assumptions in Theorem 1.2, there exists a positive constant c * > 0 such that lim sup z→∞ lim sup n→∞ e t 0 z e -t 0 {αn+z} M n > αn + zc * = 0, where as before α := ψ(t 0 ) t 0 and {αn + z} ∈ [0, 1) denotes the fractional part of αn + z. The value of c * is given in (6.14) by c * = e -t 0

  of M n . We prove the following upper bound: for all z ∈ , lim sup n→∞ e -t 0 {αn+z} M n > αn + z ≤ c * e -t 0 z . Let us start from M n > αn + z = ∃|u| = n : X u > αn + z . For any n ≥ 1 and any |u| = n, denote by u 0 = < u 1 < ... < u n = u the shortest path relating to u such that |u k | = k for any k ≤ n.

  = k) M n-k > αn + z ≤ n k=ℓ C e -t 0 (αk+z) = C e -t 0 z e -rℓ r . It follows that for any z ≥ y 0 (ǫ) and any x ∈ , lim sup n→∞ e t 0 z e -t 0 {αn+z} x M n > αn + z ≤ (c * + ǫ)φ(x) + C e 1-rℓ r . (6.21)

Lemma 7 . 1 . 1 m 1

 7111 v(x) = a∈ m 1 (x) x e -rτ 1 (τ=τ a ,τ<∞) v(a) (x ∈ ) .Let us denote by p(x, y) = x S 1 = y and P f (x) = y p(x, y) f ( y) the random walk kernel and semigroup. Let us consider the hitting timesT x := inf n ≥ 0 : S n = x , T = inf x∈ T x = inf n ≥ 0 : S n ∈ . The function φ(x) := a∈ v(a) x e -r T 1 (T =T a ,T <∞)is a solution of Pφ(x) = e r φ(x) (x) 1 (x∈ ) + 1 (x / ∈ ) .

Lemma 7 . 2 . ( 1 )

 721 For the CBRW process with multiple catalysts, the processΛ n := e -r n |u|=n φ(X u ) is a martingale.

m 1 ( 2 e

 12 S l ) m 2 (S l ) (1 -m 1 (S k-1 ) m 2 (S k-1 ) ) S k-1 ∆ n-(k-1) 2r(n-(k-1))   .

Proposition 7 . 4 (

 74 Law of large numbers). Assume the supercritical regime and (1.3). Then, on the set of non extinctionwe have Malthusian parameter defined by ρ (r) = 1 and t 0 > 0 such that ψ(t 0 ) = r.

Acknowledgements. We are very grateful to two anonymous referees for their careful readings and helpful comments on the first version of this paper.

Supported by Grant ANR-2010-BLAN-0108 supported by ANR 2010 BLAN 0125


Nhan T Nguyen

Minh Ha-Duong

Sandra Greiner

Michael Mehling

 Nhan
email: nhan@centre-cired.fr.

T Nguyen

Improving the Clean Development Mechanism Post-2012: A Developing Country Perspective

In this article, we assess the future prospects of the Clean Development Mechanism (CDM) from the perspective of a developing country, drawing on Vietnam as a case study. First, we review the performance of the CDM and describe the evolution of carbon markets on the path towards a post-2012 climate regime. Next, we place Vietnam in a post-2012 context, and assess potential project resources, challenges, and opportunities that could arise for the country from a future climate policy framework. Our analysis suggests that the CDM should remain in place and be improved to facilitate more meaningful participation by developing countries in climate mitigation efforts beyond 2012. Finally, the article sets out eight proposals that could help improve the CDM as the world progresses towards a new international climate policy framework.

I. Introduction

As one of the flexible mechanisms created under the Kyoto Protocol, 1 the Clean Development Mechanism (CDM) allows developed countries to co-finance projects realized in developing countries in exchange for certificates of greenhouse gas emission reductions. Identifying a future for this mechanism has become an urgent matter for international climate negotiations, given that the first commitment period of the Kyoto Protocol expires at the end of 2012. Also, the CDM remains the only established instrument allowing an active role for the developing world in mitigation activities. In recent years, this mechanism has attracted a lot of criticism, but has also seen a variety of proposals to improve its effectiveness. Both are discussed in this text from a developing country's perspective, based on the case of Vietnam.

Accordingly, Section II reviews the past performance and future prospects of the CDM in the context of global carbon markets. Section III reviews the performance of the CDM and its prospects from the point of view of Vietnam. It shows that, while there is a large potential for mitigation projects in the country, to date the mechanism has not been used as much as might be expected, suggesting that numerous opportunities could be created through a new international climate regime. Based on this analysis, Section IV offers eight suggestions to improve the CDM that would allow it to continue expanding post-2012 as a major instrument to finance climate mitigation activities in developing countries. Section V summarises the main conclusions.

II. The CDM and the Carbon Market

Performance of the CDM

The Clean Development Mechanism (CDM) is a flexible mechanism created by the Kyoto Protocol to the United Nations Framework Convention on Climate Change (UNFCCC). 2 A central objective of the CDM is to lower the cost of compliance with emissions reductions commitments entered by developed countries (Annex I countries, as defined in the UNFCCC); but it equally seeks to increase the level of foreign investment, promote the transfer of advanced climate-friendly technologies, and ensure the sustainable development of developing (or non-Annex I) countries.

Figure 1 shows that, from 2005 to 2010, the CDM has experienced strong and continuous growth, as expressed by the number of projects registered or in the pipeline. The first project was registered on 16 February 2005; by 14 April 2008, or three years later, the 1.000 th project had been registered. Less than two years after that, the 2.000 th registered project milestone was passed, with 2.067 registered projects registered as of since as of 1 March 2010. 3 More than 1.7 billion Certified Emission Reductions (CERs) are expected from these projects and projects in the pipeline by the end of 2012. As each CER represents one ton of CO 2 not emitted into the atmosphere, this quantity is not negligible. For instance, it is roughly equivalent to the amount emitted in one year by Japan and France together.

By early 2010, over 4.200 CDM projects were in the pipeline. Yet, the relative stagnation in the second half of 2009 apparent in Figure 1 suggests that the window of opportunity offered by Kyoto's first commitment period is closing. As one group of authors described it, the CDM has experienced a "gold rush" period between late 2005 and late 2008, with more than a hundred CDM project design documents entering the validation process each month. 4 The distribution of projects by economic sector and by country is not uniform. In January 2010, about 60% of registered projects lay in the energy industries (renewable and nonrenewable), while the transport and construction sectors were virtually absent. Over 75% of registered projects are currently being carried out in Asia. China alone represents 36% of projects, followed by India with 24% and Brazil with 8%. A limited number of projects are being implemented in other developing countries, and even fewer projects are located in leastdeveloped countries (LDCs). According to Axel Michaelowa and Benito Müller, credits issued up to the end of May 2009 indicate a very good performance of the CDM. [START_REF] Michaelowa | [END_REF] Three types of project, including projects to destroy the industrial gases N 2 O and HFC-23, outperformed expectations.

Still, the CDM has also been subject to ample criticism. Several authors have noted that industrial gas destruction projects receive high windfall profits because these gases have a higher global warming potential than CO 2 and relatively low abatement costs. [START_REF] Wara | A Realistic Policy on International Carbon Offsets[END_REF] Others have pointed out that high windfall profits prevent resources from being used more effectively elsewhere. [START_REF] Stefan | Differentiation in the CDM: Options and Impacts[END_REF] At the same time, the environmental effectiveness of the CDM has been questioned because it does not reduce emissions itself, but offsets the increase in emissions elsewhere. Whether the emissions reduced by a CDM project are "additional" to mitigation measures the host country would have implemented in the absence of the project is an issue of ongoing controversy. [START_REF] Michaelowa | Additionality Determination of Indian CDM Projects: Can Indian CDM Project Developers Outwit the CDM Executive Board?[END_REF] Concerns have also been voiced about administrative and institutional constraints linked to the CDM project cycle, such as lengthy and expensive approval procedures, or the need for complex methodologies determining whether projects actually reduce greenhouse gas emissions, and about the role of the CDM Executive Board (CDM EB) and Designated Operational Entities (DOEs). [START_REF] Streck | The Governance of the Clean Development Mechanism: The Case for Strength and Stability[END_REF] A number of improvements and reform options have been put forward to improve the CDM and scale it up for the expected needs under an international climate regime beyond 2012. Such reforms commonly focus on expanding the already existing programmatic CDM, creating a new sectoral CDM, or exploring options for a policy-based CDM. Some of these options form part of an ongoing improvement process, while others are currently still in the negotiation stage.

