Backbone decomposition of multitype superprocesses

In this paper, we provide a construction of the so-called backbone decomposition for multitype supercritical superprocesses. While backbone decompositions are fairly well-known for both continuous-state branching processes and superprocesses in the one-type case, so far no such decompositions or even description of prolific genealogies have been given for the multitype cases. Here we focus on superprocesses, but by turning the movement off, we get the prolific backbone decomposition for multitype continuous-state branching processes as an easy consequence of our results.


Introduction and main results.
Motivated by the distributional decomposition of supercritical superprocesses with quadratic branching mechanism presented in Evans and O'Connell, [10] and the pathwise decomposition of Duquesne and Winkel [5] of continuous-state branching processes (CB-processes), Berestycki et al. [3] provided a pathwise construction of the so-called backbone decomposition for supercritical superprocesses. The authors in [3] showed that the superprocess can be written as the sum of two independent processes. The first one is an initial burst of subcritical mass, while the second one is subcritical mass immigrating continuously and discontinuously along the path of a branching particle system called the backbone that we explain briefly below.
In Evans and O'Connell [10] a distributional decomposition of supercritical superprocesses with quadratic spatially independent branching mechanism, as sum of two independent processes, was given. Later Engländer and Pinsky [8] provided a similar decomposition for the spatially dependent case. In both constructions, the first process is a copy of the original process conditioned on extinction. The second process is understood as the aggregate accumulation of mass that has immigrated continuously along the path of an auxiliary dyadic branching particle diffusion which starts with a Poisson number of particles. Such embedded branching particle system was introduced as the backbone.
A pathwise backbone decomposition appears in Salisbury and Verzani [21], who consider the case of conditioning a super-Brownian motion as it exits a given domain such that the exit measure contains at least n pre-specified points in its support. There it was found that the conditioned process has the same law as the superposition of mass that immigrates in a Poissonian way along the spatial path of a branching * Department of Mathematical Sciences, University of Bath, Claverton Down, Bath, BA2 7AY, United Kingdom. Email: d.fekete@bath.ac.uk, sp2236@bath.ac.uk † Centro de Investigación en Matemáticas A.C. Calle Jalisco s/n. 36240 Guanajuato, México. E-mail: jcpardo@cimat.mx, jl.garmendia@cimat.mx. particle motion which exits the domain with precisely n particles at the pre-specified points. Another pathwise backbone decomposition for branching particle systems is given in Etheridge and Williams [9], which is used in combination with a limiting procedure to prove another version of Evan's immortal particle picture.
In this case the backbone corresponds to a continuous-time Galton-Watson process, and the general nature of the branching mechanism induces three different sorts of immigration. The continuous immigration is described by a Poisson point process of independent processes along the backbone, and the immigration mechanism is given by the so-called excursion measure which assigns zero initial mass and finite length to the immigration processes. The discontinuous immigration is provided by two sources of immigration. The first one is described again by a Poisson point process of independent processes along the backbone where the immigration mechanism is given by the law of the original process conditioned on extinction, and with initial mass randomised by an infinite measure. The second source of discontinuous immigration is given by independent copies of the original process conditioned on extinction, which are added to the backbone at its branching times, with randomly distributed initial mass that depends on the number of offspring at the branch point. In Berestycki et al. [3], a similar decomposition is provided for a class of superprocesses whose branching mechanisms satisfy the same conditions as those considered by Duquesne and Winkel. It is important to note that the authors in [3] also considered supercritical CB-processes that, with positive probability, may die out without this ever happening in a finite time. This also allows the inclusion of branching mechanisms which are associated to CB-processes with paths of bounded variation which were excluded in [5]. Kyprianou and Ren [16] look at the case of a CB-process with immigration for which a similar backbone decomposition to [3] can be given. Finally, backbone decompositions have also been considered for superprocesses with spatially dependent branching mechanisms which are local, see Kyprianou et al. [15] and Eckhoff et al. [7], and non-local, see Murillo-Salas and Pérez [18] and Chen et al. [4].
In this paper, we offer a similar construction for multitype superprocesses whose branching mechanisms are general, but with the restriction of being spatially independent and having a finite number of types. While backbone decompositions are fairly well-known for both CB-processes and superprocesses in the one-type case, so far no such decompositions or even description of prolific genealogies (i.e. those individuals with infinite line of descent) have been given for multitype processes. Here we focus on superprocesses, but by turning the movement off, we get the prolific backbone decomposition for multitype continuous-state branching processes (MCB-processes) as an easy consequence of our results.
Multitype superprocesses were first studied by Gorostiza and Lopez-Mimbela [11] for the particular case of quadratic branching. Later Li [17] extended the notion of multitype superprocesses to more general branching mechanisms (see also Section 6.2 in the monograph of Li [19]). Roughly speaking, the dynamics of the superprocesses introduced by Li are as follows. The movement of mass of a given type is a Borel process, the death and birth of mass of each type are associated with a spectrally positive Lévy process. From a given type, the creation of mass of other types is given by the law of a subordinator, and is distributed according to a discrete distribution that depends on the type. We are interested in a slightly more general superprocess where the discrete distributions are randomly chosen by a probability kernel that depends on the type. Thus the locations of non-locally displaced offspring involve two sources of randomness. One of the advantages of taking this general branching mechanism is that if there is no spatial motion, we recover the MCB-process studied by Kyprianou et al. [14], which was properly defined by Li in Example 2.2 in [19].
Kyprianou et al. [14] studied the almost sure growth of supercritical MCB-processes and implicitly described a spine decomposition. In [14], the authors show that a MCB-process conditioned to never get extinct is equal in law to the sum of an independent copy of the original process and three different sources of immigration along a spine (continuous, discontinuous and in the times when the spine jumps). More precisely, the spine is given by a Markov chain, the continuous and discontinuous immigrations are described by a Poisson point process along the spine, where MCB-processes with the original branching mechanism are immigrating with zero initial mass and with randomised initial mass, respectively. Due to the non-local nature of the branching mechanism, an additional phenomenon occurs; a positive random amount of mass immigrates off the spine each time it jumps from one state to another. Moreover, the distribution of the immigrating mass depends on where the spine jumped from and where it jumped to.
The backbone and spine decompositions are quite different. In the backbone decomposition, the object that we dress is a multitype branching diffusion while in the spine decomposition, this object is a Markov chain which does not branch. Another difference is related to the immigration processes. In the spine decomposition, these are independent copies of the original process while in the backbone decomposition they are independent copies of the process conditioned to become extinct. In other words, we can think of the backbone as all the particles that have an infinite genealogical line of descent, and of the spine as just one infinite line of descent.

Multitype superprocesses.
Before we introduce multitype superprocesses and some of their properties, we first recall some basic notation. Let ℓ ∈ N be a natural number, and set S = {1, 2, · · · , ℓ}. We denote by M(R d ), B(R d ) and B + (R d ) the respective spaces of finite Borel measures, bounded Borel functions and positive bounded Borel functions on R d . The space M(R d ) is endowed with the topology of weak convergence.
For u, v ∈ R ℓ , we introduce [u, v] = ℓ j=1 u j v j , and u·v as the vector with entries (u·v) j = u j v j . For a matrix A, we denote by A t its transpose. For any f = (f 1 , . . . , f ℓ ) t ∈ B(R d ) ℓ and µ = (µ 1 , . . . , µ ℓ ) t ∈ M(R d ) ℓ , we define Furthermore, we also use |u| := [u, u] 1/2 for the Euclidian norm of any u ∈ R ℓ , and µ := 1, µ for the total mass of the measure µ.
Suppose that for any i ∈ S, the process ξ (i) = (ξ (i) where B is an ℓ × ℓ real valued matrix such that B ij 1 {i =j} ∈ R + , {e 1 , . . . , e ℓ } is the natural basis in R ℓ , β i ∈ R + , and Π is a measure satisfying the following integrability condition We call the vectorial function ψ the branching mechanism and we also refer to Π as its associated Lévy measure.
The first result that we present here says that multitype superprocesses associated to the branching mechanism ψ and the diffusions {ξ (i) , i ∈ S} are well-defined. Its proof is based on similar arguments as those used to prove Theorem 6.4 in Li [19], for completeness we present its proof in Section 2. Proposition 1. There is a strong Markov process X = (X t , (F t ) t≥0 , P µ ) with state space M(R d ) ℓ and transition probabilities defined by The process X is called a (P, ψ)-mutitype superprocess with ℓ types and with law given by P µ for each initial configuration µ ∈ M(R d ) ℓ .
Our definition is consistent with the multitype superprocesses that appear in the literature. Indeed, we observe that the multitype superprocesses considered by Gorostiza and Lopez-Mimbela [11] are associated with the branching mechanism j , j ∈ S} is a probability distribution on S, and the spatial movement is driven by the family {ξ (i) , i ∈ S} of symmetric stable processes. Li [17] (see also Section 6.2 in [19]) introduced multitype superprocesses with spatial movement driven by Borel right processes and whose branching mechanism is of the form j , j ∈ S} is a probability distribution on S, and l(i, du), n(i, du) are measures on R + satisfying that represent the local and non-local kernels, respectively. The latter branching mechanism can be rewritten in the form of (1.1) by taking B ji : It is important to note that if the branching mechanism is given as in (1.1) and there is no spatial movement, then the associated total mass of a superprocess is a MCB-process, see for instance Example 2.2 in [19]. Indeed, it is not difficult to see that the total mass vector of a multitype superprocess is a MCB-process. Recall that an ℓ-type MCB-process Y = (Y t , t ≥ 0) with branching mechanism ψ can be characterised through its Laplace transform. If we denote by P y the law of such a process with initial state y ∈ R ℓ + , then is the unique locally bounded solution, with non-negative entries, to the system of integral equations Suppose that (X t , P µ ) t≥0 is a (P, ψ)-multitype superprocess and define the total mass vector as Y = Since the branching mechanism and the vector θ are spatially independent, the system of functions V t θ that satisfies (1.3) does not depend on x ∈ R d . In other words Recall that the previous system of equations has a unique solution, therefore V t θ = v t (θ) for any x ∈ R d . By (1.2) and the relationship between X and Y , the total mass vector is indeed a MCB-process. Since the total mass vector of a multitype superprocess is a MCB-process, we can determine its asymptotic behaviour through its first moment, similarly to the one-type case. More precisely, denote by where e i δ x denotes a measure valued vector that has unit mass at position x ∈ R d , in the i-th coordinate, and zero mass everywhere else.
Barczy et al. [1] (see Lemma 3.4) proved that the mean matrix M (t) can be written in terms of the branching mechanism ψ. In other words, for all t > 0 where the matrix B is given by where (a) + = a ∨ 0, denotes the positive part of a. Moreover, after straightforward computations (see for instance the computations after identity (2.15) in [1]) we observe that the branching mechanism ψ can be rewritten as follows In the sequel, we assume that the matrix B t is irreducible, and therefore the mean matrix M is irreducible as well. Then classical Perron-Frobenius theory guarantees that there exists a unique leading eigenvalue Γ, and right and left eigenvectors u, v ∈ R ℓ + , whose coordinates are strictly positive such that, for t ≥ 0, It is important to note, that since the branching mechanism is spatially independent, the value of Γ does not depend on the spatial variable. Moreover, Γ determines the long term behaviour of X. Indeed, employing the same terminology as in the one-type case, we call the process supercritical, critical or subcritical accordingly as Γ is strictly positive, equal to zero or strictly negative. In Kyprianou and Palau [13], the authors show that when Γ ≤ 0 the total mass goes to zero almost surely. Barczy and Pap [2] show that if Γ > 0, then the total mass process satisfies lim which is a non-zero vector. In particular, we deduce that Here, we are also interested in the case that extinction occurs in finite time. More precisely, let us define E := { X t = 0 for some t > 0}, the event of extinction and take w i : Since the branching mechanism is spatially independent, and the total mass vector of X is a MCB-process, we get that In what follows, we assume Assumption (1.9) or similar assumptions have been used in most of the cases where backbones have been constructed. For instance in [3] and [5], the authors assume Grey's condition which is equivalent to w i being finite. In [4,7,14,18] a very similar condition appears for the spatially dependent case. Assumption (1.9) is not only necessary for the construction of the multitype superprocess conditioned on extinction but also for the construction of the so-called Dynkin-Kuznetsov measure, as we will see below.
On the other hand, assumption (1.9) is not very restrictive. For instance, it is satisfied if Γ > 0 and β := inf i∈S β i > 0. Indeed from (1.7), we see that From (1.4) and the fact that the total mass is a MCB-process, it is clear that where v t (i, θ) is given by (1.5) and θ ֒→ ∞ means that each coordinate of θ goes to ∞. In other words, if we show that lim then we have that (1.9) holds. In order to prove that the above limit is finite, we introduce where u i denotes the i-th coordinate of the right eigenvector associated to Γ. Since the supremum of finitely many continuously differentiable functions is differentiable except at most countably many isolated points, we may fix t ≥ 0 such that A t (θ) is differentiable at t and select i in such a way that . Then by using (1.5) and (1.6) we can deduce that Next, we use that u is an eigenvector of B t to get By defining u := inf i∈S u i and recalling the definition of β, the previous identity implies Since, Γ, β and u are strictly positive, an integration by parts allow us to deduce that .
Finally, if we define u := sup i∈S u i , the previous computations lead to The following result is also needed for constructing the associated Dynkin-Kuznetsov measures which provide a way to dress the backbone.
Proposition 2. Suppose that condition (1.9) holds, then ψ(w) = 0. Moreover, for x ∈ R d , i ∈ S and t > 0, we have that For simplicity of exposition, the proof of this result is presented in Section 2.
As we said before, our aim is to describe the backbone decomposition of X. According to Berestycki et al. [3] a one-type supercritical superprocess can be decomposed into an initial burst of subcritical mass and three types of immigration processes along the backbone, which are two types of Poissonian immigrations and branch point immigrations. In order to use the same idea in the multitype case, we need to determine the components of this decomposition. These are the multitype branching diffusion process, that gives the prolific genealogies, and copies of the original multitype superprocess conditioned on extinction.

The multitype supercritical superdiffusion conditioned on extinction.
It is well known that under some conditions a supercritical CB-process can be conditioned to become extinct by conditioning the associated spectrally positive Lévy process to drift to −∞. Such a conditioning appears as an Esscher transform on the underlying Lévy process in the Lamperti transform, where the shift parameter is given by the largest root of the branching mechanism. Here we show that a similar result still holds in the multitype case. In particular we have the following result.
Proposition 3. For each µ ∈ M(R d ) ℓ , define the law of X with initial configuration µ conditioned on becoming extinct by P † µ , and let F t := σ(X s , s ≤ t). Specifically, for all events A, measurable with respect to F, is the unique locally bounded solution of where ψ † (λ) := ψ(λ + w) and w is given by (1.8). In other words, (X, P † µ ) is a (P, ψ † )-multitype superprocess.
For simplicity of exposition, the proof of this result is presented in Section 2.

Dynkin-Kuznetsov measure.
As we mentioned before, a key ingredient in the construction of the backbone, or even the spine decomposition for superprocesses, is the so-called Dynkin-Kuznetsov measure. It is important to note that the existence of such measures was taken for granted in most of the references that appear in the literature, in particular in [3,7,14,18]. Fortunately, from the assumptions and the way the dressing processes are constructed this omission does not play an important role on the validity of their results. Here, we provide a rigorous argument for their existence. See also Chen et al. [4] for the study of Dynkin-Kutznetsov measures for one-type superprocesses with non-local branching mechanism.
Let us denote by X the space of càdlàg paths from [0, ∞) to M(R d ) ℓ .
Proposition 4. Let X be a (P, ψ)-multitype superprocess satisfying (1.9). For x ∈ R d , there exists a measure N xe i on the space X satisfying Again, for simplicity of exposition we provide the proof of this Proposition in Section 2.
Following the same terminology as in the literature, we call {(N xe i , x ∈ R d ), i ∈ S} the Dynkin-Kuznetsov measures. We denote by N † the Dynkin-Kuznetsov measures associated to the multitype superprocess conditioned on extinction, which are also well defined (see the discussion after the proof of Proposition 4).

Prolific individuals.
Here, we consider those individuals of the superprocess who are responsible for the infinite growth of the process. In our case, we have that the so-called prolific individuals, i.e. those with an infinite genealogical line of descent, form a branching particle diffusion where the particles move according to the same motion semigroup as the superprocess itself, and their branching generator can be expressed in terms of the branching mechanism of the superprocess. Let Z = (Z t , t ≥ 0) be a multitype branching diffusion process (MBDP) with ℓ types, where the movement of each particle of type i ∈ S is given by the semigroup P (i) . The branching rate q ∈ R ℓ + takes the form where w was defined in (1.8).
The offspring distribution (p where j = (j 1 , · · · , j ℓ ). Note that p (i) j 1 ,...,j ℓ is a probability distribution. Indeed, since ψ(w) = 0, for each i ∈ S we get that where in the last row we have used the multinomial theorem, i.e. (1.14) Let F (s) = (F 1 (s), . . . , F ℓ (s)) t , s ∈ [0, 1] ℓ , be the branching mechanism of Z, which is given by where we recall that 1 denotes the vector with value 1 in each coordinate and u · v is the element-wise multiplication of the vectors u and v. The intuition behind the process Z is as follows. A particle of type i from its birth executes a P (i) motion, and after an independent and exponentially distributed random time with parameter q i dies and gives birth at its death position to an independent number of offspring with distribution {p (i) j , j ∈ N ℓ }. We call Z the backbone of the multitype superprocess X, and denote its initial distribution by ν ∈ M a (R d ) ℓ , where M a (R d ) denotes the space of atomic measures on R d . Comparing the form of the offspring distribution between the one-type case and the multitype case, the main difference is that now we are allowed to have one offspring at a branching event. However in this case, that offspring has to have a different type from its parent.

The backbone decomposition.
Our primary aim is to give a decomposition of the (P, ψ)-multitype superprocess along its embedded backbone Z. The main idea is to dress the process Z with immigration, where the processes we immigrate are copies of the (P, ψ † )-multitype superprocess. The dressing relies on three different types of immigration mechanisms. These are two types of Poissonian immigrations along the life span of each prolific individual, and an additional creation of mass at the branch points of the embedded particle system. In the first case, we immigrate independent copies of the (P, ψ † )-multitype superprocess, where the immigration rate along a particle of type i ∈ S is related to a subordinator in R ℓ + , whose Laplace exponent is given by which can be rewritten as When an individual of type i ∈ S has branched and its offspring is given by j = (j 1 , . . . , j ℓ ) ∈ N ℓ , we immigrate an independent copy of the (P, ψ † )-multitype superprocess where the initial mass has distribution η (i) (1.17) Before we state our main results, we recall and introduce some notation. Recall that X denotes the space of càdlàg paths. Similarly to the one-type case, we use an Ulam-Harris labelling to reference the particles, and we denote the obtained tree by T . For a particle u ∈ T let γ u denote the type of the particle, τ u its birth time, σ u its death time, and z u (t) its spatial position at time t (whenever τ u ≤ t < σ u ).

Definition 2.
For ν ∈ M a (R d ) ℓ , let Z be a MBDP with initial configuration ν, and let X be an independent copy of X under P † µ . We define the stochastic process Λ = (Λ t , t ≥ 0) on M(R d ) ℓ by where the processes I N † = (I N † t , t ≥ 0), I P † = (I P † t , t ≥ 0), and I η = (I η t , t ≥ 0) are independent of X and, conditionally on Z, are mutually independent. Moreover, these three processes are described pathwise as follows.
i) Continuous immigration. The process I N † is M(R d ) ℓ -valued such that where, given Z, independently for each u ∈ T such that τ u < t, the processes X (1,u,r) are countable in number and correspond to X -valued Poissonian immigration along the space-time trajectory ii) Discontinuous immigration. The process I P † is M(R d ) ℓ -valued such that where, given Z, independently for each u ∈ T such that τ u ≤ t, the processes X (2,u,r) · are countable in number and correspond to X -valued, Poissonian immigration along the space-time trajectory iii) Branch point based immigration. The process I η is M(R d ) ℓ -valued such that where, given Z, independently for each u ∈ T such that σ u ≤ t, the process X is an independent copy of X issued at time σ u with law P Y uδz u(σu) where Y u is an independent random variable with distribution η Moreover, we denote the law of the pair (Λ, Z) by P (µ,ν) .
Since Z is a MBDP and, given Z, immigrating mass occurs independently according to a Poisson point process or at the splitting times of Z, we can deduce that the process ((Λ, Z), P (µ,ν) ) is Markovian. It is important to note that the mass which has immigrated up to a fixed time evolves in a Markovian way thanks to the branching property. Now we are ready to state the main results of the paper. Our first result determines the law of the couple (Λ, Z), and in particular shows that Λ is conservative.
where V † is defined in (1.10), and exp{−U ) for x ∈ R d , and t ≥ 0. In particular, for each t ≥ 0, Λ t has almost surely finite mass.
Finally, we state the main result of this paper which, actually, is a consequence of Theorem 1. To be more precise, we consider a randomised version of the law P (ν,µ) by replacing the deterministic choice of ν in such a way that for each i ∈ S, ν i is a Poisson random measure in R d having intensity w i µ i . The resulting law is denoted by P µ .
Theorem 2. For any µ ∈ M(R d ) ℓ the process (Λ, P µ ) is Markovian and has the same law as (X, P µ ).
The remainder of this paper is devoted to the proofs of all the results presented in the Introduction.

Proofs
We first present the proofs of Propositions 1,2 and 4 which are devoted to the construction of the multitype superprocess X and its associated Dynkin-Kuznetsov measures.
Proof of Proposition 1. Recall that (P  t , t ≥ 0). We introduce Ξ = (Ξ t , t ≥ 0) a Markov process in the product space R d × S whose transition semigroup (T t , t ≥ 0) is given by where f is a bounded Borel function on R d × S. We denote the aforementioned set of functions by B(R d × S) and we use M(R d × S) for the space of finite Borel measures on R d × S, endowed with the topology of weak convergence. For each f ∈ B(R d × S), we introduce the operator , 1), · · · , f (x, ℓ))).
Recall that for a measure µ ∈ M(R d × S), we use the notation Following the theory developed in the monograph of Li [19], we observe that the operator Ψ satisfies equation (2.26) in [19], and that the assumptions of Theorems 2.21 and 5.6, in the same monograph, are fulfilled. Therefore there exits a strong Markov superprocess Z = (Z t , G t , Q µ ) with state space M(R d ×S), and transition probabilities determined by , d(y, j)).
In other words, we can define a strong Markov process X ∈ M(R d ) ℓ associated with Z and (U i ) i∈S as follows. For each i ∈ S, we define X t (i, dx) := U i Z t (dx) = Z t (dx × {i}) with probabilities P µ := Q µ , where µ = (µ 1 , · · · , µ ℓ ) ∈ M(R d ) ℓ , and each µ i = U i µ. In a similar way, there is a homeomorphism between B(R d ) ℓ and B(R d × S); that is to say for f ∈ B(R d ) ℓ we define f (x, i) = f i (x). By applying the aforementioned homeomorphisms, we deduce that (X t , P µ ) satisfies (1.2), and (1.3) has a unique locally bounded solution.
We now prove Proposition 2, which will be very useful for the existence of Dynkin-Kutznetsov measures.
Proof of Proposition 2. By (1.8) and the branching property of X we have P µ (E) = e − w,µ . (2.21) Furthermore by conditioning the event E on F t and using the Markov property, we obtain that Thus from (1.3) and the assumption (1.9) we also get that ψ(w) = 0.
For the second part of the statement, we recall the definition of the total mass vector Y = (Y t , t ≥ 0) whose entries satisfy Y t (i) = X t (i, R d ). From identity (1.4) and assumption (1.9), we know that for each i ∈ S, there exists a positive deterministic time T i such that where v t (i, θ) is given by (1.5), and we recall that θ ֒→ ∞ means that each coordinate of θ goes to ∞. Next, we define the sets S 1 := {i ∈ S : T i = 0} and S 2 := {i ∈ S : T i > 0}. For a vector y = (y 1 , · · · , y ℓ ), we denote its support by supp(y) := {i ∈ S : y i = 0}. Thus, the proof will be completed if we show that S 2 = ∅. We proceed by contradiction.
Let us assume that S 2 = ∅ and define T := inf{T i : i ∈ S 2 } which is strictly positive by definition. Take i ∈ S 2 and observe from the Markov property that By the branching property, if y is a vector such that supp(y) ⊂ S 1 then P y ( Y t = 0) > 0, for all t > 0. Therefore, we necessarily have 0 = P e i supp(Y T /2 ) ⊂ S 1 , and implicitly Hence, using the branching property again, if y is a vector such that supp(y) ∩ S 2 = ∅, we have Finally, we use the Markov property recursively and the previous equality, to deduce that for all k ≥ 1, which is inconsistent with the definitions of T and T i . In other words, S 2 = ∅. This completes the proof.
We now prove the existence of the Dynkin-Kuznetsov measures.
Proof of Proposition 4. Let us denote by M 0 (R d × S) := M(R d × S) \ {0}, where 0 is the null measure. Consider the Markov superprocess Z introduced in the previous proof. Let (Q t , t ≥ 0) and (V t , t ≥ 0) be the transition and cumulant semigroups associated with Z. By Theorem 1.36 in [19], V t has the following representation Let X + be the space of càdlàg paths t → w t from [0, ∞) to M(R d × S) having the null measure as a trap. Let (Q 0 t , t ≥ 0) be the restriction of (Q t , t ≥ 0) to M 0 (R d × S) and By Proposition 2.8 in [19], for all (x, i) ∈ E 0 the family of measures (L t (x, i, ·), t ≥ 0) on M 0 (R d × S) constitutes an entrance law for (Q 0 t , t ≥ 0). Therefore, by Theorem A.40 of [19] for all (x, i) ∈ E 0 there exists a unique σ-finite measure N (x,i) on X + such that N (x,i) ({0}) = 0, and for any 0 < t 1 < · · · < t n < ∞ It follows that for all t > 0, (x, i) ∈ E 0 , and f ∈ B(R d × S) positive, we have Recall the homeomorphism µ → (U i µ) i∈S and the definition of the superprocess X from the proof of Proposition 1. By taking the constant function f (x, i) = λ ∈ R, and using the definitions of V t , Q t , we deduce that If we take λ goes to infinity, the left hand side of the above identity converges to − log P e i δx ( X t = 0) which is finite by Proposition (2).
Next, recall that X denotes the space of càdlàg paths from [0, ∞) to M(R d ) ℓ . Then (U i ) i∈S induces an homeomorphism between X and X . More precisely, the homeomorphism U : X → X is given by w t → w t = (w t (1), · · · , w t (ℓ)) where for all i ∈ S the measure in the ith coordinate is given by w t (i, B) = w t (B × {i}). This implies that for all (x, i) ∈ R d × S we can define the measures N xe i on X given by N xe i (B) := N (x,i) (U −1 (B)). In other words, we obtain It is important to note that the Dynkin-Kuznetsov measures N † associated to the multitype superprocess conditioned on extinction are also well defined since | log P † δxe i (E)| < ∞. We now prove Proposition 3.
Proof of Proposition 3. Using (2.21), (1.9) and the Markov property, we have for Since V t (f + w) satisfies (1.3), using the definitions of V † t f and ψ † we obtain that V † t f satisfies (1.10). Recalling that ψ(w) = 0 and computing ψ(θ + w) − ψ(w), we deduce that This implies that ψ † is a branching mechanism and therefore the solution of (1.10) is unique. In other words, X under P † µ is a multitype superprocess with branching mechanism given by ψ † (θ).
In order to proceed with the proof of Theorem 1, the following two lemmas are necessary.

24)
where φ is given by (1.16) and V † t f satisfies (1.10). Proof. As the different immigration mechanisms are independent given the backbone, we may look at the Laplace functional of the continuous and discontinuous immigrations separately. For the continuous immigration, we can condition on Z, use Campbell's formula, then equation (1.11) for N † , and finally the definition of In a similar way, for the discontinuous immigration, by conditioning on Z, using Campbell's formula and the definition of V † t f we get Therefore, by putting the pieces together we obtain the following where φ(i, λ) is given by formula (1.16). The previous equation is in terms of the tree T . We want to rewrite it in terms of the multitype branching diffusion, thus Observe that the processes I N † = (I N † t , t ≥ 0), I P † = (I P † t , t ≥ 0) and I η = (I η t , t ≥ 0) are initially zero-valued P (µ,ν) -a.s. In order to study the rest of the immigration along the backbone we have the following result.
Then, e −W t(x) is a locally bounded solution to the integral system where In the latter formula B † is given by (2.23) and V † t f is the unique solution to (1.10). It is important to note that W depends on the functions f , h and g but for simplicity on exposition we suppress this dependency.
Proof. Recall that Z is a multitype branching diffusion, where the motion of each particle with type i ∈ S is given by the semigroup P (i) and its branching generator is given by (1.12). For simplicity, we denote by P where j = (j 1 , . . . , j ℓ ). On the other hand, by Proposition 2.9 in [19], we see that e −W (i) t (x) also satisfies By substituting the definitions of p So, putting the pieces together and using the definitions of q i , B † and H (i) , (see identities (1.12),(2.23) and (2.27)) we deduce that t−s (ξ (i) s , w · e −W t(ξ (i) s ) )ds , as stated. Therefore, e −W t(x) satisfies (2.26).
Proof of Theorem 1. Since X is an independent copy of X under P † µ , it is enough to show that for µ ∈ M(R d ) ℓ , ν ∈ M a (R d ) ℓ , f , h ∈ B + (R d ) ℓ , the vectorial function e −U (f ) t h(x) defined by e −U (f ,i) t h(x) := E µ,e i δx e − f ,I N † +I P † +I η t − h,Zt , is a solution to (1.19) and that this solution is unique. By its definition, it is clear that e −U (f ) Hence, if we prove that for any f , h ∈ B + (R d ) ℓ , x ∈ R d , and i ∈ S, the following identity holds we can deduce (2.30). In order to obtain (2.31), we first observe that identities (1.10) and (1.19) together with the definition of ψ † allow us to see that both left and right hand sides of (2.31) solve (1.3) with initial condition f + w · (1 − e −h ). Since (1.3) has a unique solution, namely V t (f + w · (1 − e −h )), we conclude that (2.31) holds and it is equal to V (i) t (f + w · (1 − e −h ))(x). Hence, we can finally deduce that (Λ, P µ ) is a Markov process. Moreover, we have E µ e − f ,Λt − h,Zt = e − V t(f +w·(1−e −h )),µ = E µ e − f +w·(1−e −h ),Xt , and if, in particular, we take h = 0 the above identity is reduced to This completes the proof.