Limit theorems for conditioned multitype Dawson-Watanabe processes and Feller diffusions

A multitype Dawson-Watanabe process is conditioned, in subcritical and critical cases, on non-extinction in the remote future. On every finite time interval, its distribution is absolutely continuous with respect to the law of the unconditioned process. A martingale problem characterization is also given. Several results on the long time behavior of the conditioned mass process|the conditioned multitype Feller branching diffusion are then proved. The general case is first considered, where the mutation matrix which models the interaction between the types, is irreducible. Several two-type models with decomposable mutation matrices are analyzed too.


Introduction
The paper focuses on some conditioning of the measure-valued process called multitype Dawson-Watanabe (MDW) process, and on its mass process, the well-known multitype Feller (MF) diffusion. We consider the critical and subcritical cases, in which, for any finite initial condition, the MF diffusion vanishes in finite time, that is the MDW process dies out a.s. In these cases, it is interesting to condition the processes to stay alive forever -an event which we call remote survival, see the exact definition in (3). Such a study was initiated for the monotype Dawson-Watanabe process by A. Rouault and the second author in [28] (see also [10], [9] and [8] for the study of various aspects of conditioned monotype superprocesses). Their results were a generalization at the level of measure-valued processes of the pioneer work of Lamperti and Ney ([20], Section 2), who studied the same questions applied to Galton-Watson processes. We are interested here in the multitype setting which is much different from the monotype one. The mutation matrix D introduced in (2), which measures the quantitative interaction between types, will play a crucial role. We now briefly describe the contents of the paper. The model is precisely defined in the first section. In the second section we define the conditioned MDW process, express its law as a locally absolutely continuous measure with respect to the law of the unconditioned process, write explicitly the martingale problem it satisfies and give the form of its Laplace functional; all this in the case of an irreducible mutation matrix. Since D is irreducible, all the types communicate and conditioning by remote survival is equivalent to conditioning by the non-extinction of only one type (see Remark 2.5). The third section is devoted to the long time behavior of the mass of the conditioned MDW process, which is then a conditioned MF diffusion. First the monotype case is analyzed (it was not considered in [28]), and then the irreducible multitype case. We also prove that both limits interchange: the long time limit and the conditioning by long time survival (see Theorem 3.7). In the last section we treat the same questions as in Section 3 for various reducible 2-types models. Since D is decomposable, the two types can have very different behaviors, that also depend on the precise conditioning that is considered (see Section 4.1).

The model
In this paper, we will assume for simplicity that the (physical) space is Ê. k is the number of types. Any k-dimensional vector u ∈ Ê k is denoted by (u 1 ; · · · ; u k ). 1 will denote the vector (1; . . . ; 1) ∈ Ê k . u is the euclidean norm of u ∈ Ê k and (u, v) the scalar product between u and v in Ê k . If u ∈ Ê k , |u| is the vector in Ê k with coordinates |u i |, 1 ≤ i ≤ k.
We will use the notations u > v (resp. u ≥ v) when u and v are vectors or matrices such that u − v has positive (resp. non-negative) entries.
Let C b (Ê, Ê k ) denote the space of Ê k -valued continuous bounded functions on Ê. By C b (Ê, Ê k ) + we denote the set of non-negative elements of C b (Ê, Ê k ). . . . ; f k ) ∈ C b (Ê, Ê k ). For any λ ∈ Ê k , the constant function of C b (Ê, Ê k ) equal to λ will be also denoted by λ.
A multitype Dawson-Watanabe process with mutation matrix D = (d ij ) 1≤i,j≤k is a continuous M(Ê) k -valued Markov process whose law È on the canonical space (Ω := C(Ê + , M(Ê) k ), (X t ) t≥0 , (F t ) t≥0 ) has as transition Laplace functional where U t f ∈ C b (Ê, Ê k ) + , the so-called cumulant semigroup, is the unique solution of the non-linear PDE Here, u ⊙ v denotes the componentwise product (u i v i ) 1≤i≤k of two kdimensional vectors u and v and u ⊙2 = u ⊙ u. To avoid heavy notation, when no confusion is possible, we do not write differently column and row vectors when multiplied by a matrix. In particular, in the previous equation, Du actually stands for Du ′ . The MDW process arises as the diffusion limit of a sequence of particle systems ( 1 K N K ) K , where N K is an appropriate rescaled multitype branching Brownian particle system (see e.g. [15] and [16], or [32] for the monotype model): after an exponential lifetime with parameter K, each Brownian particle splits or dies, in such a way that the number of offsprings of type j produced by a particle of type i has as (nonnegative) mean δ ij + 1 K d ij and as second factorial moment c (δ ij denotes the Kronecker function, equal to 1 if i = j and to 0 otherwise). Therefore, the average number of offsprings of each particle is asymptotically one and the matrix D measures the (rescaled) discrepancy between the mean matrix and the identity matrix I, which corresponds to the pure critical case of independent types. For general literature on DW processes we refer the reader e.g. to the lectures of D. Dawson [3] and E. Perkins [25] and the monographs [5] and [7]. Let us remark that we introduced a variance parameter c which is typeindependent. In fact we could replace it by a vector c = (c 1 ; · · · ; c k ), where c i corresponds to type i. If inf 1≤i≤k c i > 0, then all the results of this paper are still true. We decided to take c independent of the type to simplify the notation.
When the mutation matrix D = (d ij ) 1≤i,j≤k is not diagonal, it represents the interaction between the types, which justifies its name. Its non diagonal elements are non-negative. These matrices are sometimes called Metzler-Leontief matrices in financial mathematics (see [29] § 2.3 and the bibliography therein). Since there exists a positive constant α such that D + αI ≥ 0, it follows from Perron-Frobenius theory that D has a real eigenvalue µ such that no other eigenvalue of D has its real part exceeding µ. Moreover, the matrix D has a non negative right eigenvector associated to the eigenvalue µ (see e.g. [14], Satz 3 § 13.3 or [29] Exercise 2.11). The cases µ < 0, µ = 0 and µ > 0 correspond respectively to a subcritical, critical and supercritical processes. In the present paper, we only consider the case µ ≤ 0, in which the MDW dies out a.s. (see Jirina [17]).

(Sub)critical irreducible MDW process conditioned by remote survival.
Let us recall the definition of irreducibility of a matrix.
In all this section and in the next one, the mutation matrix D is assumed to be irreducible. By Perron-Frobenius' theorem (see e.g. [29] Theorem 1.5 or [14], Satz 2 §13.2, based on [27] and [13]), the eigenspace associated to the maximal real eigenvalue µ of D is one-dimensional. We will always denote its generating right (resp. left) eigenvector by ξ (resp. by η) with the normalization conventions (ξ, 1) = 1 and (ξ, η) = 1. All the coordinates of both vectors ξ and η are positive.

The conditioned process as a h-process
The natural way to define the law È * of the MDW process conditioned to never die out is by if this limit exists.
The following Theorem 2.2 proves that È * is well-defined by (3) and is a probability measure on F t absolutely continuous with respect to È. Furthermore, the density is a martingale, so that È * can be extended to ∨ t≥0 F t , defining a Doob h-transform of È (see the seminal work [22] on h-transforms and [24] for applications to monotype DW processes).
Theorem 2.2 Let È be the distribution of a critical or subcritical MDW process characterized by (1), with an irreducible mutation matrix D and initial measure m ∈ M(Ê) k \ {0}. Then, the limit in (3) exists and defines a probability measure È * on ∨ t≥0 F t such that, for any t > 0, where ξ ∈ Ê k is the unitary right eigenvector associated to the maximal real eigenvalue µ of D.
Proof of Theorem 2.2 By definition, for B ∈ F t , For any time s > 0, x s := ( X s,1 , 1 ; . . . ; X s,k , 1 ), the total mass at time s of the MDW process-a multitype Feller diffusion-is a continuous Ê k +valued process with initial value x = ( m 1 , 1 ; . . . ; m k , 1 ) characterized by its transition Laplace transform Here, u λ t = (u λ t,1 ; . . . ; u λ t,k ) := U t λ satisfies the non-linear differential system or componentwise Then, where λ ֒→ ∞ means that all coordinates of λ go to +∞. Using the Markov property of the MDW process, one obtains In the monotype case (k = 1), u λ t can be computed explicitly (see Section 3.1), but this is not possible in the multitype case. Nevertheless, one can obtain upper and lower bounds for u λ t . This is the goal of the following two lemmas, the proofs of which are postponed after the end of the proof of Theorem 2.2. Lemma 2.3 Let u λ t = (u λ t,1 ; . . . ; u λ t,k ) be the solution of (6).
(ii) Let C λ t := sup -in the subcritical case (µ < 0) and therefore sup λ∈Ê k (iv) For any λ ∈ Ê k + and t ≥ 0, The main difficulty in the multitype setting comes from the noncommutativity of matrices. For example (6) can be expressed as However, since D and A t do not commute, it is not possible to express u λ t in terms of the exponential of t 0 (D + A s ) ds. The following lemma gives the main tool we use to solve this difficulty.

Lemma 2.4
Assume that t → f (t) ∈ Ê is a continuous function on Ê + and t → u t ∈ Ê k is a differentiable function on Ê + . Then For any 1 ≤ i ≤ k, applying (5) with x = e i where e i j = δ ij , 1 ≤ j ≤ k, one easily deduces the existence of a limit in [0, ∞] of u λ t,i when λ ֒→ ∞. Moreover, by Lemma 2.3 (ii) and (iii), for any t > 0, Therefore lim θ→∞ lim λ֒→∞ u λ θ = 0 and, for sufficiently large θ, for some constant K that may depend on t but is independent of θ. Since X t , 1 < ∞ for any t ≥ 0 (see [15] or [16]), Lebesgue's dominated convergence theorem can be applied to make a first-order expansion in θ in (7). This yields that the density with respect to È of È conditioned on the nonextinction at time t + θ on F t , converges in L 1 (È) when θ → ∞ to if this limit exists. We will actually prove that This will imply that the limits in θ and in λ can be exchanged in (12) and thus completing the proof of Theorem 2.2. Subcritical case: µ < 0 As a preliminary result, observe that, since D has nonnegative nondiagonal entries, there exists α > 0 such that D + αI ≥ 0, and then exp(Dt) ≥ 0.
Since du λ t dt ≤ Du λ t , we first remark by Lemma 2.4 (applied to −u λ t ), that Second, it follows from Lemma 2.3 (iv) that Therefore, since In particular, Third, it follows from Lemma 2.3 (ii) that there exists t 0 such that 2K .
Last, there exists a positive constant K ′ such that Combining all the above inequalities, we get for any a ∈ Ê k where the constantsK may vary from line to line, but are independent of λ and t 0 . Now, let t 0 (θ) be an increasing function of θ larger than t 0 such that t 0 (θ) → ∞ when θ → ∞. By Lemma 2.3 (ii), u λ t 0 (θ) → 0 when θ → ∞, uniformly in λ ≥ 0. Then, by (16), uniformly in λ ≥ 0, which completes the proof of Theorem 2.2 in the case µ < 0. Critical case: µ = 0 The above computation has to be slightly modified. Inequality (14) becomes Therefore, the right-hand side of (16) has to be replaced by Now, using Lemma 2.3 (iii) again, it suffices to choose a function t 0 (θ) in such a way that lim θ→∞ θ sup λ≥0 u λ t 0 (θ) = 0. One can now complete the proof of Theorem 2.2 as above. 2 Therefore, for any i such that λ i > 0, u λ t,i > 0 for any t ≥ 0. Let I := {i : λ i > 0} and J := {j : λ j = 0}. By the irreducibility of the matrix D, there exist i ∈ I and j ∈ J such that d ji > 0. Therefore, for sufficiently small t > 0, and thus u λ t,j > 0 for t > 0 in a neighborhood of 0. Moreover, as long as u λ t,i > 0, for the same reason, u λ t,j cannot reach 0.
For sufficiently small ε > 0, u λ ε,i ′ > 0 and the previous argument shows that u λ ε+t,j ′ > 0 for t > 0 as long as u λ ε+t,i ′ > 0. Letting ε go to 0 yields that u λ t,j ′ > 0 for sufficiently small t > 0. Applying the same argument inductively shows that u λ t > 0 for t > 0 in a neighborhood of 0. Using (19) again, this property can be extended to all t > 0.
(ii) and (iii) As the supremum of finitely many continuously differentiable functions, t → C λ t is differentiable except at at most countably many points. Indeed, it is not differentiable at time t if and only if there exist two types i and j such that u λ t,i /ξ i = u λ t,j /ξ j and d(u λ t,i /ξ i )/dt = d(u λ t,j /ξ j )/dt. For fixed i and j, such points are necessarily isolated, and hence are at most denumerable. Fix a time t at which C λ t is differentiable and fix i such that u λ t,i = C λ t ξ i . Then where the inequality comes from the fact that D is nonnegative outside of the diagonal and where the third line comes from the specific choice of the subscript i. Therefore, Assume µ = 0. By Point (i), if λ = 0, C λ t > 0 for any t ≥ 0 (the case λ = 0 is trivial). Then, for any t ≥ 0, except at at most countably many points, Integrating this inequality between 0 and t, we get The proof of the case µ < 0 can be done by the same argument applied to t → e −µt C λ t . Inequalities (iii) are obtained in a similar way too.
> 0 for any t ≤ t 0 and thus t 0 = +∞. Letting ε go to 0 completes the proof of Lemma 2.4. 2 Remark 2.5 Since the limit in (13) is uniform in λ, one can choose in par- Therefore, Theorem 2.2 remains valid if the conditioning by the nonextinction of the whole population is replaced by the non-extinction of type i only. This property relies strongly on the irreducibility of the mutation matrix D. In Section 4, we will show that it does not always hold true when D is reducible (see for example Theorem 4.1 or Theorem 4.4). ♦

Laplace functional of È * and Martingale Problem
To better understand the properties of È * , its Laplace functional provides a very useful tool.
where the semigroup V t f is the unique solution of the PDE Let Comparing with the Laplace functional of È, the multiplicative term m,Vtf m,ξ e −µt appears in the Laplace functional of È * . In particular, the multi- where v λ t := V t λ satisfies the differential system The following theorem gives the martingale problem satisfied by the conditioned MDW process. This formulation also allows one to interpret È * as an unconditioned MDW process with immigration (see Remark 2.8 below). The term with Laplace functional m,Vtf m,ξ e −µt that we just mentioned is another way to interpret this immigration.
Theorem 2.7 È * is the unique solution of the following martingale problem: is a È * -local martingale.
Proof According to [15] (see also [6] for the monotype case), È is the unique solution of the following martingale problem : for any function F : Here the infinitesimal generator A is given by where we use the notation ∂F/∂m = (∂F/∂m i ) 1≤i≤k and Applying this to the time-dependent function Therefore, The uniqueness of the solution È * to the martingale problem (25) comes from the uniqueness of the solution of the martingale problem (26). 2 Remark 2.8 Due to the form of the martingale problem (25), the probability measure È * can be interpreted as the law of a MDW process with interactive immigration whose rate at time s, if conditioned by X s , is a random measure with Laplace functional exp −c Xs,f ⊙ξ Xs,ξ . Monotype DW processes with deterministic immigration rate were introduced by Dawson in [2]. The first interpretation of conditioned branching processes as branching processes with immigration goes back to Kawazu and Watanabe in [18], Example 2.1. See also [28] and [10] for further properties in the monotype case. ♦

Long time behavior of conditioned multitype Feller diffusions
We are now interested in the long time behavior of the MDW process conditioned on non-extinction in the remote future. Unfortunately, because of the Laplacian term in (22), there is no hope to obtain a limit of X t under È * at the level of measure (however, see [10] for the long time behavior of critical monotype conditioned Dawson-Watanabe processes with ergodic spatial motion). Therefore, we will restrict our attention to the Ê k -valued multitype Feller diffusion x t . As a first step in our study, we begin this section with the monotype case.

Monotype case
In this subsection, we first give asymptotic behavior of x t under È * (Proposition 3.1). This result is already known, but we give a proof that will be useful in the following section. We also give a new result about the exchange of limits (Proposition 3.3). Let us first introduce some notation for the monotype case. The matrix D is reduced to its eigenvalue µ, the vector ξ is equal to the number 1. Since we only consider the critical and subcritical cases, one has µ ≤ 0. The law È * of the MDW process conditioned on non-extinction in the remote future is locally absolutely continuous with respect to È (monotype version of Theorem 2.2, already proved in [28], Proposition 1). More precisely 15 Furthermore the Laplace functional of È * satisfies (see [28], Theorem 3): The total mass process x t = X t , 1 is a (sub)critical Feller branching diffusion under È. By (28) with Recall that the cumulant u λ t satisfies This yields in the subcritical case the explicit formulas In the critical case (µ = 0) one obtains (see [20] Equation (2.14)) We are now ready to state the following asymptotic result.
where this notation means that x t converges in È * -distribution to a Gamma distribution with parameters 2 and 2|µ|/c.
One can find in [19] Theorem 4.2 a proof of this theorem for a more general model, based on a pathwise approach. We propose here a different proof, based on the behavior of the cumulant semigroup and moment properties, which will be useful in the sequel.
Actually, the rate of explosion is also known: in [10] Lemma 2.1, the authors have proved that xt t converges in distribution as t → ∞ to a Gamma-distribution. This can also be deduced from (34), since u λ/t t and v λ/t t converge to 0 and 1/(1 + cλ/2) 2 respectively, as t → ∞.
For µ < 0, by (30), (32) and (33), the process x t has the same law as the sum of two independent random variables, the first one with distribution Γ(2, 2|µ| c(1−e µt ) ) and the second one vanishing for t → ∞. The conclusion is now clear. 2 The presence of a Gamma-distribution in the above Proposition is not surprising.
• As we already mentioned it appears in the critical case as the limit law of x t /t [10].
• It also goes along with the fact that these distributions are the equilibrium distributions for subcritical Feller branching diffusions with constant immigration. (See [1], and Lemma 6.2.2 in [4]). We are grateful to A. Wakolbinger for proposing this interpretation.
• Another interpretation is given in [19]. The Yaglom distribution of the process x t , defined as the limit law as t → ∞ of x t conditioned on x t > 0, is the exponential distribution with parameter 2|µ|/c (see Proposition 3.3 below, with θ = 0). The Gamma distribution appears as the size-biased distribution of the Yaglom limit (È * (x ∞ ∈ dr) = rÈ(Y ∈ dr)/ (Y ), where Y ∼ Exp( 2|µ| c )), which is actually a general fact ( [19,Th.4

.2(ii)(b)]). ♦
We have just proved that, for µ < 0, the law of x t conditioned on x t+θ > 0 converges to a Gamma distribution when taking first the limit θ → ∞ and next the limit t → ∞. It is then natural to ask whether the order of the two limits can be exchanged: what happens if one first fix θ and let t tend to infinity, and then let θ increase? We obtain the following answer. Proposition 3.3 When µ < 0, conditionally on x t+θ > 0, x t converges in distribution when t → ∞ to the sum of two independent exponential r.v. with respective parameters 2|µ| c and 2|µ| c (1 − e µθ ). Therefore, one can interchange both limits in time: Proof First, observe that, by (32) As in (7), it holds where the first (resp. the second) factor is equal to the Laplace transform of an exponential r.v. with parameter 2|µ|/c (resp. with parameter 2|µ| c (1−e µθ )). This means that It is now clear that Thus, the limits in time interchange. 2 Remark 3.4 The previous computation is also possible in the critical case and gives a similar interchangeability result. More precisely, for any θ > 0, x t explodes conditionally on x t+θ > 0 in È-probability when t → +∞. In particular, for any M > 0, ♦ Remark 3.5 One can develop the same ideas as before when the branching mechanism with finite variance c is replaced by a β-stable branching mechanism, 0 < β < 1, with infinite variance (see [2] Section 5 for a precise definition). In this case, equation (31) has to be replaced by Therefore, with a similar calculation as above, one can easily compute the Laplace transform of the limit conditional law of x t when t → ∞ and prove the exchangeability of limits: As before, this distribution can be interpreted as the size-biased Yaglom distribution corresponding to the stable branching mechanism. This conditional limit law for the subcritical branching process has been obtained in [21] Theorem 4.2. We also refer to [19] Theorem 5.2 for a study of the critical stable branching process. ♦

Multitype irreducible case
We now present the multitype generalization of Proposition 3.1 on the asymptotic behavior of the conditioned multitype Feller diffusion with irreducible mutation matrix D. Proof One obtains from (24) that Then, by Lemma 2.4, Therefore, in the critical case, v λ t vanishes for t large if λ > 0, due to the divergence of ∞ 0 inf i u λ s,i ds, which is itself a consequence of Lemma 2.3 (iii). If λ i = 0 for some type i, by Lemma 2.3 (i) and the semigroup property of t → u t , we can use once again Lemma 2.3 (iii) starting from a positive time, to prove that lim t→∞ v λ t = 0. Then, the explosion of x t in È * -probability follows directly from (23) and from the fact that lim t→∞ u λ t = 0. To prove (b), we study the convergence ofṽ λ t := e −µt v λ t when t → ∞. By (35) and Lemma 2.3 (ii) we know that t →ṽ λ t is bounded and bounded away from 0. Fix ε ∈ (0, 1) and t 0 such that ∞ t 0 sup i u λ t,i dt < ε. Then, for any t ≥ 0, and so, for any s, t ≥ 0, By Perron-Frobenius' theorem, lim t→∞ e (D−µI)t = P := (ξ i η j ) i,j and thus, when t → ∞, where the negligible term o(1) does not depend on t 0 , s, ε and λ, since ε < 1 and exp((D − µI)s) is a bounded function of s. Therefore, (ṽ λ t ) t≥0 satisfies the Cauchy criterion and converges to a finite positive limitṽ λ ∞ when t → ∞. We just proved the convergence of the Laplace functional (23) of x t under È * when t → ∞. In order to obtain the convergence in law of x t , we have to check the continuity of the limit for λ = 0, but this is an immediate consequence of lim λ→0 lim t→∞ṽ λ t = ξ. Finally, letting t go to infinity in (36), we get It follows thatṽ λ ∞ is proportional to ξ, as limit of quantities proportional to ξ. Therefore (x,ṽ λ ∞ )/(x, ξ) = (ṽ λ ∞ , 1) is independent of x and the limit law of x t too. 2 We can also generalize the exchange of limits of Proposition 3.3 to the multitype irreducible case.
Theorem 3.7 In the subcritical case, conditionally on (x t+θ , 1) > 0, x t converges in distribution when t → +∞ to a non-trivial limit which depends only on θ. Furthermore, one can interchange both limits in t and θ : Proof Following a similar computation as in the proof of Proposition 3.3, . Since one gets : This inequality, similar to (35), can be used exactly as in the proof of Theorem 3.6 (b) to prove that, for any ε, there exists t 0 large enough such that and to deduce from (37) thatũ λ t := e −µt u λ t converges as t → ∞ to a non-zero limitũ λ ∞ proportional to the vector ξ. Moreover, because of (5), u λ t is increasing with respect to each coordinate of λ. Therefore, it is elementary to check that t → limλ ֒→∞ uλ t = sup n u n1 t is also solution of the non-linear differential system (6), but only defined on (0, ∞) (recall that, by Lemma 2.3 (ii), limλ ֒→∞ uλ t < ∞ for any t > 0). Indeed, assume that b t = sup n a n t whereȧ n t = F (a n t ) for a locally Lipschitz function F . Fix t such that b t < +∞ and a small η > 0, and let F := inf |x−bt|≤η F (x) andF := sup |x−bt|≤η F (x). There exists n 0 such that, for n ≥ n 0 , |a n t − b t | ≤ η/2. Moreover, for any s in a neighborhood of t, |a n s − b t | ≤ η, where the neighborhood depends onF and F , but is uniform in n ≥ n 0 . Therefore, for sufficiently small s and for n sufficiently large, F ≤ (a n t+s − a n t )/s ≤F . Letting n → ∞, s → 0 and finally η → 0, sinceF − F → 0 when Therefore, the semigroup property of the flow of (6) implies that, for any t ≥ 0, so that e −µt limλ ֒→∞ uλ t also converges as t → ∞ to a positive limitũ ∞ ∞ , proportional to ξ too. Hence, which is independent of the initial condition x.
In order to prove the convergence in distribution as t → ∞ of x t conditionally on (x t+θ , 1) > 0 to a random variable with Laplace transform (38), it remains to prove the continuity of this expression as a function of λ for λ → 0. To this aim and also to prove the exchangeability of limits, we use the following Lemma, the proof of which is postponed at the end of the subsection. This lemma gives the main reason why the limits can be exchanged: v λ (24) as v λ t except for the initial condition given by ▽ η u λ 0 = η. Furthermore, λ →ũ λ ∞ is differentiable too and its derivative in the direction η, denoted by Since ▽ η u λ t satisfies the same differential equation as v λ t , in particular, ▽ ξ u λ t = v λ t and, with the notations of the proof of Theorem 3.6, ▽ ξũ λ ∞ =ṽ λ ∞ . It also follows from the above lemma thatũ λ ∞ is continuous as a function of λ. As a result, Finally, let us check that the limits in t and θ can be exchanged. Since limλ ֒→∞ uλ θ ∼ e µθũ∞ ∞ = e µθ (ũ ∞ ∞ , 1)ξ when θ → ∞, it follows from Lemma 3.8 that which completes the proof of Theorem 3.7. 2 Proof of Lemma 3.8 The differentiability of u λ t with respect to λ and the ODE satisfied by its derivatives are classical results on the regularity of the flow of ODEs (see e.g. Perko [26]).
Moreover, since ▽ η u λ t and v λ t are both solution of the ODE (24) (with different initial conditions), it is trivial to transport the properties of v λ t proved in the proof of Theorem 3.6 to ▽ η u λ t . In particular, e −µt ▽ η u λ t converges as t → +∞ to a non-zero vector w λ η which is proportional to ξ. We only have to check that ▽ ηũ λ ∞ exists and that w λ η = ▽ ηũ λ ∞ . Moreover, as for (35), Therefore, since exp(Dt) ≥ 0, which implies that e −µt ▽ η u λ t is uniformly bounded for t ≥ 0, λ ≥ 0 and η in a compact subset of Ê k . Now, letting t → +∞ in (37) one gets for any h ≥ 0, Letting ε → 0 (and thus t 0 → +∞) in the previous inequality, Lebesgue's convergence theorem yields Therefore,ũ λ ∞ is differentiable with respect to λ and ▽ ηũ λ ∞ = w λ η . The proof of Lemma 3.8 is completed. 2

Some decomposable cases conditioned by different remote survivals
In this section, we study some models for which the mutation matrix D is not irreducible: it is called 'reducible' or 'decomposable'. In this case, the general theory developed above does not apply. In contrast with the irreducible case, the asymptotic behavior of the MDW process and the MF diffusion depends on the type. Decomposable critical multitype pure branching processes (without motion and renormalization) were the subject of several works since the seventies. See e.g. [23,11,12,30,33,31].

A first critical model
Our first example is a 2-types DW process with a reducible mutation matrix of the form For this model type 1 (resp. type 2) is subcritical (resp. critical). Moreover mutations can occur from type 1 to type 2 but no mutations from type 2 to type 1 are allowed.
In this section we analyze not only the law È * of MDW process conditioned on the non-extinction of the whole population, but also the MDW process conditioned on the survival of each type separately.
Theorem 4.1 Let È be the distribution of the MDW process X with mutation matrix (39) and non-zero initial condition m. Let us define È * ,È * anď È * for any t > 0 and B ∈ F t by Then, all these limits exist and, for any t > 0, Proof Let us first prove (41). Using the method leading to (12), we get .
The cumulant u λ t of the mass process satisfies for any λ = (λ 1 , λ 2 ) The second equation admits as solution and u λ t,1 can be computed explicitly if λ 2 = 0: and u We then getÈ Proof We first compute the vector v λ Therefore, replacing u λ t,2 by its value obtained in (42), In particular, if λ 2 = 0, v (λ 1 ;0) t,2 = 1/2 and one gets from the explicit expression (43) of u Similarly, forv λ t :=V t λ, λ ∈ Ê 2 , we get In particular, if λ 2 = 0, using (43) again, where the last inequality is obtained by splitting the integral over the time interval [0, t] into the sum of the integrals over [0, t 2 ] and [ t 2 , t]. This implies the first part of (c). 2 We interpret this proposition as follows. Conditionally on the survival of the whole population, the weakest type gets extinct and the strongest type has the same behavior as in the critical monotype case. Conversely, conditionally on the long time survival of the weakest type, the weakest type behaves at large time as in the monotype subcritical case and the strongest type explodes.

A more general subcritical decomposable model
We consider a generalization of the previous model. The mutation matrix is now given by where α > 0 (as before) and β > 0 with β = α.
In this case, the whole population is subcritical. Here again, mutations are only possible from type 1 to type 2. If β < α, type 2 is "less subcritical" than type 1 (as in the previous case) but if α < β, type 1 is "less subcritical" than type 2. We will see below that the behavior of the various conditioned processes is strongly related to the so-called dominating type, which is the first one if α < β and the second type if β < α. Before treating separately both cases with different techniques, we define the common ingredients we need.
We now consider the system of equations which solutions are given by and We denote as before by v λ t ,v λ t orv λ t the respective solutions of (51) with initial conditions v λ 0 = ξ,v λ 0 = (1; 0) andv λ 0 = (0; 1).

Case β < α
We now identify the laws obtained by conditioning with respect to the various remote survivals.
The long time behavior of x t under È * is different from Section 4.2 and is given in the following proposition.
we have to compute lim t→∞ v λ t e βt . For the proof of (a) we remark that, from (52) and (53), v The integral can be computed explicitly and is equal, for t large, to

Thus, lim
For the proof of (b), it suffices to show that It then remains to compute the limit, for λ 2 > 0, of v

32
The first term is O(e −(α−β)t ) and goes to 0 as t → ∞. The limit of the integral can be computed as follows: Here again, this result can be interpreted as follows: for i = 1, 2, conditionally on the survival of the type i, this type i behaves as if it was alone, and the other type j explodes or goes extinct according to whether it is stronger or weaker.

Case α < β
When α < β, the greatest eigenvalue of D is µ = −α and the normalized right eigenvector to µ is ξ = (1; 0). In particular, ξ > 0, so that Lemma 2.3 does not hold and we cannot use the previous method anymore. However, in our specific example, u λ t,2 can be explicitly computed, and, by (49), u λ t,1 is solution of the one-dimensional differential equation .
This equation can be (formally) extended to the case λ 2 = ∞ as . 33 The following technical lemma gives (non-explicit) long-time estimates of the solutions of (54) that are sufficient to compute the various conditioned laws of the MDW process. We postpone its proof at the end of the subsection.
When λ 2 = 0, the computations can be made explicitly as in the proof of Proposition 4.3 (b) and give the usual Gamma limit distribution for x t,1 under È * when t → +∞.