Copenhagen and the Future of the CDM

When parties to the UNFCCC met in Copenhagen in December 2009 for the 15 th Session of the Conference of the Parties (COP15), they failed to define the details of a new global climate regime. Developing countries, notably large emerging countries, refused to enter legally binding commitment on emissions reductions for the short to medium-term, although they agreed to implement necessary mitigation actions. Additionally, they insisted that climate mitigation actions only be measured, reported, and verified to the extent that developed countries comply with their promises of financial resources for mitigation, adaptation, technology transfer and capacity building.

Many developing countries attended COP15 with expectations for an extensive reform of the CDM or adoption of new, complementary mechanisms to better support implementation of their mitigation and adaptation efforts. And indeed, considerable progress was made with a number of technical documents, including a decision on various improvements to the CDM titled "Further Guidance Relating to the Clean Development Mechanism". 10 Under this decision, the CDM Executive Board will have the ability to streamline the procedures governing registration and CER issuance for CDM projects, and provide new funding to accelerate the development of CDM projects in countries with fewer than 10 approved CDM projects in operation. Lex de Jonge, chairman of the CDM Executive Board, predicted that the reforms would serve to "enhance the efficiency of the mechanism, expand its reach, and maintain its environmental integrity." 11 But for a variety of environmental and political reasons, negotiations on future commitment periods of the Kyoto Protocol stalled at the Copenhagen climate summit, an impasse that also affects the further role of the Kyoto flexible mechanisms. Currently, the future of the Kyoto Protocol remains unclear beyond 2012, leaving the global carbon markets, including the CDM offsetting market, in a state of uncertainty.

Carbon Markets in a Post-2012 Climate Regime

At the moment, demand and supply dynamics for CERs post-2012 depend on various factors. On the supply side, these are: the fate of CERs issued if Kyoto is or is not extended; constraints on the development of new projects due to an issuance bottleneck; and new projects entering the pipeline. On the demand side, they include: the demand for CERs from the European Union beyond 2012; the introduction of mandatory cap-and-trade systems in other developed countries, including Japan and the United States, and the extent to which these allow for compliance through use of CERs or similar credits; the introduction of new project categories, such as carbon capture and storage (CCS) or reduced emissions from The market for CERs beyond 2012 is vitally linked to the future architecture of the broader carbon market. It is unclear whether and how the CDM or a CDM-like mechanism will be included in the post-2012 regime, and, if it is, what the demand for and supply of credits will be. Despite significant uncertainties at the international level, regional and domestic initiatives continue unfolding in a number of jurisdictions. In many cases, these will continue to operate independently of the progress -or lack thereof -with negotiations for a post-2012 climate agreement.

Currently, the biggest driver for the global carbon market is the European Emission Trading Scheme (EU ETS), which caps emissions of the largest point sources in Europe. It is followed in size by the CDM market, consisting of the primary, secondary, and options market for CERs. 12 The latter is closely interlinked with the EU ETS: prices for CERs are highly correlated with the prices of European Union Allowances (EUA), given that CERs are a fungible compliance unit for EU ETS participants. Given the current uncertainties at the international level and excess supply in the market for Assigned Amount Units (AAUs) in the international emissions trading system created by the Kyoto Protocol, this fungibility of CERs under the EU ETS has been considered strategic for the CDM. Importantly, the EU ETS has already been extended beyond 2012 with a 3 rd Phase (2013-2020), regardless of the fate of the international climate regime post-2012. This provides an important message to the developing world that the EU ETS, a key carbon market in the first Kyoto commitment period, will continue serving as the main driver for emissions reductions within Europe.

Carbon markets are also emerging in the United States at the regional level, and may eventually also be introduced at the federal level. Under the rules framing these markets, international credits, such as EUAs and CERs, might be eligible for compliance purposes under specified conditions. Other countries, such as Japan, Australia and New Zealand, are also actively interested in carbon markets. Over time, such national and regional markets may converge to form a global carbon market from the bottom-up, through linkages across carbon markets, both North-North and North-South. 13 Carbon markets are and will remain politically driven, as supply and demand for credits are determined to a substantial degree by political decisions. However, economic forces are a strong underlying driver of policy decisions. Current trends in the growth of increasingly integrated carbon markets may lead to a global reference price for CO 2 emissions by 2020. 14 Observers have argued that a global price for CO 2 would benefit low-income countries; 15 they would profit from a wider range of carbon-reducing technologies, and have opportunities for "leapfrogging" beyond the technologies already installed in high-income countries.

III. Vietnam: A Case Study

Vietnam as a CDM Project Host: A Large and Untapped Potential

Vietnam has significant potential for the implementation of CDM projects. 16 Most of this potential lies in the energy sector: between 1996 and 2007, the demand for electricity increased by more than 14% each year. It is expected to continue growing at a faster pace than GDP from 2010 to 2030, rising between 15% and 18% per year. In order to satisfy this rapid growth in demand, the Vietnamese government is expanding generation capacity mainly through construction of coal-fired plants. In the baseline case, simulations by Nguyen and Ha-Duong suggest that about 70 GW of power generation based on coal-fired plants could be installed by 2030 in Vietnam. [START_REF] Nguyen | CO2 Emissions Mitigation Potential in Vietnam's Power Sector[END_REF] Yet many cleaner development options exist:  Vietnam is endowed with an abundance of natural resources and geophysical conditions that can be leveraged to generate significant amounts of renewable energy, including hydropower, wind, geothermal, sun, biogas and biomass, [START_REF]Mainly in the form of residues from sugar, rice, agriculture, and wood. 19 Mainly waste from landfills, animal farms, tapioca starch[END_REF] and various forms of waste. 19  Fuel switching in a number of sectors [START_REF]Notably electricity generation, beverage, iron, steel, cement[END_REF] and improved energy efficiency in both industry and buildings could offer great opportunities for hosting CDM projects. [START_REF] Manh | Main CDM Activities in Vietnam[END_REF]  Vietnam has large onshore and offshore sedimentary basins that could provide significant potential for storing CO 2 . The country is expected to have sufficient capacity to store approximately 20 to 60 Gt of CO 2 emissions. [START_REF] Bonijoly | Potential for Capturing and Storing CO2 Emissions in Vietnam: Where is the Potential?[END_REF] [START_REF]Designing a REDD compliant benefit distribution system for Vietnam: Executive summary[END_REF] This potential is currently mostly untapped. By 26 January 2010, Vietnam only had 20 projects registered with the CDM Executive Board (CDM EB). As Figure 2 shows, that number accounts for only 1% of the total CDM projects registered within the Asia and Pacific region. This fraction is unlikely to grow in the near future; in addition to having only 1% of overall projects in the pipeline, Vietnam also accounts for only 1% of credit volume expected by 2012. 24 FIGURE 2 In a separate study, the authors have examined barriers preventing fuller implementation of CDM projects in Vietnam. We have found that regulatory barriers, barriers arising from an unfavorable business environment, difficulties in access to information, weak local capacity, and the structure of the projects portfolio constitute the main barriers to greater success of the CDM in Vietnam. We conclude that proper government policies and actions are necessary for Vietnam to benefit more fully from the CDM or any future mechanism similar to the CDM.

Vietnam's Role in a Post-2012 Climate Regime

As a developing country with significant potential for hosting CDM projects, Vietnam could bear risks if it continues pursuing this investment vehicle in the context of an uncertain post-2012 climate regime. In the event that future rules restrict developing country access to the carbon market or revoke the mechanism, Vietnam could fail to receive benefits from the implementation of CDM projects, exposing the government and investors to significant investment risks. Moreover, given current uncertainties over market prices beyond 2012, it cannot be ruled out that investors will be forced to sell emission reductions credits from CDM projects at very low prices or even prove unable to sell them altogether. Such risks are borne both by the host country government and the project investors.

As the year 2012 approaches, the window of opportunity created by the first commitment period under the Kyoto Protocol will narrow and eventually close. Project developers launching CDM projects at this point in time will run into the 2012 deadline and may hence suffer investment risks, both due to the time required for implementing a project and the increasing shortage of CER buyers. At present, most commercial CER buyers -such as EU ETS compliance buyers, Japanese buyers, and speculative funds -are not committing to a purchase of post-2012 CERs, except for a minority of buyers willing to commit to forward purchases of post-2012 CERs as part of an offer to purchase pre-2012 CERs. Currently, the most concrete opportunity for sales of post-2012 credits arises from a number of post-2012 carbon funds set up by multilateral institutions. Most of these funds assume the continued existence of a project-based GHG market, and focus on development objectives. In Vietnam, such funds have been set up by the World Bank, the Asian Development Bank, the European Investment Bank, and the Nordic Environment Finance Corporation (NEFCO), among others. 25 One of the most controversial issues addressed during the climate negotiations in Copenhagen was the adoption of emission reduction targets, both directly or indirectly, by developing countries and especially by emerging economies. While many of these countries show hesitation or openly refuse any engagement in this debate, scientific evidence suggests that meaningful participation by all parties to the UNFCCC will be needed to meet the climate challenge, even if the future policy framework remains based on the principle of common but differentiated responsibilities. Under such a framework, developing countries would take on commitments in line with their different levels of economic development. Over time, domestic policies and practices can evolve into low-carbon growth plans, with financial and technical assistance where needed.

In Vietnam, the opportunities and challenges raised by more ambitious climate policies are necessarily featured in the national development agenda, not only because Vietnam is one of the countries most vulnerable to the dangerous impacts of climate change, but also because effective climate action offers prospects for sustainable development: aided by financial support for adaptation and mitigation from developed countries, it can drive investment and clean growth, incentivize job creation, provide opportunities for poverty eradication, and help elevate the standard of living. A discussion of the potential benefits for Vietnam of a reformed CDM follows below.

A number of sectors in the Vietnamese economy offer substantial opportunities for the implementation of mitigation projects, with the energy sector accounting for the largest overall potential. This potential largely rests in the field of renewable energy, an area dominated by small-scale hydropower and wind energy, as well as in energy efficiency improvements both for industry and buildings. 26 In Vietnam, however, small-scale projects may face greater technical and financial barriers than large projects,27 suggesting that the CDM process and modalities for small-scale energy projects would need to be simplified and streamlined before the potential for such projects can be fully exploited: small-scale projects implemented under similar environmental conditions and socio-economic settings, using the same technology, methodology, and so on, should be cleared through a fast-track procedure rather than being required to go through all stages of the regular project cycle. In order to harness the potential for such projects most effectively, this process should be as standardized as possible, including establishment of emission reduction baselines for different sectors, systematic definition of positive lists for technologies, simplification of project description formats, and other measures to facilitate participation in the mechanism.

Additionally, Vietnam offers significant potential for small industrial projects and scattered mitigation actions, such as demand-side energy efficiency improvements, measures for increased boiler efficiency in small industries, biomass-fired cooking stoves, biogas digesters, solar water heating, geothermal energy, small hydropower, and wind energy. CDM Programmes of Activities (PoAs) would appear the most effective mechanism to support such actions in Vietnam, given that PoAs have already proven successful for small-scale projects to include on-grid/off-grid applications using renewable energy. More specifically, registration of renewables as PoAs would result in large benefits for electrification and poverty reduction in remote areas, where more than 70% of the Vietnamese population currently lives.

Further, the adoption of carbon capture and sequestration (CCS) technology in the Vietnamese power sector, which has high CO 2 emissions due to the large share of coal-fired generation, could offer significant potential for financial investment and technology transfer to Vietnam if large-scale CCS projects are included as eligible projects in a post-2012 climate regime. Likewise, the inclusion of REDD in the CDM, a proposition currently under discussion, might also create new opportunities for climate finance based on program funding from developed countries or through new climate mechanisms. Specifically, implementation of REDD activities offers one of the most effective ways to alleviate rural poverty and improve biodiversity conservation in Vietnam.

National Climate Policy for the Post-2012 Period

In 2008, Vietnam approved the National Target Programme to Respond to Climate Change (NTP). With the NTP, the government plots a set of specific policies and action plans for climate mitigation activities over both the short and the long term, focusing on a number of sectors that may be eligible for crediting under a climate regime beyond 2012: energy generation and industry (enhancing energy efficiency, fossil fuel switching, promoting use of renewable energy, nuclear energy, early application of CCS for cement and electricity generation, and others), agriculture, forestry and waste (a 5 million hectares reforestation program, restoration of cultivated peaty soils and degraded lands, landfill CH 4 recovery, and others), and so on. Nonetheless, the government has acknowledged the need for financial and technological support from developed countries as well as other international funding sources. 28 Coming to the Copenhagen summit, the Vietnamese government expressed its firm adherence to the principle of common but differentiated responsibilities as a premise for participation by all nations around the globe in the international efforts to mitigate climate change:

(i) Developed countries should take the lead in making strong midterm and long term commitments on GHG emissions reductions, although developing countries could make a more active contribution to the global GHG abatement efforts by elaborating and implementing measures such as National Appropriate Mitigation Actions (NAMAs) with adequate support from developed countries and the international community through flexible mechanisms enabling financial and technology transfer.

(ii) The Kyoto Protocol should be retained and expanded beyond 2012 to incorporate new provisions for large GHG emitting countries. Similar to many other developing countries, Vietnam also supports extending the CDM or creating improved CDM-like mechanism to encourage more meaningful participation from developing countries in climate activities.

With regard to the CDM, 29 the Vietnamese government will issue new legal documents this year in order to favor investment in mitigation projects from both domestic and foreign investors and project developers.

(iii) A future international climate regime should be revised to afford greater priority to the specific needs of developing countries like Vietnam, which are most vulnerable to the threat of climate change, and assist them in strengthening capacity to respond to climate change effectively. 30

IV. Improving the CDM: Eight Proposals

In spite of ongoing concerns about the mechanism's effectiveness and uncertainties about the climate regime beyond 2012, many observers believe that the CDM will prove versatile enough to survive and evolve so as to play a key function in a future international climate regime, whatever shape this may take. 31 In view of this developing country perspective, we propose eight options for reforming the CDM below.

First, the CDM should be extensively improved to reduce perceived administrative and organizational constraints: (i) at the CDM EB: actions proposed during the Copenhagen climate change summit to improve the CDM 32 should be implemented promptly, including actions to reduce delays in project registration and certification (by streamlining and simplifying administrative procedures for projects and further improving methodologies and additionality criteria), as well as actions to improve the performance of Designated Operational Entities (DOEs) (by providing regional workshops to train and empower DOEs), to the point where the EB can delegate most project reviewing work to DOEs and focus on managing the vetting process rather than projects review. (ii) at host parties: implementing necessary corrective actions by looking at international practices and lessons learnt from more advanced developing countries to alleviate major barriers and administrative constraints in host countries, as well as provision of additional support measures (for instance in the shape of guidelines, workshops and training sessions, local capacity building initiatives, and so on) by developed countries.

Second, the CDM can be reasonably extended as a major source of emissions credits for Kyoto signatories. Moreover, if developed countries are allowed to make use of offset credits, this may create an incentive for developing countries to join a new international climate agreement. For the next phase of the EU ETS and other emerging domestic carbon markets in the United States, Japan, New Zealand and Australia, restrictions on the import of CERs or similar credits from developing countries should therefore be adopted cautiously. A more specific option would be, for instance, to allow companies within developed countries to satisfy at least 10% of their national emissions commitment through CDM credits. 33 Third, wider implementation of PoAs (programmatic CDM) should be facilitated both by the CDM EB and potential host countries, as this option potentially enhances the efficiency of the CDM and offers more opportunities for small and low-income developing countries to access the CDM.

Fourth, positive lists of projects and technologies that are deemed additional by default should be adopted, helping streamline the operational process and reducing barriers both at the CDM EB and at the Designated National Authority (DNA) of host countries. Such a positive list could be based on the usage of certain technologies and both small and large scale projects.

Fifth, new project categories -such as CCS -should be included in the CDM in order to enrich the CDM project portfolio and increase the overall volume of credits. Specifically, the motivation to include CCS projects is that the developing world contributes a rapidly increasing share of global CO 2 emissions, with growing energy demand largely satisfied through coal-fired power plants; in industrialized countries, CCS technology is already being explored as an important option for deep cuts in CO 2 emissions. 34 Sixth, differentiated approaches should be considered, affording disadvantaged and vulnerable developing countries preferential treatment under the CDM. 35 projects or of potential buyers to use CERs for compliance; (iii) introducing a premium rate to incentivize investment and development of certain projects.

Seventh, the criteria for CDM approval could be expanded to allow for a "policy CDM", allowing the focus to shift from "real, verified, and permanent reductions" to climate mitigation and adaptation actions in developing countries. 36 CDM resources could then benefit a wider set of policies and activities, such as energy efficiency standards, renewable energy portfolio standards, and reductions of energy subsidies, that are not eligible within the current framework of the CDM.

Eight, and finally, the CDM should be used to encourage developing country engagement under a new proposed financing framework for nationally appropriate mitigation actions (NAMAs) that is called for by the Bali Action Plan. However, it is necessary to clearly determine how NAMAs can interact with the CDM, and what types of projects may qualify as a NAMA. This is likely easier to negotiate in tandem with an expanded CDM than by merely requesting developing countries to take on explicit emissions caps for the short and medium term.

V. Conclusion

In this article, we assessed the evolving framework of the CDM and identified possible reform proposals beyond 2012 from a developing country perspective. Acknowledging the mechanism's performance over time, the incomplete outcome of negotiations at COP-15, and the reluctance of developing countries to enter legally binding mitigation commitments in the near term, we argue that continuation of the mechanism remains an effective way to reduce greenhouse gases emissions.

An analysis of Vietnam's current climate policy and its position regarding the shape of a future international climate regime both suggest that the CDM should be retained and improved for more flexible mitigation options post-2012. Ideally, a reformed CDM would allow for more active and meaningful participation by developing countries in the global efforts to mitigate climate change, while still upholding the principle of differentiated responsibilities. Also, it can be reasonably assumed that developing countries will exert pressure in upcoming negotiations to extend the CDM and enhance its operation.

Finally, the article provides eight proposals to reform the CDM and increase its usefulness in a future international climate change framework. These range from streamlining and simplifying the CDM project cycle and extending it to include additional project categories over improving accessibility for developing countries through differentiation and capacity building to exploring completely new approaches, such as crediting of mitigation policies in addition to projects. As negotiations on the future climate regime resume in 2010, decision makers must carefully these options and also ensure that any project-based mechanism emerging from the CDM find an appropriate role alongside other market instruments and nationally appropriate mitigation actions.

  deforestation and forest degradation (REDD); the further development of the CDM in terms of governance and eligibility rules post-2012.

FiguresFigure 1 :

 1 Figures Figure 1: Growth of the Clean Development Mechanism. Source: <cdm.unfccc.int>.

Figure 2 :

 2 Figure 2: Registered CDM projects, Asia & Pacific, by host party as of 26 January 2010. Source: <cdm.unfccc.int> Number of registered CDM projects in Asia & Pacific by country

  Carbon and Climate Law Review (2009), 16. 10 Draft Decision -/CMP.5, Further Guidance Relating to the Clean Development Mechanism, December 2009, available on the Internet at <unfccc.int/files/meetings/cop_15/application/pdf/cmp5_cdm_auv.pdf> (last accessed on 1 March 2010).

	Strategies	Final	Report,	June	2008,	available	on	the	Internet	at
	<www.climatestrategies.org/component/reports/category/39/138.html> (last accessed on 1 March 2010); Grant
	Boyle, Jennifer Kirton, Rudi M. Lof, and Tanya Nayler, "Transitioning from the CDM to a Clean Development
	Fund", 3

11

NDRC Press Release, "Copenhagen Greenlights Plan to Streamline CDM", 25 December 2009, available on the Internet at <cdm.ccchina.gov.cn/english/NewsInfo.asp?NewsId=4160> (last accessed on 1 March 2010).

  12 See Karan Capoor and Philippe Ambrosi, State and Trends of the Carbon Market 2009 (Washington, DC: Climate Policy (2009), 341. 14 Point Carbon, "Carbon 2008-Post-2012 Is Now", 27 March 2008, available on the Internet at <www.pointcarbon.com/research/carbonmarketresearch/analyst/1.912721> (last accessed on 1 March 2010); in this study, Point Carbon forecasts a global carbon market worth €2 trillion by 2020 and assumes a market volume of 38 Gt and a carbon price of €50 by 2020 .

	World Bank, 2009), 5 and 31.
	13 See Michael Mehling and Erik Haites, "Mechanisms for Linking Emissions Trading Schemes." 9 Climate
	Policy (2009), 169; Andreas Tuerk, Michael Mehling, Christian Flachsland, et al., "Linking Carbon Markets:
	Concepts, Case Studies and Pathways", 9

15 Frank Ackerman, "Carbon markets and Beyond: The Limited Role of Prices and Taxes in Climate and Development Policy", G-24 Discussion Paper Series No. 53, December 2008, available on the Internet at <www.unctad.org/en/docs/gdsmdpg2420084_en.pdf> (last accessed on 1 March 2010).

  28 Vietnam, National Target Program to Respond to Climate Change, Decision No. 158/QĐ-TTg, 2 December 2008; Institute of Strategy and Policy on Natural Resources and Environment, Vietnam Assessment Report on Climate Change (Ha Noi: ISPNRE, 2009). 29 Information provided by the Ministry of Natural Resources and Environment at Side-event on the climate change in Vietnam, which was held at the Copenhagen (2009). 30 The speech of H.E. Mr. Nguyen Tan Dung, Prime Minister of the Socialist Republic of Vietnam at COP15

31

Andrew G.

Keeler

and Alexander Thompson, "Industrialized Country Mitigation Policy and Resource Transfers to Developing Countries: improving and Expanding Greenhouse Gas Offsets", Discussion Paper 08-05, available on the Internet at <belfercenter.ksg.harvard.edu/files/KeelerWeb4.pdf> (last accessed on 15 January 2010); Noriko Fujiwara, "Flexible Mechanisms in Support of a New Climate Change Regime: The Clean Development Mechanism and Beyond", CEPS Task Force Report, available on the Internet at <www.ceps.be/ceps/download/2679> (last accessed on 15 January 2010).

Expected average annual CERs from registered projects in Asia & Pacific by country

  Vietnam: A Suitable Host for the First Carbon Capture and Storage Project Registered under the CDM? Domestic coal reserves, geological potential, rapid expansion of coal-fired electricity generation and pronounced climate change vulnerability all make carbon capture and storage technology (CCS) highly interesting mitigation options for Vietnam. Indeed, the White Tiger (Bach Ho) Field project in Vietnam was an early proposal to include a CCS project under the CDM. It involves CO 2 capture from Natural Gas Combined Cycle (NGCC) plants, pipeline transport, storage in offshore/onshore oil fields and enhanced oil recovery. As the first commercial CCS project in Asia, it would have a high demonstration value, and could potentially generate emission reductions of approximately 7.7 million tCO 2 per year, facilitating the recovery of an average of 50 thousand barrels of crude oil per day. Work to include CCS in the CDM started in 2006, but has not yet been concluded as of December 2009. There are pending methodological issues, given that the technology is still evolving, and the scale is out of proportion relative to the average CDM project: out of 2236 requested and registered projects in February 2010, only 7 are larger than the White Tiger project in terms of avoided emissions.

				Vietnam 0.5%	Philippines 0.5%	South Korea
						5.4%	Sri Lanka
	Indonesia Vietnam 1% Philippines 3%	South Korea 2%	1% 0% Others Sri Lanka	0.7% Thailand 1.5% Indonesia	India		0.1%	Others 0.8%
	3%			Malaysia	14.9%
	Malaysia 2% Thailand		50% China	1.8%
	5%
	India
	33%							China 73.8%
	Box 1:

Axel Michaelowa and Benito Muller, "The Clean Development Mechanism in the Post-2012 Climate Change

See UNEP Risoe Centre on Energy, Climate and Sustainable Development (URC), "CDM Projects by Host Region", January 2010, available on the Internet at <cdmpipeline.org> (last accessed on 15 January 2010).

Climate Focus, Renewable Energy Small Power Producers in Vietnam, Final Report prepared for the Ministry of Industry and Trade and the World Bank (Rotterdam: Climate Focus, 2009).

See Nguyen et al., supra, note 21.

Ibid.

Keeler et al., supra, note 31; Michaelowa, supra, note 8.

Research for this article has been carried out at CIRED under a Ph.D. research grant offered by the Centre National de la Recherche Scientifique (CNRS) of France. All errors and opinions remain those of the authors